Past exam of the particle and nuclear physics course of the University of Cambridge / 2016 / Paper 1 / 1

In the nuclear shell model, a level with total angular momentum $j$ has $2j+1$ substates for each nucleon species. The first three levels accommodate $2+4+2=8$ protons or eight neutrons. nuclear pairing makes a filled shell or a pair of like nucleons contribute $J=0$, and an orbital has spatial parity $(-1{)}^{\ell }$.

For oxygen-17, eight protons and eight neutrons form a closed core; the remaining neutron occupies $1d_{5/2}$, so $J^P=5/2^{+}$. For nitrogen-15, eight neutrons close their shell, while the seventh proton occupies $1p_{1/2}$. Equivalently it is a proton hole relative to the closed eight-proton shell, giving $J^P=1/2^{-}$. beryllium-10 is an even-even nucleus; pairing its four protons and six neutrons gives $J^P=0^{+}$.

boron-10 needs an additional qualification. The fifth proton and fifth neutron each occupy $1p_{3/2}$. addition of angular momentum permitsThe experimentally correct ground-state assignment is $3^{+}$, corresponding to aligned unpaired $j=3/2$ contributions in this simple description. However, one-particle level ordering alone does not select it over the other couplings: their splitting requires the nuclear residual interaction. Assuming a preference for the aligned proton-neutron coupling gives the intended prediction; without that assumption, the strictly deducible result is the allowed positive-parity multiplet. The evaluated ground state is $3^{+},T=0$, with a $1^{+},T=0$ level above it.^[ref]

Use natural units and write the common light constituent quark mass as $m$, and the strange one as $M$. For two spin one-half constituents,The relative hyperfine contributions are $-3$ and $+1$. Absorb their normalization and spatial overlap into pair-dependent parameters $h_{\ell \ell },h_{\ell s},h_{ss}$. The meson hyperfine mass splitting model givesA common color-magnetic coefficient gives $h_{ij}\propto 1/(m_i{m}_j)$, but its numerical value is not needed.

In the specified Eta octet state used for the eta meson the light-flavor probability is $1/3$ and the strange-flavor probability $2/3$. Assuming a flavor-diagonal mass operator,Inserting the five mass expressions into the proposed combination makes the coefficient of $h_{\ell s}$ equal to $4-4=0$. Those of $m,M,h_{\ell \ell },h_{ss}$ are $2/3,4/3,-1,-2$. ConsequentlyThis is an identity within the assumed mass operator and octet flavor assignment. It is not an exact relation among measured masses: eta mixing and additional spin-independent binding effects are outside those assumptions.

For neutrino-electron elastic scattering, work in the laboratory frame with a stationary electron and a massless incoming neutrino of energy $E$. If the electron receives kinetic energy $T$ and momentum $\mathbf{p}$, its relativistic energy-momentum relation givesconservation of energy fixes the outgoing neutrino energy to $E-T$, and conservation of momentum fixes its momentum to $\mathbf{k}-\mathbf{p}$. Its zero mass therefore requiresSubstitution and cancellation giveFor $T>0$, this requires $\cos \theta >0$; squaring must not introduce a backward-electron branch. Squaring, cancelling $T$ and solving givesThe expression increases with ${\cos }^2\theta$ on the physical interval $0\le \theta \le \pi /2$. Maximum transfer occurs for a forward electron and a backward outgoing neutrino:The remaining $0.197\mathrm{MeV}$ is carried by the neutrino. An elastic collision cannot transfer the entire incident energy to a free massive electron.