---
title: Nuclear Composition and Ground-State Properties
module: Nuclear Properties
moduleNumber: 1
lessonNumber: 1
order: 101
summary: >
  The nucleus is a bound assembly of Z protons and N neutrons packed to a
  radius R = R0 A^(1/3) at a nearly constant density of about 10^17 kg/m^3.
  We fix the vocabulary of nuclides, derive nuclear size from mirror-nuclide
  and electron-scattering data, read the binding-energy-per-nucleon curve,
  and model it with the liquid-drop semiempirical mass formula.
topics: [Nuclear Properties]
draft: false
sources:
  - book: Tipler & Llewellyn
    ref: "Ch. 11 — Nuclear Physics; §11-1 The Composition of the Nucleus"
  - book: Tipler & Llewellyn
    ref: "§11-2 Ground-State Properties of Nuclei; Liquid-Drop Model and the Semiempirical Mass Formula"
---

The nucleus carries a charge $+Ze$ and almost the entire mass of the atom in a
region a hundred-thousandth the atomic radius. Two particles build it: the
**proton**, charge $+e$, and the **neutron**, charge zero, of nearly equal mass.
A nucleus is fixed by two integers, the proton number $Z$ and the neutron number
$N$, and the strong force binds these **nucleons** into a droplet whose radius,
density, and binding energy vary smoothly enough that a single classical
picture, the liquid drop, reproduces most ground-state properties.[^tl-composition]

> **Definition (Nucleon numbers).** A nucleus has $Z$ protons, $N$ neutrons, and
> mass number $A = Z + N$. $Z$ (the atomic number) sets the element and hence the
> chemical symbol; $N$ is the neutron number. A specific $(Z, N)$ species is a
> **nuclide**, written $^{A}\mathrm{X}$, e.g. $^{16}\mathrm{O}$ (with $Z = 8$,
> $N = 8$).

Because each element has a unique symbol, $Z$ need not be written; when it is, it
is a pre-subscript, $^{16}_{8}\mathrm{O}$. Three families of nuclides recur:

- **Isotopes**: same $Z$, different $N$ — e.g. $^{15}\mathrm{O}$ and
  $^{16}\mathrm{O}$. Chemically identical, they differ only in mass.
- **Isotones**: same $N$, different $Z$ — e.g. $^{13}\mathrm{C}_7$ and
  $^{14}\mathrm{N}_7$.
- **Isobars**: same $A$, different $Z$ — e.g. $^{14}\mathrm{C}$ and
  $^{14}\mathrm{N}$. These are the pairs connected by beta decay.

## Absence of electrons in the nucleus

Before Chadwick found the neutron in 1932, the mass $\approx A$ and charge
$\approx Z \approx \tfrac{1}{2}A$ tempted a model of $A$ protons and $A-Z$
nuclear electrons. Beta decay, which ejects electrons from nuclei, seemed to
support it. Four arguments overturn the idea.

- **Uncertainty-principle energy.** Confining an electron to $\Delta x \sim r
  \approx 10^{-14}\,\mathrm{m}$ forces $\Delta p \gtrsim \hbar/r$ and a minimum
  kinetic energy of order $100\,\mathrm{MeV}$. Beta electrons emerge with only
  $1$ to $2\,\mathrm{MeV}$, and no attractive potential deep enough to bind a
  $100\,\mathrm{MeV}$ electron to the nucleus exists.
- **No confining barrier.** The electron-nucleus Coulomb energy is negative, so
  there is no barrier to hold an electron in; a positive-energy electron would
  escape at once.
- **Magnetic moments.** Nuclear moments are of order the nuclear magneton
  $\mu_N = e\hbar/2m_p$, about $2000$ times smaller than the Bohr magneton
  $\mu_B = e\hbar/2m_e$ expected of a bound electron.
- **Statistics.** $^{14}\mathrm{N}$ has nuclear spin $I = 1$ and obeys
  Bose-Einstein statistics. A model of $14$ protons and $7$ electrons (21
  spin-$\tfrac{1}{2}$ fermions) would give half-integer spin and Fermi-Dirac
  statistics. The neutron model ($7$ protons, $7$ neutrons) gives integer spin,
  matching experiment.

The nucleon properties that settle these questions are collected below.[^tl-composition]

| Particle | Charge | Mass ($\mathrm{u}$) | Spin | Magnetic moment |
| --- | --- | --- | --- | --- |
| Proton | $+e$ | $1.007276$ | $\tfrac{1}{2}$ | $+2.79285\,\mu_N$ |
| Neutron | $0$ | $1.008665$ | $\tfrac{1}{2}$ | $-1.91304\,\mu_N$ |
| Deuteron ($^2\mathrm{H}$ nucleus) | $+e$ | $2.013553$ | $1$ | $+0.85744\,\mu_N$ |
| Electron | $-e$ | $5.4858 \times 10^{-4}$ | $\tfrac{1}{2}$ | $+1.00116\,\mu_B$ |

Both nucleons are spin-$\tfrac{1}{2}$ fermions and obey the exclusion principle.
The neutron is not a proton-electron composite: its spin is $\tfrac{1}{2}$, not
the integer such a pair would give. Each nucleon is itself three quarks bound by
the strong force, a structure taken up in particle physics.

## Nuclear size and density

Every measurement of nuclear radius agrees on one scaling: $R \propto A^{1/3}$,
so the nuclear volume is proportional to $A$ and the density is the same for all
nuclei. Three independent methods fix the coefficient.

**Mirror nuclides.** Two nuclei with $Z$ and $N$ interchanged, such as
$^{15}\mathrm{O}$ ($Z=8, N=7$) and $^{15}\mathrm{N}$ ($Z=7, N=8$), differ only in
electrostatic energy if the nuclear force is charge-independent. The
electrostatic energy of a uniformly charged sphere of charge $q$ and radius $R$
is

$$
U = \frac{3}{5}\frac{1}{4\pi\epsilon_0}\frac{q^2}{R},
$$

so the energy released when $^{15}\mathrm{O}$ positron-decays to
$^{15}\mathrm{N}$ is

$$
\Delta U = \frac{3}{5}\frac{1}{4\pi\epsilon_0}\frac{e^2}{R}\bigl[\,Z^2 - (Z-1)^2\,\bigr].
$$

Measured decay energies for $18$ mirror pairs give $R = R_0 A^{1/3}$ with
$R_0 = 1.2 \pm 0.2\,\mathrm{fm}$.

**Electron scattering.** Hofstadter bombarded nuclei with $200$–$500\,\mathrm{MeV}$
electrons (de Broglie wavelength $\sim 2.5\,\mathrm{fm}$, smaller than a heavy
nucleus) and read the nuclear charge distribution from the diffraction pattern.
The first diffraction minimum sits at $\sin\theta \approx 0.61\,\lambda/R$,
yielding the mean charge radius

$$
R = (1.07 \pm 0.02)\,A^{1/3}\ \mathrm{fm},
\qquad t = 2.4 \pm 0.3\ \mathrm{fm},
$$

where $t$ is the surface (skin) thickness over which the density falls from $90\%$
to $10\%$ of its central value.

**Fast-neutron attenuation.** The attenuation of a fast-neutron beam measures
the nuclear-force radius rather than the charge radius, giving $R_0 = 1.4\,\mathrm{fm}$.
The two coefficients differ because electrons probe charge while neutrons probe
the reach of the nuclear force.

$$
% caption: The nuclear charge density is flat in the interior and falls over a
% surface thickness t; the half-density point defines the radius R0.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % axes
  \draw[->, black] (0,0) -- (7.2,0) node[right, black!70] {$r$};
  \draw[->, black] (0,0) -- (0,3.6) node[above, black!70] {density};
  % density profile: flat then Woods-Saxon-like falloff
  \draw[acc, very thick]
    (0,3.0) -- (2.6,3.0)
    .. controls (3.4,3.0) and (3.6,0.3) .. (4.6,0.3)
    -- (6.6,0.05);
  % central density marker
  \draw[black, dashed] (0,3.0) -- (2.6,3.0);
  \node[anchor=east, black!70] at (0,3.0) {1.0};
  % 0.9 and 0.1 levels
  \draw[black, dashed] (0,2.7) -- (3.1,2.7);
  \draw[black, dashed] (0,0.3) -- (4.35,0.3);
  \node[anchor=east, black, font=\scriptsize] at (0,2.7) {0.9};
  \node[anchor=east, black, font=\scriptsize] at (0,0.3) {0.1};
  % half-density radius
  \draw[black, dashed] (3.6,0) -- (3.6,1.65);
  \node[anchor=north, black!70] at (3.6,0) {$R_0$};
  % skin thickness bracket (hand-drawn) placed in clear space below the falloff
  \draw[black] (3.1,0.9) -- (4.35,0.9);
  \draw[black] (3.1,0.8) -- (3.1,1.0);
  \draw[black] (4.35,0.8) -- (4.35,1.0);
  \node[black, anchor=east, font=\scriptsize] at (3.05,0.9) {$t$};
\end{tikzpicture}
$$

The numerical density is about $\rho \approx 10^{17}\,\mathrm{kg/m^3}$, fourteen
orders of magnitude above ordinary matter: a cubic millimeter of nuclear matter
has a mass near $2 \times 10^{5}\,\mathrm{metric\ tons}$.

> **Example (Radius of a neutron star).** A supernova core of pure neutrons has
> roughly nuclear density. For a solar mass $M = 1.99 \times 10^{30}\,\mathrm{kg}$
> and $\rho = 10^{17}\,\mathrm{kg/m^3}$, $M = \rho\,\tfrac{4}{3}\pi R^3$ gives
> $R^3 = 4.75 \times 10^{12}\,\mathrm{m^3}$, so $R \approx 16.8\,\mathrm{km}$ — a
> Sun-mass object the size of a city.

Nuclei are nearly spherical; the exceptions, mostly rare-earth nuclei
($Z = 57$–$71$), are ellipsoidal by $20\%$ or less. Departure from a sphere is
measured by the electric quadrupole moment $Q \propto 3\langle z^2\rangle -
\langle x^2 + y^2 + z^2\rangle$, positive for prolate ("watermelon") shapes and
negative for oblate ("flattened") ones.

## The line of stability

Of more than $3000$ known nuclides, only $266$ have stable ground states; the
rest decay. Plotting $N$ against $Z$ for the stable nuclides traces a **line of
stability** that follows $N = Z$ for light nuclei and bends toward $N > Z$ for
heavy ones.

$$
% caption: Stable nuclides cluster along a band that starts on the N = Z line
% and curves toward neutron excess; heavy nuclei need extra neutrons to dilute
% Coulomb repulsion.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (6.4,0) node[right, black!70] {$Z$};
  \draw[->, black] (0,0) -- (0,5.4) node[above, black!70] {$N$};
  % N = Z reference line
  \draw[black, dashed] (0,0) -- (4.6,4.6) node[right, black, font=\scriptsize] {$N=Z$};
  % stability band (curves above N=Z for large Z)
  \draw[acc, very thick]
    (0,0) .. controls (2.2,2.4) and (3.6,4.2) .. (5.2,5.2);
  \node[acc, anchor=west] at (4.0,3.5) {stable band};
  % a few representative stable dots
  \foreach \p in {(0.6,0.6),(1.4,1.5),(2.2,2.5),(3.0,3.4),(3.8,4.3)}
    \fill[black] \p circle (1.6pt);
\end{tikzpicture}
$$

The shape follows from two competing effects.

- **Exclusion principle.** Model the nucleons as $A$ particles in a shared well.
  The total kinetic energy is lowest when $N = Z$: filling separate proton and
  neutron ladders two-per-level packs more particles into low states than one
  ladder alone (protons and neutrons are distinguishable, so both can occupy
  $n=1$). This pulls the band toward $N = Z$.
- **Coulomb repulsion.** The electrostatic energy grows as $Z^2$. For large $A$,
  adding two neutrons costs less energy than adding a proton and a neutron, so
  the band drifts to neutron excess.

$$
% caption: Seven neutrons in one well (left) fill high levels; splitting into
% four neutrons and three protons (right) lets both species reuse the low levels
% and lowers the total energy, favoring N = Z.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % ---- left well: 7 neutrons ----
  \begin{scope}
    \draw[black] (0,0) -- (0,4.2);
    \draw[black] (2.2,0) -- (2.2,4.2);
    \draw[black] (0,0) -- (2.2,0);
    % levels at E1, 4E1, 9E1, 16E1 (scaled)
    \foreach \y/\lab in {0.5/1, 1.4/2, 2.6/3, 3.9/4} \draw[black] (0.1,\y) -- (2.1,\y);
    % occupancy: 2,2,2,1
    \fill[black] (0.7,0.5) circle (2pt); \fill[black] (1.5,0.5) circle (2pt);
    \fill[black] (0.7,1.4) circle (2pt); \fill[black] (1.5,1.4) circle (2pt);
    \fill[black] (0.7,2.6) circle (2pt); \fill[black] (1.5,2.6) circle (2pt);
    \fill[black] (1.1,3.9) circle (2pt);
    \node[anchor=north, black!70] at (1.1,-0.1) {7 neutrons};
    \node[anchor=south, black, font=\scriptsize] at (1.1,4.2) {$44\,E_1$};
  \end{scope}
  % ---- right well: 4 n + 3 p ----
  \begin{scope}[xshift=4.4cm]
    \draw[black] (0,0) -- (0,4.2);
    \draw[black] (2.6,0) -- (2.6,4.2);
    \draw[black] (0,0) -- (2.6,0);
    \foreach \y in {0.5,1.4,2.6} \draw[black] (0.1,\y) -- (2.5,\y);
    \draw[black] (0.1,0) -- (1.25,0);
    % neutrons left half
    \fill[black] (0.4,0.5) circle (2pt); \fill[black] (0.4,1.4) circle (2pt);
    \fill[black] (0.9,0.5) circle (2pt); \fill[black] (0.9,1.4) circle (2pt);
    % protons right half (open circles)
    \draw[thick] (1.7,0.5) circle (2pt); \draw[thick] (2.2,0.5) circle (2pt);
    \draw[thick] (1.95,1.4) circle (2pt);
    \node[anchor=north, black!70, align=center] at (1.3,-0.1) {4 neutrons\\3 protons};
    \node[anchor=south, black, font=\scriptsize] at (1.3,4.2) {$16\,E_1$};
  \end{scope}
\end{tikzpicture}
$$

Nucleons also pair with identical partners: of the $266$ stable nuclides, $159$
are even-even ($Z$ and $N$ both even) and only $4$ are odd-odd.

| $Z$ | $N$ even | $N$ odd |
| --- | --- | --- |
| Even | $159$ | $53$ |
| Odd | $50$ | $4$ |

Nuclei with $Z$ or $N$ equal to a **magic number** ($2, 8, 20, 28, 50, 82, 126$)
have unusually many stable isotopes or isotones — tin ($Z = 50$) has ten. These
are the nuclear analog of closed electron shells, taken up with the
[shell model](/nuclear-physics/nuclear-force-deuteron/nuclear-force-shell-overview).

## Binding energy

The mass of a nucleus is less than the sum of its constituent masses; the deficit,
times $c^2$, is the energy that would be needed to pull it apart.

> **Definition (Nuclear binding energy).** For a nucleus of $Z$ protons and $N$
> neutrons with atomic mass $M_A$,
> $$
> B = Z M_{\mathrm{H}}c^2 + N m_n c^2 - M_A c^2,
> $$
> using the hydrogen-atom mass $M_{\mathrm{H}}$ so the $Z$ electron masses cancel.
> Atomic (electron) binding energies, of order $\mathrm{keV}$, are negligible
> beside nuclear binding of many $\mathrm{MeV}$.

Dividing by $A$ and plotting against $A$ gives the single most important curve in
nuclear physics.

$$
% caption: Binding energy per nucleon peaks near iron (A = 56) at about 8.8 MeV;
% the near-flatness above A = 16 signals a saturated, short-range force.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (8.2,0) node[right, black!70] {$A$};
  \draw[->, black] (0,0) -- (0,4.2) node[above, black!70] {$\frac{B}{A}$ (MeV)};
  % y ticks
  \foreach \y/\lab in {1/2,2/4,3/6,3.9/8.8} {
    \draw[black] (-0.08,\y) -- (0.08,\y);
    \node[anchor=east, black, font=\scriptsize] at (-0.1,\y) {\lab};
  }
  % x ticks
  \foreach \x/\lab in {2/50,4/100,6/150,7.6/238} \node[anchor=north, black, font=\scriptsize] at (\x,-0.05) {\lab};
  % rising then slow decline curve
  \draw[acc, very thick]
    (0.15,0.2)
    .. controls (0.5,2.6) and (1.1,3.85) .. (2.3,3.95)
    .. controls (4.0,4.0) and (5.5,3.6) .. (7.6,3.15);
  % peak marker near Fe
  \fill[acc] (2.3,3.95) circle (1.6pt);
  \node[acc, anchor=south, font=\scriptsize] at (2.5,3.95) {$^{56}$Fe};
  % light-nucleus labels
  \node[black, font=\scriptsize] at (0.55,1.3) {$^2$H};
  \node[black, font=\scriptsize] at (1.15,3.55) {$^4$He};
  % fusion / fission arrows along the curve direction
  \draw[->, black] (0.9,2.2) -- (1.9,3.2) node[midway, sloped, above, font=\scriptsize] {fusion};
  \draw[->, black] (6.7,3.35) -- (4.6,3.75) node[midway, sloped, above, font=\scriptsize] {splitting};
\end{tikzpicture}
$$

Three features read off directly.

- **Near constancy** ($B/A \approx 8.3$–$8.8\,\mathrm{MeV}$ for $A > 16$)
  signals **saturation**: each nucleon binds only to its nearest neighbors. If
  every nucleon bound to all $A-1$ others, $B/A$ would grow as $A-1$, not stay
  flat.
- **The peak near** $^{56}\mathrm{Fe}$ is the reason energy is released both by
  fusing light nuclei up the left slope and by splitting heavy nuclei down the
  right, the subject of
  [nuclear reactions](/nuclear-physics/nuclear-reactions/reaction-kinematics-cross-sections).
- **Sharp light-nucleus spikes** ($^4\mathrm{He}$, $^{12}\mathrm{C}$,
  $^{16}\mathrm{O}$) mark especially tight closed-shell structures.

## The liquid-drop model

Constant density and saturation are the defining properties of a classical liquid
drop, and the analogy yields the **semiempirical (Weizsäcker) mass formula**.[^tl-liquiddrop]
Binding energy is written as a sum of terms, each with a physical origin and a
fitted coefficient.

> **Theorem (Semiempirical mass formula).** The binding energy of a nucleus is
> approximately
> $$
> B = a_V A - a_S A^{2/3} - a_C \frac{Z(Z-1)}{A^{1/3}} - a_A \frac{(N-Z)^2}{A} \pm \delta,
> $$
> and the atomic mass follows from $M(Z,A)c^2 = Z M_{\mathrm{H}}c^2 + N m_n c^2 - B$.

- **Volume term** $a_V A$: each nucleon binds to a fixed number of neighbors, so
  bulk binding scales with the nucleon count (with $A \propto$ volume).
- **Surface term** $-a_S A^{2/3}$: nucleons at the surface have fewer neighbors,
  a deficit proportional to surface area $\propto R^2 \propto A^{2/3}$. This is
  the nuclear analog of surface tension.
- **Coulomb term** $-a_C Z(Z-1)/A^{1/3}$: the electrostatic repulsion of $Z$
  protons in a sphere of radius $R \propto A^{1/3}$.
- **Asymmetry term** $-a_A (N-Z)^2/A$: the exclusion-principle cost of an
  imbalance between $N$ and $Z$, minimized at $N = Z$.
- **Pairing term** $\pm\delta$: positive for even-even nuclei (extra binding
  from paired nucleons), negative for odd-odd, zero for odd-$A$.

Representative fitted coefficients are of order $a_V \approx 15.8$,
$a_S \approx 18.3$, $a_C \approx 0.71$, $a_A \approx 23.2$, and
$\delta \approx 12\,A^{-1/2}\,\mathrm{MeV}$.[^tl-liquiddrop] The formula produces
the smooth solid curve through the $B/A$ data above, is quadratic in $Z$ at fixed
$A$ (setting up the parabolas that govern
[beta decay](/nuclear-physics/radioactive-decay/decay-law-modes)), and predicts
the fission of heavy nuclei once the Coulomb term outgrows the surface term.

$$
% caption: The four smooth terms of the mass formula: volume binding grows with
% A while surface, Coulomb, and asymmetry each subtract, leaving the observed
% B/A curve.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (7.6,0) node[right, black!70] {$A$};
  \draw[->, black] (0,0) -- (0,4.4) node[above, black!70] {$\frac{B}{A}$ (MeV)};
  % volume (constant)
  \draw[black, thick] (0.15,4.0) -- (7.2,4.0);
  \node[black, anchor=west, font=\scriptsize] at (5.3,4.15) {volume $a_V$};
  % after subtracting surface (rises then flattens toward a_V)
  \draw[black, dashed] (0.15,1.2) .. controls (1.5,3.2) and (3.5,3.7) .. (7.2,3.85);
  \node[black, anchor=west, font=\scriptsize] at (2.4,3.15) {minus surface};
  % after also Coulomb + asymmetry (net observed, peaks then falls)
  \draw[acc, very thick] (0.3,0.7) .. controls (1.4,3.4) and (2.6,3.55) .. (3.0,3.55)
    .. controls (4.6,3.55) and (6.2,2.7) .. (7.2,2.2);
  \node[acc, anchor=west, font=\scriptsize] at (3.4,3.55) {net $\frac{B}{A}$};
  \node[black, anchor=west, font=\scriptsize] at (5.4,2.55) {minus Coulomb term};
\end{tikzpicture}
$$

## Nuclear spin and magnetic moment

The nucleons' spins and orbital motions combine into a resultant angular momentum
$\vec I$, the **nuclear spin**. All even-even nuclei have $I = 0$ in the
ground state. The nuclear magnetic moment, of order the nuclear magneton
$\mu_N = e\hbar/2m_p$, couples to the electrons' angular momentum $\vec J$ to
give the atom's total angular momentum $\vec F = \vec I + \vec J$.
This coupling splits each spectral line into $(2J+1)$ or $(2I+1)$ components,
whichever is fewer — **hyperfine structure**. The splitting is smaller than the
fine-structure [spin-orbit](/atomic-physics/fine-structure-and-the-dirac-atom/spin-orbit-thomas-precession)
splitting by the ratio $\mu_N/\mu_B \approx 10^{-3}$, so counting hyperfine lines
is a direct measurement of $I$. Nuclear magnetic moments, driven to resonance in
a field, are also the basis of the
[NMR and MRI](/nuclear-physics/radiation-matter-applications/nuclear-applications-dating-medicine) taken up later.

[^tl-composition]: **Tipler & Llewellyn**, _Modern Physics_, §11-1 — The Composition of the Nucleus: nucleon properties, the arguments against nuclear electrons, and the isotope/isotone/isobar vocabulary.
[^tl-liquiddrop]: **Tipler & Llewellyn**, _Modern Physics_, §11-2 — Ground-State Properties of Nuclei; the liquid-drop model and semiempirical mass formula (volume, surface, Coulomb, asymmetry, and pairing terms) fitted to the binding-energy-per-nucleon data of Figure 11-10. The quoted coefficient values are representative fits to the Weizsäcker terms.