Spin-magnetic field interaction

The electron spin generates a magnetic moment fie which contributes to the energy in the presence of the magnetic field through the operator 

The magnetic moment of a charged particle associated with some angular momentum / is 

The solutions of the above homogeneous system of linear equations exist only when the determinant of the system equals zero, i.e. 

The energy shift due to the electron spin interacting with the magnetic field is named the Zeeman term. 

Having determined the eigenvalues, the evaluation of the wave functions will continue. Making use of interrelations between the cartesian and polar coordinates, viz. 

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The Hamiltonian term, including the electron spin-magnetic field interaction, is envisaged as... 

We have not yet considered these effects in the non-relativistic Hamiltonian (2.1) on p. 67 (e.g., no spin-spin or spin-field interactions). The effects that we are now dealing with are so small-on the order of 10 kcal/mole-that they are of no importance for most applications, including UV-VIS, IR. Raman spectra, electronic structure, chemical reactions, intermolecular interactions, etc. This time, however, the situation is different we are going to study very subtle interactions using the NMR technique, which aims precisely at the energy levels that result from spin-spin and spin- magnetic field interactions. Even if these effects are very small, they can be observed. Therefore,... 

Figure 1.2 Energy levels of an unpaired electron in a magnetic field interacting with a spin-1/2 nucleus. The arrows show two allowed transitions.

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 interaction with an external magnetic field, when L = 0, is 2p H S. Therefore the spin Hamiltonian which includes the magnetic field interaction may be written as an extension of Eq. (138)... 

The energy levels in this case are the result of the interactions between the nuclear spin magnetic fields in the molecule and an external magnetic field. We begin by way of a glance at nuclear spins. 

Another critical problem with exciting these transitions is that they are driven not by the electric field, but by the magnetic field of the radiation. There is no significant change in the electric field of the atom when we flip the nuclear spin, so the electric field of the photon cannot cause the transition to occur. The photon s magnetic field can cause the flip, but magnetic field interactions are typically much weaker than electric field interactions. We make up for this partly by using powerful rf sources. 

We first consider tlnee examples as a prelude to the general discussion of basic statistical mechanics. These are (i) non-mteracting spin-i particles in a magnetic field, (ii) non-interacting point particles in a box,... 

We now come back to the important example of two spin 1/2 nuclei with the dipole-dipole interaction discussed above. In simple physical tenns, we can say that one of the spins senses a fluctuating local magnetic field originatmg from the other one. In tenns of the Hamiltonian of equation B 1.13.8. the stochastic fiinction of time F l t) is proportional to Y2 (9,( ))/rjo, where Y, is an / = 2 spherical hannonic and r. is the... 

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