Nuclei: Radioactivity, Fission & Fusion

Nuclear Structure and Properties

The nucleus contains protons (Z) and neutrons (N), collectively called nucleons. Mass number A = Z + N. Isotopes have the same Z but different N (e.g., H-1, H-2, H-3). Isobars have the same A but different Z (e.g., C-14, N-14). Isotones have the same N but different Z (e.g., C-13, N-14, both with N=7).

Nuclear size: R = $R_{0}$ * A^($\frac{1}{3}$), where $R_{0}$ = 1.2 fm = 1.2 x $10^{-15}$ m.
Nuclear volume is proportional to A, implying constant nuclear density: $\rho$ = $3m_{p}$/(4*$\pi$*$R_{0}^{3}$) ≈ 2.3 x $10^{17}$ kg/$m^{3}$ (independent of A).

Atomic mass unit (amu): 1 u = 1.66054 x $10^{-27}$ kg = 931.5 MeV/$c^{2}$. Proton mass = 1.00728 u, neutron mass = 1.00866 u, electron mass = 0.00055 u.

Mass Defect and Binding Energy

The mass of a nucleus is always less than the sum of its constituent nucleons:

Mass defect: $\Delta_{m}$ = [Z*$m_{p}$ + (A-Z)*$m_{n}$] - $M_{nucleus}$

Binding energy: BE = $\Delta_{m}$ * $c^{2}$ = $\Delta_{m}$ * 931.5 MeV

This is the energy required to completely disassemble the nucleus into individual nucleons.

Binding energy per nucleon (BE/A): The key indicator of nuclear stability. The BE/A curve shows:

• Rises steeply for light nuclei (H-2: 1.1 MeV, He-4: 7.07 MeV)
• Peaks near A = 56 (Fe-56: 8.79 MeV — most stable nucleus)
• Gradually decreases for heavy nuclei (U-238: 7.57 MeV)
• He-4, C-12, O-16 show unusually high BE/A (magic numbers)

The curve explains both fission (heavy nuclei splitting increases BE/A) and fusion (light nuclei combining increases BE/A).

Radioactivity

Spontaneous disintegration of unstable nuclei, emitting $\alpha$, $\beta$, or $\gamma$ radiation.

Alpha decay: Nucleus emits He-4 (2 protons, 2 neutrons). Z decreases by 2, A decreases by 4. $_Z^A X \rightarrow _{Z-2}^{A-4} Y + _2^4 He$ Q-value: Q = [$M_{parent}$ - $M_{daughter}$ - $M_{alpha}$] * $c^{2}$. Alpha particles are monoenergetic for a given decay.

Beta-minus decay: Neutron converts to proton, emitting electron and antineutrino. Z increases by 1, A unchanged. $_Z^A X \rightarrow _{Z+1}^A Y + e^- + \bar{\nu}_e$ The neutrino carries variable energy, making the $\beta$ spectrum continuous.

Beta-plus decay (positron emission): Proton converts to neutron, emitting positron and neutrino. Z decreases by 1, A unchanged.

Gamma decay: Excited nucleus transitions to lower energy state, emitting a $\gamma$ photon. Z and A unchanged.

Radioactive Decay Law

$N(t) = N_0 e^{-\lambda t}$ $A(t) = A_0 e^{-\lambda t} = \lambda N(t)$

where $\lambda$ is the decay constant, A is the activity (decays per second).

Half-life: t_($\frac{1}{2}$) = ln(2)/$\lambda$ = 0.693/$\lambda$

Mean life: $\tau$ = 1/$\lambda$ = t_($\frac{1}{2}$)/0.693 = 1.44 * t_($\frac{1}{2}$)

After n half-lives: N = $\frac{N_{0}}{2}$^n, A = $\frac{A_{0}}{2}$^n

SI unit of activity: Becquerel (Bq) = 1 decay/s. Also: 1 Curie = 3.7 x $10^{10}$ Bq.

Nuclear Fission

Heavy nucleus splits into two medium-mass fragments plus neutrons, releasing energy.

$_{92}^{235}U + _0^1 n \rightarrow _{56}^{141}Ba + _{36}^{92}Kr + 3_0^1 n + 200 \text{ MeV}$

Energy released ≈ 200 MeV per fission (≈ 0.9 MeV per nucleon). Chain reaction occurs when released neutrons cause further fissions. Critical mass is the minimum mass needed for a self-sustaining chain reaction. Controlled fission uses moderators (heavy water, graphite) to slow neutrons and control rods (cadmium, boron) to absorb excess neutrons.

Nuclear Fusion

Light nuclei combine to form heavier nuclei, releasing energy.

$4 _1^1 H \rightarrow _2^4 He + 2e^+ + 2\nu_e + 26.7 \text{ MeV}$

Fusion requires extremely high temperatures (~$10^{7}$ K) to overcome Coulomb repulsion (thermonuclear reaction). Energy per nucleon from fusion (~6.7 MeV per nucleon in p-p chain) is much greater than fission (~0.9 MeV per nucleon). Fusion powers the Sun and stars. Uncontrolled fusion: hydrogen bomb. Controlled fusion remains an engineering challenge.

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