The photon a fast electron throws back

A laser that a physicist would call infrared emits photons of 1.17 electronvolts. Point it head-on at a beam of electrons carrying a gigaelectronvolt each, in the right vacuum and at the right moment, and a narrow beam of gamma rays comes out along the electrons’ direction. The hardest of them carry 17.6 million electronvolts — about the energy of the gamma rays a nucleus emits, from a photon that started out too weak to break a chemical bond. Gamma-ray sources built this way run at several laboratories, where they excite nuclei, make polarised beams for nuclear physics, and image the contents of shipping containers.

The process is ordinary scattering. An electron struck by light shakes and re-radiates it, which the cross-section that forgets the colour found to be the same for every colour of light from the ultraviolet to the X-ray, Thomson’s constant cross-section. What is not ordinary is the frame. A slow electron re-radiates light at the colour that hit it. A fast one re-radiates it at its own colour — the colour the light has in the electron’s frame — and the laboratory sees that colour shifted again by the electron’s motion. Two Doppler shifts and a bounce make the gamma ray, and following them is a clean way of seeing how far a moving mirror’s arithmetic can be trusted.

A mirror that is also a receiver #

The magnet period that comes out as an X-ray followed an electron through a row of alternating magnets. In the electron’s frame the row was contracted by $\gamma$ and rushing past, a wave that the electron scattered; back in the laboratory the forward light was shortened by a further $2\gamma$, so the magnet period was divided by $2\gamma^2$. A laser beam coming the other way is the same device with one difference: the “magnets” — the oscillating field of the light — are already moving towards the electron at the speed of light, instead of standing still in the laboratory.

That difference changes the first factor. In the electron’s frame a stationary magnet row is contracted by $\gamma$, exactly as the length that depends on when found for any object the observer moves past; a light wave approaching at $c$ is not merely contracted but Doppler shifted, and its frequency rises by $\gamma(1 + \beta)$, which is $2\gamma$ for a fast electron. So the laser photon arrives in the electron’s frame carrying $2\gamma$ times its laboratory energy. For a gigaelectronvolt electron, with $\gamma = 1957$, the 1.17-eV photon arrives as a 4.6-keV soft X-ray.

The electron, at rest in its own frame, scatters that X-ray back the way it came. Then the laboratory sees the scattered photon emitted by a source moving towards it at nearly $c$, and the Doppler factor multiplies its energy by $2\gamma$ again. The product of the two is

$$E_\gamma \approx 4\gamma^2\varepsilon,$$

where $\varepsilon$ is the laser photon’s energy. It is the same factor the shift a mirror gives twice derived for a radar reflecting from a moving car, which is no coincidence: a reflector moving at speed $v$ is a receiver and then a source, and each role contributes one Doppler factor. At a car’s speed the two factors are $1 + 2v/c$ and nobody notices the relativity. At $\gamma = 1957$ they multiply a photon’s energy by fifteen million.

Where the mirror stops being a mirror #

The mirror picture has one assumption hidden in it. A mirror is heavy: light bounces off it with its energy unchanged in the mirror’s frame. An electron is not heavy. In its own frame it scatters the photon by Compton’s rule, which a photon with a momentum derived from nothing but the photon’s momentum $h/\lambda$: a photon of energy $\varepsilon'$ bounced straight back loses some of its energy to the electron’s recoil and leaves with

$$\varepsilon'' = \frac{\varepsilon'}{1 + 2\varepsilon'/mc^2}.$$

When $\varepsilon'$ is a few kiloelectronvolts the correction is a per cent or two, and the mirror is a good mirror. When $\varepsilon'$ approaches the electron’s rest energy of 511 keV, the electron is no longer a wall — it is a billiard ball of comparable weight, and the bounce sends a large share of the energy into the electron instead of back along the beam. It is the case the kick a fast particle can give an electron met from the other side, where a heavy projectile stopped being able to bounce an electron off itself once the two met as equals in the centre-of-momentum frame.

The figure follows this through three frames. For a 0.5 and a 1 GeV electron the photon reaches the electron’s frame as a few keV, bounces with almost no loss, and comes back multiplied by about $4\gamma^2$. For a 46.6 GeV electron the same laser photon reaches the electron’s frame as 213 keV, almost half the electron’s rest energy. The recoil takes 46 per cent of it, and the second Doppler factor multiplies what is left, not what arrived.

Doing the two boosts and the bounce exactly gives one closed form. With

$$x = \frac{4\gamma\varepsilon}{mc^2},$$

which is twice the photon’s energy in the electron’s frame measured in electron rest masses, the highest energy a back-scattered photon can have is

$$E_{\max} = \frac{x}{1 + x}\,E_e,$$

where $E_e = \gamma mc^2$ is the electron’s energy. For small $x$ this is $xE_e = 4\gamma^2\varepsilon$, the mirror’s answer. For large $x$ it approaches $E_e$ and never reaches it. The photon can carry away almost all of the electron’s energy and never more than all of it, which is the conservation of energy reasserting itself through a correction the mirror picture leaves out.

The marked point is the regime of an experiment at the Stanford Linear Accelerator Center in the 1990s, known by its number, E-144, which collided the 46.6 GeV electron beam with a terawatt green laser. The mirror’s arithmetic gives 78 GeV for the back-scattered photons — 1.7 times the energy of the electron that is supposed to supply it. The exact edge is 29.2 GeV. Nothing about the scattering changed between the gigaelectronvolt case and this one; what changed is that the photon in the electron’s frame stopped being small compared with the electron, so the electron stopped behaving as an immovable reflector. The 1 GeV infrared case sits 1.8 per cent below its mirror line. The 46.6 GeV case sits at 37 per cent of it.

Angle sorts the photons by energy #

The highest energy belongs to photons scattered straight along the electron’s direction. Photons scattered at an angle get less, and the relation is close to exact and very simple: at an angle $\theta$ from the electron’s direction, the energy is the edge divided by about $1 + \gamma^2\theta^2$. The angle that matters is measured in units of $1/\gamma$, the same angle the sky that crowds into a cone found every fast source’s light squeezed into: at $\theta = 1/\gamma$ the photon carries half the edge energy, at $2/\gamma$ a fifth.

For a gigaelectronvolt electron $1/\gamma$ is 511 microradians — half a milliradian, a beam that has spread by half a millimetre a metre downstream. All of the scattered light, from the edge down to a few per cent of it, comes out in a cone a few milliradians wide, which is what makes the source useful at all. A thermal source spreads its light over the sky; this one sends it into a pencil.

The angle–energy lock is what turns a broad spectrum into a narrow one. Every photon arriving close to the axis has an energy close to the edge, so an aperture on the axis is an energy filter: passing $\gamma\theta < 0.10$ selects a band 1 per cent wide at the top of the spectrum, in a cone 52 microradians across. Nothing else in gamma-ray physics offers that. Bremsstrahlung, the gamma rays an electron makes when it is stopped in a target, has a spectrum running from zero to the electron’s energy with no relation between direction and energy worth having, and a crystal monochromator does not work at nuclear energies. The narrowness is bought with intensity — most of the photons fall outside the aperture — but a few-per-cent band of polarised gamma rays at a tunable energy is exactly what an experiment on nuclear resonances needs, and tuning it means changing the electron energy or the laser colour.

With recoil the cone widens. The relation becomes $(1 + x)/(1 + x + \gamma^2\theta^2)$, so at $x = 1.68$ the energy halves at $\gamma\theta = 1.64$ rather than at 1. The recoil takes energy away from the photons near the axis and leaves the off-axis photons, which were softer in the electron’s frame, relatively less affected.

A flat spectrum that tips towards its edge #

How many photons come out at each energy is set by the cross-section in the electron’s frame. In the soft limit it is Thomson scattering, whose angular pattern is $1 + \cos^2$ of the scattering angle — as likely to throw a photon straight back as straight forward, half as likely to throw it sideways. Transformed to the laboratory, where angle has become energy, that pattern becomes a spectrum shaped like a shallow U: as many photons per unit energy at the edge as at the bottom, half as many in the middle, $1 - 2u + 2u^2$ in the energy $u$ measured as a fraction of the edge. The back-scattered and forward-scattered photons of the electron’s frame are the top and the bottom of the laboratory spectrum, and the sideways ones are the trough.

When the photon in the electron’s frame is no longer soft, the cross-section is Klein and Nishina’s, worked out in 1929 from Dirac’s then new equation for the electron. It does two things to this spectrum. It tips it towards the high end — the hardest photons become more likely relative to the softest — and it lowers the total, because a photon comparable in energy to the electron is scattered less often than Thomson’s constant allows.

The tipping is the property a designer of a gamma-ray collider wants. A collider that made its photon beams by back-scattering laser light off two linear electron beams — proposed in detail from the early 1980s by Ilya Ginzburg, Valery Telnov and colleagues at Novosibirsk — needs as many photons as possible near the edge, so that the energy of the photon–photon collisions is well defined. The larger $x$, the more of the spectrum sits at its top and the more of the electron’s energy each photon carries.

The line the laser energy cannot cross #

So why not make $x$ as large as possible, with the hardest laser light available? The answer is in the last figure, and it comes from the photons themselves rather than from the electron. A back-scattered photon near the edge travels forward through the laser pulse that made it, meeting laser photons head-on. Two photons can make an electron–positron pair if the energy available between them reaches twice the electron’s rest energy — the same invariant-mass condition the invariant that survives a boost applied to a pair of 511-keV photons. Head-on, that condition is $4E_\gamma\varepsilon \ge (2mc^2)^2$ — the threshold arithmetic the collision that wastes most of the energy used for colliding beams, with photons for the beams.

Put the edge energy $xE_e/(1 + x)$ into it and the electron’s energy cancels out. The condition becomes $x^2 = 4(1 + x)$, whose positive root is

$$x = 2 + 2\sqrt2 = 4.83.$$

Below it, no back-scattered photon can make a pair with the laser light. Above it, the hardest ones can, and a share of the gamma-ray beam is converted into pairs before it leaves the laser pulse. The threshold depends on nothing but the recoil parameter: not on the laser intensity, not on the electron energy separately.

The two curves on that figure are the two things the recoil parameter trades. As $x$ rises from 0.018 to 4.83 the photon’s share rises from 1.8 per cent of the electron’s energy to 83 per cent; the probability of scattering at all falls from Thomson’s value to 29 per cent of it. A design sitting at $x \approx 4.8$ is therefore not a compromise between intensity and energy but a hard stop: below it, every gain in energy share is usable; above it, the gain goes into making pairs. For a 250 GeV electron beam the line falls at a laser photon of 1.26 eV, close to the wavelength of the neodymium lasers the designs assumed.

That pair-making is not a nuisance in every context. In E-144 it was the point: the same collisions of back-scattered photons with laser photons, in a field intense enough for several laser photons to act together, produced positrons from light alone — the first laboratory production of matter in collisions involving only photons, a process predicted by Gregory Breit and John Wheeler in 1934. The calculation behind the threshold here is the single-photon version of that result.

Two contractions, and what the undulator lacked #

The comparison with the undulator is worth making exactly, because it says what the laser buys. An undulator of period $\lambda_u$ makes light of wavelength $\lambda_u/2\gamma^2$ on axis: one factor of $\gamma$ from contraction of the magnet row and one factor of $2\gamma$ from the forward Doppler shift. A counter-propagating laser of wavelength $\lambda_L$ makes $\lambda_L/4\gamma^2$: the first factor is $2\gamma$ instead of $\gamma$, because the oncoming wave is not only shortened in the electron’s frame but moving into the electron at $c$, so its crests arrive twice as often again. Equivalently, the laser is an undulator whose period is half its wavelength.

That half-micrometre period is the whole attraction. An undulator’s period is set by magnets and is a centimetre or more; a laser’s is a micrometre. To make 10-keV X-rays from a 3-centimetre undulator needs electrons of several gigaelectronvolts and a storage ring a few hundred metres round. To make them from a 1-micrometre laser needs electrons of about 25 MeV, which a few metres of linear accelerator provide. Compact X-ray sources built on this principle exist, the size of a room rather than a campus, and their limitation is the one the undulator does not have: a magnet row is thousands of periods long and fixed, while the laser pulse is short, the electron bunch has to meet it at the right moment, and the scattered flux is modest because Thomson’s cross-section is small.

Length contraction was half of one factor in the undulator and is half of one factor here. Contraction alone would give the undulator’s $\gamma$; motion of the wave towards the electron supplies the other $\gamma$ that turns it into $2\gamma$; and the Doppler shift on the way back supplies the rest. That is why a laser photon gains $4\gamma^2$ where a magnet’s field gains $2\gamma^2$.

What the figures leave out #

Every figure here takes the electron and the photon to collide one at a time: a single electron, a single photon, one scattering. Three effects that matter in a real source are not in them. The electron beam has a spread of angles and energies, so the edge is smeared and the angle–energy lock is blurred by the beam’s own divergence; a source is only as monochromatic as its electron beam is parallel. The laser has a bandwidth, which smears the edge further by the same fraction. And when the laser is intense — when the dimensionless field strength that makes an electron’s quiver relativistic approaches one — an electron absorbs several laser photons at once and emits harmonics of the edge, and its effective mass in the field rises, which moves the edge down. That nonlinear regime is where E-144 operated, and none of it appears in a cross-section for one photon.

The figures also show only energies, never polarisation. Near the edge the scattered photons carry the laser’s polarisation almost entirely, which is why these beams are prized for nuclear physics, and the spectrum depends on the electron’s and laser’s spins in ways the unpolarised curves do not show. The domain of what is drawn is a free electron, one photon at a time, a laser weak enough for its field to be a perturbation, and photon energies in the electron’s frame of up to a few times its rest energy — a regime in which the Klein–Nishina formula is exact to the accuracy of quantum electrodynamics at lowest order.

Still open: how small a gamma-ray source can be made #

The limits on a laser-Compton source now are practical rather than physical: how many electrons can be put into a bunch small enough to meet a focused laser pulse, how often the meeting can be repeated, and how much laser power can be recirculated in an optical cavity so that each pulse meets thousands of bunches. Sources built on storage rings with recirculating laser cavities, and others built on linear accelerators, are pushing the flux up by orders of magnitude, aiming at gamma-ray beams bright enough to map isotopes in nuclear waste or to drive nuclear reactions selectively. Whether electrons accelerated in a plasma wave — the wave an electron rides to a gigaelectronvolt — can be made reliable and monoenergetic enough to replace the accelerator, putting a gamma-ray source on a table, is under active development, and the answer depends on the plasma accelerator’s energy spread more than on anything in the scattering.

The photon collider that motivated the design limit has not been built. It was planned as an option on a future linear electron–positron collider, and its fate is tied to that machine’s.

The scattering itself is settled. A fast electron throws a laser photon back with $4\gamma^2$ times its energy as long as the photon is soft in the electron’s frame — two Doppler factors and a bounce — and with $x/(1 + x)$ of the electron’s own energy once it is not, $x = 4\gamma\varepsilon/mc^2$: 17.6 MeV from a 1.17-eV photon on a gigaelectronvolt electron, 29.2 GeV rather than the mirror’s 78 from green light on 46.6 GeV. The electron is a mirror until its recoil is noticed, and the recoil is noticed precisely when the mirror’s promise would have broken the conservation of energy.