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 function The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function, named after Oliver Heaviside (1850–1925), the value of which is zero for negative arguments and one for positive argume ...

s of amplitude 1. It is used in

crystallography Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids. Crystallography is a fundamental subject in the fields of materials science and solid-state physics (condensed matter physics). The wo ...

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. Mathematically, :

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The coefficients of its

Fourier series A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or '' ...

are :

P_k(\Delta,x) = \frac = \Delta\cdot\mathrm(\pi k\Delta)\cdot\mathrm^.

with the

Sinc function In mathematics, physics and engineering, the sinc function, denoted by , has two forms, normalized and unnormalized.. In mathematics, the historical unnormalized sinc function is defined for by \operatornamex = \frac. Alternatively, the ...

References

{{reflist, refs= {{Cite journal , doi = 10.1107/S0108767395000365, title = On the use of crenel functions for occupationally modulated structures, journal = Acta Crystallographica Section A, volume = 51, issue = 4, pages = 529, year = 1995, last1 = Petříček , first1 = V., last2 = Van Der Lee , first2 = A., last3 = Evain , first3 = M. {{cite book, author=Malliakas, Christos D. , title=Charge Density Waves and Structural Modulations in Polytelluride Compounds, url=https://books.google.com/books?id=vSYU5Ih-W1QC&pg=PA30, year=2008, publisher=ProQuest, isbn=978-0-549-61737-2, pages=30–31 Special functions Generalized functions