---
title: The Liquid-Drop Model and Collective Deformation
module: Nuclear Models
moduleNumber: 3
lessonNumber: 2
order: 302
summary: >
  Deforming a charged liquid drop into a spheroid raises its surface energy and
  lowers its Coulomb energy; the two effects compete through the deformation
  parameter to set a stability minimum and a fission barrier. The ratio of
  Coulomb to twice the surface energy is the fissility Z-squared over A, which
  crosses one near 49 and marks the point where the sphere is unstable. The same
  surface tension that restores small deformations quantizes into collective
  vibrations, carrying the static mass formula into dynamic collective motion.
topics: [Nuclear Models]
sources:
  - book: Krane
    ref: "Ch. 13 — Nuclear Fission; §13.1 The Liquid-Drop Model of Fission"
  - book: Wong
    ref: "Ch. 6 — Nuclear Bulk Properties; §6-1, Ch. 5 The Collective Model"
  - book: Povh
    ref: "Ch. 3 — Nuclear Structure (surface deformation)"
draft: false
---

The
[semi-empirical mass formula](/nuclear-physics/nuclear-properties/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.[^krane-ld]

## 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,

$$
R(\theta) = R_0\Bigl[1 + \alpha_0 + \alpha_2 P_2(\cos\theta) + \cdots\Bigr],
$$

where $\alpha_2$ is the quadrupole deformation and $P_2(\cos\theta) =
\tfrac12(3\cos^2\theta - 1)$. Positive $\alpha_2$ stretches the drop along the
symmetry axis into a prolate (cigar) spheroid; negative $\alpha_2$ flattens it
into an oblate (disk) spheroid. The monopole term $\alpha_0$ is not free: the
nuclear fluid is incompressible, so the volume must be conserved. Integrating
$R(\theta)^3$ over solid angle and holding the result at $\tfrac43\pi R_0^3$
forces

$$
\alpha_0 = -\frac{1}{5}\alpha_2^2 + O(\alpha_2^3),
$$

a second-order correction that keeps the volume, and therefore the volume energy,
unchanged. Only the surface and Coulomb terms respond to $\alpha_2$ at leading
order.

$$
% caption: A volume-conserving quadrupole deformation stretches the sphere of
% radius R0 into a prolate spheroid; the deformation parameter alpha-2 measures
% the elongation along the symmetry axis.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % sphere
  \draw[black, thick] (2.0,2.0) circle (1.0);
  \node[black, anchor=north] at (2.0,0.85) {sphere};
  \node[black, anchor=center, font=\scriptsize] at (2.0,2.0) {$R_0$};
  % arrow
  \draw[->, black, thick] (3.4,2.0) -- (4.4,2.0);
  \node[black, anchor=south, font=\scriptsize] at (3.9,2.05) {deform};
  % prolate spheroid
  \draw[acc, very thick] (6.4,2.0) ellipse (0.8 and 1.5);
  \fill[acc!10] (6.4,2.0) ellipse (0.8 and 1.5);
  \node[acc, anchor=north] at (6.4,0.35) {prolate};
  % symmetry axis
  \draw[black, dashed] (6.4,0.3) -- (6.4,3.7);
  \node[black, anchor=west, font=\scriptsize] at (6.5,3.4) {symmetry axis};
\end{tikzpicture}
$$

## 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 $\alpha_2$,

$$
E_S(\alpha_2) = E_S^{(0)}\left(1 + \frac{2}{5}\alpha_2^2\right),
\qquad
E_C(\alpha_2) = E_C^{(0)}\left(1 - \frac{1}{5}\alpha_2^2\right),
$$

with the spherical values taken from the mass formula,

$$
E_S^{(0)} = a_s A^{2/3},
\qquad
E_C^{(0)} = a_c \frac{Z^2}{A^{1/3}}.
$$

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,

$$
\Delta E = E_S(\alpha_2) + E_C(\alpha_2) - E_S^{(0)} - E_C^{(0)}
        = \frac{\alpha_2^2}{5}\left(2 E_S^{(0)} - E_C^{(0)}\right).
$$

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

$$
% caption: Surface energy rises and Coulomb energy falls with deformation; their
% sum is a parabola in alpha-2 that curves upward for a stable nucleus, giving a
% restoring minimum at the spherical shape.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,2.0) -- (7.2,2.0) node[right, black!70] {deformation};
  \draw[->, black] (0.6,0.2) -- (0.6,4.2) node[above, black!70] {energy change};
  % surface (rising parabola)
  \draw[black, thick, dashed] (0.9,2.15)
    .. controls (2.5,2.5) and (4.0,3.2) .. (6.6,4.05);
  \node[black, anchor=south, font=\scriptsize] at (5.4,3.55) {surface (rises)};
  % coulomb (falling parabola)
  \draw[black, thick, densely dotted] (0.9,1.85)
    .. controls (2.5,1.5) and (4.0,0.8) .. (6.6,0.05);
  \node[black, anchor=north, font=\scriptsize] at (5.4,0.55) {Coulomb (falls)};
  % sum (net upward for stable)
  \draw[acc, very thick] (0.9,2.0)
    .. controls (2.6,2.15) and (4.2,2.55) .. (6.6,3.1);
  \node[acc, anchor=south, font=\scriptsize] at (4.6,2.75) {sum, stable};
  \fill[black] (0.6,2.0) circle (1.4pt);
  \node[black, anchor=east, font=\scriptsize] at (0.5,2.0) {sphere};
\end{tikzpicture}
$$

## The fissility parameter

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

$$
x \equiv \frac{E_C^{(0)}}{2 E_S^{(0)}}
       = \frac{a_c}{2a_s}\cdot\frac{Z^2}{A}.
$$

The deformation energy is $\Delta E = \tfrac{2}{5}E_S^{(0)}\alpha_2^2(1 - x)$. 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

$$
\left(\frac{Z^2}{A}\right)_{\text{crit}} = \frac{2a_s}{a_c}
   \approx \frac{2(17\ \mathrm{MeV})}{0.71\ \mathrm{MeV}}
   \approx 49,
$$

using the fitted mass-formula coefficients. The fissility is therefore
$x = (Z^2/A)/49$, and the heaviest nuclei approach it: $^{238}\mathrm{U}$ 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.[^wong-ld]

> **Worked example (Fissility of uranium-238).** With $Z = 92$ and $A = 238$,
> $$
> \frac{Z^2}{A} = \frac{92^2}{238} = 35.6,
> \qquad
> x = \frac{35.6}{49} = 0.73.
> $$
> The bracket $2E_S^{(0)} - E_C^{(0)} = 2E_S^{(0)}(1 - x) > 0$ is positive, so
> $^{238}\mathrm{U}$ is stable against small deformation and fissions only by
> tunnelling through a barrier of a few $\mathrm{MeV}$.

$$
% caption: The deformation-energy barrier against fission shrinks as Z-squared
% over A grows toward the critical value; a light nucleus keeps a high barrier
% while a heavy nucleus is nearly unstable and the barrier flattens.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0.4) -- (7.2,0.4) node[right, black!70] {deformation};
  \draw[->, black] (0.5,0.2) -- (0.5,4.2) node[above, black!70] {energy};
  % light nucleus: high barrier
  \draw[acc, very thick] (0.7,1.0)
    .. controls (1.8,1.1) and (2.6,3.6) .. (3.4,3.6)
    .. controls (4.2,3.6) and (4.8,1.4) .. (6.6,0.6);
  \node[acc, anchor=south, font=\scriptsize] at (3.4,3.6) {light: high barrier};
  % heavy nucleus: low flat barrier
  \draw[black, very thick, dashed] (0.7,1.6)
    .. controls (1.8,1.65) and (2.6,2.05) .. (3.4,2.0)
    .. controls (4.2,1.95) and (4.8,1.2) .. (6.6,0.5);
  \node[black, anchor=south, font=\scriptsize] at (5.3,1.5) {heavy: low};
  \node[black, anchor=east, font=\scriptsize] at (0.65,1.0) {sphere};
\end{tikzpicture}
$$

## From static to dynamic: collective vibrations

For a stable nucleus, $x < 1$, the deformation energy is a restoring potential
quadratic in $\alpha_2$,

$$
V(\alpha_2) = \frac{1}{2}C_2\,\alpha_2^2,
\qquad
C_2 = \frac{4}{5}E_S^{(0)}(1 - x)
    = \frac{4}{5}a_s A^{2/3}\left(1 - \frac{Z^2/A}{49}\right).
$$

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 $\alpha_2$ with frequency

$$
\hbar\omega_2 = \hbar\sqrt{\frac{C_2}{B_2}}.
$$

Quantizing this oscillator gives evenly spaced levels, the surface **phonons**.
Because $\alpha_2$ carries angular momentum $2\hbar$ 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.

$$
% caption: The surface tension acts as a restoring spring on the quadrupole
% coordinate, so the drop oscillates between prolate and oblate through the
% sphere; quantizing the oscillator gives the phonon states of the vibrational
% band.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % prolate
  \draw[very thick] (1.2,2.0) ellipse (0.55 and 1.1);
  \node[black, anchor=north, font=\scriptsize] at (1.2,0.75) {prolate};
  % sphere
  \draw[black, thick] (4.0,2.0) circle (0.85);
  \node[black, anchor=north, font=\scriptsize] at (4.0,0.95) {sphere};
  % oblate
  \draw[very thick] (6.8,2.0) ellipse (1.1 and 0.55);
  \node[black, anchor=north, font=\scriptsize] at (6.8,1.25) {oblate};
  % oscillation arrows
  \draw[<->, black, thick] (2.0,3.2) .. controls (3.0,3.6) .. (4.0,3.4);
  \draw[<->, black, thick] (4.0,3.4) .. controls (5.2,3.6) .. (6.0,3.2);
  \node[black, anchor=south, font=\scriptsize] at (4.0,3.55) {surface oscillation};
\end{tikzpicture}
$$

## 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 $\alpha_\lambda$, not the individual nucleons. This description
complements the
[shell model](/nuclear-physics/nuclear-models/shell-model-single-particle),
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](/nuclear-physics/nuclear-models/collective-model-rotations-vibrations),
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](/nuclear-physics/fission/fission-barrier-dynamics).

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

[^krane-ld]: Krane, "Introductory Nuclear Physics," Wiley (1988), §13.1;
    Bohr and Wheeler surface-deformation analysis.
[^wong-ld]: 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/](https://www.nndc.bnl.gov/).