Magnetic moments of nucleons

Question

I was comparing my notes of the nuclear physics class (undergraduate level) on magnetic moments of nucleons with the Krane's explanation.

In my notes I wrote that there are two types of magnetic moments:

The first one is the orbital one. It's written as $ mu_l=g_l l mu_n$ where l is the orbital quantum number. I also wrote that $vec{mu_l}=g_lvec{L}$ so that this vector is parallel to $vec{L}$. 
The second one is the spin one. It's written as $mu_s=g_ssmu_n$ where s is the spin quantum number, s=1/2 for nucleons. Its vectorial form is $vec{mu_s}=g_svec{S}$ so that this vector is parallel to $vec{S}$

Then, the total magnetic moment is $vec{mu_j}=vec{mu_l}+vec{mu_s}$ where $vec{mu_j}=g_jvec{J}$. The next step on the notes is about finding the value of $g_j$. I wrote that $|vec{mu_j}|=|vec{mu_l}|cos{theta}+|vec{mu_s}|cos{varphi}$ where $varphi$ is the angle between $vec{S}$ and $vec{J}$ and $theta$ is the angle between $vec{L}$ and $vec{J}$. In the next step I substitute $|vec{mu_l}|$ with $g_lhbar (l(l+1))^{1/2}$, $|vec{mu_s}|$ with $g_shbar (s(s+1))^{1/2}$ and $|vec{mu_j}|$ with $g_jhbar (j(j+1))^{1/2}$.

So here's my problem: why is $|vec{mu_l}|$ different from $mu_l$? In fact the first one it's written like $g_l|vec{L}|$ and the second one as $mu_ng_ll$. The same happens with $|vec{mu_s}|$ and $mu_s$.

Also: In my notes I wrote that $vec{mu_j}$ isn't parallel to $vec{J}$ and it is, in fact, rotating about $vec{J}$. So why $vec{mu_j}=g_jvec{J}$? Shouldn't $vec{mu_j}$ and $vec{J}$ be parallel this way?

Thank you in advance.

drvrm · Answer

Also: In my notes I wrote that $vec{mu_j}$ isn't parallel to
$vec{J}$ and it is, in fact, rotating about $vec{J}$. So why
$vec{mu_j}=g_jvec{J}$? Shouldn't $vec{mu_j}$ and $vec{J}$ be
parallel this way?

The angular momentum operators L^2 and L(z) commute with  Hamiltonian and can be measured simultaneously giving the eigenvalues l(l+1) h_bar^2  and mh_bar ,
however the other components of L namely L(x) and L(y) do not commute along with **L(z)**so they can not be measured simultaneously...meaning thereby that direction of L remains indeterminate.
so one can not talk about the specific  direction of orbital  angular momentum  L vector .
So when we describe the magnetic  moment of a nucleous
mue(j) = mue(l) + mue(s)                      (1)
then
mue(l) = g(l). mue(N). sqrt (l(l+1)) ,
where mue(N) is magnetic moment of nucleon
For neutron  as it is uncharged mue(l) will be zero and for proton g(l)=1
so  for Proton
mue(lp)= mue(N). sqrt(l(l+1))  ....(2)
As nucleons are spin 1/2 particles  the QM values of intrinsic magnetic  moment can be written as
mue(s) = g(s).  mue(N) . sqrt(s(s+1)).....        (3)
So Total  magnetic  moment  component in the j direction
mue(j) = mue(l) cos (l, j) +  mue(s) .cos (s,j)
Those Cosine terms can be calculated in terms of l, s and j values.
Moreover the last    nucleon in the  extreme  single particle  model  ( in odd A nucleus) and its state is to be considered which  determines the magnetic  moment . For even even  nucleus the resultant spin is zero.
so classical  description is not possible.though i have seen vector  model drawing of coupling of angular  momentum but
i think  all its diagrams are not measurable. when  one imposes the external  magnetic  field  then the projections along z axis   are measured.
For  details see

Atomic and Nuclear Physics, Vol-II,S.N. Ghoshal,S. Chand & Co., New Delhi >India,Second Edition 1998

Magnetic moments of nucleons

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