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From Intensities to Structural Parameters

Given a structural model such that F(hkl) can be calculated, the intensity of any peak, I, in the powder diffraction pattern can be calculated (see earlier) using the equation:

I(calc) = c jhkl L(2θ) P(2θ) A(2θ) F2(hkl)
where L, P, A are the Lorentz, polarisation, and absorption corrections, respectively. j is the multiplicty factor that depends on the reciprocal-space symmetry of the material (see earlier), and c is a scale factor that takes into account data acquisition time, etc.

To refine the structure from the powder diffraction data, one possibility is to simply minimize the least-squares quantity:

Δ = N
Σ
n=1		 { In(obs) - In(calc)}2
I(obs) is obtained by a simple summation of the individual profile points, y, from which background counts, estimated from the counts on either side of the peak, are subtracted. Δ can be minimized as a function of the crystallographic parameters using standard non-linear full-matrix least-squares techniques (as discussed later).

However, all powder patterns exhibit peak overlap, even cubic patterns where reflections such as 333 and 511 have the same d spacing due to symmetry. For most symmetries, the overlap is random and depends on the values of the lattice constants a, b, c, α, β, and γ. So in practice, it is only possible to extract the integrated intensities of groups of reflections.

The equation for Δ is therefore modified to:

Δ = N
Σ
n=1		 { In(obs) - M
Σ
m=1		 Im(calc)}2
where M is the total number of individual hkl reflections that contribute to the nth total integrated intensity In(obs) that is extracted from the powder diffraction data.

The limitations of this method are obvious: The total number of measured reflections is some product of M and N, but the total number of independent "observations" is only N. This effectively reduces the maximum number of refinable crystallographic parameters to, say, N/3. Owing to the relatively poor resolution of early powder diffractometer data (which resulted in a small value for N anway), the use of powder data for structural work was severly limited until the pioneering work by Rietveld in the late 1960's.

As an example of the difficulties resulting from this approach, the incorrect structure of hydrogen sulphide was published (fraudulently in my opinion) in Nature in 1969 using a mere 18 integrated peak intensities: the actual 2θ range contained 184 unique hkl reflections! A much later study in 1990, solved and refined the structure correctly using 35 parameters and 211 unique hkl reflections. Clearly this would not have been possible using only the 18 integrated peak intensity totals used in the earlier work. So how do modern powder methods solve this problem? The answer, as devised by Rietveld after whom the method is named, is discussed in the next few pages.

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© Copyright 1997-2006.  Birkbeck College, University of London. Author(s): Jeremy Karl Cockcroft