vi. reciprocal lattice
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VI. Reciprocal lattice. 6-1. Definition of reciprocal lattice from a lattice with periodicities in real space. Remind what we have learned in chapter 5 Pattern Fourier transform diffraction Pattern of the original pattern!. - PowerPoint PPT PresentationTRANSCRIPT
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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!
![Page 2: VI. Reciprocal lattice](https://reader033.vdocuments.mx/reader033/viewer/2022061507/568161ac550346895dd16bc4/html5/thumbnails/2.jpg)
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 ?
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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
)(
*
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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
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* 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
![Page 6: VI. Reciprocal lattice](https://reader033.vdocuments.mx/reader033/viewer/2022061507/568161ac550346895dd16bc4/html5/thumbnails/6.jpg)
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
![Page 7: VI. Reciprocal lattice](https://reader033.vdocuments.mx/reader033/viewer/2022061507/568161ac550346895dd16bc4/html5/thumbnails/7.jpg)
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
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Graphical view of reciprocal lattice!
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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!
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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
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*001
**010
**100
* ;; dcdbda
****001
*010
*100
** cbadddd lkhlkhG hklhkl
Lattice point in reciprocal space
cbar wvuuvw Lattice points in real space
Integer
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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
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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
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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
*
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(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 ** ;
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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
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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
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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
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)()(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
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)()(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
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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
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6-5. The relationship between real lattice and reciprocal lattice in cubic system :
Simple cubic Simple cubicBCC FCC
FCC BCC
Real lattice Reciprocal lattice
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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
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)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 ?
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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
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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?
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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
![Page 28: VI. Reciprocal lattice](https://reader033.vdocuments.mx/reader033/viewer/2022061507/568161ac550346895dd16bc4/html5/thumbnails/28.jpg)
2** 3ˆˆˆˆˆˆ
aazyx
azyxaa
2**** 1ˆˆˆˆˆˆ
aazyx
azyxabba
2**** 1ˆˆˆˆˆˆ
aazyx
azyxacca
2** 3ˆˆˆˆˆˆ
aazyx
azyxbb
2**** 1ˆˆˆˆˆˆ
aazyx
azyxbccb
2** 3ˆˆˆˆˆˆ
aazyx
azyxcc
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)()(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)
![Page 30: VI. Reciprocal lattice](https://reader033.vdocuments.mx/reader033/viewer/2022061507/568161ac550346895dd16bc4/html5/thumbnails/30.jpg)
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
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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
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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 ;;
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)](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
![Page 34: VI. Reciprocal lattice](https://reader033.vdocuments.mx/reader033/viewer/2022061507/568161ac550346895dd16bc4/html5/thumbnails/34.jpg)
)(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!