X-Ray Methods

 

There are three different experimental methods for x-ray diffraction that we are going to look at: The Laue method, the rotating crystal method and the powder (Debye-Scherrer) method. To help explain each method, the illuminating and very helpful Ewald construction will be explained.

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 which has the reciprocal lattice points. The incoming x-ray wave vector is drawn such that it starts at the origin of the reciprocal space and is of course is 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 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 Laue Method

 

The Laue method eases the restriction of illustrated by the figure above by using a range of wavelengths for the incoming x-rays. Hence the ewald construction is modified as follows.

 

 

Where x-rays of wave vector ko through k1 are incident on the crystal at the same angle of incidence. The figure shows the small circle of radius ko and the large circle of radius k1. All of the reciprocal lattice points in the shaded section will satisfy the diffraction condition and you will have many diffracted x-rays. In the figure above, there will be 12 diffracted waves.

 

Rotating Crystal and Debye-Scherrer Method

 

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.

 

For real 3D crystals, there will be diffaction cones 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: