Swans on Tea

The topic comes up, as it sometimes does, of the mass-energy equivalence from relativity. There are different tangents to this — what does the equivalence really mean, can you really turn energy into mass, does a photon have rest mass, what is the difference between relativistic mass and rest mass, and is the use of relativistic mass grounds for justifiable homicide, or is one compelled to stop at maiming?

E = mc² is the equation everyone knows, but what many don’t know is that the equation already assumes one is at rest. The actual equation is E² = p²c² + m²c⁴, which reduces to the more familiar form when the object is at rest. The implications of this are that photons have no mass, the mass term for massive particles doesn’t change when you move — that energy is in the kinetic term, (which renders relativistic mass moot) and also that the mass will increase if you add energy that does not appear in the kinetic term, i.e. extra energy in the center-of-momentum frame appears as mass.

The last concept showed up at Cosmic Variance recently, in the context of the mass of a spinning top

The spinning gyroscope has more energy than the non-spinning one. As a test, we can imagine extracting work from the spinning gyroscope — for example, by hooking it up to a generator — in ways that we couldn’t extract work from the stationary gyroscope. And since it has more energy, it has more mass. And the weight is just the acceleration due to gravity times the mass — so, as long as we weigh our spinning and non-spinning gyroscopes in the same gravitational field, the spinning one will indeed weigh more.

Of course, that’s a thought experiment, and what we’d really like to have is experimental confirmation of this. Macroscopic objects aren’t likely to have enough energy where one would be able to measure the change, so we have to go smaller. An atomic system will have masses that are dependent on the number of neutrons and protons in them, and these particles have a mass of around one atomic mass unit each, or roughly 1 GeV/c² of mass (i.e. the mass equivalent of a billion electron-Volts of energy). Atomic excitations are going to be on the order of 1 eV, and if our atomic system is anything other than Hydrogen or Helium (the Baltic and Mediterranean Ave of the periodic table) we have at least 10 GeV to worry about. That’s a part in 10¹⁰ change in mass. Nuclear excitations, on the other hand, are perhaps 1 MeV, and even if we have 100 nucleons, that’s still five orders of magnitude easier. So I did a quick Google on any mass measurements on isomers, since we want excited states that will last long enough to be measured, and got a recent hit: Mass Measurement Technique Uncovers New Iron Isomer

The article takes a little while to get to the point (it’s a bad press release in that regard), but they used Germanium in an accelerator to produce Iron isotopes, including Fe-65, and in the article eventually gets to the punchline

A Penning trap catches and retains charged particles in a strong magnetic field. Responding to this field, captured particles move in what’s known as a cyclotron motion, the frequency of which is directly related to the mass of the particle.
During the experiment, Bollen and his collaborators observed two distinct frequencies associated with the trapped iron-65 particles. They concluded that the heavier of the two was a previously unknown isomer of iron-65.

Bingo! Same isotope, different masses, because one is in an excited state and has extra energy. Unfortunately they don’t actually mention the results, but of course they do in the paper (in the abstract, no less): 402 keV.

The paper is “Discovery of a Nuclear Isomer in ⁶⁵Fe with Penning Trap Mass Spectrometry,” by Block, et. al. in case you’re not inclined to click on the link. (OK, the title is the same even if you are inclined to click on the link, but you get my point)

And, as for the other question that comes up in this situation: no, that isomer does not make you look fat. Besides, I love your quadrupole moment.