2094 CANDLAND, DECKER, AND VANFLEE T

G. V. Kidson, Phil. Mag. 13, 247 (1966).B. F. Dyson, T. R. Anthony, and D. Turnbull, J.

Appl. Phys. 37, 2370 (1966).H. R. Curtin, D. L. Decker, and H. B. Vanf lect,

Phys. Rev. 139, A1552 {1965).~ J. A. Weyland, D. L. Decker, and H. B. Vanf lect,

Phys. Rev. B 4 4225 (1971).F. C. Frank and D. Turnbull, Phys. Rev. 104, 617

(1956).A. E. Stern and H. Eyring, J. Phys. Chem. 44, 955

(194O).M. D. Boren, S. E. Babb, and G. J. Scott, Rev. Sci.

Instr. 36, 1456 {1965).i?These values are typical up-cycle transition pres-

sures for the Bi I-II, Tl II-III, and Ba I-II as measuredby H. B. Vanf lect and R. J. Zeto (unpublished) using amanganin gauge in a hydrostatic pressure cell.

R. K. Young, Ph. D. dissertation (Brigham YoungUniversity, 1969), p. 95 (unpublished).

L. E. Millet, Ph. D. dissertation (Brigham YoungUniversity, 1968), p. 25 (unpublished).

C. Wang, Rev. Sci. Instr. 38, 1 (1967).21H. M. Gilder (private communication).

I. C. Getting and G. C. Kennedy, J. Appl. Phys. 41,4552 0.97o).

23R. E. Hanneman, H. M. Strong, and F. P. Bundy,

in Accurate Characterization of the High Pressure En-vA"onment, edited by E. C. Lloyd (NBS, Washington,D. C. , 1971).

W. B. Alexander and L. M. Slifkin, Phys. Rev. B 1,3274 (1970).

L. W. Barr, J. N. Mundy, and F. A. Smith, Phil.Mag, 20, 389 (1969).

N. H. Nachtrieb, J. Petit, and J. Wehrenberg, J.Chem. Phys. 26, 106 (1957).

Reference 19, p. 38.L. Tewordt, Phys. Rev. 113, 438 (1959).R. A. Johnson and E. Brown, Phys. Rev. 127, 446

(1962).K. Weiser, Phys. Rev. 126, 1427 (1962).T. R. Anthony, General Electric Research and

Development Center, schenectady, New York, Report No.68-C-413, 1968 (unpublished).

J. H. Westbrook and K. T. Aust, Acta Met. 11, 1151(1963).

T. R. Anthony and D. Turnbull, Phys. Rev. 151,495 (1966).

34Table of Periodic Properties of the Elements (Sargent-Welch Scientific, Skokie, Illinois, 1968).

3 N. L. Peterson and S. J. Rothman, Phys. Rev. B 1,3264 (197O).

C. Wert and C. Zener, Phys. Rev. 76, 1169 (1949).

PHYSICA L REVIEW B VOLUME 5, NUMBER 6 15 MA RCH 1972

Symmetry of Quadrupolar Arrangements in Crystals*

J. SivardierefBrookhaven National I aboxatory, Upton, Nese Fork 11973

(Received 19 August 1971)

We carry out a new group-theoretical classification of all possible static quadrupolar arrange-ments in a crystal: We define normal quadrupolar configurations and classify them accordingto the irreducible representations of the space group of the crystal. This method is a gener-alization of the representation analysis of magnetic structures, and is equivalent to the methodrecently proposed by Fe1steiner, Litvin, and Zak.

INTRODUCTION

It is well known that the symmetry of electricand magnetic dipolar arrangements may be de-scribed from two points of view: the method ofOpechowski and Guccione based on the theory ofmagnetic groups, ' and the representation theoryof crystallographic groups. ' As discussed re-cently, both of them lead to almost equivalentclassifications of all possible dipolar arrangementsin a crystal. Moreover, the second one has athermodynamical aspect, since it is connectedwith the theory of second-order phase transi-tions, ' and at the same time has a purely geo-metrical aspect related to the idea of crypto-symmetry. '

Felsteiner, Litvin, and Zak' have recently car-ried out a classification of all possible quadrupolararrangements in a crystal following the method

of Opechowski and Guccione. In .his article, itis shown that representation theory may be usedas well; normal quadrupolar configurations aredefined and classified according to the irreduciblerepresentations of the space group of the crystal.According to the discussion given by Opechowski'for the case of dipolar arrangements, we may ex-pect the two methods to be equivalent in most cases.

NORMAL QUADRUPOLAR CONFIGURATIONS

Let us consider that each atom i of a crystal ona, given sublattice bears a quadrupolar momentcharacterized by a trace), ess symmetrical tensorq'z Ly q~~=0). These quadrupolar moments maybe electrical, as in molecular crystals (in this caseeach nonpolar molecule is punctualized in r, ), ormay be considered as magnetic as in some rare-earth compounds (PrA10„DyVO„TbVO„9. . .).

The static interaction energy E, of the quadrupoles

SY1V[1VIE TRY OF QUADRUPOLAR ARRANGEMENTS IN CRYSTALS 2095

q&=R q iR (2)

with

r,. = n(r, )+7

R being the 3 &3 matrix representing the rotation+. Knowing the transformation properties of q~(they are the same for electric and magnetic quad-rupoles) we may use the projection-operator tech-nique to find the normal quadrupolar configurations.

EXAMPLES

(i) We first consider the somewhat academic ex-ample of possible magnetic quadrupolar arrange-ments in rare-earth orthoferrites. The spacegroup is Pbn~ and the quadrupoles are on sublattices4b and 4c. We restrict ourselves to the arrange-ments which conserve the unit cell (k = 0), and we

use the notations E, 6, C, A for the linear com-binations (++++), (+-+-), (++--), (+--+); for in-

1 2 3 4stance, C„,=q„,+ q „,—q„, —q„,,The situation is very simpl. e since the X"p& are

real. one dimensional representations. Let V' bea polar or axial vector linked to the atom i, and letq'

~ transform Bs V' & V&, if V' and V& trBnsformaccording to X'p and X'pz, respectively, under arotation conserving the atom i, then q'& transformsaccording to X'p + X"p8 Using the projection-opera-tor technique, we easily get the classification of

is assumed quadratic with respect to their coordi-nates".

g ii'~ (gg Ofsgl qote q0fl 3s ~

ii' 0t8 0.'g'

At high temperatures the crystal is in a disorderedphase since all the correlations (q'„p q' i&. ) are zeroAt lower temperatures cooperative transitions to anordered quadrupolar state may appear. ' We arelooking here for all possible independent static con-figurations of the quadrupoles. A similar problemoccurs when looking for symmetry-allowed spinconfigurations in magnetic crystals.

Following a well-known procedure, we have toreduce E, to a block-diagonal form in order toput normal quadrupolar conf igurations into evidence.Such configurations must be symmetry adapted,as are vibrational modes or static dipolar configu-rations, i. e. , they must transform according tothe various representations X'» of the space group6 of the crystal; only configurations transformingaccording to the same row of the same representa-tion may be coupled.

Consequently, we shall classify l.inear combina-tions of the coordinates q~ of the quadrupolar atomsin ri in the unit cell according to the I'». Underthe operation (n ~r ) of the space group, q; be-comes

quadrupolar modes given in Table I. Each com-ponent of the tensors shown in the table transformsaccording to the corresponding X'p .

If the tensors q'~ are supposed to be nonsym-metrical, their Bntisymmetrical. part transformslike an axial vector; consequently Table I gives atthe same time the well-known classification of axialdipolar modes; for instance (for site 4b), the com-ponents of the tensor

A„„G„„C„g

transforms according to X'2, and the same is truefor the components of the axial vector [F„C„G,].This property, however, disappears if the sym-metry is higher than orthorhombic: A quadru-polar configuration and the associated axial con-figuration may belong to different representations.

(ii) We now consider the space group G= Fmmmand the site 4a(mmm), and we look for allowedquadrupolar arrangements with k= (001). All theX'» are real one dimensional representations. I etus call I'~p the representation such that I"„(2„)= I'@(2y) = —1. Introducing the F, G, C, A nota-tions as above, we find that the basis functions forX"-„p are E„„,E„, E„,A„. The corresponding sym-metry is the kernel. H of IIp thus H =Cmca (site4a, 2,/m) or Bmab (»«4a, 2, /m), which is thesymmetry group of solid I2. " Consequently wemay consider that the structure of I2 is a super-structure of Fmmm (this symmetry is obtainedwhen the Ip molecules are punctualized), due toan ordering in the direction of the molecules.

In the same way, let us consider the space groupf(4/m)mm and the site 2a, and the real one-di-mensional representation such that I'fp(1) =+1, I'pp

(4,) = 1, 1"-„p(2„)= —1. The basis function forI'pp is a„,=q„', —q,', . After a 45 rotation of the xand y axes around the z axis, the quadrupoles ofthe two atoms in the cell are

o o)

Such a quadrupolar structure, found in tetragonaly —Nz,

' has the symmetry P(4z/m)nm (site 2a,mmm) .

In the preceding examples, real one-dimensionalrepresentations were involved in the descriptionof quadrupolar structures: These structures maybe described as well by the corresponding magneticgroups' (here the antisymmetry operation changesthe signs of all the components of the quadrupoles).

2096 J. SIVARDIE RE

Quadrupolar modes Axial dipolar modes4b c 4b 4c

&xy Gx»

I;I

+„A„ l»

I A„, G„C, c,"..".)G C~A E

Al

(0 0 c)o F

lF„, C„G, F„, c,

olA E . 0 0 EG c

lo cl C„, F„A, C„, F,

G ) l 0

(C

lI 4

(0 0 AiO G

fG A 0)

TABLE I. Bipolar and quadrupolar modes in the symmetryPbnm, K=O.

the representation E of the little group F(4/m)mm.If q„=0, the reduced symmetry is Bbcm (HBr, HI). "Finally the two structures of solid hydrogen con-sidered in Ref. 8 are associated with two represen-tations of the full space group characterized by thestar "[k].= *(1,0, 0) .

Finally, we consider the situation where theatoms on the lattice simultaneously have dipolarand quadrupolar moments. According to Felsteinerand Litvin, ' the existence of the dipolar momentleads to some restrictions on the possible quadru-polar arrangements. However, the restrictionsconsidered in Ref. 12 imply simultaneous dipolarand quadrupolar ordering; this is not always thecase, in particular in magnetic crystals.

Our method enables us to study simultaneous(HCl) or successive (HBR, HI)" dipolar and quadru-

polar ordering. We first classify linear combina-tions of the dipolar and quadrupolar moment com-ponents according to the representations of thehigh-temperature space group. If I » describessome symmetry- allowed quadrupolar arrangement,dipolar arrangements described by the same I'»may occur simultaneously. If no dipolar arrange-ment is described by I'g, we classify linear com-binations of the dipolar moments according to therepresentations &)-,

&of the space group of the ordered

quadrupolar structure; each 4;; (different fromthe identity representation) describes symmetry-allowed dipolar arrangements.

CONCLUSION

0 0 G

O A

o&

(ill Finally we take the fcc lattice 6 = Em3m.Fo» = 0, (q„, q, „,q„,) transforms according to therepresentation T~ . If q„=q,„=q„,= q, the reducedsymmetry is R3m; the quadrupoles are elongatedalong a (ill) direction, as are the SH molecules inNaSH. "If q„=q,„=0, the reduced symmetry isImmy, and quadrupoles are elongated along a (110)direction.

For k= (0, 1, 0), (q„„q„)transforms according to

We have shown that the representation theoryof crystallographic groups provides a useful framefor the geometrical description of electric ormagnetic quadrupolar arrangements. The generali-zation to the description of multipolar arrange-ments is straightforward, since the components ofa multipole have transformation properties similarto those of a quadrupole [Eq. (2)]. Moreover, therepresentation theory may also describe the sym-

ry of ordered orbital configurationsso and JahnTeller transitions, and finally it enables us to studythe possible coupling between dipolar and quadru-polar ordering.

ACKNOW'LEDGMENTS

I thank Dr. M. Blume and Dr. A. B. Harris fora critical reading of the manuscript, and Dr. D.B. Litvin for clarifying correspondence.

*Work performed under the auspices of the U. S.Atomic Energy Commission.

fGuest scientist on leave from Centre d'Etudes Nu-

cleaires, Grenoble, France.W. Opechowski and R. Guccione, in Magnetism, edited

by G. T. Rado and H. Suhl (Academic, New York, 1965),

Vol. II A, p. 105.S. Alexander, Phys. Rev. 127, 420 (1962).

3E. F. Bertaut, Acta Cryst. A24, 217 (1968).4W. Opechowski, International Conference on Magnetism

(Grenoble, 1970) (unpublished); and J. Phys. C 1, 457(1971).

SYMMETRY OF QUADRUPOLAR ARRANGEMENTS IN CRYSTALS 2097

L. D. Landau and E. M. Lifschitz, in StatisticalPhysics (Pergamon, London, 1959), p. 421.

E. Dimmock, Phys. Rev. 130, 1337 (1963).A. Niggli and H. Wondratschek, Z. Krist. 115, 1

(1961).J. Felsteiner, D. B. Litvin, and J. Zak, Phys. Rev.

B 3, 2v06 (1ev1).See, for instance, K. A. Gehring„A. P. Malozemoff,

W. Staude, and R. N. Tyte, Solid State Commun. 9, 511

(1ev1).~ O. Nagai and T. Nakamura, Progr. Theoret. Phys.

(Kyoto) 24, 432 (1960).~~R. W. G. Wyckoff, Crystal Stmctm"e (Interscience,

New York, 1948).~ J. Felsteiner and D. B. Litvin, Phys. Rev. B 4, 671

{1ev1).See, for instance, M. Ito, Chem. Phys. Letters 7,

439 (1evo).

P H YSICA L RE VIEW .B VOLUME 5, NUMBER 6 15 MAHc H 1972

Mean Free Path of Electrons in a One-Dimensional Liquid Model

D. G. Blair~Theoretical Physics Institute, University of &Ibexta, Edmonton, Alberta, Canada

(Received 4 October 1971)

This paper derives the expression giving the mean free path (which is also the range oflocalized eigenstates) in the limit of small (disorder for an electron in the energy band of achain with "liquid" disorder. The method also establishes the mechanism involved, namely,the electron is scattered from one Bloch state into another by a lattice wave which is a Fouriercomponent of the disordered structure. Difficulties are resolved regarding the "average" be-havior of the amplitude of solutions satisfying one boundary condition; a theorem is proved re-lating different averages. The "distorted-coordinate" method of Gubanov is shown to yield thecorrect mean free path to first order, but to become unphysical thereafter.

I. INTRODUCTION

For the problem of an electron in a one-dimen-sional chain with "liquid" disorder, it is wellknown that the electron eigenstates at all energiesare localized, the modulus of the wave functiondecaying exponentially with distance' '; the rate ofthis decay will be called the "decrement. " Fur-thermore, when an electron wave of precise energyimpinges on a long chain, the wave decays expo-nentially with the same decrement. While nu-merical values for the decrement are available, 'it is desirable to have analytic expressions; it isalso desirable to know what is the physical mecha-nism producing the decrement in the various energyregions, since the latter information may be appli-cable in three dimensions. For "gap states" (stateswhose energy lies well within the gap of the periodiclattice of the same mean spacing), the answer issimple: The wave function already decays expo-nentially in the periodic lattice, and the same valueof the decrement holds in the liquid chain. The"local-density approximation" enables one also tounderstand energies near the gap edges. ' Thepresent paper describes how the decrement is pro-duced in the rest of the allowed band. Precisely,the paper deals with the "Bloch region, " definedin an earlier paper; the corresponding eigenstateswill be called "band states. "

In Secs. II and III a "phase" method is presented,which yields an expression for the decrement; the

latter is verified by comparison with numericalresults. The conditions of validity are also derived.Section IV resolves difficulties regarding different"average" behaviors of the amplitudes of solutionsand proves a related theorem.

In Sec. V, it is deduced from the "phase" methodthat the exponential decay is produced by scatteringof the electron from one Bloch state into anotherby a lattice wave. The significance of this resultis discussed.

Finally (Sec. VI), the method used by Gubanov 'is reexamined. His is a perturbation method basedon "distorted coordinates, " which is applicablealso in three dimensions. It is interesting to note(Sec. VI) that his method yields the correct ex-pression for the decrement, provided that oneavoids the additional approximations that Gubanovintroduced. On the other hand, while the methodyields correct results in fA'st order of perturbationtheory, it appears to become unphysical in secondorder and beyond.

II. MODEL

We consider a single electron in a chain of po-tential wells of strength -2W (using units such thata'/2m = 1), situated at the sites xo(= 0), x, , xz, ~ . ~,x„, where (x&,, —x&),„=1 (by suitable choice ofunits). Let x, „, -x& = 1+ y, ; then for "liquid"disorder the y& are independent random variables,all with the same probability distribution and havingstandard deviation 0.