A single crystal consists of a basic structural unit, a parallelepiped-shaped volume known as the unit cell, which undergoes transitional repetition in three dimensions to generate the entire crystal. The contents of the cell may be as small as a single atom or as large as a biological molecule. A family of parallel and equidistant planes in the crystal lattice is described by its Miller indices (hkl). If a, b, c are the vectors along the sides of the unit cell, the planes with Miller indices (hkl) make intercepts on these axes which are in the ratio a/h:b/k:c/l.
The reciprocal lattice is derived from the crystal lattice. Its translation vectors are defined by:
where Vc = a x b x c is the volume of the unit cell in direct (or real) space. To note are three important properties of the reciprocal space:
- The vector G = ha* + kb* + lc* in the reciprocal lattice is normal to the family of planes in the direct lattice with Miller indices (hkl).
- The magnitude of G is 2Π/dhkl, where dhkl is the interplanar spacing of the (hkl) planes.
- The value of the scalar product of a vector of the crystal lattice with a vector of the reciprocal lattice is 2Πn.
In the figure on the right, a plane incident wave of wavelength λ travelling in the direction specified by the unit vector Si is scattered by the particles located at two points, A and B. A detector is placed in the direction specified by the unit vector Sf.
Designating the position of the second scatterer relative to the first as r, we have CB = Si · r and AD =Sf · r .
The path difference is then:
q is called the scattering vector and completely characterizes the scattering geometry:
the incident and scattered beam directions and the wavelength. It is defined as q = kf- ki , with ki(f) = (2Π/λ)Si(f).
The scattering vector q is also sometimes referred to as the momentum transfer vector since ħq represents the change in the momentum by the incident neutron upon scattering.
The Bragg's law for diffraction tells that for having diffraction in one direction, the path difference must be an integer of the wave length. The condition to have diffraction thus reads:
This last equation means that the diffraction condition can be simply expressed by the scattering vector q must be a vector of the reciprocal lattice.
The diffraction pattern will therefore give direct information on the reciprocal lattice and in turn on the crystal lattice. Single-crystal neutron diffraction is a powerful tool:
- In chemistry to study molecular structures, the nature of the hydrogen bond and the structures of metal hydrides;
- In biology for the crystallography of proteins and the diffraction from fibres and membranes;
- In physics for the investigation of magnetic materials – using the interaction of the atomic magnetic moment of the material under study withthe magnetic moment of the neutron (performing then magnetic neutron diffraction).
Much crystallographic information is lost by condensing the three dimensions of reciprocal space, which are explored in single-crystal diffraction, into the single dimension of the powder pattern. However, for some materials it may not be possible to prepare a suitable single-crystal (for examples zeolites, fast-ion conductors, high-Tc superconductors…). In spite of its inherent limitations, powder diffraction can give not only the details atomic arrangement, but also the texture and reactions and processes.
R.-J. Roe, Methods of X-Ray and Neutron Scattering in Polymer Science, Oxford University Press, 2000.
B.T.M. Willis and C.J. Carlile, Experimental Neutron Scattering, Oxford University Press, 2009.