Ponderomotive force
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In
The ponderomotive force Fp is expressed by
which has units of newtons (in SI units) and where e is the electrical charge of the particle, m is its mass, ω is the angular frequency of oscillation of the field, and E is the amplitude of the electric field. At low enough amplitudes the magnetic field exerts very little force.
This equation means that a charged particle in an inhomogeneous oscillating field not only oscillates at the frequency of ω of the field, but is also accelerated by Fp toward the weak field direction. This is a rare case in which the direction of the force does not depend on whether the particle is positively or negatively charged.
Etymology
The term ponderomotive comes from the Latin ponder- (meaning weight) and the english motive (having to do with motion).[2]
Derivation
The derivation of the ponderomotive force expression proceeds as follows.
Consider a particle under the action of a non-uniform electric field oscillating at frequency in the x-direction. The equation of motion is given by:
neglecting the effect of the associated oscillating magnetic field.
If the length scale of variation of is large enough, then the particle trajectory can be divided into a slow time (secular) motion and a fast time (micro)motion:[3]
where is the slow drift motion and represents fast oscillations. Now, let us also assume that . Under this assumption, we can use Taylor expansion on the force equation about , to get:
- , and because is small, , so
On the time scale on which oscillates, is essentially a constant. Thus, the above can be integrated to get:
Substituting this in the force equation and averaging over the timescale, we get,
Thus, we have obtained an expression for the drift motion of a charged particle under the effect of a non-uniform oscillating field.
Time averaged density
Instead of a single charged particle, there could be a gas of charged particles confined by the action of such a force. Such a gas of charged particles is called
where is the ponderomotive potential and is given by
Generalized ponderomotive force
Instead of just an oscillating field, a permanent field could also be present. In such a situation, the force equation of a charged particle becomes:
To solve the above equation, we can make a similar assumption as we did for the case when . This gives a generalized expression for the drift motion of the particle:
Applications
The idea of a ponderomotive description of particles under the action of a time-varying field has applications in areas like:
- High harmonic generation
- Plasma acceleration of particles
- Plasma propulsion engine especially the Electrodeless plasma thruster
- Quadrupole ion trap
- Terahertz time-domain spectroscopy as a source of high energy THz radiation in laser-induced air plasmas
The quadrupole ion trap uses a linear function along its principal axes. This gives rise to a harmonic oscillator in the secular motion with the so-called trapping frequency , where are the charge and mass of the ion, the peak amplitude and the frequency of the radiofrequency (rf) trapping field, and the ion-to-electrode distance respectively.[5] Note that a larger rf frequency lowers the trapping frequency.
The ponderomotive force also plays an important role in laser induced plasmas as a major density lowering factor.
Often, however, the assumed slow-time independency of is too restrictive, an example being the ultra-short, intense laser pulse-plasma(target) interaction. Here a new ponderomotive effect comes into play, the ponderomotive memory effect.[6] The result is a weakening of the ponderomotive force and the generation of wake fields and ponderomotive streamers.[7] In this case the fast-time averaged density becomes for a Maxwellian plasma: , where and .
References
- General
- Schmidt, George (1979). Physics of High Temperature Plasmas, second edition. Academic Press. p. 47. ISBN 978-0-12-626660-3.
- Citations
- doi:10.1063/1.367318.
- ^ "ponderomotive". Retrieved 2023-09-27.
- ISBN 0-471-09045-X
- ^ V. B. Krapchev, Kinetic Theory of the Ponderomotive Effects in a Plasma, Phys. Rev. Lett. 42, 497 (1979), http://prola.aps.org/abstract/PRL/v42/i8/p497_1
- ^ S. R. Jefferts, C. Monroe, A. S. Barton, and D. J. Wineland, Paul Trap for Optical Frequency Standards, IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 44, NO. 2 (1995)
- ^ H. Schamel and Ch. Sack,"Existence of a Time-dependent Heat Flux-related Ponderomotive Effect", Phys. Fluids 23,1532(1980), doi:10.1063/1.863165
- ^ U. Wolf and H. Schamel,"Wake-field Generation by the Ponderomotive Memory Effect", Phys. Rev.E 56,4656(1997), doi:10.1103/PhysRevE.56.4656
Journals
- Cary, J. R.; Kaufman, A. N. (1981). "Ponderomotive effects in collisionless plasma: A Lie transform approach". Phys. Fluids. 24 (7): 1238. S2CID 56314589.
- Grebogi, C.; Littlejohn, R. G. (1984). "Relativistic ponderomotive Hamiltonian". Phys. Fluids. 27 (8): 1996. doi:10.1063/1.864855.
- Morales, G. J.; Lee, Y. C. (1974). "Ponderomotive-Force Effects in a Nonuniform Plasma". Phys. Rev. Lett. 33 (17): 1016–1019. .
- Lamb, B. M.; Morales, G. J. (1983). "Ponderomotive effects in nonneutral plasmas". Phys. Fluids. 26 (12): 3488. doi:10.1063/1.864132. Archived from the originalon September 23, 2017.
- Shah, K.; Ramachandran, H. (2008). "Analytic, nonlinearly exact solutions for an rf confined plasma". Phys. Plasmas. 15 (6): 062303. doi:10.1063/1.2926632. Archived from the originalon 2013-02-23.
- Bucksbaum, P. H.; Freeman, R. R.; Bashkansky, M.; McIlrath, T. J. (1987). "Role of the ponderomotive potential in above-threshold ionization". Journal of the Optical Society of America B. 4 (5): 760. .