Charged Particle Dynamics


Contents

  1. Introduction
  2. Uniform, Constant Electric Field
  3. Uniform, Constant Magnetic Field
  4. Time-Varying Magnetic Field
  5. Uniform, Constant Electric and Magnetic Fields
  6. Time-varying Electric Fields
  7. Nonuniform, Constant Magnetic Fields
  8. Debye-Hückel Screening Distance
  9. References

Introduction

This article treats the motion of charged particles, mainly electrons or protons, in an external electromagnetic field, when classical, nonrelativistic mechanics is valid. The results can be applied to the ionosphere, upper atmosphere, interplanetary space, dilute plasmas and similar environments, as well as to the motion of electrons and ions in electronic apparatus. The motion of charged particles has had a wide field of application, including particle accelerators, electron microscopes, magnetrons and klystrons, cathode-ray and X-ray tubes, photomultipliers and gas discharges, as well as in geophysics and plasma physics. Nevertheless, it is no longer taught to undergraduate electrical engineers and physicists, except incidentally. Since it is fundamental to a lot of interesting physics and engineering, I thought a brief account would not be out of place.

There are many excellent opportunities to make errors in what follows, chiefly with minus signs and directions of rotations and drifts, so I recommend that the reader check the results for himself. There may even be more significant errors, since I often make mistakes, as well as the less significant misprints and mistypings. I hope you will inform me of any that you detect, so I can remove them.

The fundamental equations are Newton's Second Law, mdv/dt = F, and Lorentz's force equation, F = q [E + (1/c)(v x B)]. In these equations, m is the mass of the particle in grams, and q its charge in esu (statcoulomb). The vector velocity v is in cm/s, and the vector force F is in dyne. Vectors will be represented in boldface, and their magnitudes by the same letter in normal weight. E is the electric field in stavolt/cm. B is the magnetic field in gauss, and c is the speed of light, 2.998 x 1010 cm/s. These equations can be the basis of computer programs to calculate particle trajectories in the general case, a valuable resource, but such calculations do not give a general understanding of particle motion. This article is intended to foster such an understanding as an aid to reasoning about natural phenomena.

Conversion to practical units from the Gaussian units used here is easy. An esu of charge is (10/c) coulomb. One statvolt/cm is 300 V/cm or 30,000 V/m. 10,000 gauss is 1 tesla (Wb/m2). It is least confusing to convert units before and after a calculation, not during one. The dimensions of esu are g1/2cm3/2/s, and the dimensions of the electric and magnetic fields are g1/2/cm1/2-s. These are useful for working out the dimensions of parameters we define and checking the correctness of relations.

The electron and proton have charges of equal magnitude e = 4/803 x 10-10 esu, but of opposite sign. The electronic charge is negative. The magnitude e will always be taken as positive here. The mass of the electron is 9.109 x 10-28 g, and the mass of the proton is 1836.2 times greater, 1.6726 x 10-24 g. Subscripts "e" and "p" will be used to distinguish quantities referring to the electron and proton, respectively. The ratio of charge to mass for the electron is e/m = 5.273 x 1017 esu/g, and for the proton it is 1836.2 times smaller. The quantity e/mc that often appears is the ratio of the charge in emu to mass, and is 1.759 x 107 (cm/g)1/2 for the electron.

The proton is the nucleus of the hydrogen atom 1H, the most common atom in the universe, making up more than 90% of the sun's mass, and a similar fraction of primary cosmic rays. Most of the remaining 10% is 4He, which has four times the mass of the proton and twice its charge. Its nucleus is called the alpha particle. There are also rarer isotopes of both atoms, 2H, 3H, and 3He. In the sun and other places, these atoms are found as nuclei only, positive ions, while their electrons wander freely. The universe appears to be electrically neutral on the large scale, like any common substance. The lower atmosphere is dominated by diatomic, neutral nitrogen and oxygen. Higher up, photochemistry introduces new species, such as atomic oxygen O, NO and O3, together with their positive ions formed by knocking off an electron, or negative ions formed when a wandering electron attaches itself. Still higher up, even nitrogen becomes atomic, and hydrogen and helium become noticeable. The earth is bombarded by protons and electrons from the sun (the "solar wind") and by very energetic protons from deep space. Most of the interesting things happen where the pressure is similar to a good laboratory vacuum, and mean free paths of electrons are very long.

The positron is the antiparticle to the electron, with opposite charge but the same mass. It is rare and evanescent, occurring in nature only in cosmic ray showers and in radioactivity. The association of a positron and an electron is called positronium. Formed in an excited state, it emits radiation until the particles come close enough that mutual annihilation occurs and energy and momentum are conserved by the emission of two gamma-ray photons (one photon could not conserve momentum) of 0.51 MeV energy. Antihydrogen has recently been made in the laboratory by uniting a positron with an antiproton. It has the properties of ordinary hydrogen, but is a little hard to store. Antiparticles exist for every particle (some are their own antiparticle) with the same mass but charge of opposite sign.

The conditions of the motion of the particles discussed on this page are at the other limit of the spectrum from that discussed in the article on Magnetohydrodynamics, where the accent is on the collective behavior of the particles in the plasma, which is treated as a continuum. In the present article, the particles are individuals and move without the cooperation of their mates. What distinguishes these two regimes is the ratio of the scale of the motion to the mean free path of the particles. Here, we assume that the mean free path of an electron is much larger than, for example, the radius of its circular motion around the direction of the the magnetic field. The intermediate case is much more difficult than either limit. Unfortunately, many interesting plasmas find themselves in this perplexing state.

Uniform, Constant Electric Field

Motion in a uniform, constant electric field is motion with constant acceleration a = (q/m)E, which is in the direction of the field for a positive charge q and in the opposite direction for a negative charge. Choose the x-axis in the direction of the electric field, and the y-axis so that the xy-plane contains both the electric field and the initial velocity. Since the acceleration normal to this plane is zero, and the initial velocity normal to it is zero, the motion remains in this plane. We'll consider this problem below, but first a simpler one demonstrates some important points.

If we consider the particle to be at rest at the origin at t = 0, then the subsequent motion is x = (a/2)t2, and v(t) = at. The kinetic energy of the particle is U = mv2/2 = m(at)2/2 = qEx. The kinetic energy increase proportionally to the distance travelled (and is equal to the work done upon it by the force). The momentum of the particle, p = mv = mat = qEt, increased proportionally to the time (and is equal to the impulse of the force acting upon it).

The apparent mass of a particle increases according to m = mo[1 - (v/c)2]1/2. When v/c = 1/3, the mass is 6% greater than the rest mass, and we may consider that relativistic mechanics must be used for greater velocities for an accurate result. The corresponding kinetic energy is U = mc2/9 = qEx = qV, where V is the potential drop to achieve this velocity. The maximum potential drop before relativistic effects become important is then V = c2/9(q/m). For electrons, this is 189 statvolts, or 5.7 kV. For protons, it is 107 V. We note that we must look out for relativistic effects with electrons, but ordinarily protons are safe. The kinetic energy of a particle is often quoted in units of qV, where V is in volts. This unit is 1.602 x 10-12 erg. A 5.7 keV electron is borderline relativistic, as is a 10 MeV proton. mc2 is the rest energy of the electron, about 0.51 MeV.

The general problem is shown in the diagram at the right. The parabolic trajectories shown are described with time as a parameter by the two equations x = at2/2 + voxt and y = voyt, which you should have no trouble finding. The particle is considered to be at the origin at t = 0, but this is easily changed to any desired initial position by a simple translation of axes. The coordinates of the vertex are also shown in terms of the initial velocities. A positive charge approaches from the right, is constantly decelerated and finally turned around at the vertex, then comes through the origin with the intial velocities assumed, and accelerates back to the right. A negative charge behaves in a similar way, but comes in from the left and goes through the origin before reaching the vertex. All the familiar properties of the parabola can be used in analyzing this motion.

Uniform, Constant Magnetic Field

We see from the Lorentz equation that the force exerted by the magnetic field on a particle is always normal to its velocity. Therefore, the magnetic field cannot increase the energy of the particle, and so energy is conserved during movment in a purely magnetic, time-independent field. This holds whether the field is uniform or not. We also note that div B = 0 as well, which puts restrictions on the ways the field can vary in space. In the present case, this is satisfied a fortiori, as well as curl B = 0. Incidentally, the field B does not contain any part due to the motion of the particle itself, just as the electric field in the previous section did not contain a contribution from the field of the particle itself.

Newton's Law for this problem is dv/dt = (q/mc)v x B. A suggestion of how to solve the problem is shown in the figure at the right. If we re-write the equation of motion dv/dt - ω x v = 0, where ω = -qB/mc, the left-hand side is the time rate of change of v in a coordinate system rotating with angular velocity ω, so the equation says that this time rate of change is zero. That is, the velocity v' in the rotating system is a constant. Nothing could be simpler than this! If vo is resolved into components parallel and perpendicular to B, then the parallel component is not changed by the rotation and is constant. If we choose the axis so that the rotational velocity about it is equal and opposite to the perpendicular component of the initial velocity in the nonrotating system, then this point remains fixed in the rotating system, and rotates about the axis at a fixed distance r determined by rω = vo,perp. In general, then, the particle moves on a helix of radius r and pitch 2πvo,par/ω in a uniform magnetic field. A positive charge rotates clockwise as seen facing the point of the magnetic field arrow, or anticlockwise riding on the particle looking forward.

The angular frequency ωc = |qB/mc| is called the cyclotron frequency. It is the product, in radians per second, of the charge in emu and the magnetic field in gauss. The period of one gyration about the line of force is T = 2π/ω. An electron, in a field of one gauss, has a cyclotron frequency of 1.759 x 107 radians per second, or 2.80 MHz. When quoted in herz, the straight frequency is always meant. A proton, in a field of 5000 gauss, has a cyclotron frequency of 47.9 MHz. A 10 MeV proton has a speed of 4.38 x 109 cm/s, so its radius of gyration is 249 cm. A cyclotron, as shown in the diagram at the left, with dees about 5 m in diameter, in a field of 5000 gauss, with an oscillator of frequency 47.9 MHz, could be used to accelerate protons to this energy. The protons would be injected at low velocity at the center of the dees by ionizing hydrogen there, and could be extracted at 10 MeV at the periphery of the dees, after spiralling out, receiving two kicks of, say 500 V, per revolution. Since the angular velocity is constant and independent of the particle velocity, the particles cross the gap between the dees at constant intervals. They would make 10,000 revolutions while being accelerated. A fixed-frequency cyclotron produces a continuous beam, but is limited to nonrelativistic particles. An FM (frequency-modulated) cyclotron, or synchrotron decreases the frequency as the mass of the particles increases.

Cyclotron motion, or gyration around the magnetic field, is illustrated in the diagram at the right. The same initial velocity gives loops below for a positive particle, in the direction v x B, and above for a negative particle. The kinetic energy of the cyclotron motion is U = mvo,perp2/2 = m(ωcr)2/2. If the particle moves slowly relative to the cyclotron frequency, so that many rotations are made in the time taken to move one radius r, or adiabatically, this motion is very little disturbed, except that the radius and cyclotron frequency may change as the particle moves into regions of different magnetic field. The total kinetic energy of the particle, composed of the energy of gyration and the energy in the direction of the field, remains constant as the particle drifts.

The motion of the charge q represents an average current of q/T around the particle orbit. Since the area of the orbit is S = πr2, the gyratory motion has a magnetic moment μ = (i/c)S = q2r2B/2mc2.

Time-Varying Magnetic Field

If the magnetic field varies with time, an electrical field is produced that is described by Faraday's Law, curl E = -(1/c)∂B/∂t. If we integrate this around the gyratory orbit, the average electric field E along the orbit is given by 2πrE = (πr2/c)dB/dt, or E = (r/2c)dB/dt. This is permissible if the magnetic field does not change greatly during one gyration, again the adiabatic condition. The equation of motion is qE = mdv/dt = m(d/dt)(qBr/mc), or (qr/2c)dB/dt = (d/dt)(qBr/c). Then, (r/2)dB/dt = d(Br)/dt = rdB/dt + Bdr/dt. Therefore, 0 = (r/2)dB/dt + Bdr/dt. Multiplying through by r, we get d(r2B)/dt = 0. or r2B = constant. If this is multiplied by π, we see that this means that the flux linked with the orbit Φ is a constant, and also that the magnetic moment is a constant. This is a very important result, that allows us to find the radius of the orbit at any point as the particle moves, and combined with the conservation of energy, how fast the orbit moves in the direction perpendicular to itself.

This behavior, that a change of magnetic field induces a change in the current so that the flux linkage does not change, is called diamagnetism. Since the magnetic moment is a constant, we can also find the forces acting as if on the center of gyration by the gradient of the magnetic energy, F = -grad (μB). B and μ are, of course, in opposite directions. A negative charge gyrates in the opposite sense to a positive charge, remember. This gives a force in the z-direction, the direction of the magnetic field, of -μ(dB/dz). However, before we discuss nonuniform magnetic fields, we must discuss combinations of electric and magnetic fields.

Uniform, Constant Electric and Magnetic Fields

Let us take the magnetic field in the z-direction, and let the electric field define the x-z plane. The electric field can be resolved into a component along the magnetic field, and a component perpendicular to it. Considering the Lorentz forces, we see that the only force in the z-direction is due to the component of the electric field in this direction, so this degree of freedom acts just like the case of the electric field alone, accelerating the particle in the z-direction without affecting the motion in the x-y plane, which may be the magnetic gyration just discussed. If the electric field has only a z-component, then it affects only the pitch of the helix described by the particle in an obvious way. We shall, therefore, assume that the electric field is perpendicular to the magnetic field, and points in the x-direction, for the rest of this section.

Just as in the case of the magnetic field, let us try to use the same trick, and make forces disappear by choosing an appropriate coordinate system. This is really not a trick, but the exercise of intelligence, and gives us insight. Try a coordinate system moving with velocity w = c(E x B)/B2. In the present case, we can use the simpler formula w = cE/B, since we have assumed E and B perpendicular. The new coordinates move with constant speed w along the y-axis. Let's see what happens with the equation of motion.

We have mdv/dt = q[E + (1/c)(v x B)], and will substitute v = u + w. Then, (1/c)(v x B) = (1/c)(u x B) - [B x (E x B)]/B2. When the triple cross product is expanded, it gives simply E, which cancels the original E! In the moving system, there is an electric field equal and opposite to the applied electric field. All we have left is du/dt = (q/mc)(u x B), which we then reduce to utter simplicity by transforming to a rotating system.

The velocity w is called a drift, and we have found that a force on a positive particle causes a drift of the center of gyration in a direction perpendicular to both the applied force and the magnetic field. Here the force is the electric force qE, but it obviously could be any force, since the electric field in the drifting coordinates will exert a force that cancels it. Instead of the electric field E, we could have a gravitational field g (the component perpendicular to B), and all we need do in the equations is to replace qE by mg. For example, the drift velocity w = cE/B becomes (mc/q)g/B = g/(qB/mc) = g/ωc, which is the drift velocity in a gravitational field. Be careful with the sign, however, since a charge-independent force does not reverse when the sign of the charge changes, as the electric force does. Positive and negative charges will drift in opposite directions under a gravitational force, but in the same direction under an electric force.

Let us now go back to the original problem of a charge at rest at the origin, released at t = 0. The center of gyration will move in the direction of the y-axis with constant velocity w = cE/B. It will start from a position on the x-axis, since in the moving system we must assign an intial velocity -w along the y-axis so the particle is intially at rest in the space system. This determines the radius of gyration r = (cE/B)/ωc = mc2E/qB2. We now know that the center of gyration will move on the line x = r with constant velocity w. Since the velocity of the point in contact with the y-axis will always be -w with respect to the center of gyration, its instantaneous velocity in the space system will be zero. That is, the circle of gyration will roll without slipping on the y-axis. We know that any point on the circumference of the circle will trace out a cycloid.

A negative charge will behave the same way, except that the circle of gyration must be on the other side of the y-axis so that it will roll upwards considering the reversal of the cyclotron angular velocity. The cycloid will be traced on the other side of the y-axis. If it were a gravitational force causing the drift instead of the electric force, then the direction of drift would be reversed, and the cycloid would be on the same side of the y-axis, but below the x-axis instead of above it. If the initial conditions are different, many possibilities result, and the path may be a hypocycloid or a prolate cycloid, making little loops, or just waving back and forth. There is nothing new in all this, but it may be interesting.

It is possible to integrate the two coupled equations of motion for the x and y directions directly. The equations to start with are x" = (q/m)E - ωcy', and y" = ωcx'. The primes indicate differentiation with respect to time. If the first equation is differentiated with respect to t, then the second equation can be used to eliminate y", and the resulting third-order equation integrated to x" - x"(0) = -ωc2x. Since x"(0) = -(q/m)E, the resulting second-order linear equation can be integrated by the usual methods. Then one can go back and integrate the y-equation. The final result for the present problem is x = r(1 - cos ωct), y = wt - r sin ωct. The magnitude of the velocity of the particle is v = 2w sin(ωct/2). There are quite a few ways to go astray in the algebra, so if you complete it you will appreciate the crafty method more.

You have probably noticed that since w = cE/B, it would seem possible to have a drift velocity greater than c. It is not only possible, but could easily happen. This would have no effect on the algebra, and the result would be the same even in this case. However, our analysis is not valid for velocities approaching c, and indeed we have taken c/3 as a kind of informal limit. If w ≥ c, the particle is strongly accelerated in the x-direction without very much progress in the y-direction. Any circle of gyration has no time to be completed, and only a small fraction is covered before the velocity passes our limit. If, for example, B = 1 gauss and E = 300 V/cm, then w = c. The radius of the circle of gyration for an electron will be 1704 cm. Starting from rest at the origin, the electron reaches a speed of 0.1c in 2.13 cm, at which time it has moved only 0.0356 cm in the y-direction. This should make clear what happens in this case.

Time-Varying Electric Fields

Electromagnetic waves provide time-varying electric and magnetic fields, which exert forces on charged particles. These fields may be considered as functions of ej(ωt - k·r), or plane waves, propagating in the direction k with angular frequency ω. The wavelength is 2π/k, the period 2π/ω, and the phase velocity v = ω/k. The dispersion relation is the function ω = ω(k), determined by the medium through which the wave is passing. The group velocity is vg = dω/dk, and is the velocity with which a signal group propagates, and usually the velocity of energy propagation. In free space, v = vg = c. The electric field E and the magnetic field Bb are perpendicular to k, at right angles, in phase and equal in numerical magnitude. That is, if the electric field is 1 statvolt/cm, then the magnetic field is 1 gauss. The energy flux in the wave is P = (c/4π)E x B, the Poynting vector. The electric and magnetic energy densities are equal.

The wave exerts forces qE and (q/c)v x B on a particle of charge q that oscillate rapidly, so the particle does not wander far because of the wave. Nevertheless, these small motions may have important effects. As we shall see, the excursion of the particle is about x = eE/mω2, and its v/c = (e/mcω)E. These are, of course, amplitudes of oscillation, but are characteristic. For an electron, v/c = 1.759 x 107(E/ω). For an electric field E of 1 mV/cm, and ω = 105 Hz, v/c = 0.59, and nonrelativistic mechanics is applicable. Most electromagnetic waves encountered in practice are much weaker and of higher frequency, so v/c is even smaller. Since v/c is also the ratio of the magnetic to the electric force on the particle, the magnetic force due to the wave's magnetic field can be neglected in the first approximation.

First, let us assume that there is no static magnetic filed. Then the equation of motion is dv/dt = (q/m)E. For the plane wave, differentiation with respect to time is equivalent to multiplication by jω, so we have -ω2x = (q/m)E, where we have chosen the x-axis in the direction of the electric field of amplitude E. This is easily solved for the displacement x: x = -(q/mω2)E. Since the dipole moment is q times the displacement, the susceptibility is p/E = α = -(q2/mω2). If there are n particles per unit volume, but the density is not high enough that they influence each other significantly, then the dielectric constant is κ = 1 + 4πnα = 1 - (ωp/ω)2, where ωp is the plasma frequency √(4πnq2/m).

Under the action of a time-varying electric field, then, a free particle vibrates rapidly with small amplitude, producing a dielectric constant that is slightly less than unity for frequencies higher than the plasma frequency. The plasma frequency is a paramter that appears frequently when motions in a plasma are considered. It is the frequency of electrostatic oscillations of the electrons relative to the heavy ions (which move very little) in a plasma. As the frequency of the applied field decreases, the dielectric constant passes through zero, and is then negative for frequencies less than the plasma frequency. This implies an imaginary index of refraction, which means that waves transmitted from the surface to the interior of the medium are strongly absorbed, and cannot penetrate, which means that the waves are reflected. This behavior is observed in many cases, such as shiny metals and the reflecting ionosphere.

The plasma frequency for electrons is 5.64 x 104n1/2 radian/s, or 8979n1/2 Hz. It depends only on the absolute value of the charge, and the mass. The plasma frequency is 42.9 times lower for protons than for electrons. Since the only waves that can penetrate are those with frequencies higher than the plasma frequency, the assumption that magnetic forces can be neglected is well-founded.

If, however, there is a static magnetic field B, it cannot be neglected if it is of the same or greater magnitude than the oscillating magnetic field of the wave. It makes the medium anisotropic by picking out a special direction. Wave propagation in an anisotropic or, better, aelotropic, medium is a subject of considerable complexity that will not add greatly to our understanding of particle dynamics. Therefore, we consider only the simple case of propagation in the direction of B. Now, motion in one direction perpendicular to the magnetic field will cause a force in the perpendicular direction. Let us choose the x-axis in the direction of E, the z-axis in the direction of k, and the y-axis to make a right-handed triplet. The unit vectors in the x and y directions will be represented by i and k, respectively.

The equations of motion are then x" = (q/m)E + (qB/mc)y' and y" = -(qB/mc)x', where primes indicate time derivatives. These are simple enough, and can be solved easily, but there is a better way that is worth the trouble. We can combine x and y displacements in a single expression by introducing complex vectors i ± jj. These vectors, when multiplied by aejωt, give rotating vectors in the xy-plane, since the y-component has a ±90° phase shift with respect to the x-component. The + sign gives a clockwise rotation looking into the wave (towards -z), and the - sign an anticlockwise rotation. In optics, these are called right and left circular polarization. In particle physics, they are called left and right helicity. If we write the equations of motions in terms of these rotating amplitudes, then they uncouple and we have one equation for each helicity.

In terms of the amplitude a, we find that a" ± j(qB/mc)a' = (q/m)E, where the + sign is to be taken for negative helicity, and - for positive. Note that this correspondence depends on the convention for the time dependence. Physicists often use e-iωt, which reverses the sign. Using the harmonic time dependence, and the definition of the cyclotron frequency, we get (-ω2 ± ωωc)a = (q/m)E. Therefore, a = (q/m)[ω(-ω ± ωc)]-1, and so the dielectric constant κ = 1 + ωp/[ω(-ω ± ωc)]. This gives different indices of refraction for the two helicities, so the medium is birefringent. The dielectric constant becomes zero for two frequencies. Below the smallest of these, the dielectric constant is negative for both polarizations, which are reflected. Above the larger, the dielectric constant is positive for both, so they can propagate (but at different velocities). Between the two critical frequencies, the positive helicity (left circular polarization) can propagate, but the other polarization will be reflected. These predictions are borne out by ionospheric studies.

In the ionosphere, a typical electron density is n = 104 - 106 cm-3, and a typical magnetic field is B = 0.3 gauss. This gives, roughly, ωp = 6 - 60 x 106 radian/s, and ωc = 5 x 106 radian/s.

When ω is small, the dielectric constant is about κ ≈ ω2/(ωωc), so the index of refraction is n = √κ = ωp/√(ωωc). This gives the dispersion relation (ω/k)= c/n = c√(ωωc)/ω. We note that the propagation is strongly dispersive--that is, the phase velocity depencs significantly on the frequency. From this, we find that the group velocity vg = dω/dk = 2v. Energy propagates twice as fast as the waves (just the opposite of the propagation of waves on the surface of a pond).

Suppose a lightning flash in the southern hemisphere occurs, creating a broad spectrum of electromagnetic waves. These waves may be guided by the geomagnetic field at high altitudes (we have not shown that this guiding occurs; let's assume here that it does happen) along the field to a corresponding point in the northern hemisphere. An average angular frequency of 105 radian/s (16 kHz) will have a group velocity of 3.9 x 107 cm/s if the plasma frequency is 6 x 106 radian/s, a typical value. Since the distance is about 1.5 x 109 cm, waves of this frequency will reach the other hemisphere in roughly 0.4 s. A wave of 100 radian/s, on the other hand, will have a group velocity of only 1.23 x 106 cm/s, since it is proportional to the square root of the frequency. This wave will arrive in about 13 seconds. The electron density is not large over all the path, so the actual difference in delay times will be smaller.

If a long wire is attached to a telephone, the arrival of the signal from the lightning flash will begin with a high note, then descend to a low note in times of a few seconds. This phenomenon is called a "whistler." By the rate of decrease of the frequency, the electron density can be estimated along the path, and reasonable results have been obtained.

Nonuniform Magnetic Fields

Uniform magnetic fields are usually met with only in the laboratory, or in limited regions. Nonuniform fields are more common on a large scale, such as the dipole field shown at the left. The magnetic moment of the earth's dipole field is about 8 x 1025 emu. The fields in gauss are given by the formulas in the figure. The axis of the dipole pierces the surface at the North Magnetic Pole, lat. 79°N, long. 110°W.

What we wish to study here is what happens when the field lines are curved, and the magnitude of the magnetic field changes. We assume that to a first approximation, the centers of the circles of gyration follow the magnetic lines of force, keeping normal to them. We have already seen that the magnetic moment or the flux linkage remains constant as B changes, and know that the total kinetic energy is conserved. To the next order of approximation, we find that the motion is subject to drifts in a nonuniform field with curved lines of force.

To give a feeling for what is going on, we look at a particular nonuniform field that is easy to work with, the field of a linear current i. This is certainly nonuniform, since B = 2i/cr, and the lines of force are certainly curved, with radius of curvature R = r. Choose the z-axis tangent to the circle of gyration, and directed along the field B at the point of tangency. Then, the magnetic field at a point on the z-axis is B = (2i/c)(r2 + z2)1/2. The z-component of B is then Bz = (2ir/c)(r2 + z2)1/2, the x-component is Bx = (2iz/c)(r2 + z2)1/2, and the y-component By = 0. From these expressions, it is easy to show that the derivatives of the field at the origin are ∂Bz/∂x = ∂Bx/∂z = 2i/cr2, and ∂Bz/∂z = 0. This result is consistent with the requirements that curl B = 0 and div B = 0.

The radius of curvature R of the line of force satisfies the relation dz/R = dBx/B, or 1/R = (1/B)∂Bx/∂z. This is illustrated in the diagram at the left. Substituting the known values we have just obtained in this formula, we find that R = r, as expected. We can, therefore, rely on the formula to give the radius of curvature in the general case. It means that whenever we have a gradient in the magnitude of the magnetic field, we also have curvature of the field lines at the same time.

But we can find the drift that results from a gradient in the field strength, knowing that the force is μ∂Bz/∂x, taking the x-axis along the direction of grad B. This implies a drift w = (cμ∂B/∂x)/(qB). Substituting for μ, we find that w = (1/2B)(∂B/∂x)(qBr/mc)r, or (w/vperp) = r/2R = vperp2/2Rωc. This gives the drift caused by inhomogeneity, which does not depend on the charge of the particle.

The drift due to moving along the curved line of force can be found by considering a rotating coordinate system in which the motion is rotation at a radius of R. That is, we take the origin at the center of curvature of the line of force, and let the radius follow the center of the circle of gyration. In this system, the particle experiences a centrifugal force mvpar2/R, which, again, is charge-independent. This force produces a drift w = vpar2/Rωc in the same direction as the drift due to inhomogeneity. The inhomogeneity drift is always present, at right angles to the change in magnetic field, but the centrifugal drift occurs only if the particle is helixing along the lines of force. Since the forces are charge-independent, the drift is in opposite directions for opposite signs of charge. The total drift is w =(1/Rωc)(vperp2/2 + vpar).

The kinetic energy of the cyclotron motion can be written U = (q2Φ/2πmc2)B. Since the quantity in parentheses is constant, the kinetic energy of gyration increases with B. However, the total kinetic energy, gyration plus translation along the line of force, is a constant. Therefore, as a particle moves towards a weaker field, its translational kinetic energy increases at the expense of the gyrational energy, and the movement of the gyrational center is accelerated. The opposite happens as the particle moves towards a stronger field. At some point the gyrational energy may become equal to the total kinetic energy, and at that point the particle ceases forward motion, reverses, and begins to move away towards the weaker field. The field at which the particle is reflected is easily found once its total energy is known.

The earth's dipole field is much like this, with the field growing stronger near the poles and weaker in equatorial regions. A particle "trapped" into rotation about a line of force will then oscillate from polar region to polar region. A number of such particles will form a "belt" concentrated at equatorial latitudes, distributed according to total energy and oscillating from pole to pole. The particles will drift to the east or west depending on the sign of their charge in the course of this motion. This seems to be exactly what happens in the Van Allen radiation belts, discovered by satellite probes around 1958.

We are able to say quite a bit about the motions of particles based on what we have discovered so far, and though it is mainly qualitative, we still seem to understand the reasons fairly well, and see how they would apply in different and novel circumstances. This we could not do with only numbers.

Debye-Hückel Screening Distance

Consider an electrically neutral region with equal densities no of electrons in thermal equilibrium at a temperature T, and heavy positive ions. If we place a negative charge -Ze at some point, it will repel the electrons around it, which will uncover positive charge in a certain region around it that will compensate its excess negative charge, or "screen" it so that its presence cannot be detected at larger distances. The result is achieved by competition between the electrostatic forces and diffusion driven by the concentration gradient of electrons. In equilibrium, the rate that electrons approach the charge -Ze is exactly equal to the rate that it drives them away.

Let φ be the electrostatic potential of the charge -Ze. Then, the density of electrons at any point is given by the Boltzmann factor, n = noe-eφ/kT. e/k for electrons is 3.48 x 106 K/statvolt, so at normal temperatures n will be a rather strong function of φ. Now div E = 4πρ, so div grad φ = -4πρ. Putting the unbalanced charges in this equation gives div grad φ = -4πZeδ(r) + no(1 - e-eφ/kT). If eφ << kT, which is usually the case, then the exponential can be expanded and the equation becomes linear: div grad φ - ωp2(m/kT) = -4πZeδ(r). Without the second term, the solution would be φ = -Ze/r. Let us try as a solution φ = (-Ze)e-r/d/r. In spherical coordinates, div grad φ = (1/r2)(d/dr)(r2dφ/dr), so we find div grad φ = (1/d)2φ, so the equation is satisfied, since as r → 0, φ → (-Ze)(1/r), which gives the delta function.

The quantity d = (kT/m)1/2p is called the Debye-Hückel screening distance. For electrons, it is equal to 6.90(T/no)1/2 cm. A plasma has been defined as a gas of electrons and ions in which d is much smaller than the dimensions of the plasma or of any movements that occur. For a typical hot laboratory plasma in which T = 106 K and no = 1015 cm-3, d = 2.18 x 10-4 cm, or 2.18 μm. The screening distance d can also be expressed as √3 times the ratio of the rms thermal velocity of the electrons (√(3kT/m)) to the plasma frequency.

An electrode inserted into a plasma and maintained at potential E with respect to some reference point, such as the cathode of a discharge, and arranged so that the current I supplied to the plasma can be measured, is called a Langmuir probe. A schematic E-I characteristic is shown at the right. When the potential is less than Ef, the "floating potential," the current is negative, and consists of the positive ions collected. This current is normally very small, and is exaggerated in the diagram. As Ef is approached from below, the more energetic electrons reach the probe and the current begins to increase. At Ef, the current is zero. This is the potential assumed by a nonconducting or insulated wall bordering the plasma, which is sufficiently negative to repel the electrons (or, to make the net current to the wall zero). For greater potentials, the current increases rapidly as the barrier to the electrons is lowered, and finally at Ep the maximum electron current is drawn. At this point, there is no potential barrier, and this potential is called the plasma potential. The current remains constant as the potential is raised further, until the probe becomes a secondary anode. At this point its current must be limited by the external circuit.

The potential difference Ep - Ef gives rise to the same barrier as in Debye-Hückel screening, and the thickness of the positive-ion sheath is equal to d. The sheath thickness is greater for more negative potentials, and less as the plasma potential is approached. For still greater potentials, we have an electron sheath instead, but breakdown occurs before long. From current measurements with the probe, most of the interesting plasma parameters can be obtained, such as the electron density and the electron temperature.

The first plasmas studied were the luminous positive columns of glow and arc discharges in low-pressure gases. At a total pressure of 1 mmHg, the neutral density is about 3.5 x 1016 cm-3, seldom as much as 1% ionized, which would make the electron density about 1014 cm-3. The electron temperature may be as high as 10,000K, though the neutral temperature would be much lower, around 400K. With these figures, d = 6.9 x 10-5 cm. Even if the electron density were only 1010 cm-3, d would still be no larger than 0.069 mm. The sheath, being empty of electrons, would not be luminous like the main body of the plasma.

Plasmas exist because recombination of a free electron and an ion is difficult. Radiative recombination, when a photon is emitted, is more probable but still rare. Negative ions, however, recombine readily with positive ions since there are two particles before and after the collision. Dissociative recombination of a free electron with a diatomic ion (e.g., e- + NO+ → N + O) is also probable, and plays a considerable role in the ionosphere. Although plasmas are highly conductive, there must be some voltage drop to reflect the energy required to create new electrons and ions to replace those lost by recombination, and to the walls. We see plasmas in neon advertising tubes and fluorescent lights. The orange glow in Nixie tubes is not a plasma, but a cathode glow, which is quite different.

I do not know the origin of the name "plasma." It comes eventually from the Greek plassw, meaning "mould, shape, invent," but may not have been coined directly from Greek. It is probably not related to the mineral Plasma, which is a rare, bright-green form of translucent quartz, specked with white as heliotrope is with red. It could well have come from the name of the clear fluid in blood, lymph and milk in which other particles are suspended, as in our plasmas we have electrons, ions and neutrals. Gas discharges gave rise to the whole field of electronics. Tonks and Langmuir began the systematic study of plasmas in the late 1920's. Hendrik Antoon Lorentz (1853-1928) was the father of charged particle dynamics. Plasma physics entered the limelight with Project Sherwood in 1948 for fusion power, or, as now it seems, for the extraction of sunlight from cucumbers, in the words of Jonathan Swift.

References

Here are a few classic references on the subject. Millman and Seeley use MKSA units, while the classic plasma physics books use emu. Spitzer uses a strange hybrid where the charge is in esu and appears divided by c, but otherwise emu are employed.

J. Millman and S. Seely, Electronics (New York: McGraw-Hill, 1951). Chapters 2 and 3. An excellent survey of the field, directed toward applications to electronic apparatus.

L. Spitzer, Jr., Physics of Fully Ionized Gases (New York: Interscience, 1956). Chapter 1. Supporting the study of geophysical and laboratory dilute plasmas.

J.-L. Delcroix, Introduction to the Theory of Ionized Gases (New York: Interscience, 1960). Chapter 5.


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Composed by J. B. Calvert
Created 25 October 2002
Last revised 27 October 2002