The Liquid-Drop Model and Collective Deformation

The semi-empirical mass formula models the nucleus as a charged liquid drop and reproduces the average binding energy from a volume, a surface, and a Coulomb term. Those terms were written for a sphere. Allowing the drop to change shape at fixed volume turns the static mass formula into a theory of deformation: the surface energy resists any departure from the sphere, the Coulomb energy favors it, and the balance between them sets whether the nucleus is stable, whether it can fission, and how it vibrates about its equilibrium shape. Bohr and Wheeler carried out this analysis in 1939 and it remains the backbone of the collective description of nuclei.¹

Parametrizing the shape

A small axially symmetric deformation of a sphere of radius $R_{0}$ is written as an expansion in Legendre polynomials of the polar angle,

R (θ) = R_{0} [1 + α_{0} + α_{2} P_{2} (cos θ) + \dots],

where $α_{2}$ is the quadrupole deformation and $P_{2} (cos θ) = \frac{1}{2} (3 cos^{2} θ - 1)$ . Positive $α_{2}$ stretches the drop along the symmetry axis into a prolate (cigar) spheroid; negative $α_{2}$ flattens it into an oblate (disk) spheroid. The monopole term $α_{0}$ is not free: the nuclear fluid is incompressible, so the volume must be conserved. Integrating $R (θ)^{3}$ over solid angle and holding the result at $\frac{4}{3} π R_{0}^{3}$ forces

α_{0} = - \frac{1}{5} α_{2}^{2} + O (α_{2}^{3}),

a second-order correction that keeps the volume, and therefore the volume energy, unchanged. Only the surface and Coulomb terms respond to $α_{2}$ at leading order.

A volume-conserving quadrupole deformation stretches the sphere of radius R0 into a prolate spheroid; the deformation parameter alpha-2 measures the elongation along the symmetry axis.

Surface and Coulomb energies under deformation

The surface energy is proportional to the surface area, which is smallest for a sphere; any deformation increases it. The Coulomb energy is proportional to the self-energy of the charge, which decreases when the charge spreads out over a larger dimension. Evaluating both for the quadrupole shape gives, to second order in $α_{2}$ ,

E_{S} (α_{2}) = E_{S}^{(0)} (1 + \frac{2}{5} α_{2}^{2}), E_{C} (α_{2}) = E_{C}^{(0)} (1 - \frac{1}{5} α_{2}^{2}),

with the spherical values taken from the mass formula,

E_{S}^{(0)} = a_{s} A^{2/3}, E_{C}^{(0)} = a_{c} \frac{Z ^{2}}{A ^{1/3}} .

The surface term rises and the Coulomb term falls, both quadratically in the deformation. The change in energy on deforming the sphere is their sum,

Δ E = E_{S} (α_{2}) + E_{C} (α_{2}) - E_{S}^{(0)} - E_{C}^{(0)} = \frac{α _{2}^{2}}{5} (2 E_{S}^{(0)} - E_{C}^{(0)}) .

The sign of the bracket decides everything. If $2 E_{S}^{(0)} > E_{C}^{(0)}$ the energy rises with deformation and the sphere is stable; if the inequality reverses, deformation lowers the energy and the sphere flies apart.

Surface energy rises and Coulomb energy falls with deformation; their sum is a parabola in alpha-2 that curves upward for a stable nucleus, giving a restoring minimum at the spherical shape.

The fissility parameter

Dividing the stability condition by $2 E_{S}^{(0)}$ isolates a single dimensionless number, the fissility,

x \equiv \frac{E _{C}^{(0)}}{2 E _{S}^{(0)}} = \frac{a _{c}}{2 a _{s}} \cdot \frac{Z ^{2}}{A} .

The deformation energy is $Δ E = \frac{2}{5} E_{S}^{(0)} α_{2}^{2} (1 - x)$ . For $x < 1$ the coefficient is positive and the sphere sits at a stable minimum; for $x > 1$ it is negative and the sphere is unstable against immediate fission. The critical point $x = 1$ occurs at

(\frac{Z ^{2}}{A})_{crit} = \frac{2 a _{s}}{a _{c}} \approx \frac{2 ( 17 MeV )}{0.71 MeV} \approx 49,

using the fitted mass-formula coefficients. The fissility is therefore $x = (Z^{2} / A) /49$ , and the heaviest nuclei approach it: $^{238} U$ has $Z^{2} / A = 35.6$ , so $x \approx 0.73$ , while the superheavy region near $Z = 114$ pushes $Z^{2} / A$ toward $45$ . No known nucleus reaches $x = 1$ ; every actual nucleus has a finite barrier against deformation, and fission proceeds by tunnelling through or thermal excitation over it rather than by the sphere being outright unstable.²

The deformation-energy barrier against fission shrinks as Z-squared over A grows toward the critical value; a light nucleus keeps a high barrier while a heavy nucleus is nearly unstable and the barrier flattens.

From static to dynamic: collective vibrations

For a stable nucleus, $x < 1$ , the deformation energy is a restoring potential quadratic in $α_{2}$ ,

V (α_{2}) = \frac{1}{2} C_{2} α_{2}^{2}, C_{2} = \frac{4}{5} E_{S}^{(0)} (1 - x) = \frac{4}{5} a_{s} A^{2/3} (1 - \frac{Z ^{2} / A}{49}) .

A restoring force acting on the surface makes it oscillate. Assigning the flow of nuclear matter an inertial (mass) parameter $B_{2}$ , the surface behaves as a harmonic oscillator in the collective coordinate $α_{2}$ with frequency

ℏ ω_{2} = ℏ \frac{C _{2}}{B _{2}} .

Quantizing this oscillator gives evenly spaced levels, the surface phonons. Because $α_{2}$ carries angular momentum $2ℏ$ and even parity, a single quadrupole phonon is a $2^{+}$ excitation and two phonons form a nearly degenerate $0^{+}, 2^{+}, 4^{+}$ triplet at twice the energy. The stiffness $C_{2}$ drops as $Z^{2} / A$ approaches the critical value, softening the mode and lowering the phonon energy; a nucleus whose stiffness vanishes is permanently deformed rather than vibrating about a sphere.

The surface tension acts as a restoring spring on the quadrupole coordinate, so the drop oscillates between prolate and oblate through the sphere; quantizing the oscillator gives the phonon states of the vibrational band.

Where the drop picture holds

The liquid drop describes the average, collective response of the nucleus: the smooth part of the binding energy, the deformation energy, the fission barrier, and the low-lying surface vibrations. Its variables are the collective shape coordinates $α_{λ}$ , not the individual nucleons. This description complements the shell model, which tracks single-particle levels and captures the magic numbers and ground-state spins the drop cannot see.

The two meet in the collective model, where the phonon and rotational excitations built here on the drop's surface dynamics are combined with the single-particle motion of a nucleon moving in the deformed well. The fissility and barrier developed here return in full in the theory of nuclear fission.

Regime	Fissility $x = (Z^{2} / A) /49$	Behavior
Light and medium nuclei	$x ≪ 1$	stiff sphere, high fission barrier
Heavy nuclei ( $^{238} U$ )	$x \approx 0.7$	low barrier, fission by tunnelling
Superheavy region	$x \to 0.9$	barrier a few $MeV$ , short-lived
Critical point	$x = 1$	sphere unstable, immediate fission

Krane, Introductory Nuclear Physics, Wiley (1988), §13.1; Bohr and Wheeler surface-deformation analysis. ↩
Wong, Introductory Nuclear Physics, 2nd ed., Wiley-VCH (1998), §6-1 and Ch. 5. Fitted coefficients from the AME/NNDC mass evaluation, https://www.nndc.bnl.gov/. ↩