Zeeman Hamiltonian energies

Energy level splitting in a magnetic field is called the Zeeman effect, and the Hamiltonian of eqn (1.1) is sometimes referred to as the electron Zeeman Hamiltonian. Technically, the energy of a... 

Zeeman effects 393 Zeeman energy 111, 113 Zeeman field 411 Zeeman Hamiltonian 46 Zeeman interaction 59, 79, 248 Zeeman limit 49-50 Zeolite 307, 310... 

In quantum mechanical terms the energy is given by the Hamiltonian operator, which in this case is called the nuclear Zeeman Hamiltonian... 

In the absence of an external magnetic field the Zeeman Hamiltonian provides zero energy and all the 11, Mi) levels (termed as / manifold) have the same energy. However, this may not be true for nuclei with / > V2. In this case, the non-spherical distribution of the charge causes the presence of a quadrupole moment. Whereas a dipole can be described by a vector with two polarities, a quadrupole can be visualized by two dipoles as in Fig. 1.11. 

Except for some quadrupolar effects, all the interactions mentioned are small compared with the Zeeman interaction between the nuclear spin and the applied magnetic field, which was discussed in detail in Chapter 2. Under these circumstances, the interaction may be treated as a perturbation, and the first-order modifications to energy levels then arise only from terms in the Hamiltonian that commute with the Zeeman Hamiltonian. This portion of the interaction Hamiltonian is often called the secular part of the Hamiltonian, and the Hamiltonian is said to be truncated when nonsecular terms are dropped. This secular approximation often simplifies calculations and is an excellent approximation except for large quadrupolar interactions, where second-order terms become important. 

Terms C—F can be shown not to commute with the Zeeman Hamiltonian, hence to contribute negligibly to the energy levels, but they are important in relaxation processes, as we see in Chapter 8. For this chapter we shall use the truncated Hamiltonian with terms A and B. The spin portion of this truncated Hamiltonian may look more familiar with some rearrangement of terms... 

The Zeeman Hamiltonian can be used to calculate the energy difference between the nuclear spin states. For a nucleus the energy eigenvalues can be obtained by taking the Hamiltonian operator given in Eq. 2.11 so that... 

Radford (1961, 1962) and Radford and Broida (1962) presented a complete theory of the Zeeman effect for diatomic molecules that included perturbation effects. This led to a series of detailed investigations of the CN B2E+ (v — 0) A2II (v = 10) perturbation in which many of the techniques of modern high-resolution molecular spectroscopy and analysis were first demonstrated anticrossing spectroscopy (Radford and Broida, 1962, 1963), microwave optical double resonance (Evenson, et at, 1964), excited-state hyperfine structure with perturbations (Radford, 1964), effect of perturbations on radiative lifetimes and on inter-electronic-state collisional energy transfer (Radford and Broida, 1963). A similarly complete treatment of the effect of a magnetic field on the CO a,3E+ A1 perturbation complex is reported by Sykora and Vidal (1998). The AS = 0 selection rule for the Zeeman Hamiltonian leads to important differences between the CN B2E+ A2II and CO a/3E+ A1 perturbation plus Zeeman examples, primarily in the absence in the latter case of interference effects between the Zeeman and intramolecular perturbation terms. 

The second-order contribution of the orbital Zeeman term with itself produces a term in the effective Hamiltonian which is quadratic in 5. It therefore has the same form as the diamagnetic susceptibility contribution to the energy it provides the paramagnetic or high-frequency contribution to the susceptibility of the molecular system. The resultant term in the effective Zeeman Hamiltonian is... 

The last column in Fig. 2.1 shows how an external magnetic field H affects the energies of the Ms sublevels of the electronic manifolds of a paramagnetic material. This effect is described by the Zeeman Hamiltonian... 

The eigenvalues of the Zeeman Hamiltonian (2.2.2), which are clearly proportional to the eigenvalues of the operator 4, represent the energy levels of the nucleus, given by ... 

The interaction of a dipole /itj with such field created by a dipole 112 is given by the classical Zeeman interaction energy —/iti The Hamiltonian describing the dipolar interaction can thus be written in the form... 

Weak quadrupole perturbation of magnetic levels In this case the quadrupole interaction operator (18.51) must be projected onto the coordinate system associated with the magnetic (Zeeman) hamiltonian (18.1). Since the former is actually a tensor operator, the projection introduces a more complicated angular dependence than in the converse case considered in section 1.3.2.2. The energy levels become, on the basis of first-order perturbation theory ... 

For the spin 1/2 case discussed in Section III.B.2, the spin 1/2 P and AP pairs were thought to be loosely bound. However, because of sample morphology and/ or additional gain in relaxation energy, spin pairs can also be tightly bound, as in v dimers (PD) or polaron pairs (PP) discussed in Section II. In such a case an exchange interaction term [49] is usually added to the Zeeman Hamiltonian ... 

When magnetie fields are present, the intrinsie spin angular momenta of the eleetrons S (j) and of the nuelei I(k) are affeeted by the field in a manner that produees additional energy eontributions to the total Hamiltonian H. The Zeeman interaetion of an external magnetie field (e.g., the earth s magnetie field of 4. Gauss or that of a NMR... 

Thus, the Zeeman interaction occurs only with nuclei that possess a spin greater than zero, and it yields 27+1 energy levels of separation = yB Urr. The Hamiltonian is described by... 

The Hamiltonian of a single isolated nanoparticle consists of the magnetic anisotropy (which creates preferential directions of the magnetic moment orientation) and the Zeeman energy (which is the interaction energy between the magnetic moment and an external field). In the ensembles, the nanoparticles are supposed to be well separated by a nonconductive medium [i.e., a ferrofluid in which the particles are coated with a surfactant (surface-active agent)]. The... 

For a hydrogen atom in an external field of 10,000 G, draw a figure that shows the effect on the original 1 s energy level of including first the electron Zeeman term, then the nuclear Zeeman term, and finally the hyperfine coupling term in the Hamiltonian. 

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