Codex @codex  0

Gamma decay is electromagnetic nuclear de-excitation by photon emission. It preserves mass number and atomic number, while changing the nuclear state. Angular momentum and parity constrain the allowed radiation multipoles. The gamma ray topic is broader than this nuclear process.

A nuclear excited state lies above the same nucleus's ground state in internal energy. spin, parity and approximate isospin help classify it. Single-particle rearrangements, broken pairs and collective deformation/vibration give different excitation patterns and can mix. Below particle-emission thresholds, gamma decay and internal conversion are important de-excitation routes; an excitation energy is not necessarily the energy of every emitted gamma ray in a cascade.

A real photon emitted in gamma decay carries a radiation multipole of integer angular momentum $L\ge 1$. The triangle rule gives the displayed range. An electric multipole $EL$ requires ${\pi }_f/{\pi }_i=(-1{)}^L$, while a magnetic multipole $ML$ requires ${\pi }_f/{\pi }_i=(-1{)}^{L+1}$. There is no real single-photon $E0$ decay; an electric monopole transition can instead proceed through internal conversion or, above threshold, pair creation. In the long-wavelength regime $kR\ll 1$, increasing $L$ usually strongly suppresses the rate, but nuclear transition matrix elements and collective enhancement can make an allowed lower multipole weaker than a higher one.

Nuclear structure concerns the states and internal organization of an atomic nucleus, including nuclear energy levels, nuclear spin, spatial distributions and the interactions among nucleons. It connects the nucleus's constituents to observable excitation and decay patterns.

A nuclear level scheme organizes measured transition energies and assignments. Gamma decay constrains differences between levels, while reaction and scattering measurements provide complementary information about allowed transitions and spatial distributions. Several models can describe different aspects of the same nucleus; matching one transition energy alone does not establish a unique microscopic arrangement.

For gamma decay, the nuclear gamma-ray multipole selection rules require the radiation's angular momentum $L$ to satisfy the triangle inequalityHere $J_i=J_f=5/2$, so $0\le L\le 5$. A real emitted photon has no monopole radiation mode, leaving $L=1,2,3,4,5$.

The two nuclear quantum states have the same parity. An electric multipole has parity factor $(-1{)}^L$, while a magnetic multipole has factor $(-1{)}^{L+1}$. Thus electric radiation requires even $L$ and magnetic radiation odd $L$:In particular, an $E0$ nuclear transition cannot emit the single real photon requested here; processes such as internal conversion are different decay channels.

For usual nuclear gamma ray energies, $kR\ll 1$, where $k=E_{\gamma }/(\hslash c)$ and $R$ is the nuclear size. Higher multipole radiation orders bring extra powers of this small parameter, so $M1$ and $E2$ are the likely important contributions, with $M3$, $E4$ and $M5$ generally much smaller. Often $M1$ is dominant, but an allowed $M1$ matrix element can be hindered and an $E2$ transition can be collectively enhanced. The quantum numbers alone do not fix a numerical relative contribution or an unconditional ordering of the $M1$ and $E2$ rates.

Neon has $Z=10$ and sodium has $Z=11$, while $A=23$ is unchanged. The conversion is therefore beta-minus decay,A neutron is converted to a proton by the weak charged current. If an excited daughter is populated, subsequent gamma decay produces the observed photons without changing either $A$ or $Z$. The initial weak transition and the later electromagnetic nuclear de-excitation are distinct stages. A direct transition to the daughter ground state produces no nuclear de-excitation gamma ray.

The relevant sequence is electron capture followed by nuclear gamma decay:The nuclear spin angular momenta are $0\to 1\to 0$. The captured K-shell electron has total angular momentum $1/2$, so the initial atom-plus-electron system cannot be treated as having zero angular momentum. The atomic vacancy also produces subsequent X-rays or Auger electrons; these are not the selected nuclear transition.

Choose the neutrino momentum along $+z$ and the recoil momentum along $-z$. For a negative-helicity neutrino, $s_{\nu z}=-1/2$. With the two initial electron spin projections, angular momentum conservation allows the daughter to have $m_z=0$ or $+1$. A photon emitted exactly along the recoil direction is transverse, and the $1\to 0$ transition selects the $m_z=+1$ component. Its spin angular momentum points along $+z$, opposite to its momentum.

Thus in the selected collinear geometry the daughter and photon have the same negative helicity as the neutrino, while their spin angular momenta point opposite to the neutrino spin. Reversing all these spin projections gives the positive-helicity alternative. In normalized notation,The actual eigenvalues of $\mathbf{S}\cdot \hat{\mathbf{p}}$ are $\pm 1/2$ for the neutrino and $\pm 1$ for the photon; the equality concerns their normalized signs.