7c X RayDiffraction
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Transcript of 7c X RayDiffraction
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X-RAY DIFFRACTION
X- Ray Sources
Diffraction: Braggs Law
Crystal Structure Determination
Elements of X-Ray DiffractionB.D. Cullity & S.R. StockPrentice Hall, Upper Saddle River (2001)
X-Ray Diffraction: A Practical ApproachC. Suryanarayana & M. Grant Norton
Plenum Press, New York (1998)
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For electromagnetic radiation to be diffracted the spacingin the grating should be of the same order as the wavelength
In crystals the typical interatomic spacing ~ 2-3 so thesuitable radiation is X-rays
Hence, X-rays can be used for the study of crystal structures
Beam of electrons TargetX-rays
An accelerating (/decelerating) charge radiates electromagnetic radiation
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Intensity
Wavelength ()
Mo Target impacted by electrons accelerated by a 35 kV potential
0.2 0.6 1.0 1.4
Whiteradiation
Characteristic radiation due to energy transitionsin the atom
K
K
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Target Metal Of K radiation ()
Mo 0.71
Cu 1.54
Co 1.79
Fe 1.94
Cr 2.29
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A beam of X-rays directed at a crystal interacts with theelectrons of the atoms in the crystal
The electrons oscillate under the influence of the incoming
X-Rays and become secondary sources of EM radiation
The secondary radiation is in all directions
The waves emitted by the electrons have the same frequency
as the incoming X-rays coherentThe emission will undergo constructive or destructive
interference with waves scattered from other atoms
Incoming X-rays
Secondary
emission
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Sets Electron cloud into oscillation
Sets nucleus (with protons) into oscillation
Small effect neglected
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Oscillating charge re-radiates In phase with the incoming x-rays
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BRAGGs EQUATION
d
The path difference between ray 1 and ray 2 = 2d Sin
For constructive interference: n= 2d Sin
Ray 1
Ray 2
Deviation = 2
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Incident and scatteredwaves are in phase if
Scattering from across planes is in phase
In plane scattering is in phase
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Extra path traveled by incoming wavesAY
Extra path traveled by scattered wavesXB
These can be in phase if and only ifincident= scattered
But this is still reinforced scatteringand NOT reflection
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Note that in the Braggs equation:
The interatomic spacing (a) along the plane does not appear Only the interplanar spacing (d) appearsChange in position or spacing of atoms along the plane should not affect
Braggs condition !!
d
Note: shift (systematic) isactually not a problem!
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Note: shift is actually not a problem! Why is systematic shift not a problem?
n AY YB [180 ( )] ( )AY XY Cos XY Cos
( )YB XY Cos [ ( ) ( )] [2 ]n AY YB XY Cos Cos XY Sin Sin
( )d
SinXY
[2 ] 2d
n Sin Sin d SinSin
2n d Sin
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Consider the case for which 12
Constructive interference can still occur if the difference in the path lengthtraversed by R1and R2before and after scattering are an integral multiple of the
wavelength(AY XC) = h (h is an integer)
1Cos
a
AY
2Cosa
XC hCosaCosa 21
hCosCosa 21
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Laues equations
S0incoming X-ray beam
S Scattered X-ray beam
hSSa )(0
kSSb )( 0
lSSc )(0
hCosCosa 21Generalizing into 3D
kCosCosb 43
lCosCosc 65
This is looking at diffraction from atomic arrays and not planes
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A physical picture of scattering leading to diffraction is embodied in Laues equations
Braggs method of visualizing diffraction as reflection from a set of planes is a different
way of understanding the phenomenon of diffraction from crystals
The plane picture (Braggs equations) are simpler and we usually stick to them
Hence, we should think twice before asking the question: if there are no atoms in the
scattering planes, how are they scattering waves?
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Braggs equation is a negative law
If Braggs eq. is NOT satisfied NO reflection can occur
If Braggs eq. is satisfied reflectionMAY occur
Diffraction = Reinforced Coherent Scattering
Reflection versus Scattering
Reflection Diffraction
Occurs from surface Occurs throughout the bulk
Takes place at any angle Takes place only at Bragg angles
~100 % of the intensity may be reflected Small fraction of intensity is diffracted
X-rays can be reflected at very small angles of incidence
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n= 2d Sin
n is an integer and is the order of the reflection
For Cu Kradiation (= 1.54 ) and d110= 2.22
n Sin
1 0.34 20.7 First order reflection from (110)
2 0.69 43.92Second order reflection from (110)
Also written as (220)
222 lkh
adhkl
8220
ad
2110
ad
2
1
110
220
d
d
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sin2 hkldn
In XRD nthorder reflection from (h k l) is considered as 1storder reflectionfrom (nh nk nl)
sin2 n
dhkl
sin2 nnn lkhd
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Intensity of the Scattered electrons
Electron
Atom
Unit cell (uc)
Scattering by a crystal
A
B
C
Polarization factor
Atomic scattering factor (f)
Structure factor (F)
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Scattering by an Electron
),( 00
Sets electron into oscillation
Scattered beams),( 00
Coherent(definite phase relationship)
A
The electric field (E)is the main cause for the acceleration of the electron The moving particle radiates most strongly in a direction perpendicular to its
motion The radiation will be polarized along the direction of its motion
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x
z
r
P
Intensity of the scattered beam due to an electron (I) at a point Psuch that r >>
2
2
42
4
0r
Sin
cm
eII
For a wave oscillating in z direction
For an polarized wave
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 30 60 90 120 150 180 210 240 270 300 330 360
t
Cos(t)
The reason we are able to
neglect scattering from theprotons in the nucleus
The scattered rays are also plane polarized
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2
2
42
4
0r
Sin
cm
eII
F l i d E is the measure of the amplitude of the wave
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For an unpolarized wave E is the measure of the amplitude of the waveE2= Intensity
222
zy EEE zy III
000
2
240 2 4 2
y
Py y
SineI I
m c r
IPy= Intensity at point P due to Ey
IPz= Intensity at point P due to Ez
240 2 4 2
z
Pz z
SineI I
m c r
Total Intensity at point P due to Ey& Ez
2 240 2 4 2
y z
P
Sin SineI I
m c r
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2 240 2 4 2
y z
P
Sin SineI I
m c r
2 2 2 2 2 21 1 2y z y z y zSin Sin Cos Cos Cos Cos
2 2 2 1x y zCos Cos Cos Sum of the squares of the direction cosines =1
2 2 2 22 2 1 ( ) 1 ( )y z x xCos Cos Cos Cos Hence
24
0 2 4 2
1 ( )xP
CoseI I
m c r
2
4
0 2 4 21 (2 )
PCoseI I
m c r
In terms of 2
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0
0.2
0.4
0.6
0.8
1
0 30 60 90 120 150 180
2t
[Cos(2t)]^2
In general P could lie anywhere in 3D space For the specific case of Bragg scattering:
The incident direction IOThe diffracted beam direction OPThe trace of the scattering plane BBAre all coplanar
OP is constrained to be on the xz plane
x
z
r
P
2
2
2
42
4
0
2
r
Cos
cm
eII
F l i d E is the measure of the amplitude of the wave
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For an unpolarized wave E is the measure of the amplitude of the waveE2= Intensity
222
zy EEE
zy III
000
2
242
4
02
2
42
4
0
12
rcm
eI
r
Sin
cm
eII yyPy
IPy= Intensity at point P due to Ey
IPz= Intensity at point P due to Ez
2
2
42
4
02
2
42
4
0
222
r
Cos
cm
eI
r
Sin
cm
eII zzPz
The zx plane is to the y direction: hence, = 90
1
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-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 30 60 90 120 150 180 210 240 270 300 330 360
t
Cos(t)
2
2
00
42
4 2
r
CosII
cm
eIII
zy
PzPyP
2
2
42
4
0 21
2 r
Cos
cm
eIIP
Scattered beam is not unpolarized
Forward and backward scattered intensity higher than at 90Scattered intensity minute fraction of the incident intensity
Very small number
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Polarization factorComes into being as we usedunpolarized beam
2
212
42
4
2
0 Cos
cm
e
r
IIP
0
0.2
0.4
0.6
0.8
1
1.2
0 30 60 90 120 150 180 210 240 270 300 330 360
2t
(1+Cos(2t)^2)/2
B
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B Scattering by an Atom
Scattering by an atom [Atomic number, (path difference suffered by scattering from each e, )]
Scattering by an atom [Z, (, )] Angle of scattering leads to path differences
In the forward direction all scattered waves are in phase
electronanbyscatteredwaveofAmplitudeatomanbyscatteredwaveofAmplitude
FactorScatteringAtomicf
f
)(Sin(1)
0.2 0.4 0.6 0.8 1.0
10
20
30
Schematic
)(Sin
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Coherent scattering Incoherent (Compton) scattering
Z Sin() /
B S i b A
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B Scattering by an Atom
BRUSH-UP
The conventional UC has lattice points as the vertices
There may or may not be atoms located at the lattice points
The shape of the UC is a parallelepiped (Greekparalllepipedon)in 3D
There may be additional atoms in the UC due to two reasons:
The chosen UC is non-primitive
The additional atoms may be part of the motif
C S tt i b th U it ll ( )
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C Scattering by the Unit cell (uc)
Coherent ScatteringUnit Cell (UC) is representative of the crystal structureScattered waves from various atoms in the UC interfere to create the diffraction pattern
The wave scattered from the middle plane is out of phase with the onesscattered from top and bottom planes
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d(h00)
B
Ray 1 = R1
Ray 2 = R2
Ray 3 = R3
Unit Cell
x
M
C
N
R
B
S
A
'
1R
'
2
R
'
3R
(h00) planea
a
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h
adAC h 00
::::ACMCN
xABRBS ::::
ha
xx
AC
AB
)(2 0021 SindMCN hRR
ha
x
AC
ABRBSRR 31
2
axh
hax
RR 22
31 xcoordinatefractional
a
x xhRR 231
Extending to 3D 2 ( )h x k y l z Independent of the shape of UC
Note: R1is from corner atoms and R3is from atoms in additional positions in UC
2
[2 ( )]i i h x k y l zE A f
2 ( )h k l In complex notation
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If atom B is different from atom A the amplitudes must be weighed by the respectiveatomic scattering factors (f)
The resultant amplitude of all the waves scattered by all the atoms in the UC gives thescattering factor for the unit cell The unit cell scattering factor is called theStructure Factor (F)
Scattering by an unit cell = f(position of the atoms, atomic scattering factors)
electronanbyscatteredwaveofAmplitudeucinatomsallbyscatteredwaveofAmplitudeFactorStructureF
[2 ( )]i i h x k y l z E Ae fe
2 ( )h x k y l z p
2FI
[2 ( )]
1 1
j j j j
n ni i h x k y l z hkl
n j j
j j
F f e f e
Structure factor is independent of theshapeandsize of the unit cell
For natoms in the UC
If the UC distorts so do the planes in it!!
nni)(S f l l i
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nnie )1(
)(2
Cos
ee ii
Structure factor calculations
A Atom at (0,0,0) and equivalent positions
[2 ( )]j j j ji i h x k y l z
j jF f e f e
[2 ( 0 0 0)] 0i h k l F f e f e f
22 fF F is independent of the scattering plane (h k l)
nini ee
Simple Cubic
1)( inodde
1)( inevene
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B Atom at (0,0,0) & (, , 0) and equivalent positions
[2 ( )]j j j ji i h x k y l z
j jF f e f e
1 1[2 ( 0)]
[2 ( 0 0 0)] 2 2
[ 2 ( )]0 ( )2
[1 ]
i h k l i h k l
h ki
i h k
F f e f e
f e f e f e
F is independent of the l index
C- centred Orthorhombic
Real
]1[ )( khi
efF
fF 2
0F
224fF
02 F
e.g. (001), (110), (112); (021), (022), (023)
e.g. (100), (101), (102); (031), (032), (033)
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If the blue planes are scattering in phase then on C- centering the red planes will scatter out
of phase (with the blue planes- as they bisect them) and hence the (210) reflection will
become extinct
This analysis is consistent with the extinction rules: (h + k) odd is absent
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In case of the (310) planes no new translationally equivalent planes are added on lattice
centering this reflection cannot go missing. This analysis is consistent with the extinction rules: (h + k) even is present
Body centred
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C Atom at (0,0,0) & (, , ) and equivalent positions
[2 ( )]j j j ji i h x k y l z
j jF f e f e
1 1 1[2 ( )]
[2 ( 0 0 0)] 2 2 2
[ 2 ( )]0 ( )2
[1 ]
i h k l i h k l
h k li
i h k l
F f e f e
f e f e f e
Orthorhombic
Real
]1[ )( lkhi
efF
fF 2
0F
224fF
02 F
e.g. (110), (200), (211); (220), (022), (310)
e.g. (100), (001), (111); (210), (032), (133)
F C d C bi
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D Atom at (0,0,0) & (, , 0) and equivalent positions
[2 ( )]j j j ji i h x k y l z
j jF f e f e
]1[)()()(
)]2
(2[)]2
(2[)]2
(2[)]0(2[
hlilkikhi
hli
lki
khi
i
eeef
eeeefF
Face Centred Cubic
Real
fF 4
0F
22 16fF
02 F
(h, k, l) unmixed
(h, k, l) mixed
e.g. (111), (200), (220), (333), (420)
e.g. (100), (211); (210), (032), (033)
(, , 0), (, 0, ), (0, , )
]1[ )()()( hlilkikhi eeefF
Two odd and one even (e.g. 112); two even and one odd (e.g. 122)
Mixed indices Two odd and one even (e g 112); two even and one odd (e g 122)
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Mixed indices CASE h k l
A o o e
B o e e
( ) ( ) ( )CASE A: [1 ] [1 1 1 1] 0i e i o i oe e e
( ) ( ) ( )CASE B: [1 ] [1 1 1 1] 0i o i e i oe e e
0F 02 F(h, k, l) mixed e.g. (100), (211); (210), (032), (033)
Mixed indices Two odd and one even (e.g. 112); two even and one odd (e.g. 122)
Unmixed indices CASE h k l
A o o o
B e e e
Unmixed indices
fF 4 22 16fF (h, k, l) unmixede.g. (111), (200), (220), (333), (420)
All odd (e.g. 111); all even (e.g. 222)
( ) ( ) ( )CASE A : [1 ] [1 1 1 1] 4i e i e i ee e e
( ) ( ) ( )CASE B: [1 ] [1 1 1 1] 4i e i e i ee e e
Na+ at (0 0 0) + Face Centering Translations ( 0) ( 0 ) (0 )
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E Na at (0,0,0) + Face Centering Translations (, , 0), (, 0, ), (0, , )Clat (, 0, 0) + FCT (0, , 0), (0, 0, ), (, , )
)]2
(2[)]2(2[)]
2(2[)]
2(2[
)]2
(2[)]2
(2[)]2
(2[)]0(2[
lkhi
li
ki
hi
Cl
hli
lki
khi
i
Na
eeeef
eeeefF
][
]1[
)()()()(
)()()(
lkhilikihi
Cl
hlilkikhi
Na
eeeef
eeefF
]1[
]1[
)()()()(
)()()(
khihlilkilkhi
Cl
hlilkikhi
Na
eeeef
eeefF
]1][[
)()()()( hlilkikhilkhi
ClNa eeeeffF
NaCl:Face Centred Cubic
)()()()( hlilkikhilkhi
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]1][[ )()()()( hlilkikhilkhiClNa
eeeeffF
Zero for mixed indices
Mixed indices CASE h k l
A o o e
B o e e
]2][1[ TermTermF
0]1111[]1[2:ACASE )()()( oioiei eeeTerm
0]1111[]1[2:BCASE )()()( oieioi eeeTerm
0F 02 F(h, k, l) mixed e.g. (100), (211); (210), (032), (033)
Mixed indices
Unmixed indices CASE h k li d i di
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(h, k, l) unmixed ][4)( lkhi
ClNa
effF
][4 ClNa ffF If (h + k + l) is even22 ][16 ClNa ffF
][4 ClNa
ffF If (h + k + l) is odd22 ][16 ClNa ffF
e.g. (111), (222); (133), (244)
e.g. (222),(244)
e.g. (111), (133)
Unmixed indices CASE h k l
A o o o
B e e e
4]1111[]1[2:ACASE)()()(
eieiei
eeeTerm
4]1111[]1[2:BCASE )()()( eieiei eeeTerm
Unmixed indices
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Presence of additional atoms/ions/molecules in the UC can alterthe intensities of some of the reflections
Selection / Extinction Rules
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Bravais LatticeReflections whichmay bepresent
Reflectionsnecessarilyabsent
Simple all NoneBody centred (h + k + l) even (h + k + l) odd
Face centred h, k and l unmixed h, k and l mixed
End centredh and k unmixed
C centred
h and k mixed
C centred
Bravais Lattice Allowed Reflections
SC All
BCC (h + k + l) even
FCC h, k and l unmixed
DC
h, k and l are all oddOr
all are even& (h + k + l) divisible by 4
h2+ k2+ l2 SC FCC BCC DC
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1 100
2 110 110
3 111 111 111
4 200 200 2005 210
6 211 211
7
8 220 220 220 220
9 300, 22110 310 310
11 311 311 311
12 222 222 222
13 320
14 321 321
15
16 400 400 400 400
17 410, 322
18 411, 330 411, 330
19 331 331 331
Crystal structure determination
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y
Monochromatic X-rays
Panchromatic X-rays
Monochromatic X-rays
Many s (orientations)Powder specimen
POWDERMETHOD
Single LAUETECHNIQUE
Varied by rotation
ROTATINGCRYSTALMETHOD
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POWDER METHOD
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http://www.matter.org.uk/diffraction/x-ray/powder_method.htm
Diffraction cones and the Debye-Scherrer geometry
Film may be replaced with detector
POWDER METHOD
Different cones for different reflections
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The 440 reflection is not observed
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The 331 reflection is not observed
THE POWDER METHOD
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THE POWDER METHOD
2222 sin)( lkh
2
2
2222
sin4
)( a
lkh
)(sin4
222
2
22 lkha
222
lkh
adCubic
dSin2
222
222 sin4
lkh
a
Cubic crystal
Relative Intensity of diffraction lines in a powder pattern
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Structure Factor (F)
Multiplicity factor (p)
Polarization factor
Lorentz factor
RelativeIntensity of diffraction lines in a powder pattern
Absorption factor
Temperature factor
Scattering from UC
Number of equivalent scattering planes
Effect of wave polarization
Combination of 3 geometric factors
Specimen absorption
Thermal diffuse scattering
2
1
2
1
Sin
Cos
Sin
factorLorentz
21 2CosIP
Multiplicity factor
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Lattice Index Multiplicity Planes
Cubic
(with highest
symmetry)
(100) 6 [(100) (010) (001)] (2 for negatives)
(110) 12 [(110) (101) (011), (110) (101) (011)] (2 for negatives)
(111) 12 [(111) (111) (111) (111)] (2 for negatives)
(210) 24*(210) 3! Ways, (210) 3! Ways,
(210) 3! Ways, (210) 3! Ways
(211) 24(211) 3 ways, (211) 3! ways,
(211) 3 ways
(321) 48*
Tetragonal
(with highestsymmetry)
(100) 4 [(100) (010)] (2 for negatives)
(110) 4 [(110) (110)] (2 for negatives)
(111) 8 [(111) (111) (111) (111)] (2 for negatives)
(210) 8*(210) = 2 Ways, (210) = 2 Ways,
(210) = 2 Ways, (210) = 2 Ways
(211) 16 [Same as for (210) = 8] 2 (as l can be +1 or 1)
(321) 16* Same as above (as last digit is anyhow not permuted)
* Altered in crystals with lower symmetry
Multiplicity factor
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Cubichkl hhl hk0 hh0 hhh h00
48* 24 24* 12 8 6
Hexagonal hk.l hh.l h0.l hk.0 hh.0 h0.0 00.l24* 12* 12* 12* 6 6 2
Tetragonalhkl hhl h0l hk0 hh0 h00 00l
16* 8 8 8* 4 4 2
Orthorhombichkl hk0 h0l 0kl h00 0k0 00l
8 4 4 4 2 2 2
Monoclinichkl h0l 0k0
4 2 2
Triclinichkl
2
* Altered in crystals with lower symmetry (of the same crystal class)
Polarization factor Lorentz factor
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0
5
10
15
20
25
30
0 20 40 60 80
Bragg Angle ( , degrees)
Lorentz-Polarization
factor
2
1
2
1
SinCos
SinfactorLorentz 21 2CosIP
CosSin
CosfactoronPolarizatiLorentz
2
2 21
Intensity of powder pattern lines (ignoring Temperature & Absorption factors)
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Intensity of powder pattern lines (ignoring Temperature&Absorptionfactors)
CosSin
CospFI
2
22 21
Valid for Debye-Scherrer geometry
I Relative IntegratedIntensityF Structure factorp Multiplicity factor
POINTS
As one is interested in relative (integrated) intensities of the lines constant factorsare omittedVolume of specimen me, e (1/dectector radius)
Random orientation of crystals in a with Textureintensities are modified
Iis really diffracted energy(as Intensity is Energy/area/time)
Ignoring Temperature & Absorption factors valid for lines close-by in pattern
THE POWDER METHOD
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THE POWDER METHOD
2222 sin)( lkh
2
2
2222
sin4
)( a
lkh
)(sin4
222
2
22 lkha
222
lkh
adCubic
dSin2
222
222 sin4
lkh
a
Cubic crystal
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n 2 Intensity Sin Sin2 ratio
Determination of Crystal Structure from 2versus Intensity Data
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2 Intensity Sin Sin2 ratio
1 21.5 0.366 0.134 3
2 25 0.422 0.178 4
3 37 0.60 0.362 8
4 45 0.707 0.500 115 47 0.731 0.535 12
6 58 0.848 0.719 16
7 68 0.927 0.859 19
FCC
h2+ k2+ l2 SC FCC BCC DC
1 100
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1 100
2 110 110
3 111 111 111
4 200 200 200
5 210
6 211 211
7
8 220 220 220 220
9 300, 22110 310 310
11 311 311 311
12 222 222 222
13 320
14 321 32115
16 400 400 400 400
17 410, 322
18 411, 330 411, 330
19 331 331 331
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The ratio of (h2+ K2+ l2) derived from extinction rules
SC 1 2 3 4 5 6 8
BCC 1 2 3 4 5 6 7
FCC 3 4 8 11 12
DC 3 8 11 16
Powder diffraction pattern from Al Radiation: Cu K, = 1.54056
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420
111
200
220
311
222
4
00 3
31
422
1& 2peaks resolved
Note: Peaks or not idealized peaks broadened
Increasing splitting of peaks with g
Peaks are all not of same intensity
X-Ray Diffraction: A Practical Approach, C. Suryanarayana & M. Grant Norton, Plenum Press, New York (1998)
Determination of Crystal Structure from 2versus Intensity Data
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n 2 Sin Sin2 ratio Index a (nm)1 38.52 19.26 0.33 0.11 3 111 0.404482 44.76 22.38 0.38 0.14 4 200 0.40457
3 65.14 32.57 0.54 0.29 8 220 0.40471
4 78.26 39.13 0.63 0.40 11 311 0.40480
5* 82.47 41.235 0.66 0.43 12 222 0.40480
6* 99.11 49.555 0.76 0.58 16 400 0.40485
7* 112.03 56.015 0.83 0.69 19 331 0.40491
8* 116.60 58.3 0.85 0.72 20 420 0.40491
9* 137.47 68.735 0.93 0.87 24 422 0.40494
* 1, 2peaks are resolved (1peaks are listed)
1
d
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0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0 30 60 90
t
Sin(t)
Sind
2
22
)(
Sin
Cos
d
dd
Tan
dd
Sin
Cos
d
dd
)(
Error in d spacing
14
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0
2
4
6
8
10
12
14
0 20 40 60 80 100
t
Cot(t)
Tan
dd
Sin
Cos
d
dd
)(
Error in d spacing
Error in d spacing decreases with
Applications of XRD
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Bravais lattice determination
Lattice parameter determination
Determination of solvus line in phase diagrams
Long range order
Crystallite size and Strain
More
y
CrystalS h ti f diff b t
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Diffraction angle (2)
Intensity
90 1800
Diffraction angle (2)
Intensit
y
Liquid / Amorphous solid
90 180
Diffraction angle (2)
Intensity
Monoatomic gas
Schematic of difference betweenthe diffraction patterns of various phases