Interaction Zeeman, orbital

Level-6. The most complete treatment utilizes the basis set of all free-atom terms v,L,Ml,S,Ms) for the given electronic configuration dn, and the calculation of energy levels is performed by involving the operators of the electron repulsion, CF, spin-orbit interaction, orbital-Zeeman and spin-Zeeman terms ... 

We now let the molecule interact with an external magnetic field Bo. The interaction energy, as far as the orbital is concerned, is given by the orbital Zeeman operator... 

The Hamiltonian that describes the interaction of the single magnetic center with the external magnetic field involves the spin-Zeeman term, the orbital Zeeman term, and the operator of the spin-orbit coupling, i.e.,... 

The orbital Zeeman interaction is Ms-independent lnvLSMLMs r1MB4 -i) FVL S M M g)... 

The orbital Zeeman interaction involves the matrix element, which is reduced as follows ... 

Summary. We consider the Josephson effect in a ballistic Superconductor/ Quantum Wire/ Superconductor junction. It is shown that the interplay of chiral symmetry breaking generated by Rashba spin-orbit interaction and Zeeman splitting results in the appearance of a Josephson current even in the absence of any phase difference between the superconductors. 

Here, co represents the Euler angles (diagonal matrix elements which link electronic states with A A = 0, 1, as well as purely diagonal elements. It is clearly desirable to remove the effect of these matrix elements by a suitable perturbative transformation to achieve an effective Zeeman Hamiltonian which acts only within the spin-rotational levels of a given electronic state rj, A, v), in the same way as the zero-field effective Hamiltonian in equation (7.183). 

The q = 1 components of T (L) will be neglected because they involve the mixing of excited states with the ground vibronic state, giving rise to temperature-independent paramagnetism. With this simplification the matrix elements of the orbital Zeeman interaction operating within the ground vibronic state are... 

The point to emphasise concerning this example is the reduced overlap between the ground vibronic wavefunctions associated with the two electronic states, on account of the displacement of the potential energy minima. The expectation value of any electronic operator connecting the two orbital electronic states will be similarly affected. Consider the orbital Zeeman interaction about the z axis, given by... 

The reduction in the orbital Zeeman interaction approaches zero exponentially as Ejy increases and tuo decreases. Note that the quantity Ejy depends solely on the potential energy. 

In this Hamiltonian (5) corresponds to the orbital angular momentum interacting with the external magnetic field, (6) represents the diamagnetic (second-order) response of the electrons to the magnetic field, (7) represents the interaction of the nuclear dipole with the electronic orbital motion, (8) is the electronic-nuclear Zeeman correction, the two terms in (9) represent direct nuclear dipole-dipole and electron coupled nuclear spin-spin interactions. The terms in (10) are responsible for spin-orbit and spin-other-orbit interactions and the terms in (11) are spin-orbit Zeeman gauge corrections. Finally, the terms in (12) correspond to Fermi contact and dipole-dipole interactions between the spin magnetic moments of nucleus N and an electron. Since... 

For many applications the full molecular Hamiltonian is not necessary it is sufficient to include only relevant energy terms into the model Hamiltonian. One of the model Hamiltonians is the spin Hamiltonian which includes only the angular momentum operators in their mutual interaction (orbit-orbit, spin-orbit, spin-spin interactions) as well as their interaction with a magnetic field (the Zeeman terms orbit-magnetic field and spin-magnetic field). 

The first term in the brackets is the usual kinetic energy term, while the second term produces the orbital Zeeman effect. The third term describes the second-order interactions corresponding to the atomic contribution to the susceptibility. The second term can easily be converted into the more familiar form of the Zeeman operator as follows ... 

Here we see that the spin-dependent term of the modified vector potential is in fact describing the interaction of the spin with the magnetic field, and the scalar term is describing the interaction of the orbital angular momentum with the magnetic field. These terms are the spin and orbital Zeeman terms respectively. In (15.32) there are also relativistic corrections that arise from the difference between ( ) and rlr. 

Hfi includes a nuclear Zeeman term, a nuclear dipole-dipole term, an electron-nuclear dipole term and a term describing the interaction between the nuclear dipole and the electron orbital motion. 

---