UDC 530.145.61

PHYSICS

B. K. NOVOSADOV, L. K. SAULEVICH, D. T. SVIRIDOV, Yu. F. SMIRNOV

ON THE DECOMPOSITION OF DIRECT PRODUCTS OF IRREDUCIBLE REPRESENTATIONS OF SPACE GROUPS

(Presented by Academician A. V. Shubnikov, 1 IV 1968)

In recent years, in solid-state physics, in interpreting experimental material, selection rules that take into account the full symmetry of the crystal, i.e., its space group, have been used more and more widely. Comparison of these selection rules with experimental data on absorption coefficients makes it possible to draw definite conclusions about the structure of the electronic, excitonic, and phonon spectra of a crystal. In addition to the study of direct and indirect optical transitions, these selection rules can be used in the analysis of processes of scattering of current carriers by thermal vibrations, spin waves, combination scattering of light, the Jahn—Teller effect in crystals, etc. The determination of these selection rules reduces to the decomposition of direct products of irreducible representations of the space group of the crystal into irreducible components. Numerous works have been devoted to the consideration of this question and to the physical applications of selection rules.

Lax and Hopfield (¹) considered irreducible representations of the space group \(O_h^7\) for each branch of phonon vibrations of crystals with the diamond structure. Selection rules for transitions in crystals with the zinc-blende structure (\(T_d^2\)) were found by Birman (²). G. F. Koster (³, ⁴) obtained selection rules for transitions between different irreducible sets of states and for intervalley transitions in crystals with the zinc-blende structure, taking into account spin-orbit interaction and the time-reversal operation. Birman (⁵) compared two methods of obtaining selection rules: the method using the symmetry elements of the entire space group (the full-group method), and the method using the symmetry elements that constitute a subgroup of the space group (the subgroup method).

However, in all previous works only selection rules were found, while the actual decomposition of the direct product of representations into irreducible representations, i.e., the construction from the basis functions of the direct product of such combinations that belong to definite irreducible representations, was not carried out. Meanwhile, for example, in describing exciton states in dielectric crystals it is necessary to be able to construct, from the wave functions of the particle and the hole, such combinations as correspond to the irreducible representation of the space group characterizing the exciton state. Thus, it is necessary to be able to calculate Clebsch—Gordan (C.G.) coefficients for space groups. In the work of Overhauser (⁶), devoted to the theory of excitons in alkali-halide crystals, the simplest case of these coefficients was considered for the product of two representations with quasi-momenta \(k\) and \(k'\) equal to zero, when the indicated C.G. coefficients coincide with the corresponding quantities for the point group \(O_h\). In general form, the problem of calculating C.G. coefficients for space groups has not been considered. Therefore, in the present work we shall briefly describe one possible method for calculating them.

A space group includes the operations of the translation subgroup, the operations of the point group, products of point-group operations by translations, and operations containing translations that do not belong to the translation subgroup. We shall consider symmorphic groups, which have no nontrivial screw axes or glide planes. However, the results obtained are not difficult to generalize to the case of nonsymmorphic groups, using, for example, the apparatus of loaded representations (⁷).

We shall denote the basis functions of the irreducible representation \(D^{kA}\) of the full space group (\(k\) is the vector defining the star of the representation, \(A\) is the symbol of the irreducible representation of the point group \(G_k\) of the vector \(k\)) by the symbol

\[ |k_i A \alpha\rangle; \tag{1} \]

here the index \(i\) runs over all rays of the star \(k\); \(\alpha\) enumerates the rows of the representation \(A\).

It is important to note that all basis functions corresponding to different rays of the star \(k_i\) are obtained from the function of one ray \(k\) by means of a fixed set of operations \(\sigma_i\):

\[ G(\sigma_i)|k A \alpha\rangle=|k_i A \alpha\rangle. \tag{2} \]

In the decomposition of the direct product of two representations \(D^{kA}\) and \(D^{k'B}\) into irreducible components

\[ D^{kA}\times D^{k'B}=\sum_{k''C} N_{k''C}D^{k''C} \tag{3} \]

the integers \(N_{k''C}\) are given by the formula

\[ N_{k''C}=\frac{v_3}{n_{k''}} \sum_{g\in G_{k_{i0}}\,G_{k'_{j0}}\,G_{k''}} \chi_C^*(g)\, \chi_A(\sigma_{i0}^{-1}g\sigma_{i0})\, \chi_B(\sigma_{j0}^{-1}g\sigma_{j0}), \tag{4} \]

where \(\sigma_{j0}\) is the operation which carries the vector \(k'\) into \(k'_{j0}\) in accordance with condition (3) \((\sigma_{j0}k'=k'_{j0}+B_s;\ B_s\) is a reciprocal-lattice vector satisfying the relation \(k_{i0}+k'_{j0}=k''+B_q)\); \(v_3\) is the number of stars \(k''\) in the product of the stars \(k\) and \(k'\), i.e., the number of combinations of rays \(k_i\) and \(k'_j\) that give, in the sum, the vector \(k''\), \(k_i+k'_j=k''+B_r\) *; \(n_{k''}\) is the order of the group \(G_{k''}\). The summation of products of ordinary characters of the point groups \(\chi\) must be carried out over the intersection of all three little groups \(G_{k_{i0}}, G_{k'_{j0}}\), and \(G_{k''}\).

The Clebsch–Gordan coefficients of the space group can be determined with the aid of the projection operator \(P^C_{\gamma'\gamma}\) for the point group \(G_{k''}\), which carries the function \(|k_i A\alpha'\rangle |k'_j B\beta'\rangle\) into the function \(|k''_s C\gamma'\rangle\)

\[ \langle k_i A\alpha',\, k'_j B\beta' \mid k''_s C\gamma'\rangle = \frac{ \langle k_i A\alpha',\, k'_j B\beta' \mid \sigma_s P^C_{\gamma'\gamma} \mid k_{i0}A\alpha,\, k'_{j0}B\beta\rangle }{ \langle k_{i0}A\alpha,\, k'_{j0}B\beta \mid P^C_{\gamma\gamma} \mid k_{i0}A\alpha,\, k'_{j0}B\beta\rangle^{1/2} }. \tag{5} \]

Here \(\sigma_s\) carries the vector \(k''\) into \(k''_s\) \((\sigma_s k''=k''_s+B_q)\) in accordance with condition (3),

\[ P^C_{\gamma'\gamma}=\frac{n_C}{n_{k''}}\sum_h D^C_{\gamma'\gamma}(h)\,\hat G(h), \]

\(n_C\) is the dimension of the representation \(C\); the quantities \(\alpha,\beta,k_{i0}\), and \(k'_{j0}\) in the numerator and denominator of (5) are fixed arbitrarily, but under the condition that

\[ k_{i0}+k'_{j0}=k''+B_r . \]

The numerator of (5) is computed by the formula

\[ \frac{n_C}{n_{k''}}\sum_h \langle k_i A\alpha'|\sigma_s h|k_{i0}A\alpha\rangle \langle k'_j B\beta'|\sigma_s h|k'_{j0}B\beta\rangle \langle k''_s C\gamma'|\sigma_s h|k''C\gamma\rangle = \]

\[ = \frac{n_C}{n_{k''}}\sum_h D^A_{\alpha'\alpha}(\sigma_i^{-1}\sigma_s h\sigma_{i0})\, D^B_{\beta'\beta}(\sigma_j^{-1}\sigma_s h\sigma_{j0})\, D^C_{\gamma'\gamma}(h), \tag{6} \]

* \(k_{i0}, k'_{j0}\) are one such combination.

where \(D^{A}_{\alpha'\alpha}(h')\) are elements of the matrices of irreducible representations of point groups, given, for example, in (8). The operations \(\sigma_i\) and \(\sigma_j\) are chosen from condition (3).

Although formally in (6) the summation is over all operations \(h\) of the small group \(G_{k''}\), nevertheless, in fact, only those operations which simultaneously satisfy the three conditions make a nonzero contribution:

\[ \sigma_s h k_{i0}=k_i+B_v, \]

\[ \sigma_s h k'_{j0}=k'_j+B_w, \]

\[ \sigma_s h k''=k''_s+B_q. \]

Table 1

Clebsch–Gordan coefficients of the group \(O_h^1\)
\(\langle k_i A\alpha,\ k'_j B\beta \mid k''_s C\gamma\rangle\) in the case \(kA\alpha=M_1,\)
\(k'B\beta=M_2,\ k''=\Gamma\) (\(a\)—edge of the cube)

For example, if at least one of the vectors \(k_i,\ k'_j\), or \(k''_s\) corresponds to a general point of the Brillouin zone, then in (6) only the identity operation \(h\) should be taken into account.

The denominator in (5) is calculated in an analogous way. If \(N_{h''C}>1\), then the calculation by formula (5) should be carried out with several different choices of the quantities \(k_{i0},\ k'_{j0},\ \alpha,\ \beta\), and the required number of independent sets of C.G.C. obtained in this way should be orthonormalized so that they possess the necessary orthonormality properties

\[ \sum_{k''_s C\gamma'} \langle k_i A\alpha,\ k'_j B\beta \mid k''_s C\gamma'\rangle \langle k_{i0}A'\alpha',\ k'_{j0}B'\beta' \mid k''_s C\gamma'\rangle = \delta^{\,k_{i0}A'\alpha' \, k'_{j0}B'\beta'}_{\,k_i A\alpha \, k'_j B\beta}, \]

\[ \sum_{\substack{k_i A\alpha\\ k'_j B\beta}} \langle k_i A\alpha,\ k'_j B\beta \mid k''_{s0} C\gamma\rangle \langle k_i A\alpha,\ k'_j B\beta \mid k''_s C'\gamma'\rangle = \delta^{\,k''_s C'\gamma'}_{\,k''_{s0} C\gamma}. \tag{7} \]

Using formula (5), we have calculated the C.G.C. for the cubic groups \(O_h^1\) and \(O_h^5\). As an example, Table 1 gives the C.G.C. for the symmetry group of the simple cubic lattice \(O_h^1\) for the case \(A=M_1,\ B=M_2\), and \(C=\Gamma\) (in the notation of (9)).

Institute of Crystallography
Academy of Sciences of the USSR

Moscow State University
named after M. V. Lomonosov

Received
20 III 1968

CITED LITERATURE

1. M. Lax, J. Hopfield, Phys. Rev., 124, 115 (1961).
2. J. L. Birman, Phys. Rev., 127, 1093 (1962).
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4. G. F. Karavaev, Izv. vyssh. uchebn. zaved., Fizika, No. 3, 64 (1966).
5. J. L. Birman, Phys. Rev., 150, 771 (1966).
6. A. W. Overhauser, Phys. Rev., 101, 1702 (1956).
7. G. Ya. Lyubarskii, The Theory of Groups and Its Applications in Physics, Moscow, 1958.
8. O. V. Kovalev, Irreducible Representations of Space Groups, Kiev, 1961.
9. L. P. Bouckaert, R. Smoluchowski, E. Wigner, Phys. Rev., 50, 58 (1936).