# Relationship between nuclear spin and nuclear magnetic moment?

We know that nuclear magnetic moment can be expressed in terms of the expected value for nuclear spin as:

$$\langle\mu\rangle =[g_lj+(g_s-g_l)\langle s_z\rangle]\frac{\mu_N}{\hbar}$$

(Cf. Krane), where $\vec{j}$ is the total angular momentum, $\vec{l}+\vec{s}$.

How does the expected $\langle s_z\rangle$ value relate to the $\vec{j}$-component of spin, $\langle s_j\rangle$? Krane mentions that only that value is needed, given that it remains constant.

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It would be helpful if you give more more details about your source (so far define just as "Krane"). I'm puzzled by "$\vec{j}$-component of spin $\langle s_j \rangle$" -- what do you mean? –  Slaviks Mar 12 '12 at 20:15
The source is Kenneth Krane's book 'Introductory Nuclear Physics'. $\langle s_j \rangle$ is the expected value for the $\vec{j}$-component of the spin vector, $\vec{s}$. We assume a fixed $z$ axis around which $\vec{j}$ -the sum of angular and spin momentum- rotates. In the book, it is said that if we want to measure $\langle s_z \rangle$, it suffices to know $\langle s_j \rangle$. Why? –  Xzoechord Mar 12 '12 at 20:40
I have not access to the book now, but I remember that In Krane's book the terminology is a bit sloppy. He calls spin to total angular momentum, can this be the source of your doubt? –  DaniH Mar 12 '12 at 20:47
I don't think so, total angular momentum is clearly defined as the sum of angular momentum and spin. –  Xzoechord Mar 12 '12 at 20:52
OK, I found the book. It seems that he´s applying the Wigner-Eckart theorem although it does not say it explicitly. Also, if I remember correctly there was an error in one of the formulas 5.9 although I do not remember in which one. I will try to elaborate later... –  DaniH Mar 13 '12 at 20:36

From the magnetic moment$$\bf{\mu}=\bf{\mu_L}+ \bf{\mu_S}=(g_l \bf{L}+g_s\bf{S})\frac{\mu_N}{\hbar}\qquad (1)$$ take the scalar product with $\bf{J}$ $$\bf{\mu_J} \bf{J}=(1/2(g_l +g_s)\bf{J^2}+1/2(g_l -g_s)(\bf{L^2-S^2))}\frac{\mu_N}{\hbar} \qquad (2)$$ so with commutator relation $$\mu=(1/2(g_l +g_s)j+1/2(g_l -g_s)\frac{(l-s)(l+s+1)}{j+1}) \mu_N \qquad (3)$$
since s=1/2 and j= l$\pm$1/2 you end with two possible values for $\mu$ $$\mu=(jg_l-1/2(g_l -g_s)) \mu_N \quad\text{for }j= l+1/2\\ \mu=(jg_l+(g_l -g_s)\frac{j}{2j+1}) \mu_N \quad\text{for }j= l-1/2 \qquad (4)$$
this is your $<s>$ projection in j direction.
For s $\neq$ 1/2 equation (3) still holds. But for $s>l$ the factor must be changed to the general form $\frac{l(l+1)- s (s+1)}{j+1}$