shell Model

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shell Model

Assumption of shell model

A Various assumption of the shell model are :-

Nucleon form closed subshell within the nucleus in a similar manner as the electrons do in case of atoms, Hence it is named 'shell model'
The nucleus which constitutes the nucleus is arranged in some type of shell structure. The shells get closed with a suitable number of protons and neutrons.
The extra-nuclear electrons revolve in the coulomb electrostatic field of the nucleus which is supposed to heavy and at a very large distance. The electron revolves only in specified permitted orbits.
Each nucleon moves independently inside the nucleus in a fixed orbit under the effect of central potential produced by the average interaction between remaining(A -1 ) nucleons on it.
Each nucleon is assumed to posses a spin angular momentum =$\sqrt{s(s+1)h} ,\text{ where s is the spin quantum number} =\frac 12$ and orbital quantu number =$\sqrt{l(l+1)h}$ where l is the orbital quantum number having values 0, 1, 2, 3, 4, .............. etc.

Magic Number

It is found that stable nuclei result when either number of protons or number of neutrons or both is equal to one of the numbers ;2, 8, 20, 50, 82, 126, etc these numbers are called magic number.

Similarly, the nuclei with 14, 28, 40 nucleons are slightly less stable but are more stable then other, and are called magic number.

Shell model

A nuclear shell model has been developed on the basis of above assumption.This model is produced by M.G. Mayer (1929) and is modified by HaxelJenson and Suess (1930).

The model is similar to the Bohr models of electrons in the extra-nuclear space. By analogy with the closed sub-shell and shell in the case of atoms, it is assumed that nucleons also form similar closed sub-shell and shell within the nucleus.The neutrons and protons constituting the nucleus are supposed to be arranged in some type of shell structure and these shells get closed with a suitable number of protons and neutrons. The extra-nuclear electrons are supposed to revolve in the coulomb electrostatic field of the relativity distant heavy nucleus in specified permitted orbits. from the compelling similarity of stability between the magic nuclei and the inert gasses,It is assumed that in the shell model that each nucleon has moves independently inside the nucleus in a fixed orbit under the influence of a central field of force or a central potential V(r) produced by the average interaction between all the remaining (A - 1) nucleons in it.

Harmonic Oscillator potential :-

It is assumed that the nucleons move in an average harmonic oscillator potential is given by

$$V=\frac 12 kr^2 =\frac 12 m\omega ^2 r^2\;\;\;\; \dots \dots (i)$$

where mm is the mass of the nucleon and $\omega$ the oscillator frequency. The Schrodinger's wave equation is

$$\left[ -\frac{h}{2m} \nabla ^2 +V \right]\psi =E\psi \;\;\;\;\;\; \dots \dots (ii) $$

solving above equation we find that various energy levels are given by

$$E_n=\left(N+\frac 32\right)h \omega$$

Where N=0, 1, 2, 3, .. . . . . . . and is known as oscillator quantum number. This equation shows that the energy states in the harmonic oscillator model are equally spaced. The wave function $\psi$ contains both the angular (orbital) and the radial part.

Each nucleon is spposed to have an orbital angular momentum | l | = $\sqrtl(l+1)h$ where l is the nuclear orbital quantum number havin values 0, 1, 2, 3, . . . . . . . . . etc.

Another quantum number similar to the principle quantum number of the electrostatic orbit characterizes the radial part of the nuclear wave function and is denoted by n. The value of n is 0, 1, 2, 3, . . . . . . etc.

It can be shown that the angular part of $\psi$ requires that the oscillator quantum number N is related to the orbital quantum number l and quantum number m by the relation.

$$N= 2(n -1) \= l =2n+1 -2. $$

The energy levels corresponding to each value of l for the nucleon are represented by a spectroscopic notation similar to the electron as shown

$$l\; = 0, \;1, \;2, \;3, \;4,\;5$$

$$\text{spectroscopi notation}=s,\;p,\;d,\;f,\;g,\;h$$

The nucleons are designated by making their n values followed by spectroscopic notation giving the 'l' value.

The energy levels predicted for a harmonic oscillator potential for N=0 to 4 together with the maximum number of nuclei have been calculated below.

N	n	l	state	Number of nuceons 2(2l+1)
0	1	0	1s	2
0	2 from the relation N=2n+l-2, for N=0 and n=2 is negative which is not premissible
1	1	1	1p	6
1	2 from the relation N=2n+l-2, for N=1 and n=2 is negative which is not premissible
2	1	2	1d	10
2	2	0	2s	2 total is 12
2	3 from the relation N=2n+l-2, for N=2 and n=3 is negative which is not premissible
3	1	3	1f	14
3	2	1	2p	6 total is 20
3	3 from the relation N=2n+l-2, for N=3 and n=3 is negative which is not premissible
4	1	4	1g	2
4	2	2	2d	10
4	3	0	3s	2 total is 30

The series of energy level calculated above do not agree with the observed sequence of magic number. The problem was solved by incorporating spin orbit interaction.

Lesson

Nuclear Models

Subject

Physics

Grade

Bachelor of Science

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