Crenel function

In mathematics, the crenel function is a periodic discontinuous function P(x) defined as 1 for x belonging to a given interval and 0 outside of it. It can be presented as a difference between two Heaviside step functions of amplitude 1.^[1] It is used in crystallography to account for irregularities in the occupation of atomic sites by given atoms in solids, such as periodic domain structures, where some regions are enriched and some are depleted with certain atoms.^[2]

Mathematically,

P(x)={\begin{cases}1,&x\in [-\Delta /2,\Delta /2],\\0,&x\notin [-\Delta /2,\Delta /2],\end{cases}}

Its Fourier transform coefficients are

P_{m}(\Delta ,x)={\frac {\exp(2\pi imx)\sin(\pi m\Delta )}{\pi m}}.

References

^ Attention: This template ({{cite doi}}) is deprecated. To cite the publication identified by doi:10.1107/S0108767395000365, please use {{cite journal}} (if it was published in a bona fide academic journal, otherwise {{cite report}} with |doi=10.1107/S0108767395000365 instead.
^ Malliakas, Christos D. (2008). Charge Density Waves and Structural Modulations in Polytelluride Compounds. ProQuest. pp. 30–31. ISBN 978-0-549-61737-2.

[r1-1] Attention: This template ({{cite doi}}) is deprecated. To cite the publication identified by doi:10.1107/S0108767395000365, please use {{cite journal}} (if it was published in a bona fide academic journal, otherwise {{cite report}} with |doi=10.1107/S0108767395000365 instead.

[r2-2] Malliakas, Christos D. (2008). Charge Density Waves and Structural Modulations in Polytelluride Compounds. ProQuest. pp. 30–31. ISBN 978-0-549-61737-2.

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