x-raydiffraction_copy for students
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If atom B is different from atom A p the amplitudes must be weighed by the respective
atomic scattering factors (f)T he resultant amplitude of all the waves scattered by all the atoms in the UC gives the
scattering factor for the unit cell T he unit cell scattering factor is called the Structure Factor ( F )
Scattering by an unit cell = f(position of the atoms, atomic scattering factors)
electronan byscatteredwaveof Amplitudeucinatomsall byscatteredwaveof Amplitude
Factor StructureF !!
[2 ( )]i i h x k y l z E e feN T d d d 2 ( )h x k y l zN T d d d In complex notation
2 F I w
[2 ( )]
1 1
j j j j
n ni i h x k y l zhkl
n j j
j j
f e f eN T d d d § §
Structure factor is independent of the shape and size of the unit cell
F or n atoms in the UC
If the UC distorts so do the planes in it!!
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nnie )1(!
)(2
UUU
Coseeii
!
Structure factor calculations
A Atom at (0,0,0) and equivalent positions
[2 ( )] j j j ji i h x k y l z j j f e f eN T d d d
[2 ( 0 0 0)] 0i h k l F f e f e f T � � �
22
f F ! F is independent of the scattering plane (h k l)
T T nini ee !
Simple Cubic
1)( !T inodd e
1)( !T inevene
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B Atom at (0,0,0) & (½, ½, 0) and equivalent positions
[2 ( )] j j j ji i h x k y l z
j j F f e f eN T d d d
! !1 1
[2 ( 0)][2 ( 0 0 0)] 2 2
[ 2 ( )]0 ( )2 [1 ]
i h k l i h k l
h k i
i h k
F f e f e
f e f e f e
T T
T T
� � �� � �!
! !
F is independent of the µl¶ index
C- centred Orthorhombic
Real
]1[ )( k hie f F ! T
f F 2!
0! F
22 4 f F !
02 ! F
e.g. (001), (110), (112); (021), (022), (023)
e.g. (100), (101), (102); (031), (032), (033)
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I f the blue planes are scattering in phase then on C- centering the red planes will scatter outof phase (with the blue planes- as they bisect them) and hence the (210) reflection will
become extinct
This analysis is consistent with the extinction rules: (h + k) odd is absent
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In case of the (310) planes no new translationally equivalent planes are added on latticecentering this reflection cannot go missing.This analysis is consistent with the extinction rules: (h + k) even is present
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C Atom at (0,0,0) & (½, ½, ½) and equivalent positions
[2 ( )] j j j ji i h x k y l z
j j F f e f eN T d d d
! !1 1 1
[2 ( )][2 ( 0 0 0)] 2 2 2
[ 2 ( )]0 ( )2 [1 ]
i h k l i h k l
h k l i
i h k l
F f e f e
f e f e f e
T T
T T
!
! !
B ody centredOrthorhombic
Real
]1[ )( l k hie f F ! T
f F 2!
0! F
22 4 f F !
02 ! F
e.g. (110), (200), (211); (220), (022), (310)
e.g. (100), (001), (111); (210), (032), (133)
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D Atom at (0,0,0) & (½, ½, 0) and equivalent positions
[2 ( )] j j j ji i h x k y l z
j j F f e f eN T d d d
! !
]1[ )()()(
)]2
(2[)]2
(2[)]2
(2[)]0(2[
hl il k ik hi
hl i
l k i
k hii
eee f
eeee f F
!
¼½
»¬«
!
T T T
T T T T
Face Centred Cubic
Real
f F 4!
0! F
22 16 f F !
02 ! F
(h, k, l) unmixed
(h, k, l) mixed
e.g. (111), (200), (220), (333), (420)
e.g. (100), (211); (210), (032), (033)
(½, ½, 0), (½, 0, ½), (0, ½, ½)
]1[ )()()( hl il k ik hi eee f F ! T T T
Two odd and one even (e.g. 112); two even and one odd (e.g. 122)
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M ixed indices CASE h k l
A o o eB o e e
( ) ( ) ( )CASE A : [1 ] [1 1 1 1] 0i e i o i oe e eT T T ! !( ) ( ) ( )B : [1 ] [1 1 1 1] 0i o i e i oe e eT T T ! !
0! F 02
! F (h, k, l) mixed e.g. (100), (211); (210), (032), (033)
M ixed indices T wo odd and one even (e.g. 112); two even and one odd (e.g. 122)
Unmixed indices CASE h k l
A o o o
B e e e
Unmixed indices
f F 4! 22 16 f F !(h, k, l) unmixed
e.g. (111), (200), (220), (333), (420)
All odd (e.g. 111); all even (e.g. 222)
( ) ( ) ( )CASE A : [1 ] [1 1 1 1] 4i e i e i ee e eT T T ! !( ) ( ) ( )B : [1 ] [1 1 1 1] 4i e i e i ee e eT T T ! !
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E Na + at (0,0,0) + Face Centering Translations p (½, ½, 0), (½, 0, ½), (0, ½, ½)Cl í at (½, 0, 0) + FCT p (0, ½, 0), (0, 0, ½), (½, ½, ½)
¼½
»¬«
¼½
»¬«
!
)]2
(2[)]2
(2[)]2
(2[)]2
(2[
)]2
(2[)]2
(2[)]2
(2[)]0(2[
l k h
il
ik
ih
i
Cl
hl i
l k i
k hi
i Na
eeee f
eeee f F
T T T T
T T T T
][
]1[)()()()(
)()()(
l k hil ik ihiCl
hl il k ik hi Na
eeee f
eee f F !T T T T
T T T
]1[
]1[)()()()(
)()()(!k hihl il k il k hi
l
hl il k ik hia
eeee f
eee f F T T T T
T T T
]1][[)()()()( hl il k ik hil k hi
l a eeee f f F !T T T T
NaCl:Face Centred Cubic
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]1][[ )()()()( hl il k ik hil k hil a
eeee f f F ! T T T T
Z ero for mixed indices
M ixed indices CASE h k l
A o o eB o e e
]2][1[! T er T er F
0]1111[]1[2:ACASE)()()(
!!!oioiei
eeeT ermT T T
0]1111[]1[2:B )()()( !!! oieioi eeeT er T T T
0! F 02 ! F (h, k, l) mixed e.g. (100), (211); (210), (032), (033)
M ixed indices
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(h, k, l) unmixed ][4)( l k hi
l a e f f F !T
][4!C l a
f f F I f (h + k + l) is even22 ][16!
C l a f f F
][4!C l a
f f F I f (h + k + l) is odd22 ][16!
C l a f f F
e.g. (111), (222); (133), (244)
e.g. (222),(244)
e.g. (111), (133)
Unmixed indices CASE h k l
A o o oB e e e
4]1111[]1[2:ACASE )()()( !!! eieiei eeeT er T T T
4]1111[]1[2:B )()()( !!! eieiei eeeT er T T T
Unmixed indices