Transcript Slide 1
X-RAY DIFFRACTION
X- Ray Sources Diffraction: Bragg’s Law Crystal Structure Determination
Elements of X-Ray Diffraction
B.D. Cullity & S.R. Stock Prentice Hall, Upper Saddle River (2001)
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 A accelerating charge radiates electromagnetic radiation
Mo Target impacted by electrons accelerated by a 35 kV potential White radiation K K Characteristic radiation → due to energy transitions in the atom 0.2
0.6
1.0
Wavelength ( ) 1.4
Target Metal
Mo Cu Co Fe Cr
Of K
radiation (Å)
0.71
1.54
1.79
1.94
2.29
Incident X-rays SPECIMEN Heat Fluorescent X-rays Electrons Scattered X-rays Compton recoil Coherent
From bound charges
Incoherent (Compton modified)
From loosely bound charges
Photoelectrons Transmitted beam
X-rays can also be refracted (refractive index slightly less than 1) and reflected (at very small angles)
Incoherent Scattering (Compton modified)
From loosely bound charges
Here the particle picture of the electron & photon comes handy E
1
h
1 ( 1 , 1 )
Electron knocked aside
2
E
2
h
2 ( 2 , 2 ) 2 1 0 .
0243 ( 1
Cos
2 ) No fixed phase relation between the incident and scattered waves
Incoherent
does not contribute to diffraction (Darkens the background of the diffraction patterns)
Fluorescent X-rays Vacuum Energy levels E L 3 E L 2 E L 1 E K Nucleus Knocked out electron from inner shell L 3 L 2 L 1 K Characteristic x-rays (Fluorescent X-rays)
(10 −16 s later
seems like scattering!)
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 Incoming X-rays Secondary emission
Sets Electron cloud into oscillation Sets nucleus into oscillation
Small effect
neglected
Oscillating charge re-radiates In phase with the incoming x-rays
Ray 1 Ray 2 BRAGG’s EQUATION Deviation = 2
d
The path difference between ray 1 and ray 2 = 2d Sin For constructive interference: n = 2d Sin
Incident and scattered waves are in phase if In plane scattering is in phase Scattering from across planes is in phase
Atomic Planes X A B Y 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 X A B Y But this is still reinforced scattering
and NOT reflection
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 Occurs from surface Takes place at any angle ~100 % of the intensity may be reflected Diffraction Occurs throughout the bulk Takes place only at Bragg angles 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 d 110 = 2.22 Å n 1 2 Sin 0.34
0.69
d hkl
a h
2
k
2
l
2 20.7º 43.92º First order reflection from (110) Second order reflection from (110) Also written as (220)
d
220
d
110
a a
8 2
d d
220 110 1 2
In XRD n th order reflection from (h k l) is considered as 1 from (nh nk nl) st order reflection
n
2
d hkl
sin 2
d hkl n
sin 2
d
n
h
n
k
n
l
sin
Crystal structure determination
Monochromatic X-rays Many s (orientations) Powder specimen POWDER METHOD
Pan
chromatic X-rays Single LAUE TECHNIQUE Monochromatic X-rays Varied by rotation ROTATING CRYSTAL METHOD
THE POWDER METHOD 2
dSin
d
2
h
2
a k
2
l
2
h
4
a
2 sin 2
k
2 2
l
2 (
h
2
k
2 (
h
2
k
2
l
2 )
l
2 ) 4
a
2 2 sin sin 2 2
Intensity of the Scattered electrons Scattering by a crystal A Electron B Atom C Unit cell (uc)
A Scattering by an Electron ( 0 , 0 )
Sets electron into oscillation
Coherent
(definite phase relationship)
( 0 , 0 )
Scattered beams
For an polarized wave
For a wave oscillating in z direction
z
r P
Intensity of the scattered beam due to an electron (I)
I
I
0
e
4
m
2
c
4
Sin
2
r
2
x
For an unpolarized wave E is the measure of the amplitude of the wave E 2 = Intensityc
E
2
E y
2
E z
2
I
0 2
I
0
y
I
0
z
I Py = Intensity at point P due to
E
y
I Py
I
0
y e
4
m
2
c
4
Sin
2
r
2 2
I
0
y e
4
m
2
c
4 1
r
2 I Pz = Intensity at point P due to
E
z
I Pz
I
0
z e
4
m
2
c
4
Sin
2
r
2 2 2
I
0
z e
4
m
2
c
4
Cos r
2 2
I P
I Py
I Pz
e
4
m
2
c
4
I
0
y
I
0
z Cos
2
r
2
I P
I
0 2
e
4
m
2
c
4 1
Cos
2
r
2 Scattered beam is not unpolarized
Very small number
Rotational symmetry about x axis + mirror symmetry about yz plane Forward and backward scattered intensity higher than at 90 Scattered intensity minute fraction of the incident intensity
I P
I
0
r
2
e
4
m
2
c
4 1
Cos
2 2 Polarization factor
Comes into being as we used unpolarized beam
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 Sin
( ) f Atomic Scattering Factor Amplitude Amplitude of wave scattered by an atom of wave scattered by an electron 30 20
Schematic
10 0.2
Sin
0.4
( ) 0.6
0.8
(Å −1 ) → 1.0
Z Sin( ) / Coherent scattering Incoherent (Compton) scattering
C Scattering by the Unit cell (uc)
Coherent Scattering Unit Cell (uc) 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
Ray 1 = R 1 Ray 3 = R 3 Ray 2 = R 2 (h00) plane M B A R B S N C Unit Cell a
R
2 '
R
1 '
R
3 '
x d
(h00)
AC
d h
00
a h MCN
::
AC
::
RBS
::
AB
::
x AB AC
x
x a h
R
1
R
2
MCN
2
d h
00
Sin
( )
R
1
R
3
RBS
AB
AC
x a h
R
1
R
3 2
x a h
2
h x a
2
x a
fractional coordinate
x
Extending to 3D 2 (
h x
k y
h z
)
R
1
R
3 2
h x
Independent of the shape of uc Note: R 1 is from corner atoms and R 3 is from atoms in additional positions in uc
2 (
h x
k y
h z
)
In complex notation E
Ae i
fe i
[ 2 (
h x
k y
h z
)]
If atom B is different from atom A
atomic scattering factors (f) the amplitudes must be weighed by the respective 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) F Structure Factor Amplitude of wave scattered by all atoms in uc Amplitude of wave scattered by an electron
F n hkl
j n
1
f j e i
j
j n
1
f j e i
[ 2 (
h x j
k y j
h z j
)]
I
F
2 Structure factor is independent of the
shape
and
size
of the unit cell
Structure factor calculations A Simple Cubic Atom at (0,0,0) and equivalent positions
e ni
( 1 )
n e
(
odd n
)
i
1
e
(
even n
)
i
1
e ni
e
ni
e i
e
i
Cos
( ) 2
F
f j e i
j
f j e i
[ 2 (
h x j
k y j
h z j
)]
F
f e i
[ 2 (
h
0
k
0
h
0 )]
f e
0
f F
2
f
2
F is independent of the scattering plane (h k l)
B Atom at (0,0,0) & (½, ½, 0) and equivalent positions C- centred Orthorhombic
F
f j e i
j
f j e i
[ 2 (
h x j
k y j
h z j
)]
F
f e i
[ 2 (
h
0
k
0
h
0 )]
f e i
[ 2 (
h
1 2
k
1 2
h
0 )]
f e
0
f e i
[ 2 (
h
k
2 )]
f
[ 1
e i
(
h
k
)
]
Real
F
f
[ 1
e i
(
h
k
)
]
F
2
f F
2 4
f
2
e.g. (001), (110), (112); (021), (022), (023) F
0
F
2
0
e.g. (100), (101), (102); (031), (032), (033)
F is independent of the ‘l’ index
C Atom at (0,0,0) & (½, ½, ½) and equivalent positions Body centred Orthorhombic
F
f j e i
j
f j e i
[ 2 (
h x j
k y j
h z j
)]
F
f e i
[ 2 (
h
0
k
0
h
0 )]
f e i
[ 2 (
h
1 2
k
1 2
h
1 2 )]
f e
0
f e i
[ 2 (
h
k
l
)] 2
f
[ 1
e i
(
h
k
l
)
]
Real
F
f
[ 1
e i
(
h
k
l
)
]
F
2
f F
2 4
f
2
e.g. (110), (200), (211); (220), (022), (310) F
0
F
2
0
e.g. (100), (001), (111); (210), (032), (133)
D Atom at (0,0,0) & (½, ½, 0) and equivalent positions Face Centred Cubic
F
f j e i
j
f j e i
[ 2 (
h x j
k y j
h z j
)]
F
f
e i
[ 2 ( 0 )]
f
[ 1
e i
(
h
k
)
e i
[ 2 (
h
k
2
e i
)] (
k
l
)
e i
[ 2 (
k
2
l
)]
e i
(
l
h
) ]
e i
[ 2 (
l
h
2 )] Real
(½, ½, 0), (½, 0, ½), (0, ½, ½) F
f
[ 1
e i
(
h
k
)
e i
(
k
l
)
e i
(
l
h
) ] (h, k, l) unmixed
F
4
f
(h, k, l) mixed
F
0
F
2
16
f
2
e.g. (111), (200), (220), (333), (420) F
2
0
e.g. (100), (211); (210), (032), (033)
Two odd and one even (e.g. 112); two even and one odd (e.g. 122)
E
F
Na + Cl − at (0,0,0) + Face Centering Translations at (½, 0, 0) + FCT
(½, ½, 0), (½, 0, ½), (0, ½, ½) (0, ½, 0), (0, 0, ½), (½, ½, ½)
NaCl:
f Na
e i
[ 2 ( 0 )]
e i
[ 2 (
h
2
k
)]
e i
[ 2 (
k
l
2 )]
e i
[ 2 (
l
h
2 )] Face Centred Cubic
f Cl
e i
[ 2 (
h
2 )]
e i
[ 2 (
k
2 )]
e i
[ 2 (
l
2 )]
e i
[ 2 (
h
k
l
2 )]
F
f f Na
[ 1
e i
(
h
k
)
Cl
[
e i
(
h
)
e i
(
k
)
e i
(
k
l
)
e i
(
l
)
e i
(
l
h
) ]
e i
(
h
k
l
) ]
F
f f Na
[ 1
e i
(
h
k
)
Cl
e i
e i
(
h
k
l
) [
e i
(
k
l
) (
k
l
)
e i
e i
(
l
h
) (
l
h
) ]
e i
(
h
k
) 1 ]
F
[
f Na
f Cl
e i
(
h
k
l
) ][ 1
e i
(
h
k
)
e i
(
k
l
)
e i
(
l
h
) ]
F
[
f Na
f Cl
e i
(
h
k
l
) ][ 1
e i
(
h
k
)
e i
(
k
l
)
e i
(
l
h
) ] (h, k, l) mixed
F
0
F
2
0
Zero for mixed indices
e.g. (100), (211); (210), (032), (033)
(h, k, l) unmixed
F
4 [
f Na
f Cl
e i
(
h
k
l
) ]
F
4 [
f Na
f Cl
]
F
4 [
f Na
f Cl
] If (h + k + l) is even If (h + k + l) is odd
F
2 16 [
f Na
f Cl
] 2
F
2 16 [
f Na
f Cl
] 2
Presence of additional atoms/ions/molecules in the uc (as a part of the motif ) can alter the intensities of some of the reflections
Relative Intensity of diffraction lines in a powder pattern Structure Factor (F) Scattering from uc Multiplicity factor (p) Number of equivalent scattering planes Polarization factor
I P
1
Cos
2 Effect of wave polarization Lorentz factor Combination of 3 geometric factors
Lorentz factor
1
Sin
2
Cos
1
Sin
2 Absorption factor Specimen absorption Temperature factor Thermal diffuse scattering
Multiplicity factor Lattice Cubic Index (100) Tetragonal (110) (111) (210) (211) (321) (100) (110) (111) (210) (211) (321) Multiplicity 6 12 8 24 Planes [(100) (010) (001)] ( 2 for negatives) [(110) (101) (011), ( 110) ( 101) (0 11)] ( 2 for negatives) [(111) (11 1) (1 11) ( 111)] ( 2 for negatives) (210) = 3! Ways, ( 210) = 3! Ways, (2 10) = 3! Ways, ( 2 10) = 3! Ways, 21 48 4 4 8 6 21 48 [(111) (11 1) (1 [(100) (010)] [(110) ( 11) ( 110)] 111)] ( 2 for negatives)
Polarization factor
I P
1
Cos
2 Lorentz factor
Lorentz factor
1
Sin
2
Cos
1
Sin
2
Lorentz Polarizati on factor
1
Sin Cos
2 2
Cos
25 20 15 10 5 0 0 20 40 60
Bragg Angle (
, degrees)
80
Intensity of powder pattern lines
(ignoring Temperature & Absorption factors) I
F
2
p
1
Sin
2
Cos
2 2
Cos
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 m e , e Random orientation of crystals (1/dectector radius) 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
Crystal = Lattice + Motif In crystals based on a particular lattice the intensities of particular reflections are modified
they may even go missing
Diffraction Pattern Position of the Lattice points
LATTICE
Intensity of the diffraction spots
MOTIF
Reciprocal Lattice
Properties are reciprocal to the crystal lattice
b
1 *
1
V
a
2
a
3
b i
* usuall written as
a i
*
b
2 *
1
V
a
3
a
1
b
3
* b
* 3
b
3 * 1
V
a
1
a
2
Area
(
OXMB
)
Area
(
OXMB
)
Height of Cell
1
OP
b
3 *
1
V
a
1
a
2
a
3
a
2
b
* 3 1
d
001
a
1
The reciprocal lattice is created by interplanar spacings
A reciprocal lattice vector is to the corresponding real lattice plane *
g hkl
h
b
1 *
k
b
2 *
l
b
3 * The length of a reciprocal lattice vector is the reciprocal of the spacing of the corresponding real lattice plane *
g hkl
g
*
hkl
1
d hkl
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)
a
2 (11) (21)
a
1
a
1 (10) 02 12 22 01
b
2
*
*
g
11 11 *
g
21 21 00
b
1
*
10 1
a
1 20
The reciprocal lattice has an origin!
22 (01) 12 02
a
2 (21) (11)
a
1
a
1 22 12 02 (01) 21 11 01 (21) (11) 10 (10) 20 00 (10) 01
b
2
*
00
b
1
*
11 10
Note perpendicularity of various vectors
21 20
Reciprocal lattice
geometrical
is the reciprocal of a
primitive
lattice and is does not deal with the intensities of the points
purely
Physics comes in from the following: For non-primitive cells ( decorated with motifs ( lattices with additional points) and for crystals crystal = lattice + motif) the Reciprocal lattice points have to be weighed in with the corresponding scattering power (|F hkl | 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
Examples of 3D Reciprocal Lattices
weighed in with scattering power (|F| 2 )
SC
001 011 101 111 Lattice = SC
No missing reflections
000 100 Reciprocal Lattice = SC 110 010
Figures NOT to Scale
002 022
BCC
202 222 011 101 020 Lattice = BCC 000 110 200
100 missing reflection (F = 0)
220 Weighing factor for each point
“motif”
Reciprocal Lattice = FCC
F
2 4
f
2
Figures NOT to Scale
002 022
FCC
202 222 111 020 000 Lattice = FCC 200
100 missing reflection (F = 0)
220
110 missing reflection (F = 0)
Weighing factor for each point
“motif” F
2
16
f
2 Reciprocal Lattice = BCC
Figures NOT to Scale
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
k
i then diffraction will occur only for those reciprocal lattice points that lie on the surface of the Ewald sphere
* Paul Peter Ewald (German physicist and crystallographer; 1888-1985) See Cullity’s book: A15-4
Ewald Sphere
Reciprocal Space
K i 02 2 K 01 K 00 10 20 D (41)
The Ewald Sphere touches the reciprocal lattice (for point 41)
Bragg’s equation is satisfied for 41
K
=
K
=
g
= Diffraction Vector
Diffraction cones and the Debye-Scherrer geometry
Film may be replaced with detector http://www.matter.org.uk/diffraction/x-ray/powder_method.htm
Powder diffraction pattern from Al Radiation: Cu K
,
= 1.54 Å
Note: Peaks or not idealized peaks broadend Increasing splitting of peaks with g Peaks are all not of same intensity
1 &
2 peaks resolved
5 6 7 8 9
n
1 2 3 4 Determination of Crystal Structure from 2 versus Intensity Data
2
38.52
44.76
65.14
78.26
82.47
99.11
112.03
116.60
137.47
19.26
22.38
32.57
39.13
41.235
49.555
56.015
58.3
68.735
Sin
0.33
0.38
0.54
0.63
0.66
0.76
0.83
0.85
0.93
Sin 2
0.11
0.14
0.29
0.40
0.43
0.58
0.69
0.72
0.87
12 16 19 20 24
ratio
3 4 8 11
Index
111 200 220 311 222 400 331 420 422
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
Bravais Lattice
Simple Body centred Face centred End centred Extinction Rules
Reflections which
may be
present
all (h + k + l) even h, k and l unmixed h and k unmixed
C centred
Bravais Lattice
SC BCC FCC DC
Reflections necessarily absent
None (h + k + l) odd h, k and l mixed h and k mixed
C centred
Allowed Reflections
All (h + k + l) even h, k and l unmixed h, k and l are all odd
Or
all are even
(h + k + l) divisible by 4
Determination of Crystal Structure from 2 versus Intensity Data n 2 → Intensity Sin Sin 2 ratio
The ratio of (h 2 + K 2 + l 2 ) derived from extinction rules SC BCC FCC DC 1 1 3 3 2 2 4 8 3 3 8 11 4 4 11 16 5 5 12 … 6 6 … 8 7 … …
1 2 3 4 5 6 7 2 → 21.5
25 37 45 47 58 68 Intensity Sin 0.366
0.422
0.60
0.707
0.731
0.848
0.927
Sin 2 0.134
0.178
0.362
0.500
0.535
0.719
0.859
ratio 3 4 8 11 12 16 19
FCC
h 2 + k 2 + l 2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
SC
100 110 111 200 210 211 220 300, 221 310 311 222 320 321 400 410, 322 411, 330 331
FCC
111 200 220 311 222 400
BCC
110 200 211 220 310 222 321 400 411, 330 331
DC
111 220 311 400 331
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.
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(
f
A ±
if
B ) if
h
,
k
,
l
all odd
F
= 4(
f
A -
f
B ) if
h
,
k
,
l
all even and
h
+
k
+
l
= 2n where n=odd (e.g. 200)
F
= 4(
f
A +
f
B ) if
h
,
k
,
l
all even and
h
+
k
+
l
= 2n where n=even (e.g. 400)
Applications of XRD
I P
1
Cos
2 Long range order Combination of 3 geometric factors
Lorentz factor
1
Sin
2
Cos
1
Sin
2 Crystallite size and Strain Specimen absorption Temperature factor Thermal diffuse scattering
0 0 Crystal Schematic of difference between the diffraction patterns of various phases 90 Diffraction angle (2 ) → 180 Diffraction angle (2 ) → 0 Monoatomic gas Diffraction angle (2 ) → 90 180 90 180