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  Nuclear Models, Physics tutorial

Nuclear Models:

These are just meant to explain nuclear forces in nucleus of an atom. All that is known about nuclear force is that:

i. Short range of operation of order of ≈10cm

ii. Independent of charge i.e. exists equally between proton and neutron.

iii. Strong force which can overcome Coulomb force.

iv. It is repulsive force to certain extent to prevent collapse of nucleus.

Development of nuclear models is connected with two observations i.e. stability of nuclides with number of protons or neutrons equal to any one of magic number and relation between binding energy and mass number have been utilized as tests for validity of models. These are:

i. Liquid drop model

ii. Shell model or independent particle model

iii. Collective model.

Liquid Drop Model:

In liquid drop model, nuclei are considered to act like drops of incompressible liquid, i.e., like drops of very high density (density of order of 1014kg/m3). With this point of view and using concepts from classical physics (i.e. physics of continua), concepts like surface tension and surface energy, volume, and energy predictions are made about overall behaviour of nuclides. One of predictions, as mentioned above, is about relation between binding energy and mass number for nuclides.

Using liquid drop point of view, binding energy of nuclide would be consequential of five energies, volume energy (Ev), surface energy (Es), the energy due to asymmetry (i.e., deviation from a stable configuration), (Ea), the energy due to even-odd combination of nucleons in a nuclide, (Eδ) and coulomb energy (for protons), (Ec). With this, the total binding energy for a nuclide is

B.E = Ev + Es + Ea + Eδ + Ec

The relations for Ev, Es, Ea, Eδ, Ec are

Ev = CvA

Es = -CsA2/3

Ea = -(Ca[(A-Z) - Z]2)/A = -Ca(A - 2Z)2/A

Eδ = δ/2a= for even-even nuclides;

0 = for even-odd or odd-even nuclides;

 -δ/2a = for odd-odd nuclides.

Ec = -4CcZ(Z - 1)/A1/3

Where Cv ≈ 14MeV

Cs ≈ 13.1MeV

Ca ≈ 19.4MeV

δ ≈ 270MeV

Cc ≈ 14MeV

Mass of nuclide is provided by:

Mn = (A - Z)mn + Zmp - B.E/c2                                                                                               

This relation is referred to as semi-empirical mass relation. Such predictions by liquid drop model and other predictions like predictions about fission of nuclides are in agreement with observation.

Shell Model:

In shell model, nucleons are treated as individual particles existing within potential created by nucleons of nuclide. Therefore shell model is also referred to as independent particle model. This is like treatment of electrons of atoms in atomic physics. In shell model of nuclear physics though, potential is because of both electromagnetic potential and nuclear potential. Therefore, potential that nucleon finds itself in nuclide is

V(r) = Vn(r) = -V0(1 + e-(r-R)/a) for a neutron

V(r) = Vn(r) + Ve(r) = -v0(1+e-(r-R)/a) + Ve(r) for a proton

V(r) = Ze2/4πε0Re[1 + 1/2(1-(r/Re)2)], for r < Re;

Ze2/4πε0r for r≥Re

Where V0 = 57 ± 27(A-2Z)/A MeV

(+) for protons and (-) for neutrons R = 1.25A4/3F, a constant for a nuclide a = 0.65F, a constant

With this relation for potential and assumption that for nucleons there is strong spin orbit coupling, solving Schrodinger's equation for nucleons in nuclides predicts fact that for values of Z or (A-Z = 2,8,20,28,50,82,126, and 184 there would be closed shells, that is stable nuclides in agreement with what is seen experimentally.

Collective Model:

The term collective model is utilized to involve any model which handles only with collective behavior of nucleons. In view of this, even liquid drop model can be looked on as collective model. Term is utilized with any model which takes collective effects in account.

Nuclide can have rotational energy or vibrational energy. In both cases, energies will be integer multiples of the phonon hvλ. With this, in overall modeling of structure of nuclei, first making certain suppositions about nature of nuclei, Hamiltonian for the certain model is derived. Then Hamiltonian is solved and wave function for nuclide or nucleon of interest determined. Then forecasts with wave function are compared with experimental observation. From this, model is estimated depending on degree of agreement and disagreement. Experimental observation which played the significant role in development of collective nuclear model is that of photonuclear reaction, happening of giant resonances in photonuclear reactions. For collective behaviour of nucleons, nuclides are considered to comprise of two fluids: Proton fluids and neutron fluids. Proton and neutron liquids could experience rotational and vibrational motion at their surfaces. In presence of electromagnetic fields there could be density fluctuations of density of proton Ρp(r,t), and density of neutron Ρn(r,t) and resulting dipole, quadruple, etc., resonances. This is because of fact that electromagnetic fields (and photons) react only with protons.

Additionally to the two collective behaviors, structure of nuclide can be affected by individual motion of individual nucleons comprising it. Putting the three factors together, Hamiltonian for nucleus is:

H^ = H^surface effects + H^giant resonance + H^interaction

Where H^s, Hamiltonian because of surface effects

H^gr, Hamiltonian because of giant resonance

Hamiltonian for nucleus is

H^s = H^s + H^gr + H^int & H^s = H^vib + H^rot

Thus H^ = H^vib + H^rot + H^gr + H^int

For individual particle (nucleon), Hamiltonian would be sum of Hamiltonian of collective motion, Hamiltonian of individual particle and Hamiltonian that considers interaction between individual motion and collective motion. Therefore for particle,

H^ = H^part + H^coll + H^int

In collective models, thus individual particles move in deformed shell potential and nucleus as a whole act like incompressible fluid with motions (vibratory and rotational) being influenced and affected by motion of individual particles inside nuclei.

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