VI. Reciprocal lattice6-1. Definition of reciprocal lattice from a lattice with periodicities in real spacecba ,,
Vbac
Vacb
Vcba
bacbac
acbacb
cbacba
***
***
;;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 ;0ˆˆ0ˆˆ ;1ˆˆ ;0ˆˆ0ˆˆ ;0ˆˆ ;1ˆˆ
wzvzuzwyvyuywxvxux
wzvyuxur
r uWhat 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
Vcb
cbacba
)(
*
Similarly,V
acacb
acb
)(*
Vba
bacbac
)(*
* A translation vector in reciprocal lattice is called reciprocal lattice vector
**** clbkahGhkl
*hklG
* orthogonality; orthornormal set0 ;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
ab
c
AB
C
ha k
blc
ha
kbAB
hacAC
/
The reciprocal lattice vector**** clbkahGhkl
Similarly,
ha
lcclbkahACGhkl
)( ****
0***
lccl
haahACGhkl
Therefore, ACGABG hklhkl ** ;
is perpendicular to the plane (h, k, l)*hklG
ha
kbclbkahABGhkl
)( ****
0***
kbbk
haahABGhkl
ab
c
AB
C
ha k
blc
Moreover,hkl
hkl dG 1*
interplanar spacing of the plane (h, k, l)
*
***
*
*
hklhkl
hklhkl G
clbkahkb
GG
kbd
ab
c
AB
C
ha k
blc
**
***
*
* 1
hklhklhkl
hklhkl GG
clbkahha
GG
had
**
* 1
hklhkl GGbk
kb
**
***
*
* 1
hklhklhkl
hklhkl GG
clbkahlc
GG
lcd
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 ;
dk
dk
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 **** bcacbcacbac //*
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.
Vbac
* Similarly, one getsVVacbcba
** ;
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
aaa
abcaabc
cbacba
ˆˆ
ˆ)2/sin(ˆ)2/sin(
)(*
aa
aa
3ˆ2
30cosˆ
o
bbb
bcabbca
acbacb ˆ
ˆˆ)2/sin(
ˆ)2/sin()(
*
bb3
ˆ2
ccc
cabccab
bacbac
ˆˆ
ˆ)3/2sin(ˆ)3/2sin(
)(*
cc
cc ˆ
0cosˆ
o
6-3. Interplanar spacing
hklhkl d
G 1*
2** 1
hklhklhkl d
GG
)()(1 ******2
clbkahclbkahdhkl
(i) for cubic ,orthorhombic, tetragonal systems**** acba
0 ;0 ;0 ****** cbcaba
abcaabc
acbaacb
cbacba
ˆˆ
ˆ)2/sin(||||ˆ)2/sin(||||
)(*
aaabc
abc ˆ10cos
ˆo
2** 1ˆˆ
aaa
aaaa
Similarly,
2** 1ˆˆ
bbb
bbbb
2
** 1ˆˆcc
ccccc
)()(1 ******2
clbkahclbkahdhkl
2
2
2
2
2
2
cl
bk
ah
2
2
2
2
2
21cl
bk
ah
dhkl
(ii) for the hexagonal system
2
2
2
22
3)(41
cl
ahkkh
dhkl
)()(1 ******2
clbkahclbkahdhkl
o22
** 60cos34ˆˆ
34
3
ˆ23ˆ2
aba
abb
aaba
**22 3
221
34 ab
aa
o****** 60 ; ; bacbca
0 ;0 **** cbca
2o
22**
340cos
34ˆˆ
34
3ˆ2
3ˆ2
aaaa
aaa
aaaa
2o
22**
340cos
34ˆˆ
34
3
ˆ23
ˆ2aa
bbab
bb
bbb
)()(1 ******2
clbkahclbkahdhkl
**2**2****22
21 cclbbkbahkaahdhkl
22
22
222 1
34
322
34
cl
ak
ahk
ah
2
2
2
22
3)(4
cl
akhkh
6-4. Angle between planes (h1k1l1) and (h2k2l2)cos****
222111222111 lkhlkhlkhlkh GGGG
**
**
222111
222111coslkhlkh
lkhlkh
GGGG
for the cubic system
222
22
22
221
21
21
*2
*2
*2
*1
*1
*1
//)()(cos
alkhalkhclbkahclbkah
222
22
22
21
21
21
221
221
221
////
alkhlkhallakkahh
22
22
22
21
21
21
212121
lkhlkhllkkhh
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(21 zyxa )ˆˆˆ(
21 zyxa )ˆˆˆ(
21 zyxa )ˆˆ(
21 yxa )ˆˆ(
21 zya )ˆˆ(
21 zxa
)ˆˆ(2
zxaa )ˆˆ(
2yxab
)ˆˆ(
2zyac
)ˆˆ(
2)ˆˆ(
2)ˆˆ(
2)( zyayxazxacbaV
)ˆ)ˆ(ˆ(4
)ˆˆ(2
)ˆˆ(2
2
xyzazyayxa
42
8)ˆˆˆ(
4)ˆˆ(
2)(
332 aazyxazxacbaV
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
zyayxa
cbacba
azyyx
a
zyyxa)ˆˆ()ˆˆ(
4/
)ˆˆ()ˆˆ(4
3
2
azyx
axyz ˆˆˆˆ)ˆ(ˆ
Similarly,
azyxb ˆˆˆ*
azyxc ˆˆˆ*
azyx
azyx
azyx 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
azyx
azyx
azyx ˆˆˆ,ˆˆˆ,ˆˆˆ
*a *b
*c
)()(1 ******2 clbkahclbkah
dhkl
**2****
****2**
******2
cclbcklachl
cbklbbkabhk
cahlbahkaah
2** 3ˆˆˆˆˆˆ
aazyx
azyxaa
2**** 1ˆˆˆˆˆˆ
aazyx
azyxabba
2**** 1ˆˆˆˆˆˆ
aazyx
azyxacca
2** 3ˆˆˆˆˆˆ
aazyx
azyxbb
2**** 1ˆˆˆˆˆˆ
aazyx
azyxbccb
2** 3ˆˆˆˆˆˆ
aazyx
azyxcc
)()(1 ******2 clbkahclbkah
dhkl
2
2
2222
2
2222
2 333al
akl
ahl
akl
ak
ahk
ahl
ahk
ah
)](2)(3[1 2222 hlklhklkh
a
2
222
2
2
2
2
2
2
alkh
cl
bk
ah
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
cba
CBA
100011021
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)
lkh
LKH
100011021
100012011
lkhLKH or
zacyabxaa ˆ;ˆ;ˆ
)ˆˆ(2
);ˆˆ(2
);ˆˆ(2
yxaCzxaBzyaA
cba
CBA
011101110
21
lkh
LKH
011101110
21
)(21);(
21);(
21 khLlhKlkH
LKH
lkh
111111111
LKHlLKHkLKHh ;;
)](2)(3[1 2222 HLKLHKLKH
a
)])((2))((2))((2
))(3)(3)(3[41 222
2
khlkkhlhlhlk
khlhlka
)222222(3 222 hlklhklkh
)333(2 222 hlklhklkh
]444[41 222
2 lkha
][1 2222 lkh
a
There are the same!Or
2
1
hkld
)(1 2222 lkh
a2
1
hkld222 )()()( LKHLKHLKH
KLHLHKLKH
KLHLHKLKH
KLHLHKLKH
222
222
222
222
222
222
)(2)(3 222 KLHLHKLKH
)](2)(3[11 22222 HLKLHKLKH
adhkl
We proof the other way around!