Crystallography 6
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Transcript Crystallography 6
Reciprocal lattice
How to construct reciprocal lattice
Meaning of reciprocal lattice
Relation between reciprocal lattice and diffraction
Geometrical relation between reciprocal lattice and
original lattice
How to construct a reciprocal lattice from a crystal
(1) Pick a set of planes in a crystal
parallel
Plane
set 2
d2
d1
d
d 2*
Plane set 1
Plane
set 3
d3
d1* : d 2* (1 / d1 ) : (1 / d 2 )
k
k
*
*
d1 ; d 2
d1
d2
*
1
d
d 2*
*
1
d 3*
Does it really form a
lattice?
Draw it to convince
yourself!
Example: a monoclinic crystal
Reciprocal lattice (a* and c*) on the plane
containing a and c vectors.
(b is out of the plane)
c
b
a
(-100)
(100)
c
(102)
(001)
c
(001)
(002)
(002)
a
O
a
(00-2)
c
(00-2)
(002)
(001) (101) c
(002)
O
a
O
a
d
d
*
d 001
*
d100
O
*
d10
1
*
d 00
1
O
c*
c
(002)
(00-1)
*
d 002
*
102
*
101
(10-1)
(00-2)
2D form a 3-D reciprocal lattice
*
a*
a
*
*
a * d100
; c * d 001
Lattice point in reciprocal space
*
*
*
a * d100
; b* d 010
; c * d 001
*
*
*
*
d hkl
hd100
kd 010
ld 001
ha * kb* lc *
Integer
Lattice points in real space
ruvw ua vb wc
Reciprocal lattice cells for cubic crystals:
Simple cubic:
z
a axˆ; b byˆ ; c czˆ
y abc
x
d*002
z
z
(100)
*
d100
1/ a
a
y
d
*
001
1/ a
x
d
x
x
(001)
a
(010)
y
a
*
010
1/ a
(110)
y
y
x
2a / 2
*
d110
2/a
(002)?
*
d 001
*
d100
*
d 010
*
d110
Simple cubic
Base centered cubic (BCC):
z
a axˆ; b byˆ ; c czˆ
y abc
x
*
d 200
2/a
*
d 020
2/a
*
d 002
*
d 200
*
d 020
d
*
d 002
2/a
*
d110
1 / 2a / 2 2 / a
*
110
*
d 220
*
d 011
*
d 002
*
d 011
*
d101
*
d101
*
d 200
*
d 020
d
(111)
*
d 220
(111)
*
d 002
O
*
d 222
O
*
d 020
*
d 200
(111)
*
d 222
O
*
110
(222)
*
d 220
FCC
corner
Up and down
F and B
L and R
You should do the same for a FCC and show it forms
a BCC lattice! (Homework!)
Vector: dot and cross product
v.u = |v||u|cos
Projection of v onto u and times
each other (scaler)!
vu = |v||u|sin w
|v||u|sin is the area of the
parallelogram. w v and u
v
|v|cos
u
|v|sin
u
Relationships between a, b, c and a*, b*, c*:
Monoclinic: plane y-axis (b)
c * a and c * b c * a 0 and c * b 0
c // a b
Similarly,
*
a b 0 and a c 0; a // b c
*
*
cc*
|c*|ccos,
=
cc* = 1
*
|c*|
c
*
b a 0 and b c 0; b // c a
*
: c c*.
*
= 1/d001 ccos = d001
b
c*
d001
a
Similarly, aa* = 1 and bb* =1.
c* //ab,
Define c* = k (ab), k : a constant.
cc* = 1 ck(ab) = 1 k = 1/[c(ab)]=1/V.
V: volume of the unit cell
ab
bc
ca
*
*
Similarly, one gets a
; b
c
V
V
V
*
The Weiss zone law or zone equation:
A plane (hkl) lies in a zone [uvw] (the plane contains
the direction [uvw]). d*hkl (hkl) d*hkl ruvw = 0
(ha* kb* lc* ) (ua vb wc) 0
hu kv lw 0
d*hkl ruvw d*hkl (r1 r2 ) d*hkl r2
nth plane
ruvw
Define the unit vector in the d*hkl
d*hkl
direction i,
d*hkl
i * d*hkl d hkl ; | d*hkl | 1
; ii 1
d hkl
| dhkl |
r2 i ndhkl r2 d*hkl dhkl ndhkl r2 d*hkl n
d*hkl ruvw n hu kv lw n
r1
r2
uvw
Reciprocal Lattice: Fourier transform of the spatial
wavefunction of the original lattice
wave process (e. g. electromagnetic) in the crystal
Crystal: periodic
Physical properties function of a crystal (r ) (r T)
Crystal translation vector
Periodic function Exponential Fourier Series
(r) k e2ikr (r T) k e2ikr e2ikT
k
k
e2ikT 1
T ua vb wc
u, v, w: integer
Translation vectors of the original crystal lattice
e
2ikT
1 for all T
*
*
*
k
h
a
k
b
l
c
If
h, k, l: integer
Vectors of the reciprocal lattice
k T (ha* kb* lc* ) (ua vb wc)
(hu kv lw) always integer
e2ikT 1
If k (reciprocal) lattice ; T original lattice!
Vice versa!
In crystallography
bc
ca
ab
*
*
a
; b
; c
V
V
V
*
In SSP
bc
ca
ab
*
*
a 2
; b 2
; c 2
V
V
V
*
e
2ikT
ikT
e
k (in general): momentum space vector;
G: reciprocal lattice points
Proof: the reciprocal lattice of BCC is FCC
Use primitive translation vectors only
BCC
FCC
z
x
y
corner
F and B
1
a( xˆ yˆ zˆ )
2
1
a( xˆ yˆ zˆ )
2
1
a( xˆ yˆ zˆ )
2
Up and down
L and R
1
a( xˆ yˆ )
2
1
a( yˆ zˆ )
2
1
a( xˆ zˆ )
2
BCC
1
1
1
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
a a( x y z ); b a( x y z ); c a( xˆ yˆ zˆ )
2
2
2
bc
a
V
xˆ yˆ zˆ
zˆ xˆ yˆ
*
yˆ zˆ xˆ
1
1
a2
3
b c a( xˆ yˆ zˆ) a ( xˆ yˆ zˆ) (2 yˆ 2 zˆ) ; V a / 2
2
2
4
bc 1
a
( yˆ zˆ )
V
a
*
b*
ca 1
( xˆ zˆ )
V
a
c*
ab 1
( xˆ yˆ )
V
a
The vector set is the same as the FCC primitive
translation vector.
Unit of the reciprocal lattice is 1/length.
Mathematics of Interference
Sum of two waves:
R A1 cos(t 1 ) A2 cos(t 2 )
assume A1 = A2 = A
R A[cos(t 1 ) cos(t 2 )]
1
1
cos a cos b 2 cos (a b) cos (a b)
2
2
1
1 2
R 2 A cos (1 2 ) cos( t )
2
2 2
new amplitude AR
R
Geometrical analysis of Interference
A2
AR
2
2 1
2 2
A1 A2
1
1 2
2 A cos (1 2 ) cos( t )
2
2 2
2
1 A1
2
2
1
2
A
1
t term rotation vector
1 2
2
1
A cos (1 2 )
2
Complex Wave Representation of Interference
A1 cos(t 1 ) A1ei (t 1 )
A2 cos(t 2 ) A2ei (t 2 )
R A1ei (t 1 ) A2ei (t 2 ) A1ei1 A2ei2 eit
AR2 A1ei1 A2ei2
Rˆ AR eiR
2
A1ei1 A2ei2 A1ei1 A2ei2
A12 A22 A1 A2ei (1 2 ) A1 A2ei (2 1 )
A12 A22 2 A1 A2 cos(1 2 )
assume A1 = A2 = A
AR2 2 A2 2 A2 cos(1 2 ) 2 A2 (1 cos(1 2 ))
cos2a 2 cos2 a 1
1
2
2
2 1
AR 2 A 2 cos (1 2 ) AR 2 A cos (1 2 )
2
2
Diffraction conditions and reciprocal lattices:
Theorem: The set of reciprocal lattice vectors G
determines the possible X-ray reflections.
A( x, t ) A0 cos(kx t )
k 2 /
A(r, t ) A0 cos(k r t )
| k | 2 /
r
k
k
r
k
http://en.wikipedia.org/wiki/Plane_wave
Complex exponential form
U (r, t ) A0ei (krt )
A0 cos(k r t ) iA0 sin(k r t )
A(r, t ) iA0 sin(k r t )
U (r, t ) A0ei (krt ) U0ei (krt )
U0 A0ei
Complex number
What happen to the time dependent term?
U (r, t ) U0ei t eikr
X-ray wavelength ~ 0.1 nm ~ 3x1018 1/s
Detectors get the average intensity!
Detectors measured the intensity only!
| ei t |2 1
| eiA |2 | cos A i sin A |2 cos2 A sin 2 A 1
A lot of time, examining e ikr is enough!
Path difference
Phase angle
= (2/)rsin
= kr
Similarly,
r cos( ) dV
2
r
Phase angle
k ' r
k'
k
k'
O
exp(ik r)
exp(ik ' r)
F exp(ik r )n(r ) exp(ik ' r )dV
n(r ) exp[i (k k ' ) r ]dV n(r ) exp(ik r )dV
nG exp[i (G k ) r ]dV
G
0 for G = k
otherwise, = 0
k k ' k
Fourier expansion n(r)
The diffraction condition is G = k. k + G = k’
k G k ' (k G)2 (k ' )2 k 2
(k G ) 2 k 2 2k G G 2 k 2 2k G G 2 0
G: reciprocal lattice, -G: reciprocal lattice? ____
2k G G 2 0 2k G G 2
| k | k 2 /
| G | 2 / dhkl
Bragg condition?
| k | k 1 / k
G
k
or | G | 1/ d
k
hkl
2
2k G 2 | G | cos(90 ) G 2
2
2
2d hkl sin
2 sin
d hkl
dhkl
(hkl)
plane
Bragg law
G = k.
More geometric relation between direct lattice and
reciprocal lattice:
e1, e2, e3: contravariant basis vector of R3
covariant basis vectors e1, e2, e3 (reciprocal lattice)
e2 e3
e 3 e1
e1 e 2
e1 1 2 3 ; e2 1 2 3 ; e3 1 2 3
e e e
e e e
e e e
ei and ei are not normal, but mutually orthonormal:
e i e j ij
For any vector v:
q1 v e1 ; q 2 v e 2 ; q 3 v e3
q1 v e1 ; q2 v e 2 ; q3 v e3
v can be expressed in two (reciprocal) ways:
v qi ei qi ei q1e1 q2e 2 q3e3
i
i
1
2
3
v q ei q ei q e1 q e 2 q e3
Einstein’s summation
convention, omitting
No proof here, but you can check whether these relation
is correct or not?
Use BCC or FCC lattice as examples, next page!
v qi e i qi e i q1e1 q2 e 2 q3e 3
i
i
1
2
3
v q ei q ei q e1 q e2 q e3
Use the BCC lattice as examples
1
1
1
a1 a( xˆ yˆ zˆ ); a2 a( xˆ yˆ zˆ ); a3 a( xˆ yˆ zˆ )
2
2
2
1
1
1
ˆ
ˆ
b1 ( y zˆ )
b2 ( x zˆ ); b3 ( xˆ yˆ )
a
a
a
Assume v 2 xˆ yˆ zˆ
1
a a
q1 a1 v a ( xˆ yˆ zˆ ) (2 xˆ yˆ zˆ ) a a
2
2 2
1
q2 a2 v a ( xˆ yˆ zˆ ) (2 xˆ yˆ zˆ ) 2a
2
You can check
1
q3 a3 v a ( xˆ yˆ zˆ ) (2 xˆ yˆ zˆ ) 0
the other way
2
around.
1
1
2 xˆ yˆ zˆ v a ( yˆ zˆ ) 2a ( xˆ zˆ )
a
a
q j v e j q i (ei e j ) (ei e j )q i g ij q i
where g ij ei e j
Similarly, q j v e j qi ei e j (ei e j )qi g ij qi
Prove
g ij g jk ki
ei e j ij ei g jk e k g jk ei e k g jk g ik
gij: metric tensor in direct lattice
a, b, c and , , : direct lattice parameters
(standard definition of the Bravais lattice)
g11 a a a 2 ; g12 a b ab cos ; etc.
a2
g ij ab cos
ac cos
ab cos
b2
bc cos
ac cos
bc cos
c 2
det|gij| = V2.
V 2 a 2b 2 c 2 (1 2 cos cos cos cos 2 cos 2 cos 2 )
a*
a
*
b g ij b
c*
c
a*
a*
a
*
*
1
ij
b g ij b g b
c*
c*
c
Matrix inversion:
a
M d
g
a b
d e
g h
c
f
i
b
e
h
1
c
A B
1
f
D E
det(M )
k
G H
e
A
h
b
D
h
f
k
b
G
e
c
f
c
k
T
C
A D G
1
F
B
E
H
det(
M
)
C F K
K
d f
B
g k
a c
E
g
k
d
C
g
a
F
g
a
H
d
a b
K
d e
c
f
e
h
b
h
Inverting the matrix gij.
bc sin 2 / a
c(cos cos cos ) b(cos cos cos )
abc
ij
2
g 2 c(cos cos cos )
ac sin / b
a (cos cos cos )
V
2
b
(cos
cos
cos
)
a
(cos
cos
cos
)
ab
sin
/
c
gij : metric tensor in reciprocal lattice
a*, b*, c* and *, *, *: reciprocal lattice parameters
*2
* *
*
* *
*
a
a
b
cos
a
c
cos
2
g ij a *b* cos *
b*
b*c* cos *
* *
*
* *
*
*2
a c cos b c cos
c
These two are the same!
One gets relation like
bc sin * ac sin * ab sin
a
;b
; c
V
V
V
2
bc
sin
ac
sin
abc
(cos cos cos )
* *
*
*
a b cos
cos
V
V
V2
(cos cos cos )
cos *
sin sin
*
Similarly,
(cos cos cos )
(cos cos cos )
*
cos
; cos
sin sin
sin sin
*
……………..
d-spacing of (hkl) plane for any crystal system
d
*
hkl
d
*
hkl
2 *2
2 *2
2 *2
(ha kb lc ) (ha kb lc ) h a k b l c
*
*
*
*
*
*
2hka*b* cos * 2klb*c* cos * 2lhc*a * cos *
2 2
2
2 2
2
2 2
2
1
2 b c sin
2 a c sin
2 a b sin
h
k
l
2
2
2
d hkl
V
V
V2
abc 2 (cos cos cos )
a 2bc(cos cos cos )
2hk
2kl
2
V
V2
ab 2 c(cos cos cos )
2lh
V2
Example:FCC BCC
(1) Find the primitive unit cell of the selected
structure
(2) Identify the unit vectors
1
a( xˆ yˆ )
2
1
a( yˆ zˆ)
2
1
a( xˆ zˆ)
2
1
1
1
a( xˆ yˆ zˆ) a( xˆ yˆ zˆ)
a( xˆ yˆ zˆ)
2
2
2
a
a
a
a ( xˆ zˆ)
b ( xˆ yˆ )
c ( yˆ zˆ)
2
2
2
a
a
a
V a (b c ) ( xˆ zˆ ) ( xˆ yˆ ) ( yˆ zˆ )
2
2
2
2
a
a
a
ˆ ˆ
2 ( x y ) 2 ( yˆ zˆ ) 4 ( zˆ ( yˆ ) xˆ )
a
a2
a3
a3
V a (b c ) ( xˆ zˆ) ( xˆ yˆ zˆ) 2
2
4
8
4
Volume of F.C.C. is a3. There are four atoms
per unit cell! the volume for the primitive
of a F.C.C. structure is ?
a
a
( xˆ yˆ ) ( yˆ zˆ)
b c
*
2
a 2
a3 / 4
a (b c )
a2
( xˆ yˆ ) ( yˆ zˆ )
( xˆ yˆ ) ( yˆ zˆ )
4
3
a /4
a
zˆ ( yˆ ) xˆ xˆ yˆ zˆ
a
a
Similarly,
* xˆ yˆ zˆ
* xˆ yˆ zˆ
b
c
a
a
xˆ yˆ zˆ xˆ yˆ zˆ xˆ yˆ zˆ
,
,
B.C.C.
a
a
a
See page 23
Using primitive translation vector to do the
reciprocal lattice calculation:
Case: FCC BCC
xˆ yˆ zˆ xˆ yˆ zˆ xˆ yˆ zˆ
,
,
a
a
a
*
*
c
b
* *
* *
*
*
1
(ha kb lc ) (ha kb lc )
2
d hkl
* *
* *
2 * *
h a a hka b hla c
* *
*
* *
2 *
hkb a k b b klb c
* *
* * 2 * *
hlc a klc b l c c
*
a
* * xˆ yˆ zˆ xˆ yˆ zˆ 3
a a
2
a
a
a
* * * * xˆ yˆ zˆ xˆ yˆ zˆ 1
a b b a
2
a
a
a
* * * * xˆ yˆ zˆ xˆ yˆ zˆ 1
a c c a
2
a
a
a
* * xˆ yˆ zˆ xˆ yˆ zˆ 3
b b
2
a
a
a
* * * * xˆ yˆ zˆ xˆ yˆ zˆ 1
b c c b
2
a
a
a
* * xˆ yˆ zˆ xˆ yˆ zˆ 3
c c
2
a
a
a
* *
* *
*
*
1
(ha kb lc ) (ha kb lc )
2
d hkl
3h 2 hk hl hk 3k 2 kl hl kl 3l 2
2 2 2 2 2 2 2 2 2
a
a
a
a
a
a
a
a
a
1
2 [3( h 2 k 2 l 2 ) 2( hk kl hl )]
a
h2 k 2 l 2 h2 k 2 l 2
not 2 2 2
Why?
2
a
b
c
a
(hkl) defined using unit cell!
(hkl) is defined using primitive cell!
(HKL)
Example
Find out the relation between the (hkl) and
[uvw] in the unit cell defined by a, b , c and the
(HKL) and [UVW] in the unit cell defined by
A, B, C .
A 1a 2b 0c
B 1a 1b 0c
C 0a 0b 1c
B
b
a
A 1 2 0 a
In terms of matrix B 1 1 0 b
C 0 0 1 c
A
Find out the relation between (hkl) and (HKL).
Assume there is the first plane
intersecting the a axis at
a/h and the b axis at b/k.
A
In the length of |a|, there
are h planes. In the length
of |b|, there are k planes.
B
b
2k
a
How many planes can
be inserted in the length
b/k
a/h
h
|A|? Ans. h + 2k
A/(h+2k)
H = 1h + 2k + 0l
Similarly, K = -1h + 1k +0l and L = 0h + 0k + 1l
H
K
L h k
1 1 0
H 1 2 0 h
or
l 2 1 0
K 1 1 0 k
0 0 1
L 0 0 1 l
a axˆ; b ayˆ ; c azˆ
a
a
a
A ( yˆ zˆ ); B ( xˆ zˆ ); C ( xˆ yˆ )
2
2
2
A
0 1 1 a
1
B 1 0 1 b
C 2 1 1 0 c
h 1 1 1 H
H
0 1 1 h
1
k 1 1 1 K
K 1 0 1 k
l 1 1 1 L
L 2 1 1 0 l
1
1
1
H ( k l ); K ( h l ); L ( h k )
2
2
2
h H K L; k H K L; l H K L
1
1
2
2
2
[3( H K L ) 2( HK KL HL )] 2
2
a
d hkl
1
2
2
2
2 [3( k l ) 3( h l ) 3( h k ) )
4a
2( k l )(h l ) 2( h l )(h k ) 2( k l )(h k )]
3(2h2 2k 2 2l 2 2hk 2kl 2hl)
2(h k l 3hk 3kl 3hl)
2
2
2
1
1 2
2
2
2
2 [4h 4k 4l ] 2 [h k 2 l 2 ]
4a
a
There are the same!
Or
1 2
1
2
2
(
h
k
l
)
2
2
d hkl a
(H K L) ( H K L) ( H K L)
H 2 K 2 L2 2 HK 2 HL 2 KL
2
2
H 2 K 2 L2 2 HK 2 HL 2 KL
H K L 2 HK 2 HL 2 KL
2
2
2
3( H K L ) 2( HK HL KL)
1
1
2
2
2
[
3
(
H
K
L
) 2( HK KL HL)]
2
2
d hkl a
2
2
2
We proof the other way around!
2
Interplanar spacing (defined based on unit cell)
1
h k l
Cubic:
2
d hkl
a2
1
h2 k 2 l 2
Tetragonal:
2
2
2
d hkl
a
c
1
h2 k 2 l 2
2 2 2
Orthorombic:
2
d hkl a
b
c
2
2
2
1
4 h hk k
l
2
Hexagonal:
2
2
d hkl 3
a
c
2
2
2
Get the metric tensor!
Perform the inversion of the matrix!
Comparing the inversion of the metric tensor in
direct lattice with the metric tensor in reciprocal lattice
Geometrical relation between reciprocal lattice and
direct lattice can be obtained!