VI. Reciprocal lattice

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!