plasmapy.formulary.frequencies.lower_hybrid_frequency(B: Quantity, n_i: Quantity, ion: str | int | integer | Particle | CustomParticle | Quantity, *, to_hz=False) Quantity[source]

Return the lower hybrid frequency.

Aliases: wlh_

  • B (Quantity) – The magnetic field magnitude in units convertible to tesla.

  • n_i (Quantity) – Ion number density.

  • ion (Particle) – Representation of the ion species (e.g., 'p+' for protons, 'D+' for deuterium, or 'He-4 +1' for singly ionized helium-4). If no charge state information is provided, then the ions are assumed to be singly charged.


omega_lh – The lower hybrid frequency in radians per second.

Return type:


  • TypeError – If either of B or n_i is not a Quantity, or ion is of an inappropriate type.

  • UnitConversionError – If either of B or n_i is in incorrect units.

  • ValueError – If either of B or n_i contains invalid values or are of incompatible dimensions, or ion cannot be used to identify an ion or isotope.


UnitsWarning – If units are not provided, SI units are assumed.


The lower hybrid frequency is given through the relation

\[\frac{1}{ω_{lh}^2} = \frac{1}{ω_{ci}^2 + ω_{pi}^2} + \frac{1}{ω_{ci}ω_{ce}}\]

where \(ω_{ci}\) is the ion gyrofrequency, \(ω_{ce}\) is the electron gyrofrequency, and \(ω_{pi}\) is the ion plasma frequency.

The lower hybrid frequency constitutes a resonance for electromagnetic waves in magnetized plasmas, namely for the X-mode. These are waves with their wave electric field being perpendicular to the background magnetic field. For the lower hybrid frequency, ion and electron dynamics both play a role. As the name suggests, it has a lower frequency compared to the upper hybrid frequency. It can play an important role for heating and current drive in fusion plasmas.


>>> import astropy.units as u
>>> lower_hybrid_frequency(0.2 * u.T, n_i=5e19 * u.m**-3, ion="D+")
<Quantity 5.78372...e+08 rad / s>
>>> lower_hybrid_frequency(0.2 * u.T, n_i=5e19 * u.m**-3, ion="D+", to_hz=True)
<Quantity 92050879.3... Hz>

to_hz (bool) – Set True to convert function output from angular frequency to Hz