Appendix G — Notation and Symbol Guide

This appendix collects all notation and symbol conventions used in this textbook. When a symbol has multiple standard meanings in nuclear physics, we note which convention this book follows and in which chapters the symbol appears. Where conventions differ from those common in other subfields (atomic physics, particle physics), we note the difference.

G.1 Nuclear Notation

G.1.1 Nuclide Designation

A nuclide is specified by its chemical symbol $X$, mass number $A$ (total nucleon count), and atomic number $Z$ (proton count):

$${}^A_Z X_N$$

where $N = A - Z$ is the neutron number. In practice, the subscripts $Z$ and $N$ are often omitted because the chemical symbol uniquely determines $Z$:

$${}^A X \qquad \text{e.g., } {}^{12}\text{C}, \; {}^{56}\text{Fe}, \; {}^{238}\text{U}$$

When $Z$ is needed for clarity (e.g., in nuclear reactions or when the element is unfamiliar), the full notation ${}^A_Z X$ is used:

$${}^{238}_{92}\text{U}, \quad {}^{12}_{\;6}\text{C}$$

Isomers: A nuclear isomer (metastable excited state) is designated by appending $m$ (or $m1$, $m2$ for multiple isomers) to the mass number:

$${}^{99m}\text{Tc}, \quad {}^{137m}\text{Ba}, \quad {}^{178m2}\text{Hf}$$

G.1.2 Related Terminology

G.2 Quantum Numbers

G.2.1 Nuclear State Quantum Numbers

G.2.2 Single-Particle Quantum Numbers

Shell model orbital notation: $nlj$, where $l$ is given as a spectroscopic letter. Examples: $1s_{1/2}$, $1p_{3/2}$, $1p_{1/2}$, $1d_{5/2}$, $2s_{1/2}$, $1f_{7/2}$, $1g_{9/2}$.

The maximum occupancy of an orbital $j$ is $2j + 1$ nucleons (accounting for the $m_j$ degeneracy, with each $m_j$ value accommodating one proton or one neutron due to the Pauli principle).

G.2.3 Electromagnetic Transition Quantum Numbers

G.3 Reaction Notation

G.3.1 Standard Compact Notation

Nuclear reactions are written in the compact form:

$$a(b, c)d$$

where $a$ is the target, $b$ is the projectile, $c$ is the ejectile (light outgoing particle), and $d$ is the residual nucleus. Examples:

The prime notation ($p'$, $n'$, $\alpha'$) indicates inelastic scattering where the target is left in an excited state.

G.3.2 Arrow Notation

Equivalently, reactions may be written with an arrow:

$${}^{12}\text{C} + \alpha \rightarrow {}^{16}\text{O} + \gamma$$

This form is more common in astrophysics and particle physics. When the reaction proceeds through a resonance, the intermediate state may be shown:

$$n + {}^{238}\text{U} \rightarrow {}^{239}\text{U}^* \rightarrow \text{fission fragments} + \nu n$$

G.3.3 Common Abbreviations for Reaction Types

G.4 Common Symbols

⚠️ Notational collision: The symbol $A$ is used for both mass number and activity. Context always distinguishes them: $A$ as a superscript or subscript on a nuclide symbol is the mass number; $A$ in an equation involving $\lambda N$ or measured in Bq is the activity. Where ambiguity is possible, we write activity as $\mathcal{A}$ or spell it out.

G.5 Greek Letters in Nuclear Physics

G.6 Units and Prefixes

G.6.1 Units Specific to Nuclear Physics

G.6.2 Natural Units and Conversions

In nuclear physics, the most commonly used "natural" conversion is:

$$\hbar c = 197.327 \;\text{MeV} \cdot \text{fm}$$

This allows quick conversion between energy and length scales. For example, a momentum transfer of $q = 1$ fm$^{-1}$ corresponds to $\hbar c \cdot q = 197$ MeV.

Other useful conversions:

G.7 Operator Notation

💡 Convention Note: This textbook uses the Condon-Shortley phase convention for Clebsch-Gordan coefficients and spherical harmonics. The reduced matrix elements follow the convention of Bohr and Mottelson, which differs by a factor of $\sqrt{2J_i + 1}$ from the Rose convention used in some older references. See Chapter 5 for details.

G.8 Abbreviations

G.9 A Note on Sign Conventions

Several sign and phase conventions in nuclear physics are not universal. This textbook follows these choices:

1. Isospin projection: $T_3 = (N - Z)/2$, so the neutron has $T_3 = +1/2$ and the proton has $T_3 = -1/2$. This is the convention of most nuclear physics texts (Bohr and Mottelson, Heyde, Krane). The opposite convention ($T_3 = (Z - N)/2$) is standard in particle physics.

2. Mass excess: $\Delta = M - A \cdot u$ in keV, using atomic masses (including electron masses and binding energies). This is the AME convention.

3. $Q$-value: $Q = \sum M_i(\text{initial}) - \sum M_f(\text{final})$, so that $Q > 0$ indicates an exothermic reaction.

4. Binding energy sign: $B > 0$ for bound nuclei. The binding energy is the energy required to disassemble the nucleus, and it is defined as a positive quantity.

5. Reduced matrix elements: We use the Bohr-Mottelson convention: $$\langle J_f M_f | \hat{T}^{(\lambda)}_\mu | J_i M_i \rangle = (-1)^{J_f - M_f} \begin{pmatrix} J_f & \lambda & J_i \\ -M_f & \mu & M_i \end{pmatrix} \langle J_f \| \hat{T}^{(\lambda)} \| J_i \rangle$$

6. Spherical harmonics: We follow the Condon-Shortley phase convention with the factor $(-1)^m$ included in $Y_l^m$.