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Isotone

An isotone is any of two or more nuclides that have the same number of neutrons but different numbers of protons, thereby belonging to different chemical elements.^[1] In nuclear notation, isotones are identified by their neutron number N, where the atomic mass number A = Z + N varies due to differing proton numbers Z. Examples include boron-12 (^{12}_5B, with 7 neutrons) and carbon-13 (^{13}_6C, with 7 neutrons), as well as the series ^{36}_{16}S, ^{37}_{17}Cl, ^{38}_{18}Ar, ^{39}_{19}K, and ^{40}_{20}Ca, all sharing 20 neutrons.^[1]^[2] Isotones differ from related nuclear concepts: unlike isotopes, which have the same Z but different N (e.g., uranium-235 and uranium-239), isotones maintain constant N while varying Z.^[2] In contrast to isobars, which share the same A but differ in Z and N (e.g., argon-40 and calcium-40), isotones focus solely on neutron equality. The neutron count in isotones plays a critical role in nuclear stability, as even-odd pairings of protons and neutrons influence binding energy; for instance, certain odd neutron numbers (such as 19, 21, or 35) lack stable isotones, often resulting in radioactive species.^[1]^[2] Studies of isotones are essential in nuclear physics for analyzing stability patterns, radioactive decay, and reactions in stellar nucleosynthesis.^[3]

Fundamentals

Definition

A nuclide is a species of atom characterized by the number of protons (Z) and neutrons (N) in its nucleus.^[4] Isotones are nuclides that possess the same number of neutrons (N) but differ in the number of protons (Z), thereby belonging to different chemical elements.^[1] The neutron number for a nuclide is denoted as N = A - Z, where A is the mass number (total number of protons and neutrons); thus, isotones share the same N while varying in Z.^[4] The term "isotone" was coined in 1934 by German physicist K. Guggenheimer, who adapted the term "isotope" by replacing the "p" (for proton) with "n" (for neutron) to describe nuclei with equal neutrons.^[5]

Relation to Other Nuclides

Isotones represent one of several categories in the classification of nuclides, which are atomic species defined by their proton number (Z) and neutron number (N). Unlike isotopes, which share the same Z but differ in N and thus belong to the same chemical element, isotones maintain a constant N while varying Z, resulting in different elements. Isobars, in contrast, have the same total nucleon number A (where A = Z + N) but different combinations of Z and N, leading to distinct elements with approximately equal masses. Nuclear isomers, meanwhile, possess identical Z and N but exist in different excited energy states of the nucleus, often metastable.^[6]^[4] The following table summarizes these distinctions:

Nuclide Type	Same Property	Different Properties	Notes
Isotopes	Z (protons)	N (neutrons), A (mass number)	Same element; chemical properties identical, nuclear properties vary.^[6]
Isobars	A (Z + N)	Z, N	Different elements; similar masses but distinct nuclear structures.^[4]
Isotones	N (neutrons)	Z (protons), A	Different elements; allows isolation of neutron effects in nuclear behavior.^[7]
Isomers	Z, N, A	Nuclear energy state	Same element and mass; differ in excitation, affecting decay modes.^[6]

This classification highlights how isotones emphasize the constancy of the neutron number, which is particularly useful for probing neutron-dependent aspects of nuclear forces, such as shell effects and pairing interactions, by comparing nuclei across varying proton configurations while holding N fixed.^[8]^[9] In the standard nuclide chart, which plots neutron number N along the horizontal axis and atomic number Z along the vertical axis, isotones appear as vertical lines connecting nuclides with fixed N but increasing Z. This visualization underscores their role in mapping neutron-rich regions and understanding trends in nuclear stability perpendicular to the lines of constant Z (isotopes).^[10]

Nuclear Properties

Stability and Magic Numbers

Isotones sharing the same neutron number exhibit varying degrees of nuclear stability, with particularly enhanced stability observed when the neutron number N corresponds to a magic value, such as 2, 8, 20, 28, 50, 82, or 126.^[11] These magic neutron numbers arise from the completion of neutron shells in the nuclear structure, leading to closed subshells that minimize the nucleus's energy and increase resistance to decay processes.^[12] This phenomenon is analogous to the stability of noble gases in atomic chemistry, where filled electron shells confer exceptional inertness.^[11] In the framework of the nuclear shell model, isotones with a shared N occupy similar single-particle neutron orbitals, as the neutron configuration is identical across the chain.^[12] This commonality results in comparable contributions from neutron-neutron interactions to the overall binding energy, influencing properties such as excitation spectra and decay modes.^[11] For instance, near magic N, the filled neutron shells reduce the likelihood of neutron emission or capture, thereby enhancing the nucleus's longevity compared to neighboring configurations.^[12] The neutron separation energy S_n, defined as the energy required to remove a neutron from the nucleus while keeping the proton number Z fixed, remains approximately constant along an isotone chain near shell closures.^[13] This behavior stems from the semi-empirical mass formula (SEMF), where the binding energy's volume, surface, and pairing terms dominate for fixed N, with adjustments for neutron excess yielding a relatively stable S_n expression:

S_n(A, Z) \approx a_v - \frac{2 a_s A^{-1/3}}{3} - a_a \frac{2(A - 2Z)}{A^2} + \cdots

derived by differentiating the SEMF binding energy with respect to neutron number.^[13] Such constancy underscores the shell-stabilizing effect, as S_n drops sharply beyond magic N but plateaus within the isotone sequence.^[11] Despite these shared neutron-driven features, stability along an isotone chain varies due to differences in proton number Z, which modulate the Coulomb repulsion term in the SEMF.^[13] Higher Z increases electrostatic repulsion among protons, reducing overall binding and potentially destabilizing lighter isotones in the chain, while heavier ones may approach beta-stability limits. This proton-induced variation highlights how isotonicity isolates neutron effects but cannot fully decouple them from electromagnetic interactions.^[12]

Examples of Isotones

Isotones with a fixed neutron number N = 6 provide a simple illustration of the concept, including the stable carbon-12 nucleus (^{12}\text{C}, Z = 6) and the radioactive nitrogen-13 (^{13}\text{N}, Z = 7) and oxygen-14 (^{14}\text{O}, Z = 8). The ^{13}\text{N} undergoes \beta^+ decay with a half-life of approximately 10 minutes, while ^{14}\text{O} decays via \beta^+ emission with a half-life of about 70 seconds.^[14]^[15] Another well-studied chain occurs at N = 8, featuring the stable oxygen-16 (^{16}\text{O}, Z = 8), which benefits from the magic neutron number N = 8 contributing to its stability. The neighboring fluorine-17 (^{17}\text{F}, Z = 9) is a positron emitter with a half-life of 64.5 seconds, and neon-18 (^{18}\text{Ne}, Z = 10) is unstable, decaying primarily by \beta^+ emission with a half-life of 1.67 seconds.^[16] In heavier regions, the N = 50 isotones near the tin isotopic chain highlight semi-magic stability, exemplified by the doubly magic but radioactive tin-100 (^{100}\text{Sn}, Z = 50), with a half-life of about 1 second via \beta^+ decay, and the adjacent antimony-101 (^{101}\text{Sb}, Z = 51), which is radioactive.^[17] This chain includes several stable isotones such as krypton-86, strontium-88, yttrium-89, zirconium-90, and molybdenum-92, demonstrating enhanced stability around the Z = 50 and N = 50 shell closures.^[18]^[19] These isotone chains reveal a common pattern: stability often prevails at lower Z values where neutron-proton balance is favorable, but as Z increases, proton repulsion leads to greater instability and shorter half-lives due to the Coulomb barrier's influence on nuclear binding.^[20]

Historical Development

Discovery of the Neutron and Early Concepts

In the early 20th century, atomic models struggled to account for the structure of the nucleus without incorporating neutral particles. Ernest Rutherford's 1911 experiments on the scattering of alpha particles by thin gold foil revealed that the atom's mass and positive charge were concentrated in a tiny, dense nucleus, but this model assumed the nucleus consisted primarily of protons, leaving unexplained the discrepancies in atomic masses observed in chemical elements.^[21] Frederick Soddy, building on studies of radioactive decay, proposed in 1913 that isotopes—variants of the same element with identical chemical properties but different atomic masses—arose from differences in nuclear composition while sharing the same atomic number Z, though the nature of the additional mass contributors remained unclear at the time. The breakthrough came in 1932 when James Chadwick discovered the neutron at the Cavendish Laboratory. By bombarding beryllium with alpha particles from polonium, Chadwick observed a penetrating neutral radiation that ejected protons from paraffin wax with energies indicating a particle mass approximately equal to that of the proton, confirming the existence of an uncharged constituent in the atomic nucleus.^[22] This neutral particle, termed the neutron, resolved longstanding puzzles in nuclear stability and mass, as it allowed the nucleus to be modeled as a composite of protons (determining Z) and neutrons without excessive electrostatic repulsion. The concept of neutron number N = A - Z emerged rapidly in the 1930s through experimental and theoretical advances that highlighted its role in nuclear binding. The 1932 Cockcroft-Walton accelerator experiments achieved the first artificial nuclear transmutation by accelerating protons to bombard lithium, producing two helium nuclei and demonstrating energy release consistent with mass defects, which necessitated neutrons to balance nuclear forces and masses.^[23] These mass discrepancies were formalized in Carl Friedrich von Weizsäcker's 1935 semi-empirical mass formula, which expressed nuclear binding energy as a function of both Z and N separately, incorporating terms for volume, surface, Coulomb repulsion, asymmetry between protons and neutrons, and pairing effects to predict nuclear stability.^[24] Early observations of nuclides sharing the same N appeared in analyses of radioactive decay chains during the 1930s, where beta decay processes—converting a neutron to a proton while preserving total nucleon number A—linked isobars.

Coining of the Term

The term "isotone" was coined in 1934 by German physicist Kurt Guggenheimer in his publication in the Journal de Physique et le Radium, where he adapted the word "isotope"—denoting nuclides with the same number of protons—by substituting "p" (for proton) with "n" (for neutron) to describe nuclides sharing the same neutron number.^[5] This linguistic modification emphasized the parallel role of neutrons in nuclear structure, distinct from protons' influence on atomic identity. Guggenheimer's work introduced the term alongside a proposed chart of nuclides, plotting isotopes horizontally and isotones vertically to visualize patterns in nuclear stability. Guggenheimer's coinage emerged shortly after James Chadwick's 1932 discovery of the neutron, which clarified that neutrons contribute to nuclear mass and binding without altering chemical properties, enabling the conceptualization of nuclides grouped by neutron count.^[25] In this early phase of nuclear physics, the term complemented existing nomenclature, such as "isobar" for nuclides of equal mass number, which had been proposed in 1918 by British chemist Alfred Walter Stewart in his book Recent Advances in Physical and Inorganic Chemistry.^[26] Though initially theoretical, reflecting nascent ideas of shell-like nuclear organization at neutron numbers like 50 and 82, "isotone" gained traction amid the post-World War II surge in nuclear research, aligning with the 1949 shell model formulations by Maria Goeppert-Mayer and J. Hans D. Jensen that underscored neutron shells' importance. The adoption of "isotone" was swift, appearing in scientific literature and textbooks by the late 1940s as nuclear physicists explored binding energies and stability patterns across nuclide chains. By the 1950s, advancements in particle accelerators, such as cyclotrons and early synchrotrons, shifted the term from conceptual to empirical use, facilitating the synthesis and detection of exotic isotones. For instance, in 1954, experiments identified seven new isotones with neutron numbers 150 through 156, expanding the known nuclide chart and validating theoretical predictions of neutron-driven nuclear behavior.

Applications

In Nuclear Structure Studies

In nuclear structure studies, isotones are investigated using advanced experimental techniques that leverage rare isotope beams to access exotic nuclei far from stability. Facilities such as the Facility for Rare Isotope Beams (FRIB) at Michigan State University and the Radioactive Isotope Beam Factory (RIBF) at RIKEN in Japan produce these beams through projectile fragmentation or fission, enabling the creation and study of neutron-rich or proton-rich isotones. Gamma-ray spectroscopy, often combined with beta-decay or Coulomb excitation, is employed to measure level schemes, excitation energies, and transition probabilities in these nuclei. For instance, beta-decay spectroscopy experiments at RIBF have targeted N=82 isotones like 128Pd and 130Cd to probe shell closures and collectivity near the r-process path.^[27] Similarly, ongoing FRIB experiments utilize high-intensity beams for in-beam gamma spectroscopy to explore deformation in neutron-rich isotones approaching the dripline.^[28] Theoretically, chains of isotones play a crucial role in testing predictions of the nuclear shell model by fixing the neutron number (N) and varying the proton number (Z), which isolates the effects of proton configurations on nuclear properties. In the shell model, calculations for isotonic chains, such as N=82 (Z=50–77), reproduce low-lying spectra, electromagnetic transitions, and moments using effective interactions optimized in large valence spaces. For example, quadrupole moments in these chains reflect proton-driven deformation, with shell model predictions showing systematic variations due to intruder configurations beyond the Z=50 closure. Such studies validate model parameters and reveal how proton-neutron interactions influence single-particle energies and collectivity.^[29]^[30] Key findings from isotone studies highlight multiplets with analogous level structures, where similar excitation energies across Z indicate underlying symmetries akin to isospin conservation in lighter systems. In the N=90 isotonic multiplet (Nd, Sm, Gd, Dy), microscopic models explain the near-identical deformed spectra through shared neutron configurations and X(5) symmetry, demonstrating robustness of collective modes. Extensions of isospin concepts to these systems use the formalism T = |N - Z|/2 to quantify symmetry breaking in mirror-like pairs, as seen in carbon isotones where proton decays reveal Coulomb-induced distortions in excitation patterns. These observations evidence how isospin symmetry, perturbed by electromagnetic effects, governs analog states in heavier isotones.^[31]^[32] In modern research, investigations of neutron-rich isotones uncover transitions in nuclear deformation as Z decreases, providing insights into shell evolution and the limits of stability. For N=82 isotones beyond Z=50, spectroscopy reveals a shift from spherical to prolate shapes in heavier members, driven by neutron-proton pairing and quadrupole correlations. Studies of N=20 and N=28 isotones further show how deformation parameters change abruptly near magic numbers, influencing charge radii and moments; for instance, in neutron-rich regions, β₂ values increase to ~0.3–0.4, signaling enhanced collectivity. These findings, obtained via rare beam facilities, refine global models of nuclear forces and predict behaviors in unexplored territories.^[33]^[30]

In Astrophysics and Nucleosynthesis

In the rapid proton-capture (rp) process powering type I X-ray bursts on accreting neutron stars, successive proton captures proceed along isotonic chains of constant neutron number N, as each capture increases the proton number Z while preserving N. This path is punctuated by waiting points, where the (p,γ) reaction rate slows due to low Q-values near the proton drip line, causing material to accumulate until slower β⁺ decay or electron capture advances the flow to adjacent chains. Prominent waiting points occur along the N=50 isotone chain, exemplified by ^{100}Sn (Z=50), whose uncertain half-life and masses critically determine burst energetics, light curve durations, and the final composition of nuclear ashes rich in A≈100 nuclei.^[34]^[35] In neutron-rich stellar environments, such as the neutrino-driven winds of core-collapse supernovae and the ejecta of neutron star mergers, the rapid neutron-capture (r) process generates extremely neutron-rich isotones far from stability. Neutron captures increase N while keeping Z fixed, but the flow bottlenecks at shell-closure waiting points with magic neutron numbers N=50, 82, and 126, where β⁻ decay rates are suppressed by nuclear structure effects, leading to significant abundance peaks in those isotones across a range of Z. For instance, N=82 isotones like ^{130}Cd to ^{142}Nd accumulate and shape the second r-process peak near A≈130–140, influencing the production of elements up to uranium. Recent models as of 2025 also consider additional sites like magnetar giant flares and common envelope jet supernovae for r-process nucleosynthesis, further constraining isotone abundances.^[36]^[37] Neutron star surfaces and interiors also contribute to isotone formation through accretion-induced reactions, linking to kilonova emissions.^[38]^[39] Gamma-ray spectroscopy of astrophysical sites provides direct observational constraints on isotone production and decay. In core-collapse supernovae, the characteristic γ-lines from the β⁺ decay chain of ^{56}Ni (Z=28, N=28)—primarily at 158 keV and 811 keV—dominate the early light curve, while explosive burning extends the N=28 isotone chain via (p,γ) reactions to ^{57}Cu (Z=29, N=28) and beyond, with their decay signatures informing models of silicon and iron-group yields. Similar detections in type Ia supernovae remnants and novae trace lighter isotone decays, validating nucleosynthesis simulations.^[40]^[41] Investigations of isotone nuclear properties, including precise mass measurements and β-decay rates, constrain reaction rates and branching ratios in astrophysical models, enhancing predictions for heavy element formation. In the slow neutron-capture (s) process within asymptotic giant branch stars, shell effects in neutron magic-number isotones modulate neutron-capture efficiencies and β-decay branches, affecting s-process yields from Sr to Pb as observed in solar abundances. These studies also refine r-process simulations by quantifying waiting-point impacts on third-peak elements, while extensions to Big Bang nucleosynthesis incorporate isotone data for marginal heavier-isotope contributions in non-standard scenarios.^[42]^[43]

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