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Beta Decay Calculator

Calculate beta decay kinematics and rates

About Beta Decay

Beta decay is a type of radioactive decay mediated by the weak force, involving the conversion of a neutron to a proton (or vice versa) with emission of an electron/positron and neutrino.

Decay Modes:

  • β⁻: n → p + e⁻ + ν̄ₑ (Q > 0)
  • β⁺: p → n + e⁺ + νₑ (Q > 2mₑ = 1.022 MeV)
  • EC: p + e⁻ → n + νₑ (Q > 0)

ft Values:

  • Superallowed: ft ~ 3000 s
  • Allowed: ft ~ 10³ - 10⁵ s
  • First forbidden: ft ~ 10⁵ - 10⁸ s

What Is Beta Decay?

Beta decay is a type of radioactive decay mediated by the weak nuclear force, in which a neutron transforms into a proton (or vice versa) inside an atomic nucleus, emitting an electron or positron and a neutrino or antineutrino. Discovered by Ernest Rutherford in 1899 and theoretically explained by Enrico Fermi in 1934, beta decay is responsible for the stability of atomic nuclei and drives many processes in stellar nucleosynthesis, nuclear medicine, and radiometric dating.

There are three modes: β⁻ decay (n → p + e⁻ + ν̄ₑ) where a neutron converts to a proton, increasing atomic number by 1; β⁺ decay (p → n + e⁺ + νₑ) where a proton converts to a neutron, decreasing atomic number; and electron capture (p + e⁻ → n + νₑ) where a nucleus captures an inner orbital electron, achieving the same result as β⁺ decay but with a lower energy threshold.

Beta Decay Kinematics & ft Values

The Q-value is the total energy released in the decay, shared between the electron (or positron), the neutrino, and the recoiling daughter nucleus. For β⁻ decay the maximum electron energy equals Q minus the electron rest mass (0.511 MeV); for β⁺ decay, the threshold is 2mₑc² = 1.022 MeV because a positron must be created.

Beta Decay Equations

Q = (M_parent − M_daughter)c² E_max(β⁻) = Q − mₑc² E_max(β⁺) = Q − 2mₑc² t₁/₂ = ft / f (f ≈ (Q/mₑ)^5/30)

Where:

  • Q= Q-value — total energy released (MeV)
  • mₑ= Electron rest mass = 0.511 MeV
  • ft= Comparative half-life — product of phase space factor and half-life (seconds)
  • f= Phase space factor — depends on Q-value and nuclear charge
  • t₁/₂= Half-life of the decay (seconds)

Transition Classification by ft Value

The ft value classifies beta decays by their degree of forbiddenness — essentially how much the nuclear wavefunctions overlap:

Typelog(ft)ft Range (s)Example
Superallowed3.5~3000¹⁴O, ²⁶Alᵐ
Allowed3-510³-10⁵Neutron, ³H
First forbidden5-910⁵-10⁹¹³⁷Cs, ⁹⁰Sr

How to Use This Calculator

  1. Select Decay Type: Choose β⁻, β⁺, or electron capture. The decay equation and daughter nucleus properties update automatically.
  2. Enter Parent Nucleus: Provide atomic number Z and mass number A. The calculator determines the daughter's Z, N, and A based on decay type.
  3. Enter Q-value: Input the decay energy in MeV. This computes maximum electron/positron energy, neutrino energy, and average beta energy.
  4. Optional ft Value: If you know the ft value (from nuclear data tables), enter it to estimate the half-life using the approximate phase space factor f ≈ (Q/mₑ)⁵/30. The transition type is classified based on ft magnitude.

Real-World Applications

Beta decay powers nuclear medicine. PET scans (Positron Emission Tomography) use β⁺ emitting isotopes like ¹⁸F (109.8 min half-life) — the emitted positron annihilates with an electron to produce two 511 keV gamma rays detected by the scanner. Iodine-131 (β⁻, 8-day half-life) treats thyroid cancer by delivering localized radiation. Carbon-14 dating relies on the 5,730-year β⁻ half-life of ¹⁴C to date archaeological artifacts up to ~50,000 years old.

In particle physics, superallowed beta decays provide the most precise test of the Standard Model's unitarity of the CKM quark mixing matrix. Precision measurements of ft values in nuclei like ¹⁴O and ²⁶Alᵐ constrain the Vud element, and any deviation from unitarity (|Vud|² + |Vus|² + |Vub|² = 1) would signal new physics beyond the Standard Model.

Worked Examples

Cobalt-60 β⁻ Decay

Problem:

⁶⁰Co (Z=27, A=60) undergoes β⁻ decay with Q = 2.824 MeV. Find the daughter nucleus and maximum electron energy.

Solution Steps:

  1. 1Daughter: Z = 27+1 = 28 (Nickel), A = 60 (unchanged), N = 60-28 = 32
  2. 2Decay: ⁶⁰Co → ⁶⁰Ni + e⁻ + ν̄ₑ
  3. 3Max electron energy: E_max = Q - mₑc² = 2.824 - 0.511 = 2.313 MeV
  4. 4Average energy: ~Q/3 ≈ 0.941 MeV
  5. 5⁶⁰Co is used in radiotherapy and industrial radiography

Result:

Daughter is ⁶⁰Ni (stable isotope of nickel). Maximum electron kinetic energy = 2.313 MeV — one of the highest-energy beta emitters used commercially.

Carbon-14 Dating

Problem:

¹⁴C (Z=6, N=8, Q=0.156 MeV) decays by β⁻ with ft ≈ 10⁹ s. Estimate the half-life.

Solution Steps:

  1. 1Phase space: f ≈ (0.156/0.511)⁵/30 = (0.305)⁵/30 = 0.00264/30 = 8.8×10⁻⁵
  2. 2Half-life: t₁/₂ = ft/f = 10⁹/8.8×10⁻⁵ ≈ 1.14×10¹³ s
  3. 3Convert to years: 1.14×10¹³/3.156×10⁷ ≈ 361,000 years — but actual value is ~5,730 years!
  4. 4The discrepancy arises because ¹⁴C decay is actually 'allowed' (much smaller ft ~10⁹ compared to forbidden transitions)
  5. 5Using the measured ft ≈ 10⁹ s (log ft ≈ 9), f is actually much smaller due to low Q — actual half-life 5,730 years

Result:

The Q-value phase space approximation alone doesn't capture the full physics — the ft approach combined with precise f calculations (including Coulomb corrections) yields the correct 5,730-year half-life.

Electron Capture in ⁷Be

Problem:

⁷Be (Z=4, A=7) decays via electron capture with Q = 0.862 MeV. What is the daughter nucleus and neutrino energy?

Solution Steps:

  1. 1Daughter: Z = 4-1 = 3 (Lithium), A = 7
  2. 2Decay: ⁷Be + e⁻ → ⁷Li + νₑ
  3. 3Neutrino energy: In EC, the neutrino carries essentially all the Q-value (monoenergetic)
  4. 4E_ν ≈ 0.862 MeV (minus tiny nuclear recoil energy)
  5. 5This monoenergetic neutrino signature makes ⁷Be useful for neutrino detector calibration

Result:

Daughter is ⁷Li (stable). Neutrino energy ≈ 0.862 MeV — a well-known calibration source used in solar neutrino experiments like Borexino.

Tips & Best Practices

  • ✓The average beta energy is approximately Q/3 for allowed transitions — useful for dosimetry and decay heat calculations
  • ✓β⁺ decay requires Q > 1.022 MeV (2mₑc²); below that, only electron capture can occur for proton-rich nuclei
  • ✓The ft value is the product of the statistical rate function f and the partial half-life t — it removes the Q-value and nuclear charge dependence
  • ✓Superallowed transitions (0⁺ → 0⁺, log ft ≈ 3.5) are the 'gold standard' for weak interaction physics and CKM unitarity tests
  • ✓Cobalt-60 (⁶⁰Co) β⁻ decay with Q = 2.824 MeV is one of the most widely used beta sources in industry and medicine

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Frequently Asked Questions

Unlike alpha decay (which produces discrete, monoenergetic particles), beta decay electrons/positrons have a continuous energy spectrum from near-zero up to the endpoint energy. This is because the decay energy is shared between three particles — the electron, the antineutrino, and the recoiling daughter nucleus. The neutrino carries away a variable amount of energy, and it was precisely this continuous spectrum that led Wolfgang Pauli to postulate the neutrino's existence in 1930.
β⁻ decay converts a neutron to a proton, increasing Z by 1 and emitting an electron and antineutrino. It occurs in neutron-rich nuclei. β⁺ decay converts a proton to a neutron, decreasing Z by 1 and emitting a positron and neutrino. It requires a minimum Q-value of 1.022 MeV (2mₑc²) because a positron must be created. Electron capture achieves the same nuclear transformation as β⁺ but with no minimum Q-value threshold, making it competitive for proton-rich nuclei with low decay energy.
Superallowed transitions occur between nuclear states with identical spin and parity (0⁺ → 0⁺) and nearly identical wavefunctions. The nuclear matrix element is essentially 1, making the ft value purely dependent on the vector coupling constant. These transitions provide the most precise determination of Vud in the CKM matrix. Only about 20 superallowed transitions are known, all in light nuclei (A < 70).
The Fermi function F(Z,E) accounts for the Coulomb interaction between the emitted beta particle and the daughter nucleus. For β⁻, the negatively charged electron is attracted to the positive nucleus, enhancing low-energy emission (F > 1). For β⁺, the positron is repelled, suppressing low-energy emission (F < 1). The function is essential for accurate phase space factor calculations — at low Z and high energy, F ≈ 1, but for heavy nuclei the correction can be orders of magnitude.
A free proton has mass 938.272 MeV while a neutron has mass 939.565 MeV. β⁺ decay (p → n + e⁺ + νₑ) would require the proton to gain 1.293 MeV to become a neutron, plus 1.022 MeV to create the positron — a total of 2.315 MeV deficit, making it energetically forbidden. Free neutrons, however, DO decay (n → p + e⁻ + ν̄ₑ) with a half-life of about 880 seconds because the neutron is 1.293 MeV heavier than the proton plus electron.

Sources & References

Last updated: 2026-06-06

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Editorial Note

MyCalcBuddy Editorial Team

This page is maintained as an educational calculator reference.

Source

Formula Source: University Physics

by Young & Freedman

UpdatedLast reviewed: May 2026
CheckedFormula checks are based on standard references and internal QA review.