MCGR is a program for determining total or partial radial distribution functions from one or more total structure factors measured by neutron or X-ray diffraction, by an inverse method. MCGR use a 1-dimensional version of RMC to produce a radial distribution function g(r). The corresponding structure factor A(Q) for g(r) is then compared to the measured structure factor and a decision to accept or reject g(r) is made.
The inverse method have considerabe advantages over the conventional direct methods; for instance they avoid truncation errors and allow the estimation of errors in the radial distribution function. To download click here.
Conventionally partial radial distribution functions ga b (r) are determined from the partial structure factors Aa b (r) by a Fourier transform
However if Aa b (r) is truncated or contains statistical errors then spurious oscillations are introduced into ga b (r). In addition any other errors in Aa b are redistributed in an unknown fashion in ga b . Such effects can be particularly problematic when the partial structure factors are obtained by direct separation from a set of total structure factors obtained by either neutron diffraction with isotopic substitution or X-ray diffraction with anomalous scattering. In many such cases the separation matrix is ill conditioned and errors in the total structure factor are considerably magnified in the partial structure factors.
An alternative approach is to 'generate' possible ga b (r) by some method, and then to modify these to fit the data, that is the total structure factor(s). Since ga b (r) can be generated over, as wide an r range as required there will be no truncation of the Fourier transform. In addition different sets of ga b (r) that fit the data can be generated, and from these an average and a standard deviation can be calculated, thus giving some idea of the errors.
In MCGR the ga b (r) are generates by a Monte Carlo method. The sets of ga b (r) are defined as histograms of nr points with spacing dr. The basic algorithm is described by the MCGR flowchart, see below.
In addition to the measured structure factors, some constraints can be applied to the modelling of g(r). There is always a minimum distance below which g(r) has to be zero (closest approach between atoms), MCGR can be run so that no g(r)¹ 0 below this limit will be allowed. If the coordination number of some peak in g(r) is known it can be used as a constraint. It is also possible to constrain g(r) to be positive within certain r limits (or for all r). It is also possible to use a polynomial background when fitting F(Q).
To download MCGR or other RMC related programs click here.