Engineering & Science
Bethe-Weizsäcker Semi-Empirical Nuclear Binding Energy (Liquid-Drop SEMF)
Calculate total nuclear binding energy B(A, Z) and binding energy per nucleon B / A (MeV) from Volume, Surface, Coulomb, Asymmetry, and ISEVEN/ISODD Pairing terms across the chart of nuclides.
Explain both stellar fusion AND Uranium-235 fission from five liquid-drop terms!
Carl Friedrich von Weizsäcker (1935) and Hans Bethe modeled an atomic nucleus of mass number A and proton number Z as an incompressible quantum liquid drop with five competing physical energies (standard Rohlf/Krane constants in MeV): Volume attraction (+15.75 * A), Surface tension deficit (-17.80 * A^(2/3)), Proton-proton Coulomb repulsion (-0.711 * Z * (Z - 1) / A^(1/3)), Pauli neutron-proton Asymmetry penalty (-23.70 * (A - 2*Z)^2 / A), and Spin Pairing (+11.18 / A^0.5 for even-even nuclei, 0 for odd-A, and -11.18 / A^0.5 for odd-odd)!
Look at what B / A reveals instantaneously: Iron-56 (A = 56, Z = 26, even-even) sits near the peak of nuclear stability at B = 495.38 MeV (8.8461 MeV/nucleon), whereas heavy Uranium-235 (A = 235, Z = 92, odd-A -> pairing = 0) drops to B = 1,796.50 MeV (7.6447 MeV/nucleon) due to fierce Z*(Z-1) Coulomb repulsion—releasing ~1.0 MeV per nucleon (~200 MeV per nucleus!) when U-235 fissions into medium-mass fragments!
| Nuclide (A, Z) | Nuclear Spin Parity | Total Binding Energy B (MeV) | Binding Energy B / A (MeV/nucleon) |
|---|---|---|---|
| Iron-56 (A = 56, Z = 26) | Even-Even (+pair) | 495.38 MeV | 8.8461 MeV/nucleon (Peak Stability!) |
| Uranium-235 (A = 235, Z = 92) | Odd-A (pair = 0) | 1,796.50 MeV | 7.6447 MeV/nucleon (Fissionable!) |
| Invalid Z > A Input | Unphysical Proton Count | Guarded Input Error | "Invalid Nuclide (A, Z)" |
Total Bethe-Weizsäcker Nuclear Binding Energy B(A, Z) in MeV
Combines Volume, Surface, Coulomb, Asymmetry, and ISEVEN/ISODD Pairing terms for integer nuclide (A, Z).
=IF(OR(B2 < 2, B3 < 1, B3 > B2), "Invalid Nuclide (A, Z)", ROUND((15.75 * B2) - (17.8 * (B2 ^ (2 / 3))) - ((0.711 * B3 * (B3 - 1)) / (B2 ^ (1 / 3))) - ((23.7 * ((B2 - (2 * B3)) ^ 2)) / B2) + IF(ISODD(B2), 0, IF(ISEVEN(B3), 11.18 / SQRT(B2), -11.18 / SQRT(B2))), 2))Binding Energy per Nucleon B(A, Z) / A (MeV/nucleon)
Divides total SEMF binding energy by mass number A (B2) to compare fusion and fission stability.
=IF(OR(B2 < 2, B3 < 1, B3 > B2), "Invalid Nuclide (A, Z)", ROUND(((15.75 * B2) - (17.8 * (B2 ^ (2 / 3))) - ((0.711 * B3 * (B3 - 1)) / (B2 ^ (1 / 3))) - ((23.7 * ((B2 - (2 * B3)) ^ 2)) / B2) + IF(ISODD(B2), 0, IF(ISEVEN(B3), 11.18 / SQRT(B2), -11.18 / SQRT(B2)))) / B2, 4))Try the example
Use the original values or change them to test your understanding.