MME 315
Lecture 06
X-ray Diffraction
5. Diffraction intensity, scattering and structure factor
calculation
[Ref] 1. Y. Leng, Materials Characterization, Second edition, Wiley-VCH, Verlag
GmbH & Co. KgaA, 2013. (Chapt-2)
2. B. D. Cullity, Elements of X-ray Diffraction, Addison-Wesley Publishing
Company, Inc. (Chapt-3)
Lecture Objective
This lecture introduces the X-ray diffraction intensity, scattering and determination
of diffraction plane.
Lecture Outcomes
After completion of this lecture, students should be able to
1. use beam scattering in the diffraction intensity.
2. distinguish the scattering of diffracted beam by an electron, atom and unit cell.
3. calculate the intensity of beam diffracted by different unit cell.
4. validate the planes for diffraction of a crystal system.
X-ray Diffraction Methods
Diffraction intensity
The intensity of a diffracted beam is changed, not necessarily to zero, by
any change in atomic positions, and, conversely, we can only determine
atomic positions by observations of diffracted intensities.
To establish an exact relation
between atom position and intensity is
the main purpose
The problem is complex
because of the many variables
Involved, and we will have to
proceed step by step :
x-rays are scattered first by a single
electron, then by an atom, and finally
by all the atoms in the unit cell
Scattering by an electron
An electron which has been set into oscillation by an x-ray beam is
continuously accelerating and decelerating during its motion and therefore
emits an electromagnetic wave
The scattered beam being simply the beam radiated by the electron under the
action of the incident beam
The scattered beam has the same wavelength and frequency as the incident
beam and is said to be coherent with it.
The intensity I of the beam scattered by a single electron of charge e and mass
m, at a distance r from the electron, is given by
/o = intensity of the incident beam, c = velocity of light, and
α = angle between the scattering direction and the direction of
acceleration of the electron.
We wish to know the scattered intensity at P in the
xz plane where OP is inclined at a scattering angle
of 2 to the incident beam.
On the average, Ey will be equal to Ez, since the direction
of E is perfectly random. Therefore
The intensity of these two components of the incident
beam is proportional to the square of their electric
vectors, since E measures the amplitude of the wave
and the intensity of a wave is proportional to the square
of its amplitude.
The total scattered intensity at P is obtained by
summing the intensities of these two scattered
components:
This is the Thomson
equation for the
scattering of an x-ray
beam by a
single electron.
Thomson equation gives the absolute intensity (in ergs/sq cm/sec) of the scattered
beam in terms of the absolute intensity of the incident beam.
I (2) = I0/r2 (K) (1 + Cos2(2))/2
Polarization factor
Scattering by an atom.
When an x-ray beam encounters an atom, each electron in it scatters part of the
radiation coherently in accordance with the Thomson equation.
The nucleus has an extremely large mass relative to that of the electron and
cannot be made to oscillate to any appreciable extent
The net effect is that coherent scattering by an atom is due only to the electrons
contained in that atom.
Is the wave scattered by an atom simply the sum of the waves scattered by
its component electrons?
The answer is yes, if the scattering is in the forward direction (2 = 0)
-because the waves scattered by all the electrons of the atom are then in
phase and the amplitudes of all the scattered waves can be added directly
This is not true for other directions of scattering
Partial interference occurs between
the waves scattered by A and B, with the
result that the net amplitude of the wave
scattered in this direction is less than that
of the wave scattered by the same
electrons in the forward direction.
A quantity f, the atomic scattering factor,
As increases, the waves
scattered by individual
electrons become more and
more out of phase and
fcu decreases.
At a fixed value of , f will be
smaller the shorter the
wavelength
Scattering by a unit cell
The coherent scattering occurs, not from an isolated atom, but from all the atoms
making up the crystal.
To find the intensity of the beam diffracted by a crystal as a function of atom
position, consider the beam satisfy the Bragg law
Since the intensity of a wave is proportional to the square of its amplitude, we now
need an expression for A2, the square of the absolute value of the wave vector.
We can express any scattered wave in the complex exponential form
The resultant wave scattered by all the atoms of the unit cell is called the structure
factor and is designated by the symbol F
It is obtained by simply adding together all the waves scattered by the
individual atoms
This equation may be written more compactly as
F is, in general, a complex number, and it expresses both the amplitude and phase
of the resultant wave. Its absolute value |F| gives the amplitude of the resultant
wave in terms of the amplitude of the wave scattered by a single electron.
The intensity of the beam diffracted by all the atoms of the unit cell in a direction
predicted by the Bragg law is proportional simply to |F|2 , the square of the
amplitude of the resultant beam
Structure factor F equation is a very important relation in x-ray
crystallography, since it permits a calculation of the intensity of any hkl
reflection from a knowledge of the atomic positions.
Structure-factor calculations
This expression may be evaluated without multiplication by the complex conjugate,
since (h + k) is always integral, and the expression for F is thus real and not
complex. If h and k are both even or both odd, :i.e. "unmixed," then their sum is
always even and ei(h+k) has the value 1.
