Nuclear Composition and Ground-State Properties

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.¹

Because each element has a unique symbol, $Z$ need not be written; when it is, it is a pre-subscript, ${}_8^{16}\mathrm{O}$. 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 $\approx Z\approx \frac{1}{2}A$ 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 $\Delta p\gtrsim \hslash /r$ 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 ${\mu }_N=e\hslash /2m_p$, about $2000$ times smaller than the Bohr magneton ${\mu }_B=e\hslash /2m_e$ 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-$\frac{1}{2}$ 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.¹

Both nucleons are spin-$\frac{1}{2}$ fermions and obey the exclusion principle. The neutron is not a proton-electron composite: its spin is $\frac{1}{2}$, 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

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

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

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.

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

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 $Q\propto 3⟨z^2⟩-⟨x^2+y^2+z^2⟩$, 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.

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.

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.

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.

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.

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

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.
• 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.² Binding energy is written as a sum of terms, each with a physical origin and a fitted coefficient.

• 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 $\delta \approx 12A^{-1/2}\mathrm{MeV}$.² 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), and predicts the fission of heavy nuclei once the Coulomb term outgrows the surface term.

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 ${\mu }_N=e\hslash /2m_p$, 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 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 taken up later.