05 - LEP Principles

Low Energy Plasma Instruments

References:

Q1. Retarding Potential Analyzers

In the ionosphere, mount along ram velocity, measure species densities (in like atomic mass groups)

  • Ram speed (7.5km/s) is high or supersonic relative to ion thermal speed or motion
  • Spacecraft (sc) charging is negative and small relative to motional energy
  • I-V curve has steps at \(qV_{ret} = \frac{1}{2}m(V_{sr}+V_r)^2 – q \phi_s\) ; \(\phi_s\) = sensor potential (w/r/t to plasma), \(V_{s_r}\) = sc ram speed, \(V_r\) = ion speed

Homework Q#1

  • Show that the thermal width of the steps is \(m V_{sr} V_{th}\), where \(V_{th}\) is the ion species thermal speed.
  • Show that for sensor potential of –0.8V, the step functions are at ~1.1V for H+ and ~5.5V for O+.

Figure 1. Schematic cross-sectio onf the planar RPA sensor illustrating the arrangement of internal grids required to perform the retarding potential analysis of thermal ions.

thermal width = \(\frac{1}{2} m (V_{th} + V_{s_r})^2 - \frac{1}{2} m (V_{th} - V_{s_r})^2 = m V_{s_r} V_{th}\)

step function for low temperature ions, taking \(V_{r}=0\)

\[V_{ret} = \frac{1}{2} \frac{m}{q}V_{s_r}^2 – \phi_s\]

Code 
from plasmapy.particles import Particle
import astropy.units as u
def calc_step_potential(phi, v_sr, particle: Particle):
    result = particle.mass * v_sr ** 2 / (2 * particle.charge) - phi
    return result.to(u.V)

phi = -0.8 * u.V
v_sr = 7.5 * u.km / u.s

proton = Particle('proton')
o_plus = Particle('O+')

print(f'Proton step potential: {calc_step_potential(phi, v_sr, proton)}')
print(f'O+ step potential: {calc_step_potential(phi, v_sr, o_plus)}')
Proton step potential: 1.0936161382302447 V
O+ step potential: 5.463469940275583 V

Q2. RPA/Ion Drift Meters

In the ionosphere, mounted along ram velocity, measure species velocity

  • G2 (positive) retards e.g. the lower energy H+, but allows the higher energy O+ through
  • Collimated beam comes through and falls asymmetrically (based on direction) on collectors
  • G6 (negative) suppresses electrons
  • G3-5 are grounded to remove distortions

Homework Q#2: Determine transverse velocity \(V_t\) as function of ram speed, W, D.

Figure 5. Ambient ions are accelerated through the spacecraft sheath potential and account of this acceleration decrease the apparent ion arrival angleas shown.

The ram velocity measued at the RPA can be related to the ion velocity at infinity \(V_r\) by

\[ \frac{1}{2} m V_{ram} = \frac{1}{2} m (V_{r} + V_s)^2 - q \phi_s\]

The ratio of two currents measured by the RPA is

\[ J_1 / J_2 = \frac{A_1}{A_2} = \frac{W-2 D \tan \alpha}{W+2 D \tan \alpha}\]

where \(A_1\) and \(A_2\) are the areas of the two collectors. The angle \(\alpha\) is the angle between the ion velocity and the normal to the collector surface. The ion velocity can be expressed as

\[ V_t = V_{ram} \tan \alpha \]

The log of the ratio of the fluxes, which are actually measured

\[ \log R = J_1 / J_2\]

Combining these equations gives

\[ V_t=\left[\left(V_r+V_s\right)^2-2 q \phi_s / m\right]^{\frac{1}{2}} \frac{W}{2 D}\left( 1-\log R \right) /\left(1+ \log R \right) \]

Q3. Magnetic Spectrographs

For low energy (less than a few keV) particles (left):

  • RPA provides velocity, temperature. Post-acceleration, Vpa, behind it allows refined measurement of m/q
  • Homework Q#3 Show that in LIMS: \(m/q=(B r_c)2/(2V_{pa})\), where B is magnetic field, rc particle magnetic radius

For high energy particles (right) the RPA is not used (they are hard to control with potentials):

  • Broom magnet clears electrons
  • High field bends the high energy ions
  • Ions that were not bent are assumed to be neutrals (ENAs)

LIMS on SCATHA

The energy of accelerated particle is: \[ \frac{1}{2} m v^2=q V_{p a} \]

Particles that are able to reach the detector must satisify: \[ m v^2 r_c=B q v \]

Combining Equations we get: \[ \begin{aligned} & v=\left(\frac{2 q}{m} V_{p a}\right)^{\frac{1}{2}} \\ \Rightarrow & m / q=\left(B r_c\right)^2 / 2 v_{p a} \end{aligned} \]

Q4. Electrostatic Analyzers

Electrostatic deflection “analyzes” distribution selecting particles with matching centrifugal and electrostatic forces

  • Analyser constant, \(K=R/\Delta\), where \(\Delta=R_2-R_1\); \(R=(R_2+R_1)/2\). Outer shell is at 0 Volts, inner shell at potential V.
  • Electrostatic deflection at the entrance aperture can measure incoming ions from different directions if spacecraft non-spinning
  • Homework Q#4 Show that the energy W of the particles of charge q incident on the MCP is \(W=-K q V /2\)

Particles that are able to reach the detector must satisify:

\[ m v^2 R=q E \]

where \(E\) is the electric field between \(R_2\) and \(R_1\), \(E \approx \frac{-V}{\Delta}\), so

\[ \begin{aligned} & m v^2=\frac{-R}{\Delta} q V = -K q V \\ \Rightarrow & W=\frac{1}{2} m v^2=-\frac{1}{2} K q V \end{aligned} \]

Q5. Time of Flight (post-analyzer) Mass Discrimination

Electrostatic deflection provides energy per charge: E/Q. Time of flight, t, provides the energy per mass E/M

  • Post-acceleration potential \(U_{ACC}\) is sufficient for optimal McP operation and timing electrons at foil. Measure flux at \(E/Q+ U_{ACC}\)
  • Electrons generated at carbon foil. Carbon foil results in passage of an energy fraction \(\alpha\)
  • Homework Q#5. Show \(M/Q=2a(E/Q + U_{ACC})/(d/t)^2\), where: Q=ion charge in units of elemental charge, e.

The velocity of ions after passing through the foil is

\[ v=d / t \]

Due to partial energy transfer after passing the foil

\[ \frac{1}{2} M v^2=\alpha\left(E + Q U_{ACC}\right) \]

Combining Equations we get:

\[ M / Q=2 \alpha\left(E / Q+U_{ACC}\right) /(d / t)^2 \]

References

Heelis, R. A., and W. B. Hanson. 1998. “Measurements of Thermal Ion Drift Velocity and Temperature Using Planar Sensors.” In, 61–71. American Geophysical Union (AGU). https://doi.org/10.1029/GM102p0061.