Diffraction:%20Intensity

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Diffraction Intensity (From Chapter 4 of
Textbook 2 and Chapter 9 of Textbook 1)
Electron ? atoms ? group of atoms or structure ?
Crystal (poly or single)
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Scattering by an electron
?
?
?? ?/2
P
r
?0 4??10-7 mkgC-2
by J.J. Thomson
?
a single electron charge e (C), mass m (kg),
distance r (meters)
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z
?
?
?
P
r
2?
y
O
Random polarized
x
? ?yOP ?/2
y component
? ?zOP ?/2 -2?
z component
Polarization factor
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Pass through a monochromator first (Bragg angle
?M) ? the polarization factor is ?
polarization is not complete random anymore
z
z
Random polarized
P
P?
P?
r
r?
y
2?
y
2?M
O
O
x
x
(Homework)
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M
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Atomic scattering (or form) factor a single free
electron? atoms
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x2
path different (O and dV) R-(x1 x2).
x1
dV
r
O
s
s0
2?
R
Differential atomic scattering factor (df)
Ee the magnitude of the wave from a bound
electron
Phase difference
Electron density
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Spherical integration dV dr(rd?) (rsin?d?)
rsin?d?
r 0 - ?
? 0 - ?
? 0 - 2?
dr
?
rsin(?d?)d?
d?
r
d?
?
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2
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Evaluate (S - S0)?r S - S0rcos? (S -
S0)/2 sin?.
S
S-S0
S0
2?
Let
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For n electrons in an atom
Tabulated
For ? 0, only k 0 ? sinkr/kr 1.
Number of electrons in the atom
equal to 1 bound electrons
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Anomalous Scattering
Previous derivation free electrons! Electrons
around an atom free?
k
free electron harmonic oscillator
m
Assume
Resonance frequency
Forced oscillator
Assume
Assume
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Same frequency as F(t), amplitude(?, ?0)
? ?0 ? C is ? in reality friction term exist ?
no ?
Oscillator with damping (friction ? v)
assume c m?
Assume x x0ei?t
Real part and imaginary part
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?? if ? ? 0
? ?
? E
Resonance
? X-ray frequency ?0 bounded electrons around
atoms ? ? ?0 ? electron escape ? of electrons
around an atom ? ? f ? (?f? correction term)
imaginary part correction ?f?? ? ? (linear
absorption coefficient)
f ?f ? i?f ?
real
imaginary
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Examples Si, 400 diffraction peak, with Cu K?
(0.1542 nm)
0.3
0.4
8.22
7.20
Anomalous Scattering correction
Atomic scattering factor in this case
7.526-0.20.4i 7.3260.4i
?f? and ?f? International Table for X-ray
Crystallography V.III
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Structure factor atoms ? unit cell
How is the diffraction peaks (hkl) of a structure
named?
Unit cell
How is an atom located in a unit cell affect the
h00 diffraction peak?
Miller indices (h00)
1
1?
3
3?
A
?
2
2?
S
R
B
path difference11? and 22? (NCM)
a
plane (h00)
C
N
M
why? Meaningful!
path difference 11? and 33? (SBR)
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phase difference (11? and 33?)
position of atom B fractional coordinate of a u
? x/a.
the same argument ? B x, y, z ? x/a, y/b, z/c ?
u, v, w Diffraction from (hkl) plane ?
F amplitude of the resultant wave in terms of
the amplitude of the wave scattered by a single
electron.
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N atoms in a unit cell fn atomic form factor of
atom n
F (in general) a complex number.
How to choose the groups of atoms to represent a
unit cell of a structure? 1. number of atoms in
the unit cell 2. choose the representative atoms
for a cell properly (ranks of equipoints).
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? Example 1 Simple cubic 1 atoms/unit cell
000 and 100, 010, 001, 110, 101, 011, 111
equipoints of rank 1 Choose any one will have
the same result!
for all hkl
? Example 2 Body centered cubic 2
atoms/unit cell 000 and 100, 010, 001, 110,
101, 011, 111 equipoints of rank 1 ½ ½ ½
equipoints of rank 1 Two points to choose
000 and ½ ½ ½.
when hkl is even
when hkl is odd
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? Example 3 Face centered cubic 4
atoms/unit cell 000 and 100, 010, 001, 110,
101, 011, 111 equipoints of rank 1 ½ ½ 0, ½
0 ½, 0 ½ ½, ½ ½ 1, ½ 1 ½, 1 ½ ½ equipoints of
rank 3 Four atoms chosen 000, ½ ½ 0, ½ 0 ½,
0 ½ ½.
when h, k, l is unmixed (all evens or all odds)
when h, k, l is mixed
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? Example 4 Diamond Cubic 8 atoms/unit
cell 000 and 100, 010, 001, 110, 101, 011,
111 equipoints of rank 1
½ ½ 0, ½ 0 ½, 0 ½ ½, ½ ½ 1, ½ 1 ½, 1 ½ ½
equipoints of rank 3 ¼ ¼ ¼, ¾ ¾ ¼, ¾ ¼ ¾, ¼ ¾ ¾
equipoints of rank 4 Eight atoms chosen 000, ½
½ 0, ½ 0 ½, 0 ½ ½ (the same as FCC), ¼ ¼ ¼, ¾ ¾
¼, ¾ ¼ ¾, ¼ ¾ ¾!
FCC structure factor
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when h, k, l are all odd
when h, k, l are all even and h k l 4n
when h, k, l are all even and
h k l ? 4n
when h, k, l are mixed
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? Example 5 HCP 2 atoms/unit cell 8
corner atoms equipoints of rank 1 1/3 2/3 ½
equipoints of rank 1 Choose 000, 1/3 2/3 1/2.
( 1/3 2/3 1/2)
(001)
(000)
(010)
Set h 2k/3 l/2 g
(100)
(110)
equipoints
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h 2k
l
3m 3m 3m?1 3m?1
even odd even odd
1 0 0.25 0.75
4f 2 0 f 2 3f 2
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Multiplicity Factor Equal d-spacings ? equal
?B E.g. Cubic (100), (010), (001),
(-100), (0-10), (00-1) Equivalent ?
Multiplicity Factor 6 (110), (-110),
(1-10), (-1-10), (101), (-101), (10-1),(-10-1),
(011), (0-11), (01-1), (0-1-1)
Equivalent ? Multiplicity Factor
12 lower symmetry systems ? multiplicities
?. E.g. tetragonal (100) equivalent
(010), (-100), and (0-10) not
with the (001) and the (00-1). 100 ?
Multiplicity Factor 4 001 ?
Multiplicity Factor 2
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Multiplicity p is the one counted in the point
group stereogram. In cubic (h ? k ? l)
p 48
3x2x23 48
p 24
3x23 24
p 24
3x23 24
p 12
3x22 12
p 8
23 8
p 6
3x2 6
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Lorentz factor
? dependence of the integrated peak intensities
1. finite spreading of the intensity peak 2.
fraction of crystal contributing to a diffraction
peak 3. intensity spreading in a cone
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1
2?
Imax
?
Imax/2
Intensity
?2
B
Integrated Intensity
2?
?B
2?B
?1
Diffraction Angle 2?
1?
2?
2
path difference for 11?-22? AD CB acos?2 -
acos?1 acos(?B-??) - cos (?B??)
2asin(??)sin?B 2a?? sin?B.
D
C
?1
?2
a
A
B
1?
N?
2Na?? sin?B ? ? completely cancellation (1-
N/2, 2- (N/21) )
Na
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Maximum angular range of the peak
Imax ? ?? ? 1/sin?B, Half maximum B ? 1/cos?B
(will be shown later) ? integrated intensity ?
ImaxB ? (1/sin?B)(1/cos?B) ? 1/sin2?B.
2
number of crystals orientated at or near the
Bragg angle
??
?/2-?
Fraction of crystal
r
??
?
crystal plane
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3
diffracted energy equally distributed
(2?Rsin2?B) ? the relative intensity per unit
length ? 1/sin2?B.
2?B
Lorentz factor
Lorentzpolarization factor (omitting constant)
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Absorption factor X-ray absorbed during its in
and out of the sample. Hull/Debye-Scherrer
Camera A(?) A(?) ? as ??.
Diffractometer
dID
1cm
I0
?
C
?
Incident beam I0 1cm2 incident angle ?. Beam
incident on the plate
A
x
dx
B
l
2?
a volume fraction of the specimen that are
at the right angle for diffraction b diffracted
intensity/unit volume
? linear absorption coefficient
? volume l ? dx ? 1cm ldx. actual diffracted
volume aldx Diffracted intensity Diffracted
beam escaping from the sample
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If ? ? ? ?
Infinite thickness dID(x 0)/dID(x t) 1000
and ? ? ?).
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Temperature factor (Debye Waller factor) Atoms
in lattice vibrate (Debye model)
? (1) lattice constants ? 2? ? ? (2) Intensity
of diffracted lines ? ? (3) Intensity of the
background scattering ?.
Temperature ?
u
u
d
d
low ?B
high ?B
Lattice vibration is more significant at high
?B (u/d) ? as ?B ?
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Formally, the factor is included in f as Because
F f 2 ? factor e-2M shows up What is M?
Mean square displacement Debye
h Planks constant T absolute temperature m
mass of vibrating atom ? Debye temperature of
the substance x ?/T ?(x) tabulated function
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m ? atomic weight (A)
1
e-2M
0
sin? / ?
I
TDS
2? or sin?/?
Temperature (Thermal) diffuse scattering (TDS) ?
as ? ?
I ? as ?? peak width B ? slightly as T ?
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Summary Intensities of diffraction peaks from
polycrystalline samples
Diffractometer
Other diffraction methods
Match calculation? Exactly difficult
qualitatively matched.
Perturbation preferred orientation Extinction
(large crystal)
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Example Debye-Scherrer powder pattern of Cu made
with Cu radiation
Cu Fm-3m, a 3.615 Å
1 2 3 4 5 6 7 8
line hkl h2k2l2 sin2? sin? ? (o) sin?/?(Å-1) fCu
1 2 3 4 5 6 7 8 111 200 220 311 222 400 331 420 3 4 8 11 12 16 19 20 0.1365 0.1820 0.364 0.500 0.546 0.728 0.865 0.910 0.369 0.427 0.603 0.707 0.739 0.853 0.930 0.954 21.7 25.3 37.1 45.0 47.6 58.5 68.4 72.6 0.24 0.27 0.39 0.46 0.48 0.55 0.60 0.62 22.1 20.9 16.8 14.8 14.2 12.5 11.5 11.1
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Structure Factor ?
If h, k, l are unmixed
If h, k, l are mixed
1 9 10 11 12 13 14
line F2 P Relative integrated intensity Relative integrated intensity Relative integrated intensity
line F2 P Calc.(x105) Calc. Obs. Calc.(x105) Calc. Obs. Calc.(x105) Calc. Obs.
1 2 3 4 5 6 7 8 7810 6990 4520 3500 3230 2500 2120 1970 8 6 12 24 8 6 24 24 12.03 8.50 3.70 2.83 2.74 3.18 4.81 6.15 7.52 3.56 2.01 2.38 0.71 0.48 2.45 2.91 10.0 4.7 2.7 3.2 0.9 0.6 3.3 3.9 Vs S s s m w s s
111 200 220 311 222 400 331 420
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(23 8)
p 8
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Dynamic Theory for Single crystal
Kinematical theory Dynamical theory
S0
S
K0
K0
K1
Refraction
K1
?
PRIMARY EXTINCTION
?
K2
K2
K0 K1 ?/2 K1 K2 ?/2 K0 K2 ?
destructive interference
(hkl)
K1
K2
Negligible absorption
I ? F not F2!
e electron charge m electron mass N of
unit cell/unit volume.
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Width of the diffraction peak ( 2s)
FWHM for Darwin curve 2.12s 5 arcs lt ?? lt 20
arcs