X-Ray Diffraction and Reciprocal Lattice
Up until this point, the structure of crystals have been studied and classified
because of its extreme importance to almost every mechanical and electrical property.
However, the method of physically obtaining information about the structure of crystals
has yet to be explained. This will now be done by looking at x-ray diffraction and
reciprocal lattice concepts.
The approach that will be taken in describing these concepts is to look at aspects
of x-ray diffraction first starting with the Bragg formulation and then progressing to the
more sophisticated von Laue approach, which in turn provides motivation for the
mathematical construct called the reciprocal lattice. This approach starts with the easiest
but most incomplete picture of x-ray diffraction and concludes with a very powerful
experimental and mathematical method for crystal structural analysis.
Bragg Formulation of X-Ray Diffraction
In 1913 W. H. and W. L. Bragg found that crystalline materials gave remarkably
characteristic patterns of intense reflected x-ray intensity peaks. They concluded that the
phenomenon could be accounted for if the crystal structure could be viewed as successive
planes off atoms obey two assumptions:
1. The x-rays should be specularly reflected by the ions in any one plane.
2. The reflected rays from successive planes should interfere constructively.
This simple concept is best explained by the following figure.
It is easily seen from the figure that the condition for constructive interference is:
n  2d sin  
where n is an integer and is called the diffraction order,  is the wavelength of the x-rays,
d is the spacing between the planes of atoms and  is the angle of incidence relative to the
normal of the plane of atoms. Not too much more will be said about the Bragg
formulation but we will move on to the more sophisticated and illuminating von Laue
method.
The von Laue Formulation
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.
 
 
k  
k
The path difference between the two scattered waves is       R . If this path
k
k 


difference is 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:
A e

R

R

  
i k  k  R
where R is any translation vector, and 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 tend 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 set of k  k  values. This set of 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.


The same analysis holds for the ny and nz summations resulting in peaks in the intensity
 

of the scattered x-rays only when K which was defined as k  k  is equal to a set of

values. For the S.C. lattice, K can be written as:



2
2
2
K  nx
xˆ  n y
yˆ  nz
zˆ
a
a
a

where nx, ny, nz are any integers. These values of K which are called reciprocal lattice
vectors are analogous to the translation vectors R defining the original structure and
hence can be viewed as defining a lattice, the Reciprocal Lattice!
The Structure Factor
In general, the basis at each lattice point 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 at each lattice point as follows:
where the summation over the index j is the summation over the atoms in the basis at
each lattice point with a displacement vector dj.
What the structure factor does is modify the x-ray intensity for certain reciprocal
lattice vectors. In fact, it can lead to some x-ray diffraction peaks disappearing where for
a monatomic basis the peak existed. As an example, the x-ray diffraction analysis of the
diamond structure will be assigned for homework and will illustrate this effect.