How do we know the most famous equation in physics — E = mc² — is actually correct?

The answer, of course, involves experiments. It doesn’t matter how brilliant Einstein (or anyone) is, no theory is good if it doesn’t hold up under experimental tests. The relation between mass and rest energy is no different. Indirect tests exist everywhere: without the physics underlying E = mc², nuclear and particle physics wouldn’t work at all. However, it’s challenging to measure the mass-energy relation directly, and the key is finding the mass of two systems that differ only in energy content.

The experiment: confine sulfur atoms in a trap. Bombarding those atoms with neutrons makes a new type (isotope) of sulfur, but in an “excited state”: one in which the protons and neutrons in the nucleus are bubbling around a bit more rapidly than usual. The excited state isn’t stable: it “decays” by emitting a gamma ray photon without changing its nuclear composition. In other words, before and after the emission, a given ion has the same number of protons and neutrons; the only difference is the amount of energy it possesses. Comparing the mass before and after the decay with the energy carried away by the gamma ray provides an extremely sensitive test of E = mc².

That’s the overview of the experiment performed by Simon Rainville and colleagues; their results were published in Nature in 2005, marking the most accurate experimental validation of E = mc² at the time. Obviously there’s a lot of details I skipped: you can’t just measure the mass of a single sulfur atom by slapping it on a scale, nor is it as easy to measure gamma ray energy as Star Trek would have you believe.

TheE test

The first part of the experiment is the actual nuclear physics bit: bombarding sulfur atoms with neutrons and measuring the energy of the gamma rays that are emitted. We can write the reaction as a kind of equation:  The symbols ³²S and ³³S indicate the two different isotopes of sulfur, S: while sulphur always has 16 protons, it can have several different neutron numbers. The number attached to the element name is the number of protons plus the number of neutrons, so ³²S has 16 neutrons and ³³S has 17. n is a neutron, and γ is the emitted gamma ray. The “*” in the middle term ³³S* indicates the nucleus is “excited”, temporarily existing in a higher unstable energy state.

The researchers measured the energy of the emitted gamma ray using a time-honored method called Bragg diffraction. Without going into detail, gamma rays bounce off atoms in a solid crystal in a predictable way, with the angle of scatter depending on the wavelength of the gamma ray. Since the energy of light is determined by its wavelength, measuring the angle of scatter measures the energy of the gamma ray. The situation is slightly complicated by the fact that sulphur can have more than one excited state after absorbing a neutron, so the experiment needed to combine results from several different gamma ray energies.

The mc² test

As I mentioned previously, we can’t measure the mass of an atom by placing it on a scale; microscopic mass measurements require other methods. The 2005 experiment used a Penning trap: a device using electric and magnetic fields to confine individual ions — atoms with one electron missing. (Neutral particles won’t be trapped: you need a net electric charge for the particle to feel the effects of the confining fields.) Penning traps have been used since the 1950s to measure properties of individual ions and electrons; newer experiments combine them with extreme cryogenic cooling to eliminate effects that might interfere with the measurements.

When confined in a Penning trap, charged particles follow orbits (though without a central “sun”) dictated by the electric and magnetic fields. The simplest version is just circular motion — that’s what we see in the 2005 E = mc² experiment — but under some circumstances particles can describe loopy paths, much like Ptolemaic epicycles from the old geocentric Solar System model. In any case, the rate at which the particle orbits depends on its electric charge and mass, so for our sulphur atom with one missing electron, measuring the frequency of motion is the same as measuring the mass.

The researchers actually confined two ions in the Penning trap: the more massive ³³S ion, along with a molecule consisting of a ³²S atom and a proton. Because protons and neutrons have nearly the same mass, the molecule and the ion also have nearly the same mass, but they also consist of the states of the sulphur nucleus before and after neutron bombardment.

Finally, the relevant calculations subtract the mass of the resulting ion (³³S) after the collision from the total mass (³²S plus neutron), multiplied by the speed of light squared. If E = mc² is correct, then the equivalent energy of the mass difference before and after should be exactly the same as the energy in the gamma ray emitted when the nucleus decays from the exicited state into its ground state.

In case these measurements weren’t enough, the experimenters did them twice: once for sulphur, once for silicon.  Combining the data, they found that E = mc² is accurate to at least 0.00004 percent. Again, this isn’t surprising — nearly every result in particle and nuclear physics relies on the relationship between mass and energy. However, direct tests are best whenever they’re possible, whether it’s electron-positron annihilation experiments, Penning trap measurements, or something else entirely. To me, a hardcore theoretical physicist, it’s always lovely to see experimental confirmation performed in a truly elegant way. This is science at its best: theory and experiment working in tandem.

References

• Simon Rainville et al., “A direct test of E = mc2”. Nature 438: 1096–1097. doi:10.1038/4381096a
• Fernflores, Francisco, “The Equivalence of Mass and Energy”, The Stanford Encyclopedia of Philosophy (Spring 2012 Edition), Edward N. Zalta (ed.)

(This is part 3 of my series on relativity, mass, energy, and the most famous equation in physics: E = mc². Part 1 discussed whether Einstein actually used that equation and if it matters if he did; Part 2 examined the meaning of mass and masslessness. )