Beta Decay Q-Value Calculator

The Beta Decay Q-Value Calculator estimates Q-values from tabulated nuclear masses for beta-minus, beta-plus, and electron-capture transitions.

What Is a Beta Decay Q-Value Calculator?

A beta decay Q-value calculator computes the energy released when a nucleus changes its charge by emitting a beta particle. In beta-minus decay, a neutron turns into a proton and an electron is emitted. In beta-plus decay, a proton converts to a neutron and a positron is emitted, or the nucleus captures an orbital electron (electron capture).

The calculator uses atomic masses or mass excess values to find the net energy difference between initial and final atoms. That energy, the Q-value, is shared among emitted particles, gamma rays, and recoil. You can include excited-state energies to match real decay schemes. The output gives a numeric result in MeV and shows which constants were applied.

How the Beta Decay Q-Value Method Works

Q-values come from mass–energy conservation. You compare the mass-energy of the initial neutral atom with that of the final neutral atom plus any created particles. Using atomic masses simplifies the algebra because bound electron masses largely cancel, except for specific beta-plus thresholds.

• Choose the decay mode: beta-minus (β−), beta-plus (β+), or electron capture (EC).
• Look up parent and daughter atomic masses from a trusted table (AME, NuDat).
• Apply the correct formula for the mode, including the 2 m_e term for β+ only.
• If the daughter is produced in an excited state, subtract that level energy from Q.
• Convert mass differences in atomic mass units (u) to MeV using 1 u ≈ 931.494 MeV/c².

The method ignores tiny recoil and neutrino mass effects at the first pass. For most cases, this gives results within a few keV of detailed evaluations. You can refine with electron binding energies for EC if you need sub-keV precision in heavy atoms.

Beta Decay Q-Value Formulas & Derivations

Start from energy conservation: Q equals the initial rest mass-energy minus the final rest mass-energy. Using neutral atomic masses (which include bound electrons) keeps the electron bookkeeping simple. Here are the standard working formulas with brief derivations.

• Beta-minus (β−): Q = [M_atom(parent) − M_atom(daughter)] c². Derivation: using nuclear masses gives Q = M_nuc(parent) − M_nuc(daughter) − m_e. When you switch to atomic masses, the extra orbital electron in the daughter cancels the emitted electron mass, leaving the compact expression above.
• Beta-plus (β+): Q = [M_atom(parent) − M_atom(daughter) − 2 m_e] c². The parent and daughter neutral atoms differ by one bound electron, and you must create a positron. The accounting adds a 2 m_e c² threshold (≈ 1.022 MeV) that must be exceeded.
• Electron capture (EC): Q = [M_atom(parent) − M_atom(daughter)] c². A bound electron is captured by the nucleus, and the final neutral daughter has one fewer atomic electron, so the captured electron’s mass is already included. Atomic X-rays from filling the shell hole come from this same Q.
• Excited daughter: Q* = Q − E_exc. If the daughter is born in an excited state, subtract the level energy. Any subsequent gamma carries that energy.
• Mass excess form: Q(MeV) = Δ(parent) − Δ(daughter) − mode_term, where mode_term is 0 for β− and EC, and 2 m_e c² for β+. Using mass excesses avoids handling large A·u terms explicitly.
• Unit conversion: Q(MeV) = (ΔM in u) × 931.49410242 MeV/u − (β+ only) 2 × 0.51099895 MeV.

These expressions assume the neutrino mass is negligible. Recoil kinetic energy is very small compared to MeV-scale Q-values, so it is typically ignored in the calculator’s primary result.

Inputs, Assumptions & Parameters

The calculator needs a few basic inputs and options. Keep the definitions clear so your variables map to data table values correctly. Most users will work directly with atomic masses or mass excesses in standard units.

• Parent isotope and atomic mass or mass excess (Δ). Example: 137Cs mass or Δ(137Cs).
• Daughter isotope and atomic mass or mass excess (Δ). Example: 137Ba mass or Δ(137Ba).
• Decay mode: β−, β+, or EC. This sets the formula and thresholds.
• Excited-state energy (optional): E_exc in keV or MeV to adjust Q*.
• Constants: electron mass m_e and the u-to-MeV conversion. You can select a CODATA set.

Mass values come from data libraries like AME2020/AME2024. The tool assumes neutral atoms. For EC, inner-shell binding energies are a few to tens of keV in heavy elements; the main Q is unchanged, but line energies appear in spectra. For β+ you must have Q greater than 1.022 MeV; otherwise the result will be negative and the decay is forbidden.

Step-by-Step: Use the Beta Decay Q-Value Calculator

Here’s a concise overview before we dive into the key points:

1. Select the decay mode: β−, β+, or EC.
2. Enter the parent isotope and its atomic mass or mass excess.
3. Enter the daughter isotope and its atomic mass or mass excess.
4. (Optional) Add the daughter excited-state energy if the decay feeds that level.
5. Choose the constants set (e.g., CODATA/NIST) and unit display (MeV or keV).
6. Click Calculate to compute the Q-value and see intermediate differences.

These points provide quick orientation—use them alongside the full explanations in this page.

Worked Examples

Example 1: β− decay of 137Cs → 137Ba + e− + ν̄. Use atomic masses: M(137Cs) ≈ 136.907089 u, M(137Ba) ≈ 136.905827 u. Compute ΔM = 0.001262 u. Convert: Q = 0.001262 × 931.494 ≈ 1.176 MeV. If decay goes to an excited 137Ba level at 0.662 MeV, then Q* = 1.176 − 0.662 = 0.514 MeV available to the electron, neutrino, and recoil. What this means: 137Cs can feed gamma emission from 137Ba while still giving a beta spectrum with an endpoint near 0.514 MeV for that branch.

Example 2: β+ decay of 22Na → 22Ne + e+ + ν. Use atomic masses: M(22Na) ≈ 21.994436 u, M(22Ne) ≈ 21.991385 u. ΔM = 0.003051 u ⇒ ΔM × 931.494 ≈ 2.842 MeV. Apply β+ threshold: Q(β+) = 2.842 − 2 × 0.510999 ≈ 1.820 MeV. 22Na often populates a 1.275 MeV excited state in 22Ne, so Q* ≈ 1.820 − 1.275 = 0.545 MeV for the positron and neutrino. What this means: After producing the 1.275 MeV gamma, the positron spectrum has an endpoint near 0.545 MeV, matching PET data.

Assumptions, Caveats & Edge Cases

Beta decay Q-values are robust when you use neutral atomic masses and standard constants. Still, a few details can nudge values by keV in precision work. Address these items if your use case demands fine accuracy.

• Atomic ionization state: The formulas assume neutral atoms. Highly ionized atoms shift EC capture probabilities and X-ray energies by up to keV.
• Electron binding energies: EC line energies depend on shell binding. The overall Q is unchanged, but spectral lines reflect these constants.
• Thresholds: β+ requires Q > 1.022 MeV. If not, EC may still occur with Q = ΔM c².
• Excited branches: Subtract the correct E_exc per branch. Misassigning levels leads to large errors in the beta endpoint.
• Data sources: Mixing mass tables or constants sets can add systematic offsets of a few keV.

For ultra-precise comparisons, keep a consistent constants set and the same edition of mass tables. Also note that recoil and neutrino mass are negligible for most analyses, but they do exist and can matter at the eV–keV level.

Units Reference

Q-values are energies, usually reported in MeV. Many inputs come as atomic mass differences in u. This table shows the key units and conversion constants so you can confirm each variable and result without confusion.

Read the table as follows: convert mass differences in u to MeV with the “u to MeV” factor, then apply any mode-specific constants (for β+, subtract 2 m_e c²). Keep units consistent through each step.

Tips If Results Look Off

If your Q-value seems wrong, it is usually a unit mismatch or a decay mode mix-up. Check the basics first, then refine details like excited states or constants.

• Confirm you used atomic masses, not bare nuclear masses.
• For β+, make sure you subtracted 2 m_e c² (about 1.022 MeV).
• Verify the daughter level energy and whether the branch is to ground or excited state.
• Ensure parent and daughter isotopes are the correct isobars (same A).
• Match constants and mass tables from the same release year.

If you still see a mismatch, compare against a known evaluated Q from ENSDF or AME. Differences below a few keV can arise from rounding or from newer constants.

FAQ about Beta Decay Q-Value Calculator

Why is the β+ formula different from β−?

β+ must create a positron and account for the missing bound electron in the daughter, which together add a 2 m_e c² term. β− uses atomic masses where the emitted electron is balanced by the daughter’s extra bound electron, so no mass term appears.

Can I use mass excess values instead of atomic masses?

Yes. Using Δ values is common. Q(MeV) equals Δ(parent) minus Δ(daughter) for β− and EC, and equals Δ(parent) minus Δ(daughter) minus 2 m_e c² for β+.

What if my Q-value is negative?

A negative Q means the decay mode is energetically forbidden for the ground-to-ground transition. For β+ this often indicates the parent does not exceed the 1.022 MeV threshold; electron capture may still be allowed.

How does the beta spectrum endpoint relate to Q?

The endpoint is Q* minus any gamma energy, minus small recoil. For β− and β+, the lepton pair shares Q*, so the electron or positron endpoint is slightly below Q* due to recoil and neutrino mass.

Glossary for Beta Decay Q-Value

Q-value

The net energy released by a nuclear transition, found from the mass difference between initial and final atoms times c².

Beta-minus (β−) decay

A neutron converts to a proton in the nucleus, emitting an electron and an antineutrino, and increasing Z by one.

Beta-plus (β+) decay

A proton converts to a neutron in the nucleus, emitting a positron and a neutrino, and decreasing Z by one.

Electron capture (EC)

The nucleus captures an inner-shell electron, converting a proton to a neutron and emitting a neutrino; Z decreases by one.

Atomic mass unit (u)

A mass unit based on one twelfth of the mass of a neutral 12C atom; 1 u corresponds to 931.49410242 MeV/c².

Mass excess

The difference between an atom’s mass and its mass number in atomic mass units, expressed in energy units for convenience.

Excited state energy

The energy above the ground state of a nucleus. If populated, it reduces the Q-value available to leptons.

Endpoint energy

The maximum kinetic energy of the emitted beta particle, approximately equal to the adjusted Q-value for that decay branch.

Sources & Further Reading

Here’s a concise overview before we dive into the key points:

• IAEA LiveChart of Nuclides — masses, decays, and level schemes
• NuDat 3 (NNDC) — evaluated nuclear structure and decay data
• Atomic Mass Evaluation (AME2020/2024) — comprehensive atomic masses and mass excesses
• NIST CODATA — fundamental physical constants (m_e, c, u conversions)
• Krane, K.S., Introductory Nuclear Physics — beta decay and Q-value derivations
• HyperPhysics — Beta decay overview and energetics

These points provide quick orientation—use them alongside the full explanations in this page.