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Problem with shell model and magnetic moment of Lithium6 
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#1
Jun2814, 02:45 AM

P: 70

I have a problem with the calculus of magnetic moment of Li6.
The configuration of protons is [itex]1p_{3/2}[/itex], and the neutrons' one is the same. I have to add the magnetic moment of uncoupled proton and uncoupled neutron. I use the following formula for [itex]J=l+\frac{1}{2}[/itex] (J is the particle spin): [itex]\frac{\mu}{\mu_N}=g_lJ+\frac{g_sg_l}{2}[/itex] For the proton I have: [itex]g_l=1; g_s=5.58 \rightarrow \frac{\mu}{\mu_N}=J+2.29=3.79[/itex] For the neutron I have: [itex]g_l=0; g_s=3.82 \rightarrow \frac{\mu}{\mu_N}=1.91[/itex] So the total [itex]\frac{\mu}{\mu_N}=3.791.91=1.88[/itex], exactly 1 more than the correct value, 0.88! What's wrong? 


#2
Jun2914, 11:14 AM

Physics
Sci Advisor
PF Gold
P: 6,166

Your value for the proton's magnetic moment is off by 1: it should be 2.79, not 3.79. Not sure where you are getting the values you are using to calculate it.



#3
Jun2914, 11:51 AM

P: 70

In this case, the proton's J is 3/2 and, if you insert this value in the formula, you obtain 3.79. 


#4
Jun2914, 12:16 PM

Sci Advisor
Thanks
P: 4,160

Problem with shell model and magnetic moment of Lithium6
Li6 is an oddodd nucleus, and therefore the magnetic moments predicted by the shell model are not in complete agreement with experiment.
Quoting from Preston, "Physics of the Nucleus", p323: "Turning to oddodd nuclides, the shell model would suggest simply adding the magnetic moments due to proton and neutron configurations, ignoring any interaction between the unfilled neutron and proton shells, except perhaps in the light nuclei in which neutrons and protons are filling the same shells and ispin is a good quantum number. It may be argued that, in this latter case, the neutrons and protons have precisely the same spatial motion and orientation but gfactors of opposite sign, and therefore the corrections to their freeparticle gvalues are roughly equal and opposite. Hence, despite the occurrence of interconfiguration mixing and quenching the freenucleon gfactors can be used, and μ is just the sum of the neutron and proton moments of the extreme singleparticle model. For nuclides in which neutrons and protons are filling different shells, it would seem appropriate to take the values of g_{p} and g_{n} from neighboring odd nuclides, thus allowing for interconfiguration mixing. This works quite well, and whenever the value g_{emp} obtained from empirical g's differs from g_{sp} obtained from freenucleon g's, the observed value is always much nearer g_{emp}. Some cases are shown in Table 121, where both μ_{emp} and μ_{sp} are calculated from the following formula, the only difference being the gvalues used: [tex]\mu = \frac{1}{2}\left[(g_p + g_n) + (g_p  g_n)\frac{j_p(j_p + 1)  j_n(j_n + 1)}{J + 1}\right][/tex] The table entry for Li6 has μ_{sp} = 0.6, μ_{emp} = 0.4 and μ_{obs} = 0.8. The conclusion which can be derived from our discussion is that, for the nearly spherical nuclei, which we have mainly considered, magneticmoment values are consistent with the shell model, but it is essential to include interconfiguration mixing in the ground state." (BTW, I think it would have been appropriate for you to mention that you were simultaneously posting this same question to both PF and stackexchange!) 


#5
Jun2914, 03:17 PM

Physics
Sci Advisor
PF Gold
P: 6,166




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