Shape of the atomic nucleus
The shape of the atomic nucleus depends on the variety of factors related to the size and shape of its nucleon constituents and the nuclear force holding them together. The spatial extent of the prolate spheroid nucleon is determined by root mean squared charge radius of the proton, as determined mainly by electron and muon scattering experiments, as well as spectroscopic experiments. An important factor in the internal structure of the nucleus is the nucleon-nucleon potential, which ultimately governs the distance between individual nucleons, and the radial charge density of each nuclide. The charge density of some light nuclide indicates a lesser density of nucleonic matter in the center which may have implications for a nucleonic nuclear structure. A surprising non-spherical expectation for the shape of the nucleus originated in 1939 in the spectroscopic analysis of the quadrupole moments while the prolate spheroid shape of the nucleon arises from analysis of the intrinsic quadruple moment. The simple spherical approximation of nuclear size and shape provides at best a textbook introduction to nuclear size and shape. The unusual cosmic abundance of alpha nuclides has inspired geometric arrangements of alpha particles as a solution to nuclear shapes, although the atomic nucleus generally assumes a prolate spheroid shape. Nuclides can also be discus-shaped, triaxial or pear-shaped.
Origins of nuclear shape
The atomic nucleus is composed of protons and neutrons. In the Standard Model of particle physics, nucleons are in the group called hadrons, the smallest known particles in the universe to have measurable size and shape. Each is in turn composed of three quarks. The spatial extent and shape of nucleons ultimately involves quark interactions within and between nucleons. The quark itself does not have measurable size at the experimental limit set by the electron. The size, or RMS charge radius, of the proton has a 2018 CODATA recommended value of 0.8414 fm, although values may vary by a few percent according to the experimental method employed. Nuclide size ranges up to ≈ 6 fm. The largest stable nuclide, lead-208, has an RMS charge radius of 5.5012 fm, and the largest unstable nuclide americium-243 has an experimental RMS charge radius of 5.9048 fm. The main source of nuclear radius values derives from elastic scattering experiments, but nuclear radii data also come from experiments on spectroscopic isotope shifts, β decay by mirror nuclei, α decay, and neutron scattering. Although the radius values delimit the spatial extent of the nucleus, spectroscopic and scattering experiments dating back to 1935 in many cases indicate a deviation of the nuclear charge distribution or quadrupole moment consistent with non-spherical nuclear shapes for many nuclei.Simple spherical approximation
The atomic nucleus can be visualized as a compact bundle of nucleons represented as hard-packed spheres. This depiction of the nucleus only approximates the empirical evidence for the size and shape of the nucleus. The RMS charge radius of most stable nuclides has been experimentally determined. If the nucleus is assumed to be spherically symmetric, an approximate relationship between nuclear radius and mass number is given by above = 40, where is the predicted spherical nuclear radius, is the mass number, and = 1.2 ± 0.2 fm is an experimentally-determined constant. This radius-to-mass relationship has its origin in the liquid drop model as proposed by Gamow in 1930. The graph on the right shows experimental charge radius as a function of mass number as compared to the spherical approximation. For ≤ 40, the curvilinear spherical radius contrasts with measurement; for ≥ 40, there is better agreement when = 1.0 fm.Nucleon shape
The empirical knowledge of nucleon shape originates from the study of the transition from the proton ground state N to the first excited state ∆+. Multiple studies using a variety of models have led to an expectation of non-spherical shape. The proton's RMS charge radius of 0.8414 fm only defines the spatial extent of its charge distribution, i.e. the distance from its center of mass to its farthest point. Examination of the angular dependence of the charge distribution indicates that the proton is not a perfect sphere. Model-dependent analyses of the intrinsic quadrupole moment suggests that the ground-state nucleon shape conforms to a prolate spheroid shape.The intrinsic quadrupole moment is distinct from the spectroscopic quadrupole moment, as realized more than 50 years ago. The intrinsic quadrupole moment relates to a body-fixed coordinate system that rotates with the nucleon in contrast to the spectroscopically measured quadrupole moment. While the nucleon's spectroscopic quadrupole moment is zero due to angular moment selection rules related to spin, the non-zero intrinsic quadrupole is obtained by electromagnetic quadrupole transitions between the nucleon ground N and ∆ excited states. The proton and neutron have nearly the same mass, and may be regarded as one particle, the nucleon N, with two different charge states. The proton's N ground state and ∆+ excited state have different shapes. The transition between the states supports a prolate spheroid deformation for the ground state, and an oblate spheroid deformation for the excited state.
The prolate shaped ground state reflects quark-to-quark interactions arising from the Pauli exclusion principle. The Pauli exclusion principle, sometimes referred to as the Pauli exclusion force, is a fundamental rule in quantum mechanics stating that no two fermions can occupy the same quantum state simultaneously. In the ground state, the two down quarks of a ground-state neutron are in an isospin 1 state, and simultaneously in a spin 1 state in order that the spin-isospin wave function is symmetric. The consequence of this quantum mechanical constraint is that the likelihood of finding the neutron's two identical down quarks near one another is small, but increases with distance between them. Thus, within the spatial extent of the prolate spheroid neutron, the two down quarks will most likely be at opposite ends of the prolate spheroid. A similar quantum mechanical constraint applies to the two identical up quarks of the proton. Accordingly, the spin-spin interactions between identical fermions result in finding like-flavored quarks further apart. Conversely, when the spins of a pair of unlike fermions align, such as an up-/down-quark pair within a ground-state nucleon, the nuclear force draws the particles close to other each other without violating the Pauli exclusion principle. Within the ground state neutron, this results in a picture of the spin interactions in which the two down quarks qualitatively reside on either end of the prolate nucleon structure while simultaneously attracting to the up quark in the middle. Similar spin-spin interactions play out in the proton, considered identical to the neutron but existing in a different charge state.
Electron scattering techniques pioneered by Robert Hofstadter gave the first indication of a deeper structure for the nucleon. The technique is similar in principle to Rutherford's gold foil experiment in which alpha particles are directed at a thin gold foil, but Hofstadter's use of electrons, rather than alpha particles, enabled much higher resolution. The radial charge density of the neutron in particular was shown to have a complex internal structure consisting of a positive core and a negative skin, qualitatively consistent with the neutron's quark charge distribution shown above. Hofstadter received a Nobel prize for this work in 1961, several years before Murray Gell-Mann posited the quark model in 1965.