vi. reciprocal lattice 6-1. definition of reciprocal lattice from a lattice with periodicities in...
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VI. Reciprocal lattice
6-1. Definition of reciprocal lattice from a lattice with periodicities in real spacecba
,,
V
bac
V
acb
V
cba
bac
bac
acb
acb
cba
cba
***
***
;;or
)(;
)(;
)(
Remind what we have learned in chapter 5Pattern Fourier transform diffractionPattern of the original pattern!
3-D: the Fourier transform of a function f(x,y,z)
dxdydzezyxfzyxF wzvyuxi )(2),,(),,(
Note that zzyyxxr ˆˆˆ
wwvvuuu ˆˆˆ
ux+vy+wz: can be considered as a scalarproduct of if the following conditions are met!
ur
1ˆˆ ;0ˆˆ ;0ˆˆ
0ˆˆ ;1ˆˆ ;0ˆˆ
0ˆˆ ;0ˆˆ ;1ˆˆ
wzvzuz
wyvyuy
wxvxuxwzvyuxur
r
u
What is ? Then what is ?
Consider the requirements for the basictranslation vectors of the “reciprocal lattice”
*a
0 ;0 ;1 *** acabaa
i.e. caba
** ;
Say
In other words, cba
||* )(* cbka
1)(1 ** cbkaaaaa
Vcbak
1
)(
1
V
cb
cba
cba
)(
*
Similarly,V
ac
acb
acb
)(
*
V
ba
bac
bac
)(*
＊ A translation vector in reciprocal lattice is called reciprocal lattice vector
**** clbkahGhkl
*hklG
* orthogonality; orthornormal set
0 ;0 ;1 *** acabaa
0 ;1 ;0 *** bcbbba
1 ;0 ;0 *** cccbca
* In orthorhombic, tetragonal and cubic systems,
aaa
11*
bbb
11*
ccc
11*
* is perpendicular to the plane (h, k, l) in real space
*hklG
a
b
c
AB
C
h
a k
bl
c
h
a
k
bAB
h
acAC
/
The reciprocal lattice vector**** clbkahGhkl
Similarly,
h
a
l
cclbkahACGhkl
)( ****
0***
l
ccl
h
aahACGhkl
Therefore, ACGABG hklhkl ** ;
is perpendicular to the plane (h, k, l)*
hklG
h
a
k
bclbkahABGhkl
)( ****
0***
k
bbk
h
aahABGhkl
a
b
c
AB
C
h
a k
bl
c
Moreover,hkl
hkl dG
1*
interplanar spacing of the plane (h, k, l)
*
***
*
*
hklhkl
hklhkl
G
clbkah
k
b
G
G
k
bd
a
b
c
AB
C
h
a k
bl
c
**
***
*
* 1
hklhklhkl
hklhkl
GG
clbkah
h
a
G
G
h
ad
**
* 1
hklhkl GG
bk
k
b
**
***
*
* 1
hklhklhkl
hklhkl
GG
clbkah
l
c
G
G
l
cd
or
or
Graphical view of reciprocal lattice!
How to construct a reciprocal lattice from a crystal Pick a set of planes in a crystal and using a direction and a magnitude to represent the plane
Plane set 1
Planeset 2
d1
d2
*1d
*2d
)/1(:)/1(: 21*2
*1 dddd
2
*2
1
*1 ;
d
k
d
k dd
parallel
d3
Planeset 3
*1d
*2d
*3d
Does it really form alattice?Draw it to convinceyourself!
Oa
c(001)
(002)
(00-2)
(100)
(-100)
O
*001d
*002d
*100d
*100d
Oa
c(001)
(002)
(00-2)
(101)
Oa
c(001)
(002)
(00-2)
(102) *101d
*102d
Oa
c
(002)
(002)
(00-1) (10-1)
*110d
*
a
c
a*
c*
*001
**100
* ; dcda
Example: a monoclinic crystal Reciprocal lattice (a* and c*) on the plane containing a and c vectors. (b is out of the plane) a
c
b
2D form a 3-D reciprocal lattice
*001
**010
**100
* ;; dcdbda
****001
*010
*100
** cbadddd lkhlkhG hklhkl
Lattice point in reciprocal space
cbar wvuuvw Lattice points in real space
Integer
Relationships between a, b, c and a*, b*, c*:Monoclinic: plane y-axis (b)
a
c
c*
d001
0 and 0 and **** bcacbcac
bac //*
Similarly, cbacaba // ;0 and 0 ***
acbcbab // ;0 and 0 ***
cc* = |c*|ccos, |c*| = 1/d001 ccos = d001
cc* = 1
: c c*.
b
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
bac
* Similarly, one gets
VV
acb
cba
** ;
V: volume of the unit cell
6-2. Reciprocal lattices corresponding to crystal systems in real space
(i) Orthorhombic ,tetragonal ,cubic
a
b
c
*a
*b
*c
(ii) Monoclinic
a
b
c
*a
*b
*c
*
(iii) Hexagonal
a
b
c
*a
*b
*c
60o 30o
30o
We deal with reciprocal latticeTransformation in Miller indices.
a
b
*a
*b
60o
120oc
*c
caba ** ;
cbab
** ;
a : unit vector of *a
b : unit vector of *b ba
o** 60 ba
aa
a
abca
abc
cba
cba
ˆ
ˆ
ˆ)2/sin(
ˆ)2/sin(
)(*
a
a
a
a
3
ˆ2
30cos
ˆo
bb
b
bcab
bca
acb
acb
ˆ
ˆ
ˆ)2/sin(
ˆ)2/sin(
)(*
b
b
3
ˆ2
cc
c
cabc
cab
bac
bac
ˆ
ˆ
ˆ)3/2sin(
ˆ)3/2sin(
)(*
c
c
c
c ˆ
0cos
ˆo
6-3. Interplanar spacing
hkl
hkl dG
1*
2
** 1
hkl
hklhkl dGG
)()(1 ******
2clbkahclbkah
dhkl
(i) for cubic ,orthorhombic, tetragonal systems**** acba
0 ;0 ;0 ****** cbcaba
abca
abc
acba
acb
cba
cba
ˆ
ˆ
ˆ)2/sin(||||
ˆ)2/sin(||||
)(*
aaabc
abcˆ
1
0cos
ˆo
2
** 1ˆˆ
aa
a
a
aaa
Similarly,
2
** 1ˆˆ
bb
b
b
bbb
2
** 1ˆˆ
cc
c
c
ccc
)()(1 ******
2clbkahclbkah
dhkl
2
2
2
2
2
2
c
l
b
k
a
h
2
2
2
2
2
21
c
l
b
k
a
h
dhkl
(ii) for the hexagonal system
2
2
2
22
3
)(41
c
l
a
hkkh
dhkl
)()(1 ******
2clbkahclbkah
dhkl
o
22
** 60cos3
4ˆˆ3
4
3
ˆ2
3
ˆ2
aba
ab
b
a
aba
**
22 3
2
2
1
3
4ab
aa
o****** 60 ; ; bacbca
0 ;0 **** cbca
2
o
22
**
3
40cos
3
4ˆˆ
3
4
3
ˆ2
3
ˆ2
aaaa
aa
a
a
aaa
2
o
22
**
3
40cos
3
4ˆˆ3
4
3
ˆ2
3
ˆ2
aabb
ab
b
b
bbb
)()(1 ******
2clbkahclbkah
dhkl
**2**2****2
22
1cclbbkbahkaah
dhkl
2
2
2
2
22
2 1
3
4
3
22
3
4
cl
ak
ahk
ah
2
2
2
22
3
)(4
c
l
a
khkh
6-4. Angle between planes (h1k1l1) and (h2k2l2)
cos****
222111222111 lkhlkhlkhlkh GGGG
**
**
222111
222111coslkhlkh
lkhlkh
GG
GG
for the cubic system
22
2
2
2
2
2
22
1
2
1
2
1
*
2
*
2
*
2
*
1
*
1
*
1
//
)()(cos
alkhalkh
clbkahclbkah
22
2
2
2
2
2
2
1
2
1
2
1
2
21
2
21
2
21
/
///
alkhlkh
allakkahh
2
2
2
2
2
2
2
1
2
1
2
1
212121
lkhlkh
llkkhh
6-5. The relationship between real lattice and reciprocal lattice in cubic system ：
Simple cubic Simple cubicBCC FCC
FCC BCC
Real lattice Reciprocal lattice
Example ： f.c.c b.c.c(1) Find the primitive unit cell of the selected structure(2) Identify the unit vectors
)ˆˆˆ(2
1zyxa )ˆˆˆ(
2
1zyxa )ˆˆˆ(
2
1zyxa )ˆˆ(
2
1yxa )ˆˆ(
2
1zya )ˆˆ(
2
1zxa
)ˆˆ(2
zxa
a
)ˆˆ(2
yxa
b
)ˆˆ(2
zya
c
)ˆˆ(
2)ˆˆ(
2)ˆˆ(
2)( zy
ayx
azx
acbaV
)ˆ)ˆ(ˆ(4
)ˆˆ(2
)ˆˆ(2
2
xyza
zya
yxa
42
8)ˆˆˆ(
4)ˆˆ(
2)(
332 aazyx
azx
acbaV
Volume of F.C.C. is a3. There are four atomsper unit cell! the volume for the primitiveof a F.C.C. structure is ?
4/
)ˆˆ(2
)ˆˆ(2
)( 3
*
a
zya
yxa
cba
cba
a
zyyx
a
zyyxa
)ˆˆ()ˆˆ(
4/
)ˆˆ()ˆˆ(4
3
2
a
zyx
a
xyz ˆˆˆˆ)ˆ(ˆ
Similarly,
a
zyxb
ˆˆˆ*
a
zyxc
ˆˆˆ*
a
zyx
a
zyx
a
zyx ˆˆˆ,
ˆˆˆ,
ˆˆˆ B.C.C.
See page 23
Here we use primitive cell translation vector tocalculate the reciprocal lattice.
When we are calculating the interplanar spacing,the reciprocal lattices that we chosen is different.
Contradictory?
Which one is correct?
Using primitive translation vector to do thereciprocal lattice calculation:Case: FCC BCC
a
zyx
a
zyx
a
zyx ˆˆˆ,
ˆˆˆ,
ˆˆˆ
*a *b
*c
)()(1 ******2
clbkahclbkahdhkl
**2****
****2**
******2
cclbcklachl
cbklbbkabhk
cahlbahkaah
2** 3ˆˆˆˆˆˆ
aa
zyx
a
zyxaa
2**** 1ˆˆˆˆˆˆ
aa
zyx
a
zyxabba
2**** 1ˆˆˆˆˆˆ
aa
zyx
a
zyxacca
2** 3ˆˆˆˆˆˆ
aa
zyx
a
zyxbb
2**** 1ˆˆˆˆˆˆ
aa
zyx
a
zyxbccb
2** 3ˆˆˆˆˆˆ
aa
zyx
a
zyxcc
)()(1 ******2
clbkahclbkahdhkl
2
2
2222
2
2222
2 333
a
l
a
kl
a
hl
a
kl
a
k
a
hk
a
hl
a
hk
a
h
)](2)(3[1 222
2hlklhklkh
a
2
222
2
2
2
2
2
2
a
lkh
c
l
b
k
a
h not Why?
(hkl) defined using unit cell!
(hkl) is defined using primitive cell!
(HKL)
Find out the relation between the (hkl) and [uvw] in the unit cell defined by and the (HKL) and [UVW] in the unit cell defined by .
cbaC
cbaB
cbaA
100
011
021
In terms of matrix
c
b
a
C
B
A
100
011
021
cba
, ,
CBA
, ,
a
b A
B
Example
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. In the length of |a|, there are h planes. In the length of |b|, there are k planes. How many planes can be inserted in the length |A|? Ans. h + 2k H = 1h + 2k + 0l Similarly, K = -1h + 1k +0l and L = 0h + 0k + 1l
A
Ba
b
a/hb/k
2k
hA/(h+2k)
l
k
h
L
K
H
100
011
021
100
012
011
lkhLKH or
zacyabxaa ˆ;ˆ;ˆ
)ˆˆ(2
);ˆˆ(2
);ˆˆ(2
yxa
Czxa
Bzya
A
c
b
a
C
B
A
011
101
110
2
1
l
k
h
L
K
H
011
101
110
2
1
)(2
1);(
2
1);(
2
1khLlhKlkH
L
K
H
l
k
h
111
111
111
LKHlLKHkLKHh ;;
)](2)(3[1 222
2HLKLHKLKH
a
)])((2))((2))((2
))(3)(3)(3[4
1 2222
khlkkhlhlhlk
khlhlka
)222222(3 222 hlklhklkh
)333(2 222 hlklhklkh
]444[4
1 2222
lkha
][1 222
2lkh
a
There are the same!Or
2
1
hkld
)(1 222
2lkh
a
2
1
hkld222 )()()( LKHLKHLKH
KLHLHKLKH
KLHLHKLKH
KLHLHKLKH
222
222
222
222
222
222
)(2)(3 222 KLHLHKLKH
)](2)(3[11 222
22HLKLHKLKH
adhkl
We proof the other way around!