Comments on the paper : « Allowed symmetries of soft modes in cubic crystals » by H. W. Kunert and J. Zielinski

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Comments on the paper : “ Allowed symmetries of soft modes in cubic crystals ” by H. W. Kunert and J.

Zielinski

B.L. Davies

To cite this version:

B.L. Davies. Comments on the paper : “ Allowed symmetries of soft modes in cubic crys- tals ” by H. W. Kunert and J. Zielinski. Journal de Physique, 1984, 45 (5), pp.963-968.

�10.1051/jphys:01984004505096300�. �jpa-00209830�

Comments on the paper : ^« Allowed symmetries of soft modes in cubic

crystals » by H. W. Kunert and J. Zielinski

B. L. Davies

Department ^of Applied Mathematics and Computation, University College ^{of North} Wales, Bangor ^LL57 2UW, Wales, ^U.K.

(Reçu ^{le 28 dicembre} 1983, accepté ^{le 14} février 1983 )

Résumé.

Ces commentaires concernent la partie de l’article relative au critère de Landau des transitions de

phase du second ordre dans les cristaux. On discute la méthode proposée par Lyubarskii ^{et on donne un} exemple

détaillé.

Abstract.

These comments are restricted to that part of the paper which deals with the Landau criterion for second order phase transitions in crystals. The method proposed by Lyubarskii is discussed and a detailed worked

example ^is given.

Classification Physics ^Abstracts

64.60

1. Introduction.

In a paper published ^{in this Journal, Kunert and}

Zielinski [1] have used the Landau criterion for second order phase transitions in crystals. The Landau crite- rion requires the determination of the frequency ^{of the}

unit representation ^{of a} space group G in the totally symmetrized ^cube [A’,"]’ ^{of a} space group irreducible

representation (irrep) ^Ak,ll. Unfortunately, ^{an incorrect}

formula has been given in the literature for the appli-

cation of the Landau criterion (Eq. (4) ^{of Ref.} [2]).

The original version of [I], ^{which was submitted to}

this Journal, used this formula and it led to incorrect results [3]. ^{At the} request ^{of this} Journal, ^{these com-}

ments have been prepared ^{in order to} clarify ^this point.

2. Symmetrized powers of space group irreps.

A space group irrep ^is usually expressed ^{as an induced} irrep ^{A k,,, =} Fk," T ^{G of a small} (or allowed) irrep Fk,," of the little _group G k. The reduction of [A k,,,l I

into a direct sum of induced irreps ^{is, for} general k-vectors, ^{not an} easy problem. ^{It is} part ^{of the} complex general problem of the reduction of the symmetrized

nth _power of an induced representation (symmetrized according ^{to an} irrep ^{of the} symmetric group S,,) ^into

a direct sum of induced representations. ^This general problem ^{was solved} by ^Gard [4, 5] ^{and she} applied

it to be symmetrized cubes of space group irreps giving ^an explicit prescription for their reduction ’[6].

The method of Gard [4, 5] ^was programmed ^{for space}

groups by ^the present ^author [7, 8] ^{and was used to}

reduce the symmetrized ^nth powers of all irreps (having non-trivial little co-groups Gk) for all 230 _space groups, for n ⁼ 2, 3, ^4. Gard’s definitive work, using

the full power of little group theory, ^followed by ^more

than a decade the first extensive tabulation of reduc- tions of totally symmetrized ^{cubes of} space group

irreps given by ^Birman [9] for diamond and zinc blende using ^{a full} group approach. Further references to the literature _may be found in [7].

Using ^the expression for the character [X]’ (g) ^of

the symmetrized ^cube [T]’ ^{of a} representation ^{T of a}

group G given ^on page 75 of Lyubarskii [10], ^the

Landau criterion, that the unit representation ^should

not be contained in the symmetrized ^{cube, reads :}

Now equation (4) of reference [2] appears ^to have been obtained from the above equation by simply replacing ^the character ^{x of the} representation ^of

the whole group ^G by Xkl, the character of the small

representation ^{of the} subgroup Go, ^{and the summa-}

tion carried out only ^over G’. ^{This is} not correct. A correct procedure ^{is to} replace x in the above equation

by the character of the induced representation ^{of the}

whole _space group and for the summation ^to be carried

out over the whole _group.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01984004505096300

964

In the special ^{case when} Xkl is the character of a

representation of the whole space group (so ^that

Go ⁼ Go ^{= whole} space group), i.e. when k ⁼ 0, then equation (4) of reference [2] is correct, but for

general ^{k it is} not correct (see ^Gard [6]).

In [2] the authors summarize the criteria for active

representations in second order phase transitions for space groups ^as given by Lyubarskii [10]. ^But equa- tion (4) ^of [2], ^{for the} application of the Landau crite-

rion, ^{as an} equation ^{true for} any k, ^{is not} given by Lyubarskii [10]. However, ^on page 134 of Lyu-

barskii [10] ^a very brief statement is given ^{as to how to}

determine whether or not the Landau criterion is satisfied. Unfortunately, ^the example in section 38 of

Lyubarskii [10] only ^{treats the} special ^{case when}

k ⁼ 0 (see page 144 of Lyubarskii [10]). ^{As a result}

one may be misled into believing ^{that a} formula that is true for k ⁼ 0 is also true for k =-1= ^0. It is hoped ^that

the following ^worked example ^will clarify ^this point

and show how to follow the method proposed by Lyubarskii ^{in the} general ^{case when k =-1= 0.}

3. Example.

Consider the symmorphic space group G ⁼ 0’(Pm3m)

and the totally symmetrized ^cube [AM,P]3 ^{of the matrix} irreps A",," ⁼ rM,p i ^{G of G, where} (in the notation of reference [11])

Since G is symmorphic, all elements of G are of the form { R It} ^{where R E} Oh ^{and t E} T, ^the primitive

cubic translation _group. Define the star of M, *M, ^as

follows :

where

so that

and the notation of reference [12J is used for the point

group symmetry operations. The dimension d"-l’ of

I’,", from reference [12] ^is given by :

Define basis functions for

(using the summation convention) ^and

Define bases four ^>

For given M, Jl the functions

form a basis for A’,,". (N.B. ^{From equation (10),} dM:t.,JL ⁼ dM,JL).

In order to apply the Landau criterion ^to Am," by

the method proposed by Lyubarskii, ^{it is} necessary to find the r (i.e. ^{k =} 0) component representation, Om,,", ^{in the} totally symmetrized ^cube [Am,"]’. ^{This is}

done by computing the character of the representation

carried by ^{the set of} (un-ordered) triples of functions

such that

(see ^{page 134 of} Lyubarskii [10]). ^From equations (1) ^and (5), ^{it is} easily ^{seen that the} only ^{solutions to} equa- tion (14) ^{are :}

-

But since in equation (13) triples ^{of functions} which differ only in the order of their factors are identified, ^{and all}

the solutions in equations (15) ^are permutations ^{of any one solution, then} only ^{one solution can be} taken, e.g.

In this case dM,Jl ⁼ 1, ^{and so there is} only ^one triple :

Case ii ^{= 5 ± .}

In this case dM,p ⁼ _2, and there are 23 3 ⁼ 8 such triples :

Note that from equations (17) ^and (18) the dimension dim (QM,p) of 0 ",,0 is given by :

We recall that, in the notation of reference [12],

Consider the action of { C2z 10 1 ^on t/J 1 ^where

...

where

Since dM,Jl ⁼ 1 ⁼ dim (QM,Jl) ^{in this case, we have}

where cMe and yM,.t ^are the characters of QM,.t and rM,.t respectively.

Again, ^consider the action of { Cjl I U } on §1 :

966

Therefore,

and so

Proceeding in this way, and using the character table for M on page 635 of reference [11] ^for y",", ^{the cha-}

racters (om," _may be calculated for p = I +, 2 +, 3 ±, 4 +. ^{These are} displayed ^{in table I} together ^{with the}

character table for r (see page 634 of reference [11]).

Table I.

Notes

(i) ^In the first column i ⁼ 1, 2, 3, 4 ; m ⁼ x, y, z ; p ^{= a,} b, c, d, e, f (see ^Ref. [12]).

(ii) For p ^{= 1} ±, 2 ±, 3 ±, 4 ±, 5 ±,andforallReO,

(iii) ^{For k =} T, A r ,JL ⁼ IFF,".

From table I it is clear that

so that the unit representation, A’,", of G is ^{contained in} [A","]’ ^{and in} [A","]’. The results in equation (27)

may be found on page 504 of reference [7].

Case (ii) Jl ₌ ⁵ :t.

From equation (18), 0"," is 8-dimensional with basis :

where

From _page 635 of reference [11], the small matrix representations rM,Jl, J.1 ⁼ 5 + may be found. Using equa-

tion (8) the action of {R 0 } ^{E G on} Oii,i,, (i, ^{i’, i" =} 1, 2), may be determined. For example, ^consider

{ C 4z 10 } tfJl12’ for It ^{= 5 +.}

Now and

(see ^Ref [12]).

Equations (31) ^and (32) ⁱⁿ (30) imply

Now, from reference [ 11 ],

From equations (8) ^and (34a) ^{we have}

Similarly, using equations (8), (34b, c), ^{we have}

Equations (35a, b, c) ⁱⁿ equation (33) imply ^that

Proceeding ^{in this} way, for all 4/ii,i,, ⁱⁿ (28) and for all { RJ 0 } ^E G, the matrix representation ^OM,51 may be constructed. The characters WM,.t of 0 ",,u, . ^{= 5 ±, are} given ^{in table I from} which it is clear that

The decompositions ⁱⁿ equation (37) may be found on page 504 of reference [7] ^{and so} it is verified that A’,’ +

is contained in [A M,5 +]3.

From equations (27) ^and (37) ^{it is seen that} A M,l +, A M,4+, ^{A M, 5+} are the only irreps ^{of G =} Oh (Pm3m),

labelled by ^{k =} M, ^{that do not} satisfy the Landau criterion.

4. Conclusion.

It is hoped that the above worked example ^{for the} symmorphic space group 0’(Pm3m) ^clearly illustrates the method proposed by Lyubarskii ^on page 134 of reference [10]. The method applies equally ^{well to} asymmorphic space groups. This method is only ^one

of several methods available to reduce the symmetrized

powers of _space group irreps (see [7] and references cited therein). ^{It was} precisely ^to avoid such tedious hand calculations as illustrated above, ^and pitfalls ^for

the _unwary like equation (4) of reference [2], ^that systematic, computer-generated, tabulations using ^the

definitive method of Gard [4, 5], ^were published [7, 8].

When a correct method is followed, ^as above, then

complete agreement ^with these tabulations is obtained.

968

Acknowledgments.

The author acknowledges receipt ^{of the} original version of [1] ^{from Dr. Kunert which led to the} preparation

of the above comments.

References

[1] KUNERT, ^{H. W. and} ZIELINSKI, J., ^J. Physique ⁴⁵ (3) (1984) 557.

[2] KUNERT, ^{H. and} SUFFCZY0143SKI, M., Physica ^114A (1982) 572.

[3] ^{KUNERT, H. W., Private} communication.

[4] GARD, P., ^J. Phys. ^{A 6} (1973) ^1807.

[5] BACKHOUSE, ^{N. B. and} GARD, P., ^J. Phys. ^{A 7} (1974)

1239.

[6] ^GARD, P., ^J. Phys. ^{A 6} (1973) ^1829.

[7] DAVIES, B. L. and CRACKNELL, ^A. P., ^Kronecker Product Tables, ^Volume 4, Symmetrized powers of irreducible representations of space groups

(IFI/Plenum, ^New York) ^1980.

[8] ^{DAVIES, B. L. and} CRACKNELL, ^A. P., Symmetrized fourth powers of irreducible representations of

space groups, British Library Supplementary ^Publi-

cations Scheme, Ref. No. : SUP 90047 (British Library, Lending Division, ^Boston Spa, Wetherby,

Yorkshire LS23 7BQ, England).

[9] BIRMAN, ^{J. L.,} Phys. ^{Rev. 127} (1962) ^1093.

[10] LYUBARSKII, ^G. Ya., ^The applications of group theory

in Physics (Pergamon, Oxford) ^1960.

[11] CRACKNELL, ^A. P., DAVIES, ^B. L., MILLER, ^{S. C. and} LOVE, ^W. F., Kronecker Product Tables, ^Volume 1, General introduction and tables of irreducible

representations of space groups (IFI/Plenum,

New York) ^1979.

[12] BRADLEY, ^{C. J. and} CRACKNELL, ^A. P., The mathema-

tical theory of symmetry ^{in solids} (Oxford ^Uni-

versity Press) ^1972.