11.1 Nuclear structure and properties

Atomic Nuclei Composition

Nucleon Structure and Arrangement

Atomic nuclei are made up of protons and neutrons, collectively called nucleons, held together by the strong nuclear force. The number of protons defines the element (its atomic number $Z$), while the number of neutrons affects which isotope of that element you're dealing with and how stable the nucleus is.

Nucleons are packed into an extremely dense, compact volume. The nucleus is roughly $10^{-15}$ m across, about 100,000 times smaller than the full atom with its electron cloud. A striking feature: nuclear density stays approximately constant across all nuclei, regardless of how many nucleons are present. This is a direct clue about the nature of the nuclear force.

Nuclear Models and Forces

Two complementary models describe nuclear structure:

• The liquid drop model treats the nucleus like a drop of incompressible fluid. It captures bulk properties well and forms the basis of the semi-empirical mass formula (more on that below).
• The shell model treats nucleons as occupying discrete energy levels inside a potential well, similar to how electrons fill atomic orbitals. This model explains magic numbers and spin properties.

Neither model alone gives the full picture, but together they cover most observed nuclear behavior.

The strong nuclear force has distinctive short-range characteristics:

• It's strongly attractive between nucleons at distances of about 1–3 fm
• It becomes repulsive at very short distances (below ~0.5 fm), preventing nucleons from collapsing onto each other
• It's charge-independent: the strong force between two protons, two neutrons, or a proton and neutron is essentially the same
• Its short range means each nucleon only interacts with its nearest neighbors, not with every other nucleon in the nucleus

This force must overcome the electrostatic (Coulomb) repulsion between positively charged protons. For light nuclei, the strong force wins easily. For very heavy nuclei, the cumulative Coulomb repulsion starts to compete, which is why the heaviest elements tend to be unstable.

Nuclear Properties

Mass and Energy

Nuclear mass is expressed in atomic mass units (amu), where 1 amu = 1.66054 × $10^{-27}$ kg, defined as 1/12 the mass of a carbon-12 atom.

A nucleus always has slightly less mass than the sum of its individual protons and neutrons measured separately. This difference is the mass defect ($\Delta m$), and it corresponds to the energy released when those nucleons bind together. The relationship comes from Einstein's equation:

$E_b = \Delta m \cdot c^2$

where $E_b$ is the binding energy. This is the energy you'd need to supply to completely disassemble the nucleus into free protons and neutrons.

Binding energy per nucleon ($E_b / A$) is the key measure of nuclear stability. When you plot it against mass number $A$:

• It rises steeply for light nuclei
• Peaks near $A \approx 56$ (iron-56 and nickel-62 sit at the top, around 8.8 MeV per nucleon)
• Gradually decreases for heavier nuclei

This curve explains why fusion of light nuclei and fission of heavy nuclei both release energy: both processes move nuclei toward that peak.

Charge and Size

The total nuclear charge equals $Ze$, where $Z$ is the proton count and $e$ is the elementary charge ($1.602 \times 10^{-19}$ C).

Nuclear size follows an empirical relationship:

$R = R_0 A^{1/3}$

where $R_0 \approx 1.2 \text{ to } 1.3$ fm (femtometers, $10^{-15}$ m) and $A$ is the mass number. The cube-root dependence on $A$ means the nuclear volume scales linearly with the number of nucleons. That's exactly what you'd expect if nuclear density is constant, reinforcing the liquid drop picture.

Spin and Magnetic Properties

Each proton and neutron carries an intrinsic spin of $\frac{1}{2}\hbar$. The nuclear spin $I$ results from coupling the spins and orbital angular momenta of all nucleons. Some useful patterns:

• Even-even nuclei (even $Z$, even $N$) always have $I = 0$ in their ground state
• Odd-$A$ nuclei have half-integer spin
• Odd-odd nuclei have integer spin

The nuclear magnetic moment arises because protons are charged particles with spin and orbital motion. Even neutrons contribute a magnetic moment (due to their internal quark structure). These magnetic moments are much smaller than electron magnetic moments, typically measured in nuclear magnetons ($\mu_N$).

Nuclear magnetic moments are the basis of NMR (nuclear magnetic resonance) and its medical application, MRI. When placed in an external magnetic field, nuclei with nonzero spin precess at characteristic frequencies, allowing detailed imaging and spectroscopy.

Isotopes and Nuclear Physics

Isotope Fundamentals

Isotopes are atoms of the same element (same $Z$) with different numbers of neutrons ($N$), and therefore different mass numbers ($A = Z + N$). They share essentially identical chemical properties because chemistry is governed by electron configuration, which depends on $Z$ alone. Their nuclear properties, however, can differ dramatically.

Standard isotope notation writes the mass number as a superscript and atomic number as a subscript before the element symbol: $^{A}_{Z}X$. For example, $^{12}_{6}\text{C}$ and $^{13}_{6}\text{C}$ are both carbon, but carbon-13 has one extra neutron.

Natural isotopic abundance varies widely:

• Some elements have many stable isotopes (tin has 10)
• Others have only one (fluorine exists only as $^{19}\text{F}$, gold only as $^{197}\text{Au}$)
• The weighted average of isotopic masses gives the atomic mass you see on the periodic table

Radioisotopes and Applications

Radioactive isotopes (radioisotopes) have unstable nuclei that undergo decay, emitting radiation in the process. They're used extensively across fields:

• Nuclear medicine: Technetium-99m is the most widely used medical radioisotope, with a 6-hour half-life ideal for diagnostic imaging
• Radiometric dating: Carbon-14 (half-life ~5,730 years) allows dating of organic materials up to roughly 50,000 years old
• Industrial applications: Iodine-131 serves as a tracer for leak detection; cobalt-60 is used for sterilization and radiotherapy

Isotope separation is critical for nuclear technology. Uranium enrichment, for instance, increases the proportion of fissile $^{235}\text{U}$ (natural abundance only 0.7%) relative to $^{238}\text{U}$. Methods include gaseous diffusion, gas centrifugation, and electromagnetic separation, all exploiting the small mass difference between isotopes.

Isotopes in Scientific Research

Isotope ratios serve as powerful tools across multiple disciplines:

• Nucleosynthesis: Observed isotope abundances constrain models of how elements formed, both during the Big Bang (producing deuterium, helium-3, helium-4, and lithium-7) and in stellar interiors (where fusion builds heavier elements up through iron)
• Paleoclimatology: The ratio of $^{18}\text{O}$ to $^{16}\text{O}$ in ice cores and ocean sediments tracks past temperatures, since lighter isotopes evaporate preferentially
• Geochemistry: Ratios of radiogenic isotopes (like $^{87}\text{Sr}/^{86}\text{Sr}$) trace rock formation and mantle processes

Nuclear Stability

Nuclear Chart and Stability Trends

The chart of nuclides plots all known nuclei with $Z$ (proton number) on one axis and $N$ (neutron number) on the other. Stable nuclei cluster along a narrow band called the valley of stability.

For light nuclei ($A < 40$), stable isotopes tend to have $N \approx Z$. As $Z$ increases, stable nuclei require progressively more neutrons than protons. By the time you reach lead ($Z = 82$), the ratio is about $N/Z \approx 1.5$. The extra neutrons provide additional strong-force attraction without adding Coulomb repulsion, helping to stabilize the nucleus.

Nuclei that fall outside the valley of stability are radioactive. Those with too many neutrons tend to undergo $\beta^-$ decay; those with too few neutrons undergo $\beta^+$ decay or electron capture.

Magic Numbers and Nuclear Shell Structure

Certain nucleon numbers produce exceptionally stable configurations. These magic numbers are:

$2, 8, 20, 28, 50, 82, 126$

They apply independently to protons and neutrons. The concept is analogous to closed electron shells in atomic physics (noble gas configurations), but the energy level spacing in the nuclear potential is different, which is why the magic numbers don't match.

Nuclei with a magic number of protons or neutrons show enhanced stability. Doubly magic nuclei have magic numbers for both, and they're remarkably stable:

• $^{4}\text{He}$ ($Z = 2, N = 2$)
• $^{16}\text{O}$ ($Z = 8, N = 8$)
• $^{48}\text{Ca}$ ($Z = 20, N = 28$)
• $^{208}\text{Pb}$ ($Z = 82, N = 126$)

Lead-208 is the heaviest stable nucleus, and its double magic character is a big reason why.

Factors Influencing Nuclear Stability

Several competing effects determine whether a nucleus is stable:

• Strong nuclear force: Provides short-range attraction between all neighboring nucleon pairs. Since it's short-range, its contribution scales roughly with $A$ (each nucleon interacts with a fixed number of neighbors).
• Coulomb repulsion: Acts between all proton pairs across the entire nucleus. It scales roughly as $Z(Z-1)/A^{1/3}$, growing faster than the strong force for large nuclei. This is why no stable nuclei exist beyond $Z = 82$.
• Pauli exclusion principle: Protons and neutrons are fermions, so no two identical nucleons can occupy the same quantum state. This drives the shell structure and favors roughly equal numbers of protons and neutrons (the asymmetry effect).
• Pairing effect: Nucleons tend to pair up with opposite spins, lowering energy. Even-even nuclei are more stable than odd-odd nuclei as a result.

The semi-empirical mass formula (SEMF), also called the Bethe-Weizsäcker formula, captures these effects quantitatively:

$B(A,Z) = a_V A - a_S A^{2/3} - a_C \frac{Z(Z-1)}{A^{1/3}} - a_A \frac{(A - 2Z)^2}{A} + \delta(A,Z)$

Each term has a physical origin:

• Volume term ($a_V A$): Bulk binding from the strong force
• Surface term ($-a_S A^{2/3}$): Nucleons at the surface have fewer neighbors, reducing binding
• Coulomb term ($-a_C \frac{Z(Z-1)}{A^{1/3}}$): Electrostatic repulsion between protons
• Asymmetry term ($-a_A \frac{(A-2Z)^2}{A}$): Penalizes imbalance between proton and neutron numbers
• Pairing term ($\delta$): Positive for even-even, zero for odd-$A$, negative for odd-odd nuclei

This formula does a surprisingly good job predicting binding energies across the chart of nuclides, and it's the quantitative backbone of the liquid drop model.