Atomic Nucleus

Constituents of a nucleus

• The nucleus is made of subatomic particles called protons and neutrons, together called nucleons.
• The proton is positively charged, the magnitude of its charge being equal to that of the electron's charge; its mass is about $1836$ times the mass of an electron. The neutron is electrically neutral and its mass is nearly the same as the proton's but slightly higher.
• The number of protons is the atomic number $Z$, which also equals the number of electrons in the neutral atom, so the atom as a whole is electrically neutral. The number of neutrons is $N$.
• The total number of nucleons is the mass number $A = Z + N$; it determines the mass of the nucleus and of the atom.
• A nuclide is written ${}^{A}_{Z}\text{X}$: hydrogen ${}^{1}_{1}\text{H}$, carbon ${}^{12}_{6}\text{C}$, oxygen ${}^{16}_{8}\text{O}$.
• Chemical properties are decided by the number of electrons, i.e. by $Z$ alone.

Isotopes, isobars and isotones

• The number of protons and electrons in the atoms of a given element is fixed, but the number of neutrons can vary. Hydrogen can have zero, one or two neutrons, giving ${}^{1}_{1}\text{H}$ (hydrogen), ${}^{2}_{1}\text{H}$ (deuterium) and ${}^{3}_{1}\text{H}$ (tritium).

• Isotopes have the same chemical properties, since chemistry depends only on $Z$.

Units for measuring masses

• In kilogram: $m_e = 9.109383\times10^{-31}\ \text{kg}$, $m_p = 1.672623\times10^{-27}\ \text{kg}$, $m_n = 1.674927\times10^{-27}\ \text{kg}$. The kilogram is an inconvenient unit at this scale.
• Unified atomic mass unit: one u is one twelfth of the mass of a neutral carbon atom of atomic mass $12$ in its lowest electronic state, and $1\ \text{u} = 1.6605402\times10^{-27}\ \text{kg}$. In this unit $m_e = 0.00055\ \text{u}$, $m_n = 1.008665\ \text{u}$, and the hydrogen atom is $1.007825\ \text{u}$ (the free proton is $1.00728\ \text{u}$).
• Energy equivalent: by $E = mc^2$ a mass may be quoted in $\text{eV}/c^2$, with $1\ \text{u} = 931.5\ \text{MeV}/c^2$. Then $m_e = 0.511\ \text{MeV}/c^2$, $m_p = 938.28\ \text{MeV}/c^2$ and $m_n = 939.57\ \text{MeV}/c^2$.

Size of nuclei

• Not all nuclei are the same size; the size depends on the number of nucleons, i.e. on the mass number $A$. Experimentally the radius $R_X$ of a nucleus $X$ is

$$R_X = R_0A^{1/3},\qquad R_0 = 1.2\times10^{-15}\ \text{m}$$

• So the nuclear volume $\dfrac{4}{3}\pi R_X^3 = \dfrac{4}{3}\pi R_0^3A$ is directly proportional to $A$, and $\dfrac{R_1}{R_2} = \left(\dfrac{A_1}{A_2}\right)^{1/3}$.

Density of a nucleus

• Take $m$ as the average mass of a nucleon, the proton and neutron masses being nearly equal. The mass of the nucleus is then $mA$ and the density is

$$\rho = \dfrac{mA}{\dfrac{4}{3}\pi R_X^3} = \dfrac{3mA}{4\pi R_X^3}$$

• Substituting $R_X = R_0A^{1/3}$, so that $R_X^3 = R_0^3A$, the $A$ cancels:

$$\rho = \dfrac{3m}{4\pi R_0^3} = \text{constant}$$

• Hence the density of a nucleus does not depend on its mass number and is the same for all nuclei. Putting in the numbers, $\rho = 2.3\times10^{17}\ \text{kg m}^{-3}$.
• This is enormous: osmium, the densest known element, has a density of only $2.2\times10^{4}\ \text{kg m}^{-3}$, smaller than nuclear density by thirteen orders of magnitude.
• Worked check: for ${}^{70}\text{Ge}$ of mass $69.924\ \text{u}$, $R = 1.2\times10^{-15}\times70^{1/3} = 4.945\times10^{-15}\ \text{m}$ and $\rho = \dfrac{3M}{4\pi R^3} = 2.292\times10^{17}\ \text{kg m}^{-3}$.

Nuclear forces

• Of the four fundamental forces, the one that determines the structure of the nucleus is the strong force, also called the nuclear force. It acts between protons and neutrons and is mostly attractive.
• The protons in a nucleus repel one another electrostatically. The nuclear force is much stronger than the electrostatic force at the distances between nucleons in a typical nucleus, so it overcomes the repulsion and holds the nucleus together, making it stable.
• Properties, as the board wants them listed:
• It is the strongest force among subatomic particles, about $50$ to $60$ times stronger than the electrostatic force.
• Its range is only about a few fm, and the force becomes negligible when two nucleons are separated by more than that — quite unlike the electrostatic and gravitational forces, whose range is infinite.
• It is charge independent: the force between two neutrons at a given separation is the same as that between two protons, or between a neutron and a proton, at the same separation.
• The nuclear force is not yet well understood; only these properties are known and expected in an answer.