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8/3/2019 123B_1_EE 123B W 12 lect 2 chap 2
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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
23322112 2)( vbvbvbva T!XXXX
33322113 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
lengthunitofvectorsorthogonalare,, zyx
cubealconventiontheofsidetheisa
3
3212
1aaaaV !v!
XXX The volume of the cell:
The primitive translation vectors of the reciprocal lattice:
);)(/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
lengthunitofvectorsorthogonalare,, zyx
cubealconventiontheofsidetheisa
The primitive translation vectors of the reciprocal lattice:
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