# Nuclear shell model, spin and parity predictions

1. May 1, 2016

### Incand

1. The problem statement, all variables and given/known data
Not exactly a homework problem but I tried to predict the spin and parity of (the ground states of)
$^{89}Sr$,$^{97}Zr$ and $^{137}Ba$
using the shell model and my results seem to differ from the tables.

2. Relevant equations
Parity
$\pi = (-1)^l$

Figure 4 seems to have the same values as the diagram in my book (Krane)
http://courses.chem.indiana.edu/c460/documents/SEC9ShellModel.pdf

I also found another diagram where some of the energy levels had switched place:
https://www.eng.fsu.edu/~dommelen/quantum/style_a/ntsm.html

3. The attempt at a solution
$^{89}Sr$ have $51$ neutrons. Looking at diagram the unpaired neutron is in the $1g_{7/2}$ state. The parity is then $(-1)^4=1$ so we have $7/2^+$.

$^{97}Zr$ have $57$ neutrons. The unpaired neutron is according to the diagram in an $1g_{7/2}$ state so we have $7/2^+$.

$^{137}Ba$ have $81$ neutrons. The unpaired neutron is then in a $1h_{11/2}$ state. We then have $11/2^-$.

If however I look up the values in table I have:
$^{89}Sr \; \; 5/2^+$
$^{97}Zr \; \; 1/2^+$
$^{137}Ba \; \;3/2^+$.

I'm wondering why I get different values? I suspect I have the wrong diagram, so far every prediction I did with odd number of protons have worked out but not all the ones for neutrons. Is there a difference?

2. May 1, 2016

### drvrm

i think there are a bit different placements of spin orbit splitted levels as seen by neutron and as by protons-
further the rules are general and there are exceptions- moreover the shell model is an approximation and better ones do exist....
i have seen one assignment which you may also look up....ref is given below.

with that i get your odd neutron in N= 51 placed in 2d5/2 a 5/2+ state

N=57 goes to 1/2+ as it opts for 3s1/2 state - the particle tries to go to lowest j value state-leaving option of higher j-state (it has been observed)

N=81 goes to 2d3/2 state 3/2+

ref.
PDF]Nuclear and Particle Physics
www2.ph.ed.ac.uk/~dwatts1/np3_Lecture8.pdf

Shell Model predictions. & single-particle ... J = total angular momentum (nuclear spin) π = parity ... Determine spin and parity assignments for the ground states ... Fornuclei with an odd neutron or proton shell model predicts two possible.

3. May 1, 2016

### Incand

I think I have misunderstood the shell model. I thought the electrons filled up the lower energy states first in order. I get both the examples and exercise in your link correct. But I don't understand what you did with $N=57$ and $N=81$. Is this is a general rule that they should fill the lowest $j$ state first or an exception? This doesn't seem to always be the case.

I thought they filled the lower energy levels first so the the $N=57$ nuclei would fill the $2d5/2$ level with $6$ neutrons and have one left for the $1g7/2$ level.

4. May 1, 2016

### drvrm

Let me correct myself the N=57 (101Ru) state will be 5/2 rather than 3/2 which i wrote.

i do not think its a general rule but by experimental observation of nuclear magnetic moments such behavior is seen.'

if you analyze the magnetic moments one gets a pair of Schmidt's lines for magnetic moment variation with z or N number and most of the nucleons finds a place between those lines..

the pairing energy of the nucleons also plays the role that higher j value states get filled up first and the extra neutron/proton goes to lower j-value state.

e.g. at N=33 for 61Ni the spin is found to be 3/2 and not 5/2 which should be on the basis of diagram

for N=59 (105Pd) observed spin is 5/2 and for N=61 (109Cd) also it is 5/2.
though the higher j value state 1g7/2 is first being filled in pairs.

in certain regions the deformation of the nuclear shell becomes significant and residual inter nucleon interaction plays a role.
so the shell model (as the simplest) can not present the full explanation of experimental values

Last edited: May 1, 2016
5. May 1, 2016

### Incand

Thanks for explaining! It was quite surprising to see that $101Ru$ have different spin from $97Zr$. I seem to have lot more reading to do before I understand this. It seems what I've been using is only what my book calls the "extreme independent particle" model where only the unpaired nucleon is considered. Apparently I can get better results considering the entire "incomplete" shell.