Nuclear structure Knowledge-Based Study

Study notes

1. Constituents of the Atomic Nucleus

   The atomic nucleus is the compact, positively charged core of an atom, containing almost all of its mass. It is composed of two primary subatomic particles collectively referred to as nucleons:

   • Protons: Positively charged particles, each possessing an elementary charge of +e and a rest mass of approximately 1.673 × 10-27 kg.
   • Neutrons: Electrically neutral particles with zero net charge and a rest mass slightly greater than that of a proton, approximately 1.675 × 10-27 kg.

   A specific nuclear species or nuclide is characterized by three fundamental quantities:

   • Atomic number (Z): The number of protons in the nucleus, which dictates the chemical identity of the element.
   • Neutron number (N): The number of neutrons in the nucleus.
   • Mass number (A): The total number of nucleons, defined by the relation $A = Z + N$.

   In standard notation, a nuclide of chemical symbol X is represented as $^{A}_{Z}\text{X}$. Nuclei that share the same atomic number Z but possess different neutron numbers N (and thus different mass numbers A) are designated as isotopes.

   Nuclear Dimensions and Density

   High-energy electron scattering experiments demonstrate that the nucleus does not have a sharply defined boundary, but its effective charge distribution can be approximated as a sphere. The empirical relationship describing the average nuclear radius $R$ as a function of mass number $A$ is:

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

   where $R_0$ is an experimentally determined constant typically evaluated as $1.2 \times 10^{-15}\text{ m}$ (or 1.2 fm).

   Because the volume of a sphere is proportional to $R^3$, the nuclear volume scales directly with the total number of nucleons:

   $$V = \frac{4}{3}\pi R^3 = \frac{4}{3}\pi R_0^3 A$$

   Assuming an average nucleon mass $m$, the mass density $\rho$ of nuclear matter is:

   $$\rho = \frac{m A}{V} = \frac{m A}{\frac{4}{3}\pi R_0^3 A} = \frac{3m}{4\pi R_0^3}$$

   Notice that the mass number $A$ cancels out. Consequently, to a first approximation, all atomic nuclei share an essentially constant density of approximately $2.3 \times 10^{17}\text{ kg/m}^3$, reflecting the tightly packed, incompressible nature of nuclear matter.

   Fundamental Nuclear Interactions

   Within the nucleus, positively charged protons experience mutual electrostatic repulsion given by Coulomb's law. Because protons are confined to distances on the order of femtometers, this repulsive force is extraordinarily intense. For a nucleus to remain stable, there must exist an attractive interaction powerful enough to counteract this repulsion: the strong nuclear force.

   Key Properties of the Strong Nuclear Force

   • Short Range: The force operates effectively only over femtometer scales (approximately 1 to 2 fm). Beyond roughly 2 to 3 fm, its strength drops rapidly to zero. At extremely short separations (below approximately 0.5 fm), the force becomes strongly repulsive, preventing nucleons from collapsing into one another.
   • Charge Independence: The strong interaction acts with identical strength between proton-proton (p-p), neutron-neutron (n-n), and proton-neutron (p-n) pairs in identical quantum states.
   • Saturation Property: A given nucleon interacts only with its immediate neighbors rather than with all nucleons in the nucleus, which explains why binding energy per nucleon remains relatively constant across middle-mass nuclei.

   Mass Defect and Nuclear Stability

   Precise mass measurements reveal that the rest mass of any stable bound nucleus is strictly less than the sum of the individual rest masses of its constituent free protons and neutrons. This difference is known as the mass defect ($\Delta m$):

   $$\Delta m = \left[ Z m_p + (A - Z) m_n \right] - m_{\text{nucleus}}$$

   According to special relativity, mass and energy are equivalent via $E = m c^2$. The energy equivalent of this missing mass represents the total binding energy ($E_b$) of the nucleus:

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

   Binding energy corresponds to the work required to completely disassemble a nucleus into its separate, non-interacting constituent nucleons.

   Binding Energy per Nucleon

   The measure of intrinsic nuclear stability is the binding energy per nucleon, defined as $E_b / A$:

   • For very light nuclei, $E_b / A$ is comparatively low and exhibits distinct local peaks (such as $^{4}_{2}\text{He}$).
   • The curve rises to a broad maximum around $A \approx 56\text{ to }62$ (notably iron-56 and nickel-62), where $E_b / A \approx 8.8\text{ MeV/nucleon}$, representing the most tightly bound systems.
   • For heavier nuclei, $E_b / A$ gradually decreases toward $7.6\text{ MeV/nucleon}$ due to growing long-range Coulomb repulsion among protons.

   This behavior explains why combining light nuclei (nuclear fusion) or splitting very heavy nuclei (nuclear fission) results in reaction products with higher binding energy per nucleon, thereby releasing net energy.