An important concept in plasma physics is the Debye length, which describes the screening of a charge's electrostatic potential due to the net effect of the interactions it undergoes with the other mobile charges (electrons and ions) in the system. It can be shown that, given a set of reasonable assumptions about the behaviour of charges in the plasma, the electric potential due to a "test charge", $q_\mathrm{T}$ is given by
$$
\phi = \frac{q_\mathrm{T}}{4\pi\epsilon_0 r}\exp\left(-\frac{r}{\lambda_\mathrm{D}}\right),
$$
where the electron Debye length,
$$
\lambda_\mathrm{D} = \sqrt{\frac{\epsilon_0 T_e}{e^2n_0}},
$$
for an electron temperature $T_e$ expressed as an energy (i.e. $T_e = k_\mathrm{B}T_e'$ where $T_e'$ is in K) and number density $n_0$. Rigorous derivations, starting from Gauss' Law and solving the resulting Poisson equation with a Green's function are given elsewhere (e.g. Section 7.2.2. in J. P. Freidberg, *Plasma Physics and Fusion Energy*, CUP (2008)).

The following Python code plots the shielded and unshielded Coulomb potential due to a point test charge $q_\mathrm{T} = +e$, assuming an electron temperature and density typical of a tokamak magnetic confinement nuclear fusion device.

import numpy as np from scipy.constants import k as kB, epsilon_0, e from matplotlib import rc import matplotlib.pyplot as plt rc('font', **{'family': 'serif', 'serif': ['Computer Modern'], 'size': 16}) rc('text', usetex=True) # We need the following so that the legend labels are vertically centred on # their indicator lines. rc('text.latex', preview=True) def calc_debye_length(Te, n0): """Return the Debye length for a plasma characterised by Te, n0. The electron temperature Te should be given in eV and density, n0 in cm-3. The debye length is returned in m. """ return np.sqrt(epsilon_0 * Te / e / n0 / 1.e-6) def calc_unscreened_potential(r, qT): return qT * e / 4 / np.pi / epsilon_0 / r def calc_e_potential(r, lam_De, qT): return calc_unscreened_potential(r, qT) * np.exp(-r / lam_De) # plasma electron temperature (eV) and density (cm-3) for a typical tokamak. Te, n0 = 1.e8 * kB / e, 1.e26 lam_De = calc_debye_length(Te, n0) print(lam_De) # range of distances to plot phi for, in m. rmin = lam_De / 10 rmax = lam_De * 5 r = np.linspace(rmin, rmax, 100) qT = 1 phi_unscreened = calc_unscreened_potential(r, qT) phi = calc_e_potential(r, lam_De, qT) # Plot the figure. Apologies for the ugly and repetitive unit conversions from # m to µm and from V to mV. fig, ax = plt.subplots() ax.plot(r*1.e6, phi_unscreened * 1000, label=r'Unscreened: $\phi = \frac{e}{4\pi\epsilon_0 r}$') ax.plot(r*1.e6, phi * 1000, label=r'Screened: $\phi = \frac{e}{4\pi\epsilon_0 r}' r'e^{-r/\lambda_\mathrm{D}}$') ax.axvline(lam_De*1.e6, ls='--', c='k') ax.annotate(xy=(lam_De*1.1*1.e6, max(phi_unscreened)/2 * 1000), s=r'$\lambda_\mathrm{D} = %.1f \mathrm{\mu m}$' % (lam_De*1.e6)) ax.legend() ax.set_xlabel(r'$r/\mathrm{\mu m}$') ax.set_ylabel(r'$\phi/\mathrm{mV}$') plt.savefig('debye_length.png') plt.show()

## Comments

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## Dominik Stańczak 1 year, 5 months ago

If you'd like to avoid doing the annoying unit conversions, Python has a bunch of packages like astropy.units (which I ten to use, http://docs.astropy.org/en/stable/units/ ) or unyt (which I've heard good things about) :)

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