Lecture+11+MAK +Fourier+Analysis

7/30/2019 Lecture+11+MAK +Fourier+Analysis

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PHY 3201 FIZIK KEADAAN PEPEJAL

Reciprocal Lattice

Although the Bragg law gives a simple and convenient method for

calculating the separation of crystallographic planes, further analysis is

necessary.

Fourier analysis of the periodic nature of crystal lattices reveals the

importance of a set of vectors, G, related to the lattice vectors (a1,a2,a3).

The set of vectors G is labelled the reciprocal lattice.

Each crystal lattice has an associated reciprocal lattice which makes

calculation of the intensities and positions of peaks much easier.

For a three dimensional lattice, defined by its primitive vectors

(a1,a2,a3), its reciprocal lattice can be determined by the formula

321

321 2

aaa

aab

321

132 2

aaa

aab

321

213 2

aaa

aab


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Thus b1,b2,b3 have the property biaj=2ij

where ij=1 ifi=j and ij=0 ifij

The reciprocal lattice is defined as

G = v1b1 + v2b2 + v3b3 u,v,w are integers


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Reciprocal Lattice to simple

cubic Lattice The primitive translation vectors of a simple cubic lattice may be

taken as the set The volume of the cell is

The primitive translation vectors of the reciprocal lattice are foundfrom the standard prescription

Here the reciprocal lattice is itself a simple cubic lattice, now of latticeconstant 2/a.

The boundaries of the first Brillouin zones are the planes normal tothe six reciprocal lattice vectors b1, b2, b3 at their midpoints

The six planes bound a cube of the edge 2/a and of volume (2/a)3 ;this cube is the first Brillouin zone of the simple cubic crystal lattice

zayaxa

a;

a;

a 321 3321 aaa a

zaaa

2

b;y

2

b;x

2

b 321

zaaa

b;yb;xb 121221121


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Reciprocal Lattice to body-

centered cubic Lattice The primitive translation vector of the bcc lattice are

The volume of the primitive cell is

The primitive translation vectors of the reciprocal lattice are found

from the standard prescription

By compared with the fcc system, it is just the primitive vectors of

an fcc lattice, so that an fcc lattice is the reciprocal lattice of the bcc

lattice.

The general reciprocal lattice vector is

The shortest Gs are

)

(a;)

(a;)

(a 21

32

1

22

1

1 zyxazyxazyxa 3

21

321 aaaV a

)(2b;)zx(2b;)zy(2b 321 yxaaa

)(2

;)(2

;)(2

yxa

zxa

zya

zvvyvvxvva

bvbvbvG )()()(2

213132332211


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Primitive basic vectors of the

body-centered cubic lattice

First Brillouin zone of the body-

centered cubic lattice.


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The primitive cell of the reciprocal lattice is the

parallelepiped described by b1,b2,b3. The volume of this cell in reciprocal space is

The cell contains one reciprocal lattice point, because

each of the eight corner points is shared among eight

parallelepipeds. Each parallelepiped contains one-eighth of each of eight

corner points.

Another primitive cell is the central (Wigner-Seitz) cell of

the reciprocal lattice which is the first Brillouin zone. The zone is a regular 12-faced solid, a rhombic

dodecahedron.

3

3212

2

abbb

R i l i f


7/20PHY 3201 FIZIK KEADAAN PEPEJAL

Reciprocal Lattice to face-

centered cubic Lattice

The primitive translation vector of the fcc lattice are

The volume of the primitive cell is

The primitive translation vectors of the reciprocal lattice to the fcclattice are

These are similar to primitive translation vector of a bcc lattice,so that the bcc lattice is reciprocal to the fcc lattice.

The volume of the primitive cell of the reciprocal lattice is

The shortest Gs are the eight vectors:

)(a;)(a;)(a 213212211 yxazxazya 3

41

321 aaaV a

)(2b;)zx(2b;)zyx(2b 321 zyxa

yaa

)(2

zyxa

3

a

2

4



Brillouin zones of the fcc lattice.

The cell are in reciprocal space,

and the reciprocal lattice is body-

centered.

Primitive basis vectors

of the face-centered

cubic lattice



Reciprocal lattice types for some 3D lattices:

Direct lattice Reciprocal lattice

sc sc

bcc fccfcc bcc

hcp hcp



Fourier Analysis of the Basis

If the crystal structure a lattice with a basis, then we

should take into account scattering by the atoms

which have non-equivalent positions in a unit cell.

The intensity of radiation scattered in a given Bragg

peak will depend of the extent to which the rays

scattered from these basis sites interfere with one

another.

To take into account basis atoms, first, let rewrite

the scattering amplitude at the diffraction condition,

k=G, in terms of the integral over a unit cell

where the sum is taken over all the lattice vectors T.

T Cell

TriGeTrdVnF )()(


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Taken into account that n(r+T)=n(r) and that eiG.T=1, we get

where N is the number of cells in the solid, and we defined

the structure factor

Assuming that we have s atoms in a unit cell located at

r1,r2,r3, it is convenient to write the electron concentration

n(r) as the superposition of electron concentration functionnj associated with each atom jof the cell.

Cell

riGG erdVnS )(

G

Cell

riG

T Cell

riG NSerdVnNerdVnF )()(

s

j

jj rrnrn

1

)()(


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The structure factor may now be written as

where r-rj.

We define the atom ic form factoras

By combine two equation above, we get the st ruc ture

factor of the basisin the form

S

j

iGj

riGS

j

riGjjG edVneerrdVnS

11

)()(

iGjj edVnGf )()(

j

jriGjG efS .


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The usual form of this result follows on writing for atom j:

Then, for the reflection labeled by v1,v2,v3, we get

Therefore the stru cture factor of the basisbecome

321 azayaxr jjjj

)(2

)()(

321

321332211

jjj

jjjj

zvyvxv

azayaxbvbvbvrG

j

zvyvxvi

jGjjjefvvvS)(2

321321)(


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Structure Factor of the

bcc lattice

The bcc basis referred to the cubic cell has identical

atoms at x1=y1=z1=0 and at x2=y2=z2=. Since the

atoms are identical, the atomic form factors are same,

i.e. f1=f2=f.

Therefore, the structure factor of the basis become

The value of S is zero whenever the exponential has the

value -1, which is whenever the argument is -i x (oddinteger). Thus we have

S = 0 when v1+v2+v3 = odd integer

S = 2f when v1+v2+v3 = even integer

)321(321 1)(

vvviefvvvS


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Thus, diffraction peaks will be observed, e.g., from the(110), (200), (211) planes, but not from the(100),(111),(210) planes.

The later fact is due to the destructive interference fromthe basis atoms which cancel some peaks.

For example,in the figure for the (100) plane, the phasedifference between successive planes is , so that thereflected amplitude from two adjacent planes is 1+e-i=0.


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Structure Factor of the

fcc lattice

The basis of the fcc structure referred to the cubic cellhas identical atoms at 0,0,0; 0 ; 0 ; 0. Therefore

the equation become

which is non-zero (S=4f) only if all the indices are even or

all the indices are odd.

Allowed peaks are, e.g., (111), (200), (222), (220), (131).

Thus in the fcc lattice no reflections can occur for which

the indices are partly even and partly odd.

)21()31()32(321 1)(

vvivvivvieeefvvvS


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Comparison of XRD from KCl and

KBr powders. In KCl the numbers

of electrons of K+ and Cl ions are

equal. The scattering amplitudesf(K+) and f(Cl ) are almost

exactly equal, so that the crystal

looks to x-ray as if it were a

monatomic simple cubic lattice oflattice constant a/2. Only even

integers occur in the reflection

indices when these are based on

a cubic lattice of lattice constant

a. In KBr the form factor of Br isquite different to that of K+, and all

reflections of the fcc lattice are

present.


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Summary

Various statement of the Bragg condition:

The primitive translation vectors of the reciprocal lattice

A reciprocal lattice vector has the form

The scattered amplitude in the direction k=k+k=k+G is

proportional to the geometrical structure factor :

where j runs over the s atoms of the basis, and fj is the atomic form

factor of the jth atom of the basis.

2

2;;sin2 GGkGknd

321

321 2

aaa

aab

321

132 2

aaa

aab

321

213 2

aaa

aab

332211 bvbvbvG

)(2 321 vzvyvxi

j

Gir

jG

jjjj efefS


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The first Brillouin zone is the Wigner-Seitz primitive cell

of the reciprocal lattice. Only waves whose wavevectork drawn from the origin terminates on a surface of the

Brillouin zone can be diffracted by the crystal.

Crystal lattice First Brillouin zoneSimple cubic Cubic

Body-centered cubic Rhombic dodecahedron

Face-centered cubic Truncated octahedron

Lecture+11+MAK +Fourier+Analysis

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