# nuclear structure

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## nuclear structure:

see nucleus**nucleus,**

in physics, the extremely dense central core of an atom.

**The Nature of the Nucleus**

**Composition**

Atomic nuclei are composed of two types of particles, protons and neutrons, which are collectively known as nucleons.

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## Nuclear structure

At the center of every atom lies a small, dense nucleus, which carries more than 99.97% of the atomic mass in less than 10^{-12} of its volume. The nucleus is a tightly bound system of protons and neutrons which is held together by strong forces that are not normally perceptible in nature because of their extremely short range. The small size, strong forces, and many particles in the nucleus result in a highly complex and unique quantal system that at present defies exact analysis. The study of the nucleus and the forces that hold it together constitute the field of nuclear structure physics. *See* Atomic structure and spectra, Neutron, Proton, Quantum mechanics, Strong nuclear interactions

The protons of the nucleus, being positively charged, generate a spherically symmetric electric field in which the atomic electrons orbit. The cloud of negatively charged atomic electrons normally balances the positive nuclear charge, making the atom electrically neutral. The atomic number of protons is usually denoted by *Z* and the number of neutrons, which are electrically neutral, by *N*. The total number of protons and neutrons (or nucleons) is the mass number *A* = *Z* + *N*. Isotopes have the same atomic number, *Z*, and hence are forms of the same chemical element, having the same chemical properties, but they differ in neutron number; isotones have a common number of neutrons, *N*, and isobars have the same mass number, *A*. *See* Isobar (nuclear physics), Isotone, Isotope

Nuclei have masses less than the sum of the constituents, the missing mass Δ*M* being accounted for by the binding energy Δ*Mc*^{2} (where *c* is the speed of light), which holds the nuclear system together. The characteristic energy scale is in megaelectronvolts (1 MeV = 1.6 × 10^{-13} joule). The internuclear forces generate an attractive potential field which holds the nucleus together and in which the nucleons orbit in highly correlated patterns. The volume of nuclei increases approximately linearly with mass number *A*, and the radius is roughly *R* = 1.2 × 10^{-15} · *A*^{1/3} m. *See* Nuclear binding energy

#### Size, shape, and density distributions

A variety of sophisticated techniques have been developed for precise estimates of the nuclear charge distribution, including electron scattering, the study of muonic atoms, and the laser spectroscopy of hyperfine atomic structure. An overall picture of the nuclear charge distributions emerges. The nuclear charge density saturates in the interior and has a roughly constant value in all but the lightest nuclei. The nucleus has a diffuse skin which is of nearly constant thickness.

Many nuclei are found to have nonspherical shapes. Unlike the atom, which has a spherically symmetric Coulomb field generated by the nucleus, the nuclear field is composed of a complicated superposition of short-range interactions between nucleons, and the most stable nuclear shape is the one that minimizes the energy of the system. In general, it is not spherical, and the nuclear shape is most simply described by a multipole power series, the most important term of which is the nuclear quadrupole moment. A positive quadrupole moment reflects the elongation of nuclei into a prolate or football-like shape, while a negative value reflects an oblate shape like that of Earth. *See* Nuclear moments

An accurate determination of nuclear matter distributions, that is, the distribution of both protons and neutrons in nuclei, is harder to precisely ascertain.

#### Nuclear masses and binding energies

The variation of average binding energy with mass number is approximated by the Bethe-Weizsacker mass formula, which is noteworthy for its simplicity in reproducing the overall binding energy systematics. The formula is developed by modeling the nucleus on a liquid drop. By analogy with a drop of liquid, there is an attractive volume term, which depends on the number of particles; a repulsive surface-tension term; and a term due to the mutual Coulomb repulsion of protons, which is responsible for the decrease in binding energy for heavy nuclei. The model is spectacularly successful in reproducing the overall trends in nuclear binding energies, masses, and the energetics of nuclear fission, and in predicting the limits of stability where neutrons and protons become unbound. As in the case of predicting a mean nuclear shape, a comparison of the prediction of the Bethe-Weizsacker mass formula to measured masses shows periodic fluctuations with both *N* and *Z*, which are due to the quantum shell effects. *See* Nuclear fission

#### Nuclear excited states

The small nuclear size and tightly bound nature impose very restrictive constraints on the orbits that protons and neutrons can undergo inside the system. Thus, each nucleus has a series of quantum states that particles can occupy. The Pauli principle requires that each particle have a unique set of quantum labels. Each nuclear state can then be filled with four particles: protons with internal angular momentum “up” and “down,” and likewise two neutrons. *See* Angular momentum, Energy level (quantum mechanics), Exclusion principle, Parity (quantum mechanics), Quantum numbers, Quarks, Spin (quantum mechanics)

A nucleus is most stable when all of its nucleons occupy the lowest possible states without violating this occupancy rule. This is called the nuclear ground state. During nuclear collisions the protons and neutrons can be excited from their most bound states and promoted to higher-lying unoccupied states. The process is usually very short-lived and the particles deexcite to their most stable configuration on a time scale of the order of 10^{-12} s. The energy is usually released in the form of gamma rays of well-defined energy corresponding to the difference in energy of the initial and final nuclear states. Occasionally, gamma decay is not favored because of angular momentum selection rules, and long-lived nuclear isomers result. *See* Gamma rays, Nuclear isomerism, Nuclear spectra, Selection rules (physics)

#### Nuclear models

The detailed categorization of the excitation of protons and neutrons allows a mapping of the excited states of each nucleus and determination of its quantum numbers. These data are the essential information required for development of detailed models that can describe the motion of nucleons inside nuclei. Unlike atomic molecules, where rotational, vibrational, and single-particle degrees of freedom involve different time scales and energies, the nucleus is highly complex, with rotation, vibration, and single-particle degrees of freedom being excited at similar energies and often strongly mixed. *See* Molecular structure and spectra

The measurement of static electric and magnetic moments of nuclear states and of dynamic transition moments has provided a great deal of information. Electric moments have revealed a variety of enhanced collective modes of excitation, including elongated, flattened, and pear-shaped nuclear shapes. Magnetic moments have provided detailed information on the differences between excitations involving neutrons (negative moments) and protons (positive moments).

For atoms, the solutions of the Schrödinger equation with a Coulomb potential lead to a reasonable prediction of the energies of quantized atomic states, as well as their spins, parities, and moments. Attempts to make the same progress for nuclei, using a variety of spherically symmetric geometric potentials of nuclear dimensions, failed to reproduce known properties until it was realized by M. G. Meyer and independently by J. H. Jensen in 1949 that an additional spin-orbit potential was required to reproduce the known sequence of nuclear states. A potential of this form binds states having the internal spin of the nucleons parallel to its orbital angular momentum more tightly than when they are antiparallel. The ensuing sequence of quantum shell gaps and subgaps was then correctly reproduced. The shell model has evolved rapidly, and its domain of applicability has widened from the limited regions of sphericity near doubly magic nuclei to encompass most light nuclei with *A* < 60 as well as enlarged regions around shell closures.

As valence particles are added to a closed core of nucleons, the mutual residual interactions can act coherently and polarize the nuclear system away from sphericity. The polarization effects are strongest when both valence protons and neutrons are involved. The deformed nuclear potential can then undergo collective rotation, which generally involves less energy than other degrees of freedom and thus dominates the spectrum of strongly deformed nuclei.

Nuclei undergo collective vibrations about both spherical and deformed shapes. The degree of softness of these vibrations is characterized by the excitation energy required to populate states. The distinguishing feature of vibrational excited states is that they are grouped in nearly degenerate angular momentum multiplets, each group being separated by a characteristic phonon energy.

It has been a goal of nuclear structure studies to develop models that incorporate all of the features described above in order to produce a unified nuclear picture. The development of generalized nuclear models has relevance to other fields of physics. There are many isotopes that will never be accessible in the laboratory but may exist in stars or may have existed earlier in cosmological time. The evolution of generalized models greatly increases the power to predict nuclear behavior and provides information that is required for cosmological calculations.

#### Nuclei at high excitation energies

As nuclei are excited to ever higher excitation energies, it is anticipated that shell effects will be slowly replaced by statistical, or chaotic, behavior. The number of states per megaelectronvolt of excitation energy with each spin and parity rise exponentially with increasing excitation energy until the levels become sufficiently close that they overlap and mix strongly and so become a continuum of states.

Toward the top of this energy regime, new modes of nuclear collectivity become accessible. Giant resonance states can be excited that involve compression and oscillation of the nuclear medium with vibrations of the protons and neutrons in phase (isoscalar) or beating against each other (isovector). The excitation and decay of these giant resonances can provide information about shapes of nuclei at high excitation and about the compressibility of nuclear matter. Results from giant resonance studies indicate that the shell effect persists high into the region previously thought to be statistical. *See* Giant nuclear resonances

The semiclassical statistical and hydrodynamic behavior of hot nuclear matter and its experimental, theoretical, and astrophysical aspects are of great interest at the highest nuclear energies. The influence of compression and heat on the density of nuclear matter is being investigated in order to measure a nuclear equation of state in analogy with the properties of a classical fluid. It has been suggested that the nuclear matter may undergo phase changes under compression, with high-density condensates possibly providing a new metastable state. At the highest densities and temperatures, the nucleons themselves are forced to overlap and merge, leading to a plasma of quarks and gluons that are the nucleonic constituents. *See* Quark-gluon plasma, Relativistic heavy-ion collisions