Nuclear Physics: Structure, Stability, and Radioactivity

Nuclear Physics

Atomic Nuclei: Structure and Constituents

The atomic nucleus is the dense central core of an atom, composed of protons and neutrons (collectively called nucleons). Electrons orbit the nucleus at much larger distances, making the atom mostly empty space. The nucleus contains nearly all the mass of the atom and is on the order of 10-14 m in size.

• Proton (p): Positively charged particle, mass ≈ 1 atomic mass unit (u).

• Neutron (n): Neutral particle, mass ≈ 1 u.

• Electron (e): Negatively charged, much lighter than nucleons.

• Atomic number (Z): Number of protons; defines the element.

• Neutron number (N): Number of neutrons.

• Mass number (A): Total number of nucleons, A = Z + N.

Isotopes and Isotope Notation

Isotopes are atoms of the same element (same Z) but with different numbers of neutrons (N), resulting in different mass numbers (A). Isotope notation is written as AZ, where Z is the chemical symbol and A is the mass number (e.g., 12C for carbon-12).

• Example: Uranium-238 (238U) has Z = 92 protons and N = 146 neutrons (A = 238).

Atomic Mass and Mass-Energy Equivalence

The atomic mass unit (u) is defined as 1/12 the mass of a carbon-12 atom. Using Einstein's equation, mass and energy are related by $E = mc^2$.

• 1 u = 1.6605 × 10-27 kg = 931 MeV/c2

• The atomic mass listed on the periodic table is the weighted average of naturally occurring isotopes.

Nuclear Size and Density

The radius of a nucleus is given by $R = r_0 A^{1/3}$, where $r_0 \approx 1.2 \times 10^{-15}$ m. Nuclei are extremely dense, with densities around $2.3 \times 10^{17}$ kg/m3, much greater than ordinary matter.

Nuclear Stability

There are over 3000 known isotopes, but only 252 are stable. Stability depends on the ratio of neutrons to protons. For light elements, stability occurs when N ≈ Z; for heavier elements, more neutrons are needed. No nuclei with Z > 82 (lead) are stable.

Nuclear Binding Energy

Nuclear binding energy is the energy required to separate all nucleons in a nucleus. It is calculated using the mass defect (the difference between the mass of the separated nucleons and the actual nucleus):

• $B = (Z m_H + N m_n - m_{atom}) \times (931.49 \text{ MeV/u})$

• Binding energy per nucleon: $B/A$

• Binding energy per nucleon peaks at about 8 MeV for medium-mass nuclei (A ≈ 60), indicating maximum stability.

The Strong Nuclear Force

The strong force is the fundamental force that holds nucleons together in the nucleus, overcoming the electrostatic repulsion between protons. It is short-range (acts over nuclear distances) and much stronger than the electromagnetic force within its range.

• Acts only between nucleons (not electrons).

• Responsible for nuclear stability and binding energy.

Radioactivity and Nuclear Decay

Radioactivity is the spontaneous emission of particles or electromagnetic radiation from unstable nuclei. The number of radioactive nuclei decays exponentially with time:

• $N(t) = N_0 e^{-rt} = N_0 e^{-t/\tau}$, where $\tau = 1/r$ is the mean lifetime.

• The half-life $t_{1/2}$ is the time for half the nuclei to decay: $t_{1/2} = \tau \ln(2)$.

• Activity (R) is the number of decays per second: $R = -\frac{dN}{dt} = N_0/\tau \cdot e^{-t/\tau}$

• Units: 1 becquerel (Bq) = 1 decay/s; 1 curie (Ci) = 3.7 × 1010 Bq

Radioactive Dating

Radioactive dating uses the known half-life of isotopes (e.g., 14C, 40K) to estimate the age of organic or geological samples by measuring the remaining activity or isotope ratio.

Nuclear Decay Mechanisms

• Alpha decay: Emission of a 4He nucleus (2 protons, 2 neutrons). $A^Z_X \rightarrow ^{A-4}_{Z-2}Y + \alpha^{2+} + \text{energy}$

• Beta decay: Emission of an electron (beta minus, $n \rightarrow p^+ + e^-$) or positron (beta plus, $p^+ \rightarrow n + e^+$).

• Gamma decay: Emission of high-energy photons (gamma rays) as the nucleus transitions from an excited state to a lower energy state.

Radiation	Identification	Charge	Stopped by
Alpha, α	4He nucleus	+2e	Sheet of paper
Beta, β	Electron	−e	Few mm of aluminum
Gamma, γ	High-energy photon	0	Many cm of lead

Decay Series

Many radioactive nuclei decay through a series of steps, producing a sequence of daughter nuclei until a stable nucleus is formed. This is known as a decay series.

Ionizing Radiation and Radiation Dose

Ionizing radiation (alpha, beta, gamma) has enough energy to ionize atoms and molecules, causing damage to biological tissue, including DNA. The absorbed dose is measured in grays (Gy), where 1 Gy = 1 J/kg. The biological effect is quantified by the dose equivalent (Sievert, Sv), which accounts for the type of radiation via the relative biological effectiveness (RBE):

• Dose equivalent = (absorbed dose) × RBE

• 1 Sv = 100 rem (old unit)

Medical Uses of Radiation

Radiation is used in medicine for cancer therapy (radiation therapy), imaging (e.g., PET scans with radioactive tracers), and MRI (which uses nuclear magnetic resonance of protons in tissues).

Summary of Key Equations

• $R = r_0 A^{1/3}$ (nuclear radius)

• $N(t) = N_0 e^{-rt} = N_0 e^{-t/\tau}$ (radioactive decay law)

• $N(t) = N_0 \left(\frac{1}{2}\right)^{t/t_{1/2}}$ (decay in terms of half-life)

• $B = (Z m_H + N m_n - m_{atom}) \times (931.49 \text{ MeV/u})$ (binding energy)