# Pair Distribution Functions

Atomic Pair Distribution Function (PDF) is a mathmatical equation that gives the probability of finding an atom or molecule between the distance r and r+dr. Basically what it means is sit on an atom and look at your surroundings.

Obtaining the PFD, data must first be collected from a neutron or x-ray diffraction. In, a diffraction experiment one could determine the total scattering structure function, S(Q), by measuring the intensities. The S(Q) equation is as follows:

S(Q)=(1/<b>^{2})(d?_{c}/d?)+((<b>^{2}-<b^{2}>)/<b>^{2})

b is the scattering length, the angle brackets (<>) around b indicate the average over the sample. d?_{c}/d? is indicated by the coherent single scaterring cross section of the sample which is related to the observed and normalized intensity (Egami and Billinge 2003).

From the structure function,S(Q), the PDF, G(r) is obtained through Fourier transform. The PDF is given as follows:

G(r)=4?r[?(r)-?_{0}]=(2/?)?^{?}_{0}Q(S(Q)-1)(sin(Qr)dQ)

?=3.14… or pi

?(r) is the microscopic pair density and ?_{0} is the average number density. Q is the magnitude of the scattering vector, hence, for elastic scattering Q=4?sin(?)/? with 2? being the scattering angle and ? the wavelenght of the radiation used.

### Experimentally Obtaining the PDF

As mention before, obtaining the PDF involves an experimental procedure known as powder diffraction, where nuetrons are accelerated at high velocities and shot at the specimen of study. This creates scattering upon deffraction of the neutrons.

Detectors surrounding the specimen collect the scattered neutrons and contributes to raw data.

Raw data is then used to obtain the total scattering structure function.

From the total scattering structure function the PDF is obtained

### Interpretation of PDF

The spikes represent the certainty of finding a specimen particles at a fix location. Sharp spikes represent a great chance of finding a neighboring particle at a distance between r and r+dr, called long range order. Meaning, the particles could be clustered together at short distances. Small broad peaks indicates a small chance of finding a neighboring particle at a distance between r and r+dr, meaning particles are spaced out.

### References

- Topics in functional analysis :essays dedicated to M. G. Krein on the occasion of his 70th birthday. Edited by I. Gohberg and M. Kac. Published by New York : Academic Press, 1978
- Neutron and x-ray scattering as probes of multiscale phenomena symposium held November 29-December 1, 2004, Boston, Massachusetts, U.S.A. Published by Warrendale, Pa. : Materials Research Society, c2005
- Neutron scattering in earth sciences. Edited by Hans-Rudolf Wenk. Published by Chantilly, Va. : Mineralogical Society of America, c2006