International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 1.3, pp. 64-65   | 1 | 2 |

## Section 1.3.4.2.1.6. Convolution, correlation and Patterson function

G. Bricognea

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#### 1.3.4.2.1.6. Convolution, correlation and Patterson function

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Let and be two electron densities referred to crystallographic coordinates, with structure factors and , so that

The distribution is well defined, since the generalized support condition (Section 1.3.2.3.9.7) is satisfied. The forward version of the convolution theorem implies that ifthen

If either or is infinitely differentiable, then the distribution exists, and if we analyse it asthen the backward version of the convolution theorem reads:

The cross correlation between and is the -periodic distribution defined by:If and are locally integrable,LetThe combined use of the shift property and of the forward convolution theorem then gives immediately:hence the Fourier series representation of :Clearly, , as shown by the fact that permuting F and G changes into its complex conjugate.

The auto-correlation of is defined as and is called the Patterson function of . If consists of point atoms, i.e.thencontains information about interatomic vectors. It has the Fourier series representationand is therefore calculable from the diffraction intensities alone. It was first proposed by Patterson (1934, 1935a,b) as an extension to crystals of the radially averaged correlation function used by Warren & Gingrich (1934) in the study of powders.

### References

Patterson, A. L. (1934). A Fourier series method for the determination of the components of interatomic distances in crystals. Phys. Rev. 46, 372–376.
Patterson, A. L. (1935a). A direct method for the determination of the components of interatomic distances in crystals. Z. Kristallogr. 90, 517–542.
Patterson, A. L. (1935b). Tabulated data for the seventeen plane groups. Z. Kristallogr. 90, 543–554.
Warren, B. E. & Gingrich, N. S. (1934). Fourier integral analysis of X-ray powder patterns. Phys. Rev. 46, 368–372.