langmuir 0.4.0

Functions for work on Langmuir probe theory

Programmatically accessible current-voltage characteristics for ideal and non-ideal Langmuir probes.

Installation

Install from PyPI using pip (preferred method):

pip install langmuir

Or download the GitHub repository https://github.com/langmuirproject/langmuir.git and run:

python setup.py install

Getting Started

The Langmuir library contains a collection of functions that compute the current collected by a Langmuir probe according to various theories and models. These functions take as arguments the probe geometry, for instance Cylinder, and a plasma, where a plasma may consist of one or more Species.

As an example, consider a 25mm long cylindrical probe with radius 0.255mm. The plasma consists of electrons and singly charged oxygen ions, both with a density of 1e11 and a temperature of 1000K. The current-voltage charactersitic according to OML theory is easily computed and plotted:

>>> from langmuir import *
>>> import numpy as np
>>> import matplotlib.pyplot as plt

>>> plasma = []
>>> plasma.append(Species('electron' , n=1e11, T=1000))
>>> plasma.append(Species(amu=16, Z=1, n=1e11, T=1000))

>>> Vs = np.linspace(-2, 2, 100)
>>> Is = OML_current(Cylinder(r=0.255e-3, l=25e-3), plasma, Vs)

>>> fig = plt.figure()
>>> ax = fig.add_subplot(111)

>>> ax.plot(Vs, -Is*1e6)
>>> ax.set_xlabel(r'$V_{\mathrm{p}} [\mathrm{V}]$')
>>> ax.set_ylabel(r'$I_{\mathrm{p}} [\mathrm{\mu A}]$')
>>> ax.grid(True)
>>> plt.show()

Notice that the characteristic includes all regions (ion saturation, electron retardation and electron saturation), and do not rely on approximations to the OML theory requiring the voltage to be within a certain range. What’s more, it’s easy to take into account for instance finite-radius effects, by replacing OML_current() with finite_radius_current().

Specifying a species

A species is specified using a range of keyword arguments upon initialization of a Species object. At the very least, you must specify the density and the temperature (or thermal speed):

>>> Species(n=1e11, T=1000)

Since nothing more is specified, this represents Maxwellian-distributed electrons. The full set of keywords is as follows:

Quantity	Keyword	Units	Default value
Charge	q	Coulombs	electron charge
	Z	elementary charges
Mass	m	kilograms	electron mass
	amu	atomic mass units
Density	n	partices per cubic meter
Temperature/thermal speed	T	Kelvin
	eV	electron-volts
	vth	meters per second
Spectral index kappa	kappa	dimensionless	float('inf')
Spectral index alpha	alpha	dimensionless	0

The Langmuir library supports the general Kappa-Cairns velocity distribution, but defaults to Maxwellian, which is a special case of the Kappa-Cairns distribution.Kappa, Cairns or Kappa-Cairns distributed particles are obtained by specifying kappa, alpha or both, respectively.

The charge and mass can also be specified using one of three self-descriptive flag:

• 'electron'

• 'positron'

• 'proton'

This must come before the keyword arguments, for instance:

>>> Species('proton', n=1e11, eV=0.1)

Computing fundamental plasma parameters

The Langmuir library is also convenient to use for quick computations of fundamental plasma parameters, since Species computes these upon initialization. For instance, to get the electron Debye length of a plasma with a certain density and temperature:

>>> Species(n=1e11, eV=0.1).debye
0.007433942027347403

The following member variables and methods are accessible in Species:

Member	Description
debye	The Debye length
omega_p	The angular plasma frequency
freq_p	The linear plasma frequency
period_p	The plasma period
omega_c(B)	The angular cyclotron frequency
freq_c(B)	The linear cyclotron frequency
period_c(B)	The cyclotron period
larmor(B)	The larmor radius

The latter four members are methods which take the magnitude of the magnetic flux density as an argument. In addition, every valid keyword argument is also a valid member variable:

>>> Species(n=1e11, T=1000).eV
0.08617330337217212

Finally, the total Debye length of a plasma consisting of multiple species can be obtained using the debye() function:

>>> plasma = []
>>> plasma.append(Species('electron' , n=1e11, T=1000))
>>> plasma.append(Species(amu=16, Z=1, n=1e11, T=1000))
>>> debye(plasma)
0.004879671013271479

Specifying the geometry

Langmuir supports two probe geometries, with self-descriptive names and the following signatures:

• Sphere(r)

• Cylinder(r, l)

r and l representes the radius and length, respectively, of the geometry.

Models for collected current

Langmuir comes with several models for the collected current. Each model is represented by a function which takes a geometry and a species argument. The geometry is one of the above probe geometries, and the species parameters is either a single Species object or a list of such if it is desirable to take into account the effect of all species in a plasma. Most models also depend on the potential of the probe with respect to the background plasma. The potential can either be specified in volts using th V argument, or in terms of normalized voltage e*V/(k*T) using the eta argument. If these are Numpy arrays, the output will be a Numpy array of collected currents. The normalization argument can be set to 'th', 'thmax', 'oml' to return currents normalized by the thermal current, Maxwellian thermal current, or OML current, respectively, rather than Ampére. Below is a description of all models:

• OML_current(geometry, species, V=None, eta=None, normalization=None) Current collected by a probe according to the Orbital Motion Limited (OML) theory. The model assumes a probe of infinitely small radius compared to the Debye length, and for a cylindrical probe, that it is infinitely long. Probes with radii up to 0.2 Debye lengths (for spherical probes) or 1.0 Debye lengths (for cylindrical probes) are very well approximated by this theory, although the literature is diverse as to how long cylindrical probes must be for this theory to be a good approximation.

• finite_radius_current(geometry, species, V=None, eta=None, table='laframboise-darian-marholm', normalization=None) A current model taking into account the effects of finite radius by interpolating between tabulated normalized currents. The model only accounts for the attracted-species currents (for which eta<0). It does not extrapolate, but returns nan when the input parameters are outside the domain of the model. This happens when the normalized potential for any given species is less than -25, when kappa is less than 4, when alpha is more than 0.2 or when the radius is more than 10 or sometimes all the way up towards 100 (as the distribution approaches Maxwellian). Normally finite radius effects are negligible for radii less than 0.2 Debye lengths (spheres) or 1.0 Debye lengths (cylinders).

• finite_length_current(geometry, species, V=None, eta=None, normalization=None) The Marholm-Marchand model for finite-length probes. Works for normalized voltages up to 100.

• finite_length_current_density(geometry, species, V=None, eta=None, z=None, zeta=None, normalization=None) The current per unit length according to the Marholm-Marchand model for finite-length probes. Works for normalized voltages up to 100. z is position on the probe, and zeta is position normalized by the Debye length.

• thermal_current(geometry, species, normalization=None) Returns the thermal current for the given species and geometry. The thermal current is the current the species contributes to a probe at zero potential with respect to the background plasma due to random thermal movements of particles.

• normalization_current(geometry, species) Returns the normalization current for the given species and geometry. The normalization current is the current the species would have contributed to a probe at zero potential with respect to the background plasma due to random thermal movements of particles, if the species had been Maxwellian.

As an example, the following snippet computes the normalized electron current of a probe of 3 Debye lengths radius and normalized voltage of -10:

>>> sp  = Species(n=1e11, T=1000)
>>> geo = Cylinder(r=3*sp.debye, l=1)
>>> I = finite_radius_current(geo, sp, eta=-10, normalization='th')

Notice that setting l==1 means you get the current per unit length.

Inverse problems

Sometimes the collected current of one or more probes is known and one would like to solve for one or more other parameters. The Langmuir library do not address this analytically in part due to the vast number of such inverse problems, and in part due to some characteristics not being invertible (for instance those who are of tabulated values). However, it is in principle possible to apply numerical methods of root solving, least squares, etc. along with the models in Langmuir.

Consider a cylindrical probe with known dimensions and a positive but unknown voltage collecting a current of -0.4uA in a Maxwellian plasma with known density and temperature. What is the voltage? We shall neglect the current due to ions, and define a residual function. This residual is the difference between the current collected by a probe at a given potential, and the actual collected current, and it is used by a least squares algorithm to compute the voltage:

>>> from langmuir import *
>>> from scipy.optimize import leastsq

>>> sp = Species(n=1e11, T=1000)
>>> geo = Cylinder(1e-3, 25e-3)
>>> I = -0.4e-6

>>> def residual(V):
>>>     return finite_radius_current(geo, sp, V) - I

>>> x, c = leastsq(residual, 0)
>>> print(x[0])
0.6265540484991013

The reader may verify that this voltage indeed results in the correct current. Notice also that we were in fact able to invert the model finite_radius_current, which consists of tabulated values.

A slightly more interesting inversion problem, is that of determining the ionospheric density from four cylindrical Langmuir probes with known bias voltages with respect to a spacecraft, but an unknown floating potential V0 of the spacecraft with respect to the plasma. We shall assume the bias voltages to be 2.5, 4.0, 5.5 and 7.0 volts. In the below example, we first construct the currents for such probes by assuming a floating potential and a set of plasma parameters, but we do not use this knowledge in the inversion. We do, however, make an initial guess x0 which we believe are somewhat close to the answer:

>>> from langmuir import *
>>> from scipy.optimize import leastsq

>>> geo = Cylinder(1e-3, 25e-3)
>>> V0 = -0.5
>>> V = np.array([2.5, 4.0, 5.5, 7.0])
>>> I = OML_current(geo, Species(n=120e10, T=1000), V+V0)

>>> def residual(x):
>>>     n, V0 = x
>>>     return OML_current(geo, Species(n=n, T=1500), V+V0) - I

>>> x0 = [10e10, -0.3]
>>> x, c = leastsq(residual, x0)
>>> n, V0 = x

>>> print(n)
1199899818493.931

>>> print(V0)
-0.5417515655165968

The method correctly determined the density to be 120e10. However, the floating potential V0 is off by almost ten percent. The reason is that the temperature is considered unknown, and assumed to be 1500K when solving the problem, while it is actually 1000K. Since we have four measurements (four equations) and only two unknowns, it is tempting to also include the temperature as an unknown parameter and try to solve for it. However, if this is done the least squares algorithm will fail miserably. The reason is that the set of equations arising for the attracted-species current of cylindrical probes are singular and cannot be solved for even analytically. Fortunately, both the temperature and floating potential can be eliminated from the equation when analytically solving for the density, and similarly it also works to obtain the density from the least squares algorithm. Since the floating potential and temperature represent a coupled unknown which cannot be solved for, an error in assuming one is reflected as an error in the other.

This demonstrates the usefulness as well as challenges and subtleties of solving inverse Langmuir problems.