5/15/2015 1 Solid State Physics 2. X-ray Diffraction.

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Solid State Physics

2. X-ray Diffraction

Diffraction

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Diffraction

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1 2 3sin , , , ...m

mW

Diffraction

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Diffraction using Light

http://physics.kenyon.edu/coolphys/FranklinMiller/protected/Diffdouble.html

One Slit

Two Slits

Diffraction Grating

d

m sin

Diffraction

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 The diffraction pattern formed by an opaque disk consists of a small bright spot in the center of the dark shadow, circular bright fringes within the shadow, and concentric bright and dark fringes surrounding the shadow.

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Diffraction for CrystalsPhotonsElectronsNeutrons

Diffraction techniques exploit the scattering of radiation from large numbers of sites. We will concentrate on scattering from atoms, groups of atoms and molecules, mainly in crystals.

There are various diffraction techniques currently employed which result in diffraction patterns. These patterns are records of the diffracted beams produced.

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What is This Diffraction?

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Bragg Law

nd sin2

William Lawrence

Bragg1980 - 1971

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Mo 0.07 nmCu 0.15 nmCo 0.18 nmCr 0.23 nm

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Monochromatic Radiation

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Diffractometer

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Nuts and Bolts

The Bragg law gives us something easy to use,To determine the relationship between diffractionAngle and planar spacing (which we already knowIs related to the Miller indices).

But…We need a deeper analysis to determine theScattering intensity from a basis of atoms.

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Reciprocal Lattices Simple Cubic Lattice

1 2 3ˆ ˆ ˆa x a y a za a a

1 2 32 2 2

ˆ ˆ ˆx y z

******************************************G G G

a a a

The reciprocal lattice is itself a simple cubic lattice with lattice constant 2/a.

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BCC Lattice1 1

1 22 2

13 2

ˆ ˆ ˆ ˆ ˆ ˆa ( x y z) a (x y z)

ˆ ˆ ˆ a (x y z)

a a

a

1 2 32 2 2

ˆ ˆ ˆ ˆ ˆ ˆy z x z x y

******************************************G G G

a a a

The reciprocal lattice is represented by the primitive vectors of an FCC lattice.

Reciprocal Lattices

310 1 2 3 2a a a a

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FCC Lattice

1 11 22 2

13 2

ˆ ˆ ˆ ˆ ˆ ˆ( x y z) (x y z)

ˆ ˆ ˆ (x y z)

****************************

**************G a G a

G a

1 2 3

2 2 2ˆ ˆ ˆ ˆ ˆ ˆa y z a x z a x y

a a a

The reciprocal lattice is represented by the primitive vectors of an BCC lattice.

Reciprocal Lattices

30 1 2 3a a a a

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Drawing Brillouin ZonesWigner–Seitz cell

The BZ is the fundamental unit cell in the space defined by reciprocal lattice vectors.

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Drawing Brillouin Zones

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Back to Diffraction

Diffraction is related to the electron density.Therefore, we have a...

The set of reciprocal lattice vectors determines the possible x-ray reflections.

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The difference in path length of the of the incident wave at the points O and r is sinrThe difference in phase angle is rk

sin2

r

For the diffracted wave, the phase difference is k r ****************************

So, the total difference in phase angle is r)kk(

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Diffraction Conditions Since the amplitude of the wave scattered

from a volume element is proportional to the local electron density, the total amplitude in the direction k is

dVen

dVenf

i

i

r )k(

r kk

)r(

)r(

kkk

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Diffraction Conditions When we introduce the Fourier

components for the electron density as before, we get

( k) r i ss

s

f n e dV

ks Constructive

Interference

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Diffraction Conditions

kkk

k ks

ks

nd sin2

2 2

2

(k )

or 2 k

s k

s s

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r

cell ( r ) i s

sF N n e dV NS

Diffraction Conditions For a crystal of N cells, we can write down

)rr()r(1

j

s

jjnn

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r

cell j

r

( r r )

( )j

i s

s jj

i s i s

jj

S n e dV

e n e dV

Diffraction Conditions The structure factor can now be written as

integrals over s atoms of a cell.

( )

i s

j jf n e dV

Atomic formfactor

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Diffraction Conditions Let

Then, for an given h k l reflection

1 2 3a a aj j j jr x y z

1 2 3 1 2 3r ( a a a ) ( a a a )j j j j

j j j

s h k l x y z

hx ky lz

2 j j ji hx ky lz

s jj

S f e

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Diffraction Conditions For a BCC lattice, the basis has identical

atoms at and

The structure factor for this basis is

S is zero when the exponential is i × (odd integer) and S = 2f when h + k + l is even.

So, the diffraction pattern will not contain lines for (100), (300), (111), or (221).

)0,0,0(),,( 111 zyx ),,(),,( 21

21

21

222 zyx

)1( 2 lkhiG efS

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Diffraction Conditions For an FCC lattice, the basis has identical

atoms at

The structure factor for this basis is

S = 4f when hkl are all even or all odd. S = 0 when one of hkl is either even or

odd.

0 and ,0 ,0 ,000 21

21

21

21

21

21

)1( khilhilkiG eeefS

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Structure Determination

222

lkh

ad

Simple

Cubic

2222

2

4sin lkh

a

When combined with the Bragg law:

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(degrees) sin2 ratios hkl

11.44 0.0394 1 100

16.28 0.0786 2 110

20.13 0.1184 3 111

23.38 0.1575 4 200

26.33 0.1967 5 210

29.07 0.2361 6 211

34.14 0.3151 8 220

36.53 0.3543 9 300, 221

38.88 0.3940 10 310

X-ray powder pattern determined using Cu K radiation, = 1.542 Å

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Structure Determination (310)

angstroms 88.3

104

)5420.1(3940.0

4sin

2

2

2222

2

aa

lkha

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