Key Notes
Diffraction takes place when a wave interacts with a lattice whose dimensions are of the same order of magnitude as that of the wavelength of the wave. At these dimensions, the lattice scatters the radiation, so as to either enhance the amplitude of the radiation through constructive interference, or to reduce it through destructive interference. The pattern of constructive and destructive interference yields information about molecular and crystal structure. The most commonly used radiation is X-rays, which are most strongly scattered by heavy elements. High velocity electrons behave as waves, and are also scattered by the electron clouds. Neutrons slowed to thermal velocities also behave as waves, but are scattered by atomic nuclei.
In crystallographic studies, the different lattice planes which are present in a crystal are viewed as planes from which the incident radiation can be reflected. Constructive interference of the reflected radiation occurs if the Bragg condition is met: nλ=2d sinθ. For most studies, the wavelength of the radiation is fixed, and the angle θ is varied, allowing the distance between the planes, d, to be calculated from the angle at which reflections are observed.
For a crystalline solid, the distance between the lattice planes is easily obtained from the Miller indices, and the unit cell dimensions. The relationship between these parameters can be used to modify the Bragg condition. In the simple case of a primitive cubic unit cell, the allowed values for θ as a function of h, k, and l are given by: sinθ= (h2+k2+l2)1/2 λ/2a. Some whole numbers (7, 15, 23, for example) cannot be formed from the sum of three squared numbers, and the reflections corresponding to these values of (h2+k2+l2) are missing from the series. In other unit cells, missing lines occur as a result of the symmetry of the unit cell. Simple geometric arguments show that for a body centered cubic unit cell, h+k+l must be even, and that for a face centered cubic unit cell, h, k and l must be all even or all odd for reflections to be allowed. The forbidden lines are known as systematic absences.
In the powder diffraction method, the crystalline sample is ground into a powder, so that it contains crystals which are oriented at every possible angle to the incident beam. In this way, the Bragg condition for every lattice plane is simultaneously
fulfilled, and reflections are seen at all possible values of θ.
Modern diffractometers use scintillation detectors which sweep an arc of angle 2θ around the sample, giving a measure of X-ray intensity as a function of the angle 2θ. The diffraction pattern which is obtained must be correlated with the unit cell of the sample. By obtaining the angles for which reflections occur, the ratios of the values of sin2 θ may be directly correlated to the values of h, k, and l in a process known as indexing.
Related topics Crystalline solids (A5)
Radiation for diffraction by solids
Diffraction takes place when a wave interacts with a lattice whose dimensions are of the same order of magnitude as that of the wavelength of the wave. The lattice scatters the radiation, and the scattered radiation from one point interferes with the radiation from others so as to either enhance the amplitude of the radiation (constructive interference), or to reduce it (destructive interference) (Fig. 1). The pattern of constructive and destructive interference yields information about molecular and crystal structure.
Diffraction by solids 29
Fig. 1. Constructive (a) and destructive (b) interference of two waves.
In the case of solids, this wavelength must be of the same order as the crystal lattice spacing (ca. 0.1nm), and there are three primary types of radiation which are used for structural studies of solids. The most commonly used radiation, X-rays, have wavelengths of the order of 0.15 nm, and in the course of diffraction studies are scattered by the electron density of the molecule. The heavier elements therefore have the strongest scattering power, and are most easily observed. Similarly, electrons which have been accelerated to high velocity may have wavelengths of the order of 0.02 nm, and are also scattered by the electron clouds. Fission-generated neutrons which have been slowed to velocities of the order of 1000 m s−1 also behave as waves, but are scattered by atomic nuclei. The relationship between scattering power and atomic mass is complex for neutrons. Whilst some light nuclei such as deuterium scatter neutrons strongly, some heavier nuclei, such as vanadium, are almost transparent.
The subjects covered in this topic are indifferent to the nature of the radiation used, and the arguments may be applied to all types of diffraction study.
Bragg equation
In crystallographic studies, the different lattice planes which are present in a crystal are viewed as planes from which the incident radiation can be reflected. Diffraction of the radiation arises from the phase difference between these reflections. For any two parallel planes, several conditions exist for which constructive interference can occur. If the radiation is incident at an angle, θ, to the planes, then the waves reflected from the lower plane travel a distance equal to 2d sinθ further than those reflected from the upper plane where d is the separation of the planes. If this difference is equal to a whole number of wave-lengths, nλ, then constructive interference will occur (Fig. 1). In this case, the Bragg condition for diffraction is met:
n λ=2d sinθ
In all other cases, a phase difference exists between the two beams and they interfere destructively, to varying degrees. The result is that only those reflections which meet the Bragg condition will be observed. In practice, n may be set equal to 1, as higher order reflections merely correspond to first order reflections from other parallel planes which are present in the crystal.
For most studies, the wavelength of the radiation is fixed, and the angle θ is varied, allowing d to be calculated from the angle at which reflections are observed (Fig. 2).
Fig. 2. Diffraction due to reflections from a pair of planes. The difference in path length between reflected beams a and b is equal to 2d sinθ. If this is equal to a whole number of wavelengths, nλ, then constructive interference occurs.
Diffraction by solids 31
Reflections
For a crystalline solid, the distance between the lattice planes is easily obtained from the Miller indices, and the unit cell dimensions. The simplest example is that of a primitive cubic unit cell, for which the distance between planes, d, is simply given by:
d2=a2/(h2+k2+l2)
where h, k, and l are the Miller indices and a is the length of the unit cell edge.
Substitution of this relationship into the Bragg condition yields the possible values for θ:
sinθ=(h2+k2+l2)1/2 λ/2a
Because h, k, and l are whole numbers, the sum (h2+k2+l2) also yields whole numbers, and because λ and a are fixed quantities, sin2θ varies so as to give a regular spacing of reflections. However, some whole numbers (7, 15, 23, etc.) cannot be formed from the sum of three squared numbers, and the reflections corresponding to these values of (h2+k2+l2) are missing from the series. If λ/2a is denoted A, then the values of sinθ for a simple cubic lattice are given by: A/√1, A/√2, A/√3, A/√4, A/√5, A/√6, A/√8, A/√9, A/√10, A/√11, etc. It is therefore possible to identify a primitive cubic unit cell from both the regularity of the spacings in the X-ray diffraction pattern, and the absence of certain forbidden lines.
Other unit cells yield further types of missing lines, known as systematic absences.
Simple geometric arguments show that the following conditions apply to a cubic unit cell:
Allowed reflections
Primitive cubic unit cell all h+k+l
Body centered cubic unit cell h+k+l=even
Face centered cubic unit cell h+k+l=all even or all odd
Similar, but increasingly complex, rules apply to other unit cells and identification of the systematic absences allows the unit cell to be classified.
Powder crystallography
When a single crystal is illuminated with radiation, reflections are only observed when one of the lattice planes is at an angle which satisfies the Bragg condition. In the powder diffraction method, the crystalline sample is ground into a powder, so that it effectively contains crystals which are oriented at every possible angle to the incident beam. In this way, the Bragg condition for every lattice plane is simultaneously fulfilled, and reflections are seen at all allowable values of θ relative to the incident beam (Fig. 3).