Program RFIT

This program calculates small-angle scattering (SANS or SAXS) for non-interacting particles. It covers a wide range of simple models and allows different scales and weighting of the fit to be included in the minimisation. The program also permits polydispersity and instrument resolution to be included in the calculation.

Apart from the models for isolated particles, the program can be used to fit a variety of other simple functions. In particular models with peaks in scattering either at momentum transfer (Q) equal to zero or at some finite Q can be fitted. Examples of these in practice would include a Lorentzian at Q=0 to model critical scattering, Gaussian peaks at finite Q to interpret Bragg diffraction at low Q and models of fractal behaviour such as ln(Intensity) varying linearly with ln(Q).

In some cases it may be important to include resolution in the model while for other data the effect may be insignificant. The use of the program interactively allows these effects to be rapidly visualised.

New Features. Some new features in the present version (Version 2.3 - May 1999) include (i) the ability to plot size distributions and obtain this data as an ASCII file and (ii) the calculation of g(r) as a real space distribution of density for the model. This last option is useful for comparison with other means of analysis such as indirect Fourier transforms.

The following models are presently available in the program:

• 1 - Debye model of Gaussian polymer
• 2 - Spherical particle
• 3 - Lorentzian Peak
• 4 - Gaussian Peak
• 5 - Cylinder
• 6 - Ellipsoid
• 7 - Product of 2 Debye functions
• 8 - Product of Debye and Sphere
• 9 - Fractal Log I = a Log Q + const.
• 10 - Two Gaussian Peaks

The following units are used for the various parameters used in the models:

• Radii /Angstrom. The radii are Rg for the Debye function and hard radius for the sphere model
• Polydispersity is radius or length (Angstrom)
• Ellipsoid - A/B If this is greater than 1 this corresponds to a prolate ellipsoid and if A/B is less than 1 the model is an oblate ellipsoid.
• I(q=0) is in arbitrary units
• Widths of Gaussian and Lorentzian peaks are given in Angstrom

A summary of this information can be obtained with the command I from the subsidiary command menu. See Section 3.3.

The resolution is calculated assuming that there are components of instrument broadening arising from both the wavelength spread and the finite angular resolution. The scattering is calculated by averaging the calculated I(Q) over a range of Q values appropriate to the width of the resolution function. The resolution is assumed to be a Gaussian function with a width that varies with Q. One component arises from the spread in wavelength and will be effectively constant with Q but the effect of angular divergence will vary with Q.

The resolution is defined by the following parameters:

Dimensions of diaphragms 1 and 2

This should be the diameter in mm of the source and sample diaphragms. If these are not round it should be adequate to take some other average dimension.

Collimation distance

This is the separation of the two apertures (diaphragms) that define the divergence of the incident beam. The distance is given in metres.

Sample-detector distance

This is the distance from the sample to the detector in metres. It defines the angular resolution of the detector. The program assumes that the resolution element of the detector or the rebinning interval in calculating the radial average is 1 cm. This assumption is highly unlikely for SAXS data and so some scaling of the parameters will be needed if resolution is to be included in a fit.

Wavelength

This is the wavelength of the incident beam in Angstroms. This is necessary to calculate the radius or angle for any given Q.

Wavelength Spread

This parameter is included as a variable parameter and may be fitted within the program. It is always Parameter 19. It is given in % and is the full-width at half maximum (FWHM) of an assumed triangular distribution.

Although much of this information can be found in some data files, it is common that the correct values have not been stored and it is useful to verify that the data is sensible. If studies are made of well defined spherical particles, the resolution may be very important in modifying the sharp fringes that might otherwise be observed.

Strictly the calculation of resolution broadening in this way is only appropriate to 'pin-hole' scattering cameras with a monochromatic beam. On other instruments such as time-of-flight SANS, the resolution will not take the same form. Indeed as the scattering at given Q arises from many wavelengths, it is not possible to calculate the 'resolution function' without the knowledge of the scattering distribution. In practice a reasonable approximation to the resolution may be obtained by suitable adjustment of the parameters. Data from Bonse-Hart double crystal cameras or slit geometries (e.g. Kratky cameras) will usually require separate desmearing routines.

The polydispersity can be either a log-normal distribution or a Gaussian distribution of size. The default is a log-normal distribution but this can be changed using the J and D options.

Polydispersity can be applied to all models but it may not be sensible to include this in some models (e.g. Gaussian function). For the rod, the polydispersity can be put on either the radius or the length.

The size distribution used in the fits can be seen in graphical form by using the option G from the subsidiary command menu. The data is shown as a probability distribution of radii. Scattering weights the fits according to the size of the particles, e.g. as R⁶ for spheres. The large end of the distribution is correspondingly more important in the fit and a larger range of R is used above the modal radius than below.

If it is the intention to fit a particle size distribution, care must be taken that the effects are not confused with instrument resolution or by models that intrinsically include a range of distances within particles. It may be difficult to distinguish polydisperse spheres and ellipsoids unless data is provided on an absolute scale of intensity or other information is available.

There are three options for the background:

1. Flat Background - one constant
2. Sloping background = (a + bQ)
3. A background spectrum read from a standard data file

The background spectrum can be scaled by a constant (that is a fittable parameter). It may be necessary to use such a factor if the transmission of the sample is such that only a fraction of the background should be fitted to the sample data. This facility can also be used to accomodate changing amounts of incoherent scattering.

If the model is required beyond the range of the background spectrum a simple linear extrapolation is made to either smaller or larger Q values.

The following options are available for the the scales:

Y Axis (Intensity)

1. Linear I(Q)
2. Logarithmic ln(I(Q))
3. Square Root I(Q)^0.5
4. I(Q) . Q²
5. 1 / I(Q)

X Axis

1. Linear Q
2. Square Q²
3. Logarithmic ln(Q)

These can be combined in any combination. This will allow the representation of the data while fitting in standard forms such as a Guinier or Zimm plot.

The choice of scales can also be used to influence the weighting of data points in the minimisation. See section 3.5 for more details.

The following options are available from a subsidiary menu access by J from the main FITFUN command prompt:

• B - Select a new background spectrum
• C - Change parameters for resolution calculation
• D - Change distribution type in polydispersity
• F - Fourier transform to calculate and plot r²g(r).
• G - Graph of particle size distribution
• G - List particle size distribution in an ASCII file
• I - Information about units etc.
• P - Toggle on/off inclusion of polydispersity
• R - Toggle on/off resolution broadening
• S - Store model calculation as data file
• W - Change the weighting used in least-squares fit
• X - Choose scale type for X-axis
• Y - Choose scale type for Y-axis
• E - Exit - return to main routine
• H - Help

D This allows a choice between Gaussian and log-normal distributions of particle size. The Gaussian is a symmetric while the log-normal distribution has a 'tail' of large particles. The default distribution type is a log-normal and this can only be changed by this command.

F This provides a simple Fourier integral of the model data to calculate r²g(r). It uses an integral of the model scattering function calculated without any instrument broadening or background. The model is extended to 8 times the maximum Q in the data in order to reduce errors of termination in the integral of {sin(Qr)/Qr}I(Q) Q² that runs from zero to Qmax which is taken as proportional to g(r).

Some questions with this routine allow the g(r) data to be further smoothed, replotted on different scales or output as a simple ASCII file. This option is useful if further plots or hard copy output is required. The format of this file is as described under option L.

G This provides a plot on the screen of the size distribution used in the polydispersity calculation. It indicates the range used in the calculation but the scales are rounded to some convenient numbers. At present there is no provision for hard copy output of this plot. If this is required it will be necessary to use a screen dump or to use the data obtained from option L.

L This allows the data shown in option G to be output as an ASCII file for further analysis or plotting. The format of the file is as follows:

NDATA
Run No., Ext., Title information
4 lines of parameters used in calculations
NDATA lines of R and P(R)

P & R These change the number of fitable parameters but the the current version of the FITFUN library does not allow the titles of the parameters to be updated. They will remain as on entry to the program. It may be useful to remember that the variable 20 is the polydispersity and variable 19 is the wavelength spread (% FWHM).

X & Y If the variables for the scales are changed it is essential that the data is reread using the R command from the main command menu. It is easy to reread the same data set as this will be the default data set. It will probably also be necessary to change the corresponding plot limits with the X and Y commands in the main menu.

There are three main options that are available for weighting the function to be minimised by the program. These may be selected according to the problem and the data available.

1. Chi Squared The square of the difference between the model and the data divided by the experimental error in the data is minimised for the range of points included in the fit.

2. Each Data Point Equally The square of the difference between the model and the data is minimised for each point included in the fit.

3. Scaled as Y-Axis The square of the difference between the model and the data converted to the current Y-axis scale is minimised for each point included in the fit.

If no data is available for the experimental errors, it is clearly not possible to use the first option. Although it might be thought that this is always the preferred way to produce an optimum fit to experimental data there are several reasons why this is not always the case. For example, the nature of the scattering may be such that some parts of the data are measured with far better statistical accuracy than other parts that contain important information about the sample. Under these circumstances it may be necessary to emphasize the lower intensity parts of the data in the fit.

The dependence of some models on the fit parameters may mean that the minimisation is most stable if fits are optimised using the scaled variables that effectively provide almost uniform sensitivity to each parameter that is to be fitted.

Under all circumstances it is necessary to exclude data that is known to be systematically in error from the minimisation in a fit. This will include regions that have been masked to exclude regions shadowed by a beam stop or where the detector is defective. This can be done using the Only command to select regions of interest. To aid the exclusion of masked regions in the data, it is normal for the program to exclude automatically any data point for which the experimental error is zero. This option can be switched off, either on entry to the program or in the subsidiary command menu with the option W which can be used to change other weighting options.

As with most general purpose programs it is difficult to account for correlated errors. An example of this difficulty would arise if a backround spectrum includes errors from the same detector normalisation as the sample spectrum to be fitted. The program will combine the statistical errors in these data sets as though they are independent. In practice, standard data reduction procedures will also have this problem. It is common for the errors associated with a data set to be somewhat over estimated. There seems to be no simple way around this problem unless a full history of the data reduction is available. In consequence the weighting of fits may never be truly correct in a conventional, statistical sense.

The errors in the fitted parameters that are displayed with the V command are only correct if the minimisation has terminated. If the previous message was 'Limit of Iterations reached' or 'No Minimum Found' the errors for the individual parameters can not be considered as valid!

4.1 Current Versions and Program Documentation

This program is currently available for Silicon Graphics Unix and for PC's that run DOS (or DOS prompts from Windows 3.1, Windows 9x, Windows NT etc.). The programs are functionally similar but under DOS the program will only work with full screen mode. Currently version 2.4 of rfit is available for both PC and Unix.