The gamma ray its own kind cannot absorb

Mass is a form of energy found that a body that gives out energy loses mass in proportion, E/c², whether the energy leaves as heat, as motion or as light. The mass that is missing found the same deficit in a bound nucleus. The invariant that survives a boost gave the one combination of energy and momentum every observer agrees on, and later essays weighed a box of light and found that light gravitates twice as strongly as its mass.

This essay takes the smallest possible system that gives out energy: a single excited nucleus emitting a single photon. The photon carries energy and also momentum, and the nucleus must take the opposite momentum. That recoil costs a share of the energy, so the photon arrives with slightly less than the gap between the levels — and an identical nucleus, waiting to absorb it, needs slightly more. For light from atoms the difference is far too small to matter. For gamma rays from nuclei it is enormous, it was a puzzle for decades, and the way round it, found by accident in 1958, produced the sharpest spectral line in physics.

The energy a recoil takes #

Before the emission the nucleus is at rest with a mass that includes its excitation energy. Afterwards there are two things, the nucleus in its lower state and a photon, moving apart with equal and opposite momenta. This is the two-body decay that the cone a decay cannot leave solved, with one product massless, and conservation of the invariant mass fixes the photon’s energy exactly:

$$E_\gamma = \frac{(M_i^2 - M_f^2)c^2}{2M_i} \approx E_0 - \frac{E_0^2}{2Mc^2},$$

where $E_0 = (M_i - M_f)c^2$ is the energy difference between the levels. The deficit is the kinetic energy of the recoiling nucleus, the square of the photon’s momentum over twice the nucleus’s mass. It grows as the square of the photon energy: two decades in recoil for each decade of photon energy. For visible light from an atom it is a few times ten to the minus ten electronvolts; for a gamma ray of tens of keV from a nucleus, a thousandth of an electronvolt or more.

Absorption runs the arithmetic backwards. A nucleus at rest absorbing a photon must take the photon’s momentum and recoil, and the photon must supply both the excitation energy and that recoil. So the emitted photon has $E_0 - E_R$ and the absorber needs $E_0 + E_R$: emission and absorption miss each other by $E_0^2/Mc^2$.

A small miss against a narrow line #

Whether the miss matters depends on what it is compared with, and the comparison is with the width of the line. An excited state that lives for a time τ has an energy uncertain by about ħ/τ, the relation the width that is a lifetime established, so its emission and absorption are spread over that width. If the miss is smaller, emission and absorption overlap and absorption happens; if it is larger, they do not.

For sodium’s yellow line, 2.1 electronvolts from a level that lives sixteen nanoseconds, the miss is two hundred times smaller than the natural width, and the Doppler broadening from the atoms’ thermal motion is thousands of times larger still. Sodium vapour absorbs sodium light strongly; a sodium flame held in front of a sodium lamp casts a dark shadow, which is how Robert Bunsen and Gustav Kirchhoff found that the dark lines in the Sun’s spectrum are the bright lines of the elements, reversed.

For iron-57, whose excited level at 14.4 keV lives 141 nanoseconds, the natural width is 4.7 × 10⁻⁹ eV, a part in three million million of the photon’s energy, and the miss is 3.9 × 10⁻³ eV — eight hundred and forty thousand widths. A free iron-57 nucleus cannot absorb the gamma ray of another. The only overlap comes from thermal motion: at room temperature the Doppler spread of a few thousandths of an electronvolt smears emission and absorption into broad humps that touch at their edges, and experimenters in the 1950s detected nuclear resonance absorption by heating sources to broaden the humps further, or by spinning them on a rotor to Doppler-shift the gamma rays upward by the missing amount.

A crystal that takes the kick #

Rudolf Mössbauer, a doctoral student in Heidelberg, was measuring the resonant absorption of the 129 keV gamma ray of iridium-191 in 1958, expecting it to decrease when he cooled the source and absorber, since cooling narrows the Doppler humps and should reduce their overlap. It increased. The explanation he worked out was that a nucleus bound in a crystal does not have to recoil alone.

The atoms of a crystal are held in place by their neighbours, and the crystal can take up energy only in quanta of its vibrations, the phonons that how many ways there are to vibrate counted. When a nucleus in the crystal emits a gamma ray, the momentum must go somewhere, but the energy can go two ways: into one or more phonons, as a free recoil would, or into nothing at all, the momentum being taken up by the crystal as a whole. The crystal’s mass is the mass of something like 10¹⁸ atoms or more, and a recoil against it has a kinetic energy smaller by that factor — nothing at all, for any practical purpose. Quantum mechanics gives a definite probability for each outcome, and when the recoil energy is small compared with the energy of the crystal’s vibrational quanta, the zero-phonon outcome is likely.

That probability, the recoil-free fraction, depends on how hard the nucleus is kicked compared with how stiffly it is held, and on how much the crystal is already vibrating. In the Debye model it is computed from the recoil energy and the crystal’s Debye temperature. Iron-57 in metallic iron recoils gently against a stiff lattice: more than four-fifths of its gamma rays are recoil-free even at room temperature. Iridium’s 129 keV line recoils twenty-four times harder, and only a few per cent of its emissions are recoil-free unless the crystal is cooled — which is why Mössbauer’s absorption rose when he cooled his samples, and why he received the Nobel Prize in 1961 for a discovery made while looking for something else.

The recoil-free photons carry the full transition energy, unshifted and unbroadened by the thermal motion of the atoms, because a crystal at rest as a whole does not Doppler-shift anything. They form a line of the natural width alone, a few billionths of an electronvolt on top of fourteen thousand, and an identical crystal absorbs them resonantly.

The same factor that dims an X-ray diffraction spot #

The recoil-free fraction has a twin that every crystallographer knows. When X-rays are diffracted by a crystal, the sharp Bragg spots come from scattering that leaves the crystal’s vibrations untouched — elastic scattering off the lattice as a whole — while scattering that creates or absorbs a phonon spreads into a diffuse background. The fraction of intensity left in the sharp spots is the Debye–Waller factor, $e^{-q^2\langle u^2\rangle}$, where q is the momentum the X-ray transfers and $\langle u^2\rangle$ is the mean square displacement of the atoms. The Mössbauer fraction is the same expression, with the gamma ray’s own momentum in place of q.

Both say the same thing about a crystal: it can absorb a momentum without absorbing energy only if its atoms are not displaced much, compared with the wavelength of the photon, by their vibrations. That displacement never vanishes, even at absolute zero, because the motion that cannot be stopped leaves every atom with zero-point motion; that is why iron-57’s fraction is 0.93 and not 1 at zero temperature. A stiff lattice and a long wavelength make the factor close to one. Hard gamma rays, with short wavelengths, need very stiff or very cold crystals — the reason the effect is confined to gamma rays below about a hundred keV.

A resonance seen in time #

Since the 1980s the same nuclear resonance has also been excited with X-rays from synchrotrons instead of radioactive sources. A synchrotron flash is a few picoseconds long and contains every energy across a broad band; the tiny fraction of it at exactly 14.4 keV excites iron-57 nuclei in a sample coherently, and the nuclei re-emit over the following hundred nanoseconds, long after the flash itself has passed. Timing the delayed photons measures the resonance in the time domain instead of by sweeping a Doppler drive, and any splitting of the levels appears as a beating in the decay, the quantum beat between the split lines. The method works on samples too small or under pressures too high for a radioactive source, such as iron at the conditions of the Earth’s core.

A spectrometer driven by a loudspeaker #

A line that narrow is an instrument, and the way it is scanned is the most charming thing about it. The source is mounted on a drive like a loudspeaker’s and moved towards and away from the absorber at a few millimetres a second, so that the Doppler effect shifts the gamma ray’s energy by v/c, a few parts in a hundred thousand million. A millimetre a second is ten natural widths. Sweeping the speed sweeps the energy, and counting the gamma rays that get through the absorber at each speed traces out its absorption spectrum in units of velocity.

The spectrum measures everything that shifts or splits a nuclear level by parts in a million million. In metallic iron the nucleus sits in an internal magnetic field of about thirty-three tesla, made by its own atom’s electrons, and the field splits the ground and excited levels so that six transitions are allowed, giving the six dips in the figure. In an iron compound without magnetic order the field is absent and the line is single, or split in two by an electric field gradient. The line’s position shifts with the density of electrons at the nucleus, which differs between iron’s chemical states. A Mössbauer spectrum therefore tells which form of iron a sample contains, in what proportions, and in what surroundings: a method used on minerals, catalysts, haemoglobin and, from two Mars rovers carrying small Mössbauer spectrometers, on the rocks of Mars, where it identified iron minerals that form only in water.

A tower in a stairwell #

The sharpest use of the line was its first. In 1960 Robert Pound and Glen Rebka put an iron-57 source at the top of a tower in the physics building at Harvard and an absorber at the bottom, 22.5 metres below, and measured whether the gamma rays arriving at the bottom had gained energy by falling, as the clock that runs slow lower down requires: a fractional gain of gh/c², 2.5 × 10⁻¹⁵. That is less than a hundredth of the line’s width, and the recoil miss of free nuclei, which the Mössbauer effect removed, is a hundred million times larger. They found the shift by moving the source slowly and looking for the speed at which the absorption was symmetric, and confirmed Einstein’s prediction to ten per cent, later to one.

The figure shows the other shift they had to fight. The nuclei in a crystal vibrate, and a vibrating clock runs slow by the time dilation of its motion, as the clock that runs slow because it is warm found: averaged over a vibration the first-order Doppler shift cancels and the second-order one, the relativistic time dilation, does not. Warming a crystal by one kelvin lowers its gamma ray’s energy by 2.4 × 10⁻¹⁵, almost exactly the gravitational shift over the whole tower. Pound and Rebka held source and absorber at the same temperature and corrected for what remained. The experiment that first measured gravity’s effect on light in a laboratory was, as much as anything, a measurement of temperature.

Why atoms never had the problem #

The contrast between atoms and nuclei is not that nuclei are heavier — they are the same particles, an atom’s mass being almost entirely its nucleus. It is that their photons carry so much more momentum. The recoil energy is the photon’s momentum squared over twice the mass, and the momentum is the photon’s energy over c; a gamma ray of fourteen thousand electronvolts carries seven thousand times the momentum of a visible photon and kicks fifty million times harder. Meanwhile the natural width of a level is set by its lifetime, which for nuclear and atomic levels covers a similar range. Visible photons are too gentle to matter; gamma rays are too forceful to ignore.

The same recoil is not negligible for atoms in every context. A photon with a momentum showed Compton’s X-rays shifted by the recoil of the electrons they hit, and in laser cooling, which the limit that belonged to a simpler atom followed, the recoil of a single visible photon sets the coldest temperature the simplest cooling reaches — a few hundred nanokelvin for rubidium. There the recoil is compared not with the line’s width but with the atom’s tiny thermal energy, and it is large.

Where the picture stops #

The figures treat the emitter as a free nucleus or as a nucleus in a Debye crystal, a model that replaces a real crystal’s vibrations by a smooth spectrum with a single cut-off and is good to some tens of per cent for the recoil-free fraction of simple metals. Real lattices have optical modes, anisotropy and impurities, and the fraction is measured rather than computed when it matters. The sextet is drawn with ideal Lorentzian lines of twice the natural width and the textbook positions and intensities for a thin, randomly oriented iron foil; thick absorbers broaden and saturate the lines, and the intensities change if the foil’s magnetisation is oriented. The shifts figure compares fractional sizes and ignores everything else the Harvard experiment had to control, from vibrations of the tower to the drift of the source’s speed.

The domain of the recoil argument is any emission of a photon by a body free to recoil. The Mössbauer effect’s domain is narrower: gamma rays of up to about a hundred keV, from nuclei with excited states living long enough to give narrow lines, in solids stiff enough to hold them. Iron-57 is the best of a few dozen such isotopes, which is why nearly all Mössbauer spectroscopy is done on iron and tin.

Still open: whether a nucleus can be made into a clock #

The narrowest lines are the most precise clocks. Atomic clocks now tick by optical transitions, measured to parts in 10¹⁸, and a nuclear transition would be less disturbed by stray fields, since the nucleus is shielded by its electrons and much smaller. Most nuclear transitions are far beyond the reach of lasers, but thorium-229 has an excited state only about eight electronvolts above its ground state, in the ultraviolet, and in 2024 it was excited by a laser for the first time, its frequency measured to a precision that puts a nuclear clock within reach. Whether such a clock will surpass atomic ones, and whether its comparison with them will reveal any drift in the constants of nature, to which a nuclear transition is unusually sensitive, are the questions the next decade of measurements will answer.

The habit worth carrying away is to compare a shift with the width it has to be resolved against, not with the energy it is part of. A nucleus that emits $E_0$ recoils and keeps $E_0^2/2Mc^2$; emission and absorption by free nuclei miss by $E_0^2/Mc^2$ — 3.9 meV for iron-57, 840,000 natural widths — while for sodium’s yellow line the miss is 200 times smaller than the width. Held in a crystal, the nucleus can recoil against 10¹⁸ atoms at once, and the line that results resolves gravity’s 2.5 × 10⁻¹⁵ over a 22.5 m tower. The mass that leaves with a photon is not only E/c²; it is the reason the photon arrives short.