Diffraction condition
Ftotal(r K ) = G(
r K )F(
r K ) (5.1)
G(r K ) exp 2i
r K r a +
r b + r c ( )[ ]
,, (5.2)
F(r K ) = f j (
r K )Tj (
r K )exp 2i
r K r r j( )
j=1
M (5.3)
f j (r K ) j (
r r )exp 2ir K r r ( )dv
R 3 (5.4)
Tj (r K ) g j (
r r )exp 2ir K r r ( )dv
R 3 (5.5)
M
j (r r )
j
g j (r r )
j
Laue function & Laue condition
1992
r a
r b
r c
Na
Nb
Nc
5.1
(5.2)
G(r K )
,
,
5.1
r a ,
r b ,
r c
Na,
Nb ,
Nc
G(r K ) = exp 2i
r K r a +
r b + r c ( )[ ]
=0
N c1
=0
N b1
=0
N a 1
= exp 2ir K r a ( ) exp 2i
r K
r b ( ) exp 2i
r K r c ( )
=0
N c1
=0
N b1
=0
N a1 (5.6)
exp 2ir K r a ( )
=0
N a1
=1+ exp 2ir K r a ( ) + exp 4i
r K r a ( ) +L+ exp 2 Na 1( )i
r K r a [ ]
1,
exp 2ir K r a ( )
x jj=0
n1 = 1 x
n
1 x
exp 2ir K r a ( )
=0
N a1 =
exp 2iNar K r a ( ) 1
exp 2ir K r a ( ) 1
(5.7)
Ftotal(r K )
2= G(
r K )
2F(
r K )
2 (5.8)
(5.7)
exp 2ir K r a ( )
=0
N a 1
2
=exp 2iNa
r K r a ( ) 1
2
exp 2ir K r a ( ) 1
2
=exp 2iNa
r K r a ( ) 1[ ] exp 2iNa
r K r a ( ) 1[ ]
exp 2ir K r a ( ) 1[ ] exp 2i
r K r a ( ) 1[ ]
=2 2cos 2Na
r K r a ( )
2 2cos 2r K r a ( )
=1 cos 2Na
r K r a ( )
1 cos 2r K r a ( )
=sin2 Na
r K r a ( )
sin2 r K r a ( )
(5.9)
G(K) 2 =sin2 Na
r K r a ( )
sin2 r K r a ( )
sin2 Nbr K
r b ( )
sin2 r K
r b ( )
sin2 Ncr K r c ( )
sin2 r K r c ( )
(5.10)
Laue function
sin2 Nar K r a ( )
sin2 r K r a ( )
r K r a 5.2
r K r a = h
h
1 /Na,
2 /Na,
3 /Na
limx0
sin2 Nax( )sin2 x( )
= Na2 lim
x0
sin Nax( )Nax
2x
sin x( )
2
= Na2 (5.11)
2.52.01.51.00.50.0-0.5
r K r a
sin2N
ar K r
a (
)sin
2
r K r
a (
)
Na2
~ 1Na
5.2
sin2 Nar K r a ( )
sin2 r K r a ( )
,
Na =10
Na2
1 /Na
Na
Na =10
Na2 =100
sin2 Na 3 /2Na( )[ ]sin2 3 / 2Na( )[ ]
=1
sin2 3 /2Na( )[ ]= 4.85
sin2 Na 5 /2Na( )[ ]sin2 5 / 2Na( )[ ]
=1
sin2 5 /2Na( )[ ]= 2.00
Na
Na,
Nb ,
Nc (5.10)
r K r a = hr K
r b = k
r K r c = l
h ,
k ,
l (5.12)
G(r K )
2 NaNbNc( )
2 = N 2 (5.13)
NaNbNc = N (5.12) Laue condition
lattice vectors and reciprocal lattice vectors
r a ,
r b ,
r c
r a * ,
r b * ,
r c *
r a r a * =1 r a r b * = 0 r a r c * = 0
r b r a * = 0
r b
r b * =1
r b r c * = 0
r c r a * = 0 r c r b * = 0 r c r c * =1
(5.14)
r a *
r b
r c
r a 1 (5.12)
r K
r K = hr a * + k
r b * + lr c *
h ,
k ,
l (5.15)
r a ,
r b ,
r c
r a * ,
r b * ,
r c *
r a =axayaz
,
r b =
bxbybz
,
r c =cxcycz
, (5.16)
r a * =ax*
ay*
az*
,
r b * =
bx*
by*
bz*
,
r c * =cx*
cy*
cz*
(5.17)
(5.14)
ax* ay
* az*
bx* by
* bz*
cx* cy
* cz*
ax bx cxay by cyaz bz cz
=
1 0 00 1 00 0 1
(5.18)
r a r b r c ( )
r a *r b * r c *( )
r p =pxpypz
,
r q =qxqyqz
r p r q
r p r q pyqz pzqypzqx pxqzpxqy pyqx
(5.19)
r p r p r q ( ) = px pyqz pzqy( ) + py pzqx pxqz( ) + pz pxqy
pyqx( ) = 0 (5.20)
r q r p r q ( ) = qx pyqz pzqy( ) + qy pzqx pxqz( ) + qz pxqy
pyqx( ) = 0 (5.21)
r p r q
r p
r q
r p 2 r q 2 = px2 + py
2 + pz2( ) qx2 + qy2 + qz2( )
= px2qx2 + px
2qy2 + px
2qz2 + py
2qx2 + py
2qy2 + pz
2qz2 + pz
2qx2 + pz
2qy2 + pz
2qz2 (5.22)
r p r q ( )2 = pxqx + pyqy + pzqz( )2
= px2qx2 + py
2qy2 + pz
2qz2 + 2pxqx pyqy + 2pyqy pzqz + 2pzqz pxqx (5.23)
r p r q 2 = pyqz pzqy( )2
+ pzqx pxqz( )2
+ pxqy pyqx( )2
= py2qz2 + 2pyqz pzqy + pz
2qy2 + pz
2qx2 2pzqx pxqz + px
2qz2 + px
2qy2 2pxqy pyqx + py
2qx2
(5.24)
r p 2 r q 2 = r p r q ( )2 +r p r q 2 (5.25)
r p
r q
r p r q = r p r q cos
r p r q = r p 2 r q 2 r p 2 r q 2 cos2 = r p r q sin (5.26)
r p r q
r p
r q
r p r q sin
r a ,
r b ,
r c unit cell
r a r b
r a
r b
r a
r b
r a r b
r a r b
r a
r b
1
r a r b ( ) r c
r a r b
r c
r a
r b
V = r a r b ( ) r c
r b
r c
r c
r a
V = r a r b ( ) r c =
r b r c ( ) r a = r c r a ( )
r b (5.27)
r a * =r b r c
V(5.28)
r b * =
r c r a V
(5.29)
r c * =r a
r b
V(5.30)
(5.27) - (5.30)
lattice constants
a ,
b,
c ,
,
,
r a ,
r b ,
r c
a
r a
b
r b
c
r c
r b
r c
r c
r a
r a
r b
ax ,ay ,az ,bx ,by ,bz,cx ,cy ,cz( )
a = r a = ax2 + ay
2 + az2
cos =r b r c bc
=bxcx + bycy + bzcz
bx2 + by
2 + bz2( ) cx2 + cy2 + cz2( )
a,b,c,,,( )
r a X
r b XY Y > 0
r a
r a =a00
(5.31)
r b
r b =
bcosbsin0
(5.32)
r c
r c =cxcycz
(5.33)
r c
r a
r c r a = cacos (5.34)
(5.32) (5.34)
r c r a = cxa (5.35)
cx = c cos (5.36)
r b
r c
r b r c = bc cos (5.37)
(5.33) (5.34)
r b r c = bcx cos + bcy sin
= b c cos cos + cy sin( ) (5.38)
cy =c cos cos cos( )
sin(5.39)
r c
c
cz = c2 cx
2 cy2 (5.40)
cz
V
V = axbycz (5.41)
r a * =ax*
ay*
az*
=
r b r c
V=1V
bycz bzcybzcx bxczbxcy bycx
lattice plane
r K
r K = hr a * + k
r b * + lr c *
h ,
k ,
l
r d hkl* hr a * + k
r b * + lr c * (5.42)
hkl
hkl
hkl Miller index
hkl
h ,
k ,
l
r a h,
r b k,
r c l
r p =r a h
+ xr b k
r a h
+ y
r c l
r a h
x ,
y (5.43)
r p 0 = x0
r b k
r a h
+ y0
r c l
r a h
x0 ,
y0 (5.44)
r p 0
r d hkl*
x0 ,
y0
r p 0 r d hkl* = 0
r d hkl*
hkl
r p
r d hkl*
r d hkl*
x ,
y
r p r d hkl* =1
1r d hkl*
l = 0
hk0
r a h,
r b k
r c
r p =r a h
+ xr b k
r a h
+ y
r c
x ,
y (5.45)
r p 0 = x0
r b k
r a h
+ y0
r c
x0 ,
y0 (5.46)
r p 0 r d hkl* = 0
r p r d hkl* =1
k = 0
h = 0
k = l = 0
h00
r a h
r b
r c
r p =r a h
+ xr b + yr c
x ,
y (5.47)
r p 0 = x0r b + y0
r c
x0 ,
y0 (5.48)
r p 0 r d hkl* = 0
r p r d hkl* =1
h = l = 0
h = k = 0
r d hkl*
hkl
r a * ,
r b * ,
r c *
ax*,ay
*,az*,bx
*,by*,bz
*,cx*,cy
*,cz*( )
hkl
r d hkl* = hax
* + kbx* + lcx
*( )2
+ hay* + kby
* + lcy*( )2
+ haz* + kbz
* + lcz*( )2
(5.49)
crystal structure factor and Miller indices
r r j
r r j = x jr a + y j
r b + z j
r c (5.50)
x j ,
y j ,
z j 0 1 fractional coordinate
r K
r K = hr a * + k
r b * + lr c *
h ,
k ,
l
Fhkl = f jr d hkl*( )Tj
r d hkl*( )exp 2i hx j + ky j + lz j( )[ ]
j=1
M
(5.51)
r d hkl* hr a * + k
r b * + lr c *
f jr d hkl*( ) atomic scattering factor
dhkl
= 2dhkl sinhkl Bragg
hkl
f jsinhkl
Tjr d hkl*( ) atomic displacement factor
anisotropic atomic displacement factor
g j (r r ) = 1
(2 )3/2U11/2U21/2U31/2exp X
2
2U1
Y 2
2U2
Z2
2U3
(5.52)
r p X =pXxpXypXz
,
r p Y =pYxpYypYz
,
r p Z =pZxpZypZz
r r =xyz
r r = Xr p X + Yr p Y + Z
r p Z
Tj (r K ) = g j (
r r )exp 2ir K r r ( )dv
R 3
= g j (r r )exp 2i
r K r r ( )
dXdYdZ
=1
(2 )3/2U11/2U21/2U31/2exp X
2
2U1
Y 2
2U2
Z 2
2U3
exp 2i KX X + KYY + KzZ( )[ ]
dXdYdZ
= exp 2 2 KX2U1 + KY
2U2 + KZ2U3( )[ ]
= exp 2 2 K X KY KZ( )U1 0 00 U2 00 0 U3
KXKYKZ
= exp 2 2 K x Ky Kz( )P tU1 0 00 U2 00 0 U3
P
KxKyKz
= exp 2 2 Kx Ky Kz( )U jKxKyKz
= exp 2 2r K tU j
r K ( ) (5.53)
= exp 2 2 Kx Ky Kz( )U jxx U jxy U jzxU jxy U jyy U jyzU jzx U
jyz U jzz
KxKyKz
= exp 2 2 U jxxKx2( + U jyyKy2 + U jzzKz2 + 2U jxyKxKy + 2U
jyzKyKz +2U jzxKzKx )][
(5.54)
U j11 U j12 U j13U j12 U j22 U j23U j13 U j23 U j33
=
ax* / a* ay
* / a* az* / a*
bx* /b* by
* /b* bz* /b*
cx* /c* cy
* / c* cz* / c*
U jxx U jxy U jzxU jxy U jyy U jyzU jzx U jyz U jzz
ax* / a* bx
* /b* cx* /c*
ay* / a* by* /b* cy* /c*
az* / a* bz* /b* cz* /c*
=
1 / a* 0 00 1 /b* 00 0 1 / c*
ax* ay
* az*
bx* by
* bz*
cx* cy
* cz*
U jxx U jxy U jzxU jxy U jyy U jyzU jzx U jyz U jzz
ax* bx
* cx*
ay* by
* cy*
az* bz
* cz*
1 / a* 0 00 1 /b* 00 0 1 / c*
U jxx U jxy U jzxU jxy U jyy U jyzU jzx U jyz U jzz
=
ax bx cxay by cyaz bz cz
a* 0 00 b* 00 0 c*
U j11 U j12 U j13U j12 U j22 U j23U j13 U j23 U j33
a* 0 00 b* 00 0 c*
ax ay azbx by bzcx cy cz
=
ax bx cxay by cyaz bz cz
U j11a*2 U j12a
*b* U j13a*c*
U j12a*b* U j22b
*2 U j23b*c*
U j13a*c* U j23b
*c* U j33c*2
ax ay azbx by bzcx cy cz
r K = hr a * + k
r b * + lr c *
Tj (r K )
= exp 22 U j11h2a*2 + U j22k
2b*2 + U j33l2c*2([
+2U j12hka*b* + 2U j13hla
*c* + 2U j23klb*c* )]
B11 B12 B13B12 B22 B23B13 B23 B33
= 22ax* ay
* az*
bx* by
* bz*
cx* cy
* cz*
Uxx Uxy UzxUxy Uyy UyzUzx Uyz Uzz
ax* bx
* cx*
ay* by
* cy*
az* bz
* cz*
Uxx Uxy UzxUxy Uyy UyzUzx Uyz Uzz
=122
ax bx cxay by cyaz bz cz
B11 B12 B13B12 B22 B23B13 B23 B33
ax ay azbx by bzcx cy cz
r K = hr a * + k
r b * + lr c *
T(r K ) = exp B11h
2 + B22k2 + B33l
2 + 2B12hk + 2B13hl + 2B23kl( )[ ]
c hexagonal
Uxx = Uyy ,
Uxy = 0
r a =a00
,
r b =
a / 23a / 20
,
r c =00c
r a * =1 / a1 / 3a0
,
r b * =
02 / 3a0
,
r c * =001 / c
r a * / a* =3 /21 / 20
,
r b * /b* =
010
,
r c * / c* =001
U11 U12 U13U12 U22 U23U13 U23 U33
=
3 / 2 1 / 2 00 1 00 0 1
Uxx 0 00 Uxx 00 0 Uzz
3 / 2 0 01 /2 1 00 0 1
=
3 /2 1 / 2 00 1 00 0 1
3Uxx / 2 0 0Uxx / 2 Uxx 00 0 Uzz
=
Uxx Uxx / 2 0Uxx / 2 Uxx 00 0 Uzz
a = 6.000
b = 5.000
c = 4.000
=120.0
=110.0,
=120.0
r a =ax00
r b =
bxby0
r c =c xcycz
123
200714
