Journal Articles



X-ray, Neutron, and Electron Diffraction Simulation in MOE


A. Lin
Chemical Computing Group Inc.


Introduction

MOE's diffraction simulation application generates diffraction patterns for powder, single crystal, and fiber samples using three kinds of incident radiation under various camera settings. This article describes the basic principles of diffraction, and the diffraction simulation functionality provided in MOE.

Diffraction pattern analysis plays an important role in such divers applications as solving molecular structures, identifying compounds, and the fabrication of materials. If a compound can be made to crystallize into sizeable crystals, diffraction patterns from single crystals can provide a good deal of information about the atomic structure of the compound. Many compounds, however, can only be obtained as powders. Although a powder diffraction pattern yields much less information than that generated by a single crystal, it is unique to each substance, and is therefore highly useful for purposes of identification. For materials exhibiting orientation, i.e., fibers and plates, diffraction patterns yield important information about anisotropies, and are important, for example, for controlling fabrication of drawn wires or pressed sheets.

A diffraction pattern is the 2-D picture obtained by shining short-wavelength radiation through a material. The incident radiation is scattered coherently (Thomson scattering) by the atoms making up the material, and the resultant scattered radiation generates a pattern of interference that is dependent upon the relative positioning of the atoms.

The first radiation ever used in crystal diffraction was white (broadband) X-ray radiation, in the context of studies of X-rays themselves, when it was unknown whether X-rays were particles or waves. If X-rays could be diffracted in the manner of light through an optical grating, it would be conclusive proof of their wave nature. At the same time as these first studies of X-rays were being conducted, early theories of crystal structure were being proposed in which crystals were postulated to be composed of regular sub-units. These theories led von Laue, in 1912, to suggest that a crystal could provide the "grating" needed for the X-ray experiment. Soon thereafter, the first X-ray diffraction photos were produced.

Basic Principles

Diffraction as Plane Reflections

The diffraction pattern generated from a single crystal is an array of sharp spots. To explain these spots, W. L. Bragg suggested that the mechanism of X-ray diffraction could be cast in terms of reflections from regular, parallel arrays of planes within a crystal. Only under special conditions would the reflected radiation interfere constructively and a diffracted beam be observed. The required condition is that the angle of incidence theta of the incident beam obey the following relationship:
sin theta = (n * lambd../2) * (1/d)

where lambda is the wavelength of incident radiation, d is the inter-planar spacing, and n is an integer constant. This relationship is known as Bragg's law, and theta is known as the Bragg angle. Since the angle of reflection off a plane surface is equal to the angle of incidence, the Bragg angle also defines the scattering angle: the angle between the incident and the scattered radiation is 2*theta. The Bragg angle is used to calculate both the intensity of diffraction spots and correction terms in MOE for simulating physical effects that cause observed attenuation in intensity.

Unlike the single crystal, in powder samples, crystallites of all orientations occur. Instead of a single diffracted beam being produced, a family of beams is generated, all lying on a cone with semi-angle 2*theta. Fiber samples comprise crystallites oriented in a given direction, and so generate a picture that is somewhere in between that from single crystals and that from powders.

Reciprocal Lattice

The inter-planar spacing d (which determines the condition for diffraction) of a set of parallel crystallographic lattice planes is defined by the dimensions of the crystal unit cell (for more on crystal lattices and unit cells, please see The MOE Crystal Builder from the JCCG back issues). There is an infinite number of such families of parallel planes in any given crystalline lattice, and each is associated with a particular Bragg angle theta. Each set of planes is characterized by three indices hkl, and the resulting diffracted beam is termed the hkl reflection.

d can be calculated from the three lattice vectors specifying a 3-D crystal unit cell. The inverse relationship between theta and d has given rise to the notion of a reciprocal lattice (as opposed to the "direct" lattice). In this lattice, the spacing between planes d*=1/d and the three reciprocal lattice vectors are orthogonal to the direct ones.

The concept of the reciprocal lattice turns out to be more than a mere mathematical convenience: it can be shown that a diffraction pattern is in fact a projection of the weighted 3-D reciprocal lattice into two, or in the case of powders, one dimension. Since a straight projection results in overlapped diffraction spots/peaks and an attendant loss of information, moving-film techniques have been developed to eliminate spot coincidence in 2-D diffraction patterns.

The weights on the projected reciprocal lattice points (i.e., the intensity at each point) constituting a diffraction pattern can be calculated as the sum of amplitudes of scattered radiation from every atom in a crystal unit cell. This sum is called the structure factor. The intensity of a given hkl reflection is computed as the square of the structure factor.

In MOE, both individual reflections and families of reflections can be isolated from all diffraction plots. The calculated structure factors and associated intensities can be saved to a file.

Recording Diffraction Patterns

Various instruments exist for recording diffraction patterns, and MOE provides a range of choices of camera. Powder samples are commonly recorded using either a strip of film within a cylindrical camera setting, or a diffractometer that counts the number of X-ray photons. In MOE, powder plots are presented as normalized counts.

2-D diffraction cameras come in many guises. MOE provides flat plate, cylindrical film, precession, and Weissenberg options. Precession and Weissenberg imaging techniques are more complicated as they require a moving photographic film cartridge. They offer, however, a means for photographing diffraction patterns without overlapped spots. The Buerger precession method is particularly valuable as it provides an undistorted projection of the reciprocal lattice. Its chief disadvantage is that it records less of reciprocal space than does, for instance, the Weissenberg camera.

Types of Radiation Used

Three types of radiation are commonly used for generating diffraction images: X-ray, electron, and neutron. All three are available in MOE. The most common is X-ray. The same diffraction principles apply equally to all three cases, although the mechanism of scattering differs. X-rays are scattered by the electron density cloud surrounding an atomic nucleus, electrons by the positive potential of the nucleus, and neutrons by the nucleus itself. Clearly, scattering is a function of atomic number, and the dependency is expressed in calculable atomic, electron, and neutron scattering factors, which characterize the scattering "power" of a given atomic element.

X-ray wavelengths are on the order of the interatomic distances in crystals. A beam of electrons has a much shorter wavelength, thereby providing a commensurately higher level of resolution. Electron radiation, however, has significantly less penetrating power than X-rays, and therefore requires very thin specimens. Electrons also interact with matter to a much greater extent, and therefore quantitative measurements from the images obtained may be unreliable. Neutron diffraction is similar to that of X-rays, and has a wavelength on the order of 1 Ångstrom. Neutron diffraction is superior to X-ray diffraction in two cases: where the location of light elements (e.g., hydrogen atoms) is sought, and where magnetic effects are of interest.

Correction Factors

Although, in theory, a diffraction pattern is a projection of the weighted crystal reciprocal lattice, in practice, the observed pattern is more complex, being affected by many factors including experimental setup, thermal vibration of the atoms in the crystal, internal reflections which cause interference, and so forth.

Several kinds of corrections can be modeled with readily calculable expressions and are provided as options in the MOE diffraction simulation:

  • The Lorentz geometric correction factor compensates for the geometry of the diffraction process.
  • Thermal vibrations of atoms inside the crystal cause atom scattering factors to fall off faster than in the stationary model; temperature correction factors exist to compensate for this effect.
  • The polarization of incident X-rays (particularly in the case of Synchrotron-generated rays) can be modeled.
  • Use of a crystal monochromator with X-ray radiation introduces additional polarization which can be compensated for.
  • Anomalous scattering effects are observed when the frequency of incident X-ray radiation is close to the natural absorption frequency of the atoms in the crystal, and can be modeled using a complex correction to the scattering factor.

Summary of Diffraction Simulation in MOE

MOE's diffraction simulation application provides functionality for generating powder, single crystal, and fiber diffraction plots. The three radiation sources --- X-ray, electron, and neutron --- are provided, with X-ray radiation being specified according to anode type, and with an additional optional crystal monochromator. Various correction factors --- Lorentz, polarization, anomalous scattering, temperature factor --- are available, and can be enabled or disabled.

For display purposes, five powder peak shapes are available, and four 2-D camera types. Individual reflections can be isolated, and plotting parameters can be edited.

References

Andrew, K. W., Dyson, D. J., and S. R. Keown, Interpretation of Electron Diffraction Patterns, 2nd ed., 1971, Plenum Press, NY.

International Tables for X-Ray Crystallography, Vol.I, Norman F.M. Henry and Kathleen Lonsdale, Ed., 1952, The Kynoch Press, Birmingham, England.

International Tables for X-Ray Crystallography, Vol.II, John S. Kasper and Kathleen Lonsdale, Ed., 1985, D. Reidel Publishing Company, Dordrecht, Holland.

International Tables for X-Ray Crystallography, Vol.III, Caroline H. MacGillavry and Gerard D. Rieck, Ed., 1962, The Kynoch Press, Birmingham, England.

Jeffery, J.W. Methods in X-Ray Crystallography, 1971, Academic Press Inc. (London) Ltd.

Ladd, M.F.C. and R.A. Palmer. Structure Determination by X-Ray Crystallography, 2nd ed., 1985, Plenum Press, New York.

Lipson, H.S. Crystals and X-Rays, 1970, Springer-Verlag New York Inc., New York.

Stout, George H. and Lyle H. Jensen. X-Ray Structure Determination, 2nd ed., 1989, John Wiley & Sons, Inc., New York.

Woolfson, Michael M. An Introduction to X-Ray Crystallography, 2nd ed., 1997, Cambridge University Press, U.K.