Past exam of the particle and nuclear physics course of the University of Cambridge / 2012 / Paper 1 / 4

For projectile velocity $v$ and a resting target, let $M=m_x+m_X$, $\mu =m_x{m}_X/M$, and $V=m_{x}v/M$. Subtract the kinetic energy of the centre of mass from the laboratory kinetic energy:In the centre-of-momentum frame the two momenta are opposite with equal magnitude, soFor masses quoted in $\mathrm{MeV}/c^2$, use $p^{\ast }c=\sqrt{2(\mu c^2)T^{\ast }}$. These are the nonrelativistic relations stipulated for this question; no relativistic correction should be silently mixed into the numerical results.

The partial decay width ${\Gamma }_x$ is the energy width associated with decay of the compound nucleus back into the entrance channel $x+X$; by time reversal it also controls formation from that channel. The partial decay width ${\Gamma }_y$ describes the selected exit channel $y+Y$. The total decay width sums all energetically accessible channels, including these: $\boxed{\Gamma =\sum _i{\Gamma }_i=\hslash /\tau }$. The branching fraction for channel $i$ is ${\Gamma }_i/\Gamma$. Widths have energy units, whereas ${\Gamma }_i/\hslash$ is a decay probability per unit time. The isolated-resonance formula assumes approximately constant widths over the resonance and neglects a nonresonant interfering background.

An unstable isolated level has survival amplitude $a(t)=\theta (t)e^{-i{E}_{0}t/\hslash }{e}^{-\Gamma t/(2\hslash )}$, because its survival probability falls as $e^{-\Gamma t/\hslash }$. Its energy-domain amplitude is proportional toTaking the modulus squared gives the denominator of the Breit-Wigner resonance cross sectionAt $E-E_0=\pm \Gamma /2$ the resonant factor is half its peak, so $\Gamma$ is the full width at half maximum. exponential decay is the isolated-level approximation; energy-dependent thresholds or overlapping levels need a more general lineshape.

Unpolarized projectile and target spins supply $(2J_x+1)(2J_X+1)$ equally weighted spin orientations. An isolated compound-nucleus level of spin $J$ has $2J+1$ degenerate magnetic substates. Sum over these resonance substates and average over the entrance spins to obtainThis resonance spin statistical factor is a statistical spin weight, not always a probability smaller than one: orbital angular momentum also participates in forming the resonance. The partial-wave resonance cross section carries the corresponding angular-momentum multiplicity. Polarized beams or unresolved overlapping spin channels require a different average or a sum over channels.

Using the supplied masses,Therefore $\boxed{p_{\alpha }^{\ast }=205.790\mathrm{MeV}/c\simeq 206\mathrm{MeV}/c}$.

For resonance channel threshold conversion, equate the compound-state energy measured from the two different channel thresholds. The proton-plus-nitrogen threshold exceeds the alpha-plus-carbon threshold byThus $T_p^{\ast }=T_{\alpha }^{\ast }-\Delta =2.113770\mathrm{MeV}$ andThe positive centre-of-momentum energy verifies that the proton channel is open. These numbers use the stipulated nonrelativistic kinematics and rounded nuclear masses.

The gamma width is $0.10\mathrm{eV}=10^{-7}\mathrm{MeV}$, so the remaining proton width is ${\Gamma }_p=0.20-0.15-10^{-7}=0.0499999\mathrm{MeV}$. At the peak of the same resonance the total-width denominator and gamma exit width cancel. alpha particles and carbon-12 have spin zero; the proton and the stated nitrogen level each have spin $1/2$. Hence $g_{\alpha }/g_p=4$, independently of the unspecified resonance spin. The proton reduced mass and momentum areThe same-resonance entrance-channel cross-section ratio is thereforeThe ratio compares each reaction at its own laboratory resonance energy. Factors of ${\hslash }^2$ restoring the cross section's area units cancel. Only the supplied nitrogen spin magnitude is used: its printed positive parity is immaterial to this spin average, so no orbital-parity inference is made from that assignment.