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Scattering conditions (Laue conditions)
()
IK
re
dr
iK r
G
G
iG r
∝
=
⇒
−⋅
⋅
∫
∑
ρ
ρρ
3
2
e
G
G
iG K r
∝
∑
∫
−−
⋅
3
2
(
∗
)
cf.:
δ
π
xe
d
k
e
d
k
ikx
i
kx
==
−∞
∞
−∞
∞
∫
∫
1
2
2
(wave packet with
∆
p
→
∞
⇒
∆
x
→
0)
ed
r
VG
K
V
⋅
∫
=
=
≅
3
0
for
otherwise
⇒
K
≡
k

k
0
(
∗
)
⇒
IG
G
∝ ρ
2
Scattering condition:
G
=
K
38
No absorption
⇒
()
ρρ
re
G
G
iG r
=
∑
−⋅
real
() ()
ρ
rr
G
G
=⇒
=
−
**
and with
G
G
G
G
G
G
I
−
=
=
∝
*
2
Gh
g
k
g
l
g
=++
12
3
II
hkl
hkl
=
(Friedel Rule)
Consequences:
Reflex pattern always has inversion center
E.g.: 3fold axis in lattice
⇒
6fold axis in reflex pattern
K = G
K = k
 k
0
k = k
0
= 2
π/λ
λ
continuum or crystal
rotation!
Fig. 3.4. Ewald construction. A point of the reciprocal lattice is chosen as arbitrary
origin. The vector
k
0
is drawn to point towards the origin. All points on a sphere
with radius
k
0
, centered around the starting point of
k
0
obey the condition
k = k
0
for
elastic scattering. The Laue condition
K = G
is obeyed for all reciprocal lattice
points intersecting the “Ewald sphere“.
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Miller indices hkl
Fig. 3.5. (a) Illustration of Miller indices defined by intersections of the lattice planes with the
coordinate axes in terms of integer multiples
m, n, o
of the base vectors. Here
m
=1,
n
=2,
o
=2.
The Planes (
hkl
) are spanned by
ma
1
 na
2
,
oa
3
 na
2
.(b) The dashed planes are equivalent to the
solid ones.
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This note was uploaded on 01/19/2010 for the course MATERIALS M504 taught by Professor Adelung during the Spring '02 term at Uni Kiel.
 Spring '02
 Adelung

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