Objectives

1
Objectives

• By the end of this section you should
• understand the concept of diffraction in crystals
• be able to derive and use Braggs law
• know how X-rays are produced
• know the typical emission spectrum for X-rays,
  the source of white radiation and the K? and K?
  lines
• know about Compton scattering

2
Diffraction - an optical grating
Path difference XY between diffracted beams 1 and
2 sin? XY/a ? XY a sin ?
For 1 and 2 to be in phase and give constructive
interference, XY ?, 2?, 3?, 4?..n? so a
sin ? n? where n is the order of
diffraction
3
Consequences maximum value of ? for
diffraction sin? 1 ? a ? Realistically, sin
?lt1 ? a gt ? So separation must be same order as,
but greater than, wavelength of light. Thus for
diffraction from crystals Interatomic distances
0.1 - 2 Å so ? 0.1 - 2 Å X-rays, electrons,
neutrons suitable
4
Diffraction from crystals
?
5
Beam 2 lags beam 1 by XYZ 2d sin ? so 2d sin
? n? Braggs Law
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e.g. X-rays with wavelength 1.54Å are reflected
from planes with d1.2Å. Calculate the Bragg
angle, ?, for constructive interference.
? 1.54 x 10-10 m, d 1.2 x 10-10 m, ??
n1 ? 39.9 n2 X (n?/2d)gt1
2d sin ? n?
We normally set n1 and adjust Miller indices, to
give 2dhkl sin ? ?
7
Use Braggs law and the d-spacing equation to
solve a wide variety of problems
2d sin ? n? or 2dhkl sin ? ?
8
Example of equivalence of the two forms of
Braggs law
Calculate ? for ?1.54 Å, cubic crystal, a5Å 2d
sin ? n?
(1 0 0) reflection, d5 Å n1, ?8.86o n2, ?17.
93o n3, ?27.52o n4, ?38.02o n5, ?50.35o n6,
?67.52o no reflection for n?7
(2 0 0) reflection, d2.5 Å n1, ?17.93o n2, ?
38.02o n3, ?67.52o no reflection for n?4
9
Combining Bragg and d-spacing equation
X-rays with wavelength 1.54 Å are reflected
from the (1 1 0) planes of a cubic crystal with
unit cell a 6 Å. Calculate the Bragg angle, ?,
for all orders of reflection, n.
d 4.24 Å
10
d 4.24 Å
n 1 ? 10.46 n 2 ? 21.30 n 3
? 33.01 n 4 ? 46.59 n 5 ?
65.23
(1 1 0) (2 2 0) (3 3 0) (4 4 0) (5 5 0)
2dhkl sin ? ?
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X-rays and solids

• X-rays - electromagnetic waves
• So X-ray photon has E h?
• X-ray wavelengths vary from .01 - 10Å those used
  in crystallography have frequencies 2-6 x 1018Hz

Q. To what wavelength range does this frequency
range correspond?
c ??
?max 1.5 Å ?min 0.5 Å
12
In the classical treatment, X-rays interact with
electrons in an atom, causing them to oscillate
with the X-ray beam. The electron then acts as a
source of an electric field with the same
frequency ? Electrons scatter X-rays with no
frequency shift
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Production of X-rays
14
Two processes lead to two forms of X-ray emission
? Electrons stopped by target kinetic energy
converted to X-rays continuous spectrum of
white radiation, with cut-off at short ?
(according to h?½mv2) Wavelength not
characteristic of target ? Incident electrons
displace inner shell electrons, intershell
electron transitions from outer shell to inner
shell vacancy. line spectra Wavelength
characteristic of target
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Each element has a characteristic wavelength. For
copper, the ? are CuK?1 1.540 Å CuK?2 1.544
Å CuK? 1.39 Å
Typical emission spectrum
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Many intershell transitions can occur - the
common transitions encountered are
2p (L) - 1s (K), known as the K? line 3p (M) - 1s
(K), known as the K? line
(in fact K? is a close doublet, associated with
the two spin states of 2p electrons)
17
Copper K? X-rays have a wavelength of 1.54 Å and
are produced when an electron falls from the L
shell to a vacant site in the K shell of a copper
atom. Calculate the difference in the energy
levels between the K and L shells of copper.
E h? ? c/ ? (3 x108) / (1.54 x 10-10)
1.95 x 1018 Hz E h? 6.626 x 10-34 x 1.95 x
1018 1.29 x 10-15 J 8 keV
18
Some radiation is also scattered, resulting in a
loss of energy and hence, Eh?, shorter
frequency and, c ? ?, longer wavelength. The
change in frequency/wavelength depends on the
angle of scattering. This effect is known as
Compton scattering It is a quantum effect -
remember classically there should be no frequency
shift
19
Calculate the maximum wavelength shift predicted
from the Compton scattering equation.
4.85 x 10-12 m 0.05Å
20
Filter
As well as characteristic emission spectra,
elements have characteristic absorption
wavelengths
e.g. copper
21
We want to choose an element which absorbs K?
and high energy/low ? white radiation but
transmits K?
e.g. Ni K absorption edge 1.45 Å
As a general rule use an element whose Z is one
or two less than that of the emitting atom
22
Monochromator
Choose a crystal (quartz, germanium etc.) with a
strong reflection from one set of lattice planes,
then orient the crystal at the Bragg angle for K?1
? 1.540 Å 2dhklsin?
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Example A monochromator is made using the (111)
planes of germanium, which is cubic, a5.66Å.
Calculate the angle at which it must be oriented
to give CuK?1 radiation
d3.27Å
?2d sin?
13.62
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Summary

• Crystals diffract radiation of a similar order of
  wavelength to the interatomic spacings
• We model this diffraction by considering the
  reflection of radiation from planes - Braggs
  Law
• X-rays are produced by intershell transitions -
  e.g. L-K (K?) and M-K (K?)
• The interaction of X-rays with matter produces a
  small wavelength shift (Compton scattering)
• Filters can be used to eliminate K? radiation
  monochromators are used to select K?1 radiation.