The Hume-Rothery rules for Structurally Complex Alloy Phases

Uichiro Mizutani

Toyota Physical & Chemical Research Institute, Japan

The Hume-Rothery rules for

Structurally Complex Alloy Phases

The 3rd Euroschool 2008, Ljubljana, SloveniaComplex Metallic Alloys: Surfaces and Coatings

May 29, 2008

OUTLINE

1. Prior fundamental knowledge for the discussion of phase competition inCMAs

2. What is the Hume-Rothery electron concentration rule?

3. Two different definitions of electron concentration: VEC versus e/a

4. Historical survey

5. Why do we need the first-principles FLAPW band calculations?

6. Why have we chosen a series of gamma-brasses?

7. Hume-Rothery stabilization mechanism

8. Different stabilization mechanisms in Al8V5 and Ag5Li8 gamma-brasses

9. Stability mechanism in the Al-TM-Cu-Si (TM=Fe and Ru) approximants

10. 2kF versus Kp condition. Issues on VEC versus e/a. Which is more criticalparameter to discuss the phase stability of CMA phases?

The curve !o(r ) represents the lowest energy of electrons with the wave vector k=0 while the curve!kin represents an average kinetic energy per electron. !I represents the ionization energy neededto remove the outermost 3s electron in a free Na metal to infinity and !c is the cohesive energy.The position of the minimum in the cohesive energy gives an equilibrium interatomic distance ro.

Concept of a cohesive energy in a solid

!c: cohesive energy

:ionization energy

RnAtPo

144

Bi

208

Pb

197

Tl

181

Hg

67

Au

365

Pt

565

Ir

670

Os

783

Re

782

W

837

Ta

781

Hf

611

La

434

Ba

179

Cs

80

Xe

15

I

107

Te

193

Sb

259

Sn

301

In

247

Cd

112

Ag

286

Pd

380

Rh

555

Ru

639

TcMo

658

Nb

720

Zr

610

Y

423

Sr

164

Rb

83

Kr

11

Br

118

Se

206

As

289

Ge

374

Ga

269

Zn

130

Cu

338

Ni

428

Co

424

Fe

414

Mn

288

Cr

396

V

511

Ti

469

Sc

379

Ca

176

K

91

Ar

7.7

Cl

135

S

277

P

332

Si

448

Al

322

Mg

148

Na

108

Ne

2.1

F

84

O

251

N

477

C

711

B

561

Be

322

Li

159

AcRaFr

Cohesive energies (kJ/mole) of elements in periodic table

Cohesive energy represent an energy needed to separate all atoms in a solid at absolute zero into theassembly of neural atoms. The values underlined refer to those either at 298.15 K or at the melting point.The values are in the units of kJ/mole converted from the values in the units of cal/mole. C.Kittel,“Introduction to Solid State Physics”, (Third Edition, John Wiley & Sons, New York, 1967), Chapter 3.

Ce 460

!

Pr 373

!

Nd 323

!

Pm

!

Sm 203

!

Eu 174

!

Gd 399

!

Tb 393

!

Dy 297

!

Ho 293

!

Er 322

!

Tm 247

!

Yb 151

!

Lu 427

!

Th 572

!

Pa 527

!

U 522

!

Np 440

!

Pu 385

!

Am 251

!

Cm

!

Bk

!

Cf

!

Es

!

Fm

!

Md

!

!

Lw

!

!

No Lr

The number of intermediate phases increases from unity up to five as the heat of mixing increases in a negativedirection from zero, -37 and -75 kJ/mole in!"#$!%&'()*!+*+,-./!0F.R.de Boer, R.Boom, W.C.M.Mattens, A.R.Miedemaand A.K.Niessen, “Cohesion in Metals”, (North-Holland 1988)]

The number of intermediate phases increases, as "H at xB=0.5 increases.

Neighboring phases are competing within +10kJ/mole.

"H=0 kJ/mole

"H=-37 kJ/mole

"H=-75 kJ/mole

A B

He

at of m

ixin

g, "

H/ kJ

.mol-1

Construction of the Fermi sphere. The reciprocal space is quantized in units of 2#/L in the kx-,ky- and kz-directions and is made up of cubes with edge length 2#/L as indicated in the figure.Electrons of up and down spins occupy the corner of each cube or integer set (nx, ny, nz) inaccordance with the Pauli principle while making nx

2+ny2+nz

2 as low as possible. The spherewith radius kF represents the Fermi sphere.

2 ! /L

kx

ky

kz

kF

2 !

L2 !

L

2 !

L

Electron theory of metals is an essential ingredient to deepen

understanding of an alloy phase competition

Fermi sphere

The free electron model

!h2

2m

"

# $ %

& '2( = E(

E =h2

2m

!

" # $

% (kx2

+ ky2

+ kz2) =

h2k2

2m

!

" = eir k #

r r

kx=2!n

x

L(n

x= 0, ±1, ±2,±3 " " " ")

plane wave

!

kF = 3" 2e /a

Va

#

$ %

&

' (

)

* +

,

- .

1/ 3

e/a: electron concentration defined as an average valency of constituent

atoms in an alloy.

Departure from the free electron model

Nature tries to lower the electronic energy of a systemby expelling electrons at the Fermi level. This is nothingbut the formation of a true or pseudogap at the Fermilevel, as indicated above.

Three representations for the

description of the electronic structure

E-k relations (dispersion relation)

Fermi surface

Density of states (DOS)

kx

ky

kz

0

kF

E

kx0

EF

kF

EEF!"#$%

0

N(E

)

!

DOS(E) = " E

-50

-40

-30

-20

-10

0

Gain

in e

lect

ronic

ene

rgy,

"U

/kJ

.mole

-1

5004003002001000

Width of energy gap, "E/ meV

psuedogap immediately below EF

pseudogap near the bottom of the valence band

A gain in the electronic energy amounts to several 10 kJ/mol, which is large

enough to stabilize a given phase among competing phases

Examples to stabilize a system by formation of (pseudo)gap at EF

1. A structurally complex phase by increasing the number ofatoms in a unit cell

2. A superconductor by forming the superconducting gap

3. A quasi-one-dimensional organic molecular metal byintroducing new modulations through deformation of thelattice. This is known as the Peierls transition.

Kinetic energy increases in this directionand becomes the maximum at the Fermilevel.

EEF!"#$%

0 7

0.2N(EF)free

50~500 meV

DO

S

(eV)energy

Covalent bondings12rbital hybridization between neighboring atoms

Metallic bonding (FsBz interaction)13ong-range interaction throughout a crystal

!

b

<550>

2d~4 Å

$F~4 Å

Electrons having the wave length $F resonate with aset of lattice planes with lattice spacing 2d.

What about the mechanism for the formation of the pseudogap?

!

"F =2#

kF

$1

e /a( )1/ 3

Orbital hybridization effect would occurnot only in crystals but also in non-crystalline systems like liquid metalsand amorphous alloys.

Orbital hybridizations

The FsBz interaction is unique in well-orderedsystems like crystals and quasicrystals

Anti-Bonding energy level

Bonding energy level HOMO

LUMO

EFAl-3p Mn-3d

Fermi surface-Brillouin zone

(FsBz) interaction

The concept of the Brillouin zoneStationary waves are formed and an energy gap opens, when the electron wave length

matches the lattice periodicity.

a

a

x

x

!

sin "x /a( )[ ]2

!

cos "x /a( )[ ]2

The Brillouin zone is a polyhedron bounded byplanes, which are formed by perpendicularlybisecting the relevant reciprocal lattice vectorslike G=<110> in the reciprocal space. An energygap opens across each zone face.

{002}

{111}

II. fcc, N=4, truncated octahedron with 14 zone faces

G2=3 or 4

{110} N

HP!

I. bcc, N=2, dodecahedron with 12 zone faces

G2=2

G2=2

The Brillouin zone of Three Complex Alloy Phases

III. Quasicrystal, N is infinite, polyhedron with 60 planes

Inte

nsi

ty (

arb.u

nit

s)

807060504030202! (deg.)

111000

1/2

(311111)

111100

211111 221001

322101

322111 332002

The larger the number of atoms in the unit cell, the more the number of zone faces, across

which an energy gap opens. This in turn results in a deeper pseudogap on the DOS. As a

result, the electronic energy can be more efficiently lowered by increasing the number of

atoms in the unit cell.

I. gamma-brass, N=52, polyhedron with 36 planes

{411}

{330}

G2=18

(710)

(710)

(710)

(710)

(550)

(550)

(550)

(550)

(543)(543)

(543)

(543)

(543)

(543)

(543)(543)

(543)

(543)

(543)(543)

(543) (543)

(543)

(710)

(543)

(543)

II. 1/1-1/1-1/1 approximant, N=160, polyhedron with 84 planes

G2=50

Hume-Rothery (1926) pointed out a tendency for a definite crystal structure to occur at aparticular electron concentration e/a. Mott and Jones (1936) made its first interpretation interms of the Fermi surface-Brillouin zone interaction on the basis of the nearly free electronmodel.

What is the Hume-Rothery electron concentration rule!

{411}

{330}

H.Jones (1905-1986)N.F.Mott (1905-1996)

William Hume-Rothery (1899-1968)

There exist two different definitions for electron concentration:

e/a versus VEC

Slater-Pauling curve"Saturation magnetization against VEC

Matthias ruleSuperconducting transition temperature against VEC

5.9 5.95 6 6.05 6.1

-100

-50

0

50

100

Fe2(V1-xTix)AlFe2(V1-xZrx)Al

(Fe1-xCox)2VAl(Fe1-xPtx)2VAl

Fe2V(Al1-xSix)Fe2V(Al1-xGex)

-150

300 K

Fe2VAl

Fe2(V1-xMox)AlFe2(V1-xWx)Al

!"#$!%

&"#$!%

electron concentration VEC

Se

eb

eck

co

eff

icie

nt

(µV

/K)

Y.Nishino and U.Mizutani (2005)

p- and n-type thermoelectric alloys

Fe2VAl doped with various elements

!

VEC =8"2+5"1+ 3"1

4= 6.0

Various physical properties fall on a universal curve, when plotted

against the VEC (number of electrons per atom in the valence band)

Electron concentration in the Hume-Rothery electron concentration rule is not

VEC but should be e/a (averaged number of valencies of constituent atoms).

!

e /a(Cu5Zn

8) =1"5+2"8

13=21

13

e /a(Cu9Al

4) =1"9+ 3" 4

13=21

13

!

VEC(Cu5Zn

8) =11"5+12"8

13=151

13

VEC(Cu9Al

4) =11"9+ 3" 4

13=111

13

e/a

VECZnII

CdII

AlIII

GaIII

SnIV

1CuI

AgI

AuI3/2

7/41.4

21/13

(b)

N

H

P%

H!

(110)

(a)

L

X%

W K

!

(111)

!

(002)

Model of Jones for the interpretation of the Hume-Rotheryrule on "/# phase transformation in Cu-Zn system

Brillouin zone of fcc

Brillouin zone of bcc

(c)

Ufc

c-U

bc

c (1

02

eV

ato

m-1

)

e/a

H.Jones, Proc.Phys.Soc. A49 (1937) 250

Difficulties in model of Jones

Discovery of the neck in the Fermi surface of fcc-Cu by Pippard in 1957

Ignorance of the Cu-3d band

Hume-Rothery commented in 1961 that “the work of the last ten years has made the theory of alloystructures appear less satisfactory than was the case twenty-five years ago. It is now definitelyestablished that the assumption of a spherical Fermi surface for pure Cu is quite unjustified…”.

{111}

{110}

!

"U =U fcc #Ubcc = {N(E) fcc # N(E)bcc}EdE0

EF$

Current understanding of fcc/bcc phase transformation of Cu on the

basis of the first-principles FLAPW electronic structure calculations in

combination with density functional theory

!

U = "i

i

# $1

2

n(r)n( % r )

r $ % r drd % r && + n(r) "

XCn(r)( )$µ

XCn(r)( )[ ]& dr

Total energy of a system at absolute zeroW.Kohn and L.J.Sham, Phys.Rev. 140 (1965) A1133

One-electron band structureenergy due to both valence and

core electrons

exchange-correlation energyaverage interaction energy

of electrons known as the

Hartree term

!

"h2

2m#2 +veff (r)

$

% &

'

( ) *i(r) = +i*i(r)

!

"i

is the solution of the effective one-electron Schrödinger equation of a system given by

!

Uvalence

= D(E)(E " E0)dE

Ebottom

EF

# This is a quantity evaluated by Jones and

corresponds to the contribution from the

valence electrons in the first term.

electron-electron interaction term

FLAPW valence band structure and van-Hove singularities in fcc-Cu

due to contact with {111} zone planes

due to contact with {200} zone planes

due to overlap across{111} zone planes

Fcc-Cu

Evidence for the neck

against {111} zone planes

Bcc-Cu

The first-principles band calculations prove that the

ignorance of the Cu-3d band was a vital failure in

the model of Jones. This led him to a wrong location

of van-Hove singularities and overestimation of

their sizes in both fcc and bcc Cu.

!

"U = Nbcc (E)(E #Eo#bcc )dE # N fcc (E)(E #Eo# fcc )dEEo

EF

$Eo

EF

$

You can see that the van-Hove singularities are negligibly small in both fcc- and bcc-Cu.The band structure energy difference arises essentially from the difference in Cu-3dbands and amounts to about 10 kJ/mole in favor of the bcc structure.

+41.19-1857872.647-1857913.837Uelectron-electron (kJ/mole)

-7.37-2483807.338-2483799.964core electron energy Ucore

(kJ/mole)

-31.455226.47695257.9289valence-band structureenergy Ubs (kJ/mole)

+2.37-4336453.49-4336455.852total energy U (kJ/mole)

2.86393.6048lattice constant (A)

"U=Ubcc-Ufcc

(kJ/mole)bcc-Cufcc-Cu

!

U = "i

i

# $1

2

n(r)n( % r )

r $ % r drd % r && + n(r) "

XCn(r)( )$µ

XCn(r)( )[ ]& dr

' ( )

* + ,

=Ubs +Ucore

+Uelectron$electron

"U =(-31.45)+(-7.37) +41.19=+2.37 kJ/mole

The stability of fcc Cu is correctly predicted, only if the electron-electron term is

properly evaluated.

!

Ubcc

>Ufcc

!

Ubcc

<Ufcc

!

Ubcc

>Ufcc

We believe the situation in complex alloy phases

to be different from that in fcc and bcc Cu,

where van-Hove singularities are too small to

affect the total energy.

As is clear from the argument above, the interpretation of &/'phase transformation in the Cu-Zn system is still far from our goal.

---First-principles LMTO band calculations---The origin of the pseudogap was attributed to the Mn-3d/Al-3p orbital hybridizations

A pseudogap across the Fermi level plays a key role in

stabilizing a complex alloy phase

T.Fujiwara, Phys.Rev.B 40 (1989) 942

Al-Mn approximant

Mn-3d/Al-3p bonding states

Mn-3d/Al-3p anti-bonding states

EF

!

b

<550>

2d~4 Å

$F~4 Å

Resonance of electrons having the wave length $F

with a set of lattice planes with spacing 2d.

We are interested in e/a-dependent phase stability

!

"F =2#

kF

$1

e /a( )1/ 3

Orbital hybridization effect would occurnot only in crystals but also in non-crystalline systems like liquid metalsand amorphous alloys.

Orbital hybridizations

FsBz interaction is unique in well-orderedsystems like crystals and quasicrystals

Anti-Bonding energy level

Bonding energy level HOMO

LUMO

EFAl-3p Mn-3d

FsBz interaction

An increase in the number of atoms in unit cell accompanies an increase in

Brillouin zone planes

V. Quasicrystal, N is infinite, polyhedron with 60 planes

The gamma-brass is complex enough to producea sizeable pseudogap at the Fermi level but isstill simple enough to perform the FLAPW bandcalculations with an efficient speed.

{110} N

HP!

I. bcc, N=2, dodecahedron with 12 planes

G2=2

{002}

{111}

II. fcc, N=4, truncated octahedron with 14 planes

G2=3 or 4

III. gamma-brass, N=52, polyhedron with 36 planes

{411}

{330}

G2=18

(710)

(710)

(710)

(710)

(550)

(550)

(550)

(550)

(543)(543)

(543)

(543)

(543)

(543)

(543)(543)

(543)

(543)

(543)(543)

(543) (543)

(543)

(710)

(543)

(543)

IV. 1/1-1/1-1/1 approximant, N=160, polyhedron with 84 planes

G2=50

Inner Tetrahedron (IT)

Outer Tetrahedron (OT)

Octahedron (OH)

Cubo Octahedron (CO)

(b)

(a)

26-atom cluster

The 26-atom cluster forms either bcc or CsCl-structure and contains 52 atoms in the unit cell.

Gamma-brasses with the space group P43m and I43m

ZnII

CdII

AlIII

GaIII

InIII

SnIV

1CuI

AgI

AuI21/13

Our objective is to explore if all gamma-brasses listed here

are stabilized via the same mechanism at e/a=21/13

In literature, all isostructural gamma-brassesabove had been implicitly assumed to bestabilized at e/a=21/13 or the valency of thepartner element to be forced to take a valueto fulfill the total e/a equal to 21/13.

Empirical Hume-Rothery e/a rule forgamma-brasses in the past

1.6

1.6

1.6

1.6

1.46

DOS of Cu9Al4 gamma-brass

The FsBz interaction is really responsible for the formation of the pseudogap?

3.0

2.5

2.0

1.5

1.0

0.5

0.0

DO

S (

sta

tes/e

V.a

tom

)

-10 -8 -6 -4 -2 0 2 4

Energy (eV)

Cu9Al4 gamma-brass

EF

pseudogap

Cu-3d

!

"k (E, r) = C(k+Ghkl )#k+Ghkl(E

Ghkl

$ ,r)

!

"k+Ghkl(E,r) : FLAPW basis function

Principle for the extraction of the FsBz interaction from the first-principles

FLAPW (Full-potential Linearized Augmented Plane Wave method) band calculations

!

r " a #k(r) = Almk

ul(El, r) + Blmk ˙ u l(El, r)[ ]Ylm($, %)

lm

&

!

r " a #k(r) = exp(ik $r)

Adoption of spherically symmetric muffin-tin potential

!

"k (E, r) = C(k +Ghkl )exp i(k +Ghkl ) #r[ ]Ghkl

$

EF

!

1

21,1, 0( )

(1)

FLAPW-Fourier Analysis

N

%

{110}

H

k

We extract the electronic state k+Ghkl having the largest Fourier coefficient in wavefunction (1) at symmetry point N with the energy eigen-value near the Fermi level. Thisis nothing but the extraction of the set of lattice planes resonating with electron waves.

Cu5Zn8 and Cu9Al4 gamma-brasses in group 1

?Ir2Zn1121/13Cu9Ga4

21/13?

Ag5Li831.60?Co2Zn1121/13Cu9Al4

?Mn3In1.60?Fe2Zn1121/13Au5Zn8

1.46?Al8V51.60?Pd2Zn1121/13Au5Cd8

?Pt2Zn111.60?Ni2Zn11

2

21/13Cu5Cd8

?Mn2Zn1121/13Au9In421/13Ag5Zn8

?Ni2Cd1121/13Ag9In421/13Ag5Cd8

?Ni2Be11

2

21/13Cu9In4

1

21/13Cu5Zn8

1

e/agamma-

brasse/a

gamma-brass

e/agamma-brass

E-k dispersions and DOS derived from FLAPW for Cu5Zn8 gamma-brass

5 4 3 2 1 0-1

0-8

-6-4

-20

24

Energ

y (

eV

)

Cu5Zn8 gamma-brass

DOS (states/eV.atom)

EF

Cu-3d

Zn-3d

pseudogap

R.Asahi, H.Sato, T.Takeuchi and U.Mizutani, Phys.Rev. B 71 (2005) 165103

Energ

y (e

V)

E4E3

E2E1

EFEF

N

%

EF

Energ

y (e

V)

E6E5

E4E3

EF

E8E7

E2E1

E-k dispersions and DOS derived from FLAPW for Cu9Al4 gamma-brass

5 4 3 2 1 0-1

0-8

-6-4

-20

24

Energ

y (

eV

)

Cu9Al4 gamma-brass


EF

Cu-3

dpseudogap

0.001

2

4

0.01

2

4

0.1

2

4

1

!|Ci G

hkl/2|2

16012080400

|G0|2=h

2+k

2+l

2

Cu9Al4 gamma-brass

E=-0.41 eV

2

3

4

6

7

8

9

0.001

2

4

0.01

2

4

0.1

2

4

1

!| Ci G

hkl/2|2

16012080400

|G0|2=h

2+k

2+l

2

E=-0.62 eV

2

3

4

6

8

Cu5Zn8 gamma-brass

1: {211}, 2: {310}, 3: {321}, 4: {411}+{330}, 5: {332}, 6: {510}+{431}, 7: {521}, 8: {530}+{433}, 9: {611}+{532}

!

IG2

=18/ Inext =123

!

IG2

=18/ I

next= 45

!

"k(E, r) = C(k

N+G

hkl)exp i(k

N+G

hkl) #r[ ]

Ghkl

$

18 18

N

The Fourier coefficient is extremely large at G2=18. This implies that electron waves near

the Fermi level exclusively resonate with the set of {411} and {330} lattice planes with

G2=18, resulting in the formation of the pseudogap.

FLAPW-Fourier spectra for the wave function outside the MT sphere at

the point N at energies immediately below the Fermi level

Evaluation of the e/a value by means of the FLAPW-Fourier method

---Hume-Rothery plot---

This technique is now extended to a whole valence band. We need to do this only in the irreduciblewedge of the Brillouin zone. The wedge is divided into 200 elements and the electronic state (ki+G)having the largest Fourier coefficient for the wave function at energy E in the i-th element isextracted. This is done for all elements over i=1 to 200 and an average value of (ki+G) av iscalculated. In this way, a new single-branch dispersion relation E-(ki+G) av for electrons extendingoutside the MT sphere is derived in the extended zone scheme.

We have so far limited ourselves to the extraction of the largest plane wave component of the FLAPW wavefunction only near the Fermi level and only at the symmetry point N.

!

k+GE" #

iki+G

Ei=1

N=200

$

Once the E-(k+G) is determined, the Fermi diameter 2kF is obtained from the value of 2(k+G) at EF.

The e/a is immediately calculated by inserting theFermi diameter 2kF into the relation below:

!

(e/a)total ="

3N2kF( )

3

The variance ( must be small to validate this approach.

N=52:number of atoms in the unit cellirreducible wedge

E-{2(k+G)}2 relations for itinerant electrons for Cu5Zn8 and Cu9Al4 gamma-brasses


(2kF)2=18.47 for Cu5Zn8

(2kF)2=18.45 for Cu9Al4

!

(e/a)total ="

3N2kF( )

3

N=52

30

25

20

15

10

5

0

!2 (

E)

-10 -8 -6 -4 -2 0 2 4Energy (eV)

EF

Zn-3d

Cu-3d

Zn-3d Cu-3de/a=1.60 for bothgamma-brasses

?Ir2Zn1121/13Cu9Ga4

21/13?

Ag5Li831.60?Co2Zn1121/13Cu9Al4

?Mn3In1.60?Fe2Zn1121/13Au5Zn8

1.46?Al8V51.60?Pd2Zn1121/13Au5Cd8

?Pt2Zn111.60?Ni2Zn11

2

21/13Cu5Cd8



?Ni2Be11

2

21/13Cu9In4

1

21/13Cu5Zn8

1

e/agamma-

brasse/a

gamma-brass

e/agamma-brass

Ekman (1931) studied the TM-Zn gamma-brasses and proposed that they obey

the Hume-Rothery electron concentration rule with e/a=1.60, provided that the

valency of the TM element is zero.

W.Ekman, Z.Physik.Chem. B 12 (1931) 57

Studies of TM2Zn11 (TM=Ni, Pd, Fe, Co) gamma-brasses in group 2

FLAPW-derived E-k relations and DOS for Ni2Zn11 gamma-brass

E1

E2pseudogap

Ni-3d

Zn-3d

5 4 3 2 1 0-1

0-8

-6-4

-20

24

En

erg

y (

eV

)

Ni2Zn11 gamma-brass

EF


EF

0.001

0.01

0.1

1

!|Chkl|2

16012080400

h2+k

2+l

2

E= -1.09 eV|G|2=18

5

4

3

2

1

0

DO

S (

sta

tes/e

V.a

tom

)

-10 -8 -6 -4 -2 0 2 4

Energy (eV)

Ni2Zn11 gamma-brass

EF

FLAPW-Fourier spectra just below and above the

pseudogap in Ni2Zn11 gamma-brass

G2=18 resonance is active inNi2Zn11 gamma-brass.

0.001

0.01

0.1

1

!| Chkl|2

16012080400

h2+k

2+l

2

E= +0.05 eV|G|2=18

FLAPW-derived E-k relations and DOS for Pd2Zn11 gamma-brass

pseudogap

Pd-4d

Zn-3d

5 4 3 2 1 0-1

0-8

-6-4

-20

24

En

erg

y (

eV

)

Pd2Zn11 gamma-brass

EF


EF


pseudogap in Pd2Zn11 gamma-brass

5

4

3

2

1

0

DO

S (

sta

tes/e

V.a

tom

)

-10 -8 -6 -4 -2 0 2 4Energy (eV)

Pd2Zn11 gamma-brass

EF

G2=18 resonance is active inPd2Zn11 gamma-brass.

0.001

0.01

0.1

1

!|Chkl|2

16012080400

h2+k

2+l

2

E=-0.11 eV|G|2=18

0.001

0.01

0.1

1

!|Chkl|2

16012080400|G|

2=h2+k2+l2

E=-1.14 eV|G|2=18

FLAPW-derived E-k relations and DOS for Co2Zn11 gamma-brass

Zn-3d

pseudogap

Co-3d

5 4 3 2 1 0-8

-40

4

En

erg

y (

eV

)

EF

Co2Zn11 gamma-brass


EF

5

4

3

2

1

0

DO

S (

sta

tes/e

V.a

tom

)

-8 -4 0 4Energy (eV)

EF

Co2Zn11 gamma-brass


pseudogap in Co2Zn11 gamma-brass

The G2=18 resonance still

survives only above the Fermi

level in Co2Zn11.

0.001

0.01

0.1

1

!|Chkl|2

12080400

h2+k

2+l

2

E= +0.91 eV|G|2=18

0.001

0.01

0.1

1

!|Chkl|2

12080400

|G|2=h

2+k

2+l

2

E= -0.20 eV|G|

2=18

FLAPW-derived E-k relations and DOS for Fe2Zn11 gamma-brass

5 4 3 2 1 0-1

0-8

-6-4

-20

24

Energ

y (

eV

)

Fe2Zn11 gamma-brass

EF


pseudogap

Fe-3d

Zn-3d

EF

5

4

3

2

1

0

DO

S (

sta

tes/e

V.a

tom

)

-10 -8 -6 -4 -2 0 2 4Energy (eV)

Fe2Zn11 gamma-brass

EF


pseudogap in Fe2Zn11 gamma-brass

0.001

0.01

0.1

1

!| Chkl|2

16012080400

h2+k

2+l

2

E=+0.025 eV|G|2=18

0.001

0.01

0.1

1

!| Chkl|2

16012080400

h2+k

2+l

2

E=+1.04 eV|G|

2=18

The G2=18 resonance still

survives only above the Fermi

level in Fe2Zn11.

(2kF)2=19.36 for Ni2Zn11

=19.27 for Pd2Zn11

(2kF)2=19.5 for Co2Zn11

=20.0 for Fe2Zn11

!

(e/a) total ="

3N2kF( )

3

N=52

e/aNi-Zn=1.72e/aPd-Zn=1.70

!

(e/a) total ="

3N2kF( )

3

N=52

e/aCo-Zn=1.73e/aFe-Zn=1.80

E-(k+G)2 relations for itinerant electrons for TM2Zn11 (TM=Ni, Pd, Co, Fe) gamma-brasses


11.38511.53811.711.78.511.6VEC

0.700.260.070.150.970.96(e/a)TM

1.801.731.701.721.601.60(e/a)total

20.019.519.2719.3618.4518.47(2kF)2 deduced

form the H-R plot

!!18181818|G|2 deduced

from the FLAPW-Fourier method

XXA pseudogapformed below EF

Fe2Zn11Co2Zn11Pd2Zn11Ni2Zn11Cu9Al4Cu5Zn8

Gamma-brasses in group 1 obey the Hume-Rothery e/a law with e/a=21/13

The Hume-Rothery stabilization mechanism refers to the mechanism, in which apseudogap is formed across the Fermi level as a result of electron waves resonatingwith a particular set of lattice planes and thereby the particular e/a value is specified.

group (1) group (2)

Summary for gamma-brasses in groups 1 and 2

2?Ir2Zn1121/13Cu9Ga4

21/13?

Ag5Li831.73Co2Zn111.60Cu9Al4

?Mn3In1.80Fe2Zn1121/13Au5Zn8

1.46?Al8V51.70Pd2Zn1121/13Au5Cd8

?Pt2Zn111.72Ni2Zn1121/13Cu5Cd8



?Ni2Be11

2

21/13Cu9In4

1

1.60Cu5Zn8

1

e/agamma-

brasse/a

gamma-brass

e/agamma-brass

Documents

The Hume-Rothery rules for Structurally Complex Alloy Phases