Clebsch-Gordan coefficients for X x X and R x R in beta tungsten

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Clebsch-Gordan coefficients for X x X and R x R in beta tungsten

M. Suffczynski, H. Kunert

To cite this version:

M. Suffczynski, H. Kunert. Clebsch-Gordan coefficients for X x X and R x R in beta tungsten. Journal

de Physique, 1978, 39 (11), pp.1187-1192. �10.1051/jphys:0197800390110118700�. �jpa-00208858�

CLEBSCH-GORDAN COEFFICIENTS FOR X x X

AND R x R IN BETA TUNGSTEN

M. SUFFCZYNSKI

Instytut Fizyki ^{PAN, Al Lotników} 32/46, ^{02-668 Warszawa, Poland}

and H. KUNERT

Instytut Fizyki Politechniki Pozna0144skiej, ul. Piotrowo 3, ^60-965 Pozna0144, ^Poland

(Reçu ^{le 6 décembre} 1977, ^{révisé le 9 mars} 1978, accepté ^{le 20 mars} 1978)

Résumé.

Les coefficients de Clebsch-Gordan sont calculés pour les représentations irréductibles X x X, ^{R x} R de la structure bêta-tungstène.

Abstract.

The Clebsch-Gordan coefficients are calculated for irreducible representations

X x X and R ^x R in beta tungsten ^structure.

Classification

Physics ^Abstracts

61. 50E

1. Introduction.

The binary intermetallic com-

pounds having A3B composition ^{and the} p-tungsten

A-15 structure are of great theoretical interest and

outstanding practical importance. They ^{include mate-}

rials with the highest temperatures of transition to the

superconducting ^state that have been observed, ^such

as Nb3Ge, Nb3Sn, Nb3Al, V3Ga, V3Si, ^etc.

It is known that the A-15 structure ^{is liable to lattice} instabilities, defects, ^{etc. and that the} effects of disorder

along the chains in the A-15 structure dominate its

superconducting properties [1-4].

A simplified theory ^of symmetry changes ^{in second-}

order phase transitions, ^lattice instabilities, ^etc. applied

to V3Si ^was proposed by ^Birman [5] and Gorkov [6]

and discussed in [7, 8]. Expérimental ^results relating

structural instability ^and superconductivity ^{have been} compiled ^{in several} papers [4, 9].

In several compounds ^{of the A-15 structure the} phonons become soft at the M point leading ^{to the}

lattice instability ^and eventually ^{to a} distortive phase

transition [10].

Recently Jaric and Birman [11], using group theore- tical methods, considered second-order phase ^transi-

tions in A-15 compounds.

Recently also much attention has been directed to

the Clebsch-Gordan (CG) coefficients of _{the space} group representations. ^The computation ^{of CG}

coefficients for space groups ^can be done by ^several

methods [12-22]. ^In particular ^Birman and Beren-

son [23] and Birman [1 S] have shown that the elements of the first order scattering ^{tensor are} precisely ^certain

CG coefficients or prescribed linear combinations of them, ^and the elements of the second order tensor are

particular ^{sums of} products ^{of CG} coefficients. The matrix elements of the effective Hamiltonian are

products ^of appropriate ^CG coefficients multiplied by symmetrized tensorial field quantities, ^{as shown} by

Birman and Ting-Kuo ^Lee [24].

For a calculation of the Clebsch-Gordan coeffi- cients or scattering ^{tensors, an} elaboration of the selection rules is a necessary first step. The selection rules for the double space group Of ^{of the} 03B2-tungsten

or A-15 structure are determined in [25].

This _paper deals with the problem ^of constructing

basis functions for the representations ^{whiclr are}

contained in the Kronecker product of two irreducible space group representations. ^We compute ^{here the} CG coefficients of the representations ^X Q ^{X and} R p ^{R for the} space group 0’(Pm3n). ^{The CG}

coefficients for M (D ^{M in} 03B2-tungsten ^{structure have}

yet ^{to be} published.

2. Method using ^matrix éléments of small _repre- sentations.

For the irreducible representations ^of

the crystallographic space group defined by ^wave

vectors kt1’ k03C3, k§,, satisfying ^{wave vector selection}

rules

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0197800390110118700

1188

the CG coefficients

are defined as coefficients between the basis fonctions

pv03C3..,,, ^and 03C8kYâ a‘ Yâ °’l ‘ ^{of the} representations ^of

dimensions 1" and 1 x l’ respectively :

Berenson and Birman [17] ^and Berenson et crl. [ 8]

have shown that the CG coefficients for space groups

can be calculated from the small irreducible _repre- sentations dkl , dk’l‘, dk"1" of the little wave vector groups.

We have according ^{to Birman} [14], Berenson and Birman [17]. ^Berenson, Itzkan and Birman [18] :

Here h is the order of the wave vector group of k"

and the summation runs over

i.e., the intersection of the wave vector groups of k, k’ and k" satisfying equation (1).

The CG coefficients defined in equation (4) ^consti-

tute the (111) block of coefficients. In equation (4)

we set a

à, a’ =.à’, ^{a" =} a", compute ^the righthand

side of equation (4) ^{and, if it is non-zero, say for} a.= ao" a’ = aÓ" a"

a’o’, ^{we determine} U1C101ClhlC1(’.

We choose its phase ^{so as to have this U real. We now}

fix à, «a-’, a" and let a, a’, a" range ^over all allowable values. Thus doing summations on the righthand ^side

of equation (4), ^we get ^the (111) block of the CG matrix. If _y > 1, ^we have to select _ao, a’, a" ^which

are different from the first set, yielding ^{a différent non-}

zero diagonal coefficient. Setting a

ao.. a’ = ab..

y

a§, ^{and let a,} a’, a" range ^over all allowed values,

we ,get ^{a second set} of CG coefficients which are

orthogonal ^{to the} first,set. ^The procedure ^is repeated

until all (111) blocks ofCG coefficients are determined.

The (66’ a") blocks of coefficients are determined by

matrix multiplication ^{from the U 111 block :}

where k + k’

k" are canonical wave vectors and

Here { çz 1 03A3} ^{is one} space group operation ^which

rotates the ( 111 ) block into the (uu’ a") ^{block so that}

3. Conclusions.

Recently ^much attention has been directed to the highest symmetry points ^{of the}

Brillouin zone for the A-15 structure. In particular,

Birman et al. [24] used the CG coefficients to calculate the matrix elements k.p of the effective Hamiltonian.

Ting-Kuo ^{Lee et al.} [26] introduced a three-dimen- sional k. p model for the electronic structures of.the A-15 compounds ^based upon ^a six-dimensional irreducible representation R4 ^{of the} space group 03

In order to determine energy band dispersion ^rela-

tions, they ^{used a} generalized k. p theory called the method of invariants by ^{Bir and Picus} [27]. ^{The same}

results can be obtained by calculating ^{the CG coef-}

ficients and determining KK"la’’ ^for R4 ^x R4.

According ^to Birman et al. [24] ^{we have}

where _al’’ is independent ^{of a, a’, and} ,K k" ^{are sym-} metry-adapted components of tensorial field quanti-

ties.

4. Description of tables.

Table 1 lists coordinates of the symmetry points. ^{Table Il} presents ^{the cano-} nical wave vectors, LWVSRs [28], and intersections.

TABLE 1

TABLE II

TABLE III

TABLE IV

We present in tables III-VI the stars of the wave vectors and the symmetry operations ^for calculating

the CG coefficients. Tables VII-XI and XII list CG coefficients for X x X and R x R of space group

Of (Pn3m). ^{In tables VII-XII we use the} symbols

which have the following meanings :

The symmetrized, ⁱⁿ square brackets, ^and antisym-

metrized _squares of the ,irreducible representations

can be seen explicitly ⁱⁿ the tables of CG coefficients.

TABLE V

TABLE VI

For all the cases in table VI we have :

TABLE VII

1190

To label the irreducible representations ^{of the}

group 0’ ^{we use Miller and Love} [291 ^{labels and} gene- rators of the corresponding representations, ^{and their}

numbers of the space group symmetry operators listed for cubic groups in table ^{1 on} p. 123 [29]. ^In

tables VIII, X, XI, ^{XII the entries not written} expli- citly ^{are zero.}

One of the authors (H. K.) ^was working ^under Project ^No. 1.5.6.04 coordinated by ^Inst. Exper.

Phys. ^{of Warsaw} University.

TABLE VIII

TABLE IX

TABLE X

TARLE XI

1192

TABLE XII

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