# Induced scalar electric potential

1. Jul 5, 2011

### nykon

Hi Forum!

I have got a question about the induced scalar potential. I will present the problem from beginning.

Lets say we have a Poisson's equation in form:

$\epsilon \nabla^2 \phi = -4\pi \varrho(r,t)$

where $\epsilon$ is the dielectric constant. By use of the Fourier transform:

$f(r,t) = \int \frac{d^3q}{(2\pi)^3} \int_{-\infty}^{\infty} \frac{d\omega}{2\pi} e^{i(qr-\omega t)}f(q, \omega)$,

(where $q$ is the momentum, and $\omega$ is the energy) one can write:

$\phi_{q, \omega}= \frac{4\pi \varrho_{q, \omega}}{q^2 \epsilon}$

where $\varrho_{q, \omega} = 2\pi Z \delta(\omega - q v)$, v is the velocity of the incident ion.

Now if we assume that the incident ion is moving through the electron gas we can write the induced scalar potential in form:

$\phi^{induced}_{q, \omega}= \frac{8\pi^2 Z}{q^2} \delta(\omega - q v)(\frac{1}{\epsilon} - 1)$

Now the energy loss per unit time W is:

$W = - Z v E^{induced}, \qquad E^{induced} = -\nabla \phi^{induced}$

the final result is:

$W = \int \frac{d^3q}{(2\pi)^3} \int_{-\infty}^{\infty} \frac{d\omega}{2\pi} 2 \omega Z Im[-\phi^{induced}]$,

My question is: why I need the imaginary part in the final result of the induced scalar potential? The result is taken from the "dynamic screening of ions in condensed matter" written by Echenique, Flores and Ritschie. I just do not understand the last formula. I will be gratefull for any tip or advise.

Best regards,
nykon

2. Jul 5, 2011

### Bill_K

The dielectric constant is complex, right? The ion is moving through an electron gas, which is a conducting medium. And W, the energy loss per unit time, is due to resistive dissipation. Anyway, I'm thinking that the previous line where W is given in terms of E should have an imaginary part on it also.

3. Jul 5, 2011

### nykon

Thats why I am confused. I really don't know why in the "final" equation the $Im(-\phi_{q,w})$ appears. Why we "cut" the Re part? These text book is available online, but I guess one has to pay for it.

Another publication where one can find these formulas is: "Interaction of Slow Ions with Matter" written by Echenique, Nagy and Arnau http://onlinelibrary.wiley.com/doi/...+July+from+10-12+BST+for+monthly+maintenance"

I have found a loot of publications from 1955 to 2002 where people use just the same notation, "way of thinking" and even steps are similar.

Last edited by a moderator: Apr 26, 2017