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,

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

The coefficients of its Fourier series are

${\displaystyle P_{k}(\Delta ,x)={\frac {\exp(2\pi i\,kx)\sin(\pi k\Delta )}{\pi k}}=\Delta \cdot \mathrm {sinc} (\pi k\Delta )\cdot \mathrm {e} ^{2\pi i\,kx}.}$

with the Sinc function.

References

1. Petříček, V.; Van Der Lee, A.; Evain, M. (1995). "On the use of crenel functions for occupationally modulated structures". Acta Crystallographica Section A. 51 (4): 529. Bibcode:1995AcCrA..51..529P. doi:10.1107/S0108767395000365.
2. Malliakas, Christos D. (2008). Charge Density Waves and Structural Modulations in Polytelluride Compounds. Michigan State University. Department of Chemistry. pp. 30–31. ISBN 978-0-549-61737-2.