Spin-Hamiltonian concept

If the electric quadrupole splitting of the 7 = 3/2 nuclear state of Fe is larger than the magnetic perturbation, as shown in Fig. 4.13, the nij = l/2) and 3/2) states can be treated as independent doublets and their Zeeman splitting can be described independently by effective nuclear g factors and two effective spins 7 = 1/2, one for each doublet [67]. The approach corresponds exactly to the spin-Hamiltonian concept for electronic spins (see Sect. 4.7.1). The nuclear spin Hamiltonian for each of the two Kramers doublets of the Fe nucleus is ... 

The spin-Hamiltonian concept, as proposed by Van Vleck [79], was introduced to EPR spectroscopy by Pryce [50, 74] and others [75, 80, 81]. H. H. Wickmann was the first to simulate paramagnetic Mossbauer spectra [82, 83], and E. Miinck and P. Debmnner published the first computer routine for magnetically split Mossbauer spectra [84] which then became the basis of other simulation packages [85]. Concise introductions to the related modem EPR techniques can be found in the book by Schweiger and Jeschke [86]. Magnetic susceptibility is covered in textbooks on molecular magnetism [87-89]. An introduction to MCD spectroscopy is provided by [90-92]. Various aspects of the analysis of applied-field Mossbauer spectra of paramagnetic systems have been covered by a number of articles and reviews in the past [93-100]. 

The spin-Hamiltonian formalism is a crutch in the sense that it is a parameterized theory, but it provides a common theoretical frame for the various experimental techniques with a minimum number of adjustable parameters that describe the essential physics of the system under investigation. Even more important is the fact that the same parameters can be derived relatively easily from quantum chemical calculations. Therefore, theoreticians appreciate the concept as a convenient place to rest in the analysis of experimental data by theoretical means [123, 124]. 

The EPR spectrum is a reflection of the electronic structure of the paramagnet. The latter may be complicated (especially in low-symmetry biological systems), and the precise relation between the two may be very difficult to establish. As an intermediate level of interpretation, the concept of the spin Hamiltonian was developed, which will be dealt with later in Part 2 on theory. For the time being it suffices to know that in this approach the EPR spectrum is described by means of a small number of parameters, the spin-Hamiltonian parameters, such as g-values, A-values, and )-values. This approach has the advantage that spectral data can be easily tabulated, while a demanding interpretation of the parameters in terms of the electronic structure can be deferred to a later date, for example, by the time we have developed a sufficiently adequate theory to describe electronic structure. In the meantime we can use the spin-Hamiltonian parameters for less demanding, but not necessarily less relevant applications, for example, spin counting. We can also try to establish... 

The spin Hamiltonian is an artificial but useful concept. It is possible that more than one spin Hamiltonian will fit the data. Further, we should note that in solving Eq. (48), we start with pure + and — spin functions and talk about the upper and lower states as being pure spin states. This is not the true case for the ion, as has already been noted in Eq. (38). As regards the Zeeman interaction, however, the final state behaves as a pure spin state, except that we must assign g values different from that of the free electron. 

This concept of the spin Hamiltonian was first advanced and developed by Abragam and Pryce (13,14). More recently the problem has been treated in a more general fashion by Koster and Statz (75), using group-theory arguments. 

Under conditions at which NMR experiments are usually performed, the interactions between the nucleus and the electromagnetic fields present in its environment (including the interactions with electrons, other nuclei, other ions, and so on) are well described using the concept of the nuclear spin Hamiltonian CHmiciear)- This Hamiltonian contains only terms that depend on the orientation of the nuclear spin and, therefore, its matrix representation is usually given in the m) basis, which corresponds to eigenstates of the Zeeman Hamiltonian (Hz). It is convenient to write the nuclear spin Hamiltonian in the form ... 

This section gives an explanation of the different terms of the static spin Hamiltonian. The concept of orientation selection by selective m.w. excitation, which is central to many pulse EPR experiments on disordered systems, is explained. 

The concept of a spin Hamiltonian is thus central to this discussion, in which it plays a twofold role From an experimental viewpoint, effective spin Hamiltonians are used to convey a description of the experimental spectral behaviors in terms of numerical values of the magnetic parameters thus, the structural and dynamical information on the system under examination is summarized and encoded into these empirical parametCTS. From the viewpoint of computational spectroscopy, the spin Hamiltonian is first of all decomposed into a set of individual operators corresponding to specific physical effects. Once suitable theoreticaFcomputational descriptions are established for these operators a viable link is obtained between computed and observed spectral parameters. In the case of NMR spectroscopy, a general formulation of the spin Hamiltonian is the following ... 

The one-electron additivity of the mean-field Hamiltonian gives rise to the concept of spin orbitals for any additive bi fact, there is no single mean-field potential different scientists have put forth different suggestions for over the years. Each gives rise to spin orbitals and configurations that are specific to the particular However, if the difference between any particular mean-field model and the fiill electronic... 

The so-called HORROR experiment by Nielsen and coworkers [26] introduced continuous rf irradiation recoupling to homonuclear spin-pairs and initiated the later very widely used concept of /-encoded recoupling. Using a irreducible spherical approach as described above, the HORROR experiment (Fig. 2d) is readily described as starting out with the dipolar coupling Hamiltonian in (10) and x-phase rf irradiation in the form Hrf = ncor(Ix +SX), also here without initial constraint on n. The dipolar coupling Hamiltonian transforms into tilted frame (rotation n/2 around Iy + Sy)... 

Recently, the concept of thermal entanglement was introduced and studied within one-dimensional spin systems [64-66]. The state of the system described by the Hamiltonian H at thermal equilibrium is p T) = exp —H/kT)/Z, where Z = Tr[exp(—7//feT)] is the partition function and k is Boltzmann s constant. As p T) represents a thermal state, the entanglement in the state is called the thermal entanglement [64]. 

The spin operators Sx, and Sg which occur in the Breit-Pauli Hamiltonian form a basis for the Lie algebra of SU(2). The concept of the electron as a spinning particle has arisen through the isomorphism between SU(2) and the angular momentum operators. This analogy is unnecessary and often undesirable. 

Point-group symmetry exists only within a particular Born-Oppen-heimer approximation. Though point-group symmetry often has little to do with spin conservation, it will be found in Section VIII that spin concepts and point-group symmetry are intermingled when a Hamiltonian involving spin interactions is considered. Also, we will find that Born-Oppenheimer approximations are important in Franck-Condon factors Franck-Condon factors are, in turn, critical in determining transition probabilities for a number of spin-forbidden processes. 

The examination takes place in two stages, one corresponding to the formal interelectronic repulsion component of the Hamiltonian HER and the second to the spin-orbit coupling term Hes. As will be pointed out, in principle, and in certain cases in practice, it is not proper to separate the two components. However, the conventional procedure is to develop HLS as a perturbation following the application of Her. That suffices for most purposes, and simplifies the procedures. Any interaction between the d- or/-electron set and any other set is ignored. It is assumed that it is negligible or can be taken up within the concept of an effective d-orbital set. 

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