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nuclear

Decay law & modes, binding energy, fission & reactors, fusion, stellar radiation & cosmology.

decay law · decay modes · binding energy & Q · fission & reactors · fusion · stellar radiation & cosmology

Decay law

Exponential decay. Activity \(A=-\dfrac{\dd N}{\dd t}=\lambda N\), \(N=N_0e^{-\lambda t}\), \(A=A_0e^{-\lambda t}\). \(t_{1/2}=\ln2/\lambda\); after \(n\) half-lives \(N=N_0/2^n\). Mean lifetime \(\tau=1/\lambda\).

Counting. Corrected count rate = measured − background; count \(C\propto A\).

Absorption/attenuation. \(I=I_0e^{-\mu x}\), half-value thickness \(x_{1/2}=\ln2/\mu\).

Decay modes

Alpha. \(^{A}_{Z}X\to{}^{A-4}_{Z-2}Y+{}^{4}_{2}\alpha\).

Beta minus. \(n\to p+e^-+\bar\nu_e\), \(Z\to Z+1\).

Beta plus. \(p\to n+e^++\nu_e\), \(Z\to Z-1\). Electron capture: \(p+e^-\to n+\nu_e\).

Gamma. Excited nucleus emits \(\gamma\); \(A,Z\) unchanged.

Decay chains. Apply conservation (\(A\), \(Z\)) at each step.

Binding energy & Q

Mass defect. \(\Delta m=Zm_p+(A-Z)m_n-m_{\rm nucleus}\). Binding energy \(E_b=\Delta mc^2\); binding per nucleon \(E_b/A\).

Reaction energy. \(Q=(m_{\rm initial}-m_{\rm final})c^2\); exothermic if \(Q\gt0\). Use atomic masses: electron masses cancel when total \(Z\) equal. \(1\,\mathrm u=931.5\ \mathrm{MeV}\,c^{-2}\), so \(Q(\mathrm{MeV})=\Delta m(\mathrm u)\times931.5\).

Fission & reactors

Fission. Heavy nucleus + neutron \(\to\) fragments + neutrons + energy. Binding-energy curve: fission releases energy because products have larger \(E_b/A\).

Fuel & power. Power \(P=(\text{fissions s}^{-1})\,E_{\rm fission}\). Fuel nuclei number \(N=(m/M)N_A\); total energy \(E=NE_{\rm fission}\).

Chain reaction. Multiplication factor \(k=N_{\rm neutrons,next}/N_{\rm neutrons,current}\): subcritical \(k\lt1\), critical \(k=1\), supercritical \(k\gt1\).

Reactor. Moderator reduces neutron kinetic energy; control rods absorb neutrons; coolant transfers heat.

Fusion

Fusion. Light nuclei combine; energy from increased \(E_b/A\): \(Q=\Delta mc^2\). Coulomb barrier \(E\sim k_eZ_1Z_2e^2/r\); high \(T\) and tunnelling enable fusion. Plasma thermal energy scale \(\avg K=\tfrac32kT\).

Stellar core. Hydrostatic balance; radiation pressure + gas pressure resist gravity.

Stellar radiation & cosmology

Luminosity & brightness. \(L=P=4\pi R^2\sigma T^4\) for blackbody surface. Apparent brightness/flux \(b=L/(4\pi d^2)\). Wien \(\lambda_{\max}T=b_W\); colour gives surface \(T\).

Parallax. \(d(\mathrm{pc})=1/p(\mathrm{arcsec})\), \(1\,\mathrm{pc}=3.09\times10^{16}\ \mathrm m\).

Magnitudes. \(m_1-m_2=-2.5\log_{10}(b_1/b_2)\); absolute magnitude is at \(10\,\mathrm{pc}\).

Main sequence. Approximate \(L\propto M^{3\text{–}4}\), lifetime \(t\propto M/L\). Mass-loss luminosity: \(L=-\dd(mc^2)/\dd t\).

Hubble/redshift. \(z=\Delta\lambda/\lambda\approx v/c\), \(v=H_0d\) for small \(z\).