The transition probability is governed by the value of the D(1) D(2) E S4x S-1 C2x Uyz Uyz y z 2d 2d 4x matrix element D(3) D(4) E S4y S-1 C2y Uxz xz z x 2d 2d 4y k( f ), ( f ), m( f ), µ( f )|P(k(p), (p), m(p))|k(i), (i), m(i), µ(i).
D(5) D(6) E S4z S-1 C2z Uxy xy x y 2d 2d 4z (20) W1 W1 2 2 2 0 0 0 0 The transition is said to be allowed by symmetry if the triple W2 W2 2 - 2 - 2 0 0 0 0 direct (Kronecker) product W3 W5 1 - i 1 -i - (k( f ), ( f )) (k(p), (p)) (k(i), (i)) (21) W4 W6 1 - i -1 i - W5 W3 1 - i -1 i contains the identity irrep of G. This condition can be W6 W4 1 - i 1 -i rewritten in one of three following forms:
W7 W7 2 0 0 -2i 0 0 0 (k(p), (p)) (k(i), (i)) (k( f ), ( f )) = 0, (22) (k( f ), ( f )) (k(i), (i)) (k(p), (p)) = 0, (23) of little co-groups GW( j) = D( j). Since kW( j) = -kW( j) and 2d kW(2) -kW(1) we take the irreps of the little group GW(2) (k( f ), ( f )) (k(p), (p)) (k(i), (i)) = 0. (24) from Ref.  as the irreps of the little group GW (see (25) (1) Whatever the form of the selection rules, it is necessary and the remark thereunder). Besides as kW(2) -kW(1), to find the direct product of two (or three (21)) irreps of the total sets of single- and double-valued irreps of little the space group G (complex conjugate irreps are also irreps groups GW and GW are complex conjugate, but the (1) (2) of G).
elements of GW(2) = IGW(1)I-1 which are isomorphic to We discuss now the procedure of selection rules’ generthe elements of GW according to (6) may differ from (1) ation using projective irreps of point groups. To illustrate the coset representatives in decomposition (4) by some each step of this procedure we have chosen the small irreps lattice translations. This may change the numbering of the of the little group GW in the space group O7 given in h irreps of the little group GW (and projective irreps of the (2) Tables . Note that translations an are mapped in Ref.  corresponding little co-group FW(2)) with respect to those by the factor exp(ikan). According to general definition of GW(1) (of FW(1), see Table 1; W3(D(2)) =W5 (D(1)), for 2d 2d example).
ta (r) (r - an) =exp(-ikan) (r), (25) n Taking the form (22) of selection rules, we consider the Kronecker product of the irreps of the space group G we choose the translations an to be mapped by the factor exp(-ikan). This choice does not affect the notations of [G] (i) [G]( k(p)) [G]( k(i)) (26) (p) Ref.  for small irreps in the case when k is equivalent (p) (i) to -k or refers the small irreps [Gk] of Ref.  to the wave whose basis vectors are the products vector -k in other cases (when k and -k are different vectors of the same star or belong to different stars).
|k(p), (p), m(p) · |k(i), (i), m(i), n l (1) The star W consists of six vectors: W =(1, 0, 2), (2) (3) (4) W =(1, 2, 0), W =(2, 1, 0), W =(0, 1, 2), (n = 1,... s(p); l = 1,... s(i);
(5) (6) W =(0, 2, 1), and W =(2, 0, 1) (in units of /a along Cartesian axes with a being the lattice constant). The m(p) = 1,... t(p); m(i) = 1,... t(i)), (27) little group GW has two single-valued ([GW ], = 1, 2) ( j) ( j) where s(p), s(i) are the numbers of rays in the stars k(p), and five double-valued ([GW( j) ], = 3, 4, 5, 6, 7) small k(i); t(p), t(i) are the dimensions of small irreps [Gk(p)](p) irreps [9,12], which are unambiguously related (see n (i) (i) (p) (i) Section 2) to the corresponding projective irreps [FW( j)]j and [Gk ] of little groups Gk and Gk, respectively.
n l l In the case of the W point in the BZ for the of little co-groups FW = D( j) (see Talbe 1). As the ( j) 2d h characters (and matrices) of the elements (R|vR) GW space group O7, the basis (27) of the Kronecker ( j) product (26) for (p) = (i) = 1 (k(p) = k(i) = W ;
and R FW( j) are the same, we use the notations W [GW( j)] ( = 1-7) of small irreps of the little s(p) = s(i) = 6, [G]11 [G]( W(p)) [G]( W(i))) consists of 1 groups GW( j) also for the corresponding projective irreps (2 6) (2 6) =144 vectors.
Физика твердого тела, 2003, том 45, вып. A point group approach to selection rules in crystals Table 2. Types of wave vectors in Kronecker products The set of wave vectors k( f ) (28) contains all the rays n,l of [G]( W ) [G](j W ) of all the irreducible stars appearing in the Kronecker i product (26) and may be arranged in a table similar to Table 2. Rows and columns of this table are numbered W(1i) W(2i) W(3i) W(4i) W(5i) W(6i) by the rays of the irreducible stars k(p) and k(i) (by n ( f ) (2 f ) (1 f ) (6 f ) (7 f ) W(1p) X(x f ) and l), respectively. The representatives of all the irreducible ( f ) (3 f ) (4 f ) (5 f ) (8 f ) W(2p) X(x f ) stars in (26) appear in any row of this table. Indeed, (2 f ) (3 f ) ( f ) (10 f ) (9 f ) W(3p) X(y f ) all the rows of the table (n = 2,... s(p) in (28)) may (1 f ) (4 f ) ( f ) (11 f ) (12 f ) W(4p) X(y f ) be obtained from the first one (n = 1) by applying the (6 f ) (5 f ) (10 f ) (11 f ) W(5p) X(z f ) ( f ) (7 f ) (8 f ) (9 f ) (12 f ) ( f ) operations R(p) transforming the wave vector k(p) into k(p) n n W(6p) X(z f ) (n = 2,... s(p)). Under symmetry operations R(p), the set n R e m a r k. The wave vector stars XX and consist of vectors: X( j f ) of wave vectors k(i) of the star k(i) remains unchanged ( j f ) l (i = x, y, z ; (2,0,0), (0,2,0), (0,0,2)) and ( j = 1-12; (110), (110), and the irreducible stars formed by wave vectors k( f ) may (110), (110), (101), (110), (101), (101), (011), (011), (011), (011)), 1,l respectively (in the units of /a along Cartesian axes, a being the lattice change their representatives but can neither disappear nor constant).
give raise to new irreducible stars or change the number of each star representatives. The same consideration is Table 3. Types of wave vectors in Kronecker products valid for the columns. Finally, all the rows (columns) of of (k1) (k2) (k1, k2 =, X, L, W) the whole table contain as many representatives of each irreducible star as the first row (column). Therefore, all X(xi) X(yi) X(zi) L(1i) L(2i) L(3i) L(4i) the necessary information about wave vector selection rules ( f ) X(xp) X(z f ) X(y f ) L(3 f ) L(4 f ) L(1 f ) L(2 f ) for the Kronecker product (26) is contained in any row ( f ) L(1p) L(3 f ) L(4 f ) L(2 f ) X(z f ) X(x f ) X(y f ) (column) of the corresponding table. The wave vector (2 f ) (1 f ) (12 f ) (11 f ) (10 f ) (9 f ) W(1p) W(2 f ) selection rules for all the symmetry points of BZ of the space group O7 are represented in Table 3. The latter W(1i) W(2i) W(3i) W(4i) W(5i) W(6i) h is composed of the first rows of Tables being similar to (3 f ) (4 f ) (6 f ) (5 f ) X(xp) W(2 f ) W(1 f ) Table 2 and corresponding to Kronecker products, (12 f ) (10 f ) (8 f ) (6 f ) (4 f ) (2 f ) L(1p) X X, L L, W W, X, L, W, ( f ) (2 f ) (1 f ) (6 f ) (7 f ) W(1p) X(x f ) X L, X W, L W.
At the next step of the selection rules’ generation one R e m a r k. The wave vector stars L and consist of vectors: L( j f ) ( j f ) needs to find the irreducible components of the reducible ( j = 1-4; (1,1,1), (1, 1, 1), (1, 1, 1), (1, 1, 1)) and ( j = 1-6;
(1,0,0), (1, 0, 0), (0,1,0), (0, 1, 0), (0,0,1), (0, 0, 1)) (in the units of /a reps for each star satisfying wave vector selection rules.
along Cartesian axes, a being the lattice constant).
Let k( f ) k( f ) be a wave vector of some irreducible m n,l star (m is fixed). The set of t(p)t(i) basis functions (27) with this wave vector forms the space 1 of some Decomposing the reducible rep [G]11 of G, one finds all projective rep [Fk ]Kr [Fk Fk ]Kr of the little co-group ( f ) (p) (i) the irreducible stars k( f ) contained in the reducible star m n l of [G]11 and the small irreps [Gk ]j of little groups Gk Fk Fk Fk or small rep [Gk ]Kr [Gk Gk ]Kr ( f ) ( f ) ( f ) (p) (i) ( f ) (p) (i) m n m n l l contained in the rep [G]11. A star of s(p)s(i) wave vectors of the little group Gk( f ) Gk(p) Gk(i) and m n l k( f ) = k(p) + k(i) + Bn,l n,l n l [Fk ]Kr =[Gk ]Kr Fk. (29) ( f ) ( f ) ( f ) m m m (n = 1,... s(p), l = 1,... s(i))(28) The characters of the projective rep [Fk( f )]Kr of the coof basis functions (27) splits into irreducible stars and gives m group Fk( f ) in the space 1 are the products of the characters wave vector selection rules (the vector Bn,l is a reciprocal m lattice vector which may be zero). of the co-group Fk(p) and Fk(i) irreps subduced on the con l For Kronecker products [G]( W(p)) [G]( W(i)) of two group Fk ( f ) 1 m irreps of O7 at the W point of BZ, the wave vector h selection rules (28) give three irreducible stars (, X and, ([F ] ) ([F ]Kr) ([F ] ) (i) ( f ) (p) km kn (p) kl (i) (R) = (R) (R), see Table 2). It is easy to see, rewriting the vectors of W -star in components of the reciprocal lattice primitive R Fk ( f ). (30) (1) m translations. For example, as W =(1/2, 3/4, 1/4) and (2) (1) (2) W =(1/2, 1/4, 3/4), one obtains W + W =(1, 1, 1) The multiplication of two projective irreps of the group (1) (1) (p) (i) ( ), W + W =(1, 3/2, 1/2) ( X(x)). Thus, n l Fk with factor systems (k ) and (k ) gives a projective ( f ) m products (27) are partitioned in such a way that ( f ) m rep of the same group with the factor system (k ) (as 4 6 = 24 of them corresponds to, 4 6 = 24 to X and 4 4 6 = 96 to points of the BZ. k( f ) k( f ) = k(p) + k(i) + Bnl, see also Section 2).
m n nl l 3 Физика твердого тела, 2003, том 45, вып. 1378 V.P. Smirnov, R.A. Evarestov, P. Tronc The group Fk (Gk ) either coincides with the little co- Table 4. Characters of the Kronecker product ( f ) ( f ) m m [D(1)]1 [D(2)]1 and the characters of some ordinary group Fk( f ) (little group Gk( f )) or is a subgroup of it 2d 2d m m irreps of the point group D(1) 2d Fk Fk (Gk Gk ). (31) ( f ) ( f ) ( f ) ( f ) m m m m D(1) E S4x S-1 C2x Uyz Uyz y z 2d 4x When Fk( f ) = Fk( f ) then the projective rep (30) can be 4 2 2 0 0 0 0 m m decomposed into the irreps in the usual way, the characters a1 1 1 1 1 1 1 1 of irreps being taken from Tables of small irreps of little a2 1 1 1 1 -1 -1 -1 -groups (for example ). This possibility appears, for e 2 0 0 -2 0 0 0 instance, when k(p) = 0 or k(i) = 0.
n1 l If Fk( f ) Fk( f ), the co-group Fk( f ) is decomposed into left m m m cosets of Fk( f ) Table 5. Kronecker products [D(1)]i [D(2)]j of projective single2d 2d m and double-valued irreps of the little co-group D2d in terms of orw dinary irreps of the point group D2d and subduction of ordinary Fk( f ) = RiFk( f ), R1 = E, irreps of Oh point group onto the point group D(1) m m 2d i=W1 W2 W3 W4 W5 W6 W Ri Fk, Ri Fk for i = 2,..., w. (32) ( f ) / ( f ) m m W1 a1, a2, e b1, b2, e 1 2 1 2 1, The operators Ri change both wave vectors k(p) and k(i) W2 b1, b2, e a1, a2, e 2 1 2 1 1, n l but leave their sum unchanged modulo the reciprocal lattice W3 1 2 a2 b1 a1 b2 e vector. This means that the space 1 transforms under the W4 2 1 b1 a2 b2 a1 e operaitons Ri into linearly independent spaces i = Ri 1, W5 1 2 a1 b2 a2 b1 e and W6 2 1 b2 a1 b1 a2 e w W7 1, 2 1, 2 e e e e a1, a2, b1, b = i (33) + + + + + + + + i=[Oh]i( O7) h 1 2 3 4 5 6 7 being the space of the rep of the group Fk( f ) induced by the [Oh]i D2d a1 b2 a1, b2 a2, e b1, e 2 1 1, m rep [Fk( f )]Kr of its subgroup Fk( f ) Fk( f ) - - - - - - - m m m [Oh]i( O7) h 1 2 3 4 5 6 7 [Oh]i D2d b1 a2 a2, b1 b2, e a1, e 1 2 1, [Fk ]Kr =[Fk ]Kr Fk. (34) ( f ) ( f ) ( f ) m m m Further, the small rep [Gk ]Kr =[Fk ]Kr Gk is con( f ) ( f ) ( f ) m m m tained in the Kronecker product (26) which is the subject based on the Frobenius reciprocity theorem (see Section 2).
([F ]Kr) ( f ) km of our consideration. The characters (g) (g Fk( f )) Such possibility arises in the two following cases:
m a) when the rep [Fk( f )]Kr is irreducible itself, i. e. its of this projective induced rep of Fk (or induced small rep ( f ) m m characters satisfy the condition of Gk( f )) can be calculated in the usual way m ([F ]Kr) |([F])(g)|2 = nF (37) ( f ) km (g) = i(Kr)(g)(35) gF i where nF is the order of F (see Section 3.2);
where b) when the subduction of the irreps of the co-group Fk ( f ) m 0, if g-1ggi Fk( f );
/ onto the group Fk( f ) gives directly the irreps of Fk( f ).
i m m m i(Kr)(g) = (36) ( f ) km ([F ]Kr)(g-1ggi), if g-1ggi Fk( Besides, the irreps of Fk( f ) can be taken from Ref.  f ).
i i m m where the characters of the standard form for all the projective irreps with all the possible factor systems for all As the characters of projective irreps of Fk (of small irreps ( f ) m the crystallographic point groups are given.
of Gk( f )) are known (taken from Ref. , for example), m (W(i)) In our example of the [G]( W(p)) [G]1 Kronecker the projective rep of [Fk ]Kr can be decomposed on the ( f ) m product, the following intersections of point groups irreducible components in the same way as it is made for are considered: F FW FW (D(1) D(2) = D(1)) for (1) (2) ordinary reps of point groups.
2d 2d 2d If the projective irreps [Fk( f )]j with the same factor system -component, FX FW FW (D(1) D(1) = D(x)) for (x) (1) (1) 2d 2d 2d m as the projective irreps of the co-group Fk( f ) are known, X-component, and F FW FW (D(1) D(4) = Cs ) for (1) (4) 2d 2d m there is a more simple procedure of the rep decomposition -component (see Table 2 or 3).
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