Program written by D.I. Svergun, A. Semyenuk.

Documentation written by D.I. Svergun, A. Semyenuk and M.J. Gajda. Post all your questions about GNOM to the ATSAS Forum.

© ATSAS Team, 1991-2009

GNOM is an indirect transform program for small-angle scattering data processing. It reads in one-dimensional scattering curves (possibly smeared with instrumental distortions) and evaluates the particle distance distribution function P(r) (for monodisperse systems) or the size distribution function D(R) (for polydisperse systems). The main equations relating the scattering intensity to the distribution functions and describing the smearing effects one can find in the text-books (e.g. [1]).

GNOM is similar to the known programs of Glatter [2], Moore [3] and Provencher [4]. There are, however, some points, which make GNOM more convenient in application. GNOM is a dialogue program; the user instructions presented here explain how to answer the questions properly. The versions E4.0 and higher make use of the "configuration file", where some or all the parameters can be pre-defined; this allows to work with GNOM in different modes, ranging from fully interactive to batch mode. The names of the parameters in the following description refer to the configuration file names.

The algorithms used in the program are described elsewhere [5-7]. They are based on the Tikhonov's regularization technique [8]; several subroutines published in [9] are modified and used in GNOM. The value of the regularization parameter is estimated by the program automatically using the perceptual criteria [10].

Use AUTOGNOM for automatic command-line driven functionality.

The program searches for a distribution function in real space (characteristic function for monodisperse systems or size distribution function for polydisperse systems). These functions are related to the experimental intensity by corresponding Fourier transform followed by some convolutions representing smearing effects (if the latter exist). The integral equation is represented in a form of a linear system and solved with respect to the distribution function. The matrix of the system is called "design matrix". To evaluate it, the user must specify several parameters:

The main question, typical for solving ill-posed problems is how to stabilize the solution properly (here: how to find a positive regularization parameter ALPHA, which demands smoothness of the solution). The larger ALPHA, the more attention is paid to the smoothness of the distribution function and the less attention - to fitting the experimental data.

This search was done in the earlier GNOM versions by the generalized discrepancy method [8] using successive calculations with different ALPHA values. The ALPHA was chosen so as to fulfill the well-known χ-square criterion: the sum of deviations of the smeared intensity, corresponding to the given ALPHA, from the experimantal values, weighted with the standard deviations, should be equal to 1 + AN1. Here the value of AN1 is calculated by the program and represents the degree of unconformity of the problem (it equals to zero if NS=NR and increases with the ratio NS/NR). If AN1 is too large (AN1 > 1) the warning is printed. This warning, in principle, means that the input data/parameters are to be checked; however, it does not hinder following calculations.

The discrepancy criterion alone, however, in many cases can not ensure the correctness of the solution. Standard deviations may not be known or may be wrong. Another criterion which can also be used is the stability of the solution with respect to changing ALPHA. It can serve for manual [2], or automatic [6] search, but sometimes fails as well.

In the EN.N versions the computer is instructed to get a plausible solution basing on several criteria, like its smoothness, stability, absence of systematic deviations, etc. Some of them are to be fulfilled by increasing ALPHA, the others - by its decreasing. A compromise is to be found. It is done as follows.

Each criterion is expressed mathematically. For a given solution one calculates the value of the criterion and compares with its ideal value. All estimates of the parameters have the form of Gaussians P(i) = e-[(A(i) - B(i))/σ(i)]2 where:

(values A(i) and σ(i) are specified a priori). Then the total quality estimate is calculated as

TOTAL = P = ΣP(i)*W(i) / ΣW(i)

where:

Therefore, all P(i) and P are between 0.0 and 1.0. The program searches for a maximum of the total quality estimate P.

The following criteria are used for estimates:

The user is asked to enter the ALPHA value. If this value is already known to the user (e.g., from previous calculations with similar data), he can type the negative ALPHA, then the solution with -ALPHA will be evaluated. A positive ALPHA means automatic search of ALPHA by maximizing the total quality estimate as described below; no further manual refinement is possible (this is convenient for using GNOM in the batch mode). Default case (ALPHA=0) means automatic search accompanied by a graphic representation (plot lg(ALPHA) vs TOTAL estimate). This plot would help the user to correct ALPHA manually in the cases when the automatic search breaks down (see Practical advices).

The search of ALPHA works as follows. First the program finds the theoretical maximum (highest) ALPHA value for the given problem [7] (in graphic representation the Y-axis is drawn through this value). Then it tries to maximize the total estimate with the given set of A(i), σ(i) and W(i). The default set is:

Note that parameters OSCILL, STABIL and SYSDEV are considered to be most important.

The program evaluates first the grid of the TOTAL values for ALPHA < 1000*highest ALPHA (indicated as open circles in the plot). Then the golden section search is performed to maximize the estimate (characters '+' on the plot).

Note: when OSCILL is too large while SYSDEV already reasonable (here: OSCILL>13, SYSDEV>0.5), the value of TOTAL is put to zero. If zero value is encountered, the program stops evaluating the grid. This is made in order to avoid useless computations for too small ALPHA values.

After the solution is found, the user may either accept it, or change the weight/width of distribution for any criterion, or try to get a new maximum estimate, or change the ALPHA manually. When you increase the distribution width for a given parameter, larger deviations from the expected values are allowed for this parameter. When you increase the weight of a parameter, it plays a more significaht role in the estimate, therefore the solution maximizing the estimate will be shifted to fulfill the corresponding criterion. The changed set of parameters can be stored onto an ASCII disk file and then used The solution can also be plotted on the screen. The intensity plot presents together the experimental data (astericks), the desmeared (solid) curve and the corresponding smeared (dotted) curve. Note that for the cases JOB=3, 4 and 5 the experimental curves are modified. Namely, for JOB=3, I(s) is multiplied by s2, for JOB=4 by s, for JOB=5 divided by (2*J1(s*RAD56)/s*RAD56)2 (the latter is the cross-section factor of a cylinder, J1(x) Bessel function of the 1-st order). These modified curves are are plotted and printed onto the output file.

The distribution plot shows the obtained correlation or size distribution function. Radius of gyration and zero angle intensity are also evaluated in real space using the P(r) function. Note that for the case JOB=2 the values in real and reciprocal space may not coincide, because I(0) will be evaluated as the NDIM-th moment of D(R), and Rg as sqrt((NDIM+2)th moment)/(2*I(0)))again for a similar object by entering its name when prompted 'File containing expert parameters' (parameter EXPERT). In this file user may also edit all the values (including ideal estimations).

After the soluton is accepted by the user, the errors propagation can be estimated. This is made via a series of Monte-Carlo simulations. The standard deviations in the p(r) function as well as in the radius of gyration and zero-angle intensity are evaluated. The random sequence is initiated by the four-digit value of current time (min:sec), so that slightly different values of error estimates are obtained on exactly the same data set (the number of Monte-Carlo generations is 100).

Note: the Monte-Carlo routines were rewritten for the versions E3.3 and then 4.1 therefore some differences may also exist between the results of different versions.

Graphic routines in the program use MS-Fortran 5.1 graphic package (DOS/Windows version) or public domain Gnuplot package (UNIX version.)

On UNIX machines, a possibility exists to use the public domain Gnuplot graphic package. It is used not as a separate interactive program, but as a graphic library. GNOM creates a sequence of data/command files in the /tmp directory which are then executed by Gnuplot. To use Gnuplot version of GNOM, Gnuplot version 3.0 and higher must be installed and the gnuplot executables should be in the PATH variable.

After the job is finished, the user is asked whether to run the program again (parameter NEXTJOB). Answering No stops the program. Yes means new job. Same means that the new data set will be calculated under the same conditions as the previous one (with the same integral kernel).

Answering a question mark ? followed by 'Enter' to each question allows to get a short help on the parameter.

Runtime output format is identical to format of output file. It contains first alpha search history, then regularized curve, and finally P(r) distribution.

Regularized curve is printed in five columns:

Chord distribution function is printed in three columns:

This chapter has arisen as a result of practical applications of GNOM and is based mainly on the typical user questions.

GNOM is intended to be as user-friendly as possible, that is, it normally runs OK using the default answers. In particular, default value of the number of real space points is highly recommended to use.

The units in real and reciprocal space must be reciprocal to each other. Thus, if the momentum transfer is given in reciprocal Ångströms (Å-1 for λ in Ångströms), the particle sizes in real space are to be specified also in Ångströms. If you want particle sizes in miles, the input file has to contain momentum transfer in reciprocal miles. The units of the parameters describing beam divergency should be the same as for the momentum transfer.

All the weighting functions are normalized so that the desmeared curve is on the same scale as the smeared one. In parcitular, as the wavelength effects are always relative, the wavelength distribution gets reduced to λ=1. This does not mean that the program works only for this wavelength, because the wavelength information is already contained in the s-units.

One more thing about the wavelength effects: in X-ray experiments with the conventional sources (X-ray tubes) they are normally negligible.

The rectangular slit geometry assumed for the 1D-detectors is normally a good approximation for practical applications. It is valid, of course, both for linear multi-channel detectors and single channel moving detectors. Note that the slit-height function, if non-symmetric, can be symmetrized, because of the properties of the slit-smearing integral. Asymmetries in the slit-width function, if exist, can be neglected, as the slit-width effects are normally small. The case of "infinitely long slit" can be covered, e.g., by putting AH=Smax, LH=0, where Smax is the momentum transfer of the last data point. For a more detailed description of the smearing effects see, e.g. Ref[1], Ch.9.

If you specify the maximum size so that Dmax*Smin (Smin the momentum transfer of the first data point) is greater than π=3.1416926, a warning is printed. This means that the first data point is beyond the first sampling point and the information content may be not sufficient to obtain a reliable solution. However, the warning does not prevent the program from continuing.

Default settings of the EXPERT parameters are also valid for the most of normal systems. However, in some cases it is advisable to change the default set in order to ensure reliable automatic search of ALPHA. Typical examples are:

The program may fail to evaluate the constant for the second run in the trivial case with no smearing because of poor stability of the design matrix in this case. The users are kindly asked to correct the value manually.

The program explores the range (103 * highest ALPHA) > ALPHA > (10-18 * highest ALPHA)

to maximize the estimate. If the total estimate is monotonous in this range, the warning is printed. In such cases the user can perform manual search, or restart the maximization from the current point. Important note: normally, the plausible range of ALPHA is approximately 10-6 * (highest ALPHA).

If the solution found by the program is by far not acceptable, this just means that the maximization procedure (golden section search in logarithmic ALPHA scale) is trapped by a local minimum. One can restart the maximization from an ALPHA in a plausible range.

Another reason of getting a non-acceptable solution could be a non-consistency of the data and the assumptions. In this case the user will get the warning about AN1 and, normally, rather low value of the maximum estimate.

By default, the program puts P(RMIN)=0, P(RMAX)=0. If these conditions are not applied, non-zero P(r)'s can exist at the ends of the real space interval. Sometimes by P(0) one can take a constant background into account (note that for JOB=3 P(0) is normally non-zero). If you do not know Dmax, it is better first not to fix P(RMAX), because this value will help to judge whether the chosen interval is correct (if, for example, RMAX is too small, P(r) is systematically higher than zero in the vicinity of RMAX).

Model calculations show that the evaluation of the size distribution function of cylinders (JOB=5) is a very ill-posed problem, because the scattering curves from long cylinders are rather flat and it is not so easy to distinguish between them in the scattering curve, as e.g. for polydisperse system of spheres. Therefore for JOB=5 it is very important to have good measurements in the beginning of the scattering curve where scattering from cylinders of different length differs significantly from each other.

Input file (parameter INPUT1) is a sequential ASCII file containing the experimental data. The first line is always treated as a problem title (comment). Then the program searches for the first data line. Valid data lines should contain momentum transfer, non-zero intensity, and, optionally, standard deviation in a free format (that is, separated by blanks or commas). After this all valid lines are treated as data; invalid line or end-of-file are considered as the end of the input stream. The maximum number of data points is 1024.

This way to read data is rather flexible, allowing to skip additional headers and to use the data with or without the standard deviations.

With modern devices one can measure lots of data points with tiny angular step. This is not always good, and sometimes might become harmful when using indirect methods, because each angular point gives rise to a separate equation, which are, in fact, linearly dependent. Note that the angular step δ(s)=Dsamp/6, where Dsamp=(π)/Dmax is a sampling distance, is claimed to be optimum. To optimize the performance, GNOM joins neighbouring data points when it finds the number of points to be unreasonably high.

The program can handle simultaneously two data sets corresponding to the same sample but recorded with different conditions (see below). The second input file (parameter INPUT2) should have the same structure. Default answer to the question 'Input file name, second run' is no second run.

Note, that the momentum transfer is assumed to be s = 4 (π) sin (Θ) / (λaverage), where Θ is a half of the scattering angle.

Example of the input file :

If some standard deviations (or all of them) are not specified, negative or equal to zero, the user will be prompted for their estimation (parameter DEVIAT). A positive number means constant relative deviation per point. If the user enters zero (default answer), the program will estimate the errors automatically with the help of a polynomial smoothing procedure.

Giving 3 means that for each point with a non-positive standard deviation a relative deviation 0.03*I(s) is assumed, where I(s) is the intensity in this point.

Some or all of the parameters used by the program can be specified in the configuration file named 'gnom.cfg'. The first line of the file is always treated as a comment. The structure of a valid information line is shown below:

Here NAME is the name of the parameter (all names are fixed), the letter in the 9-th column shows the type of the parameter: (C - character, I - integer, R - real). The value of the parameter is to be put between the square brackets (not more than 35 characters). If the line is not valid (e.g., ] is missing, or the parameter name is wrong), it is skipped and the warning is printed. When GNOM is started, it first tries to find and read the configuration file in the current directory. If the local file named 'gnom.cfg' does not exist, it tries to read the global configuration file:

If GNOM finds a non-blank value between the square brackets for a parameter in the configuration file, this parameter is assumed to be already defined, therefore, it will not be prompted interactively (the other parameters will be asked as usual). The read-in configuration table is printed on the screen ( ... means that the variable is not defined). The user can either accept the values or switch to the interactive mode. GNOM assumes the interactive mode also when neither local nor global configuration file has been found.

Specific values for some parameters :

The example of the global configuration file is shown below. In this file only a few variables are fixed, and the program will not ask for them. More specific configuration files can be put in the local working directories. They can be configured in such a way, that the program will ask only a few questions or even run in the batch mode. Examples of the local configuration files can be found in the distribution package.

General configuration file

The configuration file may contain arbitrary number of parameters. Not all of them have to appear in the file: the missing parameters will be treated as non-defined. It is advisable, however, to end the configuration file with the NEXTJOB parameter. If NEXTJOB is 'Y' or 'S', the next lines will describe the parameters for the new job, and so on. The last line in this case should be:

If you run the NEXTJOB, GNOM keeps the values which were used in the previous job as program defaults (for character variables only first 12 characters are kept).

Example: using the general configuration file shown above, the default output file will be 'gnom.out' (program default); if you enter another name, say,'mygnom.out' and then answer "Y" to the NEXTJOB question, default output file will be 'mygnom.out'.

The form factor of particles should be supplied in a separate file. The first line of this file should contain three values in free format:

This should be followed by the form factor values - NFORM of values in 8E10.3 format, NFORM.22) 46 mg/ml, small angles (