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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!
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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
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