# Nuclear Physics

Nuclear properties:  A nucleus is composed of $A$ nucleons ($Z$ protons and $N$ neutrons).  All nuclei have about the same density.  The radius of a nucleus with mass number $A$ is given approximately by equation $R=R_0\sqrt[3]A$ ($R_0=1.2\times10^{-15}\,\mathrm{m}$).  A single nuclear species of a given $Z$ and $N$ is called a nuclide.  Isotopes are nuclides of the same element (same $Z$) that have different number of neutrons.  Nuclear masses are measured in atomic mass units.  Nucleons have an angular momentum and a magnetic moment.

Nuclear binding and structure:  The mass of a nucleus is always less than the mass of the protons and neutrons within it.  The mass difference multiplied by $c^2$ gives the binding energy $E_{\mathrm{B}}=(ZM_{\mathrm{H}}+Nm_{\mathrm{n}}-\,_Z^AM)c^2$.  The binding energy for a given nuclide is determined by the nuclear force, which is short range and favors pairs of particles, and by the electrical repulsion between protons.  A nucleus is unstable if $A$ or $Z$ is too large or if the ratio $N/Z$ is wrong.  Two widely used models of the nucleus are the liquid-drop model and the shell model; the latter is analogous to the central-field approximation for atomic structure.

Radioactive decay:  Unstable nuclides usually emit an alpha-particle (a $\,_2^4\mathrm{He}$ nucleus) or a beta-particle (an electron) in the process of change to another nuclide, sometimes followed by a gamma-ray photon.  The rate of decay of an unstable nucleus is described by the decay constant $\lambda$, the half-life $T_{1/2}$, or the lifetime $T_{\mathrm{mean}}$: $T_{\mathrm{mean}}=\frac1{\lambda}=\frac{T_{1/2}}{\ln2}=\frac{T_{1/2}}{0.693}$.  If the number of nuclei at time $t=0$ is $N_0$ and no more are produced, the number at time $t$ is given by equation $N(t)=N_0e^{-\lambda t}$.

Biological effects of radiation:   The biological effect of any radiation depends on the product of the energy absorbed per unit mass and the relative biological effectiveness (RBE), which is different for different radiations.

Nuclear reactions:  In a nuclear reaction, two nuclei or particles collide to produce two new nuclei or particles.  Reactions can be exoergic or endoergic.  Several conservation law, including charge, energy, momentum, angular momentum, and nucleon number, are obeyed.  Energy is released by the fission of a heavy nucleus into two lighter, always unstable, nuclei.  Energy is also released by the fusion of two light nuclei into a heavier nucleus.