Exploring Nuclear Energy Levels: Shell Model

[roshan2004]

I am trying to explain nuclear energy levels in terms of Shell model and I hope that you guys would assist me through this-
1. The potential which exists inside the nucleus is assumed to be harmonic oscillator potential and it's value is obviously given by $$V(r)=\frac{1}{2}m\omega ^2r^2$$ However in my friend's note it is given as$$V(r)= -V_{o}+\frac{1}{2}m\omega ^2r^2$$ I don't know whether my friend's note is correct or not as I couldnot understand his equation at all.

 

[Meir Achuz]

The constant V_0 is correct in that simple model, but it doesn't affect the order of the energy levels.
It is there because the potential is ot necessarily zero at the origin.

 

[Orion1]

independent particle nuclear shell model...


That potential is part of the independent particle nuclear shell model.

Harmonic oscillator potential:
$$V(r) = \frac{1}{2} m \omega^2 r^2$$

According to Wikipedia, a 'more realistic' potential is the Woods–Saxon potential.

Woods–Saxon potential:
$$V(r) = -\frac{V_0}{1 + \exp(\frac{r - R}{a})}$$

Reference:
http://gcm.ac.in/downloads/elearning/Nuclear Shell Model.pdf"
http://en.wikipedia.org/wiki/Nuclear_shell_model#Deforming_the_potential"
http://en.wikipedia.org/wiki/Woods_Saxon_potential"
http://hyperphysics.phy-astr.gsu.edu/hbase/nuclear/shell.html#c3"

 

[roshan2004]

So, which one will be better for me to use, the first one or the second one?

 

[Meir Achuz]

The HO potential has simple wave functions and energy levels, helpful in identifying shell model states. The W-S potential has to be treated numerically. You do have to add a spin-orbit term (L.S)
to the HO to get the correct energy levels and states.

 

[roshan2004]

Now, I am trying to find the nuclear energy levels without spin orbit coupling which goes like this-(correct me if I have made some mistakes here-)
The Schrodinger's equation for the Quantum harmonic oscillator is given by-
$$-\frac{\hbar^2}{2m}\triangledown ^2\psi (r)+\frac{1}{2}m\omega ^2r^2\psi(r)=E\psi(r)$$
Solving this equation we have the quantum energy levels of the harmonic oscillator given by
$$E_{n}=(n+\frac{3}{2})\hbar\omega$$
On solving the radial and angular parts of this equation we get the quantum numbers $$n_{r}$$ and l which are related to n as $$n=2n_{r}+l-2$$ where $$n_{r}=1,2,3...$$ and l=0,1,2... are known as s,p,d,f,g... (orbitals of the nucleon)
The occupancy of the nucleons in nucleus is given by 2(2l+1) and I calculated different states and occupancy as
for n=0 I got 1s state with occupancy 2
for n=1 I got states 1s and 1p with occupancy 6
for n=2 I got states 1d and 2s with occupancy 12
for n=3 I got states 1f and 2p with occupancy 20
for n=4 I got states 1g,2d and 3s with occupancy 30
Now the problem is I just couldnot draw these energy levels like the one we have in the case of energy levels of nucleons after spin orbit coupling

 

[Meir Achuz]

The L.S coupling makes the J=L+S state lie lower than the J=L-S state. This changes the numbers because the ten 2d states split into 6 and then 4. By n= 2 (or maybe n=3), the 3d_5/2 lies lower than the 2p_1/2.
The common counting of states lists them so that what you call
1s and 1p for your n=1 are called 2s and 2p, and so on.

 

[Orion1]

nuclear shell model quantum energy level...


The equation that I derived including spin orbit coupling is...

Nuclear spin quantum number:
$$s = \frac{1}{2}$$

Nuclear shell model quantum energy level:
$$\boxed{E_{nljs} = \hbar \omega \left[ \left( (2n_r + l - 2) + \frac{3}{2} \right) + C_1 \cdot l(l + 1) + \frac{C_2}{2} \left( j(j + 1) - l(l + 1) - s \left( s + 1 \right) \right) \right]}$$

Nuclear shell model constants:
$$C_1 = -0.0225$$ - spherical harmonic constant
$$C_2 = -0.1$$ - spin orbit coupling constant

Reference:
http://gcm.ac.in/downloads/elearning/Nuclear Shell Model.pdf"
http://en.wikipedia.org/wiki/Principal_quantum_number"
http://en.wikipedia.org/wiki/Angular_momentum#Relation_to_spherical_harmonics"
http://en.wikipedia.org/wiki/Total_angular_momentum"
http://en.wikipedia.org/wiki/Spin_quantum_number"
http://en.wikipedia.org/wiki/Spin–orbit_interaction#Evaluating_the_energy_shift"

 

[roshan2004]

Thanks,but the suggestions given by you guys is regarding spin orbit coupling. I am trying to draw only the energy levels without spin orbit coupling from the states and occupancy I have calculated.

 

[Meir Achuz]

Then your post #4 is it, but it won't give the right magic numbers.

 

[Orion1]

Thanks,but the suggestions given by you guys is regarding spin orbit coupling. I am trying to draw only the energy levels without spin orbit coupling from the states and occupancy I have calculated.

Even without spin orbit coupling there is a spherical harmonic constant resulting from degeneracy of states with different orbital angular_momentum quantum numbers.

Principal quantum number:
$$N = (2n_r + l - 2)$$

$$n_r$$ - radial quantum number
$$l$$ - orbital angular_momentum quantum number

Nuclear shell model quantum energy level without spin orbit coupling:
$$\boxed{E_{nl} = \hbar \omega \left[ \left( (2n_r + l - 2) + \frac{3}{2} \right) + C_1 \cdot l(l + 1) \right]}$$

$$C_1 = -0.0225$$ - spherical harmonic constant

Integration via substitution:
$$N_N = (N + 1)(N + 2) = ((2n_r + l - 2) + 1)((2n_r + l - 2) + 2) = (2n_r + l)(2n_r + l - 1)$$

Number of occupied quantum states:
$$\boxed{N_N = (2n_r + l)(2n_r + l - 1)}$$

Reference:
http://gcm.ac.in/downloads/elearning/Nuclear Shell Model.pdf"