Vibration effects in crystals

FIGURE 13.3. (a) The probability density function as a function of atomic displacement, and (b) its Fourier transform, the displacement parameter (temperature factor) measured in terms of sin /A. 

The falloff in the intensity of Bragg reflections as a function of temperature, It, may be expressed, in the simplest case (that of cubic crystals composed of only one type of atom) by the exponential function 

As a result, Equation 13.1 may be rewritten for a scattering factor /t at temperature T (without the factor of 2, because the intensity is related to / ) as 

This is the form used for calculating structure factors. One of the terms in Equation 13.4, B or (7, is generally used as the displacement parameter by crystallographers, and some form of this parameter is refined in the least-squares procedure. 

Dunitz wrote of these equations Debye s paper, published only a few months after the discovery of X-ray diffraction by crystals, is remarkable for the physical intuition it showed at a time when almost nothing was known about the structure of solids at the atomic level. Ewald described how The temperature displacements of the atoms in a lattice are of the order of magnitude of the atomic distances The result is a factor of exponential form whose exponent contains besides the temperature the order of interference only [h,k,l, hence sin 9/M]. The importance of Debye s work, as stressed by Ewald,was in paving the way for the first immediate experimental proof of the existence of zero-point energy, and therewith of the quantum statistical foundation of Planck s theory of black-body radiation.  

---