# Atomic nucleus The **atomic nucleus** is the small, positively charged core at the centre of every [[Atom|atom]], made of [[Proton|protons]] and [[Neutron|neutrons]] and holding essentially all of the atom's mass in about one part in 10¹⁴ of its volume. It was not deduced from theory but forced on physics by an experiment whose result nobody expected: a beam of [[Alpha_particle|alpha particles]] fired at a thin metal foil, which should have passed through a diffuse pudding of charge almost undeflected, occasionally came straight back. In the microsim below the reader fires alphas at a single nucleus and sets two things: the alpha energy E, from 1 to 10 MeV, and the target's [[Atomic_number|atomic number]] Z, with presets for [[Gold|gold]] (Z = 79) and [[Aluminium|aluminium]] (Z = 13). Every trajectory is a hyperbola drawn in closed form from its impact parameter b through `tan(theta/2) = k·Z1·Z2·e²/(2·E·b)`, so the fan of 200 rays needs no integration at all.[^rutherford][^spec-p48] Three readouts answer. The paths themselves show that a large deflection requires a small b. The count against angle follows `1/sin⁴(theta/2)`, which is why the backward hemisphere is so sparse — at 5 MeV on gold the yield at 10° exceeds the yield at 90° by a factor of about 4,300.[^rutherford][^derived-an] And the distance of closest approach for a head-on hit, `d = k·Z1·Z2·e²/E`, reads 45.5 fm for a 5 MeV alpha on gold, against a nuclear radius of `R = 1.2·A^(1/3)` = 7.0 fm.[^os-nuclear][^derived-an] The alpha stops more than six radii short of the surface. That single comparison is the reason a point charge describes the data perfectly, and the reason the nucleus had to be very small. On the [[Physics]] flagship this article serves the *Nuclear and particle* section of Part IV — Branches and fields, and it is the root of the nuclear family: [[Nuclear_fission]] splits the object described here, and [[Nuclear_chain_reaction]] and [[Nuclear_reactor]] chain those splittings together. ## History By 1909 the accepted picture of the atom was a sphere of positive charge with electrons embedded in it, and in such an atom no single encounter could turn a fast alpha particle through a large angle. Hans Geiger and Ernest Marsden nevertheless found alphas reflected backwards from a metal foil, a result Rutherford later described as being as incredible as a shell bouncing off tissue paper.[^geiger] In 1911 Rutherford published the analysis that made sense of it: if the atom's positive charge and nearly all its mass sit in a region very much smaller than the atom, a single [[Coulomb's_law|Coulomb]] encounter can produce any deflection at all, and the number scattered through an angle theta falls as `1/sin⁴(theta/2)`.[^rutherford-1911] The formula's agreement with the counts was the evidence; the nucleus was its conclusion. The composition took two further decades. The proton was identified as the [[Hydrogen_atom|hydrogen]] nucleus and as a constituent of other nuclei, but a nucleus of Z protons alone had the wrong mass by roughly a factor of two, and the difference was patched with the assumption of extra protons neutralised by electrons inside the nucleus. James Chadwick removed the patch in 1932 by identifying the neutron, a neutral particle of almost the proton's mass, in the radiation emitted when [[Beryllium|beryllium]] was bombarded with alphas.[^chadwick] From that point the nucleus had its modern composition — Z protons and N neutrons, with mass number A = Z + N — and [[Nuclear_physics|nuclear physics]] could begin in earnest. ### Etymology The word comes from the Latin *nucleus*, a kernel or little nut, the diminutive of *nux*, nut. The term had already been used in biology for the kernel of a cell before Rutherford's model gave it the meaning it now carries in physics: the small hard centre around which everything else is arranged. ## Principles A nucleus is specified by two integers. Z, the number of protons, fixes the [[Chemical_element|element]] and therefore the chemistry; N, the number of neutrons, fixes which [[Isotope|isotope]] of that element it is; and A = Z + N is the count of nucleons. The nucleus is bound, which means its measured mass is *less* than the sum of its parts, and the missing mass is the binding energy through `dE = dm·c²`.[^murphy-be] The arithmetic is worth doing once. For ⁵⁶[[Iron|Fe]], the constituent masses sum to 56.46340 amu while the measured nuclide is 55.934942 amu; the defect of 0.528447 amu, at 931.49432 MeV per amu, is 492.25 MeV of binding, or 8.79 MeV per nucleon.[^murphy-be] Plotted against A, binding energy per nucleon rises steeply from 1.11 MeV for ²H, reaches 7.07 MeV at ⁴He and 7.68 MeV at ¹²C, peaks near 8.79 MeV around ⁵⁶Fe, and then declines slowly to 7.59 MeV at ²³⁵U.[^murphy-be] The shape of that curve is the whole of nuclear energetics: everything to the left of iron releases energy by fusing and everything to the right by splitting, and [[Nuclear_binding_energy|the curve]] is the reason both [[Nuclear_fusion|fusion]] and fission are exothermic at their respective ends.[^murphy-be][^manual10] ## Composition and shape Scattering experiments give the nucleus a radius that follows a strikingly simple rule, `R = r0·A^(1/3)` with r0 ≈ 1.2 fm.[^os-nuclear] Because R goes as the cube root of A, the volume goes as A itself, and the number of nucleons per unit volume is the same in every nucleus — nuclear matter does not compress under the addition of more nucleons the way an electron cloud does. Putting numbers to it: one nucleon of 1.66×10⁻²⁷ kg in a sphere of radius 1.2 fm gives a [[Density|density]] of 2.3×10¹⁷ kg/m³, some 2×10¹⁴ times that of water.[^derived-an][^os-nuclear] For gold, A = 197 gives R = 7.0 fm; for aluminium, A = 27 gives exactly 3.6 fm.[^derived-an] That constant density is what makes the liquid-drop picture natural, and it is also what the sim exploits. The alpha in the sim never reaches the surface at gold's Z: the Coulomb barrier for an alpha touching a gold nucleus, `k·Z1·Z2·e²/(R_alpha + R_Au)` with R_alpha = 1.9 fm, is about 25.6 MeV, far above the 1–10 MeV the slider allows, so the trajectory is pure Coulomb from start to finish and the closed-form hyperbola is exact.[^derived-an][^rutherford] Switch the target to aluminium and the arithmetic changes character. There the barrier is only 6.8 MeV, so at 10 MeV the alpha overtops it, touches nuclear matter, and the `1/sin⁴(theta/2)` law breaks at large angles.[^derived-an] Historically that breakdown was not a failure but a measurement: the energy at which Rutherford scattering stops working is a direct reading of nuclear size. Not every nucleus is a sphere. Many mid-shell and heavy nuclei are permanently deformed, usually prolate, and their rotational spectra are the evidence; a deformed nucleus also fissions more readily than a spherical one, which matters at the far right of the binding curve.[^krane] ## Forces Two forces of very different character compete inside the nucleus. The [[Coulomb's_law|Coulomb]] repulsion between protons is long-ranged and grows with the number of proton pairs, roughly as Z(Z − 1), so it becomes more damaging the larger the nucleus. The [[Nuclear_force|nuclear force]] that binds nucleons is strongly attractive but reaches only 1–2 fm, about the separation of two touching nucleons, and it saturates: a nucleon deep inside a large nucleus interacts only with its neighbours, not with every other nucleon.[^krane] Saturation is exactly what produces the constant density and the nearly constant binding energy per nucleon above A ≈ 20. It also explains the shape of the stability valley. In light nuclei the most stable arrangement has N ≈ Z, but as Z rises the Coulomb term grows faster than the nuclear term, and stability requires progressively more neutrons — from N = Z at ¹²C to N/Z ≈ 1.5 at [[Uranium|uranium]]. Past Z ≈ 83 no arrangement is stable at all, and every nucleus decays by [[Alpha_decay|alpha]] or [[Beta_decay|beta]] emission or, at the extreme, by [[Spontaneous_fission|spontaneous fission]].[^krane] The same competition, read in reverse, is why an alpha approaching a nucleus feels only repulsion until it is nearly touching, and why [[Quantum_tunnelling|tunnelling]] through that barrier — rather than climbing it — is how alpha decay proceeds. ## Halo nuclei and nuclear force range limits Because the nuclear force has a finite range, the A^(1/3) rule has to fail somewhere, and it fails most spectacularly at the neutron drip line. In certain very neutron-rich light nuclei, one or two neutrons are bound so weakly — by a few hundred keV rather than the usual several MeV — that their wavefunction extends far outside the range of the force that holds them, forming a diffuse halo around a compact core. ¹¹[[Lithium|Li]] is the standard example: the `R = 1.2·A^(1/3)` rule predicts 2.7 fm, while the measured matter radius is very much larger, comparable to that of a nucleus several times its mass.[^halo][^derived-an] Halo nuclei matter here for what they show about the force rather than for what they are. A halo exists because a weakly bound particle in a short-range potential spends most of its time in the classically forbidden region outside it — a purely quantum-mechanical statement about tunnelling, not about nuclear structure. They mark the outer edge of what "nucleus" means, and they set the practical limit of every model in the next section, all of which assume a well-defined surface.[^halo] ## Nuclear models No single model describes the nucleus. The force between nucleons is not known in closed form, the many-body problem is not solvable for A in the tens, and the result is a family of pictures, each exact in a different limit and each with a domain where it is useless. The three below are the ones that survive because each explains a class of facts nothing else does.[^krane] ### Cluster model Some nuclei behave as though built from bound sub-units rather than from loose nucleons, and the sub-unit is almost always the alpha particle — unsurprisingly, since ⁴He is exceptionally tightly bound at 7.07 MeV per nucleon among its light neighbours.[^murphy-be] ⁸Be is the clean case: it is unbound and lives only about 10⁻¹⁶ s, decaying promptly into two alphas, which is precisely what a two-cluster description predicts.[^krane] Cluster structure is why [[Triple-alpha_process|the triple-alpha process]] in stars has to proceed through a resonance, and why alpha decay is the dominant decay mode for heavy nuclei: the emitted particle is, in a real sense, already assembled inside. ### Liquid drop model The oldest quantitative model treats the nucleus as an incompressible charged drop, and it works because nuclear matter really does have constant density and a surface. The [[Semi-empirical_mass_formula|semi-empirical mass formula]] writes the binding energy as a volume term proportional to A, minus a surface term in A^(2/3), minus a Coulomb term in `Z(Z−1)/A^(1/3)`, minus an asymmetry term in `(A − 2Z)²/A`, plus a pairing term — five coefficients fitted once to the whole chart of nuclides.[^krane] With those five numbers the model reproduces the binding energy of hundreds of nuclei to within about 1 %, and it delivers the fission barrier, the Coulomb repulsion of two fragments, and the stability valley as by-products. What it cannot do is equally informative. The formula is smooth in Z and N, so it misses every shell effect; the binding energies of ⁴He and ¹⁶O, where clustering and closed shells dominate, sit well above the curve it draws. The drop is a model of bulk nuclear matter, and it stops working exactly where the nucleus stops being bulk. ### Shell models and other quantum models Nuclei with 2, 8, 20, 28, 50, 82 or 126 protons or neutrons are anomalously tightly bound, anomalously abundant, and reluctant to absorb another nucleon — the nuclear magic numbers.[^krane] The pattern is the same kind of fact as the noble gases in the [[Periodic_table|periodic table]], and it has the same kind of explanation: nucleons occupy quantised levels in an average potential made by all the others, and a filled level is stable. The nuclear sequence is not the atomic one, and reproducing it required adding a strong spin–orbit coupling to the potential, the step that earned Maria Goeppert Mayer and J. Hans D. Jensen a share of the 1963 Nobel Prize in Physics.[^krane] ²⁰⁸[[Lead|Pb]], with Z = 82 and N = 126, is doubly magic and is the heaviest stable nucleus. Shell and drop are complementary rather than rival. The collective model grafts them together — single-particle levels moving inside a deformable drop — and it is that hybrid, not either parent, that describes the deformed rotational bands and the fission barriers of the actinides.[^krane] ## See also - [[Rutherford_scattering_experiments]] — the measurement the sim reproduces - [[Nuclear_force]] - [[Proton]] - [[Neutron]] - [[Nuclear_physics]] - [[Nuclear_binding_energy]] - [[Semi-empirical_mass_formula]] - [[Isotope]] ## Notes - The Rutherford relation on this page is the non-relativistic, fixed-target, point-charge result. It assumes the target nucleus is infinitely massive, the projectile is a point charge, and the only interaction is Coulomb. All three assumptions are visibly good for a few-MeV alpha on gold and all three begin to fail for a light target at the top of the sim's energy range. - Radii quoted from `R = 1.2·A^(1/3)` are charge-distribution radii of the kind scattering measures. Different probes — electrons, neutrons, pions — measure slightly different radii, and r0 is quoted anywhere between 1.1 and 1.4 fm depending on the definition. - Sources for every figure on this page are collected under *References* below. ## References [^rutherford-1911]: Rutherford, Ernest (1911). "The Scattering of α and β Particles by Matter and the Structure of the Atom." *Philosophical Magazine*, Series 6, 21: 669–688. The paper introduces the small massive central charge and derives the angular distribution of scattered alphas. [^geiger]: Geiger, Hans; Marsden, Ernest (1909). "On a Diffuse Reflection of the α-Particles." *Proceedings of the Royal Society A* 82 (page to pin). The large-angle reflections that the plum-pudding atom could not produce. No Portal Book covers the Geiger–Marsden experiments. [^chadwick]: Chadwick, James (1932). "Possible Existence of a Neutron." *Nature* 129 (page to pin). The identification of a neutral nucleon of approximately the proton's mass in the radiation from alpha-bombarded beryllium, which fixed the modern Z/N composition of the nucleus. [^rutherford]: The Rutherford scattering relation `tan(theta/2) = k·Z1·Z2·e²/(2·E·b)` and the differential cross-section proportional to `1/sin⁴(theta/2)` are standard closed forms; they are taken here from Krane, Kenneth S. *Introductory Nuclear Physics* (Wiley, 1988), chapter on nuclear reactions and scattering (page to pin), and are consistent with the treatment in Portal Book 079, *University Physics Volume 3* (OpenStax, 2016), Chapter 10 *Nuclear Physics*, pp. 441–492 (page to pin). No Portal Book in this cluster prints the derivation. [^os-nuclear]: OpenStax (Sanny, Jeff; Ling, Samuel, eds.) (2016). *University Physics Volume 3*, Chapter 10 *Nuclear Physics*, pp. 441–492 (page to pin): the nuclear radius rule `R = r0·A^(1/3)` with r0 ≈ 1.2 fm, the resulting constant nuclear density, and the mass-defect definition of binding energy. https://openstax.org/details/books/university-physics-volume-3 [^murphy-be]: Murphy, Thomas (2021). *Energy and Human Ambitions on a Finite Planet*, Chapter 15: `dE = dm·c²` with c = 2.99792458×10⁸ m/s, p. 266; the mass defect and binding energy per nucleon, pp. 267–269; Table 15.5 binding energies and per-nucleon values — ²H 2.22 MeV / 1.11, ⁴He 28.29 / 7.07, ¹²C 92.16 / 7.68, ⁵⁶Fe 492.25 / 8.79, ²³⁵U 1,783.85 / 7.59 — pp. 267–268; the ⁵⁶Fe worked example, parts 56.46340 amu minus measured 55.934942 amu = 0.528447 amu = 492.25 MeV, p. 268; and the conversion 1 amu = 931.49432 MeV/c², p. 267. https://open.umn.edu/opentextbooks/textbooks/energy-and-human-ambitions-on-a-finite-planet [^krane]: Krane, Kenneth S. *Introductory Nuclear Physics* (Wiley, 1988), chapters on nuclear properties, nuclear models and nuclear decay (pages to pin): the 1–2 fm range and saturation of the nucleon–nucleon force; the N/Z drift of the stability valley and the end of stability above Z ≈ 83; permanent prolate deformation and rotational bands; the semi-empirical mass formula's volume, surface, Coulomb, asymmetry and pairing terms; alpha clustering and the prompt two-alpha breakup of ⁸Be; the magic numbers 2, 8, 20, 28, 50, 82 and 126; the spin–orbit term that reproduces them and the 1963 Nobel Prize in Physics awarded in part to Maria Goeppert Mayer and J. Hans D. Jensen for it; and the collective model that joins shell structure to the deformable drop. No Portal Book in this cluster treats nuclear models, so these are cited to a standard text rather than assigned Portal Book pages. [^halo]: Tanihata, Isao, et al. (1985). Interaction cross-section and matter-radius measurements on light neutron-rich nuclei, *Physical Review Letters* (volume and page to pin), the measurements from which the ¹¹Li halo was inferred; with Krane, *Introductory Nuclear Physics* (page to pin) for the underlying statement that a weakly bound nucleon in a short-range potential has most of its probability outside the potential. No Portal Book covers halo nuclei or the drip lines. [^manual10]: Wikitube MicroSim Guide, sub-manual 10 *Earth, Energy and Environment*, §4.2 "Binding energy per nucleon: why fusion and fission both pay": the binding-energy curve peaking near ⁵⁶Fe at 8.79 MeV per nucleon with ⁶²Ni marginally higher at 8.795, the statement that fusion pays on the low-A side and fission on the high-A side, and the pitfall that Table 15.5 uses atomic masses with electrons included. [^spec-p48]: Matter & Energy Cluster contract, `_registry/plans/PHYSICS_SECTIONS.md` row P48: the sim concept for this page — alphas on a nucleus of charge Z, each hyperbola drawn in closed form from its impact parameter; the reader's two controls, alpha energy E from 1 to 10 MeV and Z with an Au (79) preset and an Al (13) preset; the fan of 200 rays, the count against theta as `1/sin⁴(theta/2)`, and the closest-approach distance compared with `R = 1.2·A^(1/3)`; and the element placement, Au. [^derived-an]: Computed for this article from the equations and constants cited above, with `k·e² = 1.43996 MeV·fm`: closest approach for a 5 MeV alpha on gold, `d = 1.43996 × 2 × 79 / 5 = 45.5 fm`, against `R(Au-197) = 1.2 × 197^(1/3) = 6.98 fm`, a ratio of 6.5; `R(Al-27) = 1.2 × 27^(1/3) = 3.60 fm` and `R(He-4) = 1.905 fm`; touching-barrier energies `k·Z1·Z2·e²/(R1 + R2)` of 25.6 MeV for alpha + Au and 6.81 MeV for alpha + Al, so that the sim's 1–10 MeV range stays entirely Coulombic on gold but crosses the barrier on aluminium; the yield ratio `[sin(45°)/sin(5°)]⁴ = 4.33×10³` between 10° and 90°; nuclear density `1.66054×10⁻²⁷ kg ÷ (4/3)π(1.2×10⁻¹⁵ m)³ = 2.29×10¹⁷ kg/m³`, i.e. 2.3×10¹⁴ times the density of water; and `R(Li-11) = 1.2 × 11^(1/3) = 2.67 fm` as the rule's prediction for the halo case. ## External links - [*University Physics Volume 3*](https://openstax.org/details/books/university-physics-volume-3) (OpenStax, 2016), Chapter 10 *Nuclear Physics* — the nuclear-size and binding-energy treatment behind this page - [*Energy and Human Ambitions on a Finite Planet*](https://open.umn.edu/opentextbooks/textbooks/energy-and-human-ambitions-on-a-finite-planet) (Murphy, 2021), Chapter 15 — the binding-energy table and the mass-defect arithmetic - The Wikipedia pair's *External links* section lists the evaluated nuclear-data libraries and chart-of-nuclides services from which measured masses and radii are drawn. <!-- MATTERSIM:BEGIN g33 — Matter & Energy Cluster microsim (framework build, specs/sims/Atomic_nucleus.json); do not hand-edit inside --> **Microsim — three.js (Wikitube framework):** *Atomic nucleus* <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/matter/Atomic_nucleus.html" data-title="Atomic nucleus"></div> *Built from `MICROSIM_GUIDE/specs/sims/Atomic_nucleus.json`; part of the [[PORTAL_Matter|Matter portal]] spine (section sims and See-also variants).* <!-- MATTERSIM:END --> ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Atomic_nucleus) : [Wikitube](https://en.wikitube.io/wiki/Atomic_nucleus) · pinned revision [1371660523](https://en.wikipedia.org/w/index.php?oldid=1371660523) · 2026-09-11 ## Previous hub tags Hubs: `Life_Physics`. Portals: [[PORTAL_Physics]]. --- *Matter & Energy Cluster child articles, wave 1 · 2026-09-11 · drafted · Physics row P48 · sim pending (matter/Atomic_nucleus).*