The test of mass and energy that needs a ruler

Mass is a form of energy closes with the sharpest direct test of its title. In 2005 a team led by Simon Rainville, with the Penning-trap group of David Pritchard at MIT and the gamma-ray spectroscopists of the Institut Laue–Langevin in Grenoble, compared the mass lost when a nucleus captures a neutron with the energy of the gamma rays it emits, for silicon and for sulphur, and found them equal to four parts in ten million. That essay uses the result to argue that mass and energy are one quantity. This one takes the experiment apart, because what had to be true and what had to be measured for the comparison to mean anything is a small education in how a fundamental law is tested at all.

The law is simple to state as a balance. A silicon-28 atom and a free neutron, at rest, together have a certain mass. A silicon-29 atom at rest has less. When the neutron is captured, the difference leaves as gamma rays. If $E = mc^2$ holds, the energy of the gamma rays, plus whatever the atom keeps as recoil, is the mass difference times $c^2$. Weigh both sides; measure the gamma rays; compare.

A mass that goes missing as light #

Every neutron capture loses mass. The amount, the separation energy of the neutron in the heavier nucleus, is a few MeV for most nuclei and carries the shape of the curve of binding energy: nuclei in the middle of the periodic table hold their last neutron most tightly.

The six points in the figure lie exactly on the line $\Delta m = E/c^2$, but only because the energies were computed from the masses. That is how almost every “confirmation” of $E = mc^2$ in nuclear physics works: tables of atomic masses, measured to high precision, are used to predict the energies of reactions, and the predictions are borne out. The agreement is excellent and it tests the equation, but indirectly and at modest precision, because the reaction energies are measured with detectors calibrated against other reactions, whose energies were in turn computed from masses. The circle has to be broken by measuring the energy in units that owe nothing to mass.

The deuteron, the first point on the line, has a historical place in this. The gamma ray emitted when a neutron is captured by a proton, 2.2246 MeV, was measured by crystal diffraction in the 1990s and used, with $E = mc^2$ assumed, to determine the mass of the neutron itself: a free neutron cannot be weighed in a trap, since it carries no charge to circle a magnetic field. The 2005 experiment chose nuclei whose atoms could be weighed on both sides.

Weighing an atom by how fast it circles #

A Penning trap holds a single ion in a strong uniform magnetic field, with a weak electric field to stop it escaping along the field lines. The ion circles the field at the cyclotron frequency $qB/2\pi m$ — the frequency the force that does no work derives for any charge in a magnetic field — and the frequency is measured from the tiny image current the circling ion induces in the trap’s electrodes.

Frequency is the most precisely measurable quantity in physics, because it can be counted. An ion held for minutes completes hundreds of millions of revolutions, and the frequency can be fixed to parts in $10^{11}$. The magnetic field is the troublesome part, since it drifts and is never known to that precision; but the ratio of two ions’ frequencies in the same field is the inverse ratio of their masses, and the field cancels. The MIT trap alternated between ions of the two species, or held them simultaneously, so that both saw the same field.

The mass lost in the silicon capture is three ten-thousandths of the silicon-29 atom. A mass ratio measured to $10^{-11}$ gives that difference to better than a part in ten million, and in practice the experiment reached a few parts in $10^{8}$ on the mass differences. The weighing is not the weak link. That is the first surprise, given that a mass difference is a small difference of two large numbers: it is small, but not small enough to defeat a technique that counts revolutions.

Measuring a gamma ray with a ruler #

The other side is the energy of the gamma rays, and here the difficulty is of a different kind. Gamma-ray detectors measure energies by the charge a gamma ray liberates in a crystal of germanium, to a few parts in ten thousand, and they are calibrated against known lines. To measure an energy without reference to any other energy, it has to be converted into something that can be compared directly with the metre: a wavelength.

The wavelength of an 8.47 MeV gamma ray is 0.146 picometres, a thousandth of the spacing between atoms in a crystal. A perfect silicon crystal diffracts it all the same, by the Bragg condition that everything has a wavelength applies to any wave meeting a lattice: reflection from the crystal’s planes happens only at the angle whose sine is the wavelength divided by twice the plane spacing. For gamma rays of several MeV that angle is a few hundred microradians, a few hundredths of a degree. Measuring the energy to four parts in ten million means measuring that angle to a hundred and fifty picoradians.

The GAMS spectrometers at Grenoble do this with two perfect silicon crystals and optical interferometers that read the rotation of the crystal mounts, the gamma rays coming from a target placed close to the core of the institute’s research reactor, where the neutron flux is enormous. The angle is only half the problem; the other half is the spacing of the planes, in metres. That is measured by X-ray interferometry: a crystal is cut so that X-rays diffracted by it produce fringes whose spacing is the lattice spacing, and those fringes are counted against a laser of known wavelength as one part of the crystal is moved. The chain runs from the gamma ray to an angle, from the angle and a lattice spacing to a wavelength, and from the lattice spacing to the laser and so to the metre.

The energy the atom keeps #

The balance is not simply the mass lost against the gamma rays. A gamma ray carries momentum as well as energy, and the atom that emits it recoils.

The gamma ray’s momentum is $E/c$, and the silicon-29 atom must carry away the same momentum in the opposite direction, with kinetic energy $E^2/2Mc^2$. For a gamma ray of 8.47 MeV that is 1.33 keV, one part in about six thousand four hundred of the energy released. The same recoil is why, as the gamma ray its own kind cannot absorb explains, a nucleus generally cannot reabsorb the gamma ray an identical nucleus emitted: the emitted photon is short of the transition energy by exactly the recoil, and the absorber needs that much extra. In the comparison, the recoil is a correction four hundred times larger than the precision sought, and it has to be computed, not measured.

It is computed from relativistic kinematics: from conservation of energy and momentum for a massive atom and a massless photon, with the photon’s momentum taken as $E/c$. That relation is itself a prediction of relativity, the one that a photon with a momentum traces through the Compton effect. So a test of $E = mc^2$ cannot be performed without first assuming another piece of the same theory. It does not make the test circular, because the recoil correction is small and its dependence on the theory is weak — if the photon’s momentum were wrong by a part in ten thousand, the recoil would be wrong by the same fraction, which is a part in ten million of the energy, just at the limit. But it shows how a law is tested in practice: against the background of others, each tested separately, that it is logically entangled with.

The recoil also has to be computed for the intermediate steps. The capture usually emits several gamma rays in a cascade through the excited states of silicon-29, and each one recoils the atom in a different direction; the experiment measured enough of the lines of each cascade that their sum, with recoils, gave the total energy released.

What had to be right #

Putting the pieces side by side shows where the precision was spent.

The result, combining silicon and sulphur, was $1 - \Delta m c^2/E = (-1.4 \pm 4.4) \times 10^{-7}$: zero, to an uncertainty of four parts in ten million, the tightest direct test of the equation. The uncertainty is dominated by the gamma-ray energies — by the angles, the lattice spacing and the knowledge of the cascade — not by the masses, which had an order of magnitude to spare. That inverts the naïve expectation that the “small difference of large numbers” would be the problem. The large numbers are measured by counting, and the small energy has to be measured by a ruler.

What the comparison compares #

There is a final subtlety, and it changed meaning in 2019. The masses are measured in atomic mass units, as ratios of ion frequencies. The energies are measured in joules, or electronvolts, through a wavelength in metres and Planck’s constant: $E = hc/\lambda$. To compare them, a conversion between the atomic mass unit and the joule is needed, and that conversion is itself a product of measured constants: the atomic mass unit in kilograms, and Planck’s constant in joule-seconds. In 2005 both were measured quantities, and the combination actually needed — the ratio of Planck’s constant to the atomic mass unit — was known to better than a part in a hundred million from experiments with atoms recoiling from absorbed photons in interferometers, a third precision measurement entirely separate from the first two.

Since 2019 the kilogram has been defined by fixing Planck’s constant exactly, and the atomic mass unit in kilograms has become a measured quantity whose value depends on that definition. Read in today’s units, the 2005 comparison is a statement that four independent measurements — two frequency ratios, a crystal’s angles, a crystal’s lattice spacing against a laser — and two constants close a loop to four parts in ten million. A failure of $E = mc^2$ would have shown up as a failure of the loop to close. So would an error in any one of the measurements, which is why it is a test of all of them at once, and why it is reassuring rather than merely confirming that the loop closed.

That is the general structure of precision tests of fundamental laws. Nothing is tested alone. The box of light that weighs something derives $E = mc^2$ from the momentum of light and the conservation of the centre of mass, and every step of the experiment rests on relations of the same kind — the photon’s momentum in the recoil, the cyclotron frequency’s dependence on mass, the wave relation between energy and wavelength. The test asks whether all of them can be true together with the equation. They can, to four parts in ten million.

What a failure would have meant #

It is worth asking what kind of departure the experiment could have found, because “$E = mc^2$ is wrong” is not a single hypothesis. The equation can be read as the statement that the rest energy of a system is its invariant mass times $c^2$ — the quantity the invariant that survives a boost builds from energy and momentum together. In that reading it is a consequence of the Lorentz symmetry of spacetime, and testing it is testing whether a nucleus’s inertia responds to the energy stored in it in the way that symmetry requires. A failure would mean that energy stored in the strong nuclear field contributes to inertia differently from other energy, or that photons and massive bodies obey slightly different energy–momentum relations.

Both alternatives have relatives elsewhere. If nuclear binding energy contributed differently to inertia, it would very probably also contribute differently to weight, and the binding energy that has to fall too traces how tests of free fall bound that possibility for gravitational mass; the 2005 comparison bounds it for inertial mass, by a different route. If photons obeyed a slightly different relation between energy and momentum, the recoil correction would change and so would the Compton effect; such a modification is among those that searches for a violation of Lorentz symmetry look for in astrophysical gamma rays and in clock comparisons. A result of $(-1.4 \pm 4.4) \times 10^{-7}$ excludes the strong-interaction version of the first at that level for the binding energies of light nuclei.

There is also the simplest interpretation, which the result rules out most firmly. If mass and energy were related by any constant other than $c^2$ — if the speed in the equation were some speed other than light’s, as a theory that distinguished a “speed of inertia” from a speed of light might have it — the comparison would have failed by the square of the ratio. It agrees with $c^2$ to two parts in ten million in the speed, a speed measured by its role in inertia rather than by timing anything.

Why silicon and sulphur #

The choice of nuclei was dictated by both halves of the experiment. On the trap side, the atoms on both sides of the capture had to form singly charged ions that could be trapped and compared, and their masses had to be measurable against a common reference; silicon and sulphur are light, their isotopes are stable, and their ions are well behaved. On the gamma-ray side, the capture had to happen readily — silicon-28 and sulphur-32 have convenient cross-sections for slow neutrons — and the gamma-ray cascades had to be simple enough that the total energy could be assembled from a few lines, each bright enough to measure by diffraction. Two nuclei rather than one gave a check that nothing depended on the particular cascade.

The deuteron, the simplest capture of all, could not serve, because one side of its balance is a free neutron, which cannot be trapped and weighed; it is the reason the neutron’s mass is known only through $E = mc^2$, rather than as a test of it. The atoms chosen for the 2005 experiment were the simplest for which the whole balance could be measured.

Masses from the tables, angles from one set of planes #

The masses are those of the atomic mass evaluation, and the separation energies in the figures are computed from them, so the first figure shows the line the experiment tested rather than the experiment’s own data. The uncertainties in the second figure are representative of the published analysis, not a reproduction of its error budget, which separates many contributions from the crystal, the angle interferometers, the cascade and the trap. The Bragg angles are first-order for the silicon (220) planes; the spectrometers use particular orders and crystal geometries chosen for each line. The trap field is a representative 8.5 tesla; the frequencies quoted are free cyclotron frequencies, which in a real trap are reconstructed from the measured motional frequencies. Electron binding energies, which make atomic and nuclear mass differences differ slightly, cancel here because the atoms on both sides have the same number of electrons, and the figures ignore them.

Still open: whether the loop can be closed a hundred times tighter #

The gamma-ray side is the bottleneck, and improving it means better crystals, better angle interferometry and better knowledge of the lattice spacing, all of which are pushed by other needs: the silicon spheres used to count atoms for the kilogram, and X-ray interferometry for length metrology. Penning traps have moved on to parts in $10^{12}$. A comparison at a few parts in $10^{9}$ would test the equation at a level where some theories of how gravity or quantum effects might modify the energy–momentum relation would begin to be constrained, and would test the consistency of the international system of units at the same time. Whether the crystal measurements can reach that is not yet clear.

The habit worth carrying away is to ask what a test assumes. The 2005 comparison weighed silicon-28 plus a neutron against silicon-29 as cyclotron frequencies, measured the 8.47 MeV of gamma rays as angles of a few hundred microradians, added a recoil of 1.33 keV computed from the photon’s momentum, and found $\Delta m c^2$ equal to $E$ to four parts in ten million. A law is tested by asking whether everything else that has to be true can be true with it.