Ponderomotive force

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Classical motion of a trapped ion in a radiofrequency (rf) quadrupole (Paul) trap. A quadrupole electric field is displayed for reference, which oscillates at a given frequency <math>\omega</math>. The blue line represents the ion path in the transversal (or radial) direction of a linear trap, while the orange line is the secular (slow) motion resulting from the ponderomotive force due to the electric field onto the ion. Micromotion is the fast oscillatory motion around the secular motion <ref>D. J. Berkeland; J. D. Miller; J. C. Bergquist; W. M. Itano; D. J. Wineland (1998). "Minimization of ion micromotion in a Paul trap". Journal of Applied Physics. American Institute of Physics. 83 (10): 5025. doi:10.1063/1.367318.</ref>

In physics, a ponderomotive force is a nonlinear force that a charged particle experiences in an inhomogeneous oscillating electromagnetic field. It causes the particle to move towards the area of the weaker field strength, rather than oscillating around an initial point as happens in a homogeneous field. This occurs because the particle sees a greater magnitude of force during the half of the oscillation period while it is in the area with the stronger field. The net force during its period in the weaker area in the second half of the oscillation does not offset the net force of the first half, and so over a complete cycle this makes the particle move towards the area of lesser force.

The ponderomotive force Fp is expressed by

<math>\mathbf{F}_{\text{p}}=</math><math>-\frac{e^2}{4 m \omega^2}</math><math>\nabla</math><math>(E^2)</math>

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).<ref>"ponderomotive". Retrieved 2023-09-27.</ref>

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 <math>\omega</math> in the x-direction. The equation of motion is given by:

<math>\ddot{x}=g(x)\cos(\omega t),</math>

neglecting the effect of the associated oscillating magnetic field.

If the length scale of variation of <math>g(x)</math> is large enough, then the particle trajectory can be divided into a slow time (secular) motion and a fast time (micro)motion:<ref>Introduction to Plasma Theory, second edition, by Nicholson, Dwight R., Wiley Publications (1983), ISBN 0-471-09045-X</ref>

<math>x=x_0+x_1</math>

where <math>x_0</math> is the slow drift motion and <math>x_1</math> represents fast oscillations. Now, let us also assume that <math>x_1 \ll x_0</math>. Under this assumption, we can use Taylor expansion on the force equation about <math>x_0</math>, to get:

<math>\ddot{x}_0+\ddot{x}_1=\left[g(x_0)+x_1 g'(x_0)\right]\cos(\omega t)</math>
<math>\ddot{x}_0 \ll \ddot{x}_1</math>, and because <math>x_1</math> is small, <math> g(x_0) \gg x_1 g'(x_0) </math>, so
<math>\ddot{x}_1=g(x_0)\cos(\omega t)</math>

On the time scale on which <math>x_1</math> oscillates, <math>x_0</math> is essentially a constant. Thus, the above can be integrated to get:

<math>x_1=-\frac{g(x_0)}{\omega^2} \cos(\omega t)</math>

Substituting this in the force equation and averaging over the <math>2\pi / \omega</math> timescale, we get,

<math>\ddot{x}_0=-\frac{g(x_0)g'(x_0)}{2 \omega^2}</math>
<math>\Rightarrow \ddot{x}_0=-\frac{1}{4 \omega^2}\left.\frac{d}{dx}\left[g(x)^2\right]\right|_{x=x_0}</math>

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 plasma. The distribution function and density of the plasma will fluctuate at the applied oscillating frequency and to obtain an exact solution, we need to solve the Vlasov Equation. But, it is usually assumed that the time averaged density of the plasma can be directly obtained from the expression for the force expression for the drift motion of individual charged particles:<ref>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</ref>

<math>\bar{n}(x)=n_0 \exp \left[-\frac{e}{\kappa T} \Phi_{\text{P}} (x)\right]</math>

where <math>\Phi_{\text{P}}</math> is the ponderomotive potential and is given by

<math>\Phi_{\text{P}} (x)=\frac{ m}{4 \omega^2} \left[g (x)\right]^2</math>

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:

<math>\ddot{x}=h(x)+g(x)\cos(\omega t)</math>

To solve the above equation, we can make a similar assumption as we did for the case when <math>h(x)=0</math>. This gives a generalized expression for the drift motion of the particle:

<math>\ddot{x}_0=h(x_0)-\frac{g(x_0)g'(x_0)}{2 \omega^2}</math>

Applications

The idea of a ponderomotive description of particles under the action of a time-varying field has applications in areas like:

The quadrupole ion trap uses a linear function <math> g(x) = x</math> along its principal axes. This gives rise to a harmonic oscillator in the secular motion with the so-called trapping frequency <math>\Omega \propto \frac{qV}{m \omega d_0^2}</math>, where <math>q, m, V, \omega, d_0</math> 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 <ref>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)</ref>. 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 <math> \Phi_P</math> 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.<ref>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</ref> The result is a weakening of the ponderomotive force and the generation of wake fields and ponderomotive streamers.<ref>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</ref> In this case the fast-time averaged density becomes for a Maxwellian plasma: <math>\bar n(x,t)= n_0 e^{-\Psi} [1 + \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{+\infty} dv e^{-v^2/2} M(x,v,t)]</math>, where <math> M(x,v,t):=\int_{-\infty}^t d\tau\partial_\tau \Psi(x-v(t-\tau),\tau)</math> and <math> \Psi(x,t):=\frac {e}{\kappa T}\Phi_P(x,t)</math>.

References

General
  • Schmidt, George (1979). Physics of High Temperature Plasmas, second edition. Academic Press. p. 47. ISBN 978-0-12-626660-3.
Citations
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Journals

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