Quantum Nature of the Nucleus

Chadwick fired alpha particles at beryllium, producing a neutral radiation that knocked protons out of paraffin wax. Using conservation of momentum and energy, he showed it was a neutral particle of mass close to the proton's: the neutron.

An alpha particle is $^{4}_{2}\text{He}$, so the nucleus loses 2 protons and 2 neutrons: $Z \to Z - 2$, $A \to A - 4$.

$15$ years is $15/5 = 3$ half-lives, so the fraction remaining is $\left(\tfrac{1}{2}\right)^3 = \tfrac{1}{8}$.

The strong force is attractive and far stronger than electrostatic repulsion, but acts only over about $10^{-15}\text{ m}$. It binds protons and neutrons together despite proton-proton repulsion, and is carried by gluons.

$n \to p + e^- + \bar{\nu}_e$. A neutron becomes a proton, so $Z \to Z + 1$ while the nucleon count $A$ stays the same.

The binding energy is the energy holding the nucleus together, equal to the energy needed to pull it apart into free nucleons. It comes from the mass defect: $BE = \Delta m\,c^2$.

$24$ hours is $2$ half-lives, so $80 \times \left(\tfrac{1}{2}\right)^2 = 80 \times \tfrac{1}{4} = 20\text{ g}$.

Alpha decay removes $^{4}_{2}\text{He}$: $A \to 238 - 4 = 234$ and $Z \to 92 - 2 = 90$, giving thorium-234.

A gamma ray carries away energy but no charge and no nucleons, so $Z$ and $A$ are unchanged. The nucleus simply drops from an excited state to a lower-energy state.

$BE = \Delta m\,c^2 = (5.0\times10^{-28})(3.0\times10^8)^2 = (5.0\times10^{-28})(9.0\times10^{16}) = 4.5\times10^{-11}\text{ J}$.

Splitting a heavy nucleus produces medium-mass fragments with higher binding energy per nucleon, releasing the difference as energy. The free neutrons released can trigger a chain reaction.

Fusing light nuclei (hydrogen into helium) gives products with higher binding energy per nucleon, releasing energy. It requires extreme temperature and pressure to overcome the protons' electrostatic repulsion.

$32$ days is $32/8 = 4$ half-lives. Activity halves each half-life, so $640 \times \left(\tfrac{1}{2}\right)^4 = 640 \times \tfrac{1}{16} = 40\text{ Bq}$.

$\tfrac{1}{32} = \left(\tfrac{1}{2}\right)^5$, so $5$ half-lives have passed in $25$ years. The half-life is $25/5 = 5.0$ years.

The most tightly bound nuclei sit near iron-56. Light nuclei fusing and heavy nuclei splitting both produce nuclei nearer the peak, with higher binding energy per nucleon, releasing the difference. Elements heavier than iron cannot release energy by fusion.

$BE = \Delta m\,c^2 = 0.0304 \times 931.5 = 28.3\text{ MeV}$.

Helium-4 has $A = 4$ nucleons, so the binding energy per nucleon is $\dfrac{28.3}{4} = 7.1\text{ MeV}$.

Alpha decay: $^{238}_{\ 92}\text{U} \to {}^{234}_{\ 90}\text{Th}$ ($A - 4$, $Z - 2$). Then beta-minus: $^{234}_{\ 90}\text{Th} \to {}^{234}_{\ 91}\text{Pa}$ ($A$ unchanged, $Z + 1$). Final: $A = 234$, $Z = 91$.

$Q = +\tfrac{2}{3} + \tfrac{2}{3} - \tfrac{1}{3} = +\tfrac{3}{3} = +1e$. (A neutron, udd, gives $+\tfrac{2}{3} - \tfrac{1}{3} - \tfrac{1}{3} = 0$.)

Force carriers are bosons: the photon (electromagnetic), gluon (strong), W and Z (weak), and Higgs (mass). Electrons, quarks and neutrinos are fermions (matter particles); the proton is a composite of three quarks (a hadron), not a fundamental boson.

A neutron (udd) becomes a proton (uud) when one down quark converts to an up quark. This is mediated by the weak force through a $W^-$ boson, which then decays into an electron and an antineutrino.

A neutron-rich nucleus reduces its neutron excess by $\beta^-$ decay ($n \to p$), which raises $Z$ and lowers $N$, moving it down toward the band of stability.

Accelerators give charged particles enormous kinetic energy; in a collision that energy can convert into the mass of brand-new particles ($E = mc^2$). This is how many particles, including the Higgs boson, were discovered.

$\tfrac{1}{8} = \left(\tfrac{1}{2}\right)^3$, so $3$ half-lives have passed: $3 \times 5700 = 17{,}100$ years.

$E = \Delta m\,c^2 = (3.5\times10^{-28})(9.0\times10^{16}) = 3.15\times10^{-11}\text{ J}$. In MeV: $\dfrac{3.15\times10^{-11}}{1.6\times10^{-13}} \approx 197\text{ MeV}$, close to the $\sim 200\text{ MeV}$ released per uranium-235 fission.