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Lecture 2 continued

Chapter 2January 12, 2012

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Homework #2

due 1/19 Thursday Chapter 22.1 interplanar spacing

2.2 primitive cell calculations volume,vectors, brillouin zone

2.3 scattered radiation pattern, amplitude

and width2.4 examine scattering linewidth F, K, G

2.5 structure factor

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Diffraction We study the crystal structure using diffraction by

electrons, neutrons, X-ray photons.

E-beam elastically scatters from atomic lattice Reflected beams constructive interference

10

1 10 1000.1

1

Wavelength

()

Photon energy, keV

Neutron energy, 0.01eV

Electron energy, 100eV

X-ray photon

Neutrons

Electrons

These particleshave appropriatewavelength toresolve atoms

=5000 >> a=5

E=hR=hc/PP=hc/E=

X-rays: E=10 - 50 keVh ! 6.62x1034

J s

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Scattering Amplitude, F=NSWhat do we know already?

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Scattering Amplitude

X-Ray diffraction schematic and data

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Constructive Interference

Constructive interference occurs when reflected beampath lengths differ by n

2dsin=n Bragg Law

dsin

d

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Constructive Interference

Constructive interference occurs when reflected beampath lengths differ by n

2dsin=n Bragg Law

dsin

d

cos(k(r d)) ! cos(kr)k(r d) ! kr kd! kr 2Tn

kd! 2TP nd! nP

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Reciprocal Lattice Vectors

Crystal has 2 lattices Crystal lattice, vectors have dimensions of [length]

Reciprocal lattice, vectors have dimension of [1/length] every position in Fourier space may have a meaning as a

description of a wave

Microscope image is real space

Diffraction pattern is reciprocal spaceaX

n(x)

x

G

a a aa

0 a

T2a

T4

a

T4 a

T2

bX

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Diffraction Conditions

reciprocal space

v

kvr| Difference in phase angle J

P

Tsin

2r!

|Jsinr Difference in path length over dV

Incoming beam~ eikr Outgoing beam

~ eikr

O

r

rsin Crystal specimen

k

X

'kX

dV

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Diffraction Conditions contd.

For the diffracted wave, the difference in phase angleis: (vk vr! (N

rkk XXX

!( )'(J

The total difference in phase angle is:

])'(exp[ rkkiXXX

! Phase factor

of the wave scattered from dV at rrelative to the wave scatteredfrom a volume element at the origin O.

'kX

kX

kX

('kkkXXX

!(

Scattering vector measures

change in wavevector

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Diffraction Conditions Ewald SphereEwald sphere determines which lattice planes (represented by the grid points on

the reciprocal lattice) will result in a diffracted signal for a given wavelength, , ofincident radiation.

The incident wave on the crystal has a wavevectorKi =2 / .

kGXX

(!

Bragg condition:

The diffracted wave has a wave vector Kf.

If no energy is gained or lost (it is elastic) then

Kf has the same length as Ki. K = 0

The difference between the wave-vectors ofdiffracted and incident wave is defined asscattering vectorK = Kf Ki.

Since Ki and Kf have the same length thescattering vector must lie on the surface of asphere of radius 2 / . This sphere is called

the Ewald sphere.

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Diffraction Conditions mapreal to reciprocal

Reciprocal lattice vectors G determines possible reflection We want an expression to relate Bragg Condition to G.

Next define G using reciprocal lattice vectors

332211 bvbvbvGXXXX !

321

32

1

2

aaa

aa

b XXX

XXX

vv

!T

Translates reciprocal to real

321

13

2

2

aaa

aa

b XXX

XXX

vv

!

T

321

21

3

2

aaa

aa

b XXX

XXX

vv

!

T

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Diffraction Conditions contd.

F! dVn(

vr) exp[i(

v

kv

k') vr]! NS

G

(! )exp()( rkirdVn XXXIntegral volume Electron number Phase factor

Specify n(r) for atomic lattice G dependence !G

GrGinrn

XX

TTX)exp()(

(! GG rkGidVnF XX

XXX

])(exp[Reciprocal lattice vector

Scattering vector:kX

(

:GX

kGXX

(!G

VnF X!For

Scattering Amplitude:

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Diffraction Conditions Ewald Sphere

Now complete the description for diffraction Bragg Law Diffraction conditions

k = incident wave intersect crystal @ lattice point Ewald sphere intersect lattice @ point

k is reflected wave G connects

kX

GX

'kX

U U21

2

1 2

1

2

We want to express these

conditions in terms of d vs.

crystal wave

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Elastic Scattering G vector

Energy is conserved 'kk !

GkXX

!('kGkXXX

!2'2

)( kGk

XXX

!''2 kkGGGkkkXXXXXXXX

!2'22 2 kGGkk !

XX22

02 ! GGkXX

22 GGk !XX

GG

XX

!Express G in terms of d, k in terms of

Where d is distance between (hkl) planes normal to G 0!dGXX

'ww JJ !

'kkkXXX

!(

2'2 kk !

2'2 kk !

dG

T2!

P

T2!k

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Elastic Scattering contd.

321 blbkbhGXXXX

!

Ghkld

T2)( !

dG

T2!

Rewrite 22 GGk !XX

2)2

(sin22

2dd

TU

TP

T!

d

TU

P

T 2sin

22 !

PU !sin2d Bragg condition

Here we considered diffraction conditions based on reciprocal lattice.

Reciprocal lattice vector:

Distance between (hkl) planes normal to G:

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Laue Equations also derive E.S.

Laue equations relate to real lattice to reciprocal lattice

321 ,aabTTT

B

ijji ab TH2!TT

132,aabTTT

B 213 ,aabTTT

B

332211 bvbvbvGTTTT

!

321

321

2

aaa

aab XXX

XXX

v

v!

T

321

132

2

aaa

aab XXX

XXX

v

v!

T

321

21

3

2

aaa

aab XXX

XXX

v

v!

T

Axis vectors of the reciprocal lattice:

Each vector is orthogonal to two axis vectors of the crystal lattice:

Thus: jiij !! if1H

jiij {! if0H

Points in the reciprocal lattice are:

are integers, is a reciprocal lattice vector321 ,, vvv G

T

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Laue condition contd.

Thus can be rewritten in terms ofkGXX

(,

GakaXXXX

!( 11

13322111 2)( vbvbvbva T!XXXX



Relates back to Ewald Sphere

321 ,, aaaXXX

kX

GX

'kX

U U21

2

But why a.G?

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Brillouin Zone

Provide geometrical, visual interpretation of

Use Wigner-Seitz primitive cell in reciprocal space.

22 GGk !XX

1. Draw lines to connect central lattice point to surrounding lattice point.

2. Draw new lines (planes) which bisect these connecting linesperpendicularly.

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Brillouin Zone contd.

A similar construction describes the diffraction conditions

1kX

1 2

OC

D

CGX

2

1

DGX

2

1

2kX

CGX

2

1

CGX

O C

Normal to plane 1 from O

DGX

O D

2kX Satisfies

1kX

Incident vector satisfies *

*

*2)

2

1()

2

1( GGk

XXX!

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Brillioun Zone Oblique lattice

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Brillouin Zone contd.

All incident waves with wave vectork terminating onbisecting plane will satisfy Bragg condition

The central square is:

a primitive cell of thereciprocal lattice

A Wigner-Seitz cell of thereciprocal lattice

The first Brillouin zone

the smallest volume entirelyenclosed by planes that arethe perpendicular bisectors ofthe reciprocal lattice vectorsdrawn from the origin.

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Reciprocal Lattice to sc Lattice

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Reciprocal Lattice to sc Lattice

The primitive translation vectors of a sc lattice:

;1 xaa !X

;2

yaa !X

.3

zaa !X

lengthunitofvectorsorthogonalare,, zyx

The volume of the cell:

The primitive translation vectors of the reciprocal lattice:

;)/2(1 xab T!X

;)/2(2 yab T!X

;)/2(3

zab T!X

The reciprocal lattice is a sc lattice, with lattice constant 2/a.

The first Brillouin zone of sc crystal lattice is a cube ofedge 2/a and of volume (2/a)3

3

321aaaaV !v! XXX

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BCC Real and Reciprocal

'kk !'ww JJ ! 2'2 kk !

va1

va2

v

a3

b1b2

b3

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Reciprocal Lattice to bcc Lattice

);(2

12

zyxaa !X

);(2

13

zyxaa !X

The primitive translation vectors of a bcc lattice:

);(2

11 zyxaa !X


cubealconventiontheofsidetheisa

3

3212

1aaaaV !v!

XXX The volume of the cell:


);)(/2(1 zyab ! TX

);)(/2(2 zxab ! TX

);)(/2(3 yxab ! TX

The reciprocal lattice is a fcc lattice.

The general reciprocal lattice vector is:

];)()())[(/2(213132332211zvvyvvxvvabvbvbvG !! T

XXXX

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Reciprocal Lattice to fcc Lattice

);(2

12 zxaa !X

);(2

13

yxaa !X

The primitive translation vectors of a fcc lattice:

);(2

11 zyaa !X


cubealconventiontheofsidetheisa


vb1 !2T

a( x y z); );(

22 zyx

ab ! TX );(2

3zyx

ab ! TX

The reciprocal lattice is a bcc lattice.

The volume of the cell: 3321

4

1aaaaV !v!

XXX

The volume of the

primitive cell: V !

vb

1vb2v

vb3

!2T

4

a3

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FCC Real and Reciprocal

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Fourier Analysis of Basis Integrated contribution of electron density (over V) to

scattering amplitude:

G

cell

GNSrGirdVnNF XX

XXX!! )exp()(

Number of cells Amplitude Phase relation between crystaland scattering beam

Structure factor

Where n(r) is a superposition of electron concentrations nj from atom j in a cell

!s

j

jj rrnrn1

)()(XXX

s: number of atoms in cell

j

O

jrX

rX

Single cell

Note that F and S include both real and reciprocal information.

Real Space!!

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Structure Factor

! jjjG rGirrdVn

S)exp()(

XXXXStructure factor

)exp()()exp( VVXXXXX

! GidVnrGi jj

j

jrrXXX

!Vwhere

)exp()( VVXXX

!

GidVnf

jj

!j

jjGrGifS )exp(XX

Atomic form factor

Structure factor of the basis

321 azayaxr jjjjXXXX

!

)()(321332211 azayaxbvbvbvrG jjjjXXXXXXXX

! )(2321 jjj zvyvxv ! T

Describes a specificreflection condition

Describes crystalin real space

)](2exp[)(321321 jjj

j

jGzvyvxvifvvvS ! TXso

Scattered intensity2

GSXw can be complexGSX

What does SG indicate? - it identifies crystal symmetries

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SG for bcc lattice

bcc basis has identical atoms at3121111

azayaxrXXXX

!

3222122 azayaxrXXXX

!

)](2exp[)( 3212,1

321 jjj

j

jG zvyvxvifvvvS ! ! T)]

2

1

2

1

2

1(2exp[1

32121 vvviff ! T

)]}(exp[1{321vvvifSG ! T

f1=f2 identical atoms

)000(),,( 111 !zyx

)2

1

2

1

2

1(),,( 222 !zyx

2rX

j=2

j=1O

For bcc basis SG=0 when the exponential = -1

integer;oddhenw0321

!! vvvSGinteger;evenwhen2

321!! vvvfSG

J ! TVJ ! 2T

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SG for bcc lattice contd.

Consider metallic Na bcc basis The diffraction pattern DOES NOT include

Not (100), (300), (111), (221), Has (110), (200),

TJ 2!(Constructive interference:In bcc, second atomic plane causes phase difference => destructive

1st plane

2nd plane

3rd plane

Total Phasedifference 2

a

(100), (200)

(100)

T2'!kkXX

Howdoweanalyzea crystalgrownonthe(001) crystalplane?

SG(001)

! f{1 exp[iT (v3

! 1)] ! 1 1}! 0

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SG for fcc lattice

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123B_1_EE 123B W 12 lect 2 chap 2