plasma_plots.theory.parameters

Plasma parameters in SI units: frequencies, lengths, speeds and Struphy’s units.

Densities are in m⁻³, magnetic fields in T, temperatures in eV (k_B T / e, so 1 eV is 11604.5 K) unless a function says otherwise, and the results are in SI units: angular frequencies in rad/s, lengths in m, speeds in m/s. Ions have the mass mass_number × m_p (proton masses, as in Struphy and the NRL Plasma Formulary) and the charge charge_number × e.

Constants

CODATA 2018 values, the same as Struphy’s struphy.physics.physics.ConstantsOfNature:

• elementary_charge (C), electron_mass and proton_mass (kg),
• vacuum_permittivity ε₀ (F/m), vacuum_permeability μ₀ (N/A²), speed_of_light c (m/s), boltzmann_constant k_B (J/K),
• ev_to_joule (J per eV) and ev_to_kelvin (K per eV).

Conventions

• Thermal speed: v_th = √(T/m) by default (the standard deviation of a 1-D Maxwellian, the NRL and Struphy convention); [thermal_speed()][thermal_speed] also gives √(2T/m) and the mean speed.
• Cyclotron frequency: Ω = q|B|/m, which has the sign of the charge you pass. The default charge is +e, so the default is the magnitude; pass charge=-elementary_charge for the signed electron frequency.

References

J. D. Huba, NRL Plasma Formulary (Naval Research Laboratory, 2019).

F. F. Chen, Introduction to Plasma Physics and Controlled Fusion, 3rd ed. (Springer, 2016).

Examples

The electron plasma frequency and Debye length of a 1e19 m⁻³, 100 eV plasma:

>>> print(
...     f"{plasma_frequency(1e19):.3e} rad/s, {debye_length(1e19, 100.0):.3e} m"
... )
1.784e+11 rad/s, 2.351e-05 m

Attributes

Name	Description
STRUPHY_UNIT_SYMBOLS	No description.
boltzmann_constant	No description.
electron_mass	No description.
elementary_charge	No description.
ev_to_joule	No description.
ev_to_kelvin	No description.
proton_mass	No description.
speed_of_light	No description.
vacuum_permeability	No description.
vacuum_permittivity	No description.

Functions

Name	Description
alfven_speed	Compute the Alfvén speed v_A = B / √(μ₀ n m_i), with m_i = mass_number × m_p.
cyclotron_frequency	Compute the cyclotron (gyro-) frequency Ω = q|B|/m.
debye_length	Compute the electron Debye length λ_D = √(ε₀ T / (n e²)).
inertial_length	Compute the inertial length (skin depth) d = c / ω_p.
larmor_radius	Compute the Larmor (gyro-) radius ρ = m v⊥ / (|q| B).
lower_hybrid_frequency	Compute the lower hybrid frequency, 1/ω_LH² = 1/(Ω_i² + ω_pi²) + 1/(|Ω_e| Ω_i).
plasma_beta	Compute the plasma beta β = n T / (B²/(2μ₀)), the ratio of thermal to magnetic pressure.
plasma_frequency	Compute the plasma frequency ω_p = √(n q² / (ε₀ m)).
plasma_parameter	Compute the plasma parameter N_D = (4π/3) n λ_D³, the number of electrons in a Debye sphere.
sound_speed	Compute the ion sound speed c_s = √((γ_e Z T_e + γ_i T_i) / m_i).
struphy_equation_parameters	Compute the equation parameters α, ε and κ of one Struphy species.
struphy_units	Compute the units of Struphy's normalization from its base units.
thermal_speed	Compute the thermal speed of a Maxwellian of temperature T.
upper_hybrid_frequency	Compute the upper hybrid frequency ω_UH = √(ω_pe² + Ω_e²).

STRUPHY_UNIT_SYMBOLSattributemodule attribute#

View source

STRUPHY_UNIT_SYMBOLS = {
  'x': 'm',
  'B': 'T',
  'n': 'm⁻³',
  'kBT': 'keV',
  'v': 'm/s',
  't': 's',
  'frequency': '1/s',
  'p': 'Pa',
  'rho': 'kg/m³',
  'j': 'A/m²'
}

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boltzmann_constantattributemodule attribute#

View source

boltzmann_constant = 1.380649e-23

electron_massattributemodule attribute#

View source

electron_mass = 9.1093837015e-31

elementary_chargeattributemodule attribute#

View source

elementary_charge = 1.602176634e-19

ev_to_jouleattributemodule attribute#

View source

ev_to_joule = elementary_charge

ev_to_kelvinattributemodule attribute#

View source

ev_to_kelvin = elementary_charge / boltzmann_constant

proton_massattributemodule attribute#

View source

proton_mass = 1.67262192369e-27

speed_of_lightattributemodule attribute#

View source

speed_of_light = 299792458.0

vacuum_permeabilityattributemodule attribute#

View source

vacuum_permeability = 1.25663706212e-06

vacuum_permittivityattributemodule attribute#

View source

vacuum_permittivity = 8.8541878128e-12

alfven_speedfunction#

View source

def alfven_speed(field, density, mass_number=1, relativistic=False)

Compute the Alfvén speed v_A = B / √(μ₀ n m_i), with m_i = mass_number × m_p.

Parameters

Name	Type	Default	Description
field	float or array_like	required	Magnetic field B, in T.
density	float or array_like	required	Ion density n, in m⁻³ (the mass density is n m_i; electrons are neglected).
mass_number	float or array_like	1	Ion mass in proton masses, as in Struphy. Default: 1.
relativistic	bool	False	If true, return the relativistic v_A / √(1 + v_A²/c²). Default: False.

Returns

float or numpy.ndarray

v_A, in m/s.

References

NRL Plasma Formulary: v_A = 2.18e11 × (B/G) / √(μ n_i/cm⁻³) cm/s.

Examples

>>> # deuterium, 1 T
>>> print(f"{alfven_speed(1.0, 1e20, mass_number=2):.4g} m/s")
1.542e+06 m/s

cyclotron_frequencyfunction#

View source

def cyclotron_frequency(field, mass=electron_mass, charge=elementary_charge)

Compute the cyclotron (gyro-) frequency Ω = q|B|/m.

Parameters

Name	Type	Default	Description
field	float or array_like	required	Magnetic field B, in T (its sign doesn’t matter).
mass	float or array_like	electron_mass	Particle mass m, in kg. Default: the electron mass.
charge	float or array_like	elementary_charge	Particle charge q, in C. Ω has its sign: the default +e gives the magnitude, −e the signed electron frequency (negative, for the left-handed electron gyration).

Returns

float or numpy.ndarray

Ω in rad/s.

References

NRL Plasma Formulary: f_ce = Ω_e/2π = 2.80 MHz × B/G = 28.0 GHz × B/T.

Examples

>>> print(f"{cyclotron_frequency(1.0) / (2 * np.pi):.4g} Hz")
2.799e+10 Hz
>>> print(f"{cyclotron_frequency(1.0, charge=-elementary_charge):.4g} rad/s")
-1.759e+11 rad/s

debye_lengthfunction#

View source

def debye_length(density, temperature)

Compute the electron Debye length λ_D = √(ε₀ T / (n e²)).

Parameters

Name	Type	Description
density	float or array_like	Electron density n, in m⁻³.
temperature	float or array_like	Electron temperature T, in eV.

Returns

float or numpy.ndarray

λ_D, in m. With the default thermal_speed(), λ_D = v_th / ω_pe.

References

NRL Plasma Formulary: λ_D = 7.43e2 × √(T/eV) / √(n/cm⁻³) cm.

Examples

>>> print(f"{debye_length(1e6, 1.0):.4g} m")  # 1 cm⁻³, 1 eV
7.434 m

inertial_lengthfunction#

View source

def inertial_length(density, mass=electron_mass, charge=elementary_charge)

Compute the inertial length (skin depth) d = c / ω_p.

Parameters

Name	Type	Default	Description
density	float or array_like	required	Number density n of the species, in m⁻³.
mass	float or array_like	electron_mass	Particle mass m, in kg. Default: the electron mass.
charge	float or array_like	elementary_charge	Particle charge q, in C. Default: e.

Returns

float or numpy.ndarray

d, in m. For ions, d_i = v_A / Ω_i.

References

NRL Plasma Formulary: c/ω_pe = 5.31e5 / √(n_e/cm⁻³) cm, c/ω_pi = 2.28e7 √μ / (Z √(n_i/cm⁻³)) cm.

Examples

>>> print(f"{inertial_length(1e20):.4g} m")  # electrons
0.0005314 m
>>> print(f"{inertial_length(1e20, mass=proton_mass):.4g} m")  # protons
0.02277 m

larmor_radiusfunction#

View source

def larmor_radius(field, temperature=None, perpendicular_speed=None, mass=electron_mass, charge=elementary_charge)

Compute the Larmor (gyro-) radius ρ = m v⊥ / (|q| B).

Give either the temperature, for the thermal Larmor radius with v⊥ = √(T/m), or the perpendicular speed.

Parameters

Name	Type	Default	Description
field	float or array_like	required	Magnetic field B, in T.
temperature	float or array_like	None	Temperature T, in eV; then v⊥ = √(T/m) (the NRL convention).
perpendicular_speed	float or array_like	None	The perpendicular speed v⊥, in m/s.
mass	float or array_like	electron_mass	Particle mass m, in kg. Default: the electron mass.
charge	float or array_like	elementary_charge	Particle charge q, in C (its sign doesn’t matter). Default: e.

Returns

float or numpy.ndarray

ρ, in m.

Raises

ValueError

If not exactly one of temperature and perpendicular_speed is given.

References

NRL Plasma Formulary: r_e = 2.38 × √(T_e/eV) / (B/G) cm, r_i = 1.02e2 × √μ √(T_i/eV) / (Z B/G) cm.

Examples

>>> # 1 keV electron, 1 T
>>> print(f"{larmor_radius(1.0, temperature=1e3):.4g} m")
7.54e-05 m
>>> radius = larmor_radius(2.0, perpendicular_speed=1e6, mass=proton_mass)
>>> print(f"{radius:.4g} m")
0.00522 m

lower_hybrid_frequencyfunction#

View source

def lower_hybrid_frequency(density, field, mass_number=1, charge_number=1)

Compute the lower hybrid frequency, 1/ω_LH² = 1/(Ω_i² + ω_pi²) + 1/(|Ω_e| Ω_i).

Parameters

Name	Type	Default	Description
density	float or array_like	required	Electron density n_e, in m⁻³ (the ion density is n_e / Z).
field	float or array_like	required	Magnetic field B, in T.
mass_number	float or array_like	1	Ion mass in proton masses. Default: 1.
charge_number	float or array_like	1	Ion charge number Z. Default: 1.

Returns

float or numpy.ndarray

ω_LH, in rad/s. For ω_pi ≫ Ω_i it tends to √(|Ω_e| Ω_i).

References

Chen, Introduction to Plasma Physics, section 4.11 (lower hybrid frequency).

NRL Plasma Formulary: ω_LH = [(Ω_i Ω_e)⁻¹ + ω_pi⁻²]^(−1/2).

Examples

>>> print(f"{lower_hybrid_frequency(1e20, 1.0) / (2 * np.pi):.4g} Hz")
6.237e+08 Hz

plasma_betafunction#

View source

def plasma_beta(density, temperature, field)

Compute the plasma beta β = n T / (B²/(2μ₀)), the ratio of thermal to magnetic pressure.

For several species, add their β (or pass the summed pressure n T of all species).

Parameters

Name	Type	Description
density	float or array_like	Number density n, in m⁻³.
temperature	float or array_like	Temperature T, in eV.
field	float or array_like	Magnetic field B, in T.

Returns

float or numpy.ndarray

β (dimensionless).

References

NRL Plasma Formulary: β = 8π n k T / B² = 4.03e-11 n T / B² (cgs, T in eV, B in G).

Examples

>>> print(f"{plasma_beta(1e20, 1e4, 5.0):.4f}")  # 10 keV, 5 T
0.0161

plasma_frequencyfunction#

View source

def plasma_frequency(density, mass=electron_mass, charge=elementary_charge)

Compute the plasma frequency ω_p = √(n q² / (ε₀ m)).

Parameters

Name	Type	Default	Description
density	float or array_like	required	Number density n of the species, in m⁻³.
mass	float or array_like	electron_mass	Particle mass m, in kg. Default: the electron mass.
charge	float or array_like	elementary_charge	Particle charge q, in C (its sign doesn’t matter). Default: e.

Returns

float or numpy.ndarray

ω_p in rad/s (divide by 2π for Hz).

References

NRL Plasma Formulary: f_pe = ω_pe/2π = 8.98 kHz × √(n_e / cm⁻³).

Examples

>>> print(f"{plasma_frequency(1e18) / (2 * np.pi):.4g} Hz")
8.979e+09 Hz
>>> # deuterons
>>> print(f"{plasma_frequency(1e20, mass=2 * proton_mass):.4g} rad/s")
9.309e+09 rad/s

plasma_parameterfunction#

View source

def plasma_parameter(density, temperature)

Compute the plasma parameter N_D = (4π/3) n λ_D³, the number of electrons in a Debye sphere.

Parameters

Name	Type	Description
density	float or array_like	Electron density n, in m⁻³.
temperature	float or array_like	Electron temperature T, in eV.

Returns

float or numpy.ndarray

N_D (dimensionless). A plasma is weakly coupled for N_D ≫ 1.

References

NRL Plasma Formulary: (4π/3) n λ_D³ = 1.72e9 × (T/eV)^(3/2) / √(n/cm⁻³).

Examples

>>> print(f"{plasma_parameter(1e6, 1.0):.3g}")  # 1 cm⁻³, 1 eV
1.72e+09

sound_speedfunction#

View source

def sound_speed(electron_temperature, ion_temperature=0.0, mass_number=1, charge_number=1, electron_gamma=1.0, ion_gamma=3.0)

Compute the ion sound speed c_s = √((γ_e Z T_e + γ_i T_i) / m_i).

Parameters

Name	Type	Default	Description
electron_temperature	float or array_like	required	Electron temperature T_e, in eV.
ion_temperature	float or array_like	0.0	Ion temperature T_i, in eV. Default: 0 (cold ions).
mass_number	float or array_like	1	Ion mass in proton masses. Default: 1.
charge_number	float or array_like	1	Ion charge number Z. Default: 1.
electron_gamma	float	1.0	Adiabatic index γ_e of the electrons. Default: 1 (isothermal electrons).
ion_gamma	float	3.0	Adiabatic index γ_i of the ions. Default: 3 (1-D adiabatic ions).

Returns

float or numpy.ndarray

c_s, in m/s.

References

NRL Plasma Formulary: C_s = 9.79e5 × √(γ Z T_e / (μ eV)) cm/s.

Chen, Introduction to Plasma Physics, section 4.9 (ion acoustic waves).

Examples

>>> print(f"{sound_speed(1.0):.4g} m/s")
9787 m/s

struphy_equation_parametersfunction#

View source

def struphy_equation_parameters(units, charge_number=1, mass_number=1)

Compute the equation parameters α, ε and κ of one Struphy species.

Mirrors struphy.models.species.Species.EquationParameters, with ω_p and Ω_c of the species at the unit density and unit field: α = ω_p/Ω_c, ε = 1/(Ω_c t) and κ = ω_p t.

Parameters

Name	Type	Default	Description
units	dict	required	The units from struphy_units().
charge_number	float	1	Charge number Z of the species. Default: 1.
mass_number	float	1	Mass number A of the species, in proton masses. Default: 1.

Returns

dict

alpha, epsilon and kappa.

References

Struphy source: struphy/models/species.py (Species.EquationParameters).

Examples

>>> units = struphy_units(velocity_scale="alfvén", mass_number=1)
>>> params = struphy_equation_parameters(units)
>>> print({key: f"{value:.4g}" for key, value in params.items()})
{'alpha': '137.4', 'epsilon': '0.02277', 'kappa': '6036'}

struphy_unitsfunction#

View source

def struphy_units(x=1.0, B=1.0, n=1.0, kBT=None, velocity_scale='light', mass_number=None, charge_number=None)

Compute the units of Struphy’s normalization from its base units.

Mirrors struphy.physics.physics.Units.derive_units (the base units are those of struphy.io.options.BaseUnits in a parameter file, the velocity scale is the model’s velocity_scale and the mass and charge numbers are those of the model’s bulk species):

• velocity v: c for "light"; B / √(A m_p n μ₀) for "alfvén"; (Z e B / (A m_p)) x for "cyclotron"; √(kBT / (A m_p)) for "thermal"; 1 m/s for None,
• time t = x / v, and the frequency 1/t (normalized angular frequencies ω t are in rad/t),
• with a bulk species (mass_number given): pressure p = A m_p n v² (= B²/μ₀ for "alfvén"), mass density ρ = A m_p n, current density j = e n v.

Parameters

Name	Type	Default	Description
x	float	1.0	Unit of length, in m (BaseUnits.x). Default: 1.
B	float	1.0	Unit of magnetic field, in T (BaseUnits.B). Default: 1.
n	float	1.0	Unit of number density, in 1e20 m⁻³ as in BaseUnits.n (not in m⁻³). Default: 1.
kBT	float	None	Unit of thermal energy, in keV as in BaseUnits.kBT; needed for "thermal".
velocity_scale	('light', 'alfvén', 'cyclotron', 'thermal', None)	"light"	The model’s velocity scale ("alfven" is accepted too). Default: "light".
mass_number	float	None	Mass number A of the bulk species, in proton masses; needed except for "light" and None.
charge_number	float	None	Charge number Z of the bulk species; needed for "cyclotron".

Returns

dict

Floats, in SI units except kBT: x (m), B (T), n (m⁻³), kBT (keV or None), v (m/s), t (s), frequency (1/s), p (Pa), rho (kg/m³) and j (A/m²); the last three are None without a bulk species. STRUPHY_UNIT_SYMBOLS holds these unit names.

Raises

ValueError

If velocity_scale is unknown or a quantity it needs is missing.

References

Struphy source: struphy/physics/physics.py (Units, ConstantsOfNature) and struphy/io/options.py (BaseUnits); the “normalization” page of Struphy’s docs.

Examples

>>> units = struphy_units(
...     x=1.0, B=1.0, n=1.0, velocity_scale="alfvén", mass_number=2
... )
>>> print(f"v = {units['v']:.4g} m/s, t = {units['t']:.4g} s")
v = 1.542e+06 m/s, t = 6.484e-07 s
>>> print(f"p = {units['p']:.4g} Pa")  # B²/μ₀
p = 7.958e+05 Pa

thermal_speedfunction#

View source

def thermal_speed(temperature, mass=electron_mass, convention='sqrt(T/m)')

Compute the thermal speed of a Maxwellian of temperature T.

Parameters

Name	Type	Default	Description
temperature	float or array_like	required	Temperature T, in eV.
mass	float or array_like	electron_mass	Particle mass m, in kg. Default: the electron mass.
convention	('sqrt(T/m)', 'sqrt(2T/m)', 'mean')	"sqrt(T/m)"	"sqrt(T/m)": the standard deviation of each velocity component, as in the NRL formulary and in Struphy’s Maxwellians and velocity_scale="thermal"; "sqrt(2T/m)": the most probable speed, common in kinetic theory; "mean": the mean speed √(8T/(πm)). Default: "sqrt(T/m)".

Returns

float or numpy.ndarray

The thermal speed, in m/s.

Raises

ValueError

If convention is unknown.

References

NRL Plasma Formulary: v_Te = √(k T_e / m_e) = 4.19e7 × √(T_e / eV) cm/s.

Examples

>>> print(f"{thermal_speed(1.0):.4g} m/s")
4.194e+05 m/s
>>> print(f"{thermal_speed(1.0, convention='sqrt(2T/m)'):.4g} m/s")
5.931e+05 m/s

upper_hybrid_frequencyfunction#

View source

def upper_hybrid_frequency(density, field)

Compute the upper hybrid frequency ω_UH = √(ω_pe² + Ω_e²).

Parameters

Name	Type	Description
density	float or array_like	Electron density n_e, in m⁻³.
field	float or array_like	Magnetic field B, in T.

Returns

float or numpy.ndarray

ω_UH, in rad/s.

References

Chen, Introduction to Plasma Physics, section 4.10 (upper hybrid frequency).

Examples

>>> print(f"{upper_hybrid_frequency(1e19, 1.0):.4g} rad/s")
2.505e+11 rad/s