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

International Tables for Crystallography (2010). Vol. B, ch. 1.3, p. 101   | 1 | 2 |

## Section 1.3.4.5.1.3. The transform of an axially periodic fibre

G. Bricognea

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#### 1.3.4.5.1.3. The transform of an axially periodic fibre

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Let ρ be the electron-density distribution in a fibre, which is assumed to have translational periodicity with period 1 along z, and to have compact support with respect to the (x, y) coordinates. Thus ρ may be writtenwhere is the motif.

By the tensor product property, the inverse Fourier transform may be writtenand hence consists of `layers' labelled by l:with

Changing to polar coordinates in the (x, y) and planes decomposes the calculation of F from ρ into the following steps:and the calculation of ρ from F into:

These formulae are seen to involve a 2D Fourier series with respect to the two periodic coordinates ϕ and z, and Hankel transforms along the radial coordinates. The two periodicities in ϕ and z are independent, so that all combinations of indices (n, l) occur in the Fourier summations.