keV vs keV [Eqn]

I have noticed that our statistician collaborators are often confused by our units. (Not a surprise; I, too, am constantly confused by our units.) One of the biggest culprits is the unit of energy, [keV], which is 1000 electron Volts, for the energy acquired by an electron when it falls through an electric potential of 1 Volt:

1 [eV] ≡ 1.6021892 · 10-19 [Joule] ≡ 1.6021892 · 10-12 [erg] .

The confusion is because the same units are used to denote two separate quantities which happen to have similar magnitudes for a commonly encountered spectral model, Bremsstrahlung emission.

  1. the frequency ν, or wavelength λ, of a photon: As Planck discovered, the energy of a photon is directly related to the frequency ν,

    E = h · ν ≡ h · c / λ ,

    where h=6.6261760 · 10-27 [erg s] is Planck’s constant and c=2.9979246 · 1010 [cm s-1] is the speed of light in vaccum. When λ is given in [Ångström] ≡ 10-8 [cm], we can convert it as

    [keV] = 12.398521 / [Å] ,

    which is an extraordinarily useful thing to know in high-energy astrophysics.

  2. the temperature T of a gas or plasma: Here we look to thermodynamics, which relates the kinetic energy of random motion of particles in a gas to a gross property, the temperature of the gas,

    E = kB · T ,

    where kB = 1.3806620 · 10-16 [erg K-1] is Boltzmann’s constant. Then, a temperature in degrees Kelvin can be written in units of keV by converting it with the formula

    [keV] = 8.6173468 · 10-8 · [K] ≡ 0.086173468 · [MK] .

It is tempting to put the two together and interpret a temperature as a photon energy. This is possible for the aforementioned Bremsstrahlung radiation, where plasma at a temperature T produces a spectrum of photons distributed as e-h ν / kB T and it is possible to tie the temperature to the photon energy at the point where the numerator and denominator have the same numerical value. For example, a 1 keV (temperature) Bremsstrahlung spectrum extends out to 1 keV (photon energy). X-ray Astronomers use this as shorthand all the time, and it confuses the hell out of everybody else.

One Comment
  1. Simon Vaughan:

    On this topic Richard Feynman is supposed to have said:

    “Before I begin the lecture, I wish to apologize for something that is not my responsibility: Physicists and scientists all over the world have been measuring things in different units, and causing an enormous amount of complexity. As a matter of fact, nearly a third of what you have to learn 1 consists of different ways of measuring the same thing, and I apologize for it…

    “The physicists do something else when they want to talk about the energy of a single atom, instead of the energy of a gross amount of material. The reason is, of course, that a single atom is such a small thing that to talk about its energy in joules would be inconvenient. But instead of taking a definite unit in the same system (like 10^-20 J), they have unfortunately chosen, arbitrarily, a funny unit called an electronvolt (eV), which is the energy needed to move an electron through a potential difference of one volt, and that turns out to be about 1.6 10^-19 J. I am sorry that we do that, but that’s the way it is for the physicists.”

    07-30-2008, 1:07 pm
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