Nuclear and Particle Physics

Nuclear transition selection rules

Selection rules decide whether a transition between two nuclear states is allowed and how strongly it is hindered. For gamma emission, the multipole order L of the radiation must satisfy |J_i - J_f| \le L \le J_i + J_f, and a single photon cannot connect two 0^+ states. Electric multipoles of order L change parity by (-1)^L, whereas magnetic multipoles change it by (-1)^{L+1}. A transition with \Delta J = 2 and no parity change is therefore electric quadrupole (E2) radiation, so an E2 assignment follows directly from the spins and parities of the two levels.

For an electric dipole (E1) transition, the rank-one operator requires |\Delta J| \le 1 and a change of parity. Starting from J_i = 2 with positive parity, the reachable final states are J_f = 1, 2, 3 with negative parity. The state J_f = 0 is excluded because |2 - 0| > 1.

In beta decay, transitions are ranked by forbiddenness, which depends on \Delta J and on whether parity changes. Allowed transitions have \Delta J = 0, \pm 1 with no parity change. A first-forbidden transition has \Delta J = 0, \pm 1, \pm 2 with a parity change, and a second-forbidden transition has \Delta J = 2, 3 with no parity change. A third-forbidden transition has \Delta J = 3, 4 with a parity change. For example, a \beta^+ decay from a 3/2^- parent to a 1/2^+ daughter has \Delta J = 1 and a parity change, so it is first forbidden. The parent's spin-parity alone therefore restricts which daughter levels can be reached, and a parity mismatch is the first thing to check.

Beta decay processes

In nuclear beta decay the emitted electron shares the released energy with an antineutrino, so the electron kinetic energy is not fixed. The electron spectrum is therefore continuous, running from zero up to an end-point energy, and it vanishes at the end-point. A line spectrum or a symmetric peak is incompatible with this shape.

The three-body character of beta decay can be seen from conservation laws. A final state made only of a proton and an electron has integer total spin, whereas the neutron has spin one-half, so an additional spin-one-half antineutrino is needed to balance the angular momentum. The continuous spectrum gives the same conclusion from the energy side.

Beta processes also obey lepton-number conservation. In neutrino capture of the form \nu_e + {}^{A}_{Z}P \to {}^{A}_{Z+1}Q + e^-, the charge balances because the neutrino is neutral and the electron carries away one unit, the mass number is unchanged, and the electron lepton number +1 appears on both sides. A process that produces a positron from an incoming electron neutrino violates lepton number.

Fermi and Gamow–Teller transitions are the two allowed types. A Fermi transition has \Delta J = 0 and leaves the spin unchanged, while a Gamow–Teller transition has \Delta J = 0, \pm 1 and cannot connect 0 \to 0. A 0^+ \to 0^+ decay is therefore a pure Fermi transition, whereas a 0^+ \to 1^+ decay with no parity change is Gamow–Teller.

Conservation laws in particle reactions

A strong or electromagnetic reaction is allowed only if charge Q, baryon number B, lepton numbers, strangeness S, and the third component of isospin I_3 are all conserved, together with energy and momentum. In a charge-exchange reaction between a pion and a nucleon, charge, baryon number, strangeness and I_3 are unchanged, so such a reaction can proceed through the strong interaction.

C-parity is also a useful check for neutral mesons. A \pi^0 has C = +1, whereas a system of three photons has C = -1, so a \pi^0 cannot decay into three photons. A reaction that starts with baryon number zero, for example a baryon-antibaryon pair, cannot produce two baryons, because baryon number would change from 0 to 2.

The combination Q - S is a quick test for the quark content. Each s quark gives S = -1, each \bar s gives S = +1, and the charge follows from the quark charges. For K^- (quark content s\bar u) one finds Q - S = 0, whereas for \Xi^- (quark content dss) the value is Q - S = 1.

Strangeness is conserved in strong and electromagnetic decays but not in weak decays. The decay of a \Lambda hyperon into a proton and a pion violates strangeness conservation and is therefore a weak process, even though charge and baryon number are conserved. Lepton-flavour-conserving decays such as muon decay do not violate CPT, which is an exact symmetry. A decay into a set of final particles whose total rest energy exceeds that of the parent is forbidden by energy conservation regardless of the quantum numbers.

Quark model flavour wavefunctions

In the quark model the flavour part of a baryon wavefunction is built from its quark content. The decuplet of spin-3/2 baryons has a flavour part that is totally symmetric under the exchange of any two quarks. For a baryon with one u quark and two s quarks, the symmetric state is the sum over the three distinct orderings uss, sus and ssu with equal weight, normalised by 1/\sqrt{3}, which gives \frac{1}{\sqrt{3}}(uss + sus + ssu). A combination with a minus sign, or one that omits an ordering, is not symmetric.

Strangeness of a baryon equals minus the number of strange quarks it contains. A baryon with one s quark, such as \Sigma^{*0} (quark content uds), has S = -1. Baryons made only of u and d quarks, such as the neutron, have S = 0, and a baryon with two s quarks has S = -2.

Nuclear rotational and vibrational bands

Collective nuclear spectra fall into two families. A rotational band of a deformed nucleus has level energies E_J \propto J(J+1), so the spacings between successive levels grow in proportion to the differences of J(J+1). Such bands are typically found in deformed nuclei, including many odd-mass nuclei whose ground state has a large spin.

A vibrational spectrum is typical of nearly spherical even-even nuclei. Its low-lying levels are approximately multiples of a single quantum \hbar\omega, and the two-phonon states form a triplet of 0^+, 2^+ and 4^+ levels at roughly twice the energy of the first 2^+ level. The energy ratio and the parity of the mass number together identify which pattern a set of levels follows.

Gell-Mann–Okubo mass formula

The Gell-Mann–Okubo formula writes the baryon mass as M = M_0 + aY + b\left[I(I+1) - \frac{Y^2}{4}\right], where Y is the hypercharge and I is the isospin. For baryons the hypercharge is Y = S + B. The constant a multiplies a term linear in Y, while b multiplies the isospin-dependent combination.

If two baryon multiplets have the same hypercharge but different isospin, such as the \Sigma and \Lambda hyperons, equal masses force the isospin term to vanish, so b = 0. The mass then reduces to M = M_0 + aY. This expression does not depend on isospin, but it still varies linearly with hypercharge, so the mass does depend on Y.

Semi-empirical mass formula Coulomb term

The Coulomb term in the semi-empirical mass formula comes from the electrostatic repulsion among the Z protons. For a uniformly charged sphere of charge Ze and radius R, the electrostatic energy is E_C = \frac{3}{5}\,\frac{Z^2 e^2}{4\pi\varepsilon_0 R}. Using e^2/(4\pi\varepsilon_0) \approx 1.44 MeV fm, this becomes E_C = \frac{3}{5}\cdot\frac{1.44\,Z^2}{R} MeV with R in fm.

Given a measured Coulomb energy and radius, the relation can be inverted to find Z, which must then be rounded to the nearest integer because the proton number is a whole number. The term grows as Z^2 and decreases as the radius grows, which is why heavy nuclei are destabilised by Coulomb repulsion.

Nuclear magic numbers

Magic numbers are the proton or neutron counts at which a shell is completely filled. A simple model is a nucleon confined in a three-dimensional cubical box, whose energy depends on n_x^2 + n_y^2 + n_z^2 with positive integers n_x, n_y and n_z. Counting degenerate spatial states in order of increasing sum, and including spin degeneracy 2, the cumulative numbers of nucleons at closed shells are 2, 8, 14 and 20.

The spatial degeneracies are 1, 3, 3 and 3 for the sums 3, 6, 9 and 11. Each spatial state holds two nucleons, which gives the closures. The true nuclear shell model, which includes spin-orbit coupling, reproduces the sequence 2, 8, 20 and 28 instead, so the box model is only a rough picture of nuclear shells.

Spin-orbit doublet splitting

The spin-orbit interaction V\,\vec l\cdot\vec s splits each orbital level with l \ge 1 into two levels with j = l \pm \tfrac12. The eigenvalue is \vec l\cdot\vec s = \frac{\hbar^2}{2}\left[j(j+1) - l(l+1) - s(s+1)\right]. For j = l + \tfrac12 this gives \frac{\hbar^2}{2}\,l, and for j = l - \tfrac12 it gives -\frac{\hbar^2}{2}(l+1).

The separation of the doublet is therefore V\hbar^2(2l+1)/2 in magnitude. For a p level with l = 1 the separation is 3V\hbar^2/2, so in units of V\hbar^2/2 it equals 3. When the coefficient V is negative, as in the nuclear potential, the j = l + \tfrac12 member lies lower.

Deuteron ground state properties

The deuteron is the bound state of a proton and a neutron with J^\pi = 1^+. Its ^3S_1 component dominates, but the nucleon-nucleon tensor force mixes in a small ^3D_1 component. This admixture is the origin of the deuteron's electric quadrupole moment, about 0.286 fm^2, which is small but nonzero and shows that its charge distribution is not spherical.