Zeeman energy level, Hamiltonian

In the presence of a strong magnetic field, a nucleus possessing spin I > 0 is in one of 21 +1 equally spaced Zeeman energy levels. Each nucleus is also exposed to the influence of other nuclei, which modify the local magnetic field. Thus, the total Hamiltonian is expressed by the sum of the six individual interactions. The units for the Hamiltonians are given elsewhere.14... 

The interaction of interest in RR is the dipolar interaction, an orientationally dependent through-space spin-spin coupling, which leads to a perturbation of the CS-perturbed Zeeman energy levels. For a homonuclear two-spin system, the truncated dipolar Hamiltonian operator is given by the following ... 

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... 

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... 

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. 

Level-1. The Hamiltonian contains only the isotropic Zeeman term, and this is appropriate for an isotropic Curie paramagnet. The energy levels and the magnetic susceptibility are simple functions like xmoi = fig). 

The energy levels for the Zeeman -Quadrupolar Hamiltonian are given by... 

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 interpretation of a complex Mossbauer spectrum will obviously be simplified if the relative intensities of the various components are known. Once the energy levels of the Zeeman/quadrupole Hamiltonian have been calculated, and the spin quantum numbers for each state assigned (or appropriate linear combinations if the states are mixed), it is possible to calculate the intensities from the theory of the coupling of two angular momentum states [32, 33]. 

Urf being the selected radio frequency and H the homogeneous field applied. This original setup was then widely used for the determination of the magnetic and quadrupolar hyperfine structure (hfs) constants A and B. Hereby one has to consider that the additional magnetic field further splits the atomic energy levels now characterized by F into (2F + 1) sublevels and mixes states of the same Mp but different F values. A Zeeman term has therefore to be added to the hyperfine Hamiltonian according to... 

Sensitivity is a major issue, since investigations of materials at interfaces are limited by the intrinsically low sensitivity of NMR combined with the small amount of compound provided by a monolayer. The low sensitivity is due to the relatively small spacing of the energy levels of nuclear spins, AE. For spin I = V nuclei the spin energy levels split up in a magnetic field B = Bo z according to the Hamiltonian of the Zeeman interaction... 

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 ... 

In the cases where first-order calculation is appropriate, the effects of the Hamiltonian Hq given in Equation (2.7.22) can be readily computed as both operators 1 and commute with the main Hamiltonian Hq. The result is that the energy levels are not equally spaced as they were in the case of the Zeeman interaction only. Otherwise, the energy depends on the quantum number m and the parameter (oq in the form ... 

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