Electronic Structure of Hume-Rothery Phases

Pro!tress in Matermls Science. Vol 22. pp. 151 262. I}(J48-5500 78 0701-(}151507.5lj tt Pergamon Press kt& 1978 Printed in Great Britain

ELECTRONIC STRUCTURE OF HUME-ROTHERY PHASES'}"

T. B. Massalski and U. Mizutani~ C a r n e g i e - M e l l o n U n i v e r s i t y , P i t t s b u r g h 15213 , P e n n s y l v a n i a , U S A

(Submitted 15 Norember, 1977)

CONTENTS l. INTRODUCTION l 52 2. ALLOYS OF THE NOBLE METALS 155 3. DENSITY OF STATES CURVES 157

3.1. Construct ion of the Density of States Curves Based on 1st Principles Band Calculations 158

3.2. The Multi-Cone Model Approximations 164 4. MODELS OF THE ELECTRONIC STRUCTURE OF ALLOYS 172

4.1. Rigid Band Approximation 172 4.2. Optical Properties Related to the Band Structure 174 4.3. Band Models and Electronic Specific Heat 179

5. CONNECTIONS BETWEEN PHASE STABILITY AND ELECTRONIC STRUCTURE 183 5.1. The Model of Jones 183 5.2. Pseudopotential Considerations 185

6. ELECTRONIC SPECIFIC HEATS IN ALLOYS 191 6.1. Theory of Electronic Specific Heat 191 6.2. Many Body Effects (e-p and e-e Interaction) 193 6.3. Transit ion Metal Impurities in Normal Metals 195 6.4. Other Contributions to the Linear Term in Normal Metals 196

7. GENERAl. SURVEY OF SOME ELECTRONIC PARAMETERS IN NOBLE METAl. ALLOYS 197 7.1. Low Temperature Specific Heat Coefficients (7, 0,,, ;;) 197 7.2. Other Physical Parameters {;G,, c/a, T,.) 203

8. EI.ECTRONIC STRUCTURE OF RELATED GROUPS OF PHASES IN NOBLE METAL 206 ALLOY SYSTEMS 8.1. :t-Phase Primary Solid Solutions and lnternoble Metal Alloys 206

8.1.1. Fermi surface topography 206 8.1.2. The behavior of the electronic specific heat coefficients on alloying 211

8.2. Cubic fl-Phase Solid Solutions 218 8.3. Complex Cubic 7 and it Phases 224

8.3.1. Fermi surface Brillouin zone interactions in the ;, phase 224 8.3.2. Density of states curves and electronic specific heat data in the 7

phase 227 8.3.3. Fermi surface Brillouin zone interactions in the tt phase 232 8.3.4. Density of states curves and electronic specific heat data in the It

phase 233 8.4. Close Packed Hexagonal ¢ and e Phases Solid Solutions 235

8.4.1. Fermi surface Brillouin zone interactions 236 8.4.2. Density of states curves and electronic specific heat data General

trend 24l 8.4.3. 00.2 overlap effect 244

9. COMPETITION f'OR PHASE STABILITY IN NOBLE METAL ALLOYS 249 9.1. Thermodynamic Considerations and Phase Stability in Noble Metal

Alloys 252 9.2. Further Comments on Stability 256

APPENDIX 260 10. CONCLUSIONS, ACKNOWI.EDGEMENTS 260

+ Dedicated to the memory of Professor William Hume-Rothery, ERS. On leave of absence from Nagoya University, Nagoya 464, Japan.

151

152 P R O G R E S S IN M A T E R I A L S S C I E N C E

| . INTRODUCTION

Research on the alloying behavior of the noble metals has always been of prime importance in the theory of alloys. The early foundations for such studies were laid by Hume-Rothery, more than fifty years ago, when he pointed out connections between the observed crystal structures and electron concentration. Since then progress has continued steadily, with an ever-increasing amount of experimental information becoming available, both about the alloying behavior and individual properties of different alloy phases. Concurrently with this research, numerous attempts have been made to interpret the observed behavior in terms of various genera- lized parameters and models. The main motivation for such work is the need for isolating the major alloying factors which determine the relative stability between different competing crystal structures in phase diagrams.

Following the early work, one of the most often used alloying parameters is the electron concentration. It is related to the valence difference between the participating elements and to composition, but it may also depend on specific features of the electronic structure of any given alloy. Thus, even in its simplest form the meaning of this parameter is as yet only partially formulated; and the same is true regarding the role of electron concentration in alloying behavior. The stated principles concerning electronic interactions often differ from author to author. For example, in his 1966 summary at the Geneva conference, Hume-Rothery I1) listed four general factors of importance in relation to the electronic structure of alloys, as follows.

1. The difference between the electronegativities of the two metals. This is the most important factor, and increasing this difference increases the tendency for two atoms to unite in either liquid or solid phases.

2. A tendency for atoms of elements near the ends of the short periods and B subgroups to complete their octets of electrons, and a similar but less marked tendency to fill the d shell in the later transition elements.

3. Orbital-type restrictions. Structures whose hybrid bonding orbitals involve a very high proportion of d function may not mix with class I or class 1V elements. Similarly, atoms giving rise to almost pure p bonding may not enter structures involving other types. A high value of electronegativity difference may overcome these restrictions.

4. A tendency for definite crystal structures to occur at characteristic electron-atom ratios or electron concentrations.

The last factor is of particular prominence in the large group of alloy phases based on the three noble metals Cu, Ag and Au, and has led

E L E C T R O N I C S T R U C T U R E OF H U M E - R O T H E R Y PHASES 153

to the designation of such phases as electron phases or Hume-Rothery phases. Electron concentration in electron phases is relatively easy to define and may be expressed as the ratio of all conduction electrons to the number of atoms, e/a. In this context the d electrons are not included.

While the general importance of the above statements would meet with little dispute, the interpretation and documentation of each may differ from author to author making it possible to hold rather divergent views about the so-called "Hume-Rothery rules". For example, Heine and Weaire, ~1~ in their review dealing with the pseudopotential theory state that: "the interpretation of the Hume-Rothery rules remains in our opinion unsettled". At the same time, they emphasize that "the traditional ideas which associate a special stability of alloy phases with Brillouin zone planes touching the Fermi sphere, or Fermi surface, are simply not correct in general and that whenever such a correlation is well established it would appear to indicate at least a partial breakdown of the theory and therefore be an exception from the rule",

Regarding the electrochemical effects Phillips ~3~ states that "... most of the ionicity scales that have been proposed so far contain intrinsic inaccur- acies". However, he suggests that "the theoretical solid-state physics may have reached the level of sophistication required to elucidate chemical trends throughout the periodic table",

Finally we quote from a review by Kaufman: 141 "Definition of characteristic electron/atom ratios derived on the presumption that a single contribution exhibits a rapid variation near the phase boundary is incorrect. In particular, such approaches are not based on the concept of phase competition which offers the broadest framework for explaining and predicting the stability of metallic phases in quantitative terms".

The basis for the present review is the large amount of relatively new experimental data obtained from the measurements of the low temperature specific heats in electron phases. Such information can be related to the density of states at the Fermi level; and from there it can lead to possible connections between the density of states and stability.

We assume that the details concerning the electron phases, their occurrence in phase diagrams, crystal structures, lattice parameters and early research may be taken for granted, and these features are only briefly mentioned in Section 2 and subsequent sections. In order to review the new data on specific heats against a background of theoretical modeling, previous work and other measured properties, we discuss first the typical density of states curves that can be derived from theoretical considerations, including a new model calculation based on the multi-cone approach (Sec- tions 3 and 8). Various models of the electronic structure of alloys are considered next in relation to the available optical data in the noble metal alloys, and the electronic specific heats in the transition metal alloys (Sec- tions 4 and 5).

The various contributions to the linearly temperature dependent specific


heats are discussed in detail in Section 6, and are followed by a survey and discussion of the measurements performed on numerous fcc primary solid solutions, and the intermediate electron phases /~, #, 7, ff and E (Sec- tions 7 and 8). This information is supplemented by similar surveys and discussions of related physical properties that have a direct bearing on the electronic structure, such as the de Haas van Alphen oscillations, positron annihilation, Faraday rotation, spectroscopic data, magnetic susceptibility and numerous other features (Sections 7 and 8). The possible electronic structure of the different types of electron phases in noble metal alloys and some specific features related to the Fermi surface are then considered in detail (Section 8). Finally this is followed by an assessment of the energy of the conduction electrons as a relative contribution to the total free energy of an alloy (Section 9).

In the assessment of the various existing theoretical pictures and models of the electronic states in the noble metals and their alloys, we have followed the path of experimentalists who seek models in order to test their data. Such a path has many pitfalls and we may have ignored, simplified or misunderstood many features that others would consider important. The main purpose of our review is to tie up together much of the existing experimental information which, in one way or another, has bearing on the assessment of the relative alloy phase stability, from the point of view of the electronic structure, at least in this limited field. The selection of our topics has therefore centered on measurements and interpretations that tend to define the connections between energy, electron concentration, density of states and stability. The low temperature specific heats play a very prominent role here. We hope that the present effort will be of help to other experimentalists and also to those who may "seek data in order to test their models". In this way, hopefully, we would have bridged some gaps.

In order to facilitate cross-referencing, all equations, tables and figures in each section are identified by a common first numeral; and each section is appended with a separate list of references.

References in the text and tables are identified by a number, but occa- sionally names, or dates, are also given in brackets to facilitate the identifi- cation of different sets of experimental measurements. In order to make the text as short as possible many properties, or parameters, are referred to by a symbol rather than the proper expression. Thus electron concentration is most frequently referred to as e/a, the electronic specific heat coefficient as 7, the interaction of the Fermi surface with the Brillouin zone as FsBz, the de Haas van Alphen oscillations as dHvA, etc. Through- out the text, various density of states curves are frequently compared with the experimental electronic specific heat coefficient 7, which is plotted as a function of e/a, as determined from the respective alloy compositions. In order to allow for a direct comparison between a calculated density of states curve N(E) and 7 plotted in terms of e/a, many of the N(E)

ELECTRONIC STRUCTURE OF HUME-ROTHERY PHASES 155

curves have been integrated to provide the relation between the N(Ev)

and the corresponding e/a. In such cases, for the sake of convenience, the N(Ev) vs. e/a trend will be often shown in units of the 7 coefficient, i.e., m J/mole K 2. The details of the computer calculation utilizing the multi-cone model in Sections 3 and 8 are given in the Appendix.

REFERENCES

1. W. HUME-ROTHERY, Phase Stability in Metals and Alloys (edited by P. S. Rudman, J. Stringer and R. I. Jaffeel, p. 3 (1967), McGraw-Hill, New York.

2. V. HEINII and D. WEAIRE, Solid State Physics (edited by H. Ehrenreich, F. Seitz and D. TurnbullL 24 (1970) p. 249 Academic Press, New York, London.

3. J. C. PHILLIPS, Physics to-day 23 (1970) 23. 4. L. KAUFMAN: Prog. mater. Sei. 14 (1971) 55.

2. ALLOYS OF THE NOBLE METALS

Alloys of the noble metals Cu, Ag and Au, have traditionally constituted an attractive testing ground for the theories that relate alloy stability to the electronic structure. The main reason for this is that the position of the noble metals along the horizontal rows in the periodic table is at the border between those elements whose d bands are not yet fully filled with electrons (to the left), and those whose d bands are occupied (to the right), as shown in Fig. 2-1.

Alloying with polyvalent elements that fall to the right of the noble metals allows manipulation of the electron concentration that involves mainly the s and p conduction electrons. This has been useful in testing the possible influence on phase stability of the interaction between the Fermi surface of these conduction electrons and the Brillouin zone. A review of some earlier aspects of this subject will be found in Volume 10 of this series. (1~ Here, we need only emphasize that a typical phase diagram based on one of the noble metals may be represented schemati- cally by a sequence of phases as shown in Fig. 2-2. Following the primary

AI Si i

I .

Zn Ga Ge As Fe Co Ni [Cu I

Pt Ag Cd I n Sn Sb

Pd Au Hg TI Pb

I I I Ill IV v

FK;. 2-1. Portion of the periodic table representing alloy phases of the noble metals Cu, Ag and Au. Roman numerals indicate group valence.

156 PROGRESS IN MATERIALS SCIENCE

Cu Ag Au

0/' " , ,

fcc x \ \

i , °

electron concentrat ion

FIG. 2-2. Typical alloy phase diagram characteristic of the noble metals. Thick line boundaries represent the phases with cubic symmetry a, ]~, /1 and 7. Hcp structures are represented by if, ~ and q. The broken line shows a possible

range of hcp phases.

solid solution (~-fcc), the first intermediate phase usually possesses one of three possible crystal structures, two of which are cubic (/3-bcc or p-/~Mn structure) and one is close-packed hexagonal (~). It is possible for all three structures to occur at the same e/a, but at different temperatures, as in the Ag-A1 system (see Fig. 9-1). In most phase diagrams the subsequent intermediate phase is usually complex cubic (7), or close packed hexagonal (E and r/), as indicated in the figure. Thus, a typical alloy phase diagram of a noble metal in the range of e/a between approximately I and 2, involves an extended ~ primary solid solution followed by a set of electron phases. The electron phases with cubic structures (/~, # or 7) are, as a rule, restricted to fairly narrow and characteristic ranges of e/a, but the close-packed hexagonal structures ff and E can extend over a wide range of e/a, approximately between 1.2 and 1.9. The r/ phase occurs only if the II-B subgroup elements are zinc or cadmium, and it then constitutes the primary solid solution of a noble metal dissolved in either of these two metals. From this point of view, the E phases represent the first intermediate phase that follows the r/primary solid solution as e/a is decreased. Further details are given in Section 8.

The density of states curves characteristic of the noble metals are shown in Fig. 3-2. Since the contact between the Fermi surface and the first set of Brillouin zone planes (the {111} set) already exists in pure noble metals, the Fermi level EF is drawn past the peak corresponding to the 111 contact. The position below the Fermi level, and the form, of the d band are slightly different for each noble metal. The d band, with ten nominal electrons, involves a high density of states compared with the

E L E C T R O N I C S T R U C T U R E OF H U M E - R O T H E R Y P H A S E S 157

conduction band which is widely spread out. The top of the d band is only a few eV below the Fermi level. As pointed out in the next section, there is a considerable interaction between s and d electrons. Hence the noble metals may be regarded in part as late transition metals, when their atoms form a crystal and the electron energies become spread out into bands. ~2~

REFERENCES

1. T. B. MASSALSKI and H. W. KING, Pro,q. mater. Sci. 10 (1961) 1. 2. N. F. MOTT, Adv. Phys. 13 (1964) 325.

3. DENSITY OF STATES CURVES

As is well known, the noble metals are each situated at the right end of a transition metal row in the periodic table. When their atoms form a crystal, the outermost electrons, s or p, interact with one another and are no longer bound to the original atoms; instead they move more-or-less freely over the crystal lattice and their energies belong to the conduction band. On the other hand, the electrons situated deeply in the atoms do not interact appreciably and remain bound to the atomic cores, with the result that their energy levels remain degenerate. The d electrons in the noble metals are generally thought of as having an intermediate character. Their interaction results in the formation of a relatively narrow energy band, the density of states of which is quite high, reflecting the need for accommodating the nominal ten electrons per atom within a narrow energy range, It is well known that the d band in the noble metals or their alloys is located in the middle of the conduction band, allowing for a substantial interaction between the two bands.

The density of states is defined as the number of electronic states per atom (or volume) with energies between certain values. It can be conveniently expressed by the equation

N(Eo) = ~/'4TC 3 .~ - f f dS/lgrad Ekl, (3-1)

E E.

where the integration is carried out over a constant energy surface in k-space between Eo and Eo + AE. If fl is expressed as atomic volume, N(Eo) is given in units of states/energy.atom. In the discussion of electronic properties such as, for example, the electronic specific heats, optical properties, magnetic properties, phase stability, etc., it is often important to know the relationship between N(E) and E, i.e. the density of states curve and its changes on alloying. The N(E) curve characteristic of the d band of the transition metals has been studied more extensively than the conduction band, because of interest associated with the magnetic properties. Furthermore, the prominent features of the d band, showing peaks and valleys, can be explored relatively easily, compared with the

158 PROGRESS IN MATERIA LS SCIENCE

fine structure of the conduction band, because the latter is only about 1/10 the magnitude of the former. Thus, less information is available on the detailed structure of the conduction band. Often, the conduction band is approximated by the free electron parabolic density of states. However, in real metals, an appreciable deviation may occur from this simple form as a consequence of the Fermi surface-Brillouin zone interactions (hereafter this type of interaction will be denoted as FsBz). Furthermore, interaction can occur also with other bands, such as the s-d interaction in the noble metals. Below, we focus our attention mainly on the N(E) curve of the conduction band, particularly as it relates to the electronic specific heat coefficient.

Two approaches can be followed in order to calculate the density of states curve for the noble metals; one involves a straightforward integration through eq. (3-1), using E-k relationships derived from first principles band calculations, and another makes use of the multi-cone approximation which utilizes the geometric symmetry of the Fermi surface with respect to the corresponding Brillouin zone. Below we review the characteristic features of these two approaches.

3.1. Construction of the Density of States Curves Based on First Principles Band Calculations

In principle, the relationship between E and k for an electron in any direction can be determined by solving the Schr6dinger equation of the electron traveling in a postulated crystal potential. Different calculation procedures include the tight-binding, cellular, orthogonalized plane wave (OPW), pseudopotential, augmented plane wave (APW) and the Green function method. With the progress in the use of computers, the APW or the Green function methods are now being more frequently employed. (See, for example, the book by Loucks. ~tl)

In each case the choice of the crystal potential is crucial. For the noble metals the potential is usually constructed from two contributions; the potential which yields self consistent d functions for the free metal ion and the contribution from the conduction electron. In pure Cu, the Cho- dorow potential has been frequently employed by various authors, for example Segall, ~2) Burdick ~3) and Faulkner et al. ~4) In the case of heavier metals, such as Au, the relativistic effect must be also taken into account. Once the potential is selected, the eigenvalue of either the conduction electron, or the d electron, is a matter of solving the Schr6dinger equation. In the APW method, the energy E is usually determined for particular values of k in the Brillouin zone, while in the Green function method, the k values are often searched at a constant energy E. Regardless of the procedure, the interaction between s and d electrons is automatically taken into account.

Once the E k relations are determined in various directions in k-space,


the construction of the density of states curve is straightforward through eq. (3-1). However, since the calculated E(k) values are usually restricted only to certain particular directions, the resultant summation over the whole space may be only approximate. This may lead to a density of states curve with spurious or exaggerated, peaks and valleys caused by statistical errors, and such features may have little to do with the FsBz interactions.

For example, Kennard et al. 15) calculated two density of states curves for pure A1, using S n o w ' s ~6t first principles band calculation. The curves were computed on the basis of 256 and 2048 input E-k values, respectively. In each case, the calculated points were augmented further to 25,000 by utilizing a quadratic interpolation scheme. As can be seen from Fig. 3-1, the standard deviation is greatly reduced in the 2048 input curve, where the peaks and valleys are more specifically related to the critical interaction points associated with the Brillouin zone. Eventually, in addition to contributions from higher zones, the major peaks in the curve should correspond to the interaction with the II11} and {0021 zone planes. However, even the more detailed computation, Fig. 3-1(b), is still far from this goal, despite the fact that A1 has a relatively simple, nearly free electron, band structure. This serves to emphasize that 2048 input data points, plus an interpolation scheme, are still insufficient to reveal fine detail, and that a very considerable precision is needed in a density of states calculation before a computed curve can be compared with experimental data.

First principles band calculations have been carried out extensively for noble metals by various authors, as listed in Table 3-1. In the case of copper, Faulkner et al. ~4) calculated the E k values at a very large number of points in the Bz, sufficient to reveal the van-Hove singularities (i.e. FsBz interaction peaks in the conduction band). In the next sub-section, we consider their results in comparison with the results obtained with the multiple-cone model. Table 3-1 shows that, apart from Faulkner et al., the number of points at which the E k values have been calculated for pure Cu is too restricted to identify any detailed structure in the conduction band. Janak, ~1°) however, could identify the 111 peak, by applying the Gilet and Raubenheimer interpolation scheme (18~ to the energy band calculations of Snow and Waber. ~8~ The result is shown in Fig. 3-2.

In the case of silver, only Christensen ~15~ could locate the van-Hove singularities in the conduction band, as shown in Fig. 3-2. Here E - k values were determined at only 4272 points in the Brillouin zone, but the G R interpolation scheme was employed to increase the input data. Even with this feature, it appears certain that the rough N(E) curve in Fig'. 3-2, compared with Fig. 3-5 due to Faulkner et al., results from an insufficient number of input E - k relations. Chris tensen and Seraphin Ilvl also calculated the N(E) curve for Au, using the same approach, but the detailed structure of the conduction band cannot be identified in their overall N(E) curve, as shown in Fig. 3-2.


f , , , , , , i r A t

0.~ 256 input d a t a

0.3

0.1

0 ' * 1.0 - 0 .8 - 0 .6 - 0A

e n e r g y ( R y }

0.3

0.2

0.1

0

i i i i -02 0

i i i ~ v , , f , i , r

AI

I I * I I I i i i i I I I ~ - ~

-1.0 -0.8 -0.6 -0.4 -0.2 0 energy (Ry)

FIG. 3-1. The density of states calculation for A1 using the Snow's values of E(k) for (a) 256 and (b) 2048 input E-k values in the Brillouin zone. In each case the calculated points were increased to 25,000 by means of a quadratic

interpolation scheme. (After Kennard et al. ~5~)

In addition to the noble metals, we shall refer to the density of states curves of several simple metals such as A1, Zn, Cd, etc. The histogrammic density of states curves for pure Zn and Cd were constructed by Allen et al., (19) based on E - k relationships which Stark and Falicov (2°) obtained with a pseudopotential calculation. This calculation showed that the d bands are located below the bottom of the conduction band. Massalski et al. ~21~ smoothed out the original histograms and constructed an N(E) curve for pure Zn, in which the expected van-Hove singularities are emphasized through reference to particular E - k relations. The resulting N(E) curve is shown in Fig. 3-3. The density of states curves for Be and Mg tz2~ shown in Fig. 8-23 will be discussed later in connection with the hcp alloy phases.

Within the framework of the free electron model, the density of states of the conduction electrons can be expressed in the form:

Nfree(E ) = 6 .813 × 10 - 3 × ~ × (m*) 3/2 × E~states/eV.atom, (3-2)

Tab

le 3

-1.

Fir

st P

rinc

iple

s B

and

Cal

cula

tio

ns

for

No

ble

Met

als

Nu

mb

er o

f ca

lcul

ated

E

k

valu

es

Nob

le m

etal

s M

eth

od

A

uth

ors

C

alcu

late

d p

rop

erti

es

in t

he

Bri

llou

in z

on

e

Cu

G

reen

fu

nct

ion

S

egal

l ~21

(19

62)

E

k, F

S

AP

W

Bur

dick

~3~

(196

3)

E-k

, F

S,

N(E

) G

reen

fun

ctio

n F

aulk

ner

eta

[. ~4

~ (1

967)

F

S,

N(E

)

rrl

r"

Ag

Au

2O48

12

2510

2 (2

6066

p

oin

ts o

n e

ach

con

stan

t en

ergy

sur

face

wit

h in

terv

al o

f A

E =

0.

02-0

.2 e

V)

Gre

en f

unct

ion

Wak

oh

~71

(196

5)

E

k, F

S

AP

W

Sn

ow

an

d W

abe6

8j (

1967

) E

-k,

FS

, N

(E)

2048

A

PW

S

no

w {9

) (1

967)

E

k,

FS

, N

(E)

2048

G

R

Ja

nak

~1°

1 (1

969)

N

(E)

Sn

ow

an

d

emp

loy

ed

AP

W

Sn

ow

lllj

(19

68)

E-k

, N

(E)

Gre

en f

un

ctio

n

Jaco

bs ~

m2~

(196

81

Ek

, F

S

Gre

en f

unct

ion

Bal

ling

er a

nd

Mar

shal

l 113

~ (1

969)

E

-k,

FS

A

PW

C

hri

sten

sen

1141

(19

69)

E

k, F

S

RA

PW

C

hris

tens

en~

l 5~

(19

72)

E

k, F

S,

N(E

)

Gre

en f

un

ctio

n

Jaco

bs "

2~ (

1968

) E

-k,

FS

G

reen

fu

nct

ion

B

alli

nger

an

d M

arsh

all ~

13~

(196

9)

E-k

, F

S

AP

W

Ku

pra

tak

uln

and

F

letc

he{

16~

(19

69)

E

k, F

S,

N(E

) R

AP

W

Ch

rist

ense

n a

nd

Ser

aph

in 1~

7~ (

1971

) E

k,

FS

, N

(E)

Wab

er's

E

-k

valu

es

2048

57

6

3840

0 42

72

plus

G

R s

chem

e

576 40

00

4272

pl

us G

R

sch

eme

¢3

©

¢3

are

©

©

,.q

,.<

Abb

revi

atio

ns:

[R)A

PW

=

[rel

ativ

isti

c) a

ug

men

ted

pla

ne

wav

e m

eth

od

, E

-k =

E

k cu

rves

in

var

iou

s di

rect

ions

, F

S =

th

e F

erm

i su

rfac

e, G

R

=

Gil

et-

Rau

ben

hei

mer

in

terp

ola

tio

n s

chem

e.

162 P R O G R E S S I N M A T E R I A L S S C I E N C E

"~ 60 o

60

40

20

10

8O

6C

oc

;3,, a o z

Cu

100

E F

0.1 0 . : ~ 0.II 0..4 0-5 0 .6 0.7 energy { R y )

(o)

7 0

60

50

~3o

z

0 -0.2 0.0

Au

E F

i 0.2 0.4 0.6

energy (Ry) {b)

Ag

l- l I ~ l ~

0 O.2 0.4 0,6 O.8 1.0 1.2 1.4 energy ( R y )

"•4.5

~n

z

2~

(c)

Ag L 4 - d

,,

O.45 0,55 ~65 energy ( R y )

/v

Cff5

FIG. 3-2. The density of states curves for three noble metals; (a) Cu (Janak(~°)), (b) Au (Christensen and Seraphin (17)) and (c) Ag (Christensen~15)).

where E is the energy in units of eV and ~) is the atomic volume in (A) 3. The free electron Fermi level, E free, for different elements can be determined by integrating eq. (3-2) with m* = 1.0, and by using an appropriate value of valence. As listed in Table 3-2, the values of E fre~ do nor differ appreciably from the values of E b""d, deduced from the first principles band calculation for various elements and alloys, except for pure Au. This confirms that the Fermi level in the real metals and alloys is basically determined by the atomic volume and the number of electrons. The ratio Efree/l~'band which is proportional to a dimensionless effective mass, is com- F / Z ~ F ,

pared with the value of the mass at the bot tom of the band, mbottom , * in the last column of Table 3-2. The value of * mbouo m was deduced by fitting eq. (3-2) to the N(E) curve near the bot tom of the conduction band, obtained from the first principle band calculations. The deviation of the ratio lETfree/17band ~V / ' ~V from unity indicates the presence of perturbation; one arising from the peaks and valleys in the N(E) curve due to the FsBz


interaction, and another from electronic interband interactions, such as the s-p or s-d interactions. However, the value mbo.om* reflects only the latter effect, since no FsBz interaction exists near the bot tom of the N(E) curve. At first sight, it is surprising that a close agreement is found between the two types of effective masses for several elements. This may be due to a compensating effect of the FsBz interaction, whereby a decrease in the Fermi level due to the presence of a peak in the N(E) curve may be cancelled by an opposite effect due to the subsequently declining density of states. Thus, regardless of the presence of the FsBz interaction, the ratio ~[71"rcc ' r ' ' ' ( • ~ F /.-v or the * mb<,u .... are largely determined by the electronic interactions, being less than unity in metals such as noble metals and their alloys or Zn and Cd, where the s-d interaction is predominant, and more than unity in metals such as Be, where the s p hybridization is present. (See also discussion in Section 8.4.)

In the discussion of the electronic structure of the noble metal alloys, where no first principles band calculations are available, we shall assume an appropriate effective mass, using the above quantities as a guide, and then employ this value of effective mass in a multi-cone model calculation. These values are shown in brackets in Table 3-2. In the case of the noble metals, however, the mho,t,,m* cannot be used for this pnrpose, because of tile spilling of the d electrons into the conduction band, as will be discussed in the following subsection.

0.8

. . . 0 . 6

"6 E

0.4 A la.I

Z

0.2

electron c o n c e n t r Q t i o n

0.5 1.0 1.2 1.4 1.6 1.8 2.0 2.2 l I i i I i i i I i 1 i I

; i

? z / / Tq ' ' I

J ,

,,,, ~ -

0 I I I I I I I I [ I I

0 2 4 6 8 10 12 energy (eV)

FIG. 3-3. The smooth density of states curve for Zn, based on the histogrammic plot derived by Allen et a t . ( 1 9 ) The broken curve shows the free electron parabolic band with m* = 1.0 and e/a = 2.0. The open circle refers to tile / value of zinc, after correction for electron -phonon interaction (Section 6.2). The electron concentration scale cannot be applied to the free electron parabolic band.

(for details, see Massalski et al.~21b.


Table 3-2. The Ratio fr~ band Er /EI~ and the Value of mbo.o m * in Various Metals and Alloys

System e/a (~3) (eV) (eV) E Ffree/Evb"nd if/bottom* Ref.

Cu 1.0 11.81 7.03 9.09 0.77 4 Ag 1.0 17.06 5.50 7.5 0.73 15 Au 1.0 16.96 5.52 9.4 0.58 17 Zn 2.0 15.24 9.42 10.8 0.87 0.88 19 Cd 2.0 21.58 7.59 8.85 0.86 0.86 19 Be 2.0 8.13 14.31 11.9 1.20 1.20 22 Mg 2.0 23.23 7.09 7.1 1.0 1.0 22 A1 3.0 16.60 11.65 11.3 1.0 1.0 6 K 1.0 71.32 2.12 2.24 0.94 30 Na 1.0 37.71 3.24 3.30 0.98 30 /~ Cu-Zn 1.5 12.76 8.75 9.93 0.88 31 I' Cu-Zn 1.6 13.27 8.89 (9.5) (0.9) (0.9)

Ag-Cd 1.6 18.94 7.02 (7.5) (0.9) (0.9) u Ag-AI 1.46 16.57 7.22 (8.0) (0.9) (0.9)

ElF r~ is obtained by integrating eq. (3-2). E band refers to the Fermi level deduced from the first principles band calculation with respect to FI. if/bottom* iS determined by fitting eq. (3-2) to the first principles N(E) curve near the bottom of the conduction band. The values in the parentheses are assumed ones, judging from the various pure metal data. (See Section 8.3.)

3.2. The Multi-Cone Model Approximations

Jones t23) was the first to discuss the competition for phase stability between the ~ and fl phases in the alloy system Cu-Zn, in terms of the respective density of states curves derived for each structure from an appropriate cone model. As is well known, the energy of a conduction electron changes discontinuously across the Brillouin zone. Taking the kz axis as perpendicular to the Brillouin zone plane and passing through its center (0, 0, ko), Jones expressed the energy E of a conduction electron in the neighborhood of the center of the zone boundary (kz < ko) by the equations:

E = (h2/Zm m*). [k 2 + k~ + k~ + ~(kz - ko) 2] - AE/2

--_ E o [ X 2 + y2 + 1 + c~(z - - 1) 2] - - AE/2 (3-3)

and c~ = 1 - 4Eo/AE, (3-4)

where Eo is the energy of an electron at the point (0, 0, ko); E o ~ hZk~/2m "m* (m* is not necessarily unity, see below), AE is the energy gap across the Brillouin zone plane, and x, y and z are normalized vari- ables with respect to ko. The assumption of axial symmetry around the kz axis appears to be reasonable in view of the geometrical features of the Fermi surface in contact with or approaching the Brillouin zone. However, away from the zone boundaries, the perturbation due to lattice potential will weaken and the E - k relationship may be expected to


approach the free electron expression. Jones proposed the following equation to describe all points along the k: axis and its vicinity,

E = Eo .¥2 + y 2 __ 1 + (1 - z) 2 - 2 1 - z) 2 + (1 Z-~X)2 ' (3-5)

Equation (3-5) reduces to eq. (3-3) near the zone boundary and to the free electron expression near the origin of the zone.t The same equation was employed by Ziman, (241 who discussed various transport phenomena of the noble metal alloys.

The derivation of the density of states curve based on the cone model is simple, once the E-k relationship is decided. The Brillouin zone is divided into prisms, the zone plane forming the base, and the origin the vertex of each prism. Equation (3-5), taking the k~ axis to be the prism axis, is applied within each prism. The Fermi surface in each prism is assumed to correspond to the surface of revolution around the k_ axis• For the sake of simplicity, the polygonal base of each prism is usually replaced by a circle with equivalent area• With this replacement, however, one may not be able to map out exactly the whole 4a solid angle in k space with cones, because of small overlaps and small deficiencies occur- ring at points of contact. A misfit parameter may thus be defined as the difference between 4,r and the total of solid angles of all cones, and used to judge the approximation involved in cone fitting. The values of this parameter are listed in Table 3-3 for the various crystal structures.

The density of states of the conduction electrons can be calculated in each cone by making use of the axial symmetry of the Fermi surface around the k~ axis (the details are given in the Appendix)• The total N(E) is then written as:

N(E) = ( m n l * ~ / 2 ~ 2 h 2) E fll(k . . . . . . . . . . ill • . : - k : t i , ( 3 - 6 ) i 1

where f~ is the atomic volume, m* is the effective mass of the conduction electrons,:l: [31 is the number of equivalent cones and the index i refers to different sets of zone planes.

+Although Jones noted that eq. (3-5) gives the form for free electrons near k = 0, in fact the energy E becomes negative as kz tends to zero along the k: axis (kx = k s = 0). The same difficulty arises in Ziman's expression (see eq. (4.4) in Ziman's paper). As a result. the calculated N(E) curves have some finite states in the negative energy range, as can be seen in Fig. I of Jones ' paper. In the present analysis, eq. (3-51 is slightly modified so that E becomes zero at the origin k = 0. The modified equation is of the form.

f _ /':t(~ - 21(I - z) 2 + 1] E = E0 x 2 + v e + I(1 - z) 2 - _ % , (1 - z0- ]" (3-5a)

~.The parameter m* can be regarded as an adjustable parameter, as will be discussed later. Strictly speaking, its value should be anisotropic, reflecting a different E k relation along each direction• However, we ignore this effect here. Thus, the parameter m* plays the same role as the effective mass appearing in the free electron density of states (see

eq. (3-2)): N(E) = (~ 27r2).(2m.h2) 3 2.m,.~ 2 E 1 2.

In the present discussion, we refer to m* as a non-dimensional parameter by taking the ratio with respect to a free electron mass.

Tab

le 3

-3.

Mis

fit

Pa

ram

ete

rs i

n

the

Co

ne

M

od

el

Cry

stal

str

uct

ure

S

oli

d

ang

les

of

the

con

es¢

T

ota

l so

lid

an

gle

0

Mis

fit

pa

ram

ete

r -

[0 -

4n

]/4n

(%

)

fcc

~o11

1 =

1

.23

85

; ~o

oo2

=

0.4

47

2

8 o9

111

+

6 09

oo2

=

12.5

921

0.2

b

cc

cOll

o =

1.

0655

12

6o 1

~o =

1

2.7

86

7

1.75

7

~o33

o =

0

.42

42

; co

4~ ~

=

0.3

26

2

12tO

33o

+

24~

o411

=

1

2.9

19

2.

8 y

(fl-

Mn

) ~

22

, =

0

.35

14

; ~O

31o

=

0.1

76

7

24~

o221

+

24

co31

o =

12

.566

0.

9

t T

he

suf

fix

of

o9 r

efer

s to

th

e in

dex

o

f th

e re

spec

tiv

e zo

ne

pla

ne.

© 7. 7. r-

Z


The values k m"x and k~ '~'' are the maximum and the minimum wave vector components on a constant energy surface E, respectively. As can be understood from the geometry of the Fermi surface, the maximum k~ value always occurs along the kz axis, while the minimum k~ value occurs at the intersection between a constant energy surface and a sloping side of the cone. Hence, the k~ ~i" value depends on the solid angle of the cone, as shown in the Appendix. The Brillouin zone for the fcc structure consists of eight II111 and six 10021 zone planes. Both Jones and Ziman considered only the eight hexagonal 'Llll zone planes. The whole space was then divided into eight cones, the solid angle of each being equal to ~/2 with the [111] axis coincident with that of the cone. This approach may be referred to as the eight-cone model with /3 = 8 in eq. (3-6).

Compared with the N(E) curves derived from first principles band calculations, the N(E) curves derived from the cone model possess no peaks or valleys other than those dictated by the FsBz interactions. They are therefore of interest only as a first approximation for comparison with experimental data. Using such smooth N(E) curves Jones t23) discussed the occurrence of the phase boundary between the ~ and /3 phases in the Cu Zn system and Ziman t24~ evaluated the possible form of the N(E) curve for the noble metals, in which the Fermi level, Er, is located past the 111 contact peak in accordance with the known Fermi surface topography.

The derived decreasing slope of the N(E) curve following the contact peak in noble metals has been discussed by several a u t h o r s 125'26'27) in reference to the experimental electronic specific heat coefficients 7 in Cu- based or Ag-based alloys. However, because of the presence of the d band, several uncertainties exist, one of which is the inadequacy of the cone model itself. According to Jones and Ziman, t the parameter E0, which denotes the energy at the center of the band gap, is assumed to be given by the free electron value E0 = h2ko/2m. In the case of Cu~ this value

"~ 2 is (h-/2m).(rr,.'3/a) = 8.63 eV for the '~1111 zone plane, where a is the lattice constant. The parameter e in eq. (3-4) depends on both the E0 and the energy gap AE. Jones and Ziman used different values for AE across the {1111 zone planes (4.1 eV by Jones and 7.0 eV by Ziman). The corresponding energy at L2, (which is the zone contact energy, since it corresponds to the energy for which the Fermi surface touches the ~ 1111 zone planes), is given by (E0 - ½AE), and becomes, respectively, 6.6 and

+In using the eight-cone approach Ziman employed normalized dimensionless parameters for quantities such as the E k relations, the neck radius, the energy gap and the resulting density of states. Thus, absolute energy values, including the Fermi level and the contact energy at which the 111 peak occurs in the N(E) curve are not explicitly mentioned. However, since Ziman referred to the energy gap across the '~1111 zone planes 17.0 eV for pure Cul with a choice of m* = 1.0, we assume the effective mass in Ziman model to be that for the free electrons. This allows us to determine the absolute values of energy for a general comparison with other calculations. Any reduction in the effective mass from m* - 1 makes the derived energy gap in Ziman's approach diverge further from the values listed in Table 3-4.


5.1 eV for the two calculations. These contact energies EL2 can now be directly compared with the corresponding values derived from the first principles band calculations. As shown in Table 3-4, the EL, energy always exceeds 8 eV and is some 2-3 eV higher on the energy scale than the estimates based on the above models. The large calculated Ec2 value is undoubtedly due to the s-d interaction.

As stated earlier, eq. (3-5) describes a more or less free electron behavior for the conduction electrons except for the region near the zone boundary. Accordingly, the resulting N(E) curve is quite close to the free electron parabola, until the Fermi surface approaches the { 111 } zone planes. With one conduction electron, the Fermi level Ev in Cu occurs just above 7.0 eV, if a free electron parabolic band (m* = 1.0) is assumed. The presence of the 111 peak enhances the height of the N(E) curve, with a corresponding further slight reduction in the Fermi level. The low values of the zone contact energy implicit in the early eight-cone calculations allowed for the location of the Fermi level close to, or past the 111 peak in the N(E) curve, even with m* = 1. However, the first principles band calculations now indicate that both the zone contact energy and the Fermi energy have considerably higher values. This is undoubtedly due to the characteristic feature of the electronic structure of the noble metals, namely the strong s-d interaction.

The zone contact energy derived from the first principles band calculation makes it possible to employ presently a more suitable value of Eo (rather than the free electron value), in order to construct an N(E) curve for pure Cu on the basis of a multi-cone model. At this stage, the 10021 zone planes can be also additionally taken into account. Again, the hexagonal (111} and the square {002]. planes are all replaced by circles with equivalent areas. The misfit parameter, as listed in Table 3-3, is only 0.2°~ in this case. The choice of the parameters Eo and :~ in the cone model requires the knowledge of the E-k relations along both the [111] and [002] directions. The pertinent values for pure Cu are listed in Table 3-4. Among the different band calculations, that due to Burdick ~3~ gives all the needed information. Thus, Burdick's data are employed in the present fourteen-cone analysis.

The calculated E-k relationship corresponding to the [111] direction in the Brillouin zone and using eq. (3-5a), is shown by the heavy line in Fig. 3-4. Admittedly, this curve implicitly incorporates the existence of the s-d interaction, because it is related to parameters obtained through a first principles calculation. This assures, however, that the corresponding N(E) curve is realistically situated on the energy scale (see Fig. 3-5). As the parameter m* in eq. (3-5) is decreased, the N(E) curve reduces its height as a whole, causing the EF to raise to a higher energy. In this way, we choose an m* value to make the EF level consistent with the value obtained by Burdick. Our calculation for m * = 0.8 is plotted in Fig. 3-5 for comparison with that derived by Faulkner et al., since Bur-

Tab

le

3-4.

V

ario

us

Ban

d P

aram

eter

s fo

r P

ure

Cu

Seg

all (

2~

Bur

dick

TM

Fau

lkn

er e

t al

Y *~

B

ansi

l et

al

. (28

~ F

ree

elec

tron

mo

del

E'~

y~ (

EL

2.)

8.0

8.29

9 8.

36

8.20

6 (5

.1):~

E

]y[

(EL

,)

13.9

12

.856

--

13

.410

(1

2.1)

A

EII

t (E

L,

--

EL2

.)

5.9

4.55

7 5.

203

(7.0

) E

o-p

aram

eter

for

~ 11

t ~

pla

nes

t 10

.95

10.5

77

8.63

E~

°~2

(Ex4

.)

11.2

10

.938

11

.096

11

.171

E

~o~

(Ex,

) 15

.7

16.2

03

AE

oo2

(Ex,

-

Ex,

.)

4.5

5.26

5 E

o-p

aram

eter

for

~00

2] p

lan

es

13.4

5 13

.57

11.5

1 E

v 8.

95

8.91

9.

095

9.12

7 7.

03

t T

he

E0

-par

amet

er

in

the

cone

m

od

el

is d

efin

ed

as

rep

rese

nti

ng

the

m

ean

en

ergy

be

twee

n th

e co

nta

ct

and

o

ver

lap

en

ergi

es;

Eo

- (E

....

+

E

¢o.)

/2.

In t

he

case

of

free

ele

ctro

n m

odel

, th

e E

0 va

lue

is

give

n by

th

e fr

ee

elec

tron

en

ergy

at

th

e k-

valu

e co

rres

po

nd

ing

to

th

e ce

nter

o

f th

e re

spec

tive

zo

ne p

lane

s: E

o -

h2/2

m(n

.~3/

a)

z an

d E

o =

?f

/2m

(2rt

/a)

2 fo

r th

e ~1

11',

and

~002

~ zo

ne

plan

es,

resp

ecti

vely

. +

In

the

Z

iman

m

odet

, th

e en

ergy

g

ap

acro

ss t

he

IIil

i zo

ne

pla

ne

is

assu

med

to

be

7

eV

for

pu

re

Cu.

T

his

yiel

ds

the

valu

es

in

the

par

enth

esis

fo

r th

e co

ntac

t an

d th

e o

ver

lap

ene

rgie

s.

©

,--t

!70 P R O G R E S S IN M A T E R I A L S S C I E N C E

It,

12

I0

8

P ~6

PuPe CU ' A~'~.

a

E0 o

_ i ° °,,,,

FIG. 3-4. The E - k relationship along the [111] direction for Cu. The thin curve is due to Burdick t3~ by means of the APW method, and the heavy curve was derived in the present analysis, using eq. (3-5a). The broken curve is reproduced from eq. (3-5) with band parameters employed by Jones. ~23! Note that the E

values become negative in the vicinity of the orgin.

(13

] o

02 E

Z

0.1

, , , , , \' ' , ~ ' , , ,

PURE Cu

1/--Cone Model with Burdick's B a n d P a r a m e t e r s l x ~ Fouikr~r, Dovis and Joy 11967)

"/ i1,,)~,,. /

/ / ~ ~ %-tO %-1.34 "

I I I I I I I I I I 111 I 0 0 1 2 3 4 5 6 7 8 9 10 12

Energy(eY )

FIG. 3-5. The density of states curve for Cu, derived on the basis of the 14 cone model with Burdick's band parameters. This is compared with the curves due to Faulkner et al., ~4) and Jones. ~23) The curve due to Jones suggests a

very low Fermi energy, and zone contact energy.


dick's N(E) curve for the conduction band lacks sufficient detail to be compared with the present result.t Comparison with Faulkner et al. (4~ curve is justifiable, in view of close agreement in the respective band parameters, as given in Table 3-4.~.

It can be seen in Fig. 3-5 that the N(E) curve due to Faulkner et al. is consistently higher than our present result, although the general forms of the two curves above the 111 peak are quite similar. The reason for this may be seen by comparing the E k relation along the [111] direction in k-space between the data given by the first principles method (Burdick's original) and the eq. (3-5a) of the multi-cone model in which Burdick's band parameters have been incorporated (Fig. 3-4). Because of the presence of the d band, Burdick's E k curve is split into two in the energy range between 3.5 and 5.5eV. Near the top and the bottom of the band the relationship between E and k represents more nearly the conduction band. However, even in these regions, the E k curve derived from the first principles calculation is flatter than that derived for the multi-cone model, leading to a larger density of states through eq. (3-1). This may be taken as evidence that the s d interaction strongly affects the conduction band, even near the bottom or the top of the band. In turn, the d band itself will be reduced in height due to the same s-d interaction, pushing some d electrons into the conduction band. Thus, we must attribute the difference in N(E) between Faulkner et al. and the present cone model to d electrons spilling into the conduction band. The contribution from the d electrons amounts to approximately 24~,, of the total electron count at the Fermi level. Apart from the difference in height between the two N(E) curves, their general profile is quite similar.

Admittedly, the application of the cone model to the noble metals must be viewed with proper suspicion because of the presence of the d band in the middle of the conduction band. Recognizing this limitation, the fourteen-cone calculation appears, nevertheless, quite useful in showing the features of the FsBz interactions and the general trend of the N(E) expected in the conduction band. It will be shown below that, for the case of the intermediate alloy phases where few first principles calculations

tAn alternative way of assessing m* is to deduce the product of m*•E from the knowledge of some experimentally determined parameter, such as the cross-section of the Fermi surface neck at the zone contact. This can be done, for example, by making the calculated neck radius match the radius derived from the dHvA measurements? TM Once the product m*AE is known, one can find a set of values of m* and AE which yield the best agreement with the experimentally determined density of states at the Fermi level. The corresponding value of the Fermi energy is, in this case, below the first principles value, but lhe overall agreement is quite good. This approach gives the value of m* ~ 0.8, which is the same as the valuc deduced here.

+~Faulkner et al. ~4~ calculated the detailed structure of the conduction band in pure Cu with sufficient resolution to show the van-Hove singularities. There is however an inconsis- tency between the numerical results of their density of states p(E), in their Table I1, and their Fig. 1. Figure 1 is correct (private communicat ion from Dr Faulkner) but the p(E) values in their Table I1 must be multiplied by 1.18194 in order to be expressed in units of s ta tes /Ry.a tom


exist, the cone model provides at the moment a very useful initial guide for comparison between theoretical predictions and the available experimental data.

In the end, it may be worthwhile to mention that Ham (3°) calculated the density of states curve for alkali metals using a similar method to that developed in the present section. He employed the cone model, in which the band parameters are deduced from his first principles band calculations.

REFERENCES

1. T. L. LOUCKS, Augmented Plane Wave Method (1967) Benjamin, New York, Amsterdam. 2. B. SEGALL, Phys. Rev. 125 (1962) 109. 3. G. A. BURDICK, Phys. Rev. 129 (1963) 138. 4. J. S. FAULKNER, H. L. DAVIS and H. W. JoY, Phys. Rev. 161 (1967) 656. 5. E. B. KENNARD, D. KOSKIMAKI, J. T. WABER and F. M. MUELLER, Electric Density of

States (edited by L. H. Bennett, (1971) NBS Special Publication 323. 6. E. C. SNOW, Phys. Rev. 158 (1967) 683. 7. S. WAKOH, J. phys. Soc. Japan 20 (1965) 1894. 8. E. C. SNOW and J. T. WABER, Phys. Rev. 157 (1967) 570. 9. E. C. SNOW, Phys. Rev. 171 (1968) 785.

10. J. F. JANAK, Phys. Lett. 28A (1969) 570. 11. E. C. SNOW, Phys. Rev. 172 (1968) 708. 12. R. L. JACOBS, J. Phys. C1 (1968) 1296, 1307. 13. R. A. BALLINGER and C. A. W. MARSHALL, J. Phys. C2 (1969) 1822. 14. N. E. CHRISTENSEN, Phys. Stat. Solids 31 (1969) 635. 15. N. E. CHRlSTENSEN, Phys. Stat. Solids (b) 54 (1972) 551. 16. S. KUPRATAICULU and G. C. FLETCHER, J. Phys. C2 (1969) 1886. 17. N. E. CHRlSTI~NSEN and B. D. SERAPHIN, Phys. Rev. B4 (1971) 3321. 18. G. GILET and L. J. RAUBENr~EIMER, Phys. Rev. 144 (1966) 390. 19. P. B. ALLEN, M. L. COHEN, L. M. FALICOV and R. V. KASOWSKI, Phys. Rev. Lett. 21

(1968) 1794. 20. R. W. STARK and L. M. FALICOV, Phys. Rev. Lett. 19 (1967) 795. 21. T. B. MASSALSKI, U. MIZtJXANI and S. NOGUCHL Proc. R. Soc. A343 (1975) 363. 22. S. T. INOUE and J. YAMASHrfA, J. phys. Soc. Japan 35 (1973) 677. 23. H. JONES, Proc. phys. Soc. A49 (1937) 250. 24. J. M. ZIMAN, Adv. Phys. 10 (1961) 1. 25. U. MIZUTANL S. NOGUCHI and T. B. MASSALSKh Phys. Rev. 135 (1972) 2057. 26. B. W. VEAL and J. A. RAYNE, Phys. Rev. 130 (1963) 2156. 27. U. MIZUTAN~ and T. B. MASSALSK1, J. Phys. F5 (1975) 2262. 28. A. BANSIL, H. ErmENRElCH, L. SCnWARVZ and R. E. WATSON, Phys. Rev. B9 (1974) 445. 29. J. BEVK and T. B. MASSALSKI to be published. 30. F. S. HAM, Phys. Rev. 128 (1962) 73; ibid. 128 (1962) 89. 31. H. L. SKRIVER and N. E. CHRISTENSEN, Phys. Rev. 118 (1973) 3778.

4. MODELS OF THE ELECTRONIC STRUCTURE OF ALLOYS

4.1. Rigid Band Approximation

The subject of the electronic structure of alloys has been considered in many comprehensive papers and reviews. ~I 6) Below, we briefly mention a few general features of the proposed models, particularly as they relate to the density of states. An early model of an alloy band structure was proposed in the 1930s by Jones ~) (1937) and Mott and Jones ~8) (1936). This "rigid band" model assumes the shape of the density of states curve


of the solvent metal to be unchanged on alloying and its position on the energy scale to be fixed. The added or subtracted electrons merely change the position of the Fermi level Ev. Thus:

N(Er) = No(E ° + AE), (4-1)

where E ° is the Fermi energy of the host metal and AE is assumed to vary in response to the change in electron concentration, A(e/a). Using this approach, Mott and Jones (1936), and others, could account fairly well for a number of experimental results.

A subsequent treatment of an alloy band structure, by Friedel (3~ (1954) took into account the screening effect near the impurity atoms. When a foreign atom is dissolved in a metal, the electronic structure of the host becomes locally perturbed. Friedel derived an expression for a system of atoms with a screening potential Vp, analogous to the Thomas-Fermi approximation for free electrons:

p(E, r) = po(E - Vp, r), (4-2)

where Po (E, r) and p (E, r) represent the integrated electron densities at a position r in the pure host metal and in the alloy, respectively, with energies less than E. If the screening potential Vp is assumed to be small compared to Ev, the total density of electrons with energies less than E in the alloy can be integrated to yield:

f J • '~(E) = V - l p ( E , r ) dr = V - l - po(E- Vp, r) dr

--v-*.fpo(E,,')d,.-V-'. f?po . ) ~ Vj, d;

- - , " o ( E - V ' f Vedr t , (4-3) /

where V is the volume. Thus, the band structure preserves its form on alloying but is shifted as a whole by an average perturbation

t

A E = V - l I Veell'. (4-4) !

As a consequence of eq. (4-4) the density of states can be expressed a s

N(E) = No(E - AE), (4-5)

where AE now represents the displacement of the whole conduction band. The above solution becomes, of course, less accurate as the energy E decreases below Ev and approaches the screening energy Vp. The situation with respect to the Fermi level is as follows. When the screening potential is subject to the boundary condition of charge neutrality within a sphere


of radius R, which is considered to surround each impurity atom, the Poisson's equation can be written within the Thomas-Fermi approximation.

V2Vp( r ) = - 4 g [ p o ( E F + A E F - Vp, r) - po(Er, r)], (4-6)

which, to a first order approximation in (AE t - Vp), becomes

V 2 Vp(r) = 4~No(EF)" (Vp - AEF). (4-7)

The self-consistent solution for Vp in eq. (4-7) is easily found when a single impurity atom with excess charge Z is dissolved in a metal (R -- :c):

Z Vp(r) = -- e x p ( - qr), (4-8)

r

with the condition that A E f = 0 and q2 = 4rcN0 ( g f ) , where 1/q is the screening length. Thus, the electrons are required to adjust themselves to screen the impurity potential Vp in such a way that the region outside the screening potential range is not affected by the presence of the impurity and the Fermi level remains the same as in pure metal. As a result, in order to screen the impurity charge, the wave function must have a larger amplitude around the impurity atom than elsewhere. At first sight, the resulting perturbation might be considered to alter the band structure substantially. However, Friedel found that to a first order, the band is

only shifted as a whole to lower energies by an amount proportional to the screening field, as indicated in eq. (4-5). Its shape, however, remains unaltered. For a finite concentration (R finite), the solution of eq. (4-6) is more complicated and a change in the Fermi level is predicted on alloying:

A E f = Z q / ( q R c o s h qR - sinh qR), (4-9)

as a consequence of impurity-impurity interaction. ~3~ A good illustration of the above effects is provided by the recent UPS

data obtained by Nilsson ~9) in Ag-In alloys (Fig. 4-1). An addition of 13 at.~o In produces both a displacement of the Fermi level (by 0.4 eV), and a displacement of the Fermi level (by 0.3 eV). Hence, the total energy change on alloying, when measured from the bottom of the conduction band, is 0.7 eV, which is more or less consistent with the corresponding value of 1 eV characteristic of a rigid band.

4.2. Optical Properties Related to the Band Structure

In order to test which band model is more appropriate in a given situation, to describe the electronic structure of an alloy, it is of interest to consider optical data results in various ~ phase noble metal alloys. The measurements made by various investigators are summarized in Table 4-1. In this section, we discuss only the data for the ~ phase Cu-based alloys,


0.4 eV --I I"

o I J "~ Ag I "6 I

~'~ / / ~ A g . --" rl 0"3 eV E3

I I

/ I I I I I 1 I I I

-8 -6 -4 -2 0 Electron energy (eV)

FIG. 4-1. Schematic diagram of the density of states curve in the conduction band of Ag and :(-phase Ag In alloy. The kink originates from the contact

of the Fermi surface with the 't 111] zone planes (from Nilsson(%.

which have been most extensively studied among the three host noble

metals. Typical absorp t ion spectra in the Cu Zn alloys (1°~ are shown in

Fig. 4-2. As a general feature it can be said that the ma in absorp t ion edge near 2eV in Cu is shifted towards higher energies on alloying, while the secondary peak at about 5eV moves to lower energies. However, in

Cu-based alloy systems with higher valence solute atoms such as Ga, A1,

Table 4-1. Optical Data in Various c~ Phase Noble and Internoble Metal Alloys

Alloy system Method of measurement Authors Reference

Cu Zn calorimetric absorption Biondi and Rayne (1959) 10 Cu Ge calorimetric absorption Rayne (1961) 11 Cu Zn polarimetric absorption Pells and Montgomery 12

(1970) Cu Ga polarimetric absorption Pells and Montgomery

(1970) Cu Ge polarimetric absorption Pells and Montgomery

(t970) Cu As polarimetric absorption Pells and Montgomery

( 1970} Cu AI reflectivity Rea and DeReggi (1974) 13 Ag In reflectivity Morgan and Lynch (1968) 14 Ag Zn reflectivity Green and Muldawer 15

(1970) Ag Cd reflectivity Green and Muldawer

( 1970) Ag AI reflectivity lrani e t a / . (1971) 16 AuA1 reflectivity lrani et al. (1972) 17 Au Zn reflectivity Weiss and Muldawer 18

(1974) Cu Au reflectivity Scott and Muldawer 19

(t974)


~ 4 0 ,.m

.8 <

8O

6O

2 0 -

0 2000

I I A B C D E F G

I 3 0 0 0

I I

•t G v 30% Zinc

I J I 4000 5000 6000 7000 Wovelength (4)

A o Copper

B • 21/2% Zinc

C ~ 5 % Zinc

D A I 0% Zinc

E o 15% Zinc

F . 2 0 % Zinc

FIG. 4-2. Optical absorption data for Cu-Zn alloys in visible and ultraviolet region (from Biondi and Rayne~l°)).

Ge and As, the shift of the absorption edge is less marked. In spite of the controversies regarding the interpretation of the secondary peak, as discussed below, there seems to be little doubt that the main absorption edge can be attributed to transitions from the top of the d band to the Fermi level (L"3---, Ev): as is indicated by an arrow in the band structure of Cu in Fig. 4-3. Pells and Montgomery t121 (1970) plotted the absorption edge as a function of e/a in the alloy systems mentioned above, which is reproduced in Fig. 4-4. Here, the broken line representing the Friedel theory was derived from eq. (4-9) with an appropriate q value as determined by Biondi and Rayne i1°~ (1959). The experimental results favor the Friedel theory, compared with the Mott and Jones rigid band, but even for the former, the agreement becomes poor as the valence of the solute species decreases. Pells and Montgomery attributed this apparent solute dependence of the band shift to a thinning effect of the C u d band width, caused by a reduction in the total number of Cu atoms on alloying. This, in turn, would tend to increase the energy difference between the top of the d band and the Fermi level, provided that the center of the d band remains unchanged in its position, in fact, a narrowing of the Cu d band in the ct brass was recently confirmed, both by a band calculation (Bansil et al. 124) (1974)) and by XPS measurements (Andrews and Hiss- cott, 125) 1975).


r

/ u

J

W L. X I- K

FIG. 4-3. The band structure of Cu showing principal direct transitions (from Mueller and Phillips~23~).

The interpretation of the secondary absorption peak (or peaks) near 5 eV in Cu has been controversial and, at present, seems to be unresolved. Pells and Montgomery observed a splitting of the secondary peak on alloying and found that the dominant "lower" peak moves to lower energy faster than the "upper" peak. Earlier results of Biondi and Rayne ~1°) are similar to this, but they fail to reveal the splitting. Rayne ~1~ (1960) attributed the secondary peak to the transition from the Fermi level to the next higher band across the {11 l~j zone planes (Ev ~ L~). It can be seen from Fig. 4-3 that this transition is a non-direct (without momentum con- servation) transition, since there is no state having the Fermi energy at the L point in copper.

Lettingtonl21) (1965) associated the secondary peak with transitions from the d band to a higher band (X5 ~ X4), based on the earlier suggestions by Ehrenreich and Phillips ~221 (1962). The downward shift in energy of the secondary peak was, then, interpreted as a downward displacement of the whole conduction band, by assuming that X4, moves with the whole conduction band, while the position of the d band (Xs) remains unchanged on alloying. Furthermore, Lettington argued that a fairly small displacement of the Fermi level with respect to the d band, deduced from the shift of the main absorption edge, follows essentially a rigid band model, if the Fermi level is measured with respect to the bottom of the conduction band. With this latter condition there is essentially little difference between the two approaches, by Mott and Jones, and by Friedel. However, Pells and Montgomery considered Lettington's interpretation to be disproved by the band calculation of Mueller and Phillips ~231 (19671. Instead, they proposed that the lower and higher peaks are associated with the transitions L 2, --* L~ and L] --~ EF, respectively. The downward shift of the lower

178

3 . S

P R O G R E S S IN M A T E R I A L S S C I E N C E

i i i

> 3,0

" o e~

Rigid band

.~ ~ model

I

I / i

i i s -

/

i Zn i u - d

/

/

~ 25~ ... ~": ~ x T / C u - G a

' l / ~ ~ ~ ~ / l~Cu-Ge | . J T .T. ~ _ . . . . -

- \

. . . . . J" . . . . . . . . F'riedel model

1.0 1.1 1.2 1.3

e l e c t r o n concent ra t ion

1 . 4

FIG. 4-4. The energy corresponding to the main absorption edge as a function of e/a, as compared with the Mott-Jones rigid band model and Friedel model

(from Pells and Montgomeryt12)).

peak is, thus, considered to correspond to a decrease of the order of 1 eV in the energy gap across the {111} zone planes throughout the entire width of the ~ phase range in Cu-Zn alloys. However, the Bansil et al. ~24) calculation for the ~ phase Cu Zn alloys disagrees with this conclusion and suggests that the transitions X5 ~ X4,, or L]---* EF, are more likely to be responsible for the lower peak, while the transitions LE, ~ L~ are responsible for the higher peak.

Rea and DeReggie (13) (1972) measured optical dielectric spectra for a series of ~ phase Cu-A1 alloys and analyzed four marked threshold features A, B, C and D, as shown in Fig. 4-5 as follows: feature A represents the transition from the d band to the Fermi level, and features B and C are attributed, respectively, to non-direct and direct transitions from the Fermi level to the states at, and near, L]. Feature D involves a transition from deeply lying d states to the Fermi level. According to this analysis, feature C is assumed to be replaced by feature B at about 2 a t . ° . There is no clear indication why the non-direct transition B becomes dominant on alloying; but the analysis suggests that the Fermi level remains almost constant relative to the top of the d band, while the whole conduction band sinks at the rate of 0.1 eV per at.~o A! relative to EF. Assuming a "rigid" band situation for the sinking rate of the conduction band, the calculated increase in the [111] neck radius of the Fermi surface is in good agreement with the experimentally derived change derived from the positron annihilation measurements in this system. (See Fig. 8-1.) For

E L E C T R O N I C S T R U C T U R E O F H U M E - R O T H E R Y P H A S E S 179

i I

6 r T , i I ' ' ' '

5 - --~p~O.......O D \ ~ L .

0 1 J i i 1 I I r = , l C u 5 10

ot.%AI

Fl~3. 4-5. Location in energy of threshold-like features A, B, C and D in the room temperature optical absorption spectrum in Cu A1 alloy system. Upper right insert shows the E k relationship near the L point (from Rea and

DeReggi~l 3~).

the present, no decisive interpretation has been put forward regarding the secondary peaks; but the optical data certainly favor the Friedel model which emphasizes the band shifts on alloying.

4.3. Band Models and Electronic Spec![ic Heat

While recent progress in the various optical and spectroscopic techniques makes it possible to study effects related to energy transitions, band shifts, etc., the large bulk of the electronic specific heat data in alloys, related to the density of states, cannot be used to differentiate between an approach that considers a band shift vs. one that considers merely a shift in the Fermi level. This is because specific heat values are measured as a function of alloy concentration, i.e. (e/a), and can be translated into a function of energy only if the band form is already known. For this reason the most common way of evaluating the behavior of electronic specific heat trends in alloys is by considering an essentially unchanging (Mott and Jones) electronic band, and this works quite well in numerous cases where the essential characteristics of the band, such as its shape, appear to be altered very little by alloying.

Already some 10 years ago the electronic specific heat coefficients in the 3d, 4d and 5d transition metal alloys have been quite successfully explained using an essentially rigid band approach (see the reviews by Mott 126) (1964) and Heiniger et al. (2v) (1966)). As will be shown below, a similar interpretation is possible also for certain intermediate phases based on the noble metals. In this section, we mention only the behavior


of 7 coefficients in the 5d bcc transition metal alloys (Hf, W, Ta and Re). As can be seen from Fig. 4-6, a very good agreement exists between

values (corrected for electron-phonon interaction using McMillan's formula, see below) and the calculated density of states curves for W ~28'29~ or Ta (29'3°~. On the other hand, in some other systems, the failure of the rigid band model and a need for an adequate description of the d band, is quite evident. Ehrenreich et al. (31-33~ (1968 --- 1970) have shown, for example, that in the Cu-Ni system the rigid band approach fails to account adequately for the behavior of the Curie temperature or the electronic specific heat. They suggested a model which leaves the position of the Cu and Ni d bands essentially unchanged. Subsequently, X-ray photoelectron spectroscopy measurements by Hiifner et al. ~34) (1973) clearly demonstrated the correctness of this approach for both Cu-Ni and Ag-Pd series. The overall density of states curves indicate separate d bands characteristic of the pure metals, while the change in the density of states at the Fermi level, as deduced from 7 measurements, is fairly adequately accounted for by the CPA (coherent potential approximation) band calculations for Cu-Ni (Stocks et a/. (35)) and Ag-Pd (Stocks e t a / . , t36) 1973). These calculations indicate the emergence of virtually bound states in the dilute range of Ni in Cu or Pd in Ag, as originally proposed by Friedel t3v) (1963).

7

6

2

1

0 1

A 5

3

!E i ,i_. :~

i i i i i i i I

2 3 /0 5 6 7 8 9 10

electron concentration (e I a)

FlG. 4-6. The histogrammic plot shows the density of states values at E r obtained from the calculated N(E) curve for Ta( ) and W(---) . All experimental data are corrected for electron-phonon interaction using McMillan formula. (0 Ta-W, ~ H~Ta, ( ':' W-Re and A Ta-Re). See references (28, 29,

30) and Mamiya et al. (J. phys. Soc. Japan 28 (1970) 380.)


It seems that when the rigid band approach becomes inadequate for a description of an alloy system it is generally for one of two reasons. (1) The screening potential Vp is so large that the first order perturbation is no longer suitable within the framework of Frieders theory. According to Stern (38) (1966), the perturbation theory is appropriate only when the difference between the atomic potentials of the participating atoms is suffi- ciently small, regardless of solute concentration. Conversely, a large difference in atomic potentials results in a transfer of screening charge between unlike atoms accompanied by a breakdown of the rigid band concept. However, as the solute concentration increases and the screening clouds begin overlapping, the concept of screening may lose its intended meaning, and the rate of displacement of the Fermi level may be expected to relate more closely to the change in the nominal electron concentration, as suggested earlier by eq. (4-1t. Thus, we may expect the breakdown of the rigid band approach to be more often evident in the primary solid solutions, when the introduction of solute atoms produces a substantial initial effect, than in the concentrated alloys, when the average potential tends to be evened out. As will be described later, this idea can be supported by the behavior of the ;, coefficients in the noble metal alloys. However, the concept of a "rigid band" in a concentrated solid solution need not be taken to mean that some initial band characteristics of the pure metal components have been preserved without change. Rather, it may be considered that when a considerable charge transfer and local potential adjustment have taken place on alloying, the resulting solid solution resembles a virtual crystal and is relatively little perturbed by further alloying.

40

3O

t ~

~ 2 0

h J

Z

10

Cu

i ~ ~ I ~ I

I ~ v ~ I I

I /

/ - I

- 0 . 6 - 0 . 4 - 0 . 2 energy (Ry)

Ni _j

F

E

FIG. 4-7. The calculated density of states curves for Cu and N1 are shown, making the two Fermi levels coincide. Since the work functions for both metals are close to each other, this common energy scale is reasonably ,justified as the absolute scale measured with respect to the vacuum level. The N(E) curves

are depicted from reference (35).


(2) Each of the constituent atoms contributed by one of the alloying partners has associated with it a band of localized states (in its pure state), which may be quite different in energy compared to similar states in the other partner. In this case, one common band may not be expected on alloying, but rather the localized bands originating from each constituent will tend to appear separately in the alloy. Thus, while the rigid band model, or the virtual crystal approximation, may be qualitatively compat- ible for similar band states, as in Fig. 4-6, they may completely fail for localized states. For instance, as already mentioned above, a complete breakdown of the rigid band model is evident in the Cu-Ni and Ag-Pd alloys, where the positions of the d bands, characteristic of each original constituent, are quite different in energy, when measured with respect to the Fermi level. The results are shown in Fig. 4-7.

REFERENCES

1. Phase Stability in Metals and Alloys (edited by P. S. Rudman, J. Stringer and R. I. Jaffee), (1967) McGraw-Hill, New York.

2. T. B. MASSALSKI and H. W. KING, Prog. mater. Sci. 10 0961) 1. 3. J. FRIEDEL, Adv. Phys. 3 (1954) 446. 4. J. FRIEDEL, Trans. AIME 230 (1964) 616. 5. V. HEINE and D. WEAIRE, Solid State Physics (edited by H. Ehrenreich, F. Seitz and

D. Turnbull), 24 (1970) p. 249. Academic Press, New York, London. 6. Alloyinq Behavior and EfJects in Concentrated Solid Solutions (edited by T. B. Massalski),

(1965) Gordon & Breach, New York. 7. H. JONES, Proc. phys. Soc. A49 (1937) 250. 8. N. F. MOTT and H. JONES, The Theory of the Properties o[' Metals and Alloys (1936),

Oxford at the Clarendon Press. 9. P. O. NILSSON, Phy. Scripta I (1970) 189.

10. M. A. BLOND! and J. A. RAYNE, Phys. Rev. 115 (1959) 1522. 11. J. A. RAYNE, Phys. Rev. 121 (196l) 456. 12. G. P. PELLS and H. MONTC3OMERV, J. Phys. C3 (1970) $330. 13. R. S. REA and A. S. DEREGGI, Phys. Rev. B9 (1974) 3285; Phys. Lett. 40A (1972) 205. 14. R. M. MORGAN and D. W. LYNCH, Phys. Rev. 172 (1968) 628. 15. E. L. GREEN and L. MULDAWER, Phys. Rev. B2 (1970) 330. 16. G. B. IRANI, T. HUEN and F. WOOTEN, Phys. Rev. B3 (1971) 2385. 17. G. B. IRANI, T. HUEN and F. WOOTEN, Phys. Rev. B6 (1972) 2904. 18. D. E. WEISS and L. MULDAWER, Phys. Rev. BI0 (1974) 2254. 19. W. SCOTI and L. MULDAWER, Phys. Rer. B9 (1974) 1115. 20. J. RAYNE, The Fermi Surface (edited by W. A. Harrison and M. B. Webb), (1960) p.

266, Wiley, New York. 21. A. H. LETIINGTON, Phil. Ma 9. 11 (1965) 863. 22. H. EVmENREICH and H. R. PHILLIPS, Phys. Rev. 128 (1962) 1622. 23. F. M. MUELLER and J. C. PHILLIPS, Phys. Rev. 157 (1967) 600. 24. A. BANSlL, H. EHRENREICH, L. SCHWARTZ and R. E. WATSON, Phys. Rev. B9 (1974) 445. 25. P. T. ANDREWS and L. A. HISSCOTT, J. Phys. F5 (1975) 1568. 26. N. F. MOTT, Adv. Phys. 13 (1964) 325. 27. F. HEINIGER, E. BUCKER and J. MUELLER, Phys. Kondens. Mat. 5 (1966) 243. 28. L. F. MATTHEISS, Phys. Rev. 139 (1965) A1893. 29. I. PETROFE and C. R. VISWANATHAN, Phys. Re~:. B4 (1971) 799. 30. L. F. MATTHEISS, Phys. Rev. B1 (1970) 373. 31. B. VELICKY, S. KIRKPATRICK and H. EHRENREICH, Phys. Rev. 175 (1967) 747. 32. S. KIRKPATRICK, B. VEL1CKY, N. D. LANG and H. EHRENREICH, J. appl. Phys. 40 (1969)

1283.


33. S. K1RKPAFRICK, B. VELICKY and H. EHRENREI('H, Phys. Rer. BI (1970) 3250. 34. S. H~':VNI-:R. G. K. WER'IH~IM and J. H. WERNICK, Phys. Rec. B8 (1973) 451 I. 35. G. M. STOCKS, R. W. WILLIAMS and J. S. FAULKNER, Phys. Rer. !~ (1971) 439(/. 36. G. M. STOCKS, R. W. WILLIAMS and J. S. FAULKNER, J. Phys. F3 (1973) 1688. 37. J. FR1EDEL, Metalic Solid Solutions (edited by J. Friedel and A. Guinier), XIX (1963)

p. 1, Benjamin. New York. 38. E, A. STERN, Phys. Rev. 144 (1966) 545.

5. CONNECTIONS BETWEEN PHASE STABILITY AND

ELECTRONIC STRUCTURE

5.1. The Model o f Jones

Early theories of stability of alloy structures have readily identified the important role of the density of states curve as a possible basis for correlat- ing the extent of phase stability with the electron concentration (e/a). Jones tl} (1937), in an often quoted paper, attempted to interpret the phase competition between the ~ and fl phases in the Cu-Zn system in terms of the rigid band model for the conduction electrons. The gist of the argument was that, within the low e/a range corresponding to the a phase, the total density of states for the ~ structure is higher than for the [3 structure which follows it, because of the prominent peak in the density of states curve associated with the {1111 zone contact with the Fermi surface. Thus, the conduction electrons can be accommodated within lower total energy than for the [3 phase. As e/a increases, the 110 peak in the density of states curve of the fl phase structure becomes prominent and the energy relationship is tilted in favor of the fl phase. This type of argument does not, of course, require that the respective phases should be most stable where the peaks are most prominent, but it has often been presented as if this were a requirement.

The discussion of phase competition in terms of electronic structure was considered further by Jones 12) (1962) in terms of the relative electronic density of states curves between any two simple alloy phases. Assuming for the moment that the ranges of stability will be determined by the competition between the respective electronic energies one can write:

r U1 -- Nl (E)E dE; U2 = N2(E)E dE, (5-1) ,:0

while the corresponding total number of electrons per atom can be expressed as

(e/a)1 = NI (E ) dE; (e/a)2 = Nz(E ) dE. (5-2) JO

The first derivative of the energy difference with respect to e/a, between two phases, 1 and 2, is given by

d(A U)/d(e/a) = d( U 1 - U 2)/d(e/a) = Ev 1NI(EF 1 )dEr 1/d(e/a)

-- E F 2 N 2 ( E F z ) d E F 2 / d ( e / a ) = EF1 -- EF2, (5-3)


N(E)

T : s ~-E

AU ° , Si

, ,~

e l o

e/a

e l o

d(~J]Id(ela) I

curvature of ~U

P Q R

- 0 + a

- ~ + I

FIG. 5-1. Schematic illustration of the model of Jones for the stability competition between phases 1 and 2.

which shows that, at any given e/a, the stability difference between the two competing phases changes at a rate proportional to the difference in their respective Fermi energies. Similarly, the second derivative becomes

d2(AU) /d (e /a ) 2 = dEF1/d(e /a) - dEr2 /d (e /a ) = 1 / N I ( E v l ) - 1 /NE(Ev2) . (5-4)

It follows that the relative stability of an alloy phase will be enhanced if its density of states curve involves a large peak and a subsequently rapidly declining slope, while that of the competing phase is fairly mono- tonic in a given e/a range. This may be illustrated by a set of diagrams as in Fig. 5-1. The density of states in phase 1 increases above the free electron trend because of an imminent contact between the Fermi surface and the Brillouin zone (portion SP in Fig. 5-1), and then decreases following contact at point P (portion PR). At point R areas SPQ and QRT are equal and hence EF1 = E r 2 . The difference AU in the integrated energies has reached its largest value and the curve shows a minimum at R, which is to the right of P on the energy scale. The minimum in


energy difference curve (i.e. the largest relative gain) is reached not at the contact peak P, but at the point R on the decreasin9 slope of N(E) past the peak. A similar situation will occur for any two real phases if one has a steeply falling density of states curve (following a peak) in the range of e/a of interest, while the other is nearly free electron curve in the same range. It is clear that for the same number of electrons per atom, the corresponding Fermi levels must cross in the critical region where the two phases are competing. Since the total energy U is always an increasing function of e/a, the presence of a minimum in AU causes a change in the overall curvature of the total energy U as shown in Fig. 5-1.

5.2. Pseudopotential Considerations

In another approach, Heine and Weaire(3)(1970) have proposed that the energy difference related to the FsBz interaction (contact and overlap effects) and their contributions to the band structure energy U, are insufficient to account for the observed differences in phase stability. According to these authors, even the presence of a fairly large energy gap in the E-k curve (note, however, that a single Brillouin zone plane was assumed in this discussion), would result in only a small van Hove singularity in the density of states curve, which would tend to be further reduced and smoothed out in the computed total energy. Thus, the band structure energy is not expected to have any significant singularity in the energy range related to the FsBz interaction. This implies that, although the discussion based on Fig. 5-1 is implicitly correct, the assumed peak in the density of states curve, as well as the subsequent minimum in the energy difference curve, are unrealistically exaggerated. However, the test example by Heine and Weaire was applied only to a hypothetical situation.

In Section 9, we compute the difference AU for several cases for which reasonably realistic density of states curves can be constructed for the respective competing phases characteristic of the noble metals. The corresponding difference AU due to the FsBz interaction is, at most, some 50-100 cal/mole. This is only about 0.1°o of the total band structure energy. Nevertheless, it is this magnitude of the energy difference which must be held responsible for the Hume-Rothery rule if the Fermi surface model is basically correct.

On the other hand, Heine and Weaire state that "the traditional ideas which associate a special stability with a Brillouin zone plane touching the Fermi sphere or Fermi surface, are simply not correct in general" and that "'the success of Jones' model was a coincidence and not a demon- stration of the correctness of current ideas on the FsBz interaction". In place of the "'traditional idea", a view is expressed that the rough correla- tions of alloy structures with electron concentration probably fall within the scope of an approach based on the second order perturbation approxi-


mat ion with the aid of the pseudopotent ial formulation. There have been several at tempts to determine the stable structure for a part icular element, or to interpret the electron concentra t ion Hume-Rothe ry rule in alloys from this point of view, as opposed to the application of the Jones model.

The considerations based on pseudopotentials will now be considered briefly. Within the framework of the pseudopotent ial theory, the electron energy for a state k is expressed to the second order per turbat ion as

E(k) = hZk2/2m + (klwlk~

+ ~ S*(q)S(q)(k[wlk + q ) ( k + q[wlk)/hZ/2m[k 2 - ] k + q[2], (5-5) q

where w is the individual ion pseudopotential . Thus, the form factor ( k ] w l k + q) is independent of the structure, while the structure factor S(q) depends only on the structure. In comput ing the total energy of the system, the summat ion of eq. (5-5) is carried out over an undistorted Fermi surfacer and is combined with the electrostatic energy of positive ions embedded in a uniform electron gas (the Madelung energy). The summa- t ion of the third term over the occupied k states may be referred to as the band structure energy Eb~ and is written as

E~ = ~ S*(q)S(q)V(q), (5-8) q

where

( (k]w(r)lk + q ~ ( k + q]w(r_)[k) d3k, (5-9) F(q) = ~ 3 j (hZ/2m)Ek 2 - Ik + ql 2]

F(q) is the energy-wavenumber characteristic and is independent of the structure,

The success of the pseudopotent ial approach depends mainly on the appropr ia te evaluat ion of the form factor, i.e. the pseudopotent ial itself. Since the structure dependent term appears only as the last term in eq. (5-5), one of the advantages of the pseudopotent ial approach stems from the fact that the rather small structure dependent term can be separated

+ In Jones' model, the total electron energy is written as

U = f N(E)E dE, (5-6)

where the electron ion interaction, i.e. the FsBz interaction, is brought in through the density of states curve N(E) with the characteristic peaks and valleys. In the pseudopotential scheme, the summation of eq. (5-5) is carried out over spherical Fermi surface, since the FsBz interaction, as well as the electron electron interaction, is included via the pseudopotential w. Thus, for instance, the summation of the first term in eq. (5-5) leads simply to the kinetic energy of the free electron gas;

U = ~ h2k2/2m = INf(E)EdE, (5-7) k ~ k ~ d

which is independent of s tructure. NI(E) refers to the densi ty of s tates of free e lect rons with an appropriate atomic volume (see eq. (3-2)).


from the much larger volume dependent terms. Thus, the approach is particularly suitable for comparison of relative stability among alternative crystal structures, such as the simple structures fcc. bcc and hcp. For instance, Harrison (4) (1966) predicted the lowest energy structures at 0 K for Na, Mg and A1 to be hcp,¢ hcp and fcc, respectively, by assuming the band structure energy plus the Madelung energy, of the three alternative crystal structures for each element.

The extension of the pseudopotential approach to the discussion of alloy phase stability has also been explored further by lnglesfield (6'7) (1969a, b), Stroud and Ashcroft 18) (1971) and Krause and Morris I')~ (1974). Assuming an alloy to be composed of (1-c)N atoms of type A with ionic potential u.dr - ra) centered at the positions rx and cN atoms of type B with ionic potential u d r - ru), one can write the alloy pseudopotential matrix element W(q) in the form:

( k + q lW(r ) l k ) = W(q) = S~(q)u~x(q) + St~(q)u,~(q)

= S(q)udq) + S,,(q)(ut~(q) - u.dq)), (5-10)

where w(~) = ~ u,(," - , ' , )+ Z ".(" - ,',4

i . A I " l l

and the three structure factors are defined as

1 1 S(q) = ~ T ~ exp(iq " ri), S,,(q) = ~ ~ exp(iq " riA)

and

1 S,~(q) = ~ ~ exp(iq • ri,,). (5-12)

The summations i for A and B extend over lattice sites of A and B, respectively. S(q) is unity for the reciprocal lattice vector qo in the average lattice, and zero otherwise, if the origin of coordinates is chosen at an ion site of the Bravais lattice.++ It follows that SA(qo) = (l-c) and SB(qo) = c. Equa- tion (5-10) is now reduced to the form

W(qo) = (1 - C)UA(qO ) ~- CUI~(qo) , (5-13)

which is independent of the arrangement of A and B atoms. In the case of q :# qo, S(q) is always zero, but SA(q) and S~(q) do not vanish; the first term of eq. (5-10) in the extreme right hand side always vanishes and the matrix element becomes

W(q) = Sl,(q)'(ul,(q ) -- udq)) , (5-14)

+Sodium is k n o w n to undergo a phase t rans i t ion at ~ 40 K from the bcc to hcp modif ica- tion. The hcp s t ruc ture is considered to be the s table s t ruc ture at 0 K. ~5'

++A more compl ica ted lat t ice can be cons t ruc ted from the Bravais lat t ice by add ing a basis of a toms to each lat t ice point. The s t ruc ture factor S(q) in such s t ruc tures is zero, if q is not equa l to qo- However , even if q is equal to qo, S(q) may vanish at some special wdues. (See Har r i son 14), Sect ion 7.2.)


for any q @ qo- The band structure energy for the alloy pseudopotential (5-11) is now expressed as

Ebs = ~ Fav(qo) + ~ S*(q)SB(q)Fdif(q), (5-15) qo q ~ q o

where Edv(q) and Fdif(q) are obtained by inserting eq. (5-13) and (5-14) into (5-9), respectively. Note, however, that the band structure energy in eq. (5-8) becomes simply

Ebs = ~ F(q), q

when the total pseudopotential W(r) is inserted into eq. (5-9). The first term of eq. (5-15) represents the band energy contribution from the average potential (5-13), which is equivalent to the virtual crystal approximation. The second term, however, contains information about the particular arrangement of A and B atoms through the structure factor Sn(q). For example, Harrison ~4~ calculated the values of Sa(q) and SB(q) for a perfectly random alloy;

/ S,(q) = --SA(q) = vc(1 - c)/N, (q @ qo). (5-16)

Inglesfield ~v) directed his attention to the second term of equation (5-15) in order to discuss the stability ranges of some ordered intermetallic compounds, such as HgMg and HgzMg. The first term contribution was ignored, since the average lattice is not a major concern in the analysis. On the other hand, Stroud and Ashcroft ~8~ ignored the second term and discussed the phase stability in binary alloys Cu-A1, Li-Mg and Cu-Zn, assuming that the alloy potential is well described by a virtual crystal approximation in combination with the Linhard dielectric function. In this approximation, the alloy is treated as if it were a pure metal with a periodic potential given by a linear combination W = (1 - C)Ua + cub for the alloy A-B. The relative energies per atom are then computed for some alternative simple crystal structure such as fcc, bcc or hcp. However, the agreement with the phase diagrams for the three alloy systems under consideration was relatively poor. The lack of agreement can be attributed to several fundamental reasons; (a) the alloy pseudopotential is most likely not as simple as that proposed. Even for disordered alloys, the second term in eq. (5-15) cannot be ignored, since they are concentration as well as structure dependent; (b) the pseudopotential approach itself is inadequate for noble metals because of the presence of the d band in the middle of the conduction band, which was not evaluated; (c) the authors stress that the third term in eq. (5-5) is much larger than expected and is comparable with the first term. This suggests that the pseudopotential approach in terms of the second order perturbation becomes insufficient and that higher order terms may have to be taken into account.

The band structure energy in eq. (5-8) can be rewritten in the real space coordinates;


Eb~ = ~ N - 2F(q)exp(- i q ' ( r i - r j)) = ~ ~ Vi.d(ri -- r j) + F(q), (5-17) q i~-j

where

2 V ( r i j ) = V ( r i - r j) = ~ ~ F(q) exp ( - iq ' ( r i -- r j)) (5-18)

q

and is structure dependent, resulting from an indirect interaction between ions at different sites. The diagonal terms with i = j are included in the second term in eq. (5-17), which is independent of the structure. The function V(ri~) is often called pair-potential. Computation of this term requires a specific expression for the potential V(rij), which is sensitive to the detailed form of the pseudopotential (Harrisont4~). However, irrespective of the pseudopotential, V(r) has an asymptotic form for large values of kvr

V(r) ~ Vo cos(2kvr)/(2kvr) 3, (5-19)

where kv is the Fermi wavenumber and Vo depends on fundamental quantities in a complicated manner, but is independent of structure. Blandin ~1°~ (1965) demonstrated the equivalence of the pair-potential method with the "traditional" rigid band picture described above, by showing that the presence of the Friedel oscillation in eq. (5-19) leads to an infinite slope in the total energy when one of the reciprocal lattice vectors G is equal to 2kr. The condition kp = IG I/2 corresponds to the contact of the Fermi surface with the Brillouin zone. When Vo is positive, total electronic energy U is lowered when kr is near or greater than IGI/2. This is essentially the result as that obtained in the band approach.

Krause and Morris ~9) recently discussed the relative cohesive energies of simple metallic structures, by utilizing the asymptotic form eq. (5-19) of the pair-potential arising from the pseudopotential theory. Use of the potential allows the thermodynamically preferred structure of a simple metallic element or a solid solution at 0 K, to be determined as a function of e/a. Alloy solid solutions are treated as if they were composed of identical pseudo-atoms having average properties. Actually, this is equivalent to the virtual crystal approximation. The appropriate sums are performed exactly for the fcc, hcp and bcc structures. One of the results is reproduced in Fig. 5-2, where the most stable structures among fcc, hcp and bcc are depicted in terms of e/a and the shift 26. Here, a phase shift 26 is added to eq. (5-19) to allow the structure to depend on the Friedel shift, as well as on e/a;

V(r) = cos(2kvr + 26)/(.2kFr) 3. (5-20)

The inclusion of a phase shift has been suggested as a device for accounting for some of the effects of d electrons and Harrison argued that a phase


2"ff

0 2

electron concentration (e/a)

FIG. 5-2. The most stable structure among fcc, bcc and hcp as functions of e/a and the phase shift 26 calculated on the basis of the pair-potential approxi-

mation. (Krause and MorrisJ 9~)

shift appears in eq. (5-19), when non negligible third-order effects are present. Reasonable agreement between the model and empirical structure trends is obtained if one accepts the conclusion of Heine and Weaire (3) that 26 is small. Figure 5-2 shows that at e/a = 3.0 this range of 26 favors the fcc and hcp structures, which are empirically observed, but does not permit the bcc structure, which is not observed. At e/a = 2.0, all three structures fcc, hcp and bcc are favored over a small range of 26; all are in fact found in the divalent metals. At e/a = 1.0, the hcp and bcc structures occur with a moderate Friedel shift. Those are the structures found in monovalent alkali metals. However, the fcc structure of monovalent noble metals is not predicted by the model (see Heine and Weaire ~31, p. 388). The lack of agreement may be attributed to an incomplete treatment of the d electron contribution to the total energy. In addition, the choice of the ideal axial ratio for the hcp structure may prevent all hcp noble metal alloys from stabilizing in the electron concentration between 1.3 and 2.0. On the other hand, a phase boundary between fcc and hcp structures observed at about e/a = 2.2 in Fig. 5-2 is fairly well consistent with the experimental data. For instance, the fcc phase of A1 is stable with additions of Zn up to 66.5 at.% Zn and similarly the fcc phase of In is stable with the addition of up to 77at.}; Mg (actually, In is fct but the difference is probably slight).

On the whole, the application of the pseudopotential approach to the problem of the stability of alloy phases must be considered to be rather disappointing. The hope that the pseudopotential theory to the second term--was sufficient to deal with structure stability has not proved too fruitful and the idea that structures would be favored when lattice wave numbers came where F(q) had a large magnitude has only worked in


a few selected favorable examples, with a good deal of adjustable approximation. It seems that the second order pseudopotent ial theory may not contain all the features which determine the structure difference. ~1~

Recently Harr ison ~1 z) (1973) stressed the necessity for including the third order per turbat ion term, which explicitly introduces three-body interactions. Unlike the two-body central force interaction associated with the second term, the three-body force depends explicitly on the angles between interionic separat ions and he hoped that this is the impor tant term sensitive to structure. Indeed, Harr ison found that the three-body interaction becomes comparable in magni tude with the second order contr ibut ion when the three ions form a straight line and are separated by nearest neighbor distances. Fur thermore, assuming that the influence of the d band upon this third order term dominates the determinat ion of structures, he could qualitatively argue the correct distr ibution of cubic and hexagonal structures among the monovalent and divalent metals in the periodic table and also appropria te axial ratios among the hexagonal structures.

REFERENCES

I. H. JONES, Proe. phys. Soc. A49 (1937) 250. 2. H. JONES, J. Phys. Radium, Paris 23 (1962) 637. 3. V. HEINE and D. WEAmE, Solid State Physics Vol. 24 (edited by H. Ehrenreich, F. Seitz

and D. Turnbull) (1970) p. 249 Academic Press, New York, London. 4. W. A. HARRISON, Pseudopotentials in the Theory ~ff" Metals (1966} Benjamin, New York,

Amsterdam. 5. C. S. BARRI~TT and T. B, MASSALSKI, Structure ~fMetals (3rd edition) (1966) McGraw-Hill,

New York. 6. J. E. INGLESEIELD, J. Phys. C2 (1969) 1285. 7. J. E. INGLESHELD, J. Phys. C2 (1969} 1293. 8. D. STROUD and N. W. ASHCRO~'T, J. Phys, FI (1971} 113. 9. C. W. KRAUSE and J. W. MORRIS, JR, Acta Metall. 22 (1974) 767.

10. A. BLANDIN, Alloying Behavior and Effects in Concentrated Solid Solutions (edited by T. B. Massalski) (1965) p. 50, Gordon & Breach, New York.

11. W. A. HARRISON, private communication (1977). 12. W. A. HARRISON. Phys. Rev. B7, 2408 (1973).

6. ELECTRONIC SPECIFIC HEATS IN ALLOYS

6.1. Theory o f Electronic Specific Heat

In order to measure the trend of the density of states on alloying, the most direct parameter of interest is the electronic specific heat coefficient ? because of the well k n o w n relationship between 7 and the density of states at the Fermi level N(Er) . Over the past several years values of this coefficient and its trends on alloying, have been derived from heat capacity data obtained in the liquid helium range, in a large number of alloy systems, using the general relationship

Ct, = ?T + o~T 3 -q- 6 T 5, (6-1)


where Cv is the heat capacity at constant volume, T is the absolute temperature, and ct and 6 are lattice specific heat coefficients. The coefficient at is related to the limiting Debye temperature 0o by the relationship

o~ = (12rrg R/5)(1/Oo) 3, (6-2)

where R is the gas constant (R = 8.317J/mole.deg). From the point of view of experiment, the heat capacity is measured at constant pressure, rather than at constant volume, which is preferable for the theoretical description. The well known relationship between the two quantities Cp and Cv is given by

Cp - C v = f l 2 V T / t ¢ , (6-3)

where x, fl and V are isothermal compressibility, thermal expansion coefficient and volume, respectively. Since the difference between Cp and Cv is negligibly small at low temperatures, C, is commonly used for discussion of the low temperature specific heat data.

The electronic specific heat capacity is the temperature derivative of the internal energy Uel of a system of conduction electrons;

C~ = \ t3T ]v = ~ EN(E)f(E, Er, T) dE , (6-4)

where N(E) is the density of states and f(E, Er, T) is the Fermi-Dirac distribution function. Employing the well known expansion formula for the distribution function in the low temperature limit (kT ~ EF) (see, for example, Ziman(l)), eq. (6-4) can be expressed as

CeI = (rc2k2/3)N(Er(O))T + (28n4k~/360)K(EF(O))T 3 + .... (6-5)

where EF(0) is the Fermi energy at 0 K (note that Er in eq. (6-4) is defined at T K), kB is the Boltzmann constant and K(EF(0)) is a function involving up to third order derivative of N(E) with respect to energy E. The ratio of the second term over the first term is of the order of (1/Te) 2. Since the Fermi temperature TF is usually about 104K in normal metals and alloys, t the second term can be neglected and we can write eq. (6-5), with a fairly good approximation, as a linear function of T,

C¢~ = (rr2kE/3)N(Er(O))T. (6-6)

Thus, if the linear term can be extracted from the total heat capacity, and no other contribution is present in the linear term, the density of states at the Fermi level at 0 K is easily computed. In fact, the linear

tThe "normal" metals and alloys refer to as the state which does not involve superconductivity, magnetic contribution and any other specific behavior, which is not adequately described by the nearly free electron picture, in the temperature range of interest (i.e. 1.2 < T < 10 K). The noble metal alloys without any presence of ferromagnetic impurities are considered to be a typical example of normal metals.

E L E C T R O N I C S T R U C T U R E OF HUME-ROTHERY PHASES 193

coefficient ~ in eq. (6-1) may be determined graphically by plotting the data in the form of C/T vs. T 2, which generally results in a straight line with the intercept I' and the slope ct, provided that the ~ term is absent or small.

In experimental work ~, is usually measured in units of mJ/mole.K 2. However, the theoretical density of states curve is usually expressed in units of states/eV.atom or states/Ry.atom. The following conversion con- stants may be employed to allow direct comparison between 7 (given in mJ/mole.K 2) and N(Ev).

N(EF) = 2.647 x 1011 x 7 (states/erg.atom), = 0.4241 x 7 (states/eV.atom), = 5.7698 x 7 (states/Ry.atom).

As usual, two electrons with opposite spins are assigned in each electron state.

As has been realized over the years, the experimentally derived ?) value usually involves not only the contribution related to the conduction electrons at the Fermi level, but also some additional contributions, which sometimes modify the density of states at the Fermi level or add extra linear terms. Many-body effects, including the electron-phonon interaction and the electron-electron interaction, are well known contributions that affect the N(EF). Additional contributions to the linear term can result from magnetic interactions, imperfections or the amorphous nature of the crystal lattice. However, as far as the well characterized noble metals and alloys are concerned, the latter contributions can be usually avoided, with the result that only the many-body effects must be accounted for and subtracted, before the electronic band effects are calculated from an experimental 7exp value.

6.2. Many-Body Effects (Electron-Phonon Interaction and Electron-Electron Interaction)

The experimentally derived electronic specific heat coefficient, 7exp may be generally expressed as

~exp = '2b..d( 1 d- 2ep d- "~ee), (6-8)

where 7band is the contribution of the band structure given by eq. (6-6), and 2ep and 2ee are enhancement factors due to electron-phonon and electron-electron interaction, respectively. Quinn ~2) (1960) and Nakajima and Watabe ~a) (1963) showed that 2ep resulting from enhancement of the density of states at the Fermi level due to electron-phonon interaction varies from 0.3 to 0.6 for several alkali metals. Ashcroft and Wilkins ~4) (1965) estimated 2ep and 2e~ in simple metals Na, A1 and Pb, and found 2ee to be negligible, compared with 2ep, as can be seen from Table 6-1. In fact, as described later, independent estimates of the three quantities Yexp, 7band and 2~p


usually lead to quite satisfactory agreement with the simplified equation

~exp = 7band( 1 + 2), (6-9)

where 2 is related to electron phonon interaction alone. Under these con- ditions, the electron-phonon interaction in noble metals, which are not simple because of the presence of the d band near the Fermi level, can be estimated by comparing the experimental Yexp with the theoretically derived 7band. This leads to the values of 2 for Cu and Ag to be of the order of 0.1 and 0.05, respectively (see, for instance, Faulkner et al. ~5~ for Cu and Christensen (6~ for Ag). Recently Grimvall (7) (1970) estimated 2 values for all three noble metals by utilizing the high temperature resistivity data. The resulting range is between 0.1 and 0.15.

In superconducting materials, the electron-phonon interaction responsible for the onset of superconductivity can be estimated within the framework of the BCS theory. The superconducting transition temperature T~ can then be used to facilitate the evaluation of 2. According to McMillan ~s~ (1968), 2 can be evaluated for superconductors from the experimental values of T~ and the Debye temperature 0o;

2 = [1.04 +/~*1n(0~/1.45 T~)]/[(1 - 0.62/~*)1n(0o/1.45 T~) - 1.04], (6-10)

where/~* is the Coulomb pseudopotential originally introduced by Morel and Anderson w) (1962), which can be expressed in terms of T~, 0D and c~ (the isotope shift coefficient)

/t* = (1 - 2~)1/2/ln(0o/1.45 T~). (6-11)

Examining /~* values for several pure metals, McMillan suggested an appropriate value of #* to be 0.13 for the transition metals and 0.10 for the nearly free electron metals. Thus, for example, the experimental data plotted in Fig. 4-6 were corrected for electron-phonon interaction, using eq. (6-10) and a value o f / t * = 0.13. For Zn, McMillan found #* to be 0.12. Inserting T~ (= 0.85 K) and 0o (= 322.3 K) into eq. (6-10) we obtain 2 to be 0.415. However, if the estimate of the /~* value for Zn is changed from 0.12 to 0.10 (as for other free electron metals), 2 changes

Table 6-1. Many-Body Effects on the Electronic Specific Heat Coefficients (After Ashcroft and Wilkins ~4))

Na AI Pb

~/)ba nd/TF 1.00 1.06 1.12 2e~ 0.06 -0.01 0.00 2~p 0.18 0.49 1.05

)'b~d/TF[1 + )-~e + 2ev] 1.24 1.57 2.30 )'exp/TV 1.25 1.45 2.00

7F is the free electron value with appropriate atomic volume and electron concentration (see eq. 3-2).


from 0.415 to 0.382. The 7~xp for Zn (Massalski et al. ~°~ 1975) can now be compared with )'band derived from the band calculation (Allen et al/~ ~ 1968); 7e~p/Th~,,,J = 0.638/0.42 = 1.52 and, hence, 2 = 0.52. The agreement is reasonably good and it indicates the moderate success of McMillan's formula. It also identifies electron phonon interaction as the major contribution to 7~,,,, other than the density of states contribution.

Special caution is warranted when McMillan's formula is employed for alloy systems where the overall change in 7 is rather small or when the measured superconducting temperature T~ becomes exceedingly small, for example, tending to disappear in the middle of a phase. It is also worthwhile to bear in mind the approximations involved in eq. (6-10). Strictly speaking, the numerical coefficients in this equation are only valid for the pure metal Nb. McMillan stresses that "the formula was derived from accurate numerical solutions of the integral equations of the (accurate) theory of superconductivity with, however, a special assumption about the shape of the phonon density of states". But he suggests that since the phonon density of states of Nb is typical for fcc and bcc lattices, this introduces important error only for the strongly coupled superconductors (,;. > 1) with a widely different phonon spectrum.

Lastly, the ). values derived from eq. (6-10) tend to zero as the superconducting transition temperature T,. vanishes. This cannot be valid, since even normal metals and alloys exhibit appreciable electron phonon interaction, as stated above. Thus, the use of McMillan's formula faces some difficulty if the measured T~ values become very small (T,,. < 5 inK). Judging from the above considerations, current estimates o f / . in superconducting materials are, at best, only approximate. When the )~ correction is used in order to deduce a trend of 7b,,,,~ values on alloying, using ?,~ data, the result will be meaningful if the rate of change of ~'~, substantially exceeds possible errors involved in the estimation of )..

6.3. Trans i t ion M e t a l Impuri t ies in Norma l Me ta l s

It has been shown that very small amounts of 3d transition metal impurities in normal metals can contribute to a large increase in the measured 7 value. Hence, if the change in 7 on alloying is very small, the presence of such impurities could mask the real alloying effect. Reviewing the diffi- culties involved in the determination of 7 values in the noble metal alloys, Massalski et al. ~lz) (1966) estimated that one part per million of Fe, Co and Ni in the Cu matrix contributes approximately 0.0008, 0.00005 and 0.00002 in units of mJ/mole.K 2, by utilizing the data by Franck et al. 113~,

Crane and Zimmerman ~141 and Guthrie et al. 115~, respectively. Here, it was assumed that the observed change in 7 due to these impurities can be linearly extrapolated to very dilute limit. If the more dilute alloy data are employed (Du Chatenier and DeNoble 116~ (1966): Mizutani et al. 117t

(1972)), the same consideration yields 0.00006 for Co and less than 0.000005


for Ni. Thus, a l though the presence of Co or Ni of the order of ppm seems unimportant , a few ppm of Fe would raise the y value by an amount comparable with that expected due to alloying over the entire range of the ~ phase in noble metal alloys. In order to avoid such complications, special caut ion is needed during the prepara t ion of samples. First of all, very high purity metals, free from transit ion metal impurities, should be used as starting materials. Dur ing casting, machining and subsequent annealing, any contamina t ion should be minimized. In any event, a spec- t roscopic analysis is recommended to confirm the absence of such impurities.

6.4. Other C o n t r i b u t i o n s to the L i n e a r T e r m in the N o r m a l M e t a l s

Ahlers ~181 (1966) found a slight increase in the linear term of the low temperature specific heat where a pure Cu sample was subjected to cold work deformation. Later, Bevk t~9~ (1973) studied the same effect by employing an extremely heavily deformed pure Cu sample. Each rod was cold-rolled at r o o m temperature and the total reduct ion in thickness was approximately 96%. The resulting Y value was found to be 0 .710mJ/ mole .K 2, compared with the pure Cu value of 0.697 mJ/mole .K 2. The increase in the y value was accounted for in terms of an addit ional contr ibu- t ion to the specific heat due to the low frequency vibrations of dislocation segments. Fur thermore, it was argued that this addit ional specific heat is p ropor t iona l to the dislocation density. However, this effect becomes detectable only when extremely heavily deformat ion is applied. Thus, any residual dislocations in well annealed normal metals and alloys would not produce large enough change in the 7 value to be detectable by ordinary low temperature specific heat measurements.

REFERENCES

1. J. M. ZLMAN, Electrons and Phonons (1960) p. 100 Oxford at the Claredon Press. 2. J. J. QUINN, The Fermi Surface (edited by W. H. Harrison and M. B. Webb), (1960)

Wiley, New York. 3. S. NAKAIIMA and M. WATABE, Prog. theor. Phys., Osaka 29 (1963) 341. 4. N. W. ASHCROFT and J. W. WILKINS, Phys. Lett. 14 (1965) 285. 5. J. S. FAULKNER, H. L. DAVIS and H. W. Joy, Phys. Rev. 161 (1967) 656. 6. N. E. CHRISTENSEN, Phys. Stat. Solids (b) 54 (1972) 551. 7. G. GRIMVALL, Phys. Kondens. Mat. 11 (1970) 279. 8. W. L. MCMILLAN, Phys. Rev. 167 (1968) 331. 9. P. MOREL and P. W. ANDERSON, Phys. Rev. 125 (1962) 1263.

10. T. B. MASSALSK1, U. MIZUTANI and S. NOGUCHI, Proc. R. Soc. A343 (1975) 363. 11. P. B. ALLEN, M. L. COHEN, L. M. FALICOV and R. V. KASOWSKI. Phys. Rev. Lett. 21

0968) 1794. 12. T. B. MASSALSKI, G. SARGENT and L. L. ISACCS, Phase Stability in Metals and Alloys

(edited by P. S. Rudman, J. Stringer and R. I. Jaffee) (1967) p. 291 McGraw-Hill, New York.

13. J. P. FRANCK, F. D. MANCHESTER and D. L. MARTIN, Proc. R. Soc. A263 (1961) 494. 14. L. T. CRANE and J. E. ZIMMERMAN, Phys. Rev. 123 (1961) 113.


15. G. L. GUTHRIE, S. A. FRIEDBERG and J. E. GOLDMAN, Phys. Rev. 113 (1959) 45. 16. F. J. DU CHATENIER and J. DENOBLE, Physica, 's Gray. 32 (1966) 1097. 17. U. MIZUTANI, S. NOGUCHI and T. B. MASSALSKI, Phys. Rev. 135 (1972) 2057. 18. G. AHLERS, Rel). scient. In, strum. 37 (1966) 477. 19. J. BEVK, Phil. Mao. 28 (1973) 1379.

7. GENERAL SURVEY OF SOME ELECTRONIC

PARAMETERS IN NOBLE METAL ALLOYS

7.1. Low Temperature Specific Heat Coefficients

The low temperature specific heats have been studied in a large number of alloy systems by many investigators. Compared with the transition metals and their alloys, the ~, values in the noble metal alloys are generally about ten times smaller in magnitude and their changes on alloying, par-

FIG. 7-l. Schematic cross-section of the calorimeter assembly used in the present authors' laboratory: (a) high vacuum pumping tube, (b) radiation shields, (c) cupro-nickel shaft to operate the mechanical heat-switch, (d) electrical lead wire tube, (e) bellows, (f) Wood's-metal seal, (g) nylon string, (h) sample, (i) germanium thermometer and its holder, (j) manganin heater, (k) copper

frame and (1) mechanical heat switch.


ticularly in the c~ phases, can be of the same order of magnitude as the accuracy of measurement attainable in typical low temperature calori- meters designed to operate in the 4He range. For instance, the 7 values of pure Cu obtained by various investigators fall in the range of 0.69 and 0.70 mJ/mole.K 2, which is already large enough to mask the alloying effect on the 7 coefficient in the very dilute Cu alloys. (1) Hence, in many systems for which measurements now exist, it is still uncertain what is the exact value of the initial slope (see Section 8-1). Substantial improve- ments in calorimetric measurements have been taking place over the past several years. One of the improved methods is the adoption of a two- or three-sample system in which specimens with very similar 7 values can be compared semidifferentially. (2'3"4) The calorimeter used in the present authors' laboratory is illustrated diagramatically in Fig. 7-1.

The low temperature specific heat data, known to us in mole metals alloys, are summarized in Table 7-1. The experimentally derived electronic specific coefficients, 7~xp, (hereafter referred to as 7), for Cu- and Ag-based alloys are plotted in Fig. 7-2 as a function of e/a. The data for Au-based alloys are shown separately in Fig. 7-3. As can be seen from Figs 7-2 and 7-3, the 7 coefficients in various related phases show rather similar trends and values when plotted in terms of e/a, irrespective of the solute or solvent species, except in the dilute range of the e primary solid solutions, where the solvent dependence is clearly seen. Here, the starting values are those of pure Cu and pure Ag, and the initial slopes depend on the solvents. Nevertheless, generally speaking, within the c~ phases the 7 trends in both Cu- and Ag-based alloys seem to merge at a value of

0.9 ' ' ' ' -~ ~ Ag'Cd . . . . ~Cu-Ge u-Ge *"-~

0.8 /Cu based alloys C

0.7

0.6 Ag based alloys A ~ Ag-Zn

v ~ 0.5 "~Ag'AI = 1 p~

0.4 "~ Ag-~n \ \ , ~ Cu-Zn

\ 0.3 - - O~ phase,

0.2 I i I I i i i J i 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

electron concentration

F~G. 7-2. The trends of electronic specific heat coefficient ~, as a function of the electron concentration in Cu- and Ag-based alloys.


T a b l e 7-1.

Fitting No. of Alloy system Authors Year equation+ samples~ Ref.

l n t e r n o b l e a l loys Ag Au Green and Valladares 1966 1 5 l

Martin 1968 lIl, IV 6 2 Davis and Rayne 1972 II 9 3

Ag Cu Sargent et al. 1966 | 3 4 Au Cu (Cu3Au) Rayne 1957 1 2 5 t(_ u3Au) Martin 1968 Ill 2 6 (CuAu, CuAu 3) Martin and Waterhouse 197(I I11, IV 5 7

Detinger et al. 1972 1 3 8 Cu Au Cu Davis and Rayne 1972 I1 10 3

phase Ag AI Mizutani and Massalski 1975 l 3 9 A ~ C d Montgomery et al. 1967 1 9 10 Ag Ga Mizutani et al. 1972 1 4 11 A ~ G e Mizutani et al. 1972 I 2 11 A ~ l n Mizutani et al. 1972 I 3 11 Ag Sn Massalski and lsaacs 1965 11 6 12

Green and Culbert 1965 1 8 13 Ag Zn Green 1966 II 6 14 Au Cd Bevk et al. 1977 11 3 35 Au Ga Hoyt et al. 1976 I1 5 34 Au Sn Will and Green 1966 II 6 15 Au Zn Martin 1971 IV 4 16 Cu Ga Mizutani et al. 1972 1 7 I I Cu Ge Rayne 1958 I 8 17

Mizutani et al. 1972 1 2 11 Cu Sn Clune and Green 1966 1 5 18

Bevk and Massalski 1972 11 8 19 Cu Zn Rayne 1957 1 11 20 Cu Zn Veal and Rayne 1963 +Ill) 8 21

lsaacs and Massalski 1965 I 12 22 Clune and Green 1966 1(11) 1 23 Mizutani et al. 1972 I 6 11

fi' phase Ag Zn Cu Zn

~,, • phase Ag AI

Ag Cd Ag Sn

Ag Zn Au Cd Cu Ge Cu Zn

phase Ag Zn

p phase Ag AI

Abriata et al. 1970 11 1 24 Veal and Rayne 1962 I1 5 25

Bevk et al. 1973 11 2 26 Mizutani and Massalski 1975 II 8 9 Massalski and Mizutani 1976 II 4 27 lsaacs and Massalski 1966 I 4 28 Massalski and Mizutani 1976 II 7 27 Mizutani and Massalski 1975 II 10 29 Bevk el al. 1977 I1 2 35 Massalski and Mizutani 1976 1I 6 27 Massalski and Bevk 1976 II 4 30

Abriata et al. 1970 11 I 24

Bevk et al. 1973 II 2 26

L > . ' . l ', 22 ; 4 I~


7 phase Ag-Cd Massalski and Mizutani 1976 II 2 31 Ag-Zn Massalski and Mizutani 1976 II 2 31 Cu-Zn Veal and Rayne 1963 II 5 32

r l phase Ag-Zn Massalski et al. 1975 II 2 33 Cu-Zn Massalski and Bevk 1976 II 1 30

t(I) C/T = "~ + ~T 2. (II) C/T = y + o~T 2 + 6T 4 (III) C = H T -2 + ~tT + o~T 3 (IV) C = H T -2 + y T + ocT 3 + 3 T 5.

The pure metal is excluded when counting the number of samples. However, a single sample, which was measured twice following a different heat treatment is counted as two samples.

References

1. B. A. GREEN, JR and A. A. VALLADERES, Phys. Rev. 142 (1966) 379. 2. D. L. MARTIN, Phys. Rev. 176 (1968) 790. 3. T. K. DAVIS and J. A. RAYrqE, Phys. Rev. B6 (1972) 2931. 4. G. A. SARGENT, L. L. ISAACS and T. B. MASSALSKk Phys. Rev. 143 (1966) 420. 5. J. A. RAYNE, Phys. Rev. 1118 (1957) 649. 6. D. L. MARTIN, Can. J. Phys. (1968) 46 923. 7. D. L. MARTIN and N. WATERHOUSE, Can. J. Phys. 48 (1970) 1271. 8. W. C. DELINGER, W. R. SAVAGE and J. W. SCHWEITZER, Phys. Rev. B6 (1972) 338. 9. U. MIZUTANI and T. B. MASSALSKI, J. Phys, F5 (1975) 2262.

10. H. MONTGOMERY, G. P. PEERS and E. M. WRA¥, Proc. R. Soc. A301 (1967) 261. 11. U. MIZUTANI, S. NOGUCHI and T. B. MASSALSKX, Phys. Rev. 135 (1972) 2057. 12. T. B. MASSALSKI and L. L. ISAACS, Phys. Rev. 138 (1965) A139. 13. B. A. GREEN, JR and H. V. Ct:LBERT, Phys. Rev. 137 (1965) Al168. 14. B. A. GREEN, JR, Phys. Rev. 144 (1966) 528. 15. T. A. WILL and B. A. GREEN, JR, Phys. Rev. 150 (1966) 519. 16. D. L. MARTIN, Phys. Rev. B4 (1971) 4117. 17. J. A. RAYNE, Phys. Rev. 110 (1958) 606. 18. L. C. CLUNE and B. A. GREEN, JR, Phys. Rev. 144 (1966) 525. 19. J. BEVK and T. B. MASSALSKI, Phys. Rev. 115 (1972) 4678. 20. J. A. RAYNE, Phys. Rev. 108 (1957) 22. 21. B. W. VEAL and J. A. RAYNE, Phys. Rev. 130 (1963) 2156. 22. L. L. ISAACS and T. B. MASSALSK1, Phys. Rev. 138 (1965) A134. 23. L. C. CLUNE and B. A. GREEN, JR, Phys. Rev. 144 (1966) 525. 24. J. P. ABRIATA, O. J. BRASSAN, C. A. LUENGO and D. THOULOUZE, Phys. Rev. B2 (1970)

1464. 25. B. W. VEAL and J. A. RAYNE, Phys. Rev. 128 (1962) 551. 26. J. BEVK, J. P. ABRIATA and T. B. MASSALSK~, Acta metall. 21 (1973) 1601. 27. T. B. MASSALSKI and U. MIZUTANI, Proc. R. Soc. A351 (1976) 423. 28. L. L. ISAACS and T. B. MASSALSKI, Phys. Rev. 141 (1966) 634. 29. U. MIZUTANI and T. B. MASSALSKI, Proc. R. Soc. A343 (1975) 375. 30. T. B. MASSALSKI and J. BEVK, to be published. 31. Unpublished work. 32. B. W. VEAL and J. A. RAYNE, Phys. Rev. 132 (1963) 1617. 33. T. B. MASSALSKI, U. MIZUTANI and S. NOGUCHI, Proc. R. Soc. A343 (1975) 363. 34. R. F. HOYT, A. C. MOTA and C. A. LUENGO, Phys. Rev. B14 (1976) 441. 35. J. BEVK, T. B. MASSALSKI and U. MIZUTANI, Phys. Rev. BI6 (1977) 3456.

a b o u t 0 .70 -0 .75 m J / m o l e . K 2 as e /a exceeds a b o u t 1.2. H o w e v e r , t h e t r e n d s

of 7 in t h e ~ p h a s e s A u - b a s e d a l loys a p p e a r to c h a n g e f r o m s y s t e m t o

s y s t e m a n d a re n o t as u n i f o r m as in t h e C u - a n d A g - b a s e d a l loys . Ve ry

r e c e n t m e a s u r e m e n t s in t h e ~ p h a s e A u C d a l loys ~s) s h o w e d a r a p i d de -


0.8

0,7

_~ 0.6

E 0.5 .;,.o

0.z~

0.3

0.2 1.0

I I I I I I I 1

~ Au-Go ii, rl II?4plJ j

A,., , , , ,,,,,, zltltl?l?l ,JJnlsJzz _ trtttl/lt

I I I I I I I I I

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 electron concentrotion

FIG. 7-3. The trends of the electronic sl~ecific heat coefficient y as a function of the electron concentration in Au-based alloys. The hatched broad bands represent the possible region of the ), values in :~, ~, E, y and r/ phases Cu-

and Ag-based alloys.

crease in ~, in concentrated alloys, preceded by an initial slight increase and a broad maximum in dilute alloys at e/a near 1.04. A decrease in ? has been observed only in Au-based alloys. One of the possibilities is that this decreasing trend is due to the presence of short range order, (6~ which begins to play a role as the concentration of solute exceeds about 10~,~,. Short range order easily develops, even in samples that had been rapidly quenched from the disordered state. ~7'8) Another possibility is that the concentrated Au-based alloys reveal, more closely, the true band structure trend.

The moderate rate of increase in the 7' values in the fcc ~ phase alloys based on Cu and Ag still continues initially in the following hcp intermediate phases. The most prominent feature in all the hcp structures, involving both the ~ and E phases, is the presence of a broad peak at about 1.5 electrons per atoms, which is followed by a decreasing trend as e/a increases further. The trend in the 7 brass phasest within the relatively narrow range of e/a around 1.6, is uniquely described by a steeply falling slope, with the coefficient reaching the smallest values of all electron phases.

The measured limiting Debye temperatures for all noble metal alloys (except internoble metal alloys) are shown in Fig. 7-4 in the electron concentration range between 1.0 and 2.0. While the Debye temperatures in all a-phases seem to be clearly related to e/a, a strong dependence on both solute and solvent species is seen in the intermediate phases, contrary

+ 7 brass structure must not be confused with the 7 coefficient.


, 0 0

Cu-Zn

~ - - ~ l l ~ u - G a ~ Cu-Get. ~ ~ . ~ ~ e Ag-Zn / ~ Ag-Zn

Cu-Sn p'Cu-Zn / " ~.Y~ / /

Ag-AI Ag-Zn .

200~- Ag-~e Ag-~.a . . . . . . . JAu Zn I - ~ 5Ag-~n ~Ag-~.a ~ i; Au-Cd

I 'A : -Sn . . . . . . . .

1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

electron concent ra t ion

FIG. 7-4. The trends of the Debye temperature as a function of the electron concentration in various noble metal alloys. The alloy systems without Greek

letter symbols refer to the 7 phase.

to the trend in the ? values. The Debye temperatures in Cu-based phases are generally higher than those in the corresponding Ag-based phases. A comparison between Figs 7-2 and 7-4 indicates that the trend of the Debye temperatures in most systems resembles a mirror reflection of the trend in the 7 values, particularly in the intermediate phases. C9)

50

.d ~0

E :o 30

. . . . Cd l ~ T Ag-Zn o

zo I- Ao-Cdf ° ~'v Ag'zn .~, / / , • . ' u ~ Ag-Cd

1oI od-=n "F2' , oU" ,,ozo

c - ~ S Ag-Zn _10 I J = I i i , I i i

1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 electron concentration

Fic. 7-5. The trends of the 6 term (i.e. T 5 coefficient) as a function of the electron concentration in various noble metal alloys.

E L E C T R O N I C S T R U C T U R E OF H U M E - R O T H E R Y P HAS ES 203

The behavior of the 6 term (i.e. the coefficient of the T 5 term in eq. (6-1), sometimes called the phonon dispersion deviation from the Debye model), is shown in Fig. 7-5, again as a function of e/a, for the systems where this value has been measured. The data for the 7 phases is scarce, because the 6 term in the ~ phase noble metal alloys is generally so small that many investigators ignored the T 5 contribution completely, and employed the simpler equation C = ?,T + ~T 3. The 6 values in Fig. 7-5 also show a strong dependence on both the solute and solvent species. For instance, the large ,5 values in systems alloyed with Cd seem to result from the presence of a large fi term in Cd itself.

7.2. Other Physical Parameters (X~J, c/a, T,.)

Of general interest are some other physical parameters relevant to the electronic structure. Of these, the magnetic properties are of special importance. The magnetic susceptibility ~ was measured in the ~ phase Cu- and Ag-based alloys C~°) (Henry and Rogers (1956, 1960)), ~ phase Au-based alloys (t~ (Henry), • and q phases of Ag-Zn alloys (~1) (Noguchi et al. (1969)), ~ phase Au-In alloys ~13~ (Massalski et al. (1959)), ~ phases of Ag-In, Ag-Sn, Au-In and Au-Sn alloys <~4~ (Shimada et al. (1978)), and 7 phases of Cu-Zn, Ag-Cd, A ~ Z n and Cu-Cd alloys ~5) (Smith (1935)). These data,

20

10

-~ o E

e O 'O

- 1 0 e p<

- 2 0

-30

-40 1.0

i , , i

CU based alloys Au based alloys / lj Au-ln

: l / *,/ .. ...................... ~ ........... , , I " - . ; ~ - z .

ag ba~d o,oys.J ' ' . , * ' - / - ~ ~-Sn)

¢J Au-Sn | A

~ Ag-Zn

~ I Cu-Cd

- - d, Phase H ~ . ~ C u - Z n

v

t t i i k i i i i

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0


FIG. 7-6. The variations of magnetic susceptibility due to conduction electrons (;~,t) as a function of electron concentration. The data for the 7 phases are shown in the form of broad bands for each solvent metal Cu, Ag and Au. The data for the hcp phases are shown in terms of the actual values for each alloy, except for the data of Shimada e t al . , <14) which are shown only as trends (broken lines). The references to data for individual systems are as follows: • Au-ln; (13) [] Ag-Cd; 1151 A Ag-Zn(?); (ls~ A A~Zn(e, /~):112) C) Cu-Zn; Ils~

• Cu Cd. ~15~


1.90

1.85

1.80

1.75

,u - - 1 . 7 0 m

I

j, ~- =[ ~ 71 -

~u-Hg Au-ln Au-Sn J__ L65 i-..--_d__ ~Cu-Ga

1.6o c ~ . ~ " ~ ~ - " " " ~ . . . - ~ - ~ . A , ~Zn Ag-ln" \ ~ .~.~_Su.Zn ~/

Ag-Cd" ~ ' , , , , . ~ _ . 7 ~ Au.Z n 135

1.5( i I I

Electron concentration

FIG. 7-7. The trend of the axial ratio as a function of the electron concentration in various hop alloy systems. (After Massalski (18) (1966).)

corrected for the ionic cores,: are plotted as a function of e/a in Fig. 7-6 for all phases listed above. In the ~ phase, the trend of the Ze,, the magnetic susceptibility due to conduction electrons, again shows a clear dependence on the e/a. However, this no longer holds true for the intermediate phases.

As is well known, the total magnetic susceptibility may be expressed in three major terms; the diamagnetism from the ionic cores, the Pauli paramagnetism associated with the spins of the conduction electrons (which is proportional to the density of states at the Fermi level) and the orbital contribution gorb, due to the orbital motion of the conduction electrons. The gorb usually contributes to diamagnetism in a complicated manner, uv) In the free electron model, the magnitude of Zorb is reduced

t ln the original papers a different literature source for the magnitude of the ionic contribution for each ion is often used. In order to allow a direct comparison between the various data, the ionic correction was recalculated for systems using the values proposed by Hurd and Coodin I]6) (1967).


to just one third of the Pauli paramagnetism. Thus, the magnetic susceptibility corrected for the ionic contribution may be assumed to be simply proportional to the density of states at the Fermi level, unless the Fermi surface-Brillouin zone interaction produces an additional specific contribution through 1(orb" The lack of a common "master curve" behavior for the magnetic data in the intermediate phases seems to indicate that the Fermi surface-Brillouin zone interaction is more substantial and more specific here, compared with the ct phases.

The trend of the axial ratio, c/a, in various hcp noble metal alloys is also related to the electronic structure and has been discussed and summarized by Massalski. (18),Another property of interest is the superconducting transition temperature, T~, which was observed and measured in a number of the hcp (, E and r/ phases in noble metal alloys by Luo and Andres (19) (1970), Farrell et a/. (2°) (1964) and Matsuo et al. (2~) (1975). Very recently, the superconductivity has been observed even in the c( phase noble metal alloys in the temperature range between 4 and 400 mK. The trends of c/a and T~ are shown in Figs 7-7 and 7-8, respectively. Both properties show a strong e/a dependence.

Judging by the striking correlation between the various physical parameters, particularly the 7 values, and the electron concentration, the electronic structures of the noble metal alloys appear to have a common

900

800

7 0 0

600

500 E

400

300

2OO

100

0 1.0

Au-AI

0 ~ f Au-Sn

1.1 1.2 1.3 electron

Au-Sn

1.4 1.5 1.6 1.7 1.8 1.9 concent rotion

Ag-Zn

2.0

FIG. 7-8. The behavior of the superconducting transition temperature T¢ as a function of the electron concentration in various noble metal alloys. The references to data are as follows: ~ phase alloys (the symbol 7 is omitted in the figure), (22~ ~ phase alloysJ TM E phase Ag-Zn 12t~ and r/ phase A ~ Z n J TM


basis, as was a l r eady suspec ted by H u m e - R o t h e r y and o thers , m a i n l y o n

the basis o f phase d i a g r a m w o r k and crys ta l s t r u c t u r e s tudies . It s h o u l d

be poss ib le to in t e rp re t such overa l l b e h a v i o r in t e rms of a fairly s imp le

mode l , in wh ich the specif ic charac te r i s t i c s o f the so lven t and so lu te me ta l s

p lay a sma l l e r ro le t h a n the e l ec t ron c o n c e n t r a t i o n . In the fo l l owing sec-

t ions , we shal l e l a b o r a t e fu r the r on the p resen t u n d e r s t a n d i n g of t he elec-

t r o n i c s t r u c t u r e o f va r i ous r e l a t ed g r o u p s o f e l ec t ron phases ba sed on

the n o b l e meta ls .

REFERENCES

1. T. B. MASSALSKI, G. A. SARGENT and L. L. ISSACS, Phase Stability in Metals and Alloys (edited by P. S. Rudman, J. Stringer and R. 1. Jaffee) (1967) p. 291 McGraw-Hill, New York.

2. S. S. SHINOZAKI and A. ARROTT, Phys. Rev. 152 (1966) 611. 3. H. MONTGOMERY, G. P. PELLS and E. M. WRAY, Proc. R. Soc. A30I (1967) 261. 4. J. BEVK and T. B. MASSALSKI, Phys. Rev. B5 (1972) 4678. 5. J. BEVK, U. MIZUTANI and T. B. MASSALSKI, Phys. Rev. BI6 (1977) 3456. 6. D. L. MARTIN, Phys. Rev. 134 (1971) 4117. 7. S. C. Moss, J. appl. Phys. 35 (1964) 3547. 8. K. OHSlMA and D. WATANABE, Acta. crystallogr. A33 (1977) 784. 9. See, for example, F. HEINIGER, E. BUCHER and J. MULLER, Phys. Kondens. Mat. 5 (1966)

243. 10. W. G. HENRY and J. L. ROGERS, Phil. May. 1 (1956) 237; Can. J. Phys. 38 (1960) 908. 11. W. G. HENRY; private communication. 12. S. NOGUCHI, T. YONEYAMA and T. B. MASSALSKL J. appl. Phys. 40 (1969) 225. 13. T. B. MASSALSKI, L. MEYER and D. WEINER, Phys. Rev. l l5 (1959) 301. 14. H. SH1MADA, S. NOGUCH1 and T. B. MASSALSKI, Trans. Japan Inst. Metals 19 (1978)

138. 15. C. S. SMITH, Physics 6 (1935) 47. 16. C. M. HURD and P. COODIN, J. Phys. Chem. Solids 28 (1967) 523. 17. R. PEIERLS, Z. Phys. 80 (1933) 763. 18. See, for example, T. B. MASSALSKI, J. Phys. Radium, Paris 23 (1962) 647; also C. S.

BARRETT and T. B. MASSALSK1, The Structure of Metals (3rd edition) (1966) McGraw-Hill, New York.

19. H. L. Luo and K. ANDRES, Phys. Rev. B1 (1970) 3002. 20. D. FARRELL, J. G. PARK and B. R. COLES, Phys. Rev. Lett. 13 (1964) 328. 21. S. MATSUO, U. MIZUTANI, T. B. MASSALSKI and S. NOGUCH1, Phys. Rev. B12 (1975)

4941. 22. R. F. HOYT and A. C. MOTA, Solid St. Communs 18 (1976) 139.

8. ELECTRONIC STRUCTURE OF RELATED GROUPS

OF PHASES IN NOBLE METAL ALLOY SYSTEMS

8.1. ~t Phase Pr imary Solid Solutions and Internoble Me ta l

Solid Solutions

8.1.1. Fermi surface topography (experiments and calculations)

T h e e l ec t ron i c s t ruc tu re o f ~ phases and of i n t e r n o b l e m e t a l a l loys

has been s tud ied ex tens ive ly by m e a n s o f va r i ous t echn iques . H e r e it m a y

be r e a s o n a b l y expec ted tha t the resul ts can be re la ted to the well e s tab-

l i shed e l ec t ron i c s t ruc tu res of t he p u r e nob l e meta l s . Before we c o n s i d e r


0 Cu-Oe

• Cu-Si 2.6

.Q L2

0 it t--+ ,+ 14 ~ / -

r , + I i I J I

1,0 1.1 1.2 1.3

electron concentrot ion

FIG. 8-1. The change in the neck radius against electron concentration in the phase Cu based alloys, determined by the positron annihilation technique.

The symbols refer to the following references: O: Cu-Ge (Suzuki et al.(1%, O: CuSi (Suzuki et al.(~2~), I1: Cu-AI (Fujiwara et al.(~), [] Cu AI (Murray and McGervey(2~), 0 : Cu-AI (Thompson et al.(3)), A: Cu Zn (Becker et al. ~6~) and A: Cu Zn (Triftshauer and Stewart(5~). Solid curve is the result calculated

From the sinking-conduction band model. (After Suzuki et al.(~%

the behavior of the 7 coefficients in these systems, we shall first briefly review the electronic structure of :~ phases as established by a number of experimental techniques, as well as through band calculations.

The positron annihilation technique has been recently extensively employed to examine the topography of the Fermi surface in the c~ phase Cu based alloys; Cu-A1 (Fujiwara et al. (1~ (1968); Murray and McGervey (2~ (1970); Thompson et al. 13~ (1971); Akahane et al. ~4~ (1974)), Cu-Zn (Trifts- hauser and Stewart ~5) (1971); Becket et al. ~6) (1972); Morinaga ~7) (1972)), Cu Ga (Hasegawa et al. (s~ (1972)), Cu-Ga and Cu-Ge (McLarnon and Williams (9~ (1973)), Cu-Ge and Cu-Si (Suzuki et al. I~°l (1975)). The specific choice of Cu-based alloys results mainly from the fact that irradiation of Cu by neutrons produces an intense source of positrons from the 64Cu isotope. The neck radius of the Fermi surface can be measured from an inflection in the angular correlation curve. The known results are plotted in terms of e/a in Fig. 8-1. The neck radius related to the {1111 Fermi surface contacts increases approximately linearly with increasing e/a. The solid line in Fig. 8-1 is based on the so-called "'sinking conduction band model" described in the earlier sections, in which the whole conduction band is displaced downwards relative to the Fermi level, at a rate consistent with optical data. The linear increase in neck size was also observed by the rotating specimen method, employing positron annihilation, in the

phase Cu-Zn (Morinaga (71 (1972)) and in Cu-AI alloys (Akahane et


7.2

6.8

Io o

6.4

6.0

5.6,

S.2

10

0 Cu-Ge t ' I ' I~

• Cu-Si

• Cu-AI B.z.(Cu-Si )

B. z. (Cu-Ge)

11 12 13 electron concentration

FIG. 8-2. The change in the [100] belly radius in the c~ phase Cu-based alloys, determined by the positron annihilation technique. O: Cu-Ge (Suzuki et al.(l°)), 0: Cu-Si (Suzuki et al. (~°)) and I1: Cu-AI (Fujiwara et al.(1)). Solid curve is the result calculated from the sinking-conduction band model. {200} zone boundaries are calculated from the lattice constant data. (After Suzuki et al. (1°))

al. (4) (1974)).t The [100] belly radius also increases linearly with increasing e/a, but never reaches the {200} square zone faces, as shown in Fig. 8-2. This eliminates the possibility, considered several years ago by Hume- Rothery and Roaf (11) (1961), that the stability limit of the ct phases could in some way be associated with a steep fall in the density of states as might be expected following contact between the Fermi surface and the set of six {200} zone planes. Of course, the band calculation shown in Fig. 3-5 also leads to this conclusion. A detailed review on the application of the positron annihilation techniques to the noble metal alloys was recently published by Berko and Mader (12) (1975).

The d H v A measurements have been performed for various dilute Cu based alloys: Cu-Zn, Cu-Cd, Cu-A1 (Cholett and Templeton (~3) (1968)), Cu-Zn, Cu-A1, Cu-Ge, Cu-Si, Cu-Ni, Cu-Co, Cu-Fe, Cu-Mn, Cu-Cr (Coleridge and Templeton (~4) (197l)), Cu-A1, Cu-Ni (Templeton and Col- eridge(15) (1975)). Earlier studies (~3' 14) revealed that the change in the neck frequency is more or less linear with additions of solute atoms and that the slope in Cu-Zn, Cu-A1 and Cu Ge alloys is in quite good agreement with the rigid band model. More recent results shown in Fig. 8-3 are due to Templeton and Coleridge, (15) who studied the change in six representative extremal areas of the Fermi surface of Cu resulting from alloying with small (<200 ppm!) amounts of A1 and Ni. Again, the Fermi surface

-~ln the earlier work by Fujiwara et al/1) (1968), a pronounced deviation from a linear trend was reported at about 1.14 electrons per atom. However, new results by Akahane et al. (4) indicate no such anomaly.


100

c

v

~o

-I00

I I I I I

/ ~ / ~

x~D I I I I I

0 0.01 0.0 2

solute concentrQtion (at.%)

FIG. 8-3. Change in dHvA frequency against solute concentration m very dilute Cu AI and Cu-Ni alloy systems ×, A1; O, Ni; B, [111] belly; N, [111] neck:

D, [110] dog's bone. (After Templeton and Coleridge. ~15))

dimensions follow approximately the rigid band behavior in Cu-A1 alloys, but are dominated by the d like character of the solute in Cu-Ni alloys. The determination of such a small concentration of impurity atoms is based on residual resistivity measurements and the accuracy of the composition is claimed to be 5~o and 2~o in Cu-A1 and Cu Ni alloys, respectively.

Other miscellaneous properties relevant to the electronic structure may be listed here; the soft X-ray spectroscopy (Yeh and Azaroff (16) (1967)) for the a phase Cu-Zn, the n.m.r, data for ~ phase Cu-alloys (Rowland fly) (1960)), a-phase Ag alloys (Rowland (18} (1962)) and Ag-Au alloys (Nar- ath ~9) (1967, 1968)), the Mbssbauer effect (Lees and Flinn t2°l (1971)) for the ~ phase Cu-Sn, Cu-A1-Sn and Cu-Zn-Sn and the polar reflection Fara- day effects for the A~Au alloys (McAlister et al. (21} (1965)) and for the

phases Ag-Cd and Ag-Mg alloys (Tracy and Stern ~22) (1973)) and photoelectron spectroscopy (XPS or UPS) for the a phase Ag-In (Nilsson t23) (1970)), ~ phase Cu-Zn (Andrews and Hisscott ~24~ (1975)), and Ag-Au, Au-Cu alloys (Nilsson (25) (1970)). The results of XPS or UPS were already discussed in an earlier section.

The study of the X-ray absorption edges of the ~ phase Cu-Zn alloys ~16~


shows a decrease in the total densities of unpopulated s states with increasing e/a, in accordance with the rigid band prediction. An essentially similar result is provided by the isomer shift of 1198n nucleus in a series of Cu alloys studied by Lee and Flinn32°~ These results indicate some deviations from a linear trend in the total density of states for s like electrons, and are similar to the results of Fujiwara et al. ~1~ (1968) for the ~ phase Cu A1 alloys studied by Lee and Flinn. ~2°~ These results indicate some deviations from a linear trend in the total density of states for s like electrons and radius in Cu-A1 alloys.

Alloys based on Cu, Ag and Au have been extensively studied by the nuclear magnetic resonance measurements. Since the two stable isotopes 1°TAg and l°9Ag have spin ½, and hence no quadrupolar interactions, Ag alloys are especially well suited for the n.m.r, measurements. (18'19~ Based on his 7"1 and Knight shift data for Ag Au alloys, Narath ~191 estimated the variation of the fractional density of s character electrons on the Ag-sites as a function of Au content. This shows an ~ 30~o reduction in the s like charge density at the Ag sites on passing from pure Ag to dilute Ag in Au. Hence, a substantial transfer occurs of s like electrons towards the stronger ionic potential of the more electropositive Au ions. The Cu isotopes 63Cu and 65Cu have a spin of 3/2 and the associated presence of quadrupole interaction makes a comparable analysis rather difficult. The latest n.m.r, data may be found in the publication by Bennett et al. (26~

The polar reflection Faraday effect < PRFE > method applied to Ag-Au alloys ~2~) has shown a more or less linear change in the [-111] neck radius over the entire composition range. The same technique has shown that the Fermi surfaces of Ag-Cd and Ag Mg ~-phases ~22) vary qualitatively as one would expect from the rigid band model. An expansion of the neck radius is observed in both systems, as e/a increases.

Theoret ical band calculations have been performed for the ~ phase Cu-Zn alloys by Amar et al. ~2~) (1967) and Pant and Joshi ~28) (1969), based on the virtual crystal approximation, and Soven ~29) (1966) and Bansil et

al. ~3°) (1974), based on the average-t-matrix approximation (ATA) in combination with Korringa-Kohn Rostoker method. Amar et al. showed the downward displacement of the L2, and X4, peaks, (see Fig. 4-3) associated with the contact with the { 111} and {200} zone planes, and at the bottom of the conduction band F1, as the solute concentration increases. The downward displacements of the peaks are in good agreement with the band calculations by Soven, C29~ and Pant and Joshi. ~28) The recently developed CPA (coherent potential approximation) and ATA methods yield two sets of d bands originating from the Cu and Zn atoms in accordance with the photoelectron spectroscopy results (see, for instance, Andrews and Hisscott ~24) (1975)). Furthermore, these methods can also provide information on the damping of the electronic states due to disorder.

The comprehensive calculation by Bansil et al. ~3°~ found that the maxi-


mum broadening of the energy levels near the Fermi level is comparable with the thermal broadening of the Fermi surface at room temperature. The most interesting features on alloying are as follows: (1) the C u d band narrows on alloying, associated with the reduction in the number of Cu atoms, and the proportional increase in Zn atoms on alloying. In fact, the width of the C u d band, AE c" =- (X5 - X3), is found to decrease linearly with Zn concentration; (2) a linear downward shift of EVI (bottom of the conduction band) is observed, since the Zn potential is more attractive than that of Cu; (3) an increase in the Fermi level is found, due to the increase in e/a, but its shift is smaller than would be expected from a rigid band because of the downward shift of the bottom of the conduction band. An increase in E F - Er~ was found to be 0.8eV at 30 at."; Zn, as compared with the value of 1.2 eV expected from a rigid band model. However, when the charge transfer effect is taken into account, a further increase in (Ev - Er,) is predicted, and the final result becomes much closer to the rigid band value: (4) the change in the neck radius of the Fermi surface agrees with the positron annihilation data and the rigid band prediction. We recall here that the band structure of Cu Ni alloys can be adequately described by the CPA or ATA band models, while the rigid band description completely failed, as discussed in Section 4-3. However, in the c~ phase noble metal alloys, it appears that, as far as the Fermi surface topography is concerned, the recent band theory developed for the disordered alloys is in qualitative agreement with a rigid band description. The success of the rigid band model for the Fermi surface topography may be due to the fact that the d band for the solute atoms lies far below the Fermi level (for instance, in Cu-Zn alloys, the Zn d band is found near the bottom of the conduction band 124~) and thus the influence of the d band on the Fermi electrons becomes diminished on alloying in contrast to the Cu Ni system. In summary, all measurements and calculations described above indicate that the change in the Fermi surface topography on alloying can be fairly well described by a simple rigid band approximation, although this does not mean that the approximation itself is valid.

8.1.2. The behavior o[" the electronic specific heat coe.~cients on alloying

The detailed behavior of the 7 coefficient in the :~ phase alloys based on Cu and Ag is plotted against e/a in Figs 8-4 and 8-5, respectively. Among the various combinations of the internoble metal alloys, the low temperature specific heat data are known over the entire composition range in the systems Ag-Au and Cu-Au. As shown in Figs 8-6 and 8-7, the trends of both ;~ and 0 D in the former figure seem to be well established, whereas in the latter the trend of the ?, values is less certain than that of the 0D values.

As stated frequently before, the density of states at the Fermi level of pure Cu decreases with increasing energy (see, for example, Fig. 3-5} owing


0.7

0.7 6

0.7/,.

0 .72

o

0 3 0

~ 0.6

s ' ' I I

/ , ,

0,6 2

t c=z.

0.6 0 ~ r

0.5 8 , , 1.0 1.1 1.2


,,,, Cu-Ga et at.(1967)

0 Cu-Sn

I

1.3 1.4

FIG. 8-4. The behavior of the electronic specific heat coefficient 7 against electron concentration in the ~ phase Cu-based alloys. O: Cu-Zn (Veal and Rayne(2~)), O: Cu-Zn (Mizutani et al.(ll)), A: Cu-Ga (Mizutani et al. (~1)) and [3: Cu-Sn (Bevk and Massalski(19)). References are cited from Table 7-1. The straight line marked as Faulkner et al. is drawn from the slope of the density

of states curve shown in Fig. 3-5.

to the presence of contacts with the {111} zone planes below the Fermi level. It is now well established that all three noble metals possess this feature. However, contrary to the band structure indications, the initial slope of the 7,xp values in the ~ phase noble metal alloys is always positive, without exception. The negative deviation of the ?~xp values from a possible linear trend, as found in A ~ A u alloys, provides another puzzle, because the Fermi surface topography derived from experimental observations (as discussed in the previous subsection) may be considered to change more or less linearly over the entire composition range. Several possible explana- tions have been advanced in connection with this puzzling behavior. (a) "Soft band" and "impurity smearing" effects: in this category may be included the "non-rigid" (soft band) model of Cohen and Heine (31) (1958) and the Fermi edge smearing effect in k space due to alloying, as proposed originally by Jones (32) (1964). Cohen and Heine suggested that the energy band gaps across the { 111 } zone planes might decrease on alloying, thereby making the Fermi surface more special. However, the linear increase in the neck radius as demonstrated in Figs 8-1 and 8-3 no longer supports this idea. Jones proposed that in alloys the k vector is no longer a good quantum number and that a smearing effect in k space, due to impurity scattering, may allow some electrons below the Fermi level, but near the 111 peak, to contribute to the electronic specific heat, resulting in an enhancement of the measured coefficient. However, this model would also


0.7 6

0.74 • • ~ 0

i 072 /

o.7o

/ / i o i . 1 1 - - °

0"660 " 0 . 6 4 ~ ' I I i

1.0 1.1 1.l 1.3 1.4 electron concentration

FIG. 8-5. The behavior of the electronic specific heat coefficient y against electron concentration in the ~t phase Ag-based alloys. ©: Ag-Cd, (1°~ O: A ~ Z n , "4~ z~: Ag-ln, (1l~ A: Ag-Ga, (111 z~: A~AI, (91 KI: Ag-Ge, (11) 0 : A ~ S n ~lz) and 5~:

Ag-Sn. °3~ References are cited from Table 7-l.

require an enhancement to occur in 7 values of Ag-Au alloys where, instead, a negative deviation is observed experimentally. Furthermore, Dawber and Turner (33), who refined the earlier estimate by using a more realistic density of states curve, found that even in the presence of a smearing effect a decrease in the 7 value with increasing e/a is still indicated. (b) Electron-phonon interaction: here, it is argued that the electron-phonon interaction may change on alloying, thereby compensating for the discrepancy between the expected trend of 7 in the rigid band approximation and the experimentally observed trend of 7. For example, Haga (34) applied the Jones approach (32) to the electron-phonon interaction and argued that the enhancement factor 2 (see Section 6) should decrease on alloying. The negative deviation of the 7¢xp coefficient, from a straight line interpolation between the two pure metals in the Ag Au system, was thus attributed to the reduction of the electron-phonon interaction in the concentrated alloys. However, if this were true, there seems to be no reason why the same effect should not occur also in the ~ phase alloys. Recently, Davis and Rayne (35) (1972) pointed out that Haga's theory cannot account in a consistent manner for the behavior of their experimental 7 values in both Ag-Au and the ternary Ag-Au Cu alloys. Furthermore, Grimvall ~361 recently provided a theoretical reason why Haga's conclusion is incorrect.

Alternatively rather than a decrease in the electron-phonon interaction on alloying, Clune and Green 13v~ proposed that an increase in 2 can compensate for the discrepancies between the rigid band description and the experimental results in the ~ phase noble metal alloys. They observed an increase in the y values of Pb alloyed with T1 and Bi. Here the situation is similar to the ~ phase alloys because the band calculation for pure Pb also indicates an initially decreasing slope of N(EF) with increasing


200 \ ~ . , , ~

<~"~ 180 D \Q

160 Ag-Au " ~

070 i t o//~ r 0.68

Davis and )5 " / 0.66 ~- Rayne /~ ( "

~=' 0.62 I Green and Valladares

! 0.60 ! ~ ' i I

0 20 40 60 80 100 Ag Clt.°loAu Au

FIG. 8-6. The behavior of the electronic specific heat coefficient y and the Debye temperature 0 D in Ag Au alloy system, x : Green and Valladares (1) (1966), ½: Martin (2) (1968) and ©: Davis and Rayne 13~ (1972). References are cited

from Table 7-1.

energy. However, in the Pb-based systems, the enhancement factor 2 could be estimated from the tunneling experiments (involving superconductivity in these alloys) and the calculated increase in 2 (with an increase of e/a) was found to be sufficient to compensate for the discrepancy between the experimental 7exp trend and the band calculation.

Until recently, the approach involving superconductivity data had not been considered to be applicable to the :t phase noble metal alloys, since they were thought to be non-superconducting. However, Hoyt and Mota (3a) (1976)observed superconductivity in numerous e phase alloys, which now makes it possible to evaluate not only the 2 values of pure noble metals (by extrapolation), but also the change in 2 on alloying (see Fig. 7-8). A linear relationship is found in the plot of 2 against (e/a) 2 and the extrapolation of this trend to (e/a) = 1.0 yields the following values; 2c° = 0.16, 2Ag = 0.16 and 2Au = 0.21. Regarding the initial slope of the electronic specific heat coefficient, we can now assess both contributions in eq. (6-9), i.e. one due to the band structure and the other due to electron-phonon interaction. Assuming the rigid band model for the dilute Cu alloys, and employing the band calculation for pure Cu due to Faulkner et al., 139) the band structure contribution, d(lnN(Ev))/d(e/a) , turns out to be -0.33. The electron-phonon contribution, d(ln(1 + 2))/ d(e/a), is 0.31 for the Cu-Ga alloy system, (3s) which is the only superconducting ~ phase known in Cu based alloys. Thus, an increase in electron-


rn

320 ]~\ 280 D X 0D

o~×

200 ~.~ Cu-Au

160

0.71 I ~ i 0.70

o [

~ 0.69 ~

0.68 O

0.67 0 20 40 60 80 100 Cu at.'/, Au Au

FIG. 8-7. The behavior of the electronic specific heat coefficient ~, and the Debye temperature 0D in disordered Cu Au alloy system. D: Delinger et al. (s~ (1972), x : Davis and Rayne (31 (1972), I : Martin ~6) (1968), ©: Martin and Waterhousd 7)

(1970). References are cited from Table 7-1.

phonon interaction on alloying is only sufficient to raise the band structure slope to a less negative value, or close to zero, but is insufficient to account for the definitely positive slopes observed in the experimental plots. Similar conclusion was derived several years ago by Grimvall, ~°'41) who estimated 2 by utilizing the high temperature resistivity data. Hence, some of the puzzle is removed, but not all. (c) Band calculations Jor concentrated disordered alloys: strictly speaking, changes in the enhancement factor should be deduced for each alloy by comparing the Vexv value with the calculated density of states at the Fermi level for each particular alloy. As discussed in the preceding subsection, band structure calculations for the disordered alloys have been recently developed on the basis of the coherent potential approximation. The success of this method was verified experimentally by means of the UPS or XPS measurements, particularly for the case of d band behavior on alloying in binary systems Cu Ni and Ag-Pd. (42) As already mentioned, the same method was applied to the band structures for the 15 and 30 at.% Zn Cu alloys by Bansil et a/., (3°) but they did not calculate the corresponding density of states curve. It seems that, at the present stage, such a computation would not be accurate enough to reveal the changes in the van-Hove singularity, or the change in N(Ep), on alloying. (d) Theories of dilute alloys: some authors believe that, rather than per-


forming a band calculation for a concentrated disordered alloy, it is logi- cally easier to calculate the effect of a single impurity atom on the electronic structure of the host metal. Thus, dilute alloy theories have been developed to account for the discrepancies between Lxp and the rigid band prediction. An assessment of these theories requires a good quantitative knowledge of the experimental initial slope of the 7 coefficient for each individual alloy system.

Before discussing the theoretical treatments, we therefore consider the experimentally derived initial derivatives for the ~ phase alloys. Mizutani et al. (43) (1972) discussed the initial derivative values of dlnT/d(e/a) in terms of their volume dependence. The resulting plot of dlnT/d(e/a) vs. dlnV/ d(e/a) shows an apparent lack of volume dependence and a positive intercept at about 0.3, suggesting a departure from the rigid band behavior for both the intercept and slope. However, Bevk and Massalski (44) subsequently measured the ~ values in the :t phase Cu-Sn alloys with special attention to dilute alloys. They concluded that the initial linear trend is restricted to a fairly dilute range of e/a, as shown in Fig. 8-4. If attention is centered only on such a dilute range, the values of dln~/d(e/a) in Fig. 8-4 change to 0.29, 0.72 and 0.97 for Cu-Zn, Cu -G a and Cu-Sn alloys, respectively.t The same analysis cannot be applied to the Ag-based alloys, because the scatter of the data points in the dilute concentration range is excessive. These new initial slope values no longer indicate a linear dependence between dln~/d(e/a) and dlnV/d(e/a). When the values of dlnT/dc, instead of dlnT/d(e/a), are plotted against the valence difference Z2 - Z1 between the host and the impurity, a quadratic dependence seems to emerge as shown in Fig. 8-8. Here, the initial derivative of dilute alloys of Au in Cu is determined from Fig. 8-7 and is included, although the large error bars create a large uncertainty in determining the initial slope. This suggests that a change in the 7 value in very dilute alloys is related to the valence difference Z2 - Z~ and may be expressed as:

dT/dc = ~(Z2 - Za) 2, (8-1)

where c is the solute concentration and ~ is a numerical constant. Stern (45) (1970), emphasized that "the experimentally derived y values

may not in fact represent the variation of the density of states of the host metal, or of the alloy as a whole, on varying e/a, nor the change in elec- t ron-phonon interaction factor. Instead, one may be measuring the amount of charge attracted or retained near the impurities, referring only to the electronic states at the Fermi level". The experimental results of Dicke and Green (46) (1967) in the Al-based fcc alloys, which indicate a decrease of the 7 value with increasing Zn content and an increase with increasing Ge content, can be explained qualitatively in terms of Stern's

¢In the analysis of Mizutani et al. , (42~ the data for 0.86 at.~/o G a - C u sample was excluded, since the 7 value did not seem to fall on the general master curve. However, if a non-linear initial behavior is accepted, the initial slope should be determined using this data point.


theory in which the electron states at the Fermi level are repelled from Zn impurities and attracted to Ge impurities. Stem's theory suggests that, in the very dilute concentration range, the amount of charge attracted to solute impurities would be linearly proportional to the potential difference between the host and impurity atoms. In the case of the polyvalent impurities in noble metals, the initial slope of the 7 value is, therefore, expected to be proportional to the valence difference Z2 - Z1. Although a quadratic dependence appears to be indicated by the data discussed above, a linear relationship cannot be entirely excluded because of the large experimental errors involved, particularly in the Cu-Au alloys.

Au Zn Go Sn

u "O

C

3.0

20

1.o j ~

-1.0' ~ x

© -2.0 ' , ,

1 2 3 Z2- ZI

F~G. 8-8. The initial derivative of the electronic specific heat coefficient 7 vs. the valence differences Z 2 - Z1 between solute and solvent atoms in dilute Cu-based alloys. The experimental slopes were obtained from the following data: C u - A u by Delinger et al. ~8~ C u - Z n by Veal and Rayne ~211 and Mizutani et al., ~11~ Cu Ga by Mizutani et al. c11~ and Cu-Sn by Bevk and MassalskiJ 19~ Calculated slopes are due to Lassetter and SoveA 4"~ O. and Terakura ~48) E3.

(See the references in Table 7-1.)

Recently, Lassetter and Soven ~4v~ (1973) calculated the change in the density of states of Cu with the introduction of polyvalent impurities, based on the formalism that is exact for a single impurity in a system of muffin tin potentials. The impurity potential is calculated through a self-consistent way utilizing the Friedel sum rule. The results show a marked difference from the prediction of the rigid band model. However, a slight increase in the 7 value is predicted only for the :~ phase Cu-Zn alloys and a decrease for the solutes of higher valence such as Cu-Ga, Cu-Ge and Cu-Sn.

Terakura ~48~ (1975) carried out a similar calculation and found an increase in the 7 value for various Cu-based alloys. He calculated the change in the partial number of states with s, p, dE and d7 symmetries for 4089


points in a 1/48 section of the Brillouin zone, when a single impurity atom is introduced into a perfect Cu lattice. The results of the partial density of states at the Fermi level depend on the slope of the number of states below a given energy and, thus, the numerical differentiation seems to be quite delicate. Nevertheless, it can be seen that the large contribution that can raise the 7 value results from the tail of the resonant d states of Cu atom. Although this theory predicts an increase in the initial 7 value for various Cu alloys, no quantitative agreement is shown between the theory and the experimental initial slope in Fig. 8-8.

In summary, the overall trend of the 7 values in the ~ phase noble metal alloys based on Cu and Ag is reasonably well established and indicates a slight increase with increasing e/a. Also, the trend of the 7 values in Ag~Au alloys is fairly well established. Although numerous theories have been proposed, none seems to account satisfactorily for the overall behavior of the experimental coefficients. The ab initio band calculations developed by Lassetter and Soven ~T) and Terakura ~.8) are quite promising, because an exact calculation is made in each case for the system where only a single impurity atom is introduced into an otherwise perfect host metal. If the limited amount of information for the dilute alloys within 1 at.% solute is accepted as valid, a quadratic dependencet of dlnT/dc on Z2 - ZI emerges. In order to establish a precise form of this dependence, more accurate measurements are needed for alloys containing less than

t o/ 1 a .,,o solute in each noble metal. It seems desirable that the predictions of each theoretical approach

should be tested from the point of view of two types of experimental data; the spectroscopic and optical measurements, which chiefly provide good resolution concerning the horizontal energy axis in a density of states curve (AE), and the specific heat and magnetic information, which relates mostly to the vertical axis (i.e. AN(E)). Of course, there are some measurements, such as XPS, which emphasize both axes.

8.2. Cubic fl Phase Solid Solutions

The disordered bcc fl phase is most frequently the next phase found following the ~ phase primary solid solution in a typical phase diagram based on a noble metal, especially at relatively high temperatures. As is well known, the bcc structure is associated with a large vibrational entropy. 1.9) Hence, a V-shaped phase field is typically observed with temperature. The bottom tip of the V, where the influence of entropy is smallest and that of e/a largest, when the influence of temperature is reduced, is located in a narrow range of e/a between 1.45 and 1.5 in all known

~Dawber and Turner t331 pointed out that an additional electronic specific heat resulting from the smearing effect on alloying should be proportional to (Z2 - Z~) 2, which is compat- ible with eq. (8-1). However, as stated earlier, the effect is probably too small to account for the experimental results.


1100

10oc

900 !

8OO °C

7oo

6o0

500

400

1.30

Cu al loys 1000 Ag al loys

~ S 900 v.-. 800 '\,

"~ / i 700

' : . . . . . . :""- 600 "'., ',', -- ', ~ . '~ ' , . - - Q

(1,403) ', 1' ~ l (1,446) / L ~ - A I 40C

'~ I I (1A80) ~ Cu-Zn 30C

(1,468)

1.35 1.40 1.45 1.50 1.55 1.60


(G)

1.30

A -AI

\ " i~/__ A~-Cd \ --/-- ~1a96)

,'.~5 1.'4o 1:45 1:5o 1:55 1:6o electron concentration

( b )

FIG. 8-9. Shape of the fl-phase field plotted as a function of electron concentration: (a) in copper alloys, (b) in silver alloys. (After Massalski and King.15%

systems, as shown in Fig. 8-9. A detailed review of fl phases is given in Massalski and King) s°)

With the decrease of temperature, the disordered fl phase loses its entropy advantage and either orders or transforms to another structure. If composition invariance is preserved the transformation is either massive or martensitic. (s°) If the fl structure becomes ordered prior to transformation, the structure change is always martensitic (unless a diffusion con- trolled two-phase reaction can take place). Only in the Cu Zn and Ag Zn system can the ordered fl' phase be retained at liquid He temperatures, without transformation. Hence the amount of low temperature information that can be obtained on fl phases is very limited.

With two-valent solutes, like Zn or Cd, the characteristic e/a range near 1.5 is attained at a composition near 50 at.% solute. In this case the ordered phases possess the simple cubic CsC1 structure. With this type of order a new set of the Brillouin zone planes, represented by the {100} zone, is introduced, in addition to the original dodecahedral ', 110l zone, characteristic of the disordered bcc structure, as shown in Fig. 8-10,

As mentioned in Section 5, the electronic structure of the disordered fi phase in Cu-Zn alloys was first explored by Jones i5~ (1937), who used a cone model calculation to derive the density of states curve. More recently, the electronic structure of the stoichiometric ordered fl' phase was extensively studied with the dHvA technique, which is effective here because of good order and perfect stoichiometry (Jan et al. (52~ (1967), Karlsson ~53) (1969), Springford and Templeton 154~ (1971l, Jan and Per- rott 155~ (1972)). As may be expected, the dHvA signals disappear if the composition is displaced slightly off the stoichiometric ratio. (52~ Thus, the dHvA results are available only for one composition even though the stable ,8' phase extends over 10 at.°a in the Cu Zn system. There are experimental data on the electronic specific heats (fl' Cu Zn, Veal and Rayne t56~


(1962); fl' Ag-Zn, Abriata et al. (57) (1970)), optical properties (fl' Cu-Zn, Ag-Zn and Au-Zn, Muldawer {Ss) (1962); /3' Cu-Zn, Muldawer and Gold- man (59) (1966);/3' Au-Zn, Jan and Vishnubhatla {6°) (1967)), photoelectron spectroscopy (/3' Cu-Zn, Nilsson and Lindau ~6~) (1971)), and magnetoresis- tance (/3' Cu-Zn, Sellmyer et al. ~62) (1967)).

Band structure calculations have been performed for the ordered /3' Cu-Zn, Johnson and Amar {63) (1965);/3' Cu-Zn, Ag-Zn and Ag-Cd, Amar et al. ~64) (1966); /3' Cu-Zn, Arlinghaus {65) (1969); /3' Cu Zn, Skriver and Christensen {66) (1973);/3' Cu-Zn, Moruzzi et a/. (67) (1974)).

Of particular interest here is the electronic specific heat data due to Veal and Rayne, ~56) who also calculated the density of states curve for the/3' phase, using the 12 cone model, originally proposed by Jones, and assuming contacts of the Fermi surface with the twelve { 110} zone planes bounded by appropriate energy gaps. The decreasing trend of the 7 values which they determined experimentally seems to fit the 12 cone model quite well when the energy gap AEI~o is chosen to be 4.7eV. However, the peak due to the 110 contact is then located in the density of states curve at about 1.2 electrons per atom. The effect of the {100} zone planes was ignored.

Jan (68) (1966) used the same 12 cone model, again ignoring the {100} zone effect, and proposed the energy gap across the {110} zone planes to be 3.49 eV, in order to accommodate both the electronic specific heat

,4 i I

/

i I

( \

\

\x

\

I , # ~1

z /

#,

I I

\ i ~

\ ..'r" \ / I \

k z k

\ \

\ \

~r =- ky

/ / ' / !

\ / / ~_ . . . . . - \ t*" s ~ ' l - X /

',-Zx k, , ~.

/ ' .j_ _'.._-.~ _ _ 2 / - - .

FIG. 8-10. Brillouin zone of the ordered ]~' structure. The dodecahedron with broken lines corresponds to the Brillouin zone for the disordered bcc structure. The CsCI type ordering in ,6' structure introduces the cubic Brillouin zone

as shown by the solid line.

E L E C T R O N I C S T R U C T U R E OF H U M E - R O T H E R Y P H A S E S

Tab le 8-1. Band Parameters in the fl' Phase Noble Metal Alloys

221

Alloy system AE1 lo(eV) Ev(eV) Authors Method

fl' C u - Z n 4.7 Veal and Rayne (1962) 12-cone 3.49 Jan (1966) 12-cone

E ( M I ) - E (Ms , ) ~ 3.3 8.9 Amar et al. (1966) K K R E(M3) - E ( M v ) ~ 2.6 9.9 Arlinghaus (1969) APW E(M1) - E(Ms) ~ 3.1 9.93 Skriver and APW

Christensen (1973) fl' Ag-Zn E(M3) - E(M~,) ~ 3.3 7.9 Amar et al. (1966) KKR ff Ag Cd E(M3) - E ( M v ) ~ 5.6 7.2 Amar et al. (1966) KKR

data and the dHvA data. Such a low value of AEtlo is in good agreement with the corresponding theoretical band gap calculated by various authors as shown in Table 8-1. The above analysis should not lead to any substantial error, provided the { 100} zone planes are completely submerged below the Fermi level and the energy gap AE100 is fairly small. However, both the dHvA measurements and band calculations, confirm the presence of a hole surface centered at R in the first zone of the fl' Cu-Zn structure, as is seen in Fig. 8-11. Furthermore, the overlapping electrons across the { 100} zone planes form 6 electron "lenses", which are multiply connected through the well known 110 "necks". All band calculations strongly indicate that the overlapping energy at X for the 7th band is greatly enhanced by the effect of the s~d hybridization due to the presence of the Cu 3d band (see, for example, the energy E(X,,) in the E-k relation obtained by Skriver and Christensen~66~). Thus, the {100} zone effect in the density of states curve should not be ignored. The density of states curve from the APW band calculation by Skriver and Christense# 66~ is shown in Fig. 8-12. The sharp rise marked X4. above the copper 3d band can be interpreted as the overlap effect across the I100} zone planes, combined with the contact with the {110} zone planes, which results in a fairly fiat E k relationship along the MX direction. This contributes to a high density of states. The resulting form of the density of states curve in the vicinity of the Fermi level (for the 50 at.% alloy, i.e. e/a = 1.50) appears to be mainly determined by the overlap effect across the { 100} zone planes, superimposed on the declining slope following the {110} Fermi surface contact at M. We note that the sharp rise at X4, in the density of states curve occurs at approximately 0.8 'electrons/atom, as compared with the previously mentioned value of 1.2, deduced by Veal and Rayne. The Skriver and Christensen results are essentially confirmed in a band calculation and the density of states curve due to Murrozi et al. ~67~ (1974), done a year later. In Table 8-1 are listed the energy gap values across the { 110} zone planes and the Ev values relative to the bottom of the conduction band, deduced by the various investigators.

The density of states value in Fig. 8-12 can be plotted in the units of m J/mole K z as a function of e/a, by integrating the area under the


° o

°o~o ooOeOtOeo~ °°o- I _e'.

~.~i.~ ¸ M

" K ~ ~ i .~.~' I i . : :~,( I "

FIG. 8-11. Fermi surface topography predicted for the fl' phase Cu-Zn. The detailed shape is based on Arlinghaus' calculationJ 651 (After Jan and

Perrott. t55~)

curve, and assuming the Fermi level to occur at 9.93 eV, as deduced by the Skriver and Christensen calculation, when e/a is equal to 1.5. This attempt allows a direct comparison between the experimentally determined 7 values and the band calculation, as shown in Fig. 8-13. The two curves are remarkably parallel to one another and the 7exp trend is consistently higher than the 7band trend by about 20~o. It appears quite reasonable to attribute this difference to the electron-phonon interaction, which is of the usual order of magnitude expected for the noble metal alloys. Thus, we conclude that the behavior of the electronic specific heat coefficient in the 13' Cu Zn alloys can be satisfactorily interpreted in terms of an essentially rigid band description with a fairly constant e l ec t ro~phonon interaction (2 ~ 0.20), and the band gap across both the { 100} and ~ 110~ planes of the order of 3.5 eV.


-J .-I hi

Z O r , , t -

.J

A

200 APW CuZn

DOS

150

100

5 0 A3 ',z

x? "iJ r;,

0 0.2 0.0 0.2 0.~, 0.6 0.8

ENERGY E tRy)

FIG. 8-12. Density of states for fl' phase Cu Zn. The main van Hove singularities are indicated in the figure. The number of input E k values in the first

Brillouin zone was 404,928. (After Skriver and Christensen. ~661)

0 . 8 0 , ,

~ ' E ) ~ e x p ( V e o l Qnd Rayne)

~ 0.70

Slo'iver and Christensen )

0 . 6 0 ~

I 1

1 ./,0 1 .&5 1.50 etectron concentration

FIG. 8-13. The electronic specific heat coefficient ~, in the fl' phase Cu Zn alloy system. The 7b,,,la trend is deduced from the density of states curve in Fig. 8-12. The electron phonon enhancement factor ). corresponds to about 0.2.


Table 8-2. Classification of 7 Phase Alloys

Transition Copper Silver Gold metals

Cu-Zn 7 Ag-Li 71-3 Au-Zn 71-3 Mn-Zn 71 3 Cu Cd 7 Ag-Zn 7 Au-Cd 71-3, Yo Mn-In 71 3 Cu-Hg h 3 Ag-Cd )', 70 Au-Ga 71-3 Fe-Zn h Cu-AI 73, 71-3, Yo Ag Hg )'1-3 Au-In 71-3, 70 Co-Zn 71 3 Cu-Ga )'3, )'1-3, )'o Ag-In h-3, Yo Ni-Zn 71-3 Cu-In 71-3 N~In 71-3 Cu-Si 71-3 Ni Cd )'1-3 Cu-Sn Yo Ni Ga 71 3

Pd-Zn )'1 3 Pt-Zn 7,-3 Pt-Cd 7,-3

7 = ordered complex bcc (D8 2 type), A ~ 52 noble metal +I IB solute, 71 = ordered complex bcc (D81 type), A ~ 52 transition metal + liB solute, 73 = ordered complex bcc (D8 a type), A ~ 52 noble metal + III solute, 71-3 = ordered complex bcc (D81 3 type), A ~ 52 precise type of atomic arrangement not determined, 70 = (distorted D81_ 3 type) A ~ 46 ~ 52 or larger. (Here A is the number of atoms in the unit cell.)

8.3. Complex Cubic 7 and It Phases

In add i t i on to the ~ and fl phases, the two o ther cubic phases y and It, have complex cubic s t ructures and are l imited to qui te na r row ranges of e lec t ron concent ra t ion , 1.54-1.70 for y and 1.40-1.54 for It. They are therefore typica l e lec t ron phases. (69) Below we review their e lect ronic s truc-

ture and discuss the avai lable e lectronic specific heat d a t a in reference to mul t i - cone a p p r o x i m a t i o n calcula t ions .

8.3.1. Fermi surface-Brillouin zone interactions in the 7 phase

Whi le the s t ructures of the numerous y phases are not all identical , they are s t r ik ingly s imi lar and are charac te r ized by the large cubic unit cells. They are usual ly o rde red or par t i a l ly ordered . The precise form

of o rde r varies with the ra t io of solvent to solute a toms. Wide - range y phases are usual ly a mix ture of three general types (y = D82; Yl = D81; ?3 = D83), (69) as l isted in Tab le 8-2.

The unit cell of the y-brass (D82 structure) conta ins 52 a toms and may be cons idered to be made up of 27 unit cells of fl-brass (which would a m o u n t to 54 a toms) with 2 a toms removed from the large cell (one at the corners and one at the center) and the rest shifted somewha t in pos- i t ion. This is shown in Fig. 8-14. The C u - Z n , C u - C d , A g - Z n and A g - C d y-brasses ( formed with d ivalent solutes) are k n o w n examples of this s tructure; in the Cu-A1 system, with a t r i -va lent solute, y-brass has a s imple bcc space lat t ice with 49-52 a toms in the unit cell.

X-ray measurement s made by Wes tgren and PhragmOn {v°) showed tha t a m o n g the crystal p lanes with smal l indices, only the sets {411} and {330} give rise to s t rong X-ray reflections, which, p resumably , lead to large energy gaps across the co r r e spond ing zone planes in k-space. The " la rge


(o) ( b )

FIG. 8-14. (a) The arrangement of atoms in a cubic cell formed by stacking together 27 unit cells of bcc structure. This large cell corresponds to 54 atoms. (b) The structure of ?-brass. The small arrow on the atom indicates a direction of the displacement of the corresponding atom in (a). (After Bradley and

Thewlis.(8 %

,,o4. F _ *'

,B.t!, !,~2_m 2 / o

C3( 2.0.1)~l C40.0,2)~

B~(~ ~ ~z_~ 2 . 2 , 2 / a

C,(2.,.o1~ C3(2.o.,)~

a

FIG. 8-15. The Brillouin zone of the ? structure. Each zone plane is replaced by the circle with equal area in the present analysis.


Brillouin zone" is, therefore, constructed from these planes, as shown in Fig. 8-15. The perpendicular distance from the origin to each zone is equal to 3nx//2/a. The resulting polyhedron consists of 36 planes, 12{330} and 241411}. Its volume is 45(2n/a) ~ and the atomic volume is, of course, a3/52. Thus, the zone can contain 90 electrons per unit cell or 1.73 electrons per atom.? Since the ratio of the volume of an inscribed sphere to that of the polyhedron is 2n/5\/2 ,~ 0.89, a spherical Fermi surface would just touch the Brillouin zone at ~1.54 electrons per atom. The presence of the energy gaps will, of course, tend to distort the Fermi surface, resulting in a contact with the zone planes at a lower e/a value than the free electron value. The fact that the stable 7 phases are found in the e/a range near and above 1.6 suggests that the Fermi surface is well in contact with the zone within the whole range of stability.

The electronic properties of the ), phase alloys have been studied by various techniques: electronic specific heats, ~72'73~ magnetic susceptibility, ~74'75) electrical resistivity ~76) and thermoelectric powerF 7) Veal and Rayne tTz~ measured the electronic specific heats in the 7 phase of Cu-Zn alloys and found a rather drastic decrease in 7 values with the addition of Zn. Jones t71) assumed that a spherical Fermi surface touches all 36 zone planes at the same e/a and he derived a simple density of states curve past the point of contact by counting available states on a spherical surface whose area is decreasing as the radius is increased because 36 equal caps of increasing surface area are being subtracted. This allows the density of states to be expressed as a ratio,

N(E)/N(Eo) = 18 - lVx/e/Eo, (E > Eo), (8-2)

where Eo is the free electron energy corresponding to the contact wave vector ko = 3x/2g/a. The fit of the Veal and Rayne data to eq. (8-2) is remarkably good, although the model itself is quite unrealistic in the sense that any distortion of the Fermi surface due to contact with the Brillouin zone has been neglected. The electronic specific heats of ? phases in the systems Ag-Zn and A ~ C d have been measured recently by the present authors ~73) (Fig. ?-2), and they also show the rapid decreases in the 7 coefficient, although the numerical values in the Ag-Cd alloys are substantially higher than those in Cu-Zn and Ag-Zn alloys. We shall refer to this difference later.

The magnetic susceptibilities in various 7 phase alloys were measured many years ago by Endo ~74~ as well as by Smith. ~751 As usual, the experimental magnetic susceptibility can be corrected for ionic contribution, the corrected ZeJ value being assumed to be entirely due to the conduction electrons. The results are plotted as a function of e/a in Fig. 7-6, together

~'The vo lume of the t rue Bri l louin zone, which is a dodecahed ron associa ted wi th the bcc Bravais lattice, is 2(2~z/a) 3 and hence the large zone canno t be complete . In fact, as discussed by Jones, ~711 the A-points in Fig. 8-15 are in teror points of the t rue Bri l louin zone. Since the zone is incomplete, some electrons may lie outs ide the zone, but this does not necessari ly m e a n tha t over lap occurs across a zone plane.


with data for all other phases in noble metal alloys. The L,~ values in the 3, phases indicate a large diamagnetism which is increasing with increasing electron concentration. Although the drastic change in 7.~ is similar to that of the electronic specific heat coefficient, the presence of a large diamagnetism cannot be explained in terms of a nearly free electron model. This is because the Pauli paramagnetism, plus the Landau diamagnetism, leads to a positive value of ll2N(Ev), proportional to the density of states at the Fermi level. As was discussed in Section 7.2, Jones ~Ts~ attributed the large diamagnetism to the Peierls diamagnetism 179~ associated with the complicated orbital motion of conduction electrons, resulting from a distorted Fermi surface in the ~, phase alloys.

Because of the incompleteness of the zone, electron overlaps may occur in the middle of the ), phase range, in the neighbourhood of corner points A in Fig. 8-15. Although neither electronic specific heat nor magnetic susceptibility data indicates any strong evidence of the overlap effect, the electrical resistivity 176~ and the thermoelectric power data {77~ showed anomalies at about 65 at.% Zn, in 5, phase Cu Zn alloys, which was attributed to the overlap effect.

8.3.2. Density of states curt~es and electronic spec!tfe heat data in the 3' phase

The density of states curve for the conduction band of Cu derived from the multi-cone model showed a reasonable agreement with many features of the N(E) curve derived on the basis of the first principles approach. Since no first principles band calculation has been carried out for the 3' phase structure, it is of interest to construct as a first approximation the N(E) curve based on the multi-cone model, for comparison with the available data. As already described in Section 3.2, the cone model requires the following parameters to be specified: the solid angles associated with the zone planes, the energy gaps, the E0 value, the effective mass parameter

Tahh' ,'~-3. Band Parameters in the Cone-model for the ? Phase Alloys

i' phase Cu Zn 7 pilase Ag Cd (60.0 at)',, Zn) (60.0 atY~; Cd)

lattice constant (AI 8.8365 9.95 atomic volume (/~3) 13.2689 18.9437 k33o = 3 n \ 2/a ( A i) 1.5083 1.3395 k411 = 3~z~2/a (/k i) 1.51)83 1.3395 E,,~,. (eV) 8.6667 6.8355 AEs3o (eV) 2.01 12.01) kl','a 11 (eV) 1.29 (1.29) m* 0.9 0.9 Eo leVI 9.629 7.595 ~')33o 0.4126 (1.4126 ('~41 ~ 0.3173 0.3173 Ef (eV) 9.4(I 7.23


and the lat t ice constant . The values employed in the present analysis are l isted in Table 8-3. In the first place, the solid angle of the respect ive

zone p lane is app rox ima te ly ob ta ined by replac ing each zone p lane by an equivalent circle. The detai ls are shown in the Appendix . Secondly, the energy gap AE33 o across the {330} zone planes may be assessed by focusing our a t t en t ion on the close re la t ion between these planes and the { 110} planes in the bcc s tructure. The " idea l " 7 s t ruc ture can be envisaged to consist of 27 unit cells of the bcc lat t ice with a la t t ice cons tan t which is three t imes as large as that of each bcc unit cell. As discussed earlier, the " rea l " 7 s t ruc ture is ob ta ined from the " idea l " by r emov ing two a toms and reshuffling others. In the ideal 7 s t ructure, the 330 reflection is exact ly identical to the 110 reflection of the bcc s tructure. The s t ruc ture factors

$33o can be easily ca lcula ted for bo th " idea l " and " rea l " s t ruc tu re s t and g'ideal is are given by: o330K'ideal = 54 and ~330Crea~ = 35.2. Obvious ly , the value ,-,330

equal to the number of a toms in the unit cell, since each a tom cont r ibutes to the s t ructure factor in a const ruct ive way or in phase. However , a subs tan t ia l decrease in the s t ructure factor occurs fol lowing the d i s rup t ion of the per iod ic a r r angemen t of the lattice. Accord ing to Jones, ~78~ the ra t io of the var ious energy gaps in a given al loy may be a p p r o x i m a t e d by the ra t io of the co r r e spond ing s t ructure factors;~

Ag?real [c rea l /K ' idea l~ AK' ideal = 0.65 × AE{to. (8-3) L~330 : ~ o 3 3 0 / , . , 3 3 0 ! X t.aL, 330

Strict ly speaking, the AE~lo in eq. (8-3) should refer to a hypothe t ica l d i so rde red bcc al loy of compos i t i on in the s table 7 phase range ( ~ 60 at.~o Zn). However , accord ing to eq. (8-4), the compos i t i on dependence of the energy gap is smal l and, hence, the ca lcula ted AE~'lo for the ordered fl' C u - Z n al loy (50at.~o Zn) can be subs t i tu ted for AE~lo in eq. (8-3) (in any case AE~'lO is the only avai lable datum). As discussed in Sect ion 8.2, the band s t ruc ture of the fl' C u - Z n al loy has been extensively s tudied and the typical band pa rame te r s are l isted in Table 8-1. In the present

tThe structure factors $33o and $411 in the 7 structure were calculated to be 35.2 and 22.6, respectively, following the structure study by Bradley and Thewlis. ~8°) In Table V in Bradley and Thewlis' paper, the structure factors are listed in the form rShkd4P 2. However, Jones ~vs) and Mott and Jones ~st) apparently assumed this to be IS 12 only. Hence, the energy gaps estimated by Jones ~vs) as well as the structure factor table in the book by Mott and Jones (p. 168) should be multiplied by four in both cases. This was first pointed out by Bevk.(82)

++Jones derived a simple approximate formula for the energy gap; AEhk l = (2eZ/n2a(h2 + k 2 + /2)).(Z - - f . ) ' S h k t , (8-4)

where Shk~ is the structure factor, f, is the atomic form factor for X-rays at the value of its argument sin0/2 = n/2a, Z is the atomic number, e is the electronic charge and a is the lattice constant. The values f, for Cu and Zn are found to be 22.2 and 22.9, respectively) 831 Thus, in the Cu-Zn system, we can a s s u m e A E h k I to be weakly dependent on composition. Using eq. (8-4), we obtain approximate band gaps AE33o = 4.8 eV and AE, t~ = 3.1 eV for the 7 structure and AEll~ = 7.3eV and AE2oo = 5.5eV for pure Cu. It appears tvs)ts41 that eq. (8-4) overestimates the magnitude of the energy gaps, because of the assumption of a point charge for the periodic potential.


analysis, we use the value of AE~'lo obtained by Skriver and Chris- tensen. (66) The value of AE33o in the y structure, then becomes: AE33o = 0.65 × 3.1 = 2.01eV and the value of AE411 = ($411/$330)'- AE33o = 1.29eV.

It was shown in Section 3.2 that the parameter E0 for Cu is more realistic if taken directly from an E-k relationship based on first principles band calculation, rather than being assumed to be given by the free electron approximation. However, the lack of a band calculation for the 7 structure does not allow us to use such an Eo value directly. Thus, in the present approach the free electron Eo value is assumed to be modified by a single parameter m*, regardless of the direction of the cone axis;

Eo = Efree/m* = (h2/2m)(3rc\/2/a)2/rn * = 8.66/m*(eV). (8-5)

Furthermore, the parameter m* is assumed to be the same as the effective mass parameter in eq. (3-3) and is chosen in such a way that the resulting N(E) curve places the Fermi level in the range of 9.5-10eV, which is the expected energy level, judging from the Fermi level in various Cu- alloys as listed in Table 3-2. The m* value thus chosen is 0.9, which places the Fermi level at 9.5 eV. The resulting N(E) curve is plotted as a function of energy in Fig. 8-16 and as a function of e/a in Fig. 8-17. In the present

0.3

0 0

o 0.2

z 17.1

v I i , v f f ! i

~" phase versu'~ free electron model. F~

I I I I I I I IT

energy eV) 11

FIG. 8-16. The density of states curve for the y phase Cu Zn alloy (60.0at.% Zn) derived from the cone model. The dotted curve refers to the free electron parabolic band with the same atomic volume and the effective mass. It is clear that the cone model curve could not generate a minimum in AU (see Fig.

5-1) before the 7 phase terminates, since area B cannot outweigh area A.


0.9

0~8

0.7

06

0.5

0.3

0.2

0.1

0 0

i i i

I~111 .

g o

1 I I 0.5 1.0 1.5

e lec t ron concentration 2.0

FtG. 8-17. The N(Ev) values, derived from Fig. 8-16, are plotted as a function of electron concentration. The circles refer to the experimental 7 values

measured by Veal and RayneF 2~

analysis, the overlap effect due to incompleteness of the zone is ignored. Thus, the N(E) value becomes zero, when the large zone is full. It can be seen that the two different energy gaps across the {330} and {4111 planes give rise to two peaks in the N(E) curve, despite the fact that the distance from origin to the centers of the two different sets of zone planes is the same. Figure 8-17 clearly shows that the experimental 7 values very nearly fall on the drastic declining slope in the N(E) curve following the contact of the Fermi surface with two sets of zone planes. Therefore, the stable 7 phase is characterized by the electronic structure in which the Fermi level falls on a steep declining slope of the corresponding N(E) curve. At first sight such a drastic decrease in the density of states would be expected to accelerate the increase in the total electronic energy and, thus, should lead to an increasingly unstable structure. This seems to con- tradict the presence of the stable 7 phases. We shall refer to this feature in a later section.

As emphasized in Section 3.2, the cone model does not take the d band into account directly. However, the adjustable effective mass parameter m* certainly reflects not only the average departure from the free electron approximation due to various band effects, but also the d band influence. Strictly, the density of states curve should be constructed for each particular alloy, corresponding to its own band parameters and lattice constant. The experimental 7 value for any particular alloy could then be compared directly with the calculated N(EF) value for that alloy and would provide


some indication of the electron-phonon interaction. However, the present analysis is too approximate for such a refinement.

As can be seen from Fig. 7-2, the 7 values in 7 phase Ag Cd alloys n3) are substantially higher than those in the 7 phase Cu-Zn alloys. This puzzling behavior may be interpreted on the basis of the cone model approximation, The most obvious difference between the two alloy systems is the large difference in the lattice constant. Assuming that the band parameters, such as the energy gaps and the effective mass are the same as those employed in the Cu-Zn alloys (Table 8-3), the density of states curve for 7 phase Ag Cd is plotted as a function of e/a in Fig. 8-18, for comparison with the Cu-Zn 7-brass. An increase in the lattice constant of 13% enhances the overall N(E) curve and can qualitatively account for the large electronic specific heat coefficient in this phase. Of course, the energy gaps in the ? phase Ag-Cd alloy may not be as large as those assumed in the present analysis. Smaller energy gaps would affect mainly the peak profiles. In this analysis, the Fermi level at e/a = 1.6 corresponds to an energy of 7.5eV, which may be compared with 7.5eV for pure Ag. ~85~ The fact that the Fermi energy in pure Ag is lower than that

I.I

1.0

0.9

0.8

0.~ -o

o

.~ 0.~ E ~ c

0/-

0.3

0.2 i !

01

I i - - - T - -

./f-" ',,, I phose Cu-Zn

/ / ~ ' p

j s I /

- - t _ . [ _ _ I o 0 05 1.0 I.

electron concentrQtnon

FiG. 8-18. The N(Ev) values for the ? phase Ag-Cd alloys in comparison with those for the ? phase C u - Z n alloys. The experimental results shown by the

open circles were obtained by the present authors. ~73~


in Cu(~9.0eV), is largely due to the larger atomic volume of Ag(f~ = 17.06 A3), compared with that ofCu(f2 = 11.81 ~3) (see Table 3-2).

8.3.3. Fermi surface-Brillouin zone interactions in the # phase

The alloying behavior of the p phases with the structure of fl-manganese has received relatively little attention. The #-structure is cubic with twenty atoms per unit cell. The known/~ phases are present only in seven alloy systems; A1-Mn, Mn-Zn, Co-Zn, Cu-A1, Cu-Si, Ag-A1 and Au-A1, of which the last four are based on the noble metals and are stable only in a narrow electron concentration range between approximately 1.40-1.54. Only A1 and Si solutes, which contain no d bands, form p phases with the noble metals. This suggests a possible special relationship between the solute and solvent atoms in the p phases, in addition to the electron concentration requirement.

The large (Brillouin) zone of the p phase shown in Fig. 8-19 is based on large structure factors as given by Preston. ~86) Among various planes of low indices, the {221} and {310} planes yield the first large structure factors and form a polyhedron in k-space.t Each set of planes consists of 24 variants, resulting in a polyhedron of 48 zone planes. The volume of this zone is (971/60) (2~/a) 3 and, since the atomic volume is a3/20, the zone contains 1.62 electrons per atom. It is therefore not a true Bril- louin zone and is again incomplete. The distances of the {221} and {310} planes from the origin in k-space are 3~/a and zrx/]~O/a, respectively. The geometry of each zone plane is described in Fig. 8-19. The electron concentration for an inscribed spherical Fermi surface touching the {221} zone planes is 1.41. It follows that when the {221} contact occurs the zone will be nearly 87% occupied. Another contact for a Fermi sphere is expected to occur with the {310} set of zone planes and the corresponding electron concentration is 1.65, a value which exceeds the range of stability of the known # phases (e/a = 1.40-1.54). The above considerations are derived from the free electron model. Any departure from a spherical Fermi surface, due to the presence of finite energy gaps across the Brillouin zone, will result in zone contacts at lower values of e/a so that both sets of zone planes will be touched when the ~ phase is stable within the narrow range of composition.

The electronic structure of the p phases has seen little exploration except for the low temperature specific heat measurements in two # phase Ag-AI alloys by Bevk et al., ~87) who interpreted their results in terms of the

tln addition to the {221} and {310} zone planes, the {210} zone planes also possess a small, but finite structure factor. Assuming the relation AEzlo = ($21o/$22~) x AE2zl and using the value AE221 =4.9eV, as estimated from the Jones equation, we obtain AE21o = 0.7eV. With a much smaller value of AE221, the corresponding AEa~o becomes negligible. The electron concentration, corresponding to an inscribed Fermi sphere with '3 radius k21 o = n,,, /a, is approximately 0.58. Thus, the {210} zone planes are deeply immersed under the Fermi surface, and, with such small energy gaps, they can be ignored.

E L E C T R O N I C S T R U C T U R E OF H U M E - R O T H E R Y P HAS ES 233

C2(. ~ ^11 ~211 v-8-~ ~-

;z

A 3

7 7 2~/~~ C ~ 1"~" 0"8 ) ' ~ ' ~

311 2"¢

113 2~

.~ 7 2/[: c1( o ~ )"6"

-1 1 2 1 T ~

v .3 ~.)P,=~ AI(~'O0 a

FIG. 8-19. The Brillouin zone of the /~ phase alloy. Each zone plane in the present analysis is replaced by the circle with equal area.

1 '~ ~2x

Jones' cap model, according to which the density of states may be expressed as

N(E)/N(Eo) = 12 - ll~/E/Eo, (E > Eo). (8-6)

Here, only {221} zone planes are allowed to contact the Fermi surface and the presence of the 310 energy gap is ignored. The results indicate a remarkable agreement between uncorrected 7~xp values and an essentially free electron (no band gaps) model. Below we consider the same results again on the basis of a multi-cone model calculation.

8.3.4. Density of states curves and electronic specific heat data in the la phase

No first principles band calculation has been attempted for the p phase structure, nor for any similar structure. Thus, neither the energy gaps nor the Eo values can be assessed with some guidance to previous work. Bevk et al. ~8~) estimated the energy gaps from Jones' eq. (8-4) to be 4.5 and 3.5eV for {221} and {310] zone planes, respectively. For the Eo value, we may choose the free electron value devided by 0.8, assuming m* to


be 0.8. The solid angle for the respective zone plane is easily calculated along the lines discussed in the Appendix. The misfit parameter is found to be 0.9°,/, (Table 3-3), but readjusted values are employed in the present analysis. The resulting N(E) curve yields a large peak associated with the 221 contacts and the band is completely filled already at 1.25 electrons per atom. Hence, with this approximation, there is no available density of states in the stable composition range of the # phase (e/a = 1.4 ~ 1.5), unless the incomplete zone allows a substantial overlap into a higher zone. As mentioned previously, the Jones' eq. (8-4) usually overestimates the energy gap. Since a further substantial reduction in the m* parameter is unrealistic, a reduction in the energy gap seems to be more appropriate. Mizutani and Massalski ~8s) estimated the energy gap AEoo 2 across the {002} zone planes of the hcp ~ phase in Ag-A1 alloys to be only 0.9 eV. Although the hcp structure differs substantially from 3-Mn structure, fairly small energy gaps may be expected in the /~ phase Ag-A1 alloys. If the density of states curve is recalculated using AE221 = AE3~o = 1 eV and m* -~ 0.9, a more realistic density of states curve is obtained. The various band parameters are listed in Table 8-4. In Fig. 8-20, the total density of states curve is plotted as a function of e/a, together with the two subbands arising from the 221 and 310 type-cones. It must be noted that a total N(E) curve is derived by adding subbands at the same energy, but not at the same electron concentration. The above choice of band parameters locates the 221 peak prior to the beginning of the /~ phase, while the 310 peak is just after the termination of the phase. The experimental 7 values are found on the decreasing N(E) slope caused by the 221 contact effect. The increasing trend in N(E) due to the approach of the Fermi surface toward the 1310} zone plane may be viewed as a modify- ing influence responsible for the relatively moderately falling slope of the experimental 7 trend. The Fermi level is located at about 8.0eV corre-

Table 8-4. Band Parameters in the Cone-model for the g Phase A~AI Alloys

# phase Ag-A1 (23 at.~o AI)

lattice constant (A) 6.920 atomic volume (/~3) 16.5686 k221 = 3~/a (A l) 1.3619 k31o = n./lO/a (A J) 1.4356

221 (eV) 7.066 Efree 31o (eV) 7.851 Efree

AE221 (eV) 1.0 AE310 (eV) 1.0 m* 0.9 E~ 21 (eV) 7.851 Eo 31° (eV) 8.723 m221 0.3483 ~O31o 0.1753 E v (eV) 7.92


- - I I I

0.~ f~ A g - A t ( 23 at%At )

0.7 ,~e~, = 1.o ~v / \ zxE3~o = 1.0 eV / \

O.E .

~: 0. ~.

:~ O.Z E

l

0.:3

02 . . . . . . . . . . . . . . . . -

0.1 / " ~ 1 i

0 i I ..~ 0.5= 1.0 1.5

electron concentrat ion

FIG. 8-20. The plot of the N(EF) values for the ~ phase Ag Al alloy system, derived on the basis of the cone model. ~221]b,, d and [310lb,., ~ are contributions to the density of states curve from the respective cones. The experimental

7 values are due to Bevk et al. (87)

sponding to the electron concentration of the stable I~ region. As mentioned in the previous Subsection 8.3.2, various uncertainties in the band parameters prevent us from considering any possible overlap effects, or the electron-phonon interaction, in any detail.

8.4. Close Packed Hexagonal ~, and E Phases Solid Solutions

The hcp phases, denoted usually by the Greek letters ~, E and ~/, are the most numerous of all alloy phases of the noble metals. Their axial ratios lie relatively close to the ideal value for the close packing of spheres, as shown in Fig. 7-7. Unlike the cubic structures, which occur within characteristic ranges of e/a, the stable hcp structures can occur anywhere within the electron concentration range between approximately 1.32 and 2.0, except for a narrow range between 1.87 and 1.93.169) This wide range of e/a includes the values of 1.5 and 1.75 which were once thought to be of significance in connection with Hume-Rothery 's scheme of "3/2" and "7/4" electron compounds. However, it now appears that these particular ratios do not play any special role in the stability of the disordered hcp phases, nor is the division of these phases into ~ and E types of any special significance. The Greek symbol e is usually reserved for hcp phases in alloy systems of the noble metals (or Li) with Zn and Cd, where phases frequently follow the 7 phases in the respective phase diagrams. On the whole, however both the ~ and e phases can be considered as


essentially a part of the same wide phase field that has been interrupted by the presence of the ? phase, t89) The hcp q phases are primary solid solutions based on Zn or Cd. Since the axial ratios of Zn (c/a -- 1.856) and of Cd (c/a -- 1.886) are strikingly higher than the ideal value for the close packing of spheres (c/a = 1.633), the electronic properties of the q phases may be expected to differ from those of ( and E phases, is9'9°)

The importance of electron concentration as the major parameter con- trolling the properties and behavior of the hcp phases became clearly evident only after the relationship between the axial ratio (c/a) and electron concentration was established in detail (Raynor and Massalski ~91) (1955); Massalski and King t89) (1962)). When e/a is constant, for example in a ternary system, the c/a also remains constant. However, when e/a is allowed to change the c/a changes accordingly. In binary systems, the axial ratio trends of all known ( and E phases conform to a general pattern as shown in Fig. 7-7. Consideration of this behavior suggests a direct dependence of the structural parameters a and c on the FsBz interaction; as the electron concentration increases, the resulting contacts and overlaps of the Fermi surface with respect to different sets of zone planes cause a distortion of the Brillouin zone. This in turn affects the lattice parameters in real space. Thus, the earlier models of the electronic structure of the hcp phases have been derived mainly from the interpretation of the trends in lattice parameters, ~5°'89'92) but more recently the electronic structure has also been explored by additional techniques such as electronic specific heat t88'9°'93'94~ superconductivity measurements, t95,gm magnetic susceptibility, ~97'98) thermodynamic activity °9) and positron annihilation/1°°)

8.4.1. Fermi surface-Brillouin zone interactions

The geometry of the Brillouin zone for the hcp structure and the corresponding vertical sections through the extended and the reduced zones, are illustrated in Fig. 8-21. The dHvA data for pure hcp metals, for instance, are often interpreted in terms of the reduced zone scheme, while the low temperature specific heat data can be more conveniently discussed in terms of the extended zone. As is well known, if the extended "roofs" formed beyond the {10.0} planes by the intersection of the {10.1} planes, are removed, the resulting zone is still surrounded by energy discontinuities in all directions except along the lines of intersection between the {10.1} and {10.0} zone planes (line HL in Fig. 8-21). This smaller zone is sometimes known as the Jones zone and its electron content per atom is:

e/a = 2 - 4 \ c / L - 4 \ c / j

where c/a is the axial ratio. The Jones zone is bounded by three sets of zone planes; six {10.0}, twelve {10.1} and two {00.2} zone planes. The


(cl

O0.21overlap electron 'lens'

_ L . _ _ . ~ , / t o a 2 ) <~

/ ,,' ,,' ; ' IV

( b ) A /(00.2} L ( ~

. . . .

;l-

--t , FIG. 8-21. The Brillouin zone of the hcp structure in the extended scheme (a) and in the reduced scheme (b). The possible contours of the Fermi surface in the vertical section of the corresponding Brillouin zone are shown in (c) and (d). The shaded area corresponds to the holes in pure Zn. The hole in

(d) is known as a portion of the "monster". (After Massalski et a/. (9°))

distance from the origin to the respective zone plane in k-space is given by:

klo.o - 2~ koo.2 - 2~ and klo.1 - 1 + (8-8)

and hence depends on the axial ratio. In the range of cla higher than ,d), the {00.2} zone planes are closest to the origin, leading to the sequence koo.2 < kto.o < klo.1 which holds in the r/ phases, where c/a exceeds 1.75. The sequence klo.o < koo.2 < klo.1 holds for all ~ and e phases structures. The corresponding Jones zone holds, at most, only 1.75 electrons per atom. Therefore, overlaps of electrons from the Jones zone into higher zones are expected at relatively low values of e/a. The interpretation of the lattice spacing trends in the ~ phase Ag-based alloys, (92) whose axial ratios vary between 1.63 and 1.58, strongly suggests that overlaps of electrons across the {10.0} zone planes already occur at about 1.4 electrons per atom. The occurrence of possible overlaps across the [00.2} zone plane within the range of the e phases has been irfferred from the measurements of the lattice spacings, (89) electronic specific heat coefficient, (93) the Debye temperature, (93) the superconductivity transition temperature, (96) the magnetic susceptibility (97) and the thermodynamic activity. (9°) This is shown in Fig. 8-22. In each case the onset of electron overlaps across the {00.2} zone planes has been proposed for the range of e/a exceeding approximately 1.85 electrons per atom. All above interpretations imply the occur-


°'°I I

cD ~ 31c

296

1.59

" " 1.57 t )

1.65 2.0

A

-! -2.0 X" -&.O

-6.0

~ 0

W Tc ,_v o.1o

1.65 117 118 1.9 etectron concentration

FIG. 8-22. Behavior of various physical properties in the • phase Ag-Zn alloy system; electronic specific heat coefficient 7, the Debye temperature 0D, axial ratio c/a, magnetic susceptibility due to conduction electrons Ze], the second derivative of the free energy with respect to concentration (d2F/dXZz.) and

superconducting transition temperature T~.

rence of FsBz interactions that should be reflected also in the corresponding density of states changes on alloying.

It is of interest, at this stage, to consider the form of a density of states curve for a disordered intermediate hcp alloy in order to pinpoint the relationships between N(E) and the FsBz interactions. Unfortunately, unlike for the case.of c tand //'-brasses, no first principles band calculation has been attempted thus far. The application of a multi-cone approximation also becomes more complex here because overlaps into higher zones are significantly involved. The available density of states curves for the hcp structure are at the moment restricted to several pure metals. Allen et al. (l°a~ (1968) have derived a histogrammic density of states plot for pure Zn and Cd, while smoothed density of states curves for Be and Mg were calculated by many investigators; Herring and HillJ 1°1~ Cornwell, (1°3~ Loucks and Cutler,! 1°4t- Terrell, ~1°5) Inoue and Yamashita <1°6~ and Nilsson


1 , 0 1 1 1 1 1 1 1 1 1 1 i l l [

~ 0.5

0 0

1.0

Be

2 /, 6 8 10 12 14 energy (eV)

I I I ] I I I

"6 E

"~ 0.5 E p,=,

0 I i i i I I

1 2 3 4 5 6

energy ( eV )

FIG. 8-23. The density of states curve for Be and Mg derived by Inoue and Yamashita. (1°6) The parabolic bands are drawn, using the appropriate atomic

volume for the respective metal.

et al. (1°7) for Be, and Inoue and Yamashita (1°6~ for Mg. The density of states curves for Zn, Be and Mg are reproduced in Figs 3-3 and 8-23. These three metals may be regarded as representatives of hcp metals with widely differing axial ratios; high c/a = 1.856 for Zn, a nearly ideal c/a = 1.623 for Mg, low c/a = 1.568 for Be. As can be seen from these figures, the density of states curve for an hcp structure on the whole deviates from a parabolic band, as the energy increases, and shows a few prominent peaks, followed by a drastically declining slope. It is on this slope that the Fermi level for the divalent metals is located. However, the lower half of the curve can be fitted to a parabolic band if an appropriate effective mass is adopted: 0.88, 1.0 and 1.2 for Zn, Mg and Be, respectively. The low value of m* for Zn is most likely due to the presence of the d band close to the bot tom of the conduction band, as was suggested in Section 3. The high mass in Be can be attributed to the strong hybridization ors and p wave functions. (1°61 On the other hand, Mg is well described by a nearly free electron mass. Although all these metals have two valence electrons per atom and may be represented by relatively similar features


in the density of states curve, the Fermi levels differ, particularly between Mg and the other two metals, as is seen in Table 3-2. It was argued in Section 3 that the basic profile of the density of states curve is largely determined by the atomic volume and that the FsBz interaction, as well as the interband interaction, further perturbs the parabolic band. The larger the atomic volume, the larger is the resulting density of states and this leads to a reduction in the Fermi level. The atomic volumes of the three metals are 23.3, 8.13 and 15.2 (A) 3 for Mg, Be and Zn, respectively. The large atomic volume in Mg is certainly responsible for its low Fermi energy. Similarly, if the s-p hybridization were absent in Be, its Fermi level would reach the free electron value of 14 eV, as a consequence of its small atomic volume.

The N(E) curves, in units of mJ/mole.K z are plotted in Fig. 8-24 as a function of electron concentration for the three hexagonal metals. The high 7 values in Mg merely reflect its low Fermi energy or its large atomic volume. Nevertheless, the position of peaks and a subsequent declining slope occur more or less at the same electron concentraion for all three cases, in spite of the large difference in the axial ratios, atomic volumes and the electronic interactions, such as the s-d or s-p interactions. This strongly indicates that the main features of the respective density of states curves originate from the FsBz interactions in which e/a plays an essential role. From this we conclude that a density of states curve for a disordered hcp alloy will have essentially the same characteristic features as those of the electronic specific heat data in various hcp electron phases in terms of a general N(E) curve as indicated by the work on pure hcp metals.

1.0

0.5

0 0 1.0 2,0

electron concent rQtion

FIG. 8-24. The N(Er) values vs. electron concentration for Be, Mg and Zn, calculated from Fig. 8-23 and Fig. 3-3, respectively.


8.4.2. Density of states curves and electronic specific heat data--General trend

The experimental Y coefficients plotted in Fig. 8-25 as a function of e/a ~94) show that, irrespective of the solute or solvent species, all available 7exp values follow a very similar general trend over a wide range of electron concentrat ion. An increasing trend is evident in the lower e/a range, cul- minat ing in a broad max imum at about 1.5 electrons per atom, and followed by a decreasing trend at higher e/a values. The theoretical density of states curve for the hcp Zn, shown in units of mJ/mole .K ~ in the same figure allows a direct compar i son between a relevant calculation and the experimental data. This shows that the large peak in the theoretical curves more or less coincides with the experimental peak on the abscissa.

Massalski et al. (9°) (1975) interpreted the large peak in the N(E) curve for pure Zn as resulting from the contact between the Fermi surface and the twelve {10.1} zone planes (near the point L), combined with an additional contr ibut ion from overlaps of electrons across the six vertical {10.0} zone planes (near the point M). In this range of energy a nearly horizontal curve in the E-k relationships is indicated along the L M line in the Brillouin zone in the band calculation for pure Zn (Stark and Fali-

0.80

0.70 "6 E

0.60

0.50

0.40 0.5

; ] , L;

Ag-AI ( , )

AgCd(b) / ~ Ag.Sn cc~ Ag- Sn (hi

Ag-Zn (d) Au-Cd (') Cu'Ge(b) i Cu-Zn (')

/i ,

! : , e })ii ' l

~'-~hcp olloy phases-~ i! I I 1.0 11.5 2.0

e l e c t r o n concentration FIG. 8-25. Trends of electronic specific heat coefficients as a function of electron concentration for hcp Hume-Rothery alloys, shown against band calculation for pure Zn. Literature data are as follows: (a) Mizutani and Massalski, (9~ (b) Massalski and Mizutani, (27) (c) Isaacs and Massalski, (28~ (dj Mizutani and Massalski, (29) (e) Bevk et al. (35~ and (t) Bevk and Massalski. (3°) The above reference numbers are cited from Table 7-l. (After Massalski and Mizutani. (94~)


coy t1°8)) and also for Be (Nilsson et a/. t1°7) and so on), and for Mg (Inoue and Yamashita t1°6) (1974)). A small energy gradient in the E-k relationship can contribute to a large value in the corresponding density of states. Thus, the combination of contacts and overlaps with respect to a large number of zone planes is clearly responsible for the large peak in the N(E) curve in hcp metals. The distance of the {10.1} planes from the origin of the zone is relatively insensitive to the axial ratio, eq. (8-8). Hence, the large peak may be expected to occur at similar e/a values in most hcp structures (see Fig. 8-24). Once contact with the {10.1} planes occurs, the electrons will be allocated in the remaining hole regions of the Brillouin zone until overlaps across the {10.1} or {00.2} zone planes become possible. Until the corresponding e/a is reached, a progressive decrease in the N(E) curve is expected as is actually seen in Fig. 8-24. Based on the above interpretation the likely Fermi surface topography for a typical hcp Hume- Rothery phase may be expected to be like that shown in Fig. 8-26. The recent positron annihilation studies of the Fermi surface in the ( phase Cu-Ge alloys, by Suzuki et al., ~t°°) are entirely consistent with the conclusions drawn from the electronic specific heat data. Indeed, because of zone contacts and overlaps that are likely to occur in all hcp alloy phases, this particular group of alloys offers a most challenging research area for the positron annihilation method.

The experimental curve in Fig. 8-25, although of the same general shape as the calculated curve, is consistently displaced to higher values of N(E) by approximately 30~o.f This is undoubtedly due to the electron-phonon enhancement, suggesting a fairly constant enhancement factor, of the order of 0.2-0.3, over the wide range of electron concentration. This is in excel- lent agreement with the 2 values derived from the superconducting transition temperature Tc, (95'96) utilizing the McMillan formula (see eq. (6-10)).

Although no direct measurement of the energy band gaps exists for hcp alloy phases, their values acorss the three sets of the zone planes are likely to be small. An estimate of the energy gap across the {00.2} zone planes in the ( phase Ag-A1 and the E phase Ag-Zn alloys, from a model proposed by the present authors, t88'93) suggests a value of the order of 1 3 eV (see Section 8.4.3). With respect to the large broad peak in N(E), and the corresponding trends of the electronic specific heats, the possible FsBz interactions may be classified with respect to a parameter defined as Ato.1 = ( k F - - kao.1)/kv(%), where kxo.~ is the distance from the origin to the central point L in the [10.1] zone planes and kv is the

tAs discussed in the text and indicated by eq. (3-2), the magni tude of the 7band value is basically determined by the atomic volume. The atomic volume f~ for all hcp noble metal alloys ranges between 14 and 18(A) 3, and is similar to that for pure Zn (f~ = 15.24(A)3). In addition, in these alloys the ~ d interaction is more likely to be present than the s p interaction, suggesting an effective mass lower than unity, as is also found in pure Zn. These considerations imply that the density of states curve for the hcp electron phases will more nearly resemble that of Zn rather than the curves for Mg or Be (see Table 3-2). A similar argument was employed in Section 8.3, where the electronic structure of the 7 phase Ag-Cd alloys was compared with that of the 7 phase C u - Z n alloys.

E L E C T R O N I C S T R U C T U R E O F H U M E - R O T H E R Y P H A S E S 243

FIG. 8-26. A very l ikely Fermi surface topography in an hcp Hume-Rothery electron phase alloy. The 101 contact and 100 overlap are assumed to be

present, corresponding to case I1 (or lIl) in Fig. 8-27.

free electron Fermi radius. The parameter A~0.1 represents the relative position of the Fermi sphere with respect to contact, or overlap, of the 110.1 } zone planes. Typical Alo.1 values, corresponding to electron concentrations at the onset and termination of different hcp alloy phases, are listed in Table 8-5. Based on the free electron approach, three representative cases may be considered, as shown in Fig. 8-27. Case (I) represents a ~ phase for which both Alo.1 values, at the initial and terminal compositions, are large and negative, suggesting little or no 10.1 Fermi surface contact. Accordingly, the Yband values may be expected to increase with electron concentration as in the Au Cd ~ phase, ~t°9~ reflecting the move towards the large peak. In case 0It A lo.~ exceeds - 5°0 in negative direction at the onset of the hcp phase, but approaches zero or a small positive value, at the terminal composition. Contact between the Fermi surface and the {10.1 *, planes is then expected to occur at some value of e/a

Table 8-5. The Interaction Parameters Associated with the [10.1l Zone Planes

Akl0.1 = (kv - km~}/krC,,) System Onset Terminal Class

A u - C d - 7.92 - 7.02 1 Ag Sn - 6.04 + 0.32 I1

( C u G e - 6 . 3 2 - 2 . 7 5 I1 ~,E Ag Cd - 3.01 + 3 . 5 4 I l l

( A g AI - 3 . 2 3 +3 .11 III • A ~ Z n + 0.60 + 4.60 I I I eCu Z n + 3 . 0 3 + 4 . 5 2 I l l

The values of kr and klo.i are as follows:

[ 127r2 ]1,3 kr = [ ~ - 2 - ( e / a ) | a n d klo. l =

[ ( \ 3 } a c 3 (,, 3 ) a X / \ + 4 \ c ! )"


within the solubility range of the hcp phase. Of course, the presence of energy gaps in real alloys will tend to cause contacts between the Fermi surface and the zone planes to occur at lower values of e/a than those predicted by the free electron model, but the presence of band gaps may delay the overlap effect. The ( phases in the systems Ag-Sn and Cu-Ge appear to be examples of case (II), where the 7band values are expected (and found) to follow the large peak in the middle of each phase range. Case (III) represents a system where A10.1 at the onset of the hcp phase is only slightly negative, but becomes increasingly positive, and where the corresponding 7 values are expected to decrease over the entire range of composition because the 10.1 contacts will have occurred prior to the onset of the stable phase range. The ( phase Ag-A1, the ( and E phases in Ag-Cd, and the E phases in Cu-Zn, and Ag-Zn systems appear to satisfy this condition. (Fig. 8-25).

8.4.3. 00.2 overlap effect As shown in Fig. 8-22, various physical properties in the E phases exhibit

consistent trend reversals at approximately 1.85 electrons per atom. In each case, the onset of electron overlaps across the {00.2} zone planes has been proposed as the main cause. We consider first the possible occurrence of the 00.2 overlap effect, as deduced from the electronic specific heat studies in the e phase Ag-Zn alloys. (93) Here, the experimental values can be corrected for electron-phonon interaction by utilizing the McMil- lan's formula, since the superconducting transition temperature T~ is knownJ 96) The 7band values obtained in this way are plotted in Fig. 8-28, for comparison with the N(E) curve for pure Zn. As discussed previously, the decreasing )'ba,d trend can be considered to reflect a declining slope

( I ) ( I f ) ( h i )

F

L

F

r' M

L

F

F ° M

¢

L

Au-Cd Ag-Sn Ag-Zn

FIG. 8-27. Vertical sections through free electron spheres, corresponding to the range of electron concentrations in three typical hcp electron phases. The Fermi surface circles correspond to the onset and termination of each phase. The Brillouin zone sections are drawn to scale by utilizing the lattice constant data.

(After Massalski and Mizutani. (94))


0

0 .8

0.6

E O~ v

,J

0.,2

0 0

e n e r g y (eV) (valid only fo¢ zinc) E~ 5 6 ? 8 9 10 1 il F I I I I I

0o.1},+ ri0.o}° ~

i i I~ i

o..b I- ~ o.~

OJ.,OI - . 0.2

/ I I1 i " 1.8 ( e /o ) , 1 .9

I 1

1,0 1.2 electron concentration

ronge of J - ph?e

expected displacement o f 0 0 . 2 o v e r t o p effect between Zn and

i I I I I I I I I

1.4 1.6 1.8 2.0 Z2

FIG. 8-28. The electronic specific heat coefficient 7b~nd corrected for electron phonon interaction for the E phase Ag-Zn alloy system, in comparison with the density of states curve for pure Zn. The subscript c and o correspond to the contact and overlap energies, respectively. The energy scale applies only to the density of states of pure Zn. The relation between the electronic specific heat coefficient and the effective mass is shown in the insert. (After Mizutani

and Massalski. (93))

of the N(E) curve following the large peak due to the interaction of the Fermi surface with the {10.11 zone planes, as well as due to a progressive change in the Fermi surface contours associated with the 110.01 zone overlaps (see Fig. 8-21). A steady decrease in the density of states occurs as electrons fill the zone corners. According to Stark and Falicov, t1°8) the E k relation locates the 00.2 overlap energy in pure Zn at 8.22 eV. This is barely seen in Fig. 8-28, because the effect falls on a steeply increasing slope toward the large peak. However, in the c phase, the axial ratio and the c parameter become greatly reduced, and the corresponding onset of the 00.2 overlaps is displaced to higher energy. If it is assumed that the 00.2 overlaps contribute to a change in slope of an otherwise decreasing trend in N(E) (the broken line in the insert to Fig. 8-28), the contribution of the overlapping electrons to the density of states can be written a s : 193)

ATh~n,~ = 0.594 m*(pA(e/a)) ~, (8-9)

where p is the fraction of electrons per atom in overlapped states (inside the "lens" in the second zone), A(e/a) is the number of electrons with energy above that corresponding to (e/a)o at the onset of overlaps, i.e. A(e/a) = (e /a) - (e/a)o, and m* is the effective mass of the overlapping electrons. The fit of the calculated density of states to the experimental


]2band trend, becomes satisfactory when m* is taken as 0.4 _ 0.05. The corresponding energy gap across the {00.2} zone planes is (2.4 _+ 1.0)eV. O3~ Using this value of the band gap one can estimate that, in the ~ phase A ~ Z n alloys, the 00.2 overlaps of electrons most likely occur at an energy of ~ 10.1 eV (E0 ~ E~ tee + AEoo.2), corresponding to an electron concentration of 1.85. These values are higher than the corresponding values for pure Zn by about 1.9eV ( 1 0 . 1 - 8.2eV) in energy and about 0.5 (1.85 - 1.35) in electron concentration (see Fig. 8-28). The 00.2 overlaps produce only a barely discernible increase in the density of states, but the effect itself seems nevertheless sufficient to influence the lattice spacings, ta9~ the magnetic susceptibility, ~97) the thermodynamic activity t99) and the superconducting transition temperature/96~ Similar considerations apply also to the ~ phase in the Ag-A1 system explored recently ~88) (see the change in slope in the ( phase Ag-A1 alloys in Fig. 7-2). For comparison with the situation in the E phase Ag-Zn system, a band of spherical Fermi surface contours is shown in Fig. 8-29, drawn to scale by using the lattice constant data for Ag-A1 ~11°) and Ag-Zn. 189) Despite the fact that the electron concentration range of stability is different for each hcp phase, the relative location of the Fermi surface with respect to the corresponding Brillouin zone is quite similar. Thus, the FsBz interaction in the ( phase Ag-A1 alloys is likely to be similar to that in the E phase Ag Zn alloys and the overall decreasing trend of the 7ba°d values with increasing e/a can be accounted for by classifying this system as case (III) in Table 8-5. The small increase and change in slope in the trend of the ~band values at about 1.59 electrons per atom is interpreted as due to the onset of 00.2 overlaps. A corresponding change in slope is also observed in the behavior of the c lattice parameter. ~11°) The energy gap across the {00.2] zone plane, estimated from the assumed effective mass

phase Acj- At 148 <¢/o <182 159~clo <163

¢ phase Ag-Zn 1"67 <ela <188 I I~6 ~c/(~ <I 59

FIG. 8-29. Vertical sections through free electron Fermi spheres, corresponding to the range of electron concentrations of stable ~ and E phases, superimposed on the corresponding Brillouin zones drawn to scale by utilizing the lattice constant data. The thickness of the zone lines represents the maximum change in the k vector value due to the change in the lattice constant over the entire

range of each phase. (After Mizutani and Massalski. ~ss~)

E L E C T R O N I C S T R U C T U R E OF H U M E - R O T H E R Y P H A S E S

Table 8-6. Interaction Parameter Associated with {00.2} Zone Planes

247

Critical Interaction parameter Estimated System e/a value Akoo.'2 = [kF -- koo.E]/kf('!o) energy gap (eV)

A ~ AI 1.60 + 4.42 0.9f Ag Sn 1.60 +4.46 Ag Zn 1.85 + 6.82 2.4 + Cu Zn 1.85 +6.85 Au Zn 1.85 +6.80

t See reference (88). :~ See reference (93).

(0.3), is of the order of 0.9 eV, which is smaller than the value of 2.4 eV for the E phase Ag-Zn alloys. Massalski and Mizutani t94~ also observed a deviation from the downward slope of the ? trend in the ( phase Ag-Sn alloys at similar electron concentration (e/a = 1.6). The tendency towards the 00.2 overlaps may be assessed in general in terms of the parameter Aoo.2 = [k v - koo.2)/kF(%), whose values are listed in Table 8-6.

Finally, we note that in all E phases involving Zn or Cd as solvents with the noble metals, the number of electrons apparently needed to induce overlaps across the {00.2} planes, is larger than in the corresponding phases. Such a difference originates from the low value of the axial ratio in the e phases, as is seen in Fig. 7-7, and probably also from the larger energy gap across the {00.2} zone planes in the e phases. Although it is clear that the presence of Zn or Cd as solvents in the E phase alloys certainly plays an important role in delaying the 00.2 overlaps, and increasing the band gaps, further studies are needed before the difference between Zn or Cd, and other polyvalent alloying partners in the noble metals, such as A1 or Sn, is understood. At the moment the remarkable fact remains that hcp electron phases indicate electronic behavior that is characteristic of a nearly rigid common conduction band.

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9. COMPETITION FOR PHASE STABILITY IN

NOBLE METAL ALLOYS

In the discussion of the relative phase stability, the different forms of the free energy of formation curves are usually examined and compared among the competing phases in order to assess how this parameter changes with composi t ion and temperature. We consider here the formation of a mole of a solution from pure components in their standard states:

X A A ( S ) + XBB(S) ---' (XA, XB) AGm. (9-1)


The change in free energy AGm is the free energy of formation (or of mixing). At constant pressure, AGm can be written in the form

AG,n = AHr. - TASm, (9-2)

where AHr. is the heat of mixing (or the enthalpy of formation) and AS,. is the entropy of mixing. If the free energy of formation is obtained experimentally, the entropy of mixing and, hence, the heat of mixing, can be calculated from the temperature dependence of AGm through the well known relation ASm = -(~AGm/~T)p. In certain alloy systems, the heat of mixing is independently determined by experiment. A large amount of thermodynamic data for inorganic structures has now been collected and assessed in the books by Kubashewski et al. ~1) and Hultgren et al. ~2) Below, we summarize the general thermodynamic relations pertaining to the phase competition in normal metals and alloys.

As was shown in Section 6, the contribution to the free energy of formation due to the conduction electrons can be expressed in the form:

~ E~Jlo,

AU~I(T) = N~,m,y(E)f(E, T)E dE ,JO

lea - XA" NA(E)f(E, T )EdE + XB" JO ,JO

NB(E)f(E, T)E dE],

(9-3)

where Nalloy(E), NA(E ) and NB(E) represent the density of states for an alloy and the pure A and B components, respectively, and f(E, T) is the Fermi-Dirac distribution function. In the ordinary temperature range k T ~ EF, and eq. (9-3) is well approximated by

AUel(T) = AUe,(0) + (1/2).A7. r z, (9-4)

where AUcI(0) is the electronic energy difference at 0 K and A 7 is defined as ])alloy - - (XA)~A + XBTB)" Similarly, the entropy of mixing due to the conduction electrons is given by the relation

which reduces to

AS~ = f (AC~]T) dT,

ASel = Ay. T. (9-5)

Further, we can estimate the lattice vibrational energy, Uph, and entropy, Sv~b, on the basis of the Debye model. The results are given by the following expressions, ca) depending on the temperature range:

Uph = (9/8)'R0o + aRT; Sv~b = 3R[ln(T/OD) + 4/3],(T > 0o) (9-6)

and

Uph ~-. (9/8)'R0 D + (3/503).RTz'*T4; Svi u = (4/503) 'Rx4T 3, (T ~ OD). (9-7)


The contribution of the phonons to AGm at T > 0D is thus mainly reduced to the difference in the zero-point energy, since the second terms in eq. (9-6) are the same for both the reactants and the product, regardless of the structure (the Dulong-Petit law). Hence, the free energy of formation, and the entropy of mixing, due to the phonons at T > 0D can be expressed simply as functions of the Debye temperature:

AUph = (9R/8). [0a l loy - ( X aO A -~- XBOB) ] (9-8)

and

AS,~b = 3Rln(0 xA0x'~/0~,,oy). (9-9)

At very low temperatures, however, the corresponding values become temperature dependent and are given by:

AU,,~, = 3Rrr'*KT4/5 + ( 9 R / 8 ) ' [ O a m , ~ - (XAO a -~- XBOB)" ] (9-10)

and

where

AS,,ib = 4 R I r 4 K T 3 / 5 , (9-11)

K 3 = 1/0~,,,,,,- ( x A / o 3 + x . / 0 3 ) .

Regardless of the temperature range of interest, the free energy of formation can be written as the sum of the contributions described above, plus the Madelung energy AUM and the strain energy AUs due to the misfit of atoms with different sizes:

AG,, = AU~I(0 ) + A 7. T2/2 + AUph + AU,vt + A U s -- A 7" T 2 - T ' A S ~ o m- - T'ASvib , (9-12)

where the sixth term represents the electronic entropy contribution given by eq. (9-5), and ASco,f is the configurational entropy of the lattice. Assum- ing AUs, AUM and ASconf to be invariant with temperature, we can obtain the entropy of mixing ASm by differentiating eq. (9-12) with respect to temperature,

AS,,, = -(c')AGm/?T)p = AT" T + ASconf q- ASvi b. (%13)

Here, of course, the first term due to the electronic contribution is equivalent to eq. (9-5) and ASv~b is temperature independent only at T > 0o. The enthalpy of formation AHm is now obtained from its definition by adding TAS,,, to eq. (9-12):

AHm = AUel(0 ) + A),' T2/2 + AU M + AUph + AUs. (9-14)

According to the Debye approximation, therefore, it can be said that AHm at T > 0o is given essentially by the internal energy difference at 0 K, since the second term, AT" T2/2, will be only of the order of l0 ~ 20 cal/


mole for the noble metal alloys at ordinary temperatures and A U~,~, at T > OD reduces to the value at 0 K as indicated by eq. (9-8) and (9-10).

It is often convenient to consider the entropy of mixing AS., relative to that of an ideal solution, in which no atomic interaction is assumed to be present. For an A B alloy, the configurational entropy of an ideal solution is given by

A K'ideal = - R [ X A l n X a + XBlnXB]. (9-15) ~ c o n f

In a real alloy, we define the excess configurational entropy of mixing by

A qxs = ASt , , t . , l ( A ~'id eal ° c o n f , - - ~ c o n f -1- ASvib + m a e l ) , (9-16)

where AS,,,,~I is the experimentally determined entropy of mixing. Thus, AS~,~,,,f can be considered to provide a measure of the deviation from an ideal solution or, in other words, the degree of strength of atomic pair interaction. In the following, we shall relate the low temperature specific heat results to the thermodynamic data in various noble metal alloys and consider the relative phase stability among competing phases.

9.1. Thermodynamic Considerations and Phase

Stability in Noble Metal Alloys

Pratt C4J studied the thermodynamic properties of alloys in the Ag-Pd system and reported a negative heat of mixing and a negative excess entropy, both of which show a marked asymmetry about the equiatomic composition. He calculated the change in the band structure energy AU¢j(0), which mainly arises from the transfer of the valence electrons donated by Ag to the vacant d band in Pd, on the assumption of a rigid band model deduced from the experimental electronic specific heat data. It was found that the resulting AUto(0) curve against composition is quite similar to that of the experimentally observed AH,,. Thus, it was concluded that the asymmetric nature in AH,, corresponds to the filling of the vacant Pd d band. Later, Sato et al. ~51 measured the low temperature specific heats in Cu Pd alloys over the entire range of composition and employed the same argument as Pratt to account for the large negative heat of mixing. In fact, the computed AU¢~(0) showed a minimum at 40 at.G Pd, which agrees with the minimum observed in the experimental AHm curve at 1000 K. Sato et al. also evaluated the electronic and vibrational entropies using eq. (9-5) and (9-9). The sum of these entropies, and the ideal configurational entropy, agree well with the experimental entropy of mixing. This indicates that the excess configurational entropy AS~o~of is small in Cu-Pd system.

Abriata et al. ~6) made heat capacity measurements between 1.3 and 4.2 K on the equiatomic Ag-Zn alloy in both ordered /3' and ~o phases. The


phase stability competition at 0 K between the two phases was considered on the basis of the significant difference observed in the low temperature specific heat coefficients. The experimental values of the heat of mixing AHm were obtained earlier by Orr and Rovel ~v~ for both the (o and /T phases at 324K, and are -1535 and -1585cal/mole, respectively. This indicates that the internal energies at 0 K are extremely close, but the value for the (0 phase should be more negative than that for the fl' phase because of their relative location in the phase diagram. In fact, at 0 K, AUph in eq. (9-14) reduces to the difference in the zero point energy, AUk.p, taking the same form as in the range of T > 0o. Hence, the heat of mixing at 0 K should be quite close to the experimental value at 324 K and is composed of AUel(0), AUM, A U s and AU~.p. The difference in the zero point energy _U~.p ~ = 9R(O~, - 0~o}/8 amounts to some -116cal /mole in favor of fl'. Thus, the reason for the stability of the ~o structure should be found in AUM, AUs or AUe~(0). However, there has been no attempt to estimate these values accurately enough to explain the subtle difference in the internal energy.

Combining the low temperature specific heat data obtained by Abriata et al. with available thermodynamic data, the entropy of mixing in this system can be considered further. As mentioned by Orr and Rovel, the ~o structure can be constructed by arranging one third of the lattice sites to be filled almost exclusively by Zn atoms with the remaining sites being filled randomly at the ratio of 75 at.?~,~, Ag and 25 at.?; Zn. Following this arrangement, the configurational entropy for the ~o phase is 0.74cal/ mole.K, while that for the [J' phase is likely to be close to zero, because of the completely ordered structure. The availability of the low temperature specific heat data allows us to estimate AS~ and AS~h through eq. (9-5) and (9-9). The following sets of values are obtained:

~0 AS;onf = 0.74 cal/mole.K AS~,,f = 0 cal/mole.K

AS~"~b = -0 .04 AS~iu = 1.24

AS~] = -0.003 AS~; = 0.005.

Thus, the total entropy of mixing may be reasonably assessed by summing up the above contributions:

AS~ ° = 0.7 cal/mole.K and AS~' = 1.24 cal/mole.K,

which can be compared with the entropy of mixing of the disordered fl phase deduced from the thermodynamic data¢ 7}

AS~ = 1.71 cal/mole.K.

The magnitudes of the entropy of mixing in the respective competing phases are consistent with the order of their appearance in the phase diagram with increasing temperature at the equiatomic composition. How-


ever, this interpretation of the temperature dependence throws little light on the origin of the stability of the (° structure.

More recently, Bevk et al. (8) discussed the thermodynamic properties and competition for stability between the hcp ( and (fl-Mn) # phases in the Ag-AI system, on the basis of measured low temperature specific heat data. At temperatures above the peritectoid reaction ct + ( --o/~, the ( phase is stable, while at temperatures near absolute zero, the p phase is more stable, as shown in Fig. 9-1. Free energy curves for the ~, ( and/~ phases were constructed at two representative temperatures (721 and 500 K) by making use of available thermodynamic data. (2'9) The high temperature entropies of mixing for the ( and/a phases were computed from eq. (9-2), or the expression A S m = - ( S A G m / ~ T ) v , while ASvi b and ASet were estimated from low temperature specific heat data. It was concluded that the relatively large vibrational entropy in the ( phase is responsible for the stability of this phase at higher temperatures.

Bevk et al. (8~ also discussed the relative stability between the ( and # phases in Ag-A1 alloys. Employing thermodynamic data, they deduced that the heat of mixing of the/~ phase at 740 K is approximately 380 cal/ mole more negative than the corresponding value for the ~ phase. As

Weight Per Cent Aluminum 1 2 3 4 5 6 7 8 910 12 ,4 ,6 1820 25

96;. 5'0 . . . . , . . . .

900 ~ .

~600 20.34 '610' 566" "

GAg I or(Ag)

400

500

#-

200

Ag

8.75/ /

19.92 448'

,23.z'!

I0 20 50 40 50 60 Atomic Per Cent Aluminum

FIG. 9-1. Portion of the Ag-AI phase diagram showing the composition of the samples and respective temperatures of annealing in the low temperature

specific heat measurements by Bevk et al. (After Bevk et al. (s))


shown in eq. (9-14), this can be reasonably attributed to the difference in the internal energy at 0 K. The zero point energy difference AUz. p was found to favor the ~ phase over the tl phase. They also argued that the Madelung energy is likely to favor the ~ phase, in view of the irregular arrangement of atoms in the # structure when compared with the nearly ideal hcp ~ phase structure. Thus, they concluded that the relatively large and positive value of AH,, must arise from the electronic energy difference AUe~(0). The likely N(E) curves were then considered for both phases, which allowed to derive a possible gain of a few hundred cal/mole for AU~ in favor of the /~ phase. The stability was discussed on the basis of the Jones model: the N(E) curve of the It phase exhibits a drastically changing slope near EF (see Fig. 8-20), as compared with a moderately varying N(E) curve assumed for the ~ phase. The schematic N(E) curves are shown in Fig. 9-2, together with the trend of AUe](0) as a function

/ ..2",..'' ~ ~'ree eL

',\F i i I

E EF~ =E~F (a)

d(AU ~-~) ~ dn = EF--EF R

n-electron concentn3tion

(b ) FIG. 9-2. (a) Schematic density of states curves for the ~ and p structures in the narrow range of composition near 24 at.% AI. (b) Electronic energy difference corresponding to the density of states curves given in (a). (After Bevk

et al. (8~)


of the electron concentration. Here, of course, the drastic decrease in the N(E) curve is most likely responsible for the stability of the p phase. The above interpretation needs some refinement, however, because the more recent observation is that N(E) curves for the hcp ~ phases also involve a fairly steeply declining slope, (l°~ but the general interpretation strongly favors a model in which the FsBz interaction plays an essential role in stabilizing the p phase in the narrow e/a range.

9.2. Further Comments on Stability

In the discussion of the relative stability between two competing phases and /3, both of which consist of the same components and identical

composition, it :is more convenient to employ the following expression rather than eq. (9-3)

f? ( AU~?I~(0) = N~(E)E dE - N~(E)E dE. (9-17) d O

Although.each term amounts to some 10 5 cal/mole for the case of the noble metal alloys, the difference A U ~ J is generally reduced to the order of 10 2 eal/mole, which is merely 0.1Vo of the total electronic band structure energy. In the phase stabiliy discussion in terms of the density of states condiderations, we usually a priori assume that no other important energy difference, compared with the FsBz interaction, contributes to AUev There are, however, other factors which must clearly affect the N(E) curve as the crystal structure changes. This includes, for instance, the presence of the d band in the middle of the conduction band, as well as the magnitude of the atomic volume, which affects the shape of the parabolic part of the N(E) curve near the bottom of the band. A change in the d band configuration accompanying a structure change will most certainly change the value of Uel. According to eq. (3-2), a very small difference in the atomic volume between the competing phases easily creates an appreciable contribution to.AUe~ through the integration of eq. (9-17).t Therefore, we must emphasize that, only if all other factors cause no appreciable net effect, as a whole, on Uej, one can consider the changes in the N(E) curves associated with the FsBz interactions as holding the key role in determining the relative stability between competing phases.

As pointed out earlier, Heine and Weaire ¢13~ stated that the N(E) curves adopted by Jones ¢14) for his interpretation of the phase boundary between the e and /3 phases in the Cu-Zn system were too exaggerated, and that

-[A transformation of the crystal structure usually accompanies a small change in the atomic volume. For example, in the p and ~ phases Ag-AI alloys (24.0 at ?J~ AI), the difference in the atomic volume between two structures is estimated as 0.6% from the lattice spacing data at room temperature. "L~2) Since the total electronic energy is of the order of 105 cal/ mole and is directly proportional to the atomic volume (see eqs (3-1) and (9-17)). AU~ due to the difference in the atomic volume easily exceeds 500 cal/mole. This is already large enough to mask the effect of the FsBz interaction.


even the presence of a considerably large energy gap creates only a very small change in the total electronic energy. As shown above, this argument is quite reasonable. In order to perform a reaslistic analysis of phase com- l~etition we should employ the density of states curves based on the first principles band calculations or at least a suitable cone model with appropriate band parameters. As was shown in Figs 3-3 and 3-5, the van Hove singularities in pure Zn and Cu are discernible, and very likely contribute to relative stability, if a very small amount of energy difference is important. A calculation of the total electronic energy, using such a realistic density of states curve, shows only a very small change in slope, when plotted as a function of e/a. However, we consider that even such a small perturbation can become essential in certain cases when two similar phases are competing. Such examples are described in the preceding section. In the following discussion, we show an additional example by employing two characteristic N(E) curves for the 7 and hcp structures, both of which are constructed on the basis of information given in Figs. 8-16 and 3-3. The N(E) curves are shown in Fig. 9-3. For the sake of simplicity, the atomic volume and the d band in both structures are assumed to be the same. Hence, the lower part of each N(E) curve below 6eV may be assumed to be unimportant in evaluation of AU~I. Admittedly, the two curves in Fig. 9-3 are so constructed as to emphasize how the differences in the van Hove singularities create the difference in U~,~. The position and magnitude of the related peaks were subjected to a final adjustment so that the 7 phase stability range coincides with the narrow e/a range

0 . 3 \ x * ' j

/ /, ' / ' ,

®0.2

0.1

0 0 1 2 3 4 5 6 7 8 9 10 11

energy (eV)

FIr3. 9-3. Density of states curves for the )' and ~ phases in a Hume-Rothery alloy system. Only the effect of the FsBz interaction on the N(E) curve is included and other contributions, such as the difference in the d band and

atomic volume, are ignored.


0.3

o ~ 0.2

0.1

i i i

A y D /

L

0 I I I 0 .5 1.0 1.5

~ 3 0 0 -~ 200 ~ ; 1oo

~'o H

~ -100

-200 -300 I J I

0.5 1.0 1 .S electron concentrotion

FIG. 9-4. Phase competition between the ), and ~ Hume-Rothery phases, based on the N(E) curves in Fig. 9-3. AU is calculated from eq. (9-17).

of 1.6-1.7, where it is known to exist. The plot of A U ~ ; = U~ - U~ is shown in Fig. 9-4 as a function of e/a, together with the plot of the respective N(Ev) values. The figure shows that the ~ phase is generally more stable than 7 except for the range indicated above. We would expect that the ~ phase would appear in place of the fl phase in the Cu-Zn system, if the fl phase were suppressed by some special reason such as lack of vibrational entropy contribution.t The results show that the steep declining slope of N(E) in the 7 phase is responsible for the presence of a minimum in AUot, as expected from the model of Jones. This figure also illus- trates how a gain or loss in the density of states affects the value of AUej. For instance, the ~ phase remains stable up to 1.5 electrons per atom, although the area (C) appears as early as 1.1 electrons per atom, and is more prominent than the sum of areas (A) and (B). In this model, these areas are adjusted so that the stable range of the phase corresponds

t i t is undoubtedly for this reason that disordered martensitic structures, formed from the /~ phases, tend to be hexagonal in structure.


to the position of its large N(E) peak. If areas (A) and [B) are chosen to be slightly larger, the stable ~ phase does not occur at all or, if it does occur, it will be shifted to a higher e/a range than actually observed. Thus, Fig. 9-4 emphasizes that the typical peaks and valleys related to the FsBz interaction in Hume-Rothery electron phases can bring about a situation whereby the stability of one structure will be favored in prefer- ence to other competing structures on the basis of electronic energy alone. However, the relative differences then turn out to be no larger than about 100cal/mole, and if they are made larger by increasing the N(E) peaks of one of the structures the competing phases can never be restored to stability within the electron concentration ranges which are known from the phase diagrams. Hence the stability argument based on the density of states shows that not only the total energy difference obtained will be very small but also that it must be very small.

We must admit that, although we attempted to interpret the empirical Hume-Rothery rule regarding electron concentration in terms of the FsBz interaction, this attempt is far from being satisfactory except for some special cases, in which the structures involved are quite unique, such as the 7 and/~ structures. As was emphasized, the phase stability discussion involves a comparison of the free energy of formation to which various factors can contribute. We know that certain stability ranges strongly depend on electron concentration. Because of this, we are led to the assumption that, except for the FsBz interaction, all other factors as a whole exert no net effect on the relative competition for stability, although a change in any given factor could clearly upset this picture. This is because the peaks and valleys caused by the FsBz interaction effectively influence the total electronic energy only by approximately 0.1°~. At the present stage, further discussion of this particular effect, without the detailed knowledge of other parameters, is not worthwhile.

REFERENCES

1. O. KUBASHEWSKI, E. L. EVANS and C. B. ALCOCK, Metallurgical Thermochemistry (1967) Pergamon Press, London.

2. R. HULTGREN, R. L. ORR, P. D. ANDERSON and K. K. KELLEY. Selected Values of Thermo- dynamic Properties c)f Metals and Alloys (1963) Wiley, New York.

3. E. S. R. GOPAL, Specific Heats at Low Temperatures (1966) p. 30 Plenum Press. 4. J. N. PRATT, Trans. Faraday Soc. 56 (1960) 975. 5. Y. SATO, J. M. SIVERTSEN and E. L. "]'OTH, Phys. Rer. BI (1970) 1402. 6. J. P. ABRIATA, O. J. BRESSEN, C. A. LUENGO and D. THOUt.OUZE. Phys. Ret'. B2 (1970)

1464. 7. R. L. ORR and J. ROVEL, Acta metall. 10 (1962) 935. 8. J. BEVK, J. P. ABRIATA and T. B. MASSALSKL Acta metall. 21 (1973) 1601. 9. F. E. W1TTtG and W. SCHILLING, Z. Metallk. 50 (1959) 610.

10. T. B. MASSALSKI and U. MIZUTANI, Proc. R. Soc. A351 (1976) 423. 11. S. FAGERBERG and A. WESTGREN, Metallwirt. 14 (1935) 265. 12. T. B. MASSALSKI and B. COCKAYNE, Acta metall. 7 (1959) 762. 13. V. HEINE and D. WEArRE, Solid State Physics (edited by H. Ehrenreich, F. Seitz and

D. Turnbull), 24 (1970) Academic Press, New York, London. 14. H. JONES: Proc. phys. Soc. A49 (1937) 250.


APPENDIX

The Application of the Multi-cone Model to Noble Metal Alloys

The cone model is essentially based on solving the following equations:

E 2 = Eo[zm.,(1 - o9/2n) -2 + 2~(1 - z.,~,,) - x,,,'~(~ - 2)(1 - Zn, in) 2 + 1}/(1 C~)2}]. (A-l)

E = E0[1 + (I - Zm,x) z -- 2\./[C~(:~ -- 2)(1 - Zmax) 2 ~- 11/(1 -- ~)2], (A-2)

N(E) = 6.648 x 10 3 x m* x ~ x /3 x ko x Iz .... - z,,i,,} states/eV, atom, (A-3)

)'h,na = 1.567 x 10 -2 x m* x f2 x /3 x ko x [z ..... -Zmi .}mJ/moleK 2, (A-4)

where f~ is the atomic volume, fl is the number of equivalent zone planes, (or equivalent cones), ko is the magnitude of the k vector touching the center of a zone plane, a dimensionless parameter z represents the normalized k vector with respect to ko and runs from 0 to 1 and co is the solid angle of the cone. If there are more than two sets of zone planes, each contribution is calculated through above equations and added together. A more general equation for (A-3) is seen in eq. (3-6). At a constant energy E, the values of z,,,x and Zml n are determined from the first two equations and the solutions are inserted to (A-3) or (A-4) to obtain the density of states value at that particular energy. Once contact occurs at Zm, x= 1.0, the value Zmax is kept at unity for further increases in energy. The parameter E0 is assessed either from the E k relation of a first principles band calculation, or from the relation E0 = Efr~e/m*, as discussed in Section 3.1. The parameter ct is related to the value of E0, as well as AE, through equation ~ = 1-4Eo/AE.

The detailed geometry of each zone plane both in the y and/~ phase structures is illustrated in Figs 8-15 and 8-19, respectively. Each zone plane is replaced by a circle whose area is chosen to be equal to that of the zone plane. Now the solid angle co is obtained by the equation:

t~ = 2nil - l / \ / t a n Z O + 1], (A-5)

where 0 is the angle between the center axis and the cone edge and, thus, tan 0 is given by the ratio of the radius of the cone base over the distance from the origin to the center of the cone base. The results for various solid angles for fcc, bcc, the 7 phase and p phase structures are listed in Table 3-3. The approximation is poorest in the y structure, as is seen in Fig. 8-15. In the case of the 7 and /t phase structures, the diameter of the cone base is reduced proportionally so that the resulting total solid angle matches 4n. The following readjusted solid angles are employed in the present discussion in Section 8:

0)330 = 0.4126 and o)411 = 0.3173 for the 7 structure

and

0)221 = 0.3483 and 09310 = 0.1753 for the/~ structure.

10. CONCLUDING REMARKS

Here we attempt to summarize some of the main features of the electronic structure of Hume-Rothery phases as they appear in the light of more recent research.

Available results on the Fermi surface topography suggest that its changes on alloying are relatively simple, conforming to a general rigid band description of alloying behavior. Positron annihilation experiments, dHvA experiments, band calculations and other data support this general conclusion. In this respect the rigid band concept may be taken to mean


either the Mott and Jones description of an alloy band structure or the Friedel description, as discussed in Section 4. In the simplest form, the E vs. k curves of the pure metal suffer relatively little change on alloying and the Fermi surface merely expands to accommodate the extra electrons. Influence of the impurity atoms may, however, cause also a downward displacement in energy of the conduction band, without affecting its main shape, at a rate consistent with the observed optical data. There is no evidence that the band gaps change substantially on alloying or that they are large, at least in the concentrated alloys. When band shifts occur on alloying, the d band and the Fermi level, may remain nearly fixed at the original pure solvent positions. However, as the concentration of the solute atoms increases, the d band of the noble metal atoms becomes thinner and decreases, while the d band of the solute partner atoms, which is generally located below the bottom of the conduction band, increases in magnitude and width. This does not appear to cause any drastic changes in the Fermi surface topography in the noble metal alloys involving the B sub-group elements, but this may not be true for the Cu Ni or A ~ P d alloys.

The e l ec t ro~phonon interaction in noble metal alloys is now fairly well understood, mostly through the utilization of the available superconductivity data. The value of the enhancement factor .~ appears to increase in the ~ phases, but is more or less constant, at about 20-30~,, in the concentrated alloy phases. The same conclusion for the hcp phases and /3 phases may be obtained also from a direct comparison of a band calculation with the available electronic specific heat data.

Once the electron-phonon correction is allowed for, the experimental electronic specific heat coefficients can be compared directly with suitable alloy band models, Fermi surface topography features, and other alloying effects. On the whole, the agreement for the primary solid solution ~ (or q) phases is far from satisfactory and many puzzles remain. Electron- phonon interaction does not seem to account satisfactorily for the difference between the predicted downward slopes of the density of states in

phases and the observed upward slopes in the 7 coefficient data. Perhaps some "charging" effects occur in the dilute alloys. In any case, the data are still inadequate for the very dilute alloys (< 1 at.~i solute).

In the concentrated alloy phases, any possible charging effects appear to be smeared out and there is a remarkable agreement between the electronic specific heat data and band models utilizing a rigid band description of alloying behavior. The situation is well documented for the hcp alloy phases where the agreement is good both with respect to the likely theoretical density of states trend on alloying, as well as the predicted and observed Fermi surface topography. For the case of the /~ and 7 phases, simple rigid band models, based on a multi-cone approximation, can account very well for the observed trends and magnitudes of the 7 coefficient. The data for all intermediate phases, /~. F~. 7. ~ and E clearly indicate


that there exists a direct relationship between the electron concentration and the trend and magnitude of the density of states at the Fermi level.

Another remarkable feature is that in all Hume-Rothery intermediate phases that are restricted to narrow and characteristic ranges of e/a, such as the ~, fl or 7 phases, the density of states trend shows a pronounced downward slope. This behavior appears to be significant with respect to a simple approach to the relative stability of such phases, based on the differences in the total electronic energy at 0 K.

Considerations of the relative stability of Hume-Rothery phases raise many as yet unanswered questions. In general, it is evident that many factors should contribute to the total energy. Amongst them, an adequate assessment of the electronic energy must include perturbations due to: (1) electron-ion interactions (FsBz interaction), (2) electron-electron interactions and (3) complications arising from the superimposed presence of the d bands. Thus far the theoretical treatment of these factors has not been sufficient to deal satisfactorily with the Hume-Rothery rule. This applies both to the pseudopotential approach and to the Jones-type argument using "real" N(E) curves.

At the same time, the rather limited relative stability analysis presented in this review with respect to the intermediate phases indicates that, if only one parameter is considered, namely the electronic energy difference at 0 K, then this difference for the competing structures must not be greater than a few hundred cal/mole (~0.1Vo of the total electronic energy) or one of the phases would permanently dominate the critical ranges of electron concentration where, typically, the various crystal structures change as e/a changes. Here, strikingly, the emerging features of the density of states curves for the Hume-Rothery phases do, indeed, suggest that only such small differences in energy may be involved and that they are sufficient to produce phase transitions. In this context the original Jones concept of stability in terms of the density of states of the conduction band still remains as valid as any other approach, with the astonishing conclusion that the d band affects, or the Madelung energy, or any other terms, are of secondary importance. Yet, if such contributions were to be properly included, and if they should differ from system to system and from structure to structure, the well known and time tested relationship between e/a and stability, known as one of the Hume-Rothery rules, would surely be less simple or systematic than is observed. Thus, despite the very substantial progress of recent years, more research is still needed to resolve this dilemma.

ACKNOWLEDGEMENTS

We wish to acknowledge helpful exchange of correspondence with Pro- fessor Walter A. Harrison and many stimulating discussions with Professor Joze Bevk.

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Electronic Structure of Hume-Rothery Phases