The Structure Factor
The
Structure
Factor
Suggested Reading
P 303
312 i D G f & M H
P
ages
303
-
312
i
n
D
e
G
rae
f
&
M
c
H
enry
Pages 59-61 in Engler and Randle
1
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structure factor, Notas de estudo de Engenharia de Materiais
fator de estrutura
Tipologia: Notas de estudo
2013
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The Structure FactorThe
Structure Factor Suggested Reading
P^
303 312 i
D G
f & M H
Pages 303-312 in DeGraef & McHenryPages 59-61 in Engler and Randle
Structure Factor (
Fhkl
)
2 (^
i^ j^
i
N^
i hu^
kv^ lw
hkl^
i i F^
f e
^
^
-^ Describes how atomic arrangement (
uvw
)
(^1) i
influences the intensity of the scattered beam.It tells us which reflections (i e
peaks
hkl
) to
-^ It tells us which reflections (i.e., peaks,
hkl
) to
expect in a diffraction pattern.
Some Useful Relations^ e
i^ =
3 e i^ =^ e
5 i^ = … = -
2 i e =^ e
4 i^ =
6 e i^ = … = +
ni e = (-1)
n, where
n^ is any integer
y^
g
ni e =^ e
-ni, where
n^ is any integer ix^ e+ e
-ix^ =
cos
x
Needed for structure factor calculationsNeeded for structure factor calculations
Fhkl
for Simple Cubic
-^ Atom coordinate(s) u,v,w:– 0,0,
2 (^
) 1
i^ j^ i N^
i hu^ kv^
lw hkl^
i i F^
^ f e ^ ^
0,0,
i^ h^
k^ l 2 (
i^ h^
k^ l
F^ hkl
fe^
f
^ ^
^
No matter what atom coordinates or plane indices you substitute into theNo matter what atom coordinates or plane indices you substitute into the^ structure factor equation for simple cubic crystals, the solution is always
non-zero.
Thus, all reflections are allowed for simple cubic (primitive) structures.
Fhkl
for Face Centered Cubic
-^ Atom coordinate(s) u,v,w:– 0,0,0;
2 (^
) 1
i^ j^ i N^
i hu^ kv^
lw hkl^
i i F^
^ f e ^ ^
0,0,0; – ½,½,0;– ½,0,½;– 0,½,½.
^ ^
h^ k^
h^ l^
k^ l
i^
i^
i
i f^
f^
f^
f
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
2 0 ^
i
F^ hkl
fe^
fe^
fe^
fe
^
^
^
^
^
^
^
^
^
^
^
^
^
1
i h^ k
i h^ l^
i k^ l
F^ hkl
f^
e^
e^
e
^
^
^
^
^
Fhkl
for Face Centered Cubic
^
^ ^
^
^
1
i h^ k
i h^ l^
i k^ l
F^ hkl
f^
e^
e^
e
^
^
^
^
^
-^ Substitute in a few values of
hkl
and you will find
the following:the following:– When
h,k,l
are unmixed (i.e. all even or all odd), then
F= 4hkl^
f. [NOTE: zero is considered even] F^
0 f^
i^ d i di
(i^
bi^
ti^
f^ dd
-^ F
= 0 for mixed indices (i.e., a combination of oddhkl (^) and even).
Fhkl
for NaCl Structure – cont’d
-^ For Na:
2 (^
) 1
i^ j^ i N^
i hu^ kv^
lw hkl^
i i F^
^ f e ^ ^
^
2 (0)
(^ )^
(^ )^
(^ )
i^
i h^ k^
i h^ l^
i k^ l
f^
e^
e^
e^
e
^
^
^
^
^
^
^
(^ )^
(^ )^
(^ )^
(^ )
(^ )^
(^ )^
(^ )
Na^1
i h^ k^
i h^ l^
i k^ l
f^ e Na
e^
e^
e
f^
e^
e^
e
^
^
^
^
^
^
^
-^ For Cl:
^
^
2 (^
)^
2 (^
)^
2 (^
)
(^
)^
2
2
2
l^
k^
h
i h^ k^
i h^
l^
i^
k^ l
i h^ k^
l f^ eCl
e^
e^
e
^
^ ^
^ ^
^
^ ^
^
^
^
^
^
(^
)^
(^
)^
(^
)^
(^
)
(^
)^
( )^
(^ ) 2 2
2
(^2) ( )
2 2
i h^ k^
l^
i^
i^
i
h^ k^
h^
l^
k^ l
Cl
i h^ k^
l^
i^
i^
i
f^ e Cl
e^
e^
e
f^ e
e^
e^
e
^ ^
^ ^
^ ^
^
^
^
^
^
^
^
l^
k^
h
l^
k^
h^
These terms are all positive and even.^ ^ Whether the exponent is odd or ^ Whether
the exponent is odd or even depends solely on the remaining^ h ,
k , and
l^ in each exponent.
Fhkl
for NaCl Structure – cont’d
2 (^
) 1
i^ j^ i N^
i hu^ kv^
lw hkl^
i i F^
^ f e ^ ^
-^ Therefore
Fhkl
: ^
^
(^ )
(^ )^
(^ )
(^
)^
( )^
(^ )^
(^ )
1
i h^ k
i h^ l^
i k^ l
hkl^
Na
i h^ k
l^
i^
i^
i
Cl
F^
f^
e^
e^
e
f^
e^
e^
e^
e
^
^
^
^
^
^
^
^
^
l^
k^
h
^
f^ eCl
e^
e^
e
^
^
which can be simplified to
*^ :
^
^
(^
)^
(^ )
(^ )
(^ )
1 i h^ k
l^
i h^ k
i h^ l^
i k^ l
hkl^
Na^
Cl
F^
f^
f^ e
e^
e^
e
^ ^
^
^
^
^
^
^
(200)
100 90 80
NaClCuKα^ radiation
(220)
(^70 60) %) (^50) Intensity (I 40 30
(111)
(222)
(400)
(420)
(422)^
(600)(442)
20 10
(111)
(311)
(400)
(331)
(333)(511)^
(440)(531)
20
30
40
50
60
70
80
90
100
110
2 θ 120
10 0
Fhkl
for L
Crystal Structure 2
•^ Atom coordinate(s) u,v,w:– 0,0,0;
A^
B
2 (^
) 1
i^ j^ i N^
i hu^ kv^
lw hkl^
i i F^
^ f e ^ ^
0,0,0; – ½,½,0;– ½,0,½;
A
- 0,½,½.
^
^
^
^
^
2
2
2
2 (0)
2 2
2 2
2 2
h^ k
h^
l^
k^
l
i^
i^
i
i
^
^
^
^
^
^
^
^
2 (0)
2 2
2 2
2 2
A^
B^
B^
B
i
F^
f e^
f e^
f e^
f e
hkl
^
^
^
^
(^
)^
(^
)^
(^
)
A^
B
i h^
k^
i h^
l^
i k^
l
F^
f^
f^ e
e^
e
hkl
^
^
^
^
^
Intensity (%)^100
(111)
Example of XRD patternfrom a material with anL
crystal structure 2
A^
B
100 90 80
A^
B
80 70 6060 50 40
(200)
40 30
(220)^
(311)
2 θ^ (°)
20 10
(100)^
(110)
(210)^
(211)^
(300)(221)(310)
(222)(320) (321)
(^ )^16
20 25
30
35 40
45
50 55
60
65 70
75
80 85
90
95 100
105
110 115
0
(^ )^
(^ )
Fhkl
for MoSi
2
-^ Atom positions:^ –
Mo atoms at 0,0,0; ½,½,½Si^
t^
t 0 0^
0 0^
½ ½ ½
½ ½ ½
1/
c
-^ Si
atoms at 0,0,z; 0,0,z; ½,½,½+z; ½,½,½-z; z=1/
-^ MoSi
is actually body centered tetragonal with 2 a^ = 3.20 Å and
c^ = 7.86 Å
z
c^
c^
a c
yx
z x^ y
z xy
b
xz y
a
b
a^
b^
a b
Viewed down z-axis
a^
a
Viewed down x-axis
Viewed down y-axis
Now we can plug in different values for
h k l^
to determine the structure factor
F^ hkl
for MoSi
- cont’d 2
Now^
we can plug in different values for
h^ k l^ to
determine the structure factor.
-^ For
h k l^
^ ^
^
^
^
5(1)^
(1)
1
1
3
3
3
3
0 0^
0 0
2 ( )^
2 ( )
0 0 1 0
^
^
^
^
^
^
^
^
^
^
^
i^
i
i^
i
i hkl^
Mo^
Si
F^
f^ e^
e^
f^ e^
e^
e^
e
2 23
(^
)^
(^
(^ )^
(1^ 1)^
( 1^ 1)
^
^
^
^
^ ^
^
i^
i
Mo^
Si Mo^
si f^
e^
f^ COS
e
f^
f
-^ For
h k l^
NO REFLECTION!
Fhkl
^ ^
^
^ ^
^
^
1 1 0^
1 1 0^
1 1 0
0
2 (0)^
2 (0)
2
(0)^ (0)
2
2
(^
)^
(^
(2)^
^
^
^
^
^
^
^
^
^
^
^
^
i^
i^
i
i^
i
hkl^
Mo^
Si i^
i^
i
Mo^
Si Mo^
si F^
f^ e^
e^
f^ e^
e^
e^
e
f^
e^
f^ e^
e^ e
e
f^
f If you continue for different
h k l^
combinations
trends will emerge
this will lead you
2 POSITIVE! YOU WILL SEE A REFLECTIONFhkl
-^ If you continue for different
h^ k l^
combinations
… trends will emerge… this will lead you
to the rules for diffraction… h^ +^ k
+^ l^ = even
100
(103)
90 80
(110)
MoSi
2 CuKα^ radiation
(101)
(110)
(^50) Intensity (I 40 30 (002)^
(213)
20 10
(112)^
(200) (202)
(211)^
(116)
(206)(301) (303)
20
30
40
50
60
70
80
90
100
110
2 θ 120
10 0
(006)^
(204)^
(301)(222)
(303) (312)^
(314)