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1 1 X 1 X n ( ) ( ) (Lorentz)

2 1 X X X X ( X ) X y = sinθ y = cos θ y = A sin(2πft + δ) = A sin(ωt + δ) A f ω = 2πf () (phase) δ T = 1/f y(t) = A sin(2πt/t + δ) (1.1) ( ) ( ) λ x y(x) = A sin(2πx/λ + ǫ) (1.2) ǫ (1.1) (1.2) (1.1) (1.2) E(t) = A sin(2πt/t + δ) E(x) = A sin(2πx/λ + ǫ) 1

3 (1.1) (1.2) ( ) T λ 1/T(=f) ( ) (wave number) k = 1/λ 1/T 2π/T 1/λ 2π k = 2π/λ 2π ω 2π 2π X 2π 2π k ω Table 1 2π T f = 1/T ω = 2π/T ( ) λ k = 1/λ k = 2π/λ k = 2π/λ ω = 2π/T A ( ) δ ǫ A = 1 δ = 0 ǫ = 0 2

4 (1.1) (1.2) ( ) ( ) x (1.1) x y(x, t) = sin(ωt kx) (1.3) (1.2) y(x, t) = sin(kx ωt) (1.4) (1.3) (ωt ) (kx ) x x = (x, y, z) kx x 2π/λ k (1.3) (kx ) k x y( x, t) = sin(ωt k x) (1.5) (1.5) k x sin cos 3

5 Euler( ) e iθ = cos θ + i sinθ i sin(ωt k x) e i(ωt k x) (Schrödinger ) ( ) ( cos ) (sin ) ( ) θ θ + δ sin(θ + δ) = sinθ cos δ + cos θ sinδ e i(θ+δ) = e iθ e iδ... k y k ( x, t) = e i(ωt k x) (1.6) 2 X X X X X 4

6 X X X X X ( ) ( ) X X k 1 X k 2 X k1 e - k2 1: k 1 = k 2 = 2π/λ ψ 0 = e iωt (2.1) 5

7 2.1 r 1 (2.1) k1 k2 r 1 K = k 2 - k 1 2: r 1 ( k 1 r 1 k 2 r 1 )/(2π/λ) ( ) K = k 2 k 1 ( ) ( K r 1 )/(2π/λ) K r 1 ( + K r 1 ) r 1 ψ 0 = e iωt ψ 1 ψ 1 = e i(ωt+ K r 1 ) = e iωt e i( K r 1 ) ψ 1 = ψ 0 e i K r 1 (2.2) r 1 e i K r n ( ) n n X k 1 X k 2 K (2.2) ( ) ( ) j r j (j = 1, 2, 3,...,n) Ψ = ψ 0 n j=0 e i K r j = ψ 0 [ ] (2.3) 6

8 M (2.3) ψ 0 ( ) ( ) f M = e i K r j = M (2.4) j=0 K f M ( K) f M K ( ) f M n ( ) φ n ( r) r ρ n ( r) = φ n( r)φ n ( r) r dxdydz = d r ( ) ρ n ( r)d r = φ n( r)φ n ( r)d r d r dv dxdydz d r r n f M f M = M e i K r n = M n ρ n ( r)e i K r d r 2 ( ) ( ) K ( sinθ/λ ) [1] r ( ) 3 f M ψ 0 f M ( K ) R R e ik R 2 7

9 e ik R ψ 0 f M 1 ( ψ 0 ) 1 R e ik R ( ) P ( ) R j (j = 1, 2, 3,...,P) ( ) r j (l, m, n) = l a + m b + n c + R j l m n f j ( ) P F cell j f j R j ( ) F cell = P f j e i K R j (3.1) j=1 (3.1) ( ) f j F cell (3.1) 8

10 K j T jk F cell P F cell = f j T jk e i K R j (3.2) j=1 (3.2) ( ) T jk (3.1) (3.1) (3.2) X X ψ 0 y(ωt) y 2 (ωt) ( y (ωt)y(ωt) ) X y(ωt) = F cell ψ 0 I = ψ 0ψ 0 F cell F cell = F cell 2 ( ψ0ψ = e iωt iωt = 1 ) ( ) ψ 0 ψ ψ a N a b Nb c N c ( N a N b N c ) F cell a l ( l a ) e ik l a ( ) F = N a N b l N c e i K (l a+m b+n c) F cell m n N a = F cell e ik l a e i K m b N c l N b m n e i K n c 9

11 L 1 = e ik l a L 2 = e i K m b L 3 = e ik n c L = L 1 L 2 L 3 (Laue) X F 2 = F cell F cell L 1L 1 L 2L 2 L 3L 3 (3.3) L 1 L 3 L 1 e ik a L 1L 1 = 1 e i(na 1) K a 1 e i K a e i K a 1 ei(na 1) K a 1 e i K a e i K a = 1 cos(n a 1) K a 1 cos( K a) N a 1 = N a K a = 2θ a L 1L 1 = ( sin(n aθ a ) sin(θ a ) )2 N a θ a π L 1 L 1 N a 2 ( ) N a 2 θ a π X L 1 L 1 L 3 L 3 X (L 1 L 1) θ a π K a 2π L 2 L 2 L 3 L 3 X K a K b K c 2π a b c K X X K F cell 2 ( ) K a K c 2π a b c ( ) u v w r = u a + v b + w c a c 10

12 ( ) a = 2π( b c) a ( b c), b = 2π( c a) b ( c a), c = 2π( a b) c ( a b) a c h, k, l G = h a + k b + l c K G G = 2π/ ( ) a a = b b = c c = 2π (3.4) a b = a c = b a = b c = c a = c b = 0 (3.5) Laue K = k 2 k 1 k 2 = K + k 1 k 1 = k 2 k 2 2 = K + k 1 2 K K k 1 = 0 θ K = 2 k 1 sinθ K G K G ( G ) G 2 = 2 G k 1 G = 2 k 1 sinθ Laue 2π/ d 2π d = 2 2π λ sinθ 2d sinθ = λ Bragg Bragg n 2dsin θ = nλ X d θ X X 2dsin θ nλ 2dsin θ = nλ Bragg Bragg X n = 1 2dsin θ = λ Bragg n = 1 Bragg n 2,3,4, n = 1 11

13 3.3 Laue (=Bragg ) K G F cell j R j = (u j, v j, w j ) F cell = f j e i G R j = f j e i(h a +k b +l c ) (u j a+v j b+wj c) = f j e 2πi(hu j+kv j +lw j ) (3.6) f 1 ( K ) f 1 (0, 0, 0) f 1 F = f 1 e 2πi(0+0+0) = f 1 R 1 = (0, 0, 0) R 2 = (1/2, 1/2, 1/2) F = 2 f 1 e 2πi(hu j+kv j +lw j ) = f 1 (1 + e πi(h+k+l) ) j=1 h + k + l F = 0 R 1 = (0, 0, 0) R 2 = (1/2, 1/2, 0) R 3 = (1/2, 0, 1/2) R 4 = (0, 1/2, 1/2) F = 4 f 1 e 2πi(hu j+kv j +lw j ) j=1 = f 1 (1 + e πi(h+k) + e πi(h+l) + e πi(k+l) ) 12

14 h, k, l F = 4f 1 h, k, l F = 0 hkl ( ) ( M f M K f M K ) ( ) [1] ( ) [2] ( ) [3] ( ) [4] X (hkl) Bragg X F cell 2 (= F cell F cell ) X Bragg X X (hkl) (hkl) Bragg... 13

15 ( ) 4.2 Bragg {100} (100) (010) (001) (100) (010) (001) ( ) F cell (100) 2 Bragg 6 F cell (100) 2 {110} (110) (110) (110) (110) (101) (101) (101) (101) (110) (101) (110) (101) (011) (011) (011) (011) 12 Bragg 12 F cell (110) 2 {111} {123} 48 M hkl Bragg F hkl 2 M hkl (Lorentz) X ( ) ( N a N c ) [2] 3 hkl... hkl 14

16 0 180 y X ( y z ) x y 2θ I 1 2 (1 + cos2 (2θ)) (4.1) z k0 x e - 2 θ y 3: x-y Lorentz Bragg θ Bragg ( ) I 1 sin 2 (θ)cos(θ) (4.2) 15

17 Lorentz Lorentz Lorentz = 1 + cos2 (2θ) 2 sin 2 (θ)cos(θ) (4.3) 4.4 ( ) X 2 Lorentz (4.4) f j (4.4) [1] : (1980), p [2] : (1998), p

29 1 6 1 1 1.1 1.1 1.1( ) 1.1( ) 1.1: 2 1.2 1.2( ) 4 4 1 2,3,4 1 2 1 2 1.2: 1,2,3,4 a 1 2a 6 2 2,3,4 1,2,3,4 1.2( ) 4 1.2( ) 3 1.2( ) 1.3 1.3 1.3: 4 1.4 1.4 1.4: 1.5 1.5 1 2 1 a a R = l a l 5 R = l a +

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