THE SEMI-EMPIRICAL MASS FORMULA

The nucleus as a charged liquid drop. Five competing terms — bulk binding, a surface skin, proton repulsion, a symmetry preference, and a pairing bonus — add up to the binding energy, and their balance carves the valley of stability that beta decay slides toward. Rendered, not quoted.

source von Weizsäcker, C. F. (1935), “Zur Theorie der Kernmassen,” Z. Phys. 96, 431–458 · Bethe–Weizsäcker liquid-drop model · doi:10.1007/BF01337700 · coefficients (a_V,a_S,a_C,a_A,a_P) are a fitted set, marked AMBER

Blue team · builds & defends

3 THE MODEL

Treat A nucleons as an incompressible charged drop. Binding energy:

B(Z,A) = aVA − aSA2/3 − aCZ²/A1/3 − aA(A−2Z)²/A + δ

VOLUME rewards every nucleon (+, ∝A). SURFACE penalises the skin whose nucleons have fewer neighbours (−, ∝A2/3). COULOMB penalises proton–proton repulsion (−, ∝Z²/A1/3). ASYMMETRY penalises N≠Z (−, ∝(A−2Z)²/A). PAIRING adds δ=±aP/√A (+ even–even, − odd–odd, 0 odd–A).

5 THE LINEAGE

The drop is Gamow’s picture; Weizsäcker (1935) gave it the five terms. The Z²/A1/3 Coulomb term forces heavy nuclei to hold N > Z to stay in the valley — and it is that neutron excess that the-beta-decay unwinds, one isobar at a time, sliding down toward the the-binding-energy curve’s floor near Fe-56.

7 THE WITNESS

Live re-check of two invariants: the Coulomb term must subtract from binding, and B(Fe-56) must land within a few percent of the measured 492.25 MeV.

witness idle
The machine

4 DATA IN in ↓

Z = 26   A = 56  (iron-56)
coefficients / MeV: a_V 15.8 · a_S 18.3 · a_C 0.714 · a_A 23.2 · a_P 12.0
constants: c · u = 931.5 MeV · N_A = 6.022e23

↓   five terms   ↓

0 THE PANEL LIT

Binding energy per nucleon B/A along the valley of stability (Z = Zopt(A)) — peaks near iron.

↓   proven result   ↓

8 DATA OUT out ↓

boot pending
Red team · attacks & breaks

1 THE ADVERSARY WALL

“It is only semi-empirical — the coefficients are fitted, not derived. Change the fit and your Fe-56 number moves. So the formula proves nothing.”

Answer: the signs and A-scalings are forced by physics, not by the fit; only the five magnitudes are tuned. The liquid drop also fails at magic numbers (shell closures 2,8,20,28,50,82,126) and for light nuclei — real, disclosed limits, not fatal ones.

2 THE GRAVEYARD

Adding nucleons always adds binding.
→ surface + Coulomb terms curve B/A over: it peaks at Fe-56 (~8.8 MeV/A) then falls — that peak is why fusion below it and fission above it both release energy.

The most stable isobar at fixed A has Z = A/2.
→ only for light nuclei. Coulomb repulsion shifts Zopt = A/(2 + a_C A2/3/(2a_A)); at A=238 that is Z≈92, N≈146.

6 THE TAMPER

Planted void (disclosed): flip the Coulomb sign so proton repulsion increases binding. Heavy nuclei go absurdly overbound and the valley inverts. The WITNESS (7) catches it live.

state: intact