Nuclear Binding Energy Calculator | Mass Defect, BE/A & Bethe-Weizsäcker Formula
Compute nuclear binding energy, mass defect (Δm), binding energy per nucleon (BE/A), and Q-value for over 100 common isotopes using CODATA atomic mass data. Compares stability across isotopes and estimates BE using the semi-empirical Bethe-Weizsäcker formula.
Protons Z
26
Neutrons N
30
Mass number A
56
Mass defect Δm (u)
0.514184
Mass defect Δm (MeV/c²)
478.959
Binding energy BE (MeV)
478.959
BE per nucleon BE/A (MeV)
8.5528
Most stable region
SEMF estimate (MeV)
495.384
Δ = -16.424 MeV
CALCULATION STEPS
N = A − Z = 56 − 26 = 30
Δm = Z×mp + N×mn − M = 26×1.007276 + 30×1.008665 − 55.934942
Δm = 56.449126 − 55.934942 = 0.514184 u
BE = Δm × 931.494 = 0.514184 × 931.494 = 478.959 MeV
BE/A = 478.959 / 56 = 8.5528 MeV/nucleon
TOP 10 MOST STABLE ISOTOPES BY BE/A
| Isotope | Z | A | BE/A (MeV) | BE (MeV) |
|---|---|---|---|---|
| Fe-56 ★ | 26 | 56 | 8.5528 | 478.96 |
| Cu-63 | 29 | 63 | 8.5167 | 536.55 |
| Sr-88 | 38 | 88 | 8.5118 | 749.04 |
| Kr-84 | 36 | 84 | 8.4982 | 713.85 |
| Zn-64 | 30 | 64 | 8.4961 | 543.75 |
| Ni-58 | 28 | 58 | 8.4851 | 492.14 |
| Ar-40 | 18 | 40 | 8.3652 | 334.61 |
| K-39 | 19 | 39 | 8.3079 | 324.01 |
| Ca-40 | 20 | 40 | 8.2956 | 331.83 |
| Sn-120 | 50 | 120 | 8.2915 | 994.98 |
What Is the Nuclear Binding Energy Calculator | Mass Defect, BE/A & Bethe-Weizsäcker Formula?
Nuclear binding energy is the energy required to completely separate all nucleons (protons and neutrons) in a nucleus. It arises from the strong nuclear force, which overcomes the electrostatic repulsion between protons. The mass of a nucleus is always less than the sum of its free constituent nucleon masses — this mass defect Δm is converted to binding energy via Einstein's E = mc².
Binding energy per nucleon (BE/A) is the key metric for nuclear stability. It peaks around 8.7–8.8 MeV/nucleon for iron-56 and nickel-58, the most stable nuclei. Lighter nuclei can release energy by fusion (combining lighter nuclei releases energy because BE/A increases toward Fe-56). Heavy nuclei can release energy by fission (splitting releases energy because BE/A increases toward Fe-56 from the heavy side).
The Bethe-Weizsäcker semi-empirical mass formula (SEMF) models BE using a liquid-drop analogy. The volume term reflects the short-range nuclear force; the surface term corrects for nucleons at the surface; the Coulomb term accounts for proton repulsion; the asymmetry term penalises large N-Z imbalances; and the pairing term δ reflects the tendency of nucleons to pair.
Formula
Mass Defect: Δm = Z·mp + N·mn − M(A,Z) Binding Energy: BE = Δm × 931.494 MeV/u BE per nucleon: BE/A (MeV/nucleon) Bethe-Weizsäcker (SEMF): BE = aV·A − aS·A^(2/3) − aC·Z(Z−1)/A^(1/3) − aA·(A−2Z)²/A + δ aV = 15.75 MeV (volume term) aS = 17.8 MeV (surface term) aC = 0.711 MeV (Coulomb term) aA = 23.7 MeV (asymmetry term) δ = +11.18/√A (even-even), 0 (odd-A), −11.18/√A (odd-odd) Constants: mp = 1.007276 u, mn = 1.008665 u, 1 u = 931.494 MeV/c²
How to Use
- 1
Choose Select Isotope mode to pick from 40 common isotopes in the dropdown menu.
- 2
Or choose Manual Entry and type the proton number Z, mass number A, and atomic mass M in unified atomic mass units (u).
- 3
Read mass defect Δm in both u and MeV/c², total binding energy BE in MeV, and binding energy per nucleon BE/A in MeV/nucleon.
- 4
Compare BE to the SEMF estimate — the difference shows how well the liquid-drop model fits.
- 5
Check the stability classification (loosely bound, moderately bound, most stable, etc.) based on BE/A.
- 6
See the Top 10 Most Stable Isotopes table to compare the selected nuclide against the iron-peak region.
Select an isotope from the dropdown or enter Z, A, and atomic mass manually. The calculator shows mass defect, binding energy, BE/A, and the SEMF estimate alongside stability classification.
Example Calculation
Problem: Calculate binding energy of Fe-56 (Z=26, A=56, M=55.934942 u).
Solution:
N = 56 − 26 = 30 neutrons
Mass of free nucleons = 26 × 1.007276 + 30 × 1.008665 = 26.18918 + 30.25995 = 56.44913 u
Δm = 56.44913 − 55.934942 = 0.514188 u
BE = 0.514188 × 931.494 = 479.0 MeV
BE/A = 479.0 / 56 = 8.554 MeV/nucleon
SEMF estimate ≈ 492 MeV; Δ ≈ −13 MeV (shell effects not in SEMF)
Understanding Nuclear Binding Energy | Mass Defect, BE/A & Bethe-Weizsäcker Formula
Binding Energy per Nucleon — Key Values
| Isotope | Z | A | BE/A (MeV) | Significance |
|---|---|---|---|---|
| H-1 | 1 | 1 | 0.000 | Single proton — no binding |
| H-2 | 1 | 2 | 1.112 | Deuterium — loosely bound |
| He-4 | 2 | 4 | 7.074 | Alpha particle — doubly magic |
| C-12 | 6 | 12 | 7.680 | Key fusion product (CNO cycle) |
| O-16 | 8 | 16 | 7.976 | Double magic nucleus |
| Fe-56 | 26 | 56 | 8.790 | Near peak of curve — endpoint of exothermic fusion |
| Ni-58 | 28 | 58 | 8.732 | Highest BE/A of any nuclide |
| U-238 | 92 | 238 | 7.570 | Fissionable heavy nucleus |
Fusion and Fission Energy Release
- ›Fusion (light nuclei): D + T → He-4 + n releases ~17.6 MeV. BE/A increases from ~1 MeV (D,T) to ~7 MeV (He-4).
- ›Fission (heavy nuclei): U-235 + n → Ba + Kr + 2n releases ~200 MeV. BE/A increases from ~7.6 to ~8.5 MeV/nucleon.
- ›The iron peak (Fe-56, Ni-58) is the "valley of stability" — no further energy release by fusion or fission.
- ›Stars generate energy by fusion up to iron; heavier elements require neutron capture processes (s-process, r-process).
- ›Supernovae synthesise elements beyond iron by rapid neutron capture (r-process) during core collapse.
- ›Nuclear weapons: fission bombs use U-235 or Pu-239; thermonuclear weapons use D-T fusion boosted by fission.
Frequently Asked Questions
Why is binding energy per nucleon (BE/A) more useful than total binding energy?
Total binding energy scales roughly with A, making direct comparisons across nuclei meaningless. BE/A normalises for size, allowing a fair comparison of nuclear stability. The BE/A curve has a broad peak near A = 56–62 (iron-nickel region at ~8.8 MeV/nucleon), which explains why fusion of light nuclei and fission of heavy nuclei both release energy.
What causes the mass defect?
When nucleons are brought together from infinite separation, the strong force does work binding them. This energy, released as gamma radiation during nuclear formation, reduces the total mass of the system. Einstein's E = mc² gives BE = Δm × c². The conversion factor is 1 u = 931.494 MeV/c², so even a small mass defect corresponds to hundreds of MeV.
What are the terms in the Bethe-Weizsäcker (SEMF) formula?
Volume term (aV·A): proportional to A because each nucleon interacts with its neighbors; favours large nuclei. Surface term (−aS·A²/³): corrects for nucleons at the surface with fewer neighbors. Coulomb term (−aC·Z(Z−1)/A¹/³): electrostatic repulsion between protons. Asymmetry term (−aA·(N−Z)²/A): Pauli exclusion makes unequal N and Z energetically costly. Pairing term δ: even-even nuclei are more stable due to nucleon pairing.
Why are even-even nuclei more stable than odd-odd nuclei?
Nucleons, like electrons, tend to pair spin-up with spin-down due to the pairing interaction in the nuclear force. Even-even nuclei (even Z and even N) have all nucleons paired, gaining extra binding energy δ = +11.18/√A. Odd-odd nuclei have one unpaired proton and one unpaired neutron, losing binding energy δ = −11.18/√A. Odd-A nuclei have one unpaired nucleon; δ = 0.
How is the binding energy related to nuclear energy released in reactions?
The Q-value of a nuclear reaction equals the difference in total binding energies between products and reactants: Q = BE_products − BE_reactants. If Q > 0, the reaction is exothermic (energy is released). For fission of U-235, Q ≈ 200 MeV per event; for D-T fusion, Q ≈ 17.6 MeV per event. Fission releases more energy per reaction but fusion releases more per unit mass.
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