X-ray Scattering

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X-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
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Peak Intensities and Widths
Silicon
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Basic 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
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Basic Principles of Interaction of Waves

• For t 0
• For x 0

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Basic 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
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Superposition of Waves

• Resulting wave is algebraic sum of the amplitudes
  at each point

Small difference in phase
Large difference in phase
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Superposition of Waves
Difference in frequency and phase
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The 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.

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The Laue Equations

• Consider 1D array of scatterers spaced a apart.
• Let x-ray be incident with wavelength ?.

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Lattice Planes
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Braggs Law

• If the path AB BC is a multiple of the x-ray
  wavelength ?, then two waves will give a
  constructive interference

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Braggs 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
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Braggs Law
Silicon lattice constant
aSi 5.43 Å
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Scattering 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
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Scattering by an Atom
Cu
Atomic Scattering Factor
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Scattering 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
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Scattering 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

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Scattering by an Atom
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Scattering by a Unit Cell
For atoms A C
For atoms A B
phase
If atom B position
For 3D
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Scattering by a Unit Cell
We can write
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Scattering by a Unit Cell
Useful expressions
Hint dont try to use the trigonometric form
using exponentials is much easier
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Scattering by a Unit Cell

• Examples

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)
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Scattering by a Unit Cell

• Examples

Unit cell is base-centered
h and k unmixed
h and k mixed
(200), (400), (220) ? ? (100), (121), (300) ? ?
forbiddenreflections
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Scattering by a Unit Cell

• Examples

For body-centered cell
when (h k l ) is even
when (h k l ) is odd
(200), (400), (220) ? ? (100), (111), (300) ? ?
forbiddenreflections
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Scattering by a Unit Cell

• Examples

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) ? ?
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Scattering 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
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The 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.

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Diffracted Beam Intensity

• Structure factor
• Polarization factor
• Lorentz factor
• Multiplicity factor
• Temperature factor
• Absorption factor
• ..

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The 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)
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The 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
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The 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
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The 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
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The 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
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Diffracted Beam Intensity
where K is the scaling factor, Ib is the
background intensity, q 4sin?/? is the
scattering vector for x-rays of wavelength ?
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Diffraction 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

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Crystallite 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

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Crystallite 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

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Crystallite Size

• For angle ?B diffracted intensity is maximum
• For ?1 and ?2 intensity is 0.
• For angles ?1 gt ? gt ?2 intensity is nonzero.

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The Scherrer Equation
Subtracting
?1 and ?2 are close to ?B, so
Thus
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The 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.
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Interference 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).

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Interference 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
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Interference Function

• We define
• as deviation parameter

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Interference 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
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Interference Function
since
Converting from exp to sines
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Interference 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)
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Interference Function
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Interference Function
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Small 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
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Small 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
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Small 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
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Small 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)
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Small 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

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Reciprocal Lattice and Diffraction
It is equivalent to the Bragg law since