# Types of plasma

Two important parameters in plasma physics are the electron Debye length, $\lambda_{\mathrm{D}e}$, a measure of the distance over which charge-screening effects occur and deviations from quasi-neutrality are observed, and the number of paricles in a "Debye cube" (of side length $\lambda_{\mathrm{D}e}$), $N_\mathrm{D}$.

In terms of the electron temperature, $T_e$ (expressed as an energy) and number density, $n_e$,

$$\lambda_{\mathrm{D}e} = \sqrt{\frac{\epsilon_0 T_e}{e^2n_e}}, \quad N_\mathrm{D} = n_e\lambda_{\mathrm{D}e}.$$

The condition for an ionized gas to be considered a plasma is $N_\mathrm{D} \gg 1$: many charged particles within a Debye cube.

Below we plot lines of constant $\lambda_{\mathrm{D}e}$ and $N_\mathrm{D}$ for a range of typical values of $n_e$ and $T_e$ on a log scale and indicate the regimes corresponding to certain kinds of plasma. The code to generate this plot is given below. The lines could have been plotted as contours on a two-dimensional grid, but here we calculate them explicitly using the formulae obtained by taking the logarithm and rearranging the above expressions:

\begin{align*} \log T_e & = \frac{2}{3}\log N_\mathrm{D} + \log \left(\frac{e^2}{\epsilon_0}\right) + \frac{1}{3}\log n_e\\ \log T_e &= 2\log \lambda_{\mathrm{D}e} + \log \left(\frac{e^2}{\epsilon_0}\right) + \log n_e \end{align*}

with some extra terms to allow for unit conversions.

import numpy as np
from scipy.constants import k as kB, epsilon_0, e
import matplotlib.pyplot as plt
from matplotlib.patches import Ellipse

def calc_logTe_for_ND(logne, logND):
"""Return the log of electron temperature (in K) from ne for ND.

logne is log10 of the electron density in cm-3,
logND is log10 of the number of particles in a Debye cube.

"""

# The factor (term in log-space) here converts from log10(Joules)
# to log10(K).
fac = np.log10(kB)
# Add 6 to logne for the conversion from cm-3 to m-3.
return 2 * logND / 3 + np.log10(e**2 / epsilon_0) + (logne + 6)/3 - fac

def calc_logTe_for_lamD(logne, loglamD):
"""Return the log of electron temperature (in K) from ne for lamD.

logne is log10 of the electron density in cm-3,
loglamD is log10 of the electron Debye length in m.

"""

# The factor (term in log-space) here converts from log10(Joules)
# to log10(K).
fac = np.log10(kB)
# Add 6 to logne for the conversion from cm-3 to m-3.
return 2*loglamD + np.log10(e**2/epsilon_0) + logne + 6 - fac

fig, ax = plt.subplots()

# Electron density grid in cm-3.
logne_grid = np.array((0,30))

# The grid of log10(ND) values to plot lines for.
logND_grid = (0, 5, 10, 15)
# We need to save the logTe values for plotting the labels later.
logTe_by_ND_grid = {}
for logND in logND_grid:
logTe_by_ND_grid[logND] = calc_logTe_for_ND(logne_grid, logND)
ax.plot(logne_grid, logTe_by_ND_grid[logND], 'tab:red', alpha=0.7)

# The grid of log10(lamda_De) values to plot lines for.
loglamD_grid = (2, 0, -2, -4, -6, -8)
# We need to save the logTe values for plotting the labels later.
logTe_by_lamD_grid = {}
for loglamD in loglamD_grid:
logTe_by_lamD_grid[loglamD] = calc_logTe_for_lamD(logne_grid, loglamD)
kwargs = {'c': 'tab:blue', 'alpha': 0.7, 'ls': '--'}
ax.plot(logne_grid, logTe_by_lamD_grid[loglamD], **kwargs)

"""Add an ellipse of size width x height to the plot.

The ellipse is centred at coordinates xy and labelled.

"""

kwargs = {'fc': 'm', 'alpha': 0.7}
plasma_patch = Ellipse(xy, width, height, **kwargs)
ax.annotate(xy=xy, s=label)

# These are the plasmas we consider for illustration.
add_plasma_patch((9.5, 6.5), 1, 1, 'Solar corona')
add_plasma_patch((14.5, 4.5), 5, 1, 'Gas discharge')

# Fix the Axes limits
ax.set_xlim(0,25)
ax.set_ylim(1,12)

def annotate_line(x, y, label, xylabel=None, colour='k'):
"""Annotate a line of constant Nd or lambda_De.

The text label is rotated to coincide with the line through
(x, y) and (x, y) and centred at xylabel. If not provided,
xylabel is set to the centre of the line.

"""

rotn = np.degrees(np.arctan2(y-y, x-x))
if xylabel is None:
xylabel = ((x+x)/2, (y+y)/2)
text = ax.annotate(label, xy=xylabel, ha='center', va='center',
backgroundcolor='white', color=colour, size=9)
p1 = ax.transData.transform_point((x, y))
p2 = ax.transData.transform_point((x, y))
dy = (p2 - p1)
dx = (p2 - p1)

rotn = np.degrees(np.arctan2(dy, dx))
text.set_rotation(rotn)

# Set the Axes labels and tick labels (in scientific notation).
ax.set_xticklabels(['$10^{{{:d}}}$'.format(logne)
for logne in range(0,26,5)])
ax.set_yticklabels(['$10^{{{:d}}}$'.format(logTe)
for logTe in range(0,13,2)])
ax.set_xlabel('$n_e/\mathrm{cm^{-3}}$')
ax.set_ylabel('$T_e/\mathrm{K}$')

# Add the labels to lines of constant ND.
# xlabels is the x-coordinate of the centre of each label.
xlabels = (None, 7, None, 7)
for logND, xlabel in zip(logND_grid, xlabels):
if logND:
label = '$N_\mathrm{{D}}=10^{{{:d}}}$'.format(logND)
else:
label = '$N_\mathrm{{D}}=1$'
if xlabel is not None:
ylabel = calc_logTe_for_ND(xlabel, logND)
xylabel = (xlabel, ylabel)
else:
xylabel = None
annotate_line(logne_grid, logTe_by_ND_grid[logND], label, xylabel,
colour='tab:red')

# Add the labels to lines of constant lambda_De.
# xlabels is the x-coordinate of the centre of each label.
xlabels = (3, 5, 11, 10, 20, 20)
for loglamD, xlabel in zip(loglamD_grid, xlabels):
if loglamD:
s_val = '10^{{{:d}}}\;\mathrm{{m}}'.format(loglamD)
else:
s_val = '0'
label = '$\lambda_{{\mathrm{{D}}e}} = {}$'.format(s_val)
if xlabel is not None:
ylabel = calc_logTe_for_lamD(xlabel, loglamD)
xylabel = (xlabel, ylabel)
else:
xylabel = None

annotate_line(logne_grid, logTe_by_lamD_grid[loglamD], label, xylabel,
colour='tab:blue')

plt.savefig('plasma-types.png')
plt.show()

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