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 (DebyeScherrer) 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 xrays 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.
 
K  2k sin  
2
The x-ray intensity cones are recorded on film as shown below: