# Nuclear shell model, spin and parity predictions

• Incand
In summary, the conversation discusses the use of the shell model to predict the spin and parity of nuclei with odd numbers of neutrons. The predicted values differ from the ones in the tables and it is suggested that this may be due to differences in the placement of spin-orbit split levels as seen by neutrons and protons. The conversation also mentions that the shell model is an approximation and there may be exceptions to its rules. Further reading is suggested to better understand the discrepancies in the predicted and observed values.

## Homework Statement

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.

## Homework 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

## 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?

Incand said:
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?

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]http://www2.ph.ed.ac.uk/~dwatts1/np3_Lecture8.pdf
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.

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.

Incand said:
Is this is a general rule that they should fill the lowest jjj state first or an exception? This doesn't seem to always be the case.

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:
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.

drvrm