Playground
Adjust A and Z to see the five SEMF terms compete: volume, surface, Coulomb, asymmetry, and pairing.
Variables
| Symbol | Name | SI | Dimension | Range |
|---|---|---|---|---|
| Binding energyoutput Total nuclear binding energy | J | M·L²·T⁻² | 0 – 1e-9 | |
| Mass number Total number of nucleons | 1 | 1 | 1 – 300 | |
| Atomic number Number of protons | 1 | 1 | 1 – 120 | |
| Volume coefficient Volume term coefficient | MeV | M·L²·T⁻² | 15 – 16 | |
| Surface coefficient Surface term coefficient | MeV | M·L²·T⁻² | 17 – 19 | |
| Coulomb coefficient Coulomb repulsion term coefficient | MeV | M·L²·T⁻² | 0.6 – 0.8 | |
| Asymmetry coefficient Proton-neutron asymmetry coefficient | MeV | M·L²·T⁻² | 22 – 24 |
Deep dive
Derivation
Treat the nucleus as a charged liquid drop. Volume term (a_V·A): each nucleon interacts with nearest neighbors via the strong force, giving energy proportional to A. Surface term (−a_S·A^{2/3}): nucleons at the surface have fewer neighbors, reducing binding. Coulomb term (−a_C·Z²/A^{1/3}): proton-proton electrostatic repulsion scales as Z² divided by nuclear radius ∝ A^{1/3}. Asymmetry term (−a_A·(A−2Z)²/A): the Pauli exclusion principle penalizes unequal proton/neutron numbers. Pairing term (δ): empirical correction favoring even-even nuclei.
Experimental verification
Aston's mass spectrograph (1920s) first showed systematic mass defects. Modern atomic mass evaluations (AME2020) provide masses of ~3400 nuclides. The SEMF reproduces these to within ~1% for A > 20, confirming the liquid drop picture. Deviations at magic numbers led to the nuclear shell model.
Common misconceptions
- The formula does NOT predict magic numbers — these require the shell model.
- The pairing term δ is NOT simply +δ for even-even; it depends on whether A is odd or even, and the sign convention varies by textbook.
- The coefficients are NOT universal constants — they are fitted to experimental data and vary slightly between different fitting sets.
Real-world applications
- Predicting nuclear stability and drip lines
- Estimating fission barrier heights
- Nucleosynthesis calculations in stellar evolution
- Nuclear reactor fuel cycle analysis
Worked examples
Binding energy of ⁵⁶Fe (Z=26, A=56)
Given:
- A:
- 56
- Z:
- 26
- a_V:
- 15.8
- a_S:
- 18.3
- a_C:
- 0.714
- a_A:
- 23.2
- δ:
- 0
Find: B
Solution
B ≈ 493.4 MeV → B/A ≈ 8.81 MeV/nucleon
Binding energy of ²³⁸U (Z=92, A=238)
Given:
- A:
- 238
- Z:
- 92
- a_V:
- 15.8
- a_S:
- 18.3
- a_C:
- 0.714
- a_A:
- 23.2
- δ:
- +0.78
Find: B
Solution
B ≈ 1802 MeV → B/A ≈ 7.57 MeV/nucleon
Scenarios
What if…
- scenario:
- What if we ignore the Coulomb term?
- answer:
- Binding energy would increase monotonically with A, and there would be no upper limit on nuclear size. The Coulomb term is what makes superheavy nuclei unstable and drives fission.
- scenario:
- What if N ≠ Z for a light nucleus?
- answer:
- The asymmetry term penalizes the imbalance. For example, ¹⁴C (Z=6, N=8) has a lower binding energy than ¹⁴N (Z=7, N=7) partly due to this term, making ¹⁴C radioactive.
- scenario:
- What if a nucleus sits at a magic number?
- answer:
- Shell closures add extra binding not captured by the SEMF. For example, ²⁰⁸Pb (Z=82, N=126 — doubly magic) has ~10 MeV more binding than the SEMF predicts.
Limiting cases
- condition:
- A → large, Z ≈ A/2
- result:
- B/A peaks near A ≈ 56 then decreases
- explanation:
- The Coulomb term grows faster than the volume term for large A, reducing binding energy per nucleon and making heavy nuclei less stable.
- condition:
- Z = A/2 (symmetric nuclei)
- result:
- Asymmetry term vanishes
- explanation:
- When proton and neutron numbers are equal, the (A−2Z)² term is zero, maximizing binding for light nuclei.
- condition:
- A → small
- result:
- Surface term dominates over volume
- explanation:
- Small nuclei have a large surface-to-volume ratio, so the negative surface correction significantly reduces binding energy per nucleon.
Context
Carl von Weizsäcker & Hans Bethe · 1935
Weizsäcker proposed the liquid drop model for nuclear masses; Bethe refined it, producing a formula that predicts binding energies across the chart of nuclides.
Hook
Why is iron-56 the end of the line for stellar fusion?
Estimate the binding energy of ⁵⁶Fe (Z=26, A=56) using the SEMF.
Dimensions:
- lhs:
- B → [Energy] = MeV
- rhs:
- a_V·A → [MeV]·[1] = [MeV]; a_S·A^{2/3} → [MeV]; a_C·Z²/A^{1/3} → [MeV]·[1]/[1] = [MeV]; a_A·(A−2Z)²/A → [MeV]
- check:
- All terms have dimensions of [Energy] = MeV. ✓
Validity: Valid for nuclei with A ≳ 12 (breaks down for very light nuclei where shell effects dominate). Accuracy is ~1% for most nuclei but deviates near magic numbers (2, 8, 20, 28, 50, 82, 126). The pairing term δ(A,Z) requires separate treatment for even-even, odd-odd, and odd-A nuclei.