Density Function and Structure Factor for Crystals

In crystals this function has three-dimensional periodicity. For simplicity, here we only consider the one-dimensional projection of the three-dimensional crystal. In this case, the density function with period a is very simple 

As shown in Fig. 5.17a it consists of density maxima with a constant amplitude. The width of the peaks is governed by the thermal fluctuations of atoms, A  

According to (5.27) the amplitude of scattering F(q) for our one-dimensional crystal is given by Fourier transform of density function p(x). Since we have only the sum of cosine functions there are only discrete harmonics at wavevectors q = mqo = Inmla. The structure factor (5.25) is proportional to scattered light intensity F(q)F (q) and also consists of harmonics represented by 8-functions situated at the same wavevector values q = Inmla and having amplitude 

The correspondent X-ray picture consists of a set of narrow discrete equidistant spots dX q = mqo along the direction of periodicity. The angular spectrum of the structure factor is shown schematically in Fig. 5.17b. The amplitudes of harmonics depend mostly on the shape of the density curve and determine a number and the height of the 8-type maxima. The peak amplitudes are weakly modulated by the molecular form factor (MFF) and additionally by thermal fluctuations through the factor of Debye-Waller (DWF) for the one-dimensional case 7 oc exp(— M q /3) [1]. Here is the mean square amplitude of the ther- 

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