1. Indirect Fourier transformation in small angle scattering

2. The Glatter method of IFT

3. References

In a Fourier transform (FT), the Fourier transformed function is obtained from by:

where is defined as . can be obtained from by inverse FT:

and are inverse variables, e.g. frequency and time.

Obtaining directly requires that is well known from to , vice versa. In real experimental data this is rarely the case due to noise and limited measured range, say is known from to . Performing a FT on in the limited range may lead to systematic errors and overfitting.

An indirect Fourier transform (IFT) is a solution to this problem.

Indirect Fourier transformation in small angle scattering

In small-angle scattering on single molecules, an intensity is measured and is a function of the magnitude of the scattering vector , where is the scattered angle, and is the wavelength of the incoming and scattered beam (elastic scattering). has units 1/length. is related to the so-called pair distance distribution function via Fourier Transformation. is a (scattering weighted) histogram of distances between pairs of atoms in the molecule. In one dimensions ( and are scalars), and are related by:

where is the angle between and , and is the number density of molecules in the measured sample. The sample is orientational averaged (denoted by ), and the Debye equation ^[1] can thus be exploited to simplify the relations by

In 1977 Glatter proposed an IFT method to obtain form ,^[2] and three years later, Moore introduced an alternative method.^[3] Others have later introduced alternative and automathised methods for IFT,^[4] and automatised the process ^[5][6]

The Glatter method of IFT

This is an brief outline of the method introduced by Otto Glatter.^[2] For simplicity, we use in the following.

In indirect fourier transformation, a guess on the largest distance in the particle is given, and an initial distance distribution function is expressed as a sum of cubic spline functions evenly distributed on the interval (0,):

where are scalar coefficients. The relation between the scattering intensity and the is:

Inserting the expression for p_i(r) (1) into (2) and using that the transformation from to is linear gives:

where is given as:

The 's are unchanged under the linear Fourier transformation and can be fitted to data, thereby obtaining the coifficients . Inserting these new coefficients into the expression for gives a final . The coefficients are chosen to minimise the of the fit, given by:

where is the number of datapoints and is the standard deviations on data point . The fitting problem is ill posed and a very oscillating function would give the lowest despite being physically unrealistic. Therefore, a smoothness function is introduced:

.

The larger the oscillations, the higher . Instead of minimizing , the Lagrangian is minimized, where the Lagrange multiplier is denoted the smoothness parameter.

The method is indirect in the sense that the FT is done in several steps: .

References

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