Introduction:

Models are modes of description that scientists often employed to convey trends in observed behaviors of a specific object or concept. Observed phenomena are employed to develop models which are then tested via experiments. This can then afterwards be employed to predict the future behavior of such object. 

Nuclear Model: General Requirement

In the similar manner, quantized mode for the atom became the base for describing chemical properties of element and validating their order in the periodic system; patterns of nuclear stability, result of nuclear reaction at spectroscopy of radiation emitted via nuclei have yield information for the nucleus. In the nucleus, there are two kinds of particles: proton and neutron packed closely altogether under the affect of two main forces:

• Electrostatic force
• Nuclear force

This is worthy to note that there are numerous suggestions or proposal on models however no singular nuclear model has been capable to describe all around the nuclear phenomena.

Some general Nuclear Properties:

This is noticed that the binding energy per nucleon is nearly constant for the stable nuclei and that; the radius is proportional to the cube of the mass number. This describes or justifies fairly uniform distribution of the charge and mass all through the volume of the nucleus; it as well supports the supposition of existence of a strong short range nuclear force. 

There is an indication too, that central mass number having Z or N-values 2, 8, 20, 28, 50 and 82 appear more stable. The other uniqueness of such numbers is that if the capturing of neutron or the energy needed to discharge neutron is plotted for various parameters, it will be noticed that maxima takes place at these similar neutron number just as maxima take place for electronic ionization energy of the element He, Ne, Ar, Kr, (that is, electron number 2,8,16,32). It illustrates that some type of regular substructures exist in the nucleus.

Quantitative Energy Level:

The constituent's substructure of neutron and proton with each kind of nuclear pairs off as far as possible. The γ-emission from any particle nucleus comprises discrete value. This can then be concluded that decay, radioactive nuclei, whether α, β and γ comprises a transition between discrete quantities energy level.

The Nuclear Potential Well:

Assume that a condition whenever a neutron of low kinetic energy approaches a nucleus. As the neutron is uncharged, it is not influenced by the columbic field of the nucleus; therefore more are without interaction until it is close adequate to empower the strong nuclear force. At this point, the neutron undergoes strong attraction to the nucleus and is absorbed. Whenever the neutron is absorbed, energy is discharged and emitted in the form of a gamma-quantum. The energy of gamma γ can be computed from the known masses of reactants and products nuclides Eγ = -931.5 (M_A+1 - M_A - M_n). The energy discharged is the (neutron) binding energy of the nucleus. The net energy of the nucleus has therefore reduced. This decrease is termed as potential well. The precise shape of the well is uncertain (that is, parabolic or square) and based on the mathematical form supposed for the interaction between the incoming molecule and the nucleus.

Other Requirements (Properties):

There are other properties like the difference omission for proper understanding of models. Currently, this comprise rotational energy and angular momentum which is better stated by principal quantum number (n) (that is associated to the net energy of the system) and azimuthal quantum number (l) (that is associated to the rotational movement of the nucleus). Coupling of spin and orbital angular momentum are as well significant to the understanding of nucleus model. Different models have been stated.

The Single-Particle Shell Model:

This is known that nucleus moves around freely in a nuclear potential well that is spherically symmetric and that the energy of the nucleus differs between potential and kinetic-like harmonic oscillation. For such condition, the solution of the Schrodinger equation says ε (nucleon) = (2U_o/m²)⁶ [2(n-1) + 1]

Here, U_o = potential at radius of 20, and m = mass of the nucleus. 

The given rules are valid in the potential well that forms the basis to the model.

a) L can encompass all positive integers the value starting with 0 and independent of n.

b) The energy of the l phase increases n.

c) The nucleus enters the level by the lowest total energy independent of whether n or l is the bigger.

d) There are independent sets of levels for proton and for neutron.

e) The Pauli's principle is valid that is, the system can't have two particles by all quantum numbers by the same.

f) The spin quantum number should be taken into account.

The Collective Nuclear Model:

The single particle model supposes that the mass and the charge of the nucleus are spherical. This is only true for nuclei that have distorted shapes. The most general supposition regarding the explanation of the nucleus shape is ellipsoidal that is, cross section of the nucleus is an ellipse Bohr and Mottelson recommended that the nucleus be regarded as a highly compressed liquid undergoing rotation and vibration. Two discrete collective motions can be visualized: 

• Can assume nucleus rotates around the y-axis and also the x-axis.
• The nucleus might oscillate between prolate to oblate form (that is, irrotation) and also vibrate.

Each and every model of such collective nucleus movement has its own quantized energy. Moreover, the movement might be coupled. The model lets computation of rotational and vibration levels.

For illustration, if ²³⁸U is excited above its ground state via interaction by high energy heavy ion (that is, coulomb excitation). Three possible kinds of excitation are known:

a) Nucleus excitation in which quantum number (J) is charged to increase the nucleus to a higher energy level.

b) Vibrational excitation in which the case 'J' is unchanged through the nucleus and is increased to a higher vibrational level characterized via a specific vibrational quantum number.

c) Rotational excitation, as well featured through a specific rotational quantum number. It exhibits experimentally that rotational levels are more closely spaced and therefore transition between rotational levels comprises lower energies than de-excitation from excised nuclear or vibrational state.

In case of even-even nuclei, the rotational energy can be frequently computed for the simple expression: 

εrot = [ε²/2I_rot(n₂(n_r-1)]

Here, I_rot is the moment of the inertia as n_r to rational quantum number. The validity of this quantum based on whatever the various modes of motion can be treated independently or not which they can for strongly deformed nuclei level ²³⁸U.

The Unified Model for Deforming Nuclei:

The collective model gives good explanation of even-even nuclei, however can't account for discrepancy between observed spin and the spin value expected from simple particles shell model. This unified model perception has been developed on supposition that a nucleon shell freely in a symmetrical. Potential well, a condition which is valid only for nuclei near closed shells. The angular movement of an odd-odd defined nucleus is due to both the rotational angular momentum of the deformed core and to the angular momentum of the odd nucleon.

As a result, the energy levels for such a nucleus are dissimilar, from those of the symmetric shell models.

Sir S.G. Nilsson computed odd nuclei: as a function of the nuclear deformation β. Each and every shell model level of angular momentum J split to J +1/2 levels (Nilsson).

The Nilsson level are fairly dissimilar in all characteristics from the shell model state and their prediction of energy, angular momentum, quantum number and other properties agreed better by experimental data for the deformed nuclei as compare to those of any other model.

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