- © 2000 Mineralogical Society of America
In my role as crystallographic editor for the American Mineralogist and The Canadian Mineralogist, I examine many papers that include crystal structure refinement data and discussions about this data. It is clear to me that understanding and working with the displacement parameters is a challenge to many researchers. The purpose of this chapter is to clearly define the meaning of these parameters and the mathematics needed to interpret them. Throughout this chapter, I will provide illustrative examples from the structure of quartz refined by Kihara (1990) at a variety of temperatures.
It is well established that an atom in a crystal vibrates about its equilibrium position. This vibration can be attributed to thermal and zero-point energy. For example, in Figure 1 the anisotropic displacement parameters are drawn for the Si and O atoms in quartz at 298 K and 838 K. The displacements are significant, with maximum amplitudes for O of 0.138 Å, and 0.259 Å, at 298 K and 838 K, respectively, in the direction perpendicular to the Si-Si vector. Therefore, in a crystal structure refinement, it is imperative not only to find the mean position of an atom, but also to describe the region in space where there is a high likelihood of finding it. This region of space can be mathematically defined with a probability distribution function (p.d.f.). If we assume harmonic restoring forces between the atoms, or in other words, that the forces between the atoms are quadratic and obey Hooke’s Law, then it can be shown that the p.d.f. can be represented by a Gaussian function (Willis …