2 X-Ray Diffraction and the Reciprocal Lattice

2.1 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 .

Where and are the incident and scattered wavevectors respectively. The path difference between the two scattered waves is . If this path difference is equal to 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:

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 :

where A_R is a scattering amplitude that is the same for all and can be factored out of the summation. It is seen that significant constructive interference occurs only if for all translation vectors . This can be seen in a couple of different ways.

The first way is just to observe that the factor 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 . 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 2p. This is a very special case and only occurs for a very restricted set of 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. It is seen that only for a certain 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 that satisfies the constructive interference condition, i.e. . Hence, if is equal to any vector in the set , 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 obeying this requirement can be written as:

where , , are any integers and the reciprocal lattice basis vectors , , can be obtained by the following equations:

where , , are the primitive lattice vectors describing the structure of the crystal (i.e., the translation vectors are written as ). This result is very useful because you are able to calculate the acceptable values of 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 Lattice Structure	Real Space Lattice Constant		Reciprocal Space Lattice Structure	Reciprocal Space Lattice Constant
Simple Cubic			Simple Cubic
Body Centered Cubic			Face Centered Cubic
Face Centered Cubic			Body Centered Cubic

2.3 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 in reciprocal space (i.e., ). 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	S.C. Reciprocal Lattice	B.C.C. Reciprocal Lattice	F.C.C. Reciprocal Lattice
	1	1	1
	2		2
		2
	3

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 x-ray 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 , 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.

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 .