Hamiltonian Zeeman

If an external magnetic field, B, is applied, one has to include the Zeeman Hamiltonian... 

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... 

The application of a magnetic field to the wavefunctions obtained by the procedure described in the previous sections results in the complete removal of the degeneracy of the / multiplet, either pertaining to Kramers or non-Kramers ions, and yields a temperature-dependent population of the different 2/ + 1 components (Figure 1.2) Thus, at low temperatures, large deviations from the Curie law are observed. The effect of the magnetic field is described by the Zeeman Hamiltonian ... 

An alternative treatment of the same problem consists in writing the Zeeman Hamiltonian in the form... 

H is the total Hamiltonian (in the angular frequency units) and L is the total Liouvillian, divided into three parts describing the nuclear spin system (Lj), the lattice (Ll) and the coupling between the two subsystems (L/l). The symbol x is the density operator for the whole system, expressible as the direct product of the density operators for spin (p) and lattice (a), x = p <8> ci. The Liouvillian (Lj) for the spin system is the commutator with the nuclear Zeeman Hamiltonian (we thus treat the nuclear spin system as an ensemble of non-interacting spins in a magnetic field). Ll will be defined later and Ljl... 

Lso is the commutator with the electron spin Zeeman Hamiltonian (assuming isotropic g tensor, Hso = gS- Bo), Lrs = Lzfs (the sub-script RS stands for coupling of the rotational and spin parts of the composite lattice) is the commutator with the ZFS Hamiltonian and Lr = —ir, where is a stationary Markov operator describing the conditional probability distribution, P(QolQ, t), of the orientational degrees of freedom through ... 

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... 

The interaction of a single spin with the magnetic field (in the range 10 to 108 Hz) is described by a Zeeman Hamiltonian ... 

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. 

In such a system, the external magnetic field defines the molecular z axis. If we rotate the molecule with respect to Bo, the spin and its magnetic moment are not affected (Fig. 1.14B). However, in the molecule of Fig. 1.14A, a molecular z axis can be defined. When rotating the molecule, the orbital contribution to the overall magnetic moment changes, whereas the spin contribution is constant. The total Zeeman Hamiltonian is... 

The general Zeeman hamiltonian thus will contain two independent g values. A convenient separation of the product multiplicity may be based on spherical coupling coefficients. To this aim the U x U product space is put into correspondence with the space that results from the addition of two j — 3/2 angular momenta. Their symmetrized square yields P and F products, which subduce Tl and A2 + Tt + T2 resp. 

The two 7] products can thus be distinguished by a spherical coupling label as PTi and FT t. Now let cf create the four components of V. The appropriately coupled quantities of 7] symmetry read (ctc)PTl and (c+c)FT>. The Zeeman hamiltonian for a U representation can thus be written as follows ... 

Table 11. Isotropies of the Zeeman hamiltonian for the cubic U representation...

Now let s look at something we do not know the answer to the ideal isotropic mixing Hamiltonian. This is the ideal TOCSY mixing sequence that leads to in-phase to in-phase coherence transfer. The ideal sequence of pulses creates this average environment expressed by the Hamiltonian. The Zeeman Hamiltonian that represents the chemical shifts goes away and we have only the isotropic (i.e., same in all directions) /-coupling Hamiltonian ... 

Interactions with an applied magnetic field are particularly important for open shell free radicals, many with 2 n ground states having been studied by magnetic resonance methods. The Zeeman Hamiltonian may be written as the sum of four terms ... 

We now collect together the results of this section and summarise the Zeeman Hamiltonian as... 

We introduced the field-free nuclear Hamiltonian in section 3.10. Again by analogy with the electronic Hamiltonian, we include the effects of external magnetic fields by replacing P, by P, — Z,eA l in equation (III.248) and the effects of an external electric field by addition of the term Y,a Zae(pa, this treatment is only really justified if the nuclei behave as Dirac particles. The nuclear Zeeman Hamiltonian is thus ... 

We deal with the Zeeman Hamiltonian first. We must first choose an origin for the vector potentials this origin is completely arbitrary (indeed, this is the physical significance of gauge invariance). In our expansions above we have selected a Coulomb... 

In summary we have the total Zeeman Hamiltonian for a neutral molecule ... 

We see that there are many contributions to the Zeeman Hamiltonian, all of which must be taken into account in the construction of the effective Hamiltonian. In order to focus our attention, let us consider just one of the dipolar terms, namely, the interaction involving the orbital motion of the electrons ... 

Here, co represents the Euler angles (orbital Zeeman interaction, we see that it has off-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 cross term between X and Xr<)t can be treated in exactly the same way The result is a second-order contribution to the effective Zeeman Hamiltonian of the form... 

A recent paper [32] has suggested that the primary g-factors, gs and g L, should be defined as negative quantities so that they reveal the alignment of the magnetic dipole moment relative to the angular momentum. If this convention is adopted, the signs of the two contributions to the effective Zeeman Hamiltonian, given above, must be reversed. 

We calculate the effects of the Hamiltonian (8.105) on these zeroth-order states using perturbation theory. This is exactly the same procedure as that which we used to construct the effective Hamiltonian in chapter 7. Our objective here is to formulate the terms in the effective Hamiltonian which describe the nuclear spin-rotation interaction and the susceptibility and chemical shift terms in the Zeeman Hamiltonian. We deal with them in much more detail at this point so that we can interpret the measurements on closed shell molecules by molecular beam magnetic resonance. The first-order corrections of the perturbation Hamiltonian are readily calculated to be... 

The first level to be studied in detail by Tichten [35] was the N = 2 level of both para-Hi and ortho-H2. He measured a series of fixed-frequency magnetic resonance transitions, determining effective g- values and proving the identification of the c3nu state in the process. An effective Zeeman Hamiltonian may be written, in the space-fixed axis system,... 

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