[2, 3, 4, 5] * C s (a m k (symmetry operation E m[ 1(a ] σ m σ (symmetry element E σ {E, σ} C s 32 ( ( =, 2 =, (3 0 1 v = x 1 1 +

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1 C s C 3v k k [1] H ghg 1 = H for all g. (1 H /H *1 *1 N 1

2 [2, 3, 4, 5] * C s (a m k (symmetry operation E m[ 1(a ] σ m σ (symmetry element E σ {E, σ} C s 32 ( ( =, 2 =, (3 0 1 v = x x 2 2, (4. 1, 2 1, 2 2 k m { mẍ1 = 2kx 1 + kx 2, mẍ 2 = 2kx 2 + kx 1, 1 = 1 ( 1, = 1 ( v (5, (6 v = x x 2 1, (7 ( x1 = 1 ( x1 x 2 x 2 2 x 1 + x 2, (8 x 1 x 2 (normal coordinate 1, 2 (5 { x1 = ω 2 1 x 1 x2 = ω 2 2 x 2 (9 1 C s 3k ω 1 = m, ω k 2 = m 1 2 (9 { x1 (t = c 1 cos (ω 1 t + θ 1 (10 x 2 (t = c 2 cos (ω 2 t + θ 2 x 1, x 2 2 ( x1 v =, (2 (total representation x 2 * ω ( cos (ω 1 t + θ 1 = 0 ( x1 = c ( 2 cos θ 2 1, ( x 2 2

3 2 2(b (ferroic 1 2(c (antiferroic 3 2 ( = { 1, 2,, g }, ( i, j i j (, (group. (i ( i j k = i ( j k for i, j, k (ii E, E i = i E for i (iii i, 1 i, 1 i i = i 1 i = E ========================== C 1h (group g g = 2 (order C s 32 [6, 7, 8] E σ E E σ σ σ E 1 C 1h 2 (a, (b 2, (c σ σ σ σ C s 2 E E = E, σ E = σ, E σ = σ, σ σ = E, E (E, σ = (E, σ, σ (E, σ = (σ, E, (12a (12b E σ multiplication table C s ========================== 1, 2 (rearrangement theorem C 1h σ σ C 1h σ 2 = E, g g N = E, n N g *3 σ (3 * 4 2 d = ( 1,, d, d i=1 i i = 1, i *3 *4 n L(n, C D L(n, C n D 1 D 3

4 i = d j j i, (13 j=1 d d D ji ( = j i, (14 ( ji ( =total ( 1 0 D total (E = 0 1, (15 total 3 * σ 1 = 2, σ 2 = 1, (16 D total (σ = ( , (17 g *6 U(UU = U U = 1 D S (SDS 1 = (SDS 1 1, S 2 = g 1 Σ D (D(, (Σ S D (S 2 D( = g 1 Σ D (D ( D( D( = g 1 Σ D ( D( = g 1 Σ D ( D( = S 2. *5 1 1 (faithful representation E D total (E, σ D total (σ 1 1 *6 D 1 (S 1 D (S = S 2 D 1 (S 1, S D total (σ ( 0 1 D total (σ = ( 1, ( }{{}}{{} R W R = ( 0 1 W = 1 0, ( R W {E, σ} E 1 = + 1, σ 1 = + 1, (E, σ 1 = (+1, +1 1, (20 4

5 1 (+1, +1 1 (totally symmetric * α i α, α i i = 1,, α i=1 i α i α = 1, (24 i α i α = j=1 j α j α i α, (25 4 R W 2 (E, σ 2 = (+1, 1 2, ( direction 2 σ [??(a] invariant subspace (space of irreducible representation 1 2 (symmetry-adapted basis??(b (irreducible representation irreps. (20 1 ( 1 E 1, 1 σ 1 = (+1, +1, (22 ( 2 E 2, 2 σ 2 = (+1, 1. (23 α ( = j α i α, (26 D ji i α i, j D 1 (E = 1, D 1 (σ = 1, (27 D 2 (E = 1, D 2 (σ = 1, ( E 1 D 1 (E = +1 D 1 (σ = +1 2 D 2 (E = +1 D 2 (σ = 1 t D total (E D total (σ ( ( = = C s (8 U = 1 ( σ, (29 *7 1. 5

6 ( ( x1 x1 = U, (30 x 2 x 2 D total (σ UD total (σ U 1 = ( , (31 V= 3 2 k x k x2 2 (30 V V V = UV U 1 D total (σ UD total (σ U 1 UD total (σ U 1 UD total (σ U total (reducible 1 1, 2 (direct sum (irreducible decomposition total = 1 + 2, (32 x 1 x x 1 x 2 U D D 1, D 2, 1 1 D 1, D 2, 1 1 *8 D UDU 1 = D D (33 D D 1, D 2, D = D 1 + D 2 + (34 * D UDU 1 D α ( χ α ( = Tr [D α (] = i=1 D ii α ( index (α 3 C 1h 1 (20, ( E 1 χ 1 (E = +1 χ 1 (σ = +1 2 χ 2 (E = +1 χ 2 (σ = 1 total χ total (E = 2 χ total (σ = C 1h 123 C s = (E, σ, (35 2 D 1 = (D 1 (E, D 1 (σ, D 2 = (D 2 (E, D 2 (σ, σ (36a (36b D α D β = gδ αβ (37 g = 2 α α β β = gδ αβ (38 ( χ 1 = (+1, +1, χ 2 = (+1, 1, χ α χ β = gδ αβ, (39 6

7 α = β χ α 2 = g, (40 (39 χ α ( χ β ( = gδ αβ (41 α = β χ α ( 2 = g OT (37, (39 (reat Orthogonality Theorem=OT 1 *9 (36a (36b D ij α = ( D ij α ( 1, D ij α ( 2,, D ij α ( g, (42 (37 i α α j D ij α D kl β = g δ αβ δ ik δ jl (43 k β l β = g δ αβ δ ik δ jl (44 i = j, k = l (41 (44 D ij 2 α = g, i, j i α j α 2 g =, ( d d d 2 N = g 2 C 1h = 2 i, j i j = j i, (Abel 1 2 *9 1 D α ( A D α ( A = λe 2 D α ( D β ( d β A D α ( A = AD β ( A = 0 (α = β =n n = α n α α, (46 n α α D ([ (14] D α ([ (26] D ( = α n α D α (, (47 n α 47 χ ( = α n α χ α (, (48 χ = α nχ α, (49 39 χ α χ = β n β χ α χ β = gn α, (50 n α = 1 g χ α χ (51 n α = 1 [χ α (] χ ( (52 g

8 3 (52 total = 1 2 {χ 1 (Eχ total (E + χ 1 (σχ total (σ} {χ 2 (Eχ total (E + χ 2 (σχ total (σ} 2 = 1 2 ( ( ( 1 2 = 1 + 2, (6 v α α 1 v v = β β β v, (53 v α 44 α α β β = gδ αβ, (1 = 1 α α, v α α v = β = β = β α α β β v α α β β β β v }{{} α α β β β v β } {{ } =gδ αβ = g α v α, P α = 1 α α, (54 g 1 (54 P 1 = 1 2 (χ 1 (E E + χ 1 (σ σ = 1 (E + σ 2 1 P 1 1 = 1 ( 1 2, 2 (6 1 a 1 + b 2 P 1 (a 1 + b 2 = a 2 (E + σ 1 + b (E + σ 2 2 = a 2 ( b ( = a b 2 ( 1 2, 1 α j α (j = 1,,, (55 v = β d β l=1 l β l β v, (56 i α P i α = g i α j α (57 8

9 g = g = g β = g i α j α v i α j d l α i α j α d β d β l=1 m=1 β d β β l=1 l β m β m β l β l β v d β l=1 m=1 l β v i α j α m β l β l β v m β (a (b } g {{} = δ αβ δ imδ jl = j α v i α. Pα i v = j α v α i C s C 3v 2.2 C 3v C 3v 5(a m k 5 (a C 3v (bc 3v 5(b C 3 2π/3 σ i i = 1, 2, 3 C 3v C 3v = { E, C 3, C3 1, σ } 1, σ 2, σ 3 (58 6 g 1 = E, g 2 = C 3, g 3 = C 1 3, g 4 = σ 1, g 5 = σ 2, g 6 = σ 3 g C 3v 7 σ 2 σ 1 = g 5 g 4 = g 3 = C 1 3 σ 1 σ 2 = g 4 g 5 = g 2 = C 3 6 C 3v 3 7 C 3v 9

10 σ 1 σ 1 C 3 = σ 2 = C3 1 σ 1 σ1 1 σ 1 C 3 σ1 1 = C3 1 g 1 g 2 3 g g g 1 g 1 = g 2 (59 g 1 g 2 C 3 C 1 3 σ 1 σ 2 C 3 σ2 1 = C3 1 σ 3 C 3 σ3 1 = C3 1 C 3 C 3 C 1 3 = C 3 C 1 3 C 3 C 3 = C 3 C 3 C3 1 C 3v σ 1, σ 2, σ 3 C 3v {E}, { C 3, C 1 3 }, {σ1, σ 2, σ 3 } 3 conjugacy classes a Cl(a = {b g, b = gag 1 } ( H H E proper subgroup C 3v H {E, σ 1 }, {E, σ 2 }, {E, σ 3 }, { E, C 3, C 1 3 ghg 1 = H for all g (61 H (invariant subgroup (normal subgroup C 3v H (62 N = { E, C 3, C 1 3 C 3 E σ 1, σ 2, σ 3 } } H = {h 1, h 2, } g gh = {gh 1, gh 2, } (63 g coset representative H (left coset (rearrangement theorem gh = {h 1 g, h 2 g, } (64 H (right coset C 3v {E, σ 1 } E {E, σ 1 } = {E, σ 1 } C 3 {E, σ 1 } = {C 3, σ 3 } C3 1 {E, σ 1 } = { C3 1, σ } 2 σ 1 {E, σ 1 } = {E, σ 1 } σ 2 {E, σ 1 } = { σ 2, C3 1 } σ 3 {E, σ 1 } = {σ 3, C 3 } {gh g } = { {E, σ 1 }, { σ 2, C3 1 }, {σ3, C 3 } } (65 = H + C 3 H + C 1 3 H = H + σ 2 H + σ 3 H H {Hg g } = { {E, σ 1 }, {σ 2, C 3 }, { σ 3, C3 1 }} ( N gn = N g gn hn gn hn = ghn N = ghn 10

11 N quotient group factor group /N C 3v C 3 = { E, C 3, C3 1 } C 3v = C 3 + C 3 σ 1 C 3 C 3 σ 1 4 C 3 C 3 σ 1 C 3 C 3 C 3 σ 1 C 3 σ 1 C 3 σ 1 C 3 C 3v C 3v /C 3 C 3v /C 3 = {C 3, C 3 σ 1 } (67 /N * 10 3 * [(general point] * 12 *10 K f : K g k = f(g K g 1, g 2 g 1 g 2 = g 3 f(g 1 f(g 2 = f(g 1 g 2 = f(g 3 f homomorphism f (one-to-one f isomorphism = K (68 K f (1 f Kerf (2 /Kerf = K fundamental theorem on homomorphisms Kerf = {g f(g = K }. *11 (Schöenflies 9 A *12 * 13 [11] (Wyckoff 8 (a C 4 [ 1 4 ] (special point 8 (a 8 (a 4 4 [ 8 (b] a 1 a 2 a 3 T n = n 1 a 1 + n 2 a 2 + n 3 a 3, (69 (n 1 n 2 n 3 (space lattice 8 (c (crystallographic orbits * 14 ( ( 14 7 (hodohedral point groups * 15 (triclinic, 1 1, (monoclinic, 2 1 (orthorhombic, 2 3 tetragonal 4 1 (trigonal 3 (hexagonal 6 1 cubic 3 3 *13 orbit r = (x, y, z r r = {gr g } *14 X g X X g 1 P g Xg Burnside s lemma Burnside 1897 Frobenius 1887 the lemma that is not Burnside s *

12 (a (c (b (72 1 χ 1 (α 3 χ 1 (α 1, 0, 1, 2, 3 5 2, 3, 4, 6, (32 (11 (21 (5 2 (4 (1 (1 { (11 (10 8 (a 4 (b (c 1 2, 3, 4, 6 2π/n C n ( n α χ 1 (α = cos α, (70 * 16 (69 n = (n 1, n 2, n 3 R α n *16 [?] l l l 2l + 1 m l Y lm (θ, φ α R α R αy lm (θ, φ = e imα Y lm (θ, φ α χ l (α = Σ l l= m eimα = sin[(l α] sin( 1 2 α, (71 l = χ 1 (α = sin( 3 2 α sin( 1 = cos α, (72 α (proper rotation (improper rotation 1 1 * * 18 [ ] C 1 [1] C 2 [2] C 3 [3] C 4 [4] C 6 [6] ( ( 1 2, 3, 4, D 2 [222] D 3 [32] D 4 [422] D 6 [622] 4 T [23] O[432] S 4 [ 4] i σ h σ v C 1 C i C s (C s C 2 C 2h (C 2h C 2v C 3 C 3i C 3h C 3v C 4 C 4h (C 4h C 4v C 6 C 6h (C 6h C 6v (73 * C 1 *18 n r n = 1 r 1 r r n 1 C n C n n 1 12

13 1 σ h σ v i C 2 σ h C 2h [2/m] 2 σ d 2 i σ h, σ v σ d D 2 D 2h (D 2h D 2d D 3 D 3d D 3h (D 3d D 4 D 4h (D 4h D 6 D 6h (D 6h (74 D 2 [222] D 2d [ 42m] T O i σ h, σ v σ d T T h (T h T d O O h (O h ( [7] ( C 3 { E, C 3, C 2 3} 1 ( n C n n ( (C n n = E C n χ [(C n n ] = 1 [χ (C n ] n = 1, χ (C n 1 n χ j (C n = exp (i 2πn j, (76 j = 0, 1, 2,, n 1,.?? (76 n 1 n 1 ( , C 3 C 3 E C 3 C ε ε 2 1 ε ε (77 ε = e 2πi/n 1 + ε + ε = i ψ/ t = Hψ H i ψ / ( t = Hψ t t ψ ψ ψ Wigner 0 H ( H H 0 - (Frobenius-Schur 1 g χ ( 2 = [ {χ (E} 2 + {χ (C 3 } 2 + { χ ( C 2 } 2 ] 3 3 = 1 ( 1 + ε 2 + ε 2 = 0, 3 (

14 3.3.2 D 3 [32] C 2a 1?? 3 C 3 2 C 2a C 2b C 2c 2 C 2a 120 C 3 C 2a C 1 3 = C 2c, C 2a,C 2b,C 2c C = {C 2a, C 2b, C 2c } C D 3 C 1 = C, D 3d [ 3m] D 3 C i D 3 C i D 3d = D 3 + id 3, D 3d D i +1 1 id 3 χ (id 3 = ±χ (D 3, ± i (gerade (ungerade (79 D 3 χ (80 C (class D 3 3 {E} { C 3, C 2 3} {C2a, C 2b, C 2c } C 3 C 2a D 3 D 3d 4 D 3d D 3 id 3 g χ χ u χ χ (81 D 3 E 2C 3 3C 2 1 (A (A (E (79 * 19 χ χ ( C 1 = χ ( 1 C = χ (C, (a (b * 20 (a (b [ (79 3 3] *19 1 2C 3 {C 3, C 2 3 } 3C 2 {C 2a, C 2b, C 2c } ( ( 1, 2, (A,B,E, A 1 B 1 E 2 T 3 X g X g * (c primitive translation T n non-primitive translation τ R R i R τ R ( R i τ Ri (R i τ Ri, (82 [i = 1,, N 1 (N R ] (rotational elements (R 0 τ R0 = E (R i τ Ri Koster-Seitz τ Ri r (R i τ Ri r = R i r + τ Ri, (83 * 21 *21 τ R 14

15 4.2 R i C n T τ Cn = m n T, m < n (screw * 22 R i σ T τ σ = 1 2 T, (glide T a b c n d (symmorphic 157 (nonsymmorphic 4.3 Koster-Seitz (E T n T = {(E T n n 1, n 2, n 3 Z} (84 E, (R 1 τ R1, (R 2 τ R2,, ( R N 1 τ RN 1, (85 = {(R i τ Ri + T n R i R, n 1, n 2, n 3 Z} = {(R i τ Ri T R i R}, (86 R (R i τ Ri T = T + (R 1 τ R1 T + + ( R N 1 τ RN 1 T, (87 N T (coset/t /T = { T, (R 1 τ R1 T,, ( R N 1 τ RN 1 T }, (88 * T T 1 * 23 a 1 a 2 a 3 N 1 N 2 N 3 (E a 1 N 1 = (E a 2 N 2 = (E a 3 N 3 = (E 0, (E a 1, (E a 2, (E a 3 0 N 1 N j 1 j 2 j 3 j1 j2 ( j3 j1 χ j1 = exp i 2π j 1, N 1 3 a 1 a 2 a 3 (j 1, j 2, j 3 j χ j = χ j1 χ j2 χ j3 [ = exp 2πi ( j1 N 1 + j 2 N 2 + j 3 N 3 ], (89 N 3 1 R 0 R n = n 1 a 1 +n 2 a 2 +n 3 a 3, (E a 1 n 1 (E a 2 n 2 (E a 3 n 3 χ j (R n = ( n1 ( n2 ( n3 χ j1 χj2 χj3 [ ( j1 = exp 2πi n 1 + j 2 n 2 + j ] 3 n 2, (90 N 1 N 2 N 3 90 k χ j (R n χ k (R n = exp (ik R n (91 *23 (N (4.60 d d d2 N = N d i 1 15

16 j j k k b µ = 2π a ν a λ, (92 V (µ,ν,λ = 1, 2, 3 V = a 1 (a 2 a 3 k = j 1 N 1 b 1 + j 2 N 2 b 2 + j 3 N 3 b 3, (93 91 k R n k R n k ψ k (r 1 * 24 ψ k (r = exp (ik r u k (r (94 * 25 u k (r = j 1 b 1 + j 2 b 2 + j 3 b 3, (95 k k + (94 R n 2π k k 4.5 k (R τ R exp (ik r (R τ R (R τ R exp (ik r = exp [irk (r τ R ], * *25 ψ k (r (E R n ψ k (r = χ k (R n ψ k (r (E R n ψ k (r = ψ k (r R n ψ k (r R n = exp (ik R n ψ k (r exp [ ik (r R n ] exp [ ik (r R n ] ψ k (r R n = exp ( ik r ψ k (r exp ( ik r ψ k (r u k (r exp ( ik r ψ k (r = u k (r k k Rk, (96 τ R k (R i τ Ri R R i 1 k k. =R i k, (97 * 26 g k = {R i k. =R i k} (98 k (point group of k * 27 g k k = {(R i τ Ri T R i g k } (99 k (little group of k * 28 k g k g k k k k k k k k k (k exp (ik r k 4.6 k k g k 32 D α (R i k k D k α (R i t = exp (ik t D α (R i (100 t = τ Ri + R n *26. = *27 little cogroup of k *28 group of k 16

17 k 100 g k T k =R i k, (101. = e irik t = e ik t k R i k k = e irik t = e i t e ik t Herring * 29 k 4.7 k k k D k α D k α, (102 k 4.8 = H + gh H 2 H H [12] (1 H d (h d (g 1 hg D (h = D (gh = ( d (h 0 0 d (g 1 hg ( 0 d (ghg d (h 0 (2 d (h d (g 1 hg (103 (104 d (g 1 hg = U 1 d (hu (105 * T c U d (h d (h ±Ud (h D 4 D 4 = C 4 + C 2x C 4 C 4 1 A χ 1 (h i = 1 for all h i C 4 χ 1 (C 1 2x h ic 2x = χ 1 (h i = 1 (2 U = 1 C 4 1 D 4 χ 1 (h i = 1 ±Uχ 1 (gh i = ±1 5 2 [9, 10] Fig.1 ( T c T > T c a ρ 0 (r ( 1 T = T c T < T c 2a (doubling ρ 0 (r ρ(r = ρ 0 (r + δρ(r, (

18 0 E g ϕ 1. α ϕ α. ( ρ(r ρ(r = c 1 ϕ 1 (r + c 2 ϕ 2 (r + + c α ϕ α (r +, }{{}}{{} (108 * 30 1 * 31 0 g ϕ 1 (g 1 r = ϕ 1 (r, (109 ϕ 1 (r ( 0 a 2a a 0 α [ 2 (active representation ] * 32 (108 0 ϕ 1 (r 0 * 33 ϕ α (r α 0 * * *32 *33 0 (111 (106 ρ 0 (r = c 1 ϕ 1 (r, (110 }{{} δρ(r δρ (r = c ϕ (r, (111 α α α }{{} > 1 3 {ϕ (1 α (r,, ϕ(dα (r} (112 α (111 δρ α (r = dα c (i ϕ(i α α i=1 (r (113 ϕ (i (r α M α (r = c (i ϕ(i α α i=1 (r (114 δm(r = 0 c (i α c α T c?? c > 0 α δρ(x?? c (i α [13] c (i α ϕ (i (r α c (i α k k k k k (113 k δρ k (r = i=1 c (i k ϕ (i k (r (115 18

19 M k (r = i=1 c (i k ϕ (i k (r (116 k k i 5.3 (113 Φ ρ(r Φ ρ(r Φ[ρ] 0 [13] r f[ρ(r] ρ(r f[ρ(r] Φ[ρ] = f[ρ(r]dr, (117 f[ρ(r] 0 (106 Φ 0 [ρ 0 ] = f[ρ 0 (r]dr, (118 (106 ρ 0 (r (117 Φ[ρ] = f[ρ 0 (r + δρ(r]dr, (119 δρ(r 2 Φ[ρ] = Φ 0 + Φ 2 = f[ρ 0 (r]dr + 1 dr dr δρ(rh(r, r δρ(r, (120 2 h(r, r [h(r, r = h(r, r] [ (111] δρ(r δρ(r 1 h( 1 r, 1 r = h(r, r, (122 dr h(r, r ϕ(r = aϕ(r, (123 ĥϕ = aϕ, (124 ĥ 0 Wigner 0 ϕ ϕ α (r (123 dr h(r, r ϕ (r = a ϕ α (r, (125 α α (120 2 Φ 2 = 1 c 2 2 dr dr ϕ (rh(r, r α ϕ (r α α = a c α α 2 dr ϕ α (r 2 = a c α 2, (126 α } {{ } =1 2 2 Φ 2 = a α i=1 c (i α 2 (127 (120 4 b c α 4 = 1 Φ = Φ 0 + a α c α b c α 4 (128 h(r, r = 1 2 δ 2 Φ δρ(rδρ(r ρ(r=ρ0 (r (121 b > 0 (L Φ 19

20 a α > 0 c α = 0 a α < 0 c α = a α /b a α = 0 2 L 128 c α 111 a α (123 2 a α b * 34 2 SU(2 [2],,, [3], [4], [5] MS Dresselhaus,. Dresselhaus,and A. Jorio roup Theory:Application to the Physics of Condensed Matter, (Springer2008. [6] M. J. Burger, Elementary Crystallography, (revised printing, MIT Press, [7] F Koster, J. O. Dimmock, R.. Wheeler, and H. Statz, Properties of the Thirty-Two Point roups (MIT Press, Cambridge, Mass., [8] O. V. Kovalev, Representations of the Crystallographic Space roups, Edition 2 (ordon and Breach Science Publishers, Switzerland, [9] J.-C. Toledano and P. Toledano: Landau Theory of Phase Transitions, (World Scientific Pub, [10], [11] International Tables for Crystallography, Volume A: Space roup Symmetry, Ed. T. Hahn, Springer; 5th rev. ed [12] 78-5, p [13] Y. Onodera and Y. Tanabe: Phys. Soc. Jpn. 45 (1978 pp [1] Higgs *34 10 (L 1 10 L 2 20

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