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Welcome to The Tech Basement!
The Tech Basement is my personal knowledge base! It is organised as a wiki, but I am also using it like a blog. The subjects are programming, maths and physics, which are subjects highly related to my the PhD I'm taking in computational nuclear physics.
Check out the latest entries in The Tech Basement below!
Porter-Thomas fluctuations
Let's talk about Porter-Thomas (PT) fluctuations! To do that, we need to start talking about:
Which values to choose for gl and gs?
The gyromagnetic ratios determine how strongly the nuclear magnetic dipole moment interacts with the electromagnetic field, which directly affects the M1 transition strength. They show up in the $M1$ operator as
$$ \hat{M1} = g_l \hat{L} + g_s \hat{S}. $$
Protons and neutrons have different gyromagnetic ratios. The orbital motion of a charged particle produces a magnetic moment. Since only protons have charge, they alone contribute to the orbital part of the nuclear magnetic moment. The magnetic moment of a moving charge is given by
$$ \mu = \frac{q}{2m}L $$
(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!) TBC
Orbital contributions to the LEE
Let's get to the crux of the matter. We can analyse OBTDs all day, but what we're really wondering about is which orbitals are contributing to the enhancement in the low-energy region of the gamma strength function. Recall that in shell model calculations, the reduced transition strength is calculated by
$$ B(\sigma \lambda; \xi_i j_i \rightarrow \xi_f j_f) = \frac{1}{2 j_i + 1} \mid ( \xi_f j_f \mid \mid M_{\sigma \lambda} \mid \mid \xi_i j_i ) \mid^2, \qquad(0) $$
$$ \langle \Psi_f | \hat{O}_{\lambda \mu} | \Psi_i \rangle = \sum_{\alpha \beta} \langle \alpha | \hat{o}_{\lambda \mu} | \beta \rangle \langle \Psi_f | \hat{c}^\dagger_\alpha \hat{c}_\beta | \Psi_i \rangle. \qquad(1) $$
I so happen to be in possession of all the OBTDs and reduced matrix elements needed to re-calculate any transition of my desire. Now, what if I were to simply skip certain orbitals ($\alpha$ and $\beta$) in eq. (1)? I could for example decide to skip any mention of $0f7/2 \rightarrow 0f7/2$, or any other single-particle transitions for that matter. Re-calculating all of the transition strengths with the modified OBTD files consequently gives me the possibility to re-calculate the $M1$ GSF with the modified values. We can then directly see how the LEE is affected by the modification! That's what I call a home-run! schmack
[SOLVED] Possible bug in the OBTD calculations
So there is a possible bug that I am trying to wrap my head around. In the OBTD log files from KSHELL, the OBTDs are listed in blocks where each block represents an initial and a final state aka. one specific transition. For example
w.f. J1= 0/2( 3) J2= 2/2( 1) B(L;=>), B(L ;<=) 0.00622 0.00207 <||L||> 3 1 -0.13590 0.11892 i j OBTD <i||L||j> OBTD*<||> 1 1 0.00244 5.79655 0.01416 1 2 -0.00012 1.54919 -0.00018 1 9 0.00000 0.00000 0.00000 1 10 0.00000 0.00000 0.00000 ...
This block represents the OBTDs for the transition from the 3rd 0- state to the 1st 1- state (the parity is not listed in the block, but I know it is negative). Now, there is no guarantee that the 3rd 0- state has higher energy than the 1st 1- state. It might be so, but it is not guaranteed. If the initial state however has lower excitation energy than the final state then the OBTD indices in the blocks ($i, j$) have to be swapped in order to make it so that the final state is of lower energy.
So I made a short script which runs through all of the OBTD blocks and looks up the excitation energy of the initial and final state for the given block. The result was that 628337 of 1280000 (49.09 %) blocks had initial state excitation energy lower than the final state. Thats good, as expected, some of the final states are of lower energy and some are of higher. The puzzle however is that when I make a sub-set of transitions, selecting only the transitions which are present within the $E_\gamma = [0, 3]$ MeV region of the $M1$ GSF then only 3 of 230397 (0.00 %) transitions have initial energies lower than final. I don't understand where in the process of calculating the GSF that this kind of energy sorting happens.
Discussion