7

01111() 7.1 (ii) 7. (iii)

7.1 poit defect

d hkl d * hkl ε Δd hkl d hkl ~ Δd * hkl * d hkl (7.1) f ( ε ) 1 πσ e ε σ (7.) σ relative strai root ea square d * siθ λ (7.) Δd * cosθ Δθ λ (7.4) ε Δθ ( Δθ ) Δd * taθ ε taθ (7.5) d * ( Δθ ) ε taθ σ taθ (7.6) taθ 1/ cosθ Δθ X / cosθ + Y taθ (7.7) cosθ

( Δθ )cosθ X + Y siθ (7.8) siθ, ( Δθ )cosθ 0. X 0.1 Y 0.0 0.0 0. 0.4 0.6 0.8 1.0 si Θ 7. Williaso-Hall [100] (100) 100 010 001 Williaso-Hall Williaso-Hall 100, 00, 00,... Williaso-Hall 111,,,... Williaso-Hall

Williaso-Hall Williaso-Hall (111) ABCABCABC (i) ABCABABCABC (ii) ABCABCBACBA (i) (111) (ii) a, b, c a a, b, c a 111 c a H a + b b H b + c c H a + b + c (7.9) (7.10) (7.11) 7.4

b H c c H a H b 0 a 7.4 a H b H ( a + b ) ( b + c ) a b a c b b + b c a( a + b + c ) (7.1) V H ( a H b H ) c H a( a + b + c ) ( a + b + c ) a (7.1) b a * H H c H V H b H * c H * b + c c a + b a a c H a H V H c b + c a a a H b H V H ( a + b + c ) a a + b + c (7.14) a ( a + b + c ) ( a + b ) a a b + c (7.15) a a + b + c a (7.16)

b H b H b b c H a H c a 7.5 : A : B : C 7.5 a a H /, b b H /, c c H / a, b, c 1 V a 4 a * b * c * b c a + b + c V a c a V a b V ( a b + c ) a a + b + c a I( K) F ξ,η,ς ( K) * F ξ,η,ς ξ, η, ς ξ, η, ς { } ( K)exp π i K (ξ ξ ) a + (η η ) b + (ς ς ) c (7.17) F ξ,η,ς ( K) ξa + ηb + ςc F ξ,η,ς ( K) K h a * + k b * + l c * (7.18)

h, k, l ξ ξ + ξ, η η + η, ς ς + ς I( K) F ξ,η,ς ( K) * F ξ+ ξ,η,ς ξ, η, ς ξ,η+ η,ς + ς ( K)exp π i K ξ a + η b + ς c ξ,η,ς ξ, η, ς F ξ,η,ς ( K)F * ξ+ ξ,η+ η,ς + ς ( K) exp π i h ξ + k η + l ς (7.19) I( K) V ξ, η, ς J ξ η ς ξ, η, ς ( K)exp π i h ξ + k η + l ς (7.0) J ξ, η, ς ( K) J ξ, η, ς ( K) F ξ,η,ς ( K)F * ξ+ ξ,η+ η,ς + ς ( K) (7.1) ξ,η,ς V ξ, η, ς ξ a + η b + ς c ξ a + η b + ς c 1 ξ a + η b + ς c V ξ, η, ς 1 ς F ξ,η,ς ( K) F ξ,η,ς ( K) F ξ+ ξ,η+ η,ς ( K) J ξ, η, ς J (7.0) ς ( K) ξ η ξ, η, ς ( K) J ς ( K) I( h, k, l ) e π i h ξ e π i k η J ς ( h, k, l )e π i l ς (7.) ξ η ς

h k e π i h ξ 0, e π i ξ η k η 0 I( h, k, l ) h k I( h, k, l ) J ( h, k, l )e π i l h k (7.) A, B, C A B a + b C a + b 7.5 F A ( K) F 0 ( K) (7.4) F B ( K) F 0 ( K)exp π i K a + b (7.5) F C ( K) F 0 ( K)exp π i K a + b (7.6) A A, AB, AC P 0, P +, P P 0 + P + + P 1 (7.7) J ( K) J ( K) 1 F ( K)F * A A ( K)P 0 + F A ( K)F * B ( K)P + + F A ( K)F * C ( K)P +F B ( K)F B * ( K)P 0 + F B ( K)F C * ( K)P + + F B ( K)F A * ( K)P +F C ( K)F C * ( K)P 0 + F C ( K)F A * ( K)P + + F C ( K)F B * ( K)P (7.8) (7.4)-(7.6) F A ( K), F B ( K), F C ( K)

J ( K) 1 F ( K) π i( P 0 0 + exp h + k ) P π i( + + exp h + k ) P π i( +P 0 + exp h + k ) P π i( + + exp h + k ) P π i( +P 0 + exp h + k ) P π i( + + exp h k ) P 1 F ( K) π i( P 0 0 + exp h + k ) P π i( + + exp h k ) P π i( +P 0 + exp h + k ) P π i( + + exp h k ) P π i( +P 0 + exp h + k ) P π i( + + exp h k ) P J ( K) F 0 ( K) P 0 + P + e π i( h k )/ + P e π i( h k )/ (7.9) (7.9) P 0, P +, P α P 0 A A 0 A A A A A A B C 1 B C C B A A A A A B, B C, C A (1 α ) A C, B A, C B α P 0 0 P (1 α )α + P 0 α(1 α ) + P + (1 α )(1 α ) + P αα (7.0)

0 P 1 P + α + P (1 α ) (7.1) 0 P + + P + P 1 (7.) P 0 P 0 0 + P 1 0 + 1 α(1 α ) P 1 α(1 α ) (7.) P 0 0 1, P 1 0 0 P 0 1 + 1 1 + (1 α )i + 1 1 (1 α )i (7.4) P 0 1 + 1 α(1 α ) cos arcta (1 α ) θ arcta (1 α ) P 0 1 + 1 1 α(1 α ) 1 + 1 α(1 α ) { } (7.5) (7.4), (7.5) ( e iθ + e ) iθ cos ( θ ) (7.6) P + 1 1 1 α(1 α ) 1 i e iθ + 1+ i 1 1 1 α(1 α ) + si( θ ) cos θ P 1 1 1 α(1 α ) 1+ i e iθ + 1 i 1 1 1 α(1 α ) si( θ ) cos θ e iθ e iθ (7.7) (7.8) 0 P P 0 (7.9) P + P (7.40)

P + P (7.41) h k N N (7.9) J ( K) F 0 ( K) P 0 + P + e π i( h k )/ + P e π i( h k )/ J ( K) F 0 ( K) (7.) I( h, k, l ) J ( h, k, l )e π i l I( h, k,l H ) J ( h, k,l )e π ilh / (7.4) H l N l H l N N l, l H α l H l N (111) (111) Bragg h k N ±1 (7.9) J ( K) F 0 ( K) P 0 + P + e π i/ + P ±π i/ ( e ) (7.4) I( h, k,l H ) F 0 ( K) P 0 + P + e π i/ + P ±π i/ ( e )e π il H / F 0 ( K) 1 1+ P 0 + P + e π i/ + P ±π i/ ( e )e π il H / + P 0 + P + e π i/ + P ±π i/ ( e )e π il H / 1 F 0 ( K) 1+ P 0 + P e π i/ + P + ±π i/ ( e )e π il H / 1 + P 0 + P + e π i/ + P ±π i/ ( e )e π il H / 1 F 0 ( K) 1+ P 0 e π il H / + e π il H / + P + e π i(l H 1)/ + e π i(l H 1)/ 1 1

(7.6), (7.7), (7.8) + P e π i(l H ±1)/ + e π i(l H ±1)/ 1 F 0 ( K) 1+ P 0 cos πl H + P + 1 cos π(l 1) H 1 + P cos π(l ±1) H 1 P 0 1 + 1 α(1 α ) P + 1 1 1 α(1 α ) P 1 1 1 α(1 α ) F 0 ( K) 1+ + 1 1 cos ( θ ) + si( θ ) cos θ si( θ ) cos θ 1 + 1 α(1 α ) cos θ cos πl H 1 1 1 α(1 α ) cos π(l H 1) + 1 1 1 1 α(1 α ) cos π(l H ±1) F 0 ( K) 1+ + 1 1 cos θ + si( θ ) si( θ ) cos θ 1 + 1 α(1 α ) cos θ cos πl H 1 1 1 α(1 α ) cos πl H + 1 cos π ± si πl H 1 1 1 α(1 α ) cos πl H cos π si πl H + si( θ ) cos θ si π si( θ ) cos θ si π

F 0 ( K) 1+ + 1 1 1 + 1 α(1 α ) cos θ cos πl H 1 1 1 α(1 α ) 1 cos πl H + 1 ± si πl H 1 1 1 α(1 α ) 1 cos πl H si πl H + si( θ ) cos θ si( θ ) cos θ I( h, k,l H ) C a cos πl H + b si πl H (7.4) 0 a 0 1 (7.4) a 1 + 1 α(1 α ) cos θ 1 1 1 α(1 α ) 1 1 1 α(1 α ) 1 α(1 α ) cos θ cos θ + si( θ ) si( θ ) cos ( θ ) 0 (7.44) b 0 0 (7.45) b ± 1 1 1 α(1 α ) 1 1 1 α(1 α ) 1 α(1 α ) cos θ + si( θ ) cos θ si( θ ) si ( θ ) 0 (7.46) ± h k N ±1 (7.4)

I( h, k,l H ) C 1+ 1 α(1 α ) 1 cos( θ )cos πl H si( θ )si πl H C 1+ 1 α(1 α ) πl cos H ± θ 1 C 1+ 1 α(1 α ) πl cos H ± θ + π 1 C 1+ 1 α(1 α ) l cos π H + 1 ± θ π 1 (7.47) 1+ r cos(θ) -1+ r e iθ + e iθ 1 0 1 1+ 1 r e + 1 iθ 1 r e iθ ( 1 r e ) iθ +1 r e iθ +1 r e iθ ( 1 r e )( iθ 1 r e ) iθ 1 r e iθ ( 1 r eiθ r e iθ + r ) +1 r e iθ +1 r e iθ 1 r 1 r e iθ r e iθ + r 1+ r cosθ (7.47) α 0,1 I( h, k,l H ) C C 1 α(1 α ) 1+ 1 α(1 α ) 1 1 α(1 α ) 1 α(1 α ) cos π l H + 1 ± θ π α(1 α ) 1 α(1 α ) cos π l H + 1 ± θ π (7.48) 7.6

4 Paterso's odel for deforatio fault α 0.1 α 0. α 0. α 0.4 α 0.5 1 0-0.5 0.0 0.5 1.0 1.5.0 l H 7.6 Paterso h k 1 7.6 α h k N ±1 l H + 1 ± θ π N N' l H N π arcta (1 α ) (7.49) α 0 arcta (1 α ) π h k N ±1 l H N 1 N { N 1 y arcta x d y d x 1 1+ x d dα arcta (1 α ) 1+ 1 α arcta (1 α ) ~ π α

l H N π arcta (1 α ) ~ N 1 α 4π Δl H N π ± 1 π ~ ± α 4π arcta (1 α ) arcta (1 α ) N 1 peakl H +/ I( h, k,l )d l H H (7.47) I( h, k,l H ) C 1+ 1 α(1 α ) l cos π H + 1 ± θ π 1 peakl H +/ / I( h, k,l )d l C 1+ 1 α(1 α ) H H / 1 peakl H / C + 1 α(1 α ) 1 C + 1 α(1 α ) 1 C I ax ( h, k,l H ) C / / cos πl d l πl si π α(1 α ) / / 1 α(1 α ) 1 α(1 α ) cos πl d l peakl H /

β H peakl H +/ peakl H / I( h, k,l H )d l H I ax ( h, k,l H ) (7.50) 1 α(1 α ) 1 α(1 α ) α(1 α ) 1 1 α(1 α ) 1 1 α(1 α ) 1+ 1 α(1 α ) 1 1 α(1 α ) 1+ 1 α(1 α ) α(1 α ) 1 α(1 α ) α(1 α ) ~ 9α 4 (7.51) {111} 111, 111, 111, 111, 111, 111, 111, 111 h k l H hkl a, b, c, a, b, c H a + b a b + c b c H a + b + c a * a + b + c 4 a * + b * + a c *

b * a b + c a * b * + 4 a c H * a + b + c 1 a * + 1 b * + 1 a c * hkl h k l H K h a * + k b * + l * ch H ha * + kb * + lc * h K a h + k k K b k + l l H K c H h + k + l h K a 4 h k + l H k K b h k + l H l K c h + 4 k + l H h + k l h k l H h + k + l c * 111 h k,l H 1 l H N 1 l H 0 l H 0 l H N ±1 l H h + k + l h + k + l N h + k + l N +1 h + k + l N 1 111 111 h + k + l N 111, 111, 111, 111, 111, 111 h + k + l N +1

θ K siθ λ c H * c H * a * 7.7 7.7 c H * c H * c H * h + k + l N ±1 c * H ± 1 Δl H K ha * + kb * + lc * c H * a + b + c π arcta (1 α ) 1 a * + 1 b * + 1 a c * c H *

* d peak * d hkl ( Δl H ) c * H K K ± 1 π arcta (1 α ) c * H ± h + k + l 1 h + k + l a π ~ ± h + k + l α 4π h + k + l a K K arcta (1 α ) d * siθ λ ( Δd * Δθ )cosθ ( Δθ )d * λ taθ θ peak θ hkl d * * ( d peak hkl )taθ d * ± h + k + l taθ 1 h + k + l π ± α h + k + l taθ ~ π h + k + l arcta (1 α ) c H * 1 1 α(1 α ) β * ch H 1+ 1 α(1 α ) K Δd * β * ch H K 1 1 α(1 α ) K c * H 1+ 1 α(1 α ) K c H * 1 1 α(1 α ) h + k + l 1+ 1 α(1 α ) h + k + l a ~ α h + k + l 4 h + k + l a

Δθ Δd * taθ d * 1 1 α(1 α ) h + k + l taθ 1+ 1 α(1 α ) ( h + k + l ) α h + k + l taθ ~ h + k + l (111) 7.8 { } 111, 111 h + k + l 111 111, 111, 111, 111, 111, 111 h + k + l { } 00, 00, 00, 00, 00, 00 h + k + l 00 { } 0 h + k + l 0 0 h + k + l { 111} { 111} { } { 111} 0 { 11} 11 h + k + l 11 h + k + l 11 h + k + l

{111} {00} {0} {11} {} {400} {1} {40} {4} k siθ / λ.8 (111)