# Kronig-Penney Model with Dirac Comb

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The simplest modification of the Kronig–Penney model for electrons in a one-dimensional periodic lattice can be based on a Dirac-comb potential approximating the positive cores:

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Contributed by: S. M. Blinder (August 2022)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

In the interval , the electron is a free particle and its wavefunction can be written

.

For the cell immediately to the left of the origin, with , Bloch's theorem implies that

.

The wavefunction must be continuous at , which leads to

(condition 1).

The derivatives at are found to be

and .

Taking account of the discontinuity in the first derivatives, the second derivative can be written

,

which cancels the delta function in the potential when

(condition 2).

Eliminating and between the two conditions gives

.

This can be solved for , from which it then follows that

.

References

[1] MIT OpenCourseWare. "Band Theory of Solids" (Feb 3, 2022) ocw.mit.edu/courses/chemistry/5-62-physical-chemistry-ii-spring-2008/lecture-notes/26_562ln08.pdf.

[2] S. Rajendran. "Understanding Band Structures in Solids via Solving Schrödinger Equation for Dirac Comb." (Feb 3, 2022) saravananrajendran.weebly.com/uploads/1/0/3/9/103971060/dirac_comb.pdf.

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