Beta decay energy: the energy Q released when a nucleus beta-decays is the parent-to-daughter atomic mass difference converted at 1 u = 931.494 MeV/c2, less 1.022 MeV for beta-plus or the electron binding energy for electron capture. This free calculator returns Q in MeV and keV and tells you whether the decay is energetically allowed.
What Is the Beta Decay Energy Calculator?
The beta decay energy calculator is a free online tool built on the formula Q = (Mp - Md) × 931.494 MeV. Enter the values you already know, in whichever units suit you, and it solves for decay energy Q, or the daughter mass and shows every step of the substitution. Decay energy Q from the parent and daughter atomic masses, for beta-minus, beta-plus and electron capture.
| Symbol | Quantity | Default unit | Also accepts | Example value |
|---|---|---|---|---|
| Q | Decay energy | MeV | keV | — |
| Mp | Parent atomic mass | u | — | 21.9944374 |
| Md | Daughter atomic mass | u | — | 21.9913851 |
| Be | Electron binding energy | keV | — | 0 |
How to calculate beta decay energy
Every beta decay is paid for out of mass. Weigh the parent atom, weigh the daughter atom, and whatever mass has gone missing reappears as energy shared among the emitted particles. That missing mass is tiny — for carbon-14 it is about one part in 83,000 of the atom — so the whole calculation hinges on subtracting two large, nearly equal numbers accurately. Enter both masses to six decimal places or more, or the rounding error will swamp the answer.
The conversion factor is 1 u = 931.494 MeV/c2. For beta-minus decay that is the whole story: Q = (Mp - Md) × 931.494, because the electron bookkeeping cancels exactly when you use neutral atomic masses. The daughter gains a proton and therefore needs one more electron, and the beta particle the nucleus emits is exactly that electron.
Beta-plus decay costs an extra 1.022 MeV, which is two electron rest masses. The nucleus has to build a positron from scratch, and the daughter — now one proton lighter — is left holding a surplus electron it must shed. Two electron masses at 0.511 MeV each is where the 1.022 MeV comes from, and it is a hard threshold: if the atomic mass difference is not worth more than that, beta-plus simply cannot happen. Electron capture reaches the same daughter without paying it, because the nucleus absorbs an atomic electron that already exists. Its only correction is the binding energy of the captured electron, a few keV in light atoms and up to roughly 100 keV in heavy ones — which is why electron capture is often open when beta-plus is shut.
Read the verdict as carefully as the number. A positive Q means the decay is energetically allowed; a Q of zero or below means it is forbidden and will not occur, no matter how long you wait. Switching the mode buttons for a fixed pair of masses makes the hierarchy obvious: electron capture always yields a larger Q than beta-plus for the same pair, and the two differ by exactly 1.022 MeV. To see where that energy actually goes once the decay happens, open the beta decay simulator; to work out how quickly a sample of it disappears, use the half-life calculator.
Worked example
Take sodium-22, a positron emitter used in PET calibration sources. Its atomic mass is 21.9944374 u and its daughter neon-22 weighs 21.9913851 u, a difference of 3.0523×10-3 u. Converting, 3.0523×10-3 × 931.494 = 2.8432 MeV. That figure is the electron capture Q-value straight away. For beta-plus subtract the 1.022 MeV threshold: 2.8432 - 1.022 = 1.8212 MeV. Both channels are open, and Na-22 does indeed use both.
Contrast beryllium-7, whose Q is 0.8618 MeV. That is comfortably positive, so electron capture proceeds — but it falls short of 1.022 MeV, so the beta-plus channel is closed outright. Beryllium-7 therefore decays only by electron capture, emitting no beta particle at all. For a beta-minus example, carbon-14 at 14.003242 u decaying to nitrogen-14 at 14.003074 u gives 1.680×10-4 × 931.494 = 0.1565 MeV, or 156.5 keV — the familiar endpoint energy of the carbon-dating beta.
Why it matters
Q-values decide which nuclides exist. They set the boundaries of the valley of stability, determine which side of it a nuclide decays from, and fix the maximum energy any beta particle from that nuclide can carry — the quantity that shielding calculations, dose estimates and detector calibrations all start from. In medicine the same arithmetic picks the isotope: fluorine-18 for PET because its beta-plus Q is small enough to give a short positron range, iodine-131 for thyroid therapy because its beta-minus Q deposits energy locally. Get the Q-value and you have the ceiling on everything the decay can do.
Frequently asked questions
What units should the masses be in?
Use unified atomic mass units (u), also written amu or Da. One u is defined as one twelfth of the mass of a neutral carbon-12 atom, and it is worth 931.494 MeV/c^2. Because the mass differences in beta decay are tiny — often only a few parts in a million of the total mass — you should enter masses to at least six decimal places. Rounding the masses to four or five decimals can change the answer by more than the answer itself.
Why does beta-plus subtract 1.022 MeV?
Because atomic masses count electrons, and beta-plus decay loses two electron masses worth of energy in the bookkeeping. The nucleus creates a positron, which costs one electron rest mass, and the daughter atom is left with one electron too many for its new atomic number, so it sheds one. Two electron rest masses is 2 times 0.511 MeV, which is 1.022 MeV. That is why a nuclide can only decay by beta-plus if the parent-to-daughter atomic mass difference is worth more than 1.022 MeV.
Do I use atomic or nuclear masses?
Atomic masses. The standard Q-value formulas for beta decay are written for neutral atomic masses precisely because the electron terms then cancel neatly: for beta-minus they cancel exactly, and for beta-plus they leave the 1.022 MeV term. If you use bare nuclear masses you must add the electron masses and their binding energies back by hand, which is more work and easier to get wrong. Tabulated values from sources such as the AME atomic mass evaluation are already atomic masses.
What does a negative Q mean?
It means the decay is energetically forbidden and simply does not happen. A negative Q says the products would weigh more than the parent, so the decay would have to create energy from nothing. The calculator reports this as a verdict rather than an error, because a negative Q is a perfectly meaningful physical result — it is how you prove a nuclide is stable against that particular decay mode. A nuclide can be forbidden in one mode and allowed in another.
How do I get Q if I only know the mass excess?
Subtract the mass excesses instead of the masses. The mass excess is defined as the atomic mass minus the mass number, expressed in energy units, so the mass-number parts cancel when you take the difference between parent and daughter of the same A. That means Q for beta-minus is simply the parent mass excess minus the daughter mass excess, already in MeV or keV with no conversion factor needed. Then apply the same mode correction: subtract 1.022 MeV for beta-plus, or the binding energy for electron capture.
Why is electron capture sometimes possible when beta-plus is not?
Because electron capture does not have to pay the 1.022 MeV price. In electron capture the nucleus swallows one of its own atomic electrons instead of creating a positron, so no new particle has to be made and no spare electron has to be shed. The only correction is the binding energy of the captured electron, typically a few keV up to about 100 keV in heavy atoms. That is why beryllium-7, with a Q of only 0.8618 MeV, decays by electron capture but cannot decay by beta-plus at all.
References & formula source
- Krane — Introductory Nuclear Physics, chapter on beta decay (Q-value definitions for beta-minus, beta-plus and electron capture in terms of atomic masses).
- Serway & Jewett — Physics for Scientists and Engineers, nuclear physics chapter (decay processes and disintegration energy).
- NNDC / Brookhaven National Laboratory — NuDat evaluated nuclear structure and decay data, Q-value definitions and tabulated decay energies.
- Wang, Huang, Kondev, Audi & Naimi — The AME 2020 atomic mass evaluation, the source of the atomic masses these formulas consume.
- Further reading: Beta decay — Wikipedia