2
2.1
X-Ray Diffraction and the Reciprocal Lattice
X-Ray Diffraction
After crystal structure and related topics are reviewed, a review of x-ray diffraction
and the reciprocal lattice is useful. The Bragg description of x-ray diffraction is a
conceptually easy and insightful view of x-ray diffraction but incomplete and limiting as
well. The Bragg description will be assigned as a homework problem and we will move
on to the more useful von Laue formulation of x-ray diffraction.
The von Laue approach differs substantially from the Bragg formulation since it does
not make the assumption of specular reflection and does not require the grouping of
atoms into planes to explain the x-ray diffraction patterns. The only assumption of the
von Laue approach is that the x-rays scattered from atoms can constructively interfere
with each other. The following figure describes this basic assumption by looking at only

two identical scattering objects at two lattice points separated by a translation vector R .


Where k and k  are the incident and scattered wavevectors respectively. The path
 
 
k  
k
difference between the two scattered waves is       R . If this path difference is
k
k 


equal to m where m is any integer, then there is constructive interference and a peak in
the x-ray intensity. After substituting in the magnitude of the wave vectors, this
condition can be written as:
k  k  R  2m
A crystal is composed of many atoms and the total diffracted x-ray amplitude can be
written as the sum of plane waves with each plane wave produced as a result of scattering

by a scattering element (i.e., basis) at a position described by a translation vector R :
 AR ei k k R
  
R

where AR is a scattering amplitude that is the same for all R and can be factored out of
the summation. It is seen that significant constructive interference occurs only if
  

k  k   R  2m for all translation vectors R . This can be seen in a couple of
different ways.




  
i k k  R
The first way is just to observe that the factor e
has an amplitude of 1 and can
vary from 1 to i to –1 to –i (i.e. a circle in the complex plane centered at the origin with

modulus 1) depending on R . For any large summation involving many translation
vectors, the summation will average out to be zero except for the very special case where
the phase factor of the exponential is always and integer multiple of 2. This is a very
 
special case and only occurs for a very restricted set of k  k  values. This set of
 
k  k  values is called the reciprocal lattice. The second way to show that the
constructive interference condition is to perform the summation for a cubic lattice of
 
lattice constant a. It is seen that only for a certain k  k  values will the summation be
nonzero.
One last but important aspect of x-ray diffraction concerns the composition of the
scattering element at each lattice point. The scattering element at each lattice point (i.e.,
basis) may be composed of many atoms either identical or different. This will affect the
x-ray diffraction since the x-rays will scatter off each of the atoms in the basis at each
lattice point. Hence the equation previously discussed that describing the x-ray intensity,
i.e., the summation of the phase factors at each lattice point, must be modified to include
a summation of the phase factors at each atom in the basis. This modification is included
in what is called the structure factor and will be assigned as a homework assignment.




2.2


Reciprocal Lattice

A reciprocal lattice of a crystal structure is a set of vectors denoted by K that satisfies
 
 
the constructive interference condition, i.e. K  R  2m . Hence, if k  k  is equal to

any vector in the set K , then the constructive interference will be satisfied and a peak in

the x-ray intensity will occur. It can be shown that the set of K obeying this requirement
can be written as:




K  s1b  s2 b2  s3b3


  
where s1 , s2 , s 3 are any integers and the reciprocal lattice basis vectors b1 , b2 , b3 can
be obtained by the following equations:
 

a 2  a3
b1  2   
a1  a2  a3 
 

a3  a1
b2  2   
a1  a2  a3 
 

a a
b3  2  1 2 
a1  a 2  a3 
  
where a1 , a 2 , a 3 are the primitive lattice vectors describing the structure of the crystal





(i.e., the translation vectors R are written as R  n1a1  n2 a2  n3 a3 ). This result is very
 
useful because you are able to calculate the acceptable values of k  k  directly from the
structure of the crystal! It can be shown that the most common crystal structures have the
reciprocal lattice structures indicated in table 2.1.

Table 2.1 Transformation properties of common crystal structures
Real Space
Real Space
Reciprocal Space
Lattice Structure Lattice
Lattice Structure
Constant
Simple Cubic
Simple Cubic
a
Body Centered
Cubic
Face Centered
Cubic
2.3
a
Face Centered
Cubic
Body Centered
Cubic
a

Reciprocal
Space Lattice
Constant
2
a
4
a
4
a
Putting the concepts together
The main reason that we have introduced the reciprocal lattice is that experimentally,
we will measure aspects of the reciprocal lattice and determine that a particular crystal in
question has a B.C.C. reciprocal lattice for example. Hence we know that the real space
lattice is an F.C.C. In particular, using x-ray diffraction, we will determine the set of
lengths of the translation vectors in reciprocal space relative to the shortest translation

 
vector K o in reciprocal space (i.e., K K o ). Because each reciprocal lattice whether it
be S.C., B.C.C., or F.C.C. or otherwise has a unique set of these ratios given in the table
below, it unambiguously allows us to determine the reciprocal lattice vector.
Ratio Table

Ko

K1

K2

K3

Ko

Ko

Ko

Ko
S.C. Reciprocal
Lattice
1
B.C.C. Reciprocal
Lattice
1
2
2
3
2 2
3
3
11
3
2
2
3
F.C.C. Reciprocal
Lattice
1
2

K4

K5

K6

K7

Ko

Ko

Ko

Ko
2
5
6
4
5
3
6
2 2
19
3
3
2 5
3
7
2 2
There are three different commonly used experimental methods for x-ray diffraction:
The Laue method, the rotating crystal method and the powder (Debye-Scherrer) method.
Each method uses an illuminating and helpful construct called the Ewald construction.
The Ewald construction is a simple geometric construction to aid in visualizing the
direction of allowable diffracted x-rays. It is set up by drawing the incoming wave vector
of the x-ray on the reciprocal space lattice. The incoming x-ray wave vector is drawn
such that it starts at the origin of the reciprocal space and is of course in the direction of
the incident x-ray. Then a circle with a radius equal to the magnitude of the incoming xray wave vector is drawn with its center at the endpoint of the incoming x-ray wave
vector as shown in the figure below. For an allowable diffracted wave to exist, the
diffracted wave vector must be equal in magnitude but in a direction such that
 

k  k   K , or the difference of incoming and outgoing wavevectors must equal a
reciprocal lattice vector. This is a very restrictive condition and only reciprocal lattice
points that lie on the edge of this circle will satisfy this condition. This is all shown in the
figure below in which only one scattered x-ray is satisfies this condition and is therefore
present.
The rotating crystal and the powder (Debye-Scherrer) method will be described here
and the Laue method is explained in other books. In the rotating crystal method, there is
only one wavelength for the incoming x-rays but the crystal is rotated and hence the
reciprocal lattice is rotated. This allows for more reciprocal lattice points to satisfy the
diffraction conditions. The extreme of this is when you rotate the crystal over all possible
angles such as you would have if you had a polycrystalline material where the grains are
oriented in all directions. For example, if you crushed the crystal into small pieces, the
incoming x-rays would pass through crystals that are oriented in all directions, hence the
reciprocal lattice is rotated in all directions as illustrated in the following figure.
By measuring the angle between the incoming and outgoing wavevectors, you can
calculate the magnitude of the reciprocal lattice vectors.

 
K  2k sin  
2
The x-ray intensity cones are recorded on film as shown below where the angle  is
determined by measuring the length U on the photographic film and using the camera
constant d and the equation d    U .