X-RAY DIFFRACTIONX-RAY DIFFRACTION
X- Ray Sources
Diffraction: Bragg’s Law
Crystal Structure Determination
Elements of X-Ray DiffractionB.D. Cullity & S.R. Stock
Prentice Hall, Upper Saddle River (2001)
X-Ray Diffraction: A Practical ApproachC. Suryanarayana & M. Grant Norton
Plenum Press, New York (1998)
For electromagnetic radiation to be diffracted the spacing in the grating should be of the same order as the wavelength
In crystals the typical interatomic spacing ~ 2-3 Å so the suitable radiation is X-rays
Hence, X-rays can be used for the study of crystal structures
Beam of electrons Target X-rays
An accelerating (/decelerating) charge radiates electromagnetic radiation
Inte
nsit
y
Wavelength ()
Mo Target impacted by electrons accelerated by a 35 kV potential
0.2 0.6 1.0 1.4
White radiation
Characteristic radiation → due to energy transitions in the atom
K
K
Target Metal Of K radiation (Å)
Mo 0.71
Cu 1.54
Co 1.79
Fe 1.94
Cr 2.29
Heat
Incident X-rays
SPECIMEN
Transmitted beam
Fluorescent X-raysElectrons
Compton recoil PhotoelectronsScattered X-rays
CoherentFrom bound charges
Incoherent (Compton modified)From loosely bound charges
X-rays can also be refracted (refractive index slightly less than 1) and reflected (at very small angles) Refraction of X-rays is neglected for now.
Incoherent Scattering (Compton modified) From loosely bound charges
Here the particle picture of the electron & photon comes in handy
),( 11 Electron knocked aside
),( 22
11 hE
22 hE
)21(0243.012 Cos
2
No fixed phase relation between the incident and scattered wavesIncoherent does not contribute to diffraction
(Darkens the background of the diffraction patterns)
Vacuum
Energylevels
KE
1LE
2LE
3LE
Nucleus
K
1L
2L
3L
Characteristic x-rays(Fluorescent X-rays)
(10−16s later seems like scattering!)
Fluorescent X-raysKnocked out electron
from inner shell
A beam of X-rays directed at a crystal interacts with the electrons 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 coherent
The emission will undergo constructive or destructive interference with waves scattered from other atoms
Incoming X-raysSecondaryemission
Sets Electron cloud into oscillation
Sets nucleus (with protons) into oscillation
Small effect neglected
Oscillating charge re-radiates In phase with the incoming x-rays
BRAGG’s EQUATION
d
dSin
The path difference between ray 1 and ray 2 = 2d Sin
For constructive interference: n = 2d Sin
Ray 1
Ray 2
Deviation = 2
Incident and scattered waves are in phase if
Scattering from across planes is in phase
In plane scattering is in phase
Extra path traveled by incoming waves AY
Extra path traveled by scattered waves XB
These can be in phase if and only if incident = scattered
But this is still reinforced scatteringand NOT reflection
Note that in the Bragg’s equation: The interatomic spacing (a) along the plane does not appear Only the interplanar spacing (d) appears Change in position or spacing of atoms along the plane should not affect
Bragg’s condition !!
d
Note: shift (systematic) is actually not a problem!
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
Consider the case for which 1 2
Constructive interference can still occur if the difference in the path length traversed by R1 and R2 before and after scattering are an integral multiple of the wavelength (AY − XC) = h (h is an integer)
1Cosa
AY 2Cos
a
XC hCosaCosa 21
hCosCosa 21
Laue’s equations
S0 incoming 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
A physical picture of scattering leading to diffraction is embodied in Laue’s equations
Bragg’s 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’ (Bragg’s 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?”
Bragg’s equation is a negative law
If Bragg’s eq. is NOT satisfied NO reflection can occur
If Bragg’s eq. is satisfied reflection MAY 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
n = 2d Sin
n is an integer and is the order of the reflection
For Cu K radiation ( = 1.54 Å) and d110= 2.22 Å
n Sin
1 0.34 20.7º First order reflection from (110)
2 0.69 43.92ºSecond order reflection from (110)
Also written as (220)
222 lkh
adhkl
8
220
ad
2110
ad
2
1
110
220 d
d
sin2 hkldn
In XRD nth order reflection from (h k l) is considered as 1st order reflectionfrom (nh nk nl)
sin2n
dhkl
sin2 n n n lkhd
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)
Scattering by an Electron
),( 00 Sets electron into oscillation
Emission in ‘all’ directions
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
x
z
r
P
Intensity of the scattered beam due to an electron (I) at a point Psuch that r >>
2
2
42
4
0 r
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
Co
s(t
)
The reason we are able to neglect scattering from the protons in the nucleus
The scattered rays are also plane polarized
2
2
42
4
0 r
Sin
cm
eII
For an unpolarized wave E is the measure of the amplitude of the waveE2 = Intensity
222zy EEE zy II
I00
0
2
24
0 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
24
0 2 4 2
zPz z
SineI I
m c r
Total Intensity at point P due to Ey & Ez
2 24
0 2 4 2
y z
P
Sin SineI I
m c r
2 24
0 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 ( )x
P
CoseI I
m c r
24
0 2 4 2
1 (2 )P
CoseI I
m c r
In terms of 2
0
0.2
0.4
0.6
0.8
1
0 30 60 90 120 150 180
2t
[Co
s(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 BB’Are 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
For an unpolarized wave E is the measure of the amplitude of the waveE2 = Intensity
222zy EEE
zy III
000
2
242
4
02
2
42
4
0
12rcm
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
222r
Cos
cm
eI
r
Sin
cm
eII zzPz
The zx plane is to the y direction: hence, = 90
-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
Co
s(t
)
2
200
42
4 2
r
CosII
cm
eIII zy
PzPyP
2
2
42
40 21
2 r
Cos
cm
eIIP
Scattered beam is not unpolarized
Forward and backward scattered intensity higher than at 90 Scattered intensity minute fraction of the incident intensity
Very small number
Polarization factorComes into being as we used unpolarized beam
2
21 2
42
4
20 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+
Co
s(2
t)^
2)/
2
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
electronan by scattered waveof Amplitude
atoman by scattered waveof Amplitude
Factor Scattering Atomicf
f →
)(Sin
(Å−1) →
0.2 0.4 0.6 0.8 1.0
10
20
30
Schematic
)(Sin
Coherent scatteringIncoherent (Compton)
scattering
Z Sin() /
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 (Greek parallēlepipedon) 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 Scattering by the Unit cell (uc)
Coherent Scattering Unit Cell (UC) is representative of the crystal structure Scattered 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 ones scattered from top and bottom planes
d(h00)
B
Ray 1 = R1
Ray 2 = R2
Ray 3 = R3
Unit Cell
x
M
C
N
RB
S
A
'1R
'2R
'3R
(h00) planea
h
adAC h 00
:::: ACMCN
xABRBS ::::
haxx
AC
AB
)(2 0021SindMCN hRR
h
ax
AC
ABRBSRR
31
2
a
xh
hax
RR 2
231
xcoordinatefractionala
x xhRR 231
Extending to 3D 2 ( )h x k y l z Independent of the shape of UC
Note: R1 is from corner atoms and R3 is from atoms in additional positions in UC
2
If atom B is different from atom A the amplitudes must be weighed by the respective atomic scattering factors (f)
The resultant amplitude of all the waves scattered by all the atoms in the UC gives the scattering factor for the unit cell
The unit cell scattering factor is called the Structure Factor (F)
Scattering by an unit cell = f(position of the atoms, atomic scattering factors)
electronan by scattered waveof Amplitude
ucin atoms allby scattered waveof AmplitudeFactor StructureF
[2 ( )]i i h x k y l zE Ae fe 2 ( )h x k y l z In complex notation
2FI
[2 ( )]
1 1
j j j j
n ni i h x k y l zhkl
n j jj j
F f e f e
Structure factor is independent of the shape and size of the unit cell
For n atoms in the UC
If the UC distorts so do the planes in it!!
nnie )1(
)(2
Cosee ii
Structure factor calculations
A Atom at (0,0,0) and equivalent positions
[2 ( )]j j j ji i h x k y l zj jF f e f e
[2 ( 0 0 0)] 0i h k lF 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
B Atom at (0,0,0) & (½, ½, 0) and equivalent positions
[2 ( )]j j j ji i h x k y l zj jF f e f e
1 1[2 ( 0)][2 ( 0 0 0)] 2 2
[ 2 ( )]0 ( )2 [1 ]
i h k li 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[ )( khiefF
fF 2
0F
22 4 fF
02 F
Both even or both odd
Mixture of odd and even
e.g. (001), (110), (112); (021), (022), (023)
e.g. (100), (101), (102); (031), (032), (033)
(h + k) even
(h + k) odd
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
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
C Atom at (0,0,0) & (½, ½, ½) and equivalent positions
[2 ( )]j j j ji i h x k y l zj jF f e f e
1 1 1[2 ( )][2 ( 0 0 0)] 2 2 2
[ 2 ( )]0 ( )2 [1 ]
i h k li h k l
h k li i h k l
F f e f e
f e f e f e
Body centred Orthorhombic
Real
]1[ )( lkhiefF
fF 2
0F
22 4 fF
02 F
(h + k + l) even
(h + k + l) odd
e.g. (110), (200), (211); (220), (022), (310)
e.g. (100), (001), (111); (210), (032), (133)
D Atom at (0,0,0) & (½, ½, 0) and equivalent positions
[2 ( )]j j j ji i h x k y l zj jF f e f e
]1[ )()()(
)]2
(2[)]2
(2[)]2
(2[)]0(2[
hlilkikhi
hli
lki
khii
eeef
eeeefF
Face Centred Cubic
Real
fF 4
0F
22 16 fF
02 F
(h, k, l) unmixed
(h, k, l) mixed
e.g. (111), (200), (220), (333), (420)
e.g. (100), (211); (210), (032), (033)
(½, ½, 0), (½, 0, ½), (0, ½, ½)
]1[ )()()( hlilkikhi eeefF
Two odd and one even (e.g. 112); two even and one odd (e.g. 122)
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 16 fF (h, k, l) unmixed
e.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
E Na+ at (0,0,0) + Face Centering Translations (½, ½, 0), (½, 0, ½), (0, ½, ½) Cl− at (½, 0, 0) + FCT (0, ½, 0), (0, 0, ½), (½, ½, ½)
)]2
(2[)]2
(2[)]2
(2[)]2
(2[
)]2
(2[)]2
(2[)]2
(2[)]0(2[
lkh
il
ik
ih
i
Cl
hli
lki
khii
Na
eeeef
eeeefF
][
]1[)()()()(
)()()(
lkhilikihi
Cl
hlilkikhi
Na
eeeef
eeefF
]1[
]1[)()()()(
)()()(
khihlilkilkhi
Cl
hlilkikhi
Na
eeeef
eeefF
]1][[ )()()()( hlilkikhilkhi
ClNaeeeeffF
NaCl: Face Centred Cubic
]1][[ )()()()( hlilkikhilkhi
ClNaeeeeffF
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
(h, k, l) unmixed ][4 )( lkhi
ClNaeffF
][4 ClNa
ffF If (h + k + l) is even22 ][16
ClNaffF
][4 ClNa
ffF If (h + k + l) is odd22 ][16
ClNaffF
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
Presence of additional atoms/ions/molecules in the UC can alter the intensities of some of the reflections
Bravais LatticeReflections which
may be presentReflections
necessarily absent
Simple all None
Body 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 centredh 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
Selection / Extinction Rules
h2 + k2 + l2 SC FCC BCC DC
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, 221
10 310 310
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
Reciprocal LatticeProperties are reciprocal to the crystal lattice
32*
1
1aa
Vb
13
*2
1aa
Vb
21
*3
1aa
Vb
B
O
P
M
A
C
B
O
P
M
A
C
O
P
M
A
C
O
P
M
A
C
O
P
M
A
C
*b3
2a
1a
3a
OPCellHeight of OAMBArea
OAMBArea
aaV
bb
1
)(
)(
121
*3
*3
001
*3
1
db
The reciprocal lattice is created by interplanar spacings
** as written usuall ii ab
B
BASIS VECTORS
21*
3 to is aandab
A reciprocal lattice vector is to the corresponding real lattice plane
*3
*2
*1
* blbkbhghkl
hklhklhkl d
gg1**
The length of a reciprocal lattice vector is the reciprocal of the spacing of the corresponding real lattice plane
Planes in the crystal become lattice points in the reciprocal lattice ALTERNATE CONSTRUCTION OF THE REAL LATTICE
Reciprocal lattice point represents the orientation and spacing of a set of planes
Reciprocal Lattice
(01)
(10)(11)
(21)
10 20
11
221202
01 21
00
The reciprocal lattice has an origin!
1a
2a
1a1
1a
*11g *
21g*b2
*b1
1020
11
2212
02
01
21
00
(01)
(10)(11)
(21)
1a
2a
*b2
*b1
1a
(01)
(10)(11)
(21) Note perpendicularity of various vectors
Reciprocal lattice is the reciprocal of a primitive lattice and is purely geometrical does not deal with the intensities of the points
Physics comes in from the following:
For non-primitive cells ( lattices with additional points) and for crystals decorated with motifs ( crystal = lattice + motif) the Reciprocal lattice points have to be weighed in with the corresponding scattering power (|Fhkl|2) Some of the Reciprocal lattice points go missing (or may be scaled up or down in intensity) Making of Reciprocal Crystal (Reciprocal lattice decorated with a motif of scattering power)
The Ewald sphere construction further can select those points which are actually observed in a diffraction experiment
In crystals based on a particular lattice the intensities of particular reflections are modified they may even go missing
Crystal = Lattice + Motif
Diffraction Pattern
Position of the Lattice points LATTICE
Intensity of the diffraction spots ‘MOTIF’
There are two ways of constructing the Reciprocal Crystal:
1) Construct the lattice and decorate each lattice point with appropriate intensity
2) Use the concept as that for the real crystal
Examples of 3D Reciprocal Lattices weighed in with scattering power (|F|2)
Figures NOT to Scale
000
100
111
001
101
011
010
110
SC
Lattice = SC
Reciprocal Crystal = SC
No missing reflections
Figures NOT to Scale
000
200
222
002
101
022
020110
BCC
Lattice = BCC
Reciprocal Crystal = FCC
220
011
202
100 missing reflection (F = 0)
22 4 fF
Weighing factor for each point “motif”
Figures NOT to Scale
000200
222
002022
020
FCC
Lattice = FCC
Reciprocal Crystal = BCC
220
111
202
100 missing reflection (F = 0)110 missing reflection (F = 0)
22 16 fF
Weighing factor for each point “motif”
Ordered Solid solution
G = H TS
High T disordered
Low T ordered
470ºC
Sublattice-1
Sublattice-2
BCC
SC
In a strict sense this is not a crystal !!
Disordered Ordered
- NiAl, BCC B2 (CsCl type)
- Ni3Al, FCC L12 (AuCu3-I type)
BCC SC
BCCFCC
FCC
SC
Ordered
Ordered
There are two ways of constructing the Reciprocal Crystal:
1) Construct the lattice and decorate each lattice point with appropriate intensity
2) Use the concept as that for the real crystal
1) SC + two kinds of Intensities decorating the lattice
2) (FCC) + (Motif = 1FR + 1SLR)
1) SC + two kinds of Intensities decorating the lattice
2) (BCC) + (Motif = 1FR + 3SLR)
FR Fundamental Reflection SLR Superlattice Reflection
The Ewald Sphere
* Paul Peter Ewald (German physicist and crystallographer; 1888-1985)
organisiert von:Max-Planck-Institut für MetallforschungInstitut für Theoretische und Angewandte Physik,Institut für Metallkunde,Institut für Nichtmetallische Anorganische Materialiender Universität Stuttgart Programm
13:30 Joachim Spatz (Max-Planck-Institut für Metallforschung) Begrüßung
13:45 Heribert Knorr (Ministerium für Wissenschaft, Forschung und Kunst Baden-Württemberg Begrüßung
14:00 Stefan Hell (Max-Planck-Institut für Biophysikalische Chemie)Nano-Auflösung mit fokussiertem Licht
14:30 Antoni Tomsia (Lawrence Berkeley National Laboratory)Using Ice to Mimic Nacre: From Structural Materials to Artificial Bone
15:00 Pause Kaffee und Getränke
15:30 Frank Gießelmann(Universität Stuttgart) Von ferroelektrischen Fluiden zu geordneten Dispersionen von Nanoröhren: Aktuelle Themen der Flüssigkristallforschung
16:00 Verleihung des Günter-Petzow-Preises 2008
16:15 Udo Welzel (Max-Planck-Institut für Metallforschung) Materialien unter Spannung: Ursachen, Messung und Auswirkungen- Freund und Feind
ab 17:00 Sommerfest des Max-Planck-Instituts für Metallforschung
7. Paul-Peter-Ewald-Kolloquium
Freitag, 17. Juli 2008
The Ewald Sphere
The reciprocal lattice points are the values of momentum transfer for which the Bragg’s equation is satisfied
For diffraction to occur the scattering vector must be equal to a reciprocal lattice vector
Geometrically if the origin of reciprocal space is placed at the tip of ki then diffraction will occur only for those reciprocal lattice points that lie on the surface of the Ewald sphere
See Cullity’s book: A15-4
hklhkl Sindn 2
2
12 hkl
hklhkl
d
dSin
Draw a circle with diameter 2/ Construct a triangle with the diameter as the hypotenuse and 1/dhkl as a side (any
triangle inscribed in a circle with the diameter as the hypotenuse is a right angle triangle): AOP
The angle opposite the 1/d side is hkl (from the rewritten Bragg’s equation)
Bragg’s equation revisited
hklhklhkl d
gg1**
2
12 hkl
hklhkl
d
dSin
Radiation related information is present in the Ewald Sphere
Crystal related information is present in the reciprocal crystal
The Ewald sphere construction generates the diffraction pattern
The Ewald Sphere construction
01
10
02
00 20
2
(41)
Ki
KD
K
Reciprocal Space
K = K =g = Diffraction Vector
Ewald Sphere
The Ewald Sphere touches the reciprocal lattice (for point 41)
Bragg’s equation is satisfied for 41
(Cu K) = 1.54 Å, 1/ = 0.65 Å−1 (2/ = 1.3 Å−1), aAl = 4.05 Å, d111 = 2.34 Å, 1/d111 = 0.43 Å−1
Ewald sphere X-rays
Crystal structure determination
Monochromatic X-rays
Panchromatic X-rays
Monochromatic X-rays
Many s (orientations)Powder specimen
POWDER METHOD
Single LAUETECHNIQUE
Varied by rotation
ROTATINGCRYSTALMETHOD
THE POWDER METHOD
Cone of diffracted rays
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
The 440 reflection is not observed
The 331 reflection is not observed
THE POWDER METHOD
2222 sin)( lkh
22
2222 sin
4)(
alkh
)(sin4
2222
22 lkha
222 lkhadCubic
dSin2
222
222 sin4
lkh
a
Cubic crystal
Structure Factor (F)
Multiplicity factor (p)
Polarization factor
Lorentz factor
Relative Intensity 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
SinCos
SinfactorLorentz
21 2CosIP
Multiplicity factor
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 highest symmetry)
(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
Cubichkl hhl hk0 hh0 hhh h0048* 24 24* 12 8 6
Hexagonalhk.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 00l16* 8 8 8* 4 4 2
Orthorhombichkl hk0 h0l 0kl h00 0k0 00l8 4 4 4 2 2 2
Monoclinichkl h0l 0k04 2 2
Triclinichkl2
* Altered in crystals with lower symmetry (of the same crystal class)
Multiplicity factor
0
5
10
15
20
25
30
0 20 40 60 80
Bragg Angle (, degrees)
Lor
entz
-Pol
ariz
atio
n f
acto
r
Polarization factor Lorentz factor
2
1
2
1
SinCos
SinfactorLorentz 21 2CosIP
CosSin
CosfactoronPolarizatiLorentz
2
2 21
Intensity of powder pattern lines (ignoring Temperature & Absorption factors)
CosSin
CospFI
2
22 21
Valid for Debye-Scherrer geometry I → Relative Integrated “Intensity” F → Structure factor p → Multiplicity factor
POINTS As one is interested in relative (integrated) intensities of the lines constant factors
are omitted Volume of specimen me , e (1/dectector radius)
Random orientation of crystals in a with Texture intensities are modified I is really diffracted energy (as Intensity is Energy/area/time) Ignoring Temperature & Absorption factors valid for lines close-by in pattern
THE POWDER METHOD
2222 sin)( lkh
22
2222 sin
4)(
alkh
)(sin4
2222
22 lkha
222 lkhadCubic
dSin2
222
222 sin4
lkh
a
Cubic crystal
n 2→ Intensity Sin Sin2 ratio
Determination of Crystal Structure from 2 versus Intensity Data
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 11
5 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
2 110 110
3 111 111 111
4 200 200 200
5 210
6 211 211
7
8 220 220 220 220
9 300, 221
10 310 310
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
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
420
111
200 22
0
311
222
400 33
1
422
1 & 2 peaks resolved
Radiation: Cu K, = 1.54056 Å
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)
0
2
4
6
8
10
12
14
0 30 60 90t
1/Co
s(t)
2n d Sin
2d
d Cosd
1
2
d
d d Cos
Actually, the variation in 2 is to be seen
n 2 Sin Sin2 ratio Index a (nm)
1 38.52 19.26 0.33 0.11 3 111 0.40448
2 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
Determination of Crystal Structure from 2 versus Intensity Data
* 1 , 2 peaks are resolved (1 peaks are listed)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 30 60 90
t
Sin(t)
For the same the error in Sin with
Sind
2
22
)(
Sin
Cos
d
dd
Tan
dd
Sin
Cos
d
dd
)(
Error in d spacing
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
Bravais lattice determination
Lattice parameter determination
Determination of solvus line in phase diagrams
Long range order
Applications of XRD
Crystallite size and Strain
More
Diffraction angle (2) →
Inte
nsit
y →
90 1800
Crystal
90 1800
Diffraction angle (2) →
Inte
nsit
y → Liquid / Amorphous solid
90 1800
Diffraction angle (2) →
Inte
nsit
y →
Monoatomic gas
Schematic of difference between the diffraction patterns of various phases
Crystallite size and Strain
Bragg’s equation assumes:
Crystal is perfect and infinite
Incident beam is perfectly parallel and monochromatic
Actual experimental conditions are different from these leading various kinds of
deviations from Bragg’s condition
Peaks are not ‘’ curves Peaks are broadened
There are also deviations from the assumptions involved in the generating powder
patterns
Crystals may not be randomly oriented (textured sample) Peak intensities are
altered
In a powder sample if the crystallite size < 0.5 m
there are insufficient number of planes to build up a sharp diffraction pattern
peaks are broadened
XRD Line Broadening
Instrumental
Crystallite size
Strain
Stacking fault
XRD Line Broadening
Other defects
Unresolved 1 , 2 peaks Non-monochromaticity of the source (finite width of peak) Imperfect focusing
In the vicinity of B the −ve of Bragg’s equation not being satisfied
‘Residual Strain’ arising from dislocations, coherent precipitates etc. leading to broadening
In principle every defect contributes to some broadening
Bi
Bc
Bs
...)( SFsci BBBBFWHMB
...)( SFsci BBBBFWHMB
Crystallite size
Size > 10 m Spotty ring
(no. of grains in the irradiated portion insufficient to produce a ring)
Size (10, 0.5) Smooth continuous ring pattern
Size (0.5, 0.1) Rings are broadened
Size < 0.1 No ring pattern
(irradiated volume too small to produce a diffraction ring pattern &
diffraction occurs only at low angles)
Spotty ring
Rings
Broadened RingsDiffuse
Effect of crystallite size on SAD patterns
Single crystal
“Spotty” pattern
Few crystals in the selected region
Effect of crystallite size on SAD patterns
Ring patternBroadened Rings
Subtracting Instrumental Broadening
Instrumental broadening has to be subtracted to get the broadening effects due to the sample
1 Mix specimen with known coarse-grained (~ 10m), well annealed (strain free)
does not give any broadening due to strain or crystallite size (the only broadening is instrumental). A brittle material which can be
ground into powder form without leading to much stored strain is good. If the pattern of the test sample (standard) is recorded separately then the
experimental conditions should be identical (it is preferable that one or more peaks of the standard lies close to the specimen’s peaks)
2 Use the same material as the standard as the specimen to be X-rayed but with
large grain size and well annealed
rsci BBBBB
...)( SFsci BBBBFWHMB
For a peak with a Lorentzian profile
222ir BBB For a peak with a Gaussian profile
222 )( iir BBBBB A geometric mean can also used
Longer tail
Johann Carl Friedrich Gauss (1777-1855), painted by Christian Albrecht Jensen
http://en.wikipedia.org/wiki/Carl_Friedrich_Gauss
Hendrik Antoon LorentzOn the theory of reflection and refraction of light
University of Göttingen
Scherrer’s formula
( )cB
kB
LCos
→ Wavelength L → Average crystallite size ( to surface of specimen) k → 0.94 [k (0.89, 1.39)]
~ 1 (the accuracy of the method is only 10%)
For Gaussian line profiles and cubic crystals
0
2
4
6
8
10
12
14
0 30 60 90t
1/Co
s(t)
Strain broadening
( )s BB Tan
→ Strain in the material
Smaller angle peaksshould be used to separate Bs and Bc
0
2
4
6
8
10
12
14
0 20 40 60 80 100
t
Tan
(t)
Separating crystallite size broadening and strain broadening
scr BBB )(
CosL
kBc )( TanBs
)()(
Tan
CosL
kBr
)()( SinL
kCosBr
Plot of [Br Cos] vs [Sin]
Example of a calculation
Sample: Annealed AlRadiation: Cu k ( = 1.54 Å)
Sample: Cold-worked AlRadiation: Cu k ( = 1.54 Å)
2 →
Inte
nsit
y →
2 →
Inte
nsit
y →
40 60
40 60X-Ray Diffraction: A Practical Approach, C. Suryanarayana & M. Grant Norton, Plenum Press, New York (1998)
Annealed Al
Peak No. 2 () hkl Bi = FWHM () Bi = FWHM (rad)
1 38.52 111 0.103 1.8 10−3
2 44.76 200 0.066 1.2 10−3
3 65.13 220 0.089 1.6 10−3
Cold-worked Al
2 () Sin() hkl B () B (rad) Br Cos (rad)
1 38.51 0.3298 111 0.187 3.3 10−3 2.8 10−3 2.6 10−3
2 44.77 0.3808 200 0.206 3.6 10−3 3.4 10−3 3.1 10−3
3 65.15 0.5384 220 0.271 4.7 10−3 4.4 10−3 3.7 10−3
222ir BBB
3107.1 L
k nmLSizeGrain 90)(
end
Iso-intensity circle
Extinction Rules
Structure Factor (F): The resultant wave scattered by all atoms of the unit cell
The Structure Factor is independent of the shape and size of the unit cell; but is dependent on the position of the atoms within the cell
Structure factor calculation
Consider a general unit cell for this type of structure. It can be reduced to 4 atoms of type A at 000, 0 ½ ½, ½ 0 ½, ½ ½ 0 i.e. in the fcc position and 4 atoms of type B at the sites ¼ ¼ ¼ from the A sites. This can be expressed as:
The structure factors for this structure are:
F = 0 if h, k, l mixed (just like fcc)
F = 4(fA ± ifB) if h, k, l all odd
F = 4(fA - fB) if h, k, l all even and h+ k+ l = 2n where n=odd (e.g. 200)
F = 4(fA + fB) if h, k, l all even and h+ k+ l = 2n where n=even (e.g. 400)
Consider the compound ZnS (sphalerite). Sulphur atoms occupy fcc sites with zinc atoms displaced by ¼ ¼ ¼ from these sites. Click on the animation opposite to show this structure. The unit cell can be reduced to four atoms of sulphur and 4 atoms of zinc.
Many important compounds adopt this structure. Examples include ZnS, GaAs, InSb, InP and (AlGa)As. Diamond also has this structure, with C atoms replacing all the Zn and S atoms. Important semiconductor materials silicon and germanium have the same structure as diamond.
421 missing
Ewald sphere X-rays
(Cu K) = 1.54 Å, 1/ = 0.65 Å−1, aCu = 3.61 Å, 1/aCu = 0.28 Å−1
0.28 Å 1
0.65 Å 1
Multiplicity factor
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) 8 [(111) (111) (111) (111)] ( 2 for negatives)
(210) 24(210) = 3! Ways, (210) = 3! Ways,
(210) = 3! Ways, (210) = 3! Ways,
(211) 24
(321) 48
Tetragonal (100) 4 [(100) (010)]
(110) 4 [(110) (110)]
(111) 8 [(111) (111) (111) (111)] ( 2 for negatives)
(210) 6
(211) 24
(321) 48
