UDC

PHYSICS

Academician S. V. VONSOVSKII, V. I. CHEREPANOV,
A. N. MEN’, A. E. NIKIFOROV

GROUP-THEORETICAL CLASSIFICATION OF THE STATES OF A PAIR OF EXCHANGE-COUPLED IONS IN A CRYSTAL

In connection with the creation of optical quantum generators on the lines of pairs of exchange-coupled impurity ions in crystals ((^{1})), a group-theoretical classification of the states of the pair is of interest.

The problem of determining the possible states of a pair of interacting impurity ions in a crystal is, in principle, analogous to the problem of determining the state of a diatomic molecule ((^{2})); however, it is necessary to take into account the influence of the symmetry of the crystalline field. The symmetry group of the pair of ions A and B in the crystal (G_k) is defined as the intersection of the point group (G_C), corresponding to the midpoint (C) of the segment (AB^*), with the group (D_{\infty h}) (or (C_{\infty v}), for unlike impurity ions), whose axial axis coincides with (AB), i.e. (G_k = G_C \cap D_{\infty h}(C_{\infty v})). Since (D_{\infty h} = C_{\infty v} + I_C C_{\infty v}) (where (I_C) is inversion at the point (C)), one may write:

[
G_k = G_k^J \quad \text{(for unlike ions A and B);}
\tag{1a}
]

[
G_k = G_k^J + I_0 G_k^J \quad \text{(for like ions A and B),}
\tag{1b}
]

where (G_k^J = G_C \cap C_{\infty v}), and (I_0) is any one of the elements of the intersection (C_C \cap I_C C_{\infty v}). Equality (1) is the decomposition of the group (G_k) into cosets with respect to the invariant subgroup (G_k^J). The group (G_k^J) contains symmetry elements that do not interchange the ions A and B, while (I_0G_k^J), on the contrary, contains elements that interchange the ions A and B. On the other hand, it is easy to show that the subgroup (G_k^J) coincides with the intersection of the groups (G_A \times G_B \cap C_{\infty v}), containing only such pairs of elements (g_A) (from the group (G_A)) and (\tilde g_B) (from the group (G_B)) that transform all points of space in the same way.**

As for a molecule, the classification of states may be carried out in various ways. We shall consider the simplest one, in which one starts from the states of a pair of noninteracting ions in the crystalline field.

In contrast to a diatomic molecule, it is necessary to consider not two, but three different cases: 1) different ions, 2) identical ions in nonequivalent positions (identical nonequivalent ions), 3) identical ions in equivalent positions (identical equivalent ions).

Let us first consider the first two cases, which admit a common treatment, since in both cases (G_k = G_k^J). Let the states of the individual ions be determined by the quantum numbers (\chi_1 \Gamma_1 \mu_1) and (\chi_2 \Gamma_2 \mu_2), where (\Gamma_1\mu_1) and (\Gamma_2\mu_2) are the index and row of irreducible representations*** of the groups (G_A) and (G_B), respectively, and (\chi_1) and (\chi_2) are additional quantum numbers.

Writing the wave functions of the pair of noninteracting ions in the form of antisymmetrized (with respect to permutations of electrons between the ions) products of the wave functions of the individual ions and considering them as a basis of a representation of the group (G_k), it is easy to obtain the formula for the characters of this representation

[
X(g_A \tilde g_B) = X^{(\Gamma_1)}(g) X^{(\Gamma_2)}(\tilde g).
\tag{2}
]

* The point group (G_C) of an ideal crystal without the impurity is meant. Analogously, (G_k) is determined by Kaplyanskii and Przhevuskii (private communication).

** Such elements can only be rotations about the common symmetry axis (AB) or reflections in symmetry planes containing this axis.

*** We note that (\Gamma_i) is a double-valued representation of the double group if (N_i) (the number of electrons of the (i)-th ion) is an odd number.

Using these characters, one can decompose the given representation into irreducible representations of the group (G_k), which determines the possible terms arising in the interaction of the ions of the pair.

In the third case it is necessary to distinguish two possibilities: a) the ions of the pair are in different states ((\chi_1\Gamma_1 \ne \chi_2\Gamma_2)), b) the ions of the pair are in the same state ((\chi_1\Gamma_1 = \chi_2\Gamma_2 = \chi\Gamma)). In case a), in constructing the wave functions of the pair it is necessary to take into account degeneracy with respect to the exchange of states

Table 1

[
X(g_A\tilde g_B)=X^{(\Gamma_1)}(g)X^{(\Gamma_2)}(\tilde g)+X^{(\Gamma_2)}(g)X^{(\Gamma_1)}(\tilde g),
\qquad
X(I_0g_A\tilde g_B)=0;
\tag{3a}
]

[
X(g_A\tilde g_B)=X^{(\Gamma)}(g)X^{(\Gamma)}(\tilde g),
\qquad
X(I_0g_A\tilde g_B)=(-1)^N X^{(\Gamma)}(g\tilde g),
\tag{3b}
]

where (N) is the number of electrons in each of the ions. In the particular case when (G_A), (G_B), and (G_C) are rotation groups, the results of Wigner’s work ({}^{2}) are obtained.

As an example, let us determine the terms of pairs of identical equivalent (\mathrm{Cr}^{3+}) ions in ruby ((G_A=G_B=C_3)), when each ion of the pair is in one of the two states ({}^{4}T_2) ((\overline{2A}) or (\overline{E})) of the ground doublet, or when one of the ions of the pair is excited into one of the two states ({}^{2}T_3) ((\overline{E}^{}) or (\overline{2A}^{})) of the lowest excited doublet.*

The decomposition (1b) for pairs of orders from the first to the fifth** has the form
I. (D_3=C_3+U_2C_3). II. (C_i=C_1+I_CC_1). III. (C_2=C_1+C_2C_1). IV. (C_1=C_1). V. (S_6+I_CC_3). Table 1 gives the terms of these pairs, determined using formulas (3a) and (3b). The last column of the table gives the approximate classification of the terms.

Institute of Metal Physics
Academy of Sciences of the USSR

Ural State University
named after A. M. Gorky

Institute of Metallurgy, Sverdlovsk

Received
8 VII 1966

CITED LITERATURE

({}^{1}) A. L. Schawlow, J. E. Devlin, Phys. Rev. Lett., 6, No. 3, 96 (1961); I. Wieder, L. R. Sarles, Phys. Rev. Lett., 6, No. 3, 95 (1961). ({}^{2}) E. Wigner, E. E. Witmer, Zs. Phys., 51, 859 (1928).

* By an asterisk we distinguish excited states from ground states, since we do not write out the additional quantum numbers.

** The arrangement of pairs is meant in order of increasing distance between the ions of the pair.