A solution and solid state study of niobium complexes University of ...

Chapter 3
the diffraction pattern of the crystal. The dhkl values can be obtained when the Bragg
equation is applied to the diffraction data. 11
3.5.2 Structure factor
The structure factor, F(hkl), of any diffracted X-ray (hkl) is the quantity that expresses
both the amplitude and the phase of that reflection. The intensity of a diffracted beam
is directly related to the amplitude of the structure factor, but the phase must
normally be calculated by indirect means. In structure determination, phases are
estimated and an initial description of the positions and anisotropic displacements of
the scattering atoms is deduced. From this initial model, structure factors are
calculated and compared with those experimentally observed. Iterative refinement
procedures attempt to minimize the difference between calculation and experiment,
until a satisfactory fit has been obtained. The total scattering of all atoms in the unit
cell is given as: 12
Fhkl = ∑ exp[i2π(hxj + kyj + lzj) (3.7)
where fj = scattering factor of each of the N atoms. This relation may be recast in
terms of its amplitude, |F(hkl)|, and its phase angle, φ(hkl) or in terms of its real, A,
and imaginary, B components in the following expressions: 13
Fhkl = |Fhkl | exp [i2π φ(hkl)] (3.8)
Fhkl = A + iB (3.9)
The set of structure factors for all the reflections (hkl) are the primary quantities
necessary for the derivation of the three dimensional distribution of electron density.
When considering electron density the structure factor can be written as:
Fhkl = (x,y,z) exp [i2π(hxn + kyn + lzn)]dV (3.10)
11
R. J. D. Tilley, Crystals and Crystal Structures, Wiley and Sons, New York, 114, 2006.
12
M. F. C. Ladd, R. A. Palmer, Structure Determination by X-ray Crystallography, Plenum Press, New York, 36,
1977.
13
Y. Waseda, E. Matsubara, K. Shinoda, X-ray Diffraction Crystallography, Springer, Heidelberg, 109, 2011.
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