Electrostatics

In electric-field screening, screened Poisson equation for the electric potential ${\displaystyle \phi (\mathbf {r} )}$ is usually written as (SI units)

$${\displaystyle \left[\Delta -k_{0}^{2}\right]\phi (\mathbf {r} )=-{\frac {\rho _{\rm {ext}}(\mathbf {r} )}{\epsilon _{0}}},}$$

where ${\displaystyle k_{0}^{-1}}$ is the screening length, ${\displaystyle \rho _{\rm {ext}}(\mathbf {r} )}$ is the charge density produced by an external field in the absence of screening and ${\displaystyle \epsilon _{0}}$ is the vacuum permittivity.This equation can be derived in several screening models like Thomas–Fermi screening in solid-state physics and Debye screening in plasmas.

Three dimensions

Without loss of generality, we will take λ to be non-negative. When λ is zero, the equation reduces to Poisson's equation. Therefore, when λ is very small, the solution approaches that of the unscreened Poisson equation, which, in dimension ${\displaystyle n=3}$, is a superposition of 1/r functions weighted by the source function f:

$${\displaystyle u(\mathbf {r} )_{({\text{Poisson}})}=\iiint \mathrm {d} ^{3}r'{\frac {f(\mathbf {r} ')}{4\pi |\mathbf {r} -\mathbf {r} '|}}.}$$

On the other hand, when λ is extremely large, u approaches the value f/λ², which goes to zero as λ goes to infinity. As we shall see, the solution for intermediate values of λ behaves as a superposition of screened (or damped) 1/r functions, with λ behaving as the strength of the screening.

The screened Poisson equation can be solved for general f using the method of Green's functions. The Green's function G is defined by

$${\displaystyle \left[\Delta -\lambda ^{2}\right]G(\mathbf {r} )=-\delta ^{3}(\mathbf {r} ),}$$

where δ³ is a delta function with unit mass concentrated at the origin of R³.

Assuming u and its derivatives vanish at large r, we may perform a continuous Fourier transform in spatial coordinates:

$${\displaystyle G(\mathbf {k} )=\iiint \mathrm {d} ^{3}r\;G(\mathbf {r} )e^{-i\mathbf {k} \cdot \mathbf {r} }}$$

where the integral is taken over all space. It is then straightforward to show that

$${\displaystyle \left[k^{2}+\lambda ^{2}\right]G(\mathbf {k} )=1.}$$

The Green's function in r is therefore given by the inverse Fourier transform,

$${\displaystyle G(\mathbf {r} )={\frac {1}{(2\pi )^{3}}}\;\iiint \mathrm {d} ^{3}\!k\;{\frac {e^{i\mathbf {k} \cdot \mathbf {r} }}{k^{2}+\lambda ^{2}}}.}$$

This integral may be evaluated using spherical coordinates in k-space. The integration over the angular coordinates is straightforward, and the integral reduces to one over the radial wavenumber ${\displaystyle k_{r}}$:

$${\displaystyle G(\mathbf {r} )={\frac {1}{2\pi ^{2}r}}\;\int _{0}^{\infty }\mathrm {d} k_{r}\;{\frac {k_{r}\,\sin k_{r}r}{k_{r}^{2}+\lambda ^{2}}}.}$$

This may be evaluated using contour integration. The result is:

$${\displaystyle G(\mathbf {r} )={\frac {e^{-\lambda r}}{4\pi r}}.}$$

The solution to the full problem is then given by

$${\displaystyle u(\mathbf {r} )=\int \mathrm {d} ^{3}r'G(\mathbf {r} -\mathbf {r} ')f(\mathbf {r} ')=\int \mathrm {d} ^{3}r'{\frac {e^{-\lambda |\mathbf {r} -\mathbf {r} '|}}{4\pi |\mathbf {r} -\mathbf {r} '|}}f(\mathbf {r} ').}$$

As stated above, this is a superposition of screened 1/r functions, weighted by the source function f and with λ acting as the strength of the screening. The screened 1/r function is often encountered in physics as a screened Coulomb potential, also called a "Yukawa potential".

Two dimensions

In two dimensions: In the case of a magnetized plasma, the screened Poisson equation is quasi-2D:

$${\displaystyle \left(\Delta _{\perp }-{\frac {1}{\rho ^{2}}}\right)u(\mathbf {r} _{\perp })=-f(\mathbf {r} _{\perp })}$$

with ${\displaystyle \Delta _{\perp }=\nabla \cdot \nabla _{\perp }}$ and ${\displaystyle \nabla _{\perp }=\nabla -{\frac {\mathbf {B} }{B}}\cdot \nabla }$, with ${\displaystyle \mathbf {B} }$ the magnetic field and ${\displaystyle \rho }$ is the (ion) Larmor radius. The two-dimensional Fourier Transform of the associated Green's function is:

$${\displaystyle G(\mathbf {k_{\perp }} )=\iint d^{2}r~G(\mathbf {r} _{\perp })e^{-i\mathbf {k} _{\perp }\cdot \mathbf {r} _{\perp }}.}$$

The 2D screened Poisson equation yields:

$${\displaystyle \left(k_{\perp }^{2}+{\frac {1}{\rho ^{2}}}\right)G(\mathbf {k} _{\perp })=1.}$$

The Green's function is therefore given by the inverse Fourier transform:

$${\displaystyle G(\mathbf {r} _{\perp })={\frac {1}{4\pi ^{2}}}\;\iint \mathrm {d} ^{2}\!k\;{\frac {e^{i\mathbf {k} _{\perp }\cdot \mathbf {r} _{\perp }}}{k_{\perp }^{2}+1/\rho ^{2}}}.}$$

This integral can be calculated using polar coordinates in k-space:

$${\displaystyle \mathbf {k} _{\perp }=(k_{r}\cos(\theta ),k_{r}\sin(\theta ))}$$

The integration over the angular coordinate gives a Bessel function, and the integral reduces to one over the radial wavenumber ${\displaystyle k_{r}}$:

$${\displaystyle G(\mathbf {r} _{\perp })={\frac {1}{2\pi }}\;\int _{0}^{\infty }\mathrm {d} k_{r}\;{\frac {k_{r}\,J_{0}(k_{r}r_{\perp })}{k_{r}^{2}+1/\rho ^{2}}}={\frac {1}{2\pi }}K_{0}(r_{\perp }\,/\,\rho ).}$$