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Investigating Hot-Carrier Effects using the Backward Monte Carlo Method

A.3 Fourier Transform of the Screened Coulomb Potential

The Fourier Transform of the screened Coulomb potential

(1.12–A.11) \{begin}{align*} V(u) = \frac {\e ^{-\beta _s\,u}}{4\pi \,u} \{end}{align*}

is defined as

(1.12–A.11) \{begin}{align*} \widetilde {V}(\qv )=\int \dint ^3 u\, \frac { \e ^{-\beta _s\,u} }{4\pi \,u} \e ^{-\imag \,\qv \cdot \uv }\,. \{end}{align*}

Introducing spherical polar coordinates, with \( \qv \) defining the polar axis,

(1.12–A.11) \{begin}{align*} \qv \cdot \uv = q\,u\,\cos \theta \qquad \text {and}\qquad \cos \theta =\chi \„ \{end}{align*}

the integral can be solved as follows:

(1.12–A.11) \{begin}{align*} \widetilde {V}(\qv )&=\frac {1}{2}\int \limits ^\infty _0 \dint u \, u^2 \,\frac {\e ^{-\beta _s\,u}}{u} \int \limits ^1_{-1} \dint \chi \, \e ^{-\imag \,q\,u\,\chi } = \frac {1}{2} \int \limits ^\infty _0 \dint u \, u^2 \,\frac {\e ^{-\beta _s\,u}}{u} \left . \left [ \frac {\e ^{-\imag \,q\,u\,\chi }}{q\,u} \right ]\right |_{-1}^1\\ &= \frac {\pi }{2\,q}\int \limits ^\infty _0 \dint u \, \left [ \e ^{-(\beta _s+\imag \,q)\,u} - \e ^{-(\beta _s-\imag \,q)\,u} \right ] =\frac {1}{\beta _s^2+q^2} \{end}{align*}

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