The two beams that unravel each other

A plasma left alone oscillates. The frequency below which nothing gets in found the rate: displace the electrons of a plasma from their ions and the charge separation pulls them back, they overshoot, and they ring at the plasma frequency $\omega_p = \sqrt{ne^2/\varepsilon_0 m}$ — nine megahertz in the ionosphere, a few tens of kilohertz in the solar wind. The wave that dies with nothing to rub against found that a warm plasma rings down, without any collision, by handing the oscillation’s energy to the electrons that happen to move at its speed. Between them the two essays describe a plasma that is stable: whatever it is given, it rings, and the ringing either persists or dies.

A cold beam of electrons, moving through a background of ions at a steady speed, is stable in the same way. Its electrons can be displaced from one another along the beam and ring at the beam’s own plasma frequency, and since the whole beam is moving, the ringing is carried downstream. There are two such waves on any beam — one running a little ahead of the electrons, one dropping a little behind — and both have real frequencies at every wavelength. Nothing in a single beam grows.

Send a second beam through the first, the other way, and the pair is not stable. A ripple in the electrons’ density, of a wavelength in the right range, grows exponentially from whatever started it, takes a third of the beams’ energy of motion into an electric field within a few plasma periods of becoming visible, and then winds the two beams into a single whirlpool in phase space in which neither beam is any longer moving. This is the two-stream instability, found almost simultaneously in 1948 and 1949 by Andrew Haeff at the Naval Research Laboratory, by John Pierce and William Hebenstreit at Bell Laboratories, and by David Bohm and Eugene Gross in their theory of plasma oscillations. It is the simplest instability a plasma has, and the plainest example of a property that belongs to a pair and to neither of its members.

Two beams and four waves #

The calculation that predicts all of this is a few lines long. Take two cold electron beams, each holding half the electrons, moving at $+v_0$ and $-v_0$ through a fixed background of ions that keeps the whole neutral. Disturb the electrons by a small ripple of wavenumber $k$ and look for a disturbance that varies as $e^{i(kx - \omega t)}$.

Each beam answers the ripple’s electric field on its own. An electron in the first beam sees the field oscillate not at $\omega$ but at the Doppler-shifted $\omega - kv_0$, because it is moving through the pattern, and its bunching in response is proportional to $\omega_b^2/(\omega - kv_0)^2$, where $\omega_b$ is the plasma frequency of the beam’s own electrons. The second beam answers in the same way with $\omega + kv_0$. Poisson’s equation then asks that the bunching of the two together produce the field that caused it, which gives the dispersion relation

$$1 = \frac{\omega_b^2}{(\omega - kv_0)^2} + \frac{\omega_b^2}{(\omega + kv_0)^2}.$$

With either term alone the equation says $\omega = \pm kv_0 \pm \omega_b$: each beam carries a fast wave and a slow wave, Doppler-shifted copies of its own plasma oscillation, and all four are real. With both terms the equation is a quartic in $\omega$, and since only even powers appear it can be solved exactly. In units of $\omega_p$, the plasma frequency of all the electrons together — so that $\omega_b^2 = \omega_p^2/2$ — and writing $K = kv_0/\omega_p$, the four roots are

$$\omega^2 = \frac{(2K^2 + 1) \pm \sqrt{8K^2 + 1}}{2}.$$

The plus sign gives a fast pair of waves that is always real. The minus sign gives the slow pair, and its $\omega^2$ is negative whenever $\sqrt{8K^2+1} > 2K^2 + 1$, which is whenever $K < 1$. A negative $\omega^2$ is an imaginary frequency, and an imaginary frequency is a disturbance that does not oscillate but grows — or, for the partner root, decays — exponentially in time.

The figure shows the transition happening. Above $kv_0 = \omega_p$ the coupled slow root runs a little beneath the beams’ own slow waves, still a real oscillation; as the wavelength lengthens its frequency falls, reaches zero at exactly $kv_0 = \omega_p$, and does not come back. Below that wavenumber the slow root has become a growth rate, which rises from zero to a maximum of $\omega_p/(2\sqrt{2})$, or $0.354\,\omega_p$, at $kv_0 = \sqrt{3/8}\,\omega_p$, and falls back towards zero at long wavelengths. A disturbance at the fastest wavelength grows by a factor of $e$ in 2.8 time units of $1/\omega_p$ — less than half of one plasma period.

The arithmetic is the same arithmetic as the collapse that the disturbance that grows instead of travelling followed in a self-gravitating gas: a squared frequency, built from a restoring term and a destabilising one, that changes sign at a definite wavelength. In Jeans’s gas the destabilising term was gravity and the instability took long wavelengths. Here it is the beams’ motion, and the instability takes wavelengths longer than one beam travels in $1/\omega_p$ but not too much longer. The water waves of the even wave train that cannot stay even went the same way: a uniform state, a perturbation, and a squared growth rate that is positive over a band. What is new here is that neither term is destabilising alone. Each beam’s response is the response of a perfectly stable oscillator; the instability comes from adding two of them with opposite Doppler shifts.

Why the bunching feeds itself #

The mechanism can be followed without the algebra. Suppose the electron density has a small ripple, slightly more electrons in some places than others. The ripple makes an electric potential, low where the electrons are crowded. An electron of the first beam, crossing that pattern, slows as it climbs out of each trough of potential energy and speeds up as it falls into the next; slowed electrons spend longer in a region and so make it denser. If the beam electrons spend longest exactly where the electrons were already crowded, their crowding adds to the ripple, and the ripple’s field becomes stronger.

For one beam alone the timing is wrong. A beam’s electrons respond to a field with a delay set by their own plasma frequency, and they reach the crowded places either ahead of the ripple or behind it, which is what makes the single beam’s two waves travel at its own speed plus or minus a fixed amount. The second beam changes the timing. A ripple standing still between them, which is what the slowest root becomes when its frequency reaches zero, is crossed by both beams at the same rate in opposite directions, and each beam’s bunching can then be placed so that it deepens the troughs the other beam is responding to. The ripple does not travel: in the frame in which the two beams are symmetric, the growing disturbance is a standing pattern of charge whose amplitude rises in place.

The energy comes from the beams’ drift. A ripple that slows electrons in one place and speeds them up in another changes their kinetic energy by an amount that does not average to zero once the slowed electrons have been allowed to bunch, because more electrons are being slowed than sped up. The difference is the field energy the ripple gains. Nothing else could pay for it. A beam at rest has no drift to give, and a single beam can always be viewed from the frame in which it is at rest; two beams moving through each other cannot, and their relative motion is free energy that no change of frame removes.

A narrower band for a thinner beam #

Symmetric beams are the cleanest case and not the most common. In most plasmas that carry one, the beam is a small fraction of the electrons, moving through a much larger population that is nearly at rest — electrons accelerated by a solar flare streaming outward through the solar wind, or a beam injected into a laboratory plasma. The same calculation, with one population at rest and one moving at $v_0$, gives a quartic with odd powers in it, which is no longer solvable in a line but is solved numerically in the next figure for beams holding half, a tenth and a hundredth of the electrons.

For equal streams the curve is the symmetric one, with the wavenumber doubled because $v_0$ is now the speed of one beam relative to the other rather than relative to their midpoint. As the beam thins the curve does two things. Its peak falls, and it moves towards one value of the wavenumber, $kv_0 = \omega_p$. That value has a plain meaning. A wave at the plasma frequency of the background, with wavenumber $\omega_p/v_0$, travels at phase speed $\omega_p/k = v_0$: it is the background’s own oscillation, moving at exactly the beam’s speed. A thin beam can drive only the wave that keeps pace with it, so that its electrons see an almost steady field and have time to bunch in it — the same matching of speeds that the frequency that gets an answer found behind every resonance, here between a moving population and a wave.

The growing wave in fact travels slightly slower than the beam, and that is where its energy comes from. A wave running a little behind the beam’s electrons is overtaken by them; electrons that overtake a crest are slowed as they climb it and bunch on its rising side, where they are pushed back, so on balance the beam gives up speed. The wave can take energy from a beam only if it is slower than the beam, which is why the unstable band always lies near the beam’s own speed and ends where the wave would have to run faster.

A millionth of the electrons is enough #

The less obvious feature of that figure is how slowly the peak falls. A beam holding a tenth of the electrons grows at three-quarters of the rate of two equal streams, and a beam holding a hundredth still grows at more than a third. The next figure carries the beam density down seven orders of magnitude.

The growth falls as the cube root of the beam’s density. Near $kv_0 = \omega_p$ the background’s response is nearly resonant, its term in the dispersion relation almost cancelling the 1 on the other side, and the small beam term has to balance only what is left. Expanding about the resonance turns the quartic into a cubic in the frequency shift, with the beam density standing alone on one side, and the three roots of a cubic are the cube roots of unity times the cube root of that density. One of them has a positive imaginary part of $\sin 60° = \sqrt{3}/2$ times its size, which is where the $\sqrt{3}$ in the law comes from. The dispersion relation follows the law to 2 per cent for beams below one electron in ten thousand and to 4 per cent at one in a thousand, and departs from it only as the beam becomes a sizeable fraction of the plasma: at a tenth the law overestimates the growth by 19 per cent, because the expansion about the resonance assumed the beam’s term was small.

So a beam carrying one electron in a million grows a wave by a factor of $e$ in 23 plasma periods. In the solar wind at the Earth’s distance, where the plasma frequency is around 30 kilohertz, that is under a millisecond. Electrons accelerated by a solar flare and streaming out along the magnetic field are a beam of exactly this kind, far thinner than the wind they cross; the Langmuir waves they drive are converted, partly, into radio waves at the local plasma frequency and its harmonic, and because the plasma frequency falls as the beam travels outward into thinner plasma, the radio emission sweeps down in frequency, from hundreds of megahertz near the Sun to tens of kilohertz near the Earth, within minutes. These are the type III radio bursts, and they were the first direct evidence that beams in space drive this instability.

The same law covers a case that does not look like a beam at all. When a current flows through a plasma, the electrons drift relative to the ions. If the drift is faster than the electrons’ thermal speed, the electrons are a cold beam and the ions are the “background”, with a plasma frequency smaller than the electrons’ by the square root of the mass ratio. The weak-beam law with the mass ratio in place of the density ratio gives $(\sqrt{3}/2^{4/3})(m_e/m_i)^{1/3}\,\omega_p$, which is 0.056 of the electron plasma frequency for hydrogen: the Buneman instability, which Oscar Buneman found in 1958 as the most violent way a current can be stopped. If the drift is slower than the electrons’ thermal speed, the current-driven instability is the gentler ion-acoustic one that the sound a plasma carries on its electrons’ heat ended on, which needs hot electrons and grows far more slowly.

There is a limit on all of this, and it is the temperature of the beam. Every figure so far treats the beam as cold, every electron at exactly $v_0$. A real beam has a spread of speeds, and once the spread exceeds about $(n_b/n_p)^{1/3} v_0$ — the width in velocity of the resonance a cold beam excites — the beam’s electrons no longer act together. The growth then comes from the slope of the beam’s velocity distribution at the wave’s speed: the bump-on-tail instability, the mirror image of Landau damping, whose rate is proportional to the beam density rather than to its cube root. For a beam of a millionth that spread is a hundredth of $v_0$, which is why the cold-beam law is an upper bound and the kinetic rate is what most beams in space actually drive.

Following every electron #

The dispersion relation says when the ripple grows and how fast. It cannot say what happens when the ripple is no longer small, because it was derived by assuming that it was. For that the electrons have to be followed one by one.

The simulation drawn at the top of this essay does that in the simplest way available. Twenty thousand electrons, ten thousand in each beam, are placed evenly along a periodic box exactly one fastest-growing wavelength long, $2\pi/(\sqrt{3/8}) = 10.26$ units of $v_0/\omega_p$, so that only the fastest mode and its harmonics fit. Each step, the electrons’ charge is gathered onto a grid of 64 cells, Poisson’s equation gives the field, and each electron is pushed by the field at its own position. The beams start at exactly $\pm v_0$ with their positions rippled by one part in ten thousand of the box.

For eighteen units of time almost nothing is visible. The ripple is growing — the second figure below shows it rising as a straight line on a logarithmic axis, at a fitted rate of $0.345\,\omega_p$ against the predicted $0.354$ — but it starts so small that it takes several e-foldings to show. By $\omega_p t = 18$ both beams are visibly rippled in velocity, each slowed on one side of the box and sped up on the other, in opposite senses. By 26 the ripple has overturned: each beam has been bent so far that its fastest electrons have caught up with its slowest, and the two beams are being drawn around one point in the middle of the box.

The growth stops for a reason that the linear theory does not contain. A wave of amplitude $E$ that stands still is a row of troughs, and an electron moving slowly enough relative to it is trapped in one, rocking back and forth at the bounce frequency $\omega_B = \sqrt{ekE/m}$. Growth stops when the trapped electrons complete a swing as fast as the wave grows, because then they start to hand back the energy they took on the first half of each swing. When the wave’s amplitude first peaks, at $\omega_p t = 25.2$, the bounce frequency is $0.59\,\omega_p$, half as much again as the growth rate, and the trap holds any electron within $\pm 1.92\,v_0$ of zero velocity — both beams entirely. From then on every electron is circulating in the same set of troughs.

Where the drift goes #

The trapping does not undo what the instability did, and the energy budget shows why.

During the growth the field and the electrons’ motion about their beams’ mean rise together, and roughly equally, because the energy of a plasma wave is shared between its electric field and the coherent sloshing of the electrons in it. Both are paid for out of the drift. The field peaks at 37 per cent of the drift energy and then oscillates as the trapped electrons swing, giving energy back and taking it again with each bounce. Each swing is a little less coherent than the last. Electrons near the bottom of a trough bounce faster than those near its edge, so after a few swings the electrons that started together are at different phases of their bounce, and the vortex winds itself into the filaments visible at $\omega_p t = 40$. Between 45 and 60 the field holds 8 per cent of the original drift energy, the beams’ mean velocities have fallen to almost nothing — they keep 1 per cent of their drift — and the remaining 91 per cent is in the electrons’ motion about the mean, now disordered rather than sloshing. Total energy is conserved by the calculation to 0.31 per cent throughout.

The phase-space picture is the same filamentation that Landau damping produced, run in reverse. In that essay a wave’s energy went into structure in the electrons’ velocities too fine for a field to see; here the beams’ drift goes through the field into structure of the same kind. Nothing in the equations is irreversible. Reversing every velocity at $\omega_p t = 60$ would, in an exact calculation, unwind the vortex and return the two beams to $\pm v_0$. But no field can reach into the filaments to extract the energy again, and a single collision, or a small amount of noise, destroys the information that a reversal would need. For any practical purpose the beams have been heated: a population of electrons with a definite drift has become one with a spread of speeds about zero, without one collision between them.

Amplifiers that were mostly not built #

The instability was first wanted rather than feared. Haeff and the Bell group were looking for a microwave amplifier without the delicate circuits that a klystron or a travelling-wave tube needs: two electron beams of slightly different speed, sent through one another, would grow any small signal impressed on them, and the signal could be taken off at the far end. Such double-stream amplifiers were built and did amplify, but the signal grew from the beams’ own noise as readily as from the input, and the tubes were never competitive with the travelling-wave tube, whose principle is the same with the second beam replaced by a slow electromagnetic wave guided along a helix. The coupling of a beam to a slower wave, which the field at which no electron reaches the anode found bunching electrons on a magnetron’s cavities, is the instability put to work.

Elsewhere it is a nuisance. Intense electron beams in accelerators, in fusion experiments and in the plasma behind a laser striking a solid are subject to it whenever they pass through other electrons, and it limits how far such a beam can be sent before it is slowed and spread. The beam that stops pushing itself apart found a beam’s own repulsion weakening with speed; the two-stream instability is what happens to it when it meets other charges instead.

What the figures leave out #

Every figure is one-dimensional, electrostatic and nonrelativistic, with cold beams, fixed ions and no magnetic field. Fixing the ions is the mildest of these: a proton is 1,836 times heavier than an electron, and in the ten plasma periods the run lasts the ions of a hydrogen plasma would move only a few per cent of a wavelength. Over a longer time they would answer the heated electrons, and the plasma left behind — hot electrons, cold ions, and electrons screening every ion bunch over a Debye length as the long-range force that does not reach described — is exactly the one in which ion sound survives. Each simplification hides something. In two or three dimensions, counter-streaming beams are also unstable to disturbances across their motion, which break them into filaments of current that repel one another magnetically — the filamentation or Weibel instability, which in many astrophysical shocks grows faster than the two-stream one and makes the magnetic fields that the shock then needs. A magnetic field along the beams leaves the one-dimensional result unchanged but adds oblique modes coupled to the electrons’ gyration. Warm beams, as above, cut the growth at short wavelengths and turn the cold-beam instability into the kinetic one. And the periodic box allows only the fastest wavelength and its harmonics; in a long plasma many wavelengths grow at once, and the single vortex of the first figure becomes a chain of vortices that merge as they age.

The simulation is small. Twenty thousand electrons are enough to follow a single wavelength but give a noise floor that a larger run would push down, and a starting ripple of one part in ten thousand was chosen so that the growth could be measured over four decades before saturation.

Still open: why the beams behind solar radio bursts last so long #

The arithmetic above contains a problem that is now sixty years old. A flare’s electron beam, a millionth of the solar wind’s density, should drive Langmuir waves that grow by a factor of $e$ in under a millisecond; within a few hundred kilometres of leaving the Sun the waves should have taken most of the beam’s drift, and the beam should have spread into a plateau and stopped. Yet type III bursts are seen to drift continuously in frequency all the way from the corona to the Earth’s orbit, which means the beam is still there and still driving waves 150 million kilometres out. Peter Sturrock posed the difficulty in 1964 and it is called Sturrock’s dilemma. The usual answers are that the beam relaxes to a plateau that is marginally stable and travels with its waves, reabsorbing them at its back as it emits them at its front; that the solar wind’s density fluctuations refract the waves out of resonance before they can drain the beam; and that the waves are scattered into other waves. Which of these dominates, and where, is being tested against the electron distributions and electric fields that spacecraft close to the Sun now measure directly, and the answer appears to change with distance.

The linear statement is short. Two cold electron beams through each other have a root of the dispersion relation that grows for every wavenumber below $\omega_p/v_0$, fastest at $kv_0 = \sqrt{3/8}\,\omega_p$ at a rate $\omega_p/(2\sqrt{2})$; a beam a fraction $n_b/n_p$ of a plasma grows at $(\sqrt{3}/2^{4/3})(n_b/n_p)^{1/3}\,\omega_p$, so even one electron in a million drives a wave within 23 plasma periods; and once the wave traps the electrons its energy runs on into their spread, leaving 91 per cent of the drift as disorder and 1 per cent as drift. Two stable beams make an unstable pair, because their relative motion is energy that no frame of reference removes.