Title: X-ray Scattering
1X-ray Scattering
- Peak intensities
- Peak widths
Second Annual SSRL Workshop on Synchrotron X-ray Scattering Techniques in Materials and Environmental Sciences Theory and Application
Tuesday, May 15 - Thursday, May 17, 2007
2Peak Intensities and Widths
Silicon
3Basic Principles of Interaction of Waves
- Periodic wave characteristics
- Frequency ? number of waves (cycles) per unit
time ?cycles/time. - ? 1/sec Hz.
- Period T time required for one complete cycle
T1/?time/cycle. T sec. - Amplitude A maximum value of the wave during
cycle. - Wavelength ? the length of one complete cycle.
? m, nm, Å.
A simple wave completes cycle in 360 degrees
4Basic Principles of Interaction of Waves
5Basic Principles of Interaction of Waves
- Consider two waves with the same wavelength and
amplitude but displaced a distance x0. - The phase shift
Waves 1 and 2 are 90o out of phase
Waves 1 and 2 are 180o out of phase
When similar waves combine, the outcome can be
constructive or destructive interference
6Superposition of Waves
- Resulting wave is algebraic sum of the amplitudes
at each point
Small difference in phase
Large difference in phase
7Superposition of Waves
Difference in frequency and phase
8The Laue Equations
- If an X-ray beam impinges on a row of atoms, each
atom can serve as a source of scattered X-rays. - The scattered X-rays will reinforce in certain
directions to produce zero-, first-, and
higher-order diffracted beams.
9The Laue Equations
- Consider 1D array of scatterers spaced a apart.
- Let x-ray be incident with wavelength ?.
10Lattice Planes
11Braggs Law
- If the path AB BC is a multiple of the x-ray
wavelength ?, then two waves will give a
constructive interference
12Braggs Law
- The incident beam and diffracted beam are always
coplanar. - The angle between the diffracted beam and the
transmitted beam is always 2?. - Rewrite Braggs law
Since sin? ? 1
For n 1
UV radiation ? ? 500 Å Cu K?1 ? 1.5406 Å
For most crystals d 3 Å
? ? 6 Å
13Braggs Law
Silicon lattice constant
aSi 5.43 Å
14Scattering by an Electron
- Elementary scattering unit in an atom is electron
- Classical scattering by a single free electron
Thomson scattering equation
The polarization factor of an unpolarized primary
beam
If r few cm
Another way for electron to scatter is manifested
in Compton effect. Cullity p.127
1 mg of matter has 1020 electrons
15Scattering by an Atom
Cu
Atomic Scattering Factor
16Scattering by an Atom
- Scattering by a group of electrons at positions
rn
Scattering factor per electron
Assuming spherical symmetry for the charge
distribution ? ?(r ) and taking origin at the
center of the atom
For an atom containing several electrons
f atomic scattering factor
Calling Z the number of electrons per atom we
get
17Scattering by an Atom
- The atomic scattering factor f Z for any atom
in the forward direction (2? 0) I(2?0) Z2 - As ? increases f decreases ? functional
dependence of the decrease depends on the details
of the distribution of electrons around an atom
(sometimes called the form factor) - f is calculated using quantum mechanics
18Scattering by an Atom
19Scattering by a Unit Cell
For atoms A C
For atoms A B
phase
If atom B position
For 3D
20Scattering by a Unit Cell
We can write
21Scattering by a Unit Cell
Useful expressions
Hint dont try to use the trigonometric form
using exponentials is much easier
22Scattering by a Unit Cell
Unit cell has one atom at the origin
In this case the structure factor is independent
of h, k and l it will decrease with f as sin?/?
increases (higher-order reflections)
23Scattering by a Unit Cell
Unit cell is base-centered
h and k unmixed
h and k mixed
(200), (400), (220) ? ? (100), (121), (300) ? ?
forbiddenreflections
24Scattering by a Unit Cell
For body-centered cell
when (h k l ) is even
when (h k l ) is odd
(200), (400), (220) ? ? (100), (111), (300) ? ?
forbiddenreflections
25Scattering by a Unit Cell
For body centered cell with different atoms
when (h k l) is even
Cs
Cl
when (h k l) is odd
(200), (400), (220) ? ? (100), (111), (300) ? ?
26Scattering by a Unit Cell
- The fcc crystal structure has atoms at (0 0 0),
(½ ½ 0), (½ 0 ½) and (0 ½ ½)
- If h, k and l are all even or all odd numbers
(unmixed), then the exponential terms all equal
to 1 ? F 4 f - If h, k and l are mixed even and odd, then two
of the exponential terms will equal -1 while one
will equal 1 ? F 0
16f 2, h, k and l unmixed even and odd 0, h, k
and l mixed even and odd
27The Structure Factor
- The structure factor contains the information
regarding the types ( f ) and locations (u, v, w)
of atoms within a unit cell. - A comparison of the observed and calculated
structure factors is a common goal of X-ray
structural analyses. - The observed intensities must be corrected for
experimental and geometric effects before these
analyses can be performed.
28Diffracted Beam Intensity
- Structure factor
- Polarization factor
- Lorentz factor
- Multiplicity factor
- Temperature factor
- Absorption factor
- ..
29The Polarization Factor
- The polarization factor p arises from the fact
that an electron does not scatter along its
direction of vibration - In other directions electrons radiate with an
intensity proportional to (sin a)2
The polarization factor (assuming that the
incident beam is unpolarized)
30The Lorentz - Polarization Factor
- The Lorenz factor L depends on the measurement
technique used and, for the diffractometer data
obtained by the usual ?-2? or ?-2? scans, it can
be written as - The combination of geometric corrections are
lumped together into a single Lorentz-polarization
(Lp) factor
The effect of the Lp factor is to decrease the
intensity at intermediate angles and increase the
intensity in the forward and backwards directions
31The Temperature Factor
- As atoms vibrate about their equilibrium
positions in a crystal, the electron density is
spread out over a larger volume. - This causes the atomic scattering factor to
decrease with sinq/l (or S 4psinq/l) more
rapidly than it would normally.
The temperature factor is given by
where the thermal factor B is related to the mean
square displacement of the atomic vibration
Scattering by C atom expressed in electrons
This is incorporated into the atomic scattering
factor
32The Multiplicity Factor
- The multiplicity factor arises from the fact that
in general there will be several sets of hkl
-planes having different orientations in a
crystal but with the same d and F 2 values - Evaluated by finding the number of variations in
position and sign in ?h, ?k and ?l and have
planes with the same d and F 2 - The value depends on hkl and crystal symmetry
- For the highest cubic symmetry we have
p100 6
p110 12
p1118
33The Absorption Factor
- Angle-dependent absorption within the sample
itself will modify the observed intensity
Absorption factor for infinite thickness specimen
is
Absorption factor for thin films is given by
where µ is the absorption coefficient, t is the
total thickness of the film
34Diffracted Beam Intensity
where K is the scaling factor, Ib is the
background intensity, q 4sin?/? is the
scattering vector for x-rays of wavelength ?
35Diffraction Real Samples
- Up to this point we have been considering
diffraction arising from infinitely large
crystals that are strain free and behave like
ideally imperfect materials ( x-rays only
scattered once within a crystal) - Crystal size and strain affect the diffraction
pattern - we can learn about them from the diffraction
pattern - High quality crystals such as those produced for
the semiconductor industry are not ideally
imperfect - need a different theory to understand how they
scatter x-rays - Not all materials are well ordered crystals
36Crystallite Size
- As the crystallites in a powder get smaller the
diffraction peaks in a powder pattern get wider. - Consider diffraction from a crystal of thickness
t and how the diffracted intensity varies as we
move away from the exact Bragg angle - If thickness was infinite we would only see
diffraction at the Bragg angle
37Crystallite Size
Suppose the crystal of thickness t has (m 1)
planes in the diffraction direction. Let say ?
is variable with value ?B that exactly satisfies
Braggs Law
- Rays A, D, , M makes angle ?B
- Rays B, , L makes angle ?1
- Rays C, , N makes angle ?2
38Crystallite Size
- For angle ?B diffracted intensity is maximum
- For ?1 and ?2 intensity is 0.
- For angles ?1 gt ? gt ?2 intensity is nonzero.
39The Scherrer Equation
Subtracting
?1 and ?2 are close to ?B, so
Thus
40The Scherrer Equation
- More exact treatment (see Warren) gives
Scherrers formula
Suppose ? 1.54 Å, d 1.0 Å, and ? 49o for
crystal size of 1 mm, B 10-5 deg. for crystal
size of 500 Å, B 0.2 deg.
41Interference Function
- We calculate the diffraction peak at the exact
Bragg angle ?B and at angles that have small
deviations from ?B. - If crystal is infinite then at ? ? ?B intensity
0. - If crystal is small then at ? ? ?B intensity ? 0.
It varies with angle as a function of the number
of unit cells along the diffraction vector (s
s0). - At deviations from ?B individual unit cells will
scatter slightly out of phase. - Vector (s s0)/? no longer extends to the
reciprocal lattice point (RLP).
42Interference Function
- ?(1) gt ?B(1) for 001 and ?(2) gt ?B(2) for 002
- If Hhkl H is reciprocal lattice vector then (s
s0)/? ? H.
Real space
Reciprocal space
43Interference Function
- We define
- as deviation parameter
44Interference Function
- In order to calculate the intensity diffracted
from the crystal at ? ? ?B, the phase differences
from different unit cells must be included. - For three unit vectors a1, a2 and a3
ni particular unit cells Ni total number of
unit cells along ai
From the definition of the reciprocal lattice
vector
45Interference Function
since
Converting from exp to sines
46Interference Function
- Calculating intensity we lose phase information
therefore
interference function
Maximum intensity at Bragg peak is F2N2 Width of
the Bragg peak ? 1/N N is a number of unit cells
along (s - s0)
47Interference Function
48Interference Function
49Small crystallites (nanocrystallites)
- Consider our sample as any form of matter in
which there is random orientation. - This includes gases, liquids, amorphous solids,
and crystalline powders. - The scattered intensity from such sample
where
takes all orientations
50Small crystallites (nanocrystallites)
- Average intensity from an array of atoms which
takes all orientations in space
Debye scattering equation
where
It involves only the magnitudes of the distances
rmn of each atom from every other atom
51Small crystallites (nanocrystallites)
- Consider material consisting of polyatomic
molecules. - It is not too dense there is complete
incoherency between the scattering by different
molecules. - Intensity per molecule
- Lets take a carbon tetrachloride as example.
- It is composed of tetrahedral molecules CCl4.
- Then
correction factor
C Cl4
52Small crystallites (nanocrystallites)
- For tetrahedral CCl4 molecule
- The intensity depends on just one distance r(C
Cl). - Peaks and dips do not require the existence of a
crystalline structure. - Certain interatomic distances that are more
probable than others are enough to get peaks and
dips on the scattering curve.
Intensity (in e.u. per molecule) for a CCl4 gas
in which the C Cl distance is r 1.82 Å.
(Warren)
53Small crystallites (nanocrystallites)
- Lets treat crystal as a molecule
- FCC has 14 atoms
- 8 at corners of a cube
- 6 at face center positions
- a is the edge of a cube
54Reciprocal Lattice and Diffraction
It is equivalent to the Bragg law since