Graduate Secular Approximation of Dipole-Dipole Hamiltonian

[TeamMate]

TL;DR

Derivation of the secular approximation of the dipole-dipole-Hamiltonian

Hey folks,

I'm looking for a derivation of the secular approximation of the dipole-dipole Hamiltonian at high magnetic fields. Does anybody know a reference with a comprehensive derivation or can even provide it here?

Given we have the dipolar alphabet, I'd like to understand (in the best case using equations), why only the term A is relevant in every case, term B is relevant for like-spins (and can be dropped for unlike spins), while terms C-F can be always dropped. I know that there are qualitative descriptions of the approximation, I couldn't find but any rigorous derivation (e.g. showing that only I_z commutes with the Zeeman-Hamiltonian).

I'm looking fowrads to your suggestions and your help!
Thanks a lot!

 

[vanhees71]

First of all you have to formulate your problem in a concise way. That's often more than half towards its solution. How do you think should we know, what you mean by terms A-F if you don't clearly define them? Which book/paper are you looking at?

 

[TeamMate]

Dear vanhees71,

Thanks for the quick answer, and sorry for the brevity. I thought my question was just trivial for the experts out here^^ Please find some more details below. The infos that I have so far have been taken from (see page 63 therein): https://www.google.de/books/edition...ance+and+its+applications&printsec=frontcover

The dipolar Hamiltonian for two electrons given in the form of the dipolar alphabet with the terms A-F is

Therein, S_A and S_B are the spin operators of electrons A and B; x, y, and z refer to the cartesian coordinates; S⁺ and S^- are the raising and lowering operators, respectively. g_A and g_B are the g-factors of electrons A and B, β_e is Bohr's magneton, µ0 is the magnetic field constant, and ħ is the reduced Planck-constant. A complete derivation of these equations can be found, e.g., here:

A is called the secular term, B the pseudo-secular term, and C-F are non-secular terms. The following considerations hold true at high magnetic fields, i.e. if the Zeeman-interaction energy is much larger than the dipolar coupling energy. According to the literature (for reference, see above), A is always of relevance as it commutes with the Zeeman Hamiltonian (that's what I'd like to show somehow). B is of relevance for "like"-spins (i.e. electrons with identical g-values) but can be dropped for "unlike"-spins (i.e. different g-values); also here, B seems to commute with the Zeeman-Hamiltonian for "like"-spins, but not for "unlike"-spins. C-F are non-secular and thus do not commute with the Zeeman-Hamiltonian.

Any help in showing that A is secular, B is pseudo-secular, and C-F are non-secular is very much appreciated!
Thank you!