Experimentally, the Bragg law can be applied in two different ways:

By using x-rays of known  and measuring the angle  we can
determine the spacing d of various planes in a crystal: this is called
structure analysis (X-ray diffraction or XRD)

Alternatively, we can use a crystal with planes of known spacing d,
measure  and thus determine the wavelength  of the radiation:
this is called x-ray spectroscopy (x-ray fluorescence) (XRF)
• The instrument for studying materials by measurements of the way in
which they diffract x-rays of known wavelength  is called the
diffractometer.
• The x-ray diffractometer is the most important tool for performing
diffraction analysis of materials.
• In an x-ray diffractometer, the film of a powder camera (DebyeScherrer) is replaced by a movable counter.
• All diffractometers have the following components:
1-2

X-ray source

X-ray beam conditioning devices

Sample and detector rotation

Radiation detector and associated electronics

Control and data acquisition system.
• The components are arranged about a circle (diffractometer circle)
which lies in a plane called the diffractometer plane.
• Both the x-ray source and the detector lie on the circumference of the
circle.
• The angle between the plane of specimen and x-ray source is = , the
Bragg angle.
1-3
C
T
F
Figure 1: X-ray
Diffractometer (schematic)
1-4
Counter
Diverging
Slits
X-ray Tube
Sample
Receiving
Slits
2
Figure 1: X-ray Diffractometer (Siemens at UTM)
1-5
Figure 2
1-6
• The angle between the projection of x-ray source and the detector is =
2 .
• The x-ray source is fixed, and the detector moves through a range of
angles.
• The radius of the focusing circle is not constant but increases as the
angle 2 decreases. The 2 measurement is typically from 0o to about
170 o.
• The choice of range depends on the crystal structure of the material (if
known) and the time you want to spend obtaining the diffraction pattern
• For an unknown specimen a large range of angles is often used
because the positions of the reflections are known, at least not yet!
1-7
• The incident x-ray beam is defined by a set of Soller slits (divergence
slits), Figure 2
• The sample sits at the centre of the diffractometer.
• The Bragg beam from the sample is defined by a second set of slits
(receiving slits)
• The slits (divergence and receiving) consist of a series of closely
spaced parallel metal plates that define and collimate the incident x-ray
beam (make the beam parallel)
• The slits are typically about 30 mm long and 0.05 mm thick, and the
distance between them is about 0.5 mm.
• The slits are usually made of a metal with a high atomic number such
as Mo, or Ta (because of their high absorption capacities)
1-8
OPERATION OF A DIFFRACTOMETER
• From Figure 1, x-rays are produced at the target, T, of the x-ray tube.
• These x-rays are usually filtered to produce monochromatic radiation,
collimated (to produce a beam composed of perfectly parallel rays) and
then hit the specimen at C.
• The x-rays diffracted by the specimen are focused through a slit F onto
the counter.
• As the counter moves on the diffractometer circle through an angle 2,
the specimen rotates through an angle .
• The x-ray quanta are converted into electrical pulses by an x-ray
detector. The detector output is sent to a counter
1-9
• The counter counts the number of current pulses / unit time, and this
number is directly proportional to the intensity (energy) of the diffracted
x-ray beam.
• A typical diffraction pattern for aluminium produced in a diffractometer
is shown in Figure 3
• There is fundamental difference between the operation of a powder
camera and a diffractometer.
• In a powder camera, all diffraction lines are recorded simultaneously
and variations in the intensities of the incident x-ray beam during
analysis can have no effect of the relative line intensities in the
diffraction pattern.
1-10
Figure 3: X-ray diffraction pattern of Aluminium
1-11
• With a diffractometer, diffraction lines are recorded one after the other,
and thus, it is imperative to keep the incident beam constant when
relative intensities are measured.
EXAMINATION OF A STANDARD X-RAY DIFFRACTION PATTERN
• An example of a typical x-ray diffraction pattern of Aluminium is shown
in Figure 3
• The pattern consists of a series of peaks, which are also called
reflections. The peak intensity is plotted vs. measured diffraction angle,
2.
1-12
• Each peak corresponds to x-rays diffracted from a specific set of
planes in the specimen, and these peaks are of different heights
(intensities).
• The intensity is proportional to the number of x-rays of a particular
energy that have been counted by the detector for each angle 2.
• For diffraction analysis, we use the relative intensities of peaks
because measuring absolute intensity is very difficult.
• The position of the peaks in an x-ray diffraction pattern depends on the
crystal structure (shape and size of the unit cell) of the material.
• For low values of 2 each reflection appears as a single peak. For
higher values of 2 each reflection consists of a pair (two) peaks,
which correspond to diffraction of the K1 and K2 wavelengths.
1-13

Diffraction patterns from cubic materials
distinguished from those of non-cubic materials.
can
usually
be

Figure 5 shows the calculated diffraction patterns of the four cubic
crystal structures.
SC
110
BCC
200
FCC
111
Diamond
111
200
211
220
220
220
222
310
311
311
222
321
400
400
Figure 5: Comparison of x-ray diffraction patterns from cubic materials
1-14
Forbidden Reflections
h22 +l
+ k2 2 + l2
h2 +k
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
Primitive
PrimitiveCubic
cubic
100
110
111
200
210
211
220
221/300
310
311
222
320
321
400
FCC cubic
Face-centred
111
200
220
311
222
400
BCC cubic
Body-centred
110
200
211
220
310
222
321
400
1-15
INDEXING THE DIFFRACTION PATTERN
• Knowing the crystal structure of a material is central to understanding
the behaviour of materials under stress, alloy formation and phase
transformations.
• The size and shape of the unit cell determine the angular positions of
the diffraction peaks, and the arrangement of the atoms within the unit
cell determines the relative intensities of the peaks.
• It is therefore possible to calculate the size and shape of the cell from
the angular positions of the peaks and the atom positions in the unit
cell from the intensities of the diffraction peaks.
1-16
• Indexing the pattern involves assigning the correct Miller indices to
each peak in the diffraction pattern
• It is important to remember that correct indexing is done only when all
the peaks in the diffraction pattern are accounted for and no peaks
expected for the structure are missing from the pattern
• An example of indexing a pattern from a material with a cubic structure
is presented here.
• Two methods can be used to index a diffraction a pattern
1-17
METHOD 1:

Diffraction will occur when Bragg law is satisfied:
  2d sin 

The interplanar spacing d for a cubic material is given by:
a
d hkl 

h2  k 2  l 2
Combining the above equations results in:
d2 
a
h k l
2
2
2

2
4 sin 2 
1-18

Which gives:
Sin 2 

h2  k 2  l 2 
4a 2
2

Since 2 / 4a2 is constant, sin 2 is proportional to (h2 + k2 + l2),

As  increases, planes with higher Miller indices will diffract.

Writing the above equation for two different planes and diving by the
minimum plane, we get:
sin 2 1
sin  2
2

h12  k12  l12
h22  k 22  l 22
1-19
Example: indexing of Aluminium diffraction pattern by method 1
1-20
Example: indexing of Aluminium diffraction pattern by method 1
1. Identify the peaks
2. Determine sin2
3. Calculate the ratio sin2 / sin2min and multiply by the
appropriate integers (1, 2, or 3)
4. Select the result from step (3) that yields h2 + k2 + l2 as an
integer
5. Compare results with the sequences of h2 + k2 + l2 values to
identify the Bravais lattice
6. Calculate lattice parameter.
1-21
1-22
1-23
1-24
• The bravais lattice can be identified by noting the systematic presence
(or absence) of reflections in the diffraction pattern.
• The Table below illustrates the selection rules for cubic lattices.
• According to these rules, the (h2 + k2 + l2) values for the different cubic
lattices follow the sequence:
Simple cubic :
1,2,3,4,5,6,8,9,10,11,12,13,14,16,….
BCC
:
2,4,6,8,10,12,14,16,18,...
FCC
:
3,4,8,11,12,16,19,20,24,27,32,…
1-25
Bravais lattice
Reflections present for
Reflections absent for
Primitive (simple cubic)
All
None
Body Centered Cubic
(BCC)
h + k + l = even
h + k + l = odd
Face Centered Cubic
(FCC)
h, k, l = unmixed (all
even or all odd)
h, k, l = mixed
1-26
CALCULATION OF THE LATTICE PARAMETER

The lattice parameter,a, can be calculated from:
Sin 2 


h2  k 2  l 2 
4a 2
2
Rearranging gives

 2  2
 h  k2  l2
a  
2 
 4 sin  
2

1-27
METHOD 2:

This method can be used to index the diffraction pattern from materials with
a cubic structure. From:
Sin 2 


h2  k 2  l 2 
4a 2
2
Since 2 / 4a2 is constant for any pattern and which we will call A, we can
write:

sin 2   A h 2  k 2  l 2

1-28

In a cubic system, the possible (h2 + k2 + l2) values are: 1, 2, 3, 4, 5,
6, 8, …. (even though all may not be present in every type of cubic
lattice).

The observed sin2 values for all peaks in the pattern are therefore
divided by the integers 1, 2, 3, 4, 5, 6, 8, to obtain a common quotient,
which is the value of A, corresponding to (h2 + k2 + l2) =1.

We can then calculate the lattice parameter from the value of A using
the relationship:
A
2
4a
2
or
a

2 A
1-29
Note that 0.1448 is also common in 1, 2, 3, 4, 5, 6, BUT absent in 8
It can only be FCC
1-30