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--- science:phd-notes:2025-03-10-gl-gs [2025/03/10 14:28] – Initial commit jon-dokuwiki
+++ science:phd-notes:2025-03-10-gl-gs [2026/09/02 16:35] (current) – Expand on gl gs info jon-dokuwiki
@@ Line 4: / Line 4: @@
 $$
-\hat{M1} = g_l \hat{L} + g_s \hat{S}.
+\hat{M1} = g_L \hat{L} + g_S \hat{S}.
 $$
@@ Line 10: / Line 10: @@
 $$
-\mu = \frac{q}{2m}L
+\mu = \frac{q}{2m}L.
 $$
-(continue with completing this text and look at what happens to the M1 GSF when the orbital g-factors change, calculate some magnetic moments for different spin g-factors and see what makes most sense!)
+To compensate for various shortcomings of shell model calculations, like truncations and inert cores, it is possible to set the orbital g-factor for neutron to a non-zero number. This is is done in some cases for the sdpfsdg-mu interaction (like here where it is set to $-0.1$: [[https://doi.org/10.1103/csx6-6g5k|https://doi.org/10.1103/csx6-6g5k]]). More commonly, the spin g-factors for protons and neutrons are often "quenched" in shell model calculations, in which case the free $g_s$ values of $5.585$ and $-3.826$ for protons and neutrons respectively, are multiplied with a factor that is typically between $0.7$ and $1$ depending on the interaction. See [[https://git.joncloud.no/GaffaSnobb/nuclear-shell-model-interactions|the nuclear shell model interaction overview]] for details.
-TBC
+What we can do is to artificially set the g-factors to zero to see what effect is has on the M1 strength function.
+{{ :science:phd-notes:gsf_m1_gl_gs_comparison.png?1200 |}}
+In dark red and in orange we have "only $g_l$" and "only $g_s$" where I have set $g_s = 0$ and $g_l = 0$ respectively. In the low energy region of approx $0-2$ MeV we see that the $L$ and $S$ term play almost exactly the same role. As the gamma energy increases past $2$ MeV however, we see that the "only $g_s$" part is increasingly doing the full duty of producing the M1 strength.
+$L$ and $S$ are vector quantities which means that calculating the $L$ and $S$ parts separately and them adding them together does not work. That's because to get the GSF we have to calculate
+$$
+B(M1) = \frac{|(f|\hat{M1}|i)|^2}{2J_i + 1}
+$$
+and expanding that square we get
+$$
+|(f|\hat{M1}|i)|^2 = (|(f|g_L\hat{L}|i) + (f|g_S\hat{S}|i)|)^2
+$$
+and dubbing the first term $M_L$ and the second term $M_S$ we get
+$$
+= M_L^2 + M_S^2 + 2M_L M_S
+$$
+where the cross-term $2M_L M_S$ is the important part. Naively summing $M_L^2 + M_S^2$ is exactly the green line in the figure. From 0 to 5.2 MeV the naive sum underestimates the properly summed M1 strength meaning that there is constructive interference between the $L$ and $S$ terms that is not included, aka $2M_L M_S$ is positive. While beyond $5.2$ MeV it is the opposite.
+{{ :science:phd-notes:interference-angle.png?1200 |}}
+In the figure above I have calculated the average interference angle between $L$ and $S$. If you visualise a simple xy-plane with two vectors in it, you can imagine that if the vectors are at 90 degrees their $2M_L M_S$ is equal to zero. If the angle is less than 90 then the cross-term is positive aka. constructive interference and vice versa for angles greater than 90.