Appendix B — Nuclear Data Tables

This appendix provides reference nuclear data for nuclei that appear throughout the textbook. All values are from the 2020 Atomic Mass Evaluation (AME2020, Wang et al., Chinese Physics C 45, 030003, 2021), NUBASE2020 (Kondev et al., Chinese Physics C 45, 030001, 2021), and the Evaluated Nuclear Structure Data File (ENSDF) maintained by the National Nuclear Data Center at Brookhaven National Laboratory. Binding energies are total binding energies; atomic masses include electrons. Half-lives labeled "stable" mean no observed decay mode.

Table B.1: Properties of Selected Nuclei

The following table lists properties of nuclei referenced in this textbook. Columns are: element symbol with mass number, proton number $Z$, neutron number $N$, mass number $A$, ground-state spin and parity $J^\pi$, total binding energy $B$ in MeV, binding energy per nucleon $B/A$ in MeV, and half-life $t_{1/2}$. Binding energies are from AME2020. Spin-parity assignments are from ENSDF.

$^*$ For $^{99\text{m}}$Tc, the binding energy listed is for the ground state of $^{99}$Tc. The isomeric state lies 142.68 keV above the ground state.

$^\dagger$ Binding energy for $^{294}$Og is estimated from systematics (AME2020 extrapolation). Only three atoms have been confirmed.

Notes on selected nuclei: - $^{4}$He ($\alpha$ particle): Doubly magic ($Z = N = 2$). Exceptionally tightly bound; emitted in alpha decay. - $^{12}$C: Doubly magic in the $p$-shell. The Hoyle state (0$^+_2$, 7.654 MeV) is critical for stellar nucleosynthesis (Chapter 22). - $^{56}$Fe: Often cited as "most stable nucleus," but $^{62}$Ni actually has the highest $B/A$. The distinction is discussed in Chapter 4. - $^{56}$Ni: Doubly magic ($Z = N = 28$); produced in core-collapse supernovae, decays to $^{56}$Fe via $^{56}$Co. - $^{62}$Ni: Highest $B/A$ of any nuclide (8.7945 MeV). See Chapter 4. - $^{99\text{m}}$Tc: Most widely used medical radioisotope. The "m" denotes a metastable (isomeric) nuclear state (Chapter 27). - $^{132}$Sn: Doubly magic ($Z = 50$, $N = 82$), radioactive. Important r-process waiting point (Chapter 23). - $^{152}$Dy: First superdeformed band discovered (Chapter 8). - $^{157}$Gd: Highest thermal neutron capture cross section among stable nuclei ($\sigma_{\text{th}} = 254{,}000$ b). Used in reactor control (Chapter 26). - $^{208}$Pb: Doubly magic ($Z = 82$, $N = 126$). The benchmark nucleus for nuclear structure. - $^{209}$Bi: Formerly considered the heaviest stable nuclide; alpha decay observed in 2003. - $^{225}$Ac: Promising alpha-emitter for targeted alpha therapy (Chapter 27). - $^{294}$Og: Heaviest element confirmed (oganesson, $Z = 118$). Three events observed (Chapter 11).

Table B.2: Semi-Empirical Mass Formula Parameters

The Bethe-Weizsacker semi-empirical mass formula (SEMF) for the nuclear binding energy is (Chapter 4):

$$B(Z,N) = a_V A - a_S A^{2/3} - a_C \frac{Z(Z-1)}{A^{1/3}} - a_{\text{sym}}\frac{(N-Z)^2}{A} + \delta(A,Z)$$

where the pairing term is:

$$\delta(A,Z) = \begin{cases} +a_P A^{-1/2} & \text{even-even} \\ 0 & \text{odd-}A \\ -a_P A^{-1/2} & \text{odd-odd} \end{cases}$$

Table B.2: Commonly used SEMF parameter sets (all values in MeV)

Set 1: K.S. Krane, Introductory Nuclear Physics (1988). Set 2: J.W. Rohlf, Modern Physics from α to Z⁰ (1994). Set 3: Fit to AME2020 data for $A \geq 20$, minimizing $\chi^2$ of binding energy residuals. The parameters vary depending on the mass range fitted, the form of the pairing and shell correction terms, and the fitting methodology. See Chapter 4 for the fitting procedure and code (semf_fit.py).

The SEMF achieves an RMS deviation of approximately 2.5 MeV for $A \geq 20$ (about 0.3% of $B$ for heavy nuclei). The largest deviations occur at magic numbers, where the actual binding energy exceeds the SEMF prediction by up to 10 MeV.

Table B.3: Magic Numbers and Doubly Magic Nuclei

Nuclear Magic Numbers

The magic numbers in nuclear physics are nucleon numbers at which the nuclear binding energy is anomalously large, nucleon separation energies show sharp drops, and the first excited state energy is anomalously high. They correspond to closed shells in the nuclear mean-field potential with spin-orbit splitting (Chapter 6).

Proton magic numbers: 2, 8, 20, 28, 50, 82, (114$^*$)

Neutron magic numbers: 2, 8, 20, 28, 50, 82, 126, (184$^*$)

$^*$ Predicted but not experimentally confirmed as traditional magic numbers. Evidence suggests enhanced stability near $Z = 114$ and $N = 184$ in the superheavy region (Chapter 11), but the term "magic" may not apply in the classical sense due to deformation effects.

Doubly Magic Nuclei

A doubly magic nucleus has both $Z$ and $N$ equal to magic numbers. All have $J^\pi = 0^+$ ground states.

$^a$ $^{48}$Ca has been observed to undergo double beta decay ($t_{1/2} = 6.4 \times 10^{19}$ yr).

Subshell closures (not traditional magic numbers but exhibiting enhanced stability): $Z = 40$ (Zr), $N = 40$, $Z = 64$ (Gd), $N = 56$, $N = 64$. These are important for understanding nuclear structure far from the valley of stability (Chapter 10).

Table B.4: Shell Model Filling Order

The following table gives the single-particle orbit filling order in the nuclear shell model with a Woods-Saxon potential plus spin-orbit coupling (Chapter 6). Each orbit is labeled by the quantum numbers $n l_j$ (where $n$ is the radial quantum number starting from $n = 0$ in the harmonic oscillator convention used here) and its degeneracy is $2j + 1$. The cumulative nucleon number gives the total at each shell closure.

Complete filling order through $N = 184$

Notes: - The filling order above $N = 126$ is model-dependent and sensitive to the details of the spin-orbit potential and Coulomb effects (for protons). The order shown is representative of typical Woods-Saxon calculations but should not be regarded as definitive for superheavy elements. - Within each major shell, the spin-orbit partners ($l + \frac{1}{2}$ and $l - \frac{1}{2}$) are split, with the $j = l + \frac{1}{2}$ orbit lower in energy. The $j = l + \frac{1}{2}$ orbit from each harmonic oscillator shell is depressed into the shell below, creating the observed magic numbers (Chapter 6). - For protons, the Coulomb interaction shifts levels relative to neutrons: high-$l$ orbits are more tightly bound (Coulomb energy is lower for orbits with probability concentrated away from the nuclear surface), and the predicted proton magic numbers above $Z = 82$ differ from the neutron sequence.

Table B.5: Natural Radioactive Decay Series

Four natural decay series connect heavy radioactive nuclides through chains of alpha and beta decays to stable (or nearly stable) end products. The mass number $A$ of every member of a series satisfies $A = 4n + k$ with a fixed value of $k$ (because alpha decay changes $A$ by 4, and beta decay does not change $A$).

Uranium Series ($4n + 2$)

$$^{238}\text{U} \xrightarrow{\alpha,\; 4.468 \times 10^9\;\text{yr}} {^{234}\text{Th}} \xrightarrow{\beta^-,\; 24.1\;\text{d}} {^{234}\text{Pa}} \xrightarrow{\beta^-,\; 6.70\;\text{h}} {^{234}\text{U}} \xrightarrow{\alpha,\; 2.455 \times 10^5\;\text{yr}} {^{230}\text{Th}}$$

$$\xrightarrow{\alpha,\; 7.538 \times 10^4\;\text{yr}} {^{226}\text{Ra}} \xrightarrow{\alpha,\; 1600\;\text{yr}} {^{222}\text{Rn}} \xrightarrow{\alpha,\; 3.824\;\text{d}} {^{218}\text{Po}} \xrightarrow{\alpha,\; 3.10\;\text{min}} {^{214}\text{Pb}}$$

$$\xrightarrow{\beta^-,\; 26.8\;\text{min}} {^{214}\text{Bi}} \xrightarrow{\beta^-,\; 19.9\;\text{min}} {^{214}\text{Po}} \xrightarrow{\alpha,\; 164\;\mu\text{s}} {^{210}\text{Pb}} \xrightarrow{\beta^-,\; 22.2\;\text{yr}} {^{210}\text{Bi}}$$

$$\xrightarrow{\beta^-,\; 5.01\;\text{d}} {^{210}\text{Po}} \xrightarrow{\alpha,\; 138.4\;\text{d}} \boxed{^{206}\text{Pb}\;\text{(stable)}}$$

The $^{222}$Rn (radon) in this chain is the primary source of natural indoor radioactivity (Chapter 29).

Thorium Series ($4n$)

$$^{232}\text{Th} \xrightarrow{\alpha,\; 1.405 \times 10^{10}\;\text{yr}} {^{228}\text{Ra}} \xrightarrow{\beta^-,\; 5.75\;\text{yr}} {^{228}\text{Ac}} \xrightarrow{\beta^-,\; 6.15\;\text{h}} {^{228}\text{Th}} \xrightarrow{\alpha,\; 1.912\;\text{yr}} {^{224}\text{Ra}}$$

$$\xrightarrow{\alpha,\; 3.66\;\text{d}} {^{220}\text{Rn}} \xrightarrow{\alpha,\; 55.6\;\text{s}} {^{216}\text{Po}} \xrightarrow{\alpha,\; 0.145\;\text{s}} {^{212}\text{Pb}} \xrightarrow{\beta^-,\; 10.64\;\text{h}} {^{212}\text{Bi}}$$

$^{212}$Bi branches: - 64.06%: $\xrightarrow{\alpha,\; 60.6\;\text{min}} {^{208}\text{Tl}} \xrightarrow{\beta^-,\; 3.053\;\text{min}} \boxed{^{208}\text{Pb}\;\text{(stable)}}$ - 35.94%: $\xrightarrow{\beta^-,\; 60.6\;\text{min}} {^{212}\text{Po}} \xrightarrow{\alpha,\; 0.299\;\mu\text{s}} \boxed{^{208}\text{Pb}\;\text{(stable)}}$

The thorium series terminates at doubly magic $^{208}$Pb.

Actinium Series ($4n + 3$)

$$^{235}\text{U} \xrightarrow{\alpha,\; 7.038 \times 10^8\;\text{yr}} {^{231}\text{Th}} \xrightarrow{\beta^-,\; 25.5\;\text{h}} {^{231}\text{Pa}} \xrightarrow{\alpha,\; 3.276 \times 10^4\;\text{yr}} {^{227}\text{Ac}}$$

$^{227}$Ac branches: - 98.62%: $\xrightarrow{\beta^-,\; 21.77\;\text{yr}} {^{227}\text{Th}} \xrightarrow{\alpha,\; 18.7\;\text{d}} {^{223}\text{Ra}}$ - 1.38%: $\xrightarrow{\alpha,\; 21.77\;\text{yr}} {^{223}\text{Fr}} \xrightarrow{\beta^-,\; 22.0\;\text{min}} {^{223}\text{Ra}}$

$$^{223}\text{Ra} \xrightarrow{\alpha,\; 11.43\;\text{d}} {^{219}\text{Rn}} \xrightarrow{\alpha,\; 3.96\;\text{s}} {^{215}\text{Po}} \xrightarrow{\alpha,\; 1.781\;\text{ms}} {^{211}\text{Pb}}$$

$$\xrightarrow{\beta^-,\; 36.1\;\text{min}} {^{211}\text{Bi}} \xrightarrow{\alpha,\; 2.14\;\text{min}} {^{207}\text{Tl}} \xrightarrow{\beta^-,\; 4.77\;\text{min}} \boxed{^{207}\text{Pb}\;\text{(stable)}}$$

Neptunium Series ($4n + 1$)

This series does not occur naturally because its longest-lived member, $^{237}$Np ($t_{1/2} = 2.144 \times 10^6$ yr), has a half-life much shorter than the age of the Earth. It is produced artificially in nuclear reactors.

$$^{237}\text{Np} \xrightarrow{\alpha,\; 2.144 \times 10^6\;\text{yr}} {^{233}\text{Pa}} \xrightarrow{\beta^-,\; 27.0\;\text{d}} {^{233}\text{U}} \xrightarrow{\alpha,\; 1.592 \times 10^5\;\text{yr}} {^{229}\text{Th}} \xrightarrow{\alpha,\; 7340\;\text{yr}} {^{225}\text{Ra}}$$

$$\xrightarrow{\beta^-,\; 14.9\;\text{d}} {^{225}\text{Ac}} \xrightarrow{\alpha,\; 9.920\;\text{d}} {^{221}\text{Fr}} \xrightarrow{\alpha,\; 4.8\;\text{min}} {^{217}\text{At}} \xrightarrow{\alpha,\; 32.3\;\text{ms}} {^{213}\text{Bi}}$$

$^{213}$Bi branches: - 97.80%: $\xrightarrow{\beta^-,\; 45.6\;\text{min}} {^{213}\text{Po}} \xrightarrow{\alpha,\; 3.72\;\mu\text{s}} {^{209}\text{Pb}} \xrightarrow{\beta^-,\; 3.25\;\text{h}} \boxed{^{209}\text{Bi}\;\text{(quasi-stable)}}$ - 2.20%: $\xrightarrow{\alpha,\; 45.6\;\text{min}} {^{209}\text{Tl}} \xrightarrow{\beta^-,\; 2.16\;\text{min}} \boxed{^{209}\text{Bi}\;\text{(quasi-stable)}}$

Note that $^{225}$Ac, which appears in this series, is the parent isotope for targeted alpha therapy (TAT) in nuclear medicine (Chapter 27). Its four sequential alpha decays deliver approximately 28 MeV of alpha energy to a tumor cell.

Summary of Decay Series

Data Sources and Recommended Databases

The data in this appendix are drawn from the following evaluated databases:

• AME2020: Wang, M. et al., "The AME 2020 atomic mass evaluation." Chinese Physics C 45, 030003 (2021). The definitive source for atomic masses and binding energies.
• NUBASE2020: Kondev, F.G. et al., "The NUBASE2020 evaluation of nuclear physics properties." Chinese Physics C 45, 030001 (2021). Nuclear properties including half-lives, spins, parities, and decay modes.
• ENSDF: Evaluated Nuclear Structure Data File, maintained by the National Nuclear Data Center (NNDC), Brookhaven National Laboratory. Accessed via www.nndc.bnl.gov. Comprehensive evaluated nuclear structure data including level schemes, gamma-ray energies, transition rates, and moments.
• ENDF/B-VIII.0: Brown, D.A. et al., Nucl. Data Sheets 148, 1 (2018). Evaluated nuclear reaction data, including cross sections, angular distributions, and fission yields. Used primarily in Chapters 17–21 and 26.
• TENDL-2021: Koning, A.J. et al., "TENDL: Complete Nuclear Data Library for Innovative Nuclear Science and Technology." Nucl. Data Sheets 155, 1 (2019). TALYS-based evaluated data library with comprehensive uncertainty quantification.

For the most current values, readers should consult the live databases. Nuclear data are continuously updated as new measurements are published. Appendix E provides a guide to online nuclear data resources and how to query them programmatically using the toolkit developed in this textbook.

All binding energies are rounded to the nearest keV (three decimal places in MeV). Half-lives are given to the precision justified by current measurements. For the most precise values, consult AME2020 and NUBASE2020 directly.