Magnetic interaction Hamiltonian

A review of the systematics of the spin hamiltonian, magnetic interactions and all possible magnetic structures relating to rare earth compounds may be found in the article by Bertaut (1972). 

The low-temperature Mossbauer spectra of the spinel type oxides, NiCr204 [14,18] (Fig. 7.6b) and NiFe204 [3, 18], have been found to exhibit combined magnetic dipole and electric quadrupole interaction (Fig. 7.7). For the evaluation of these spectra, the authors have assumed a small quadrupolar perturbation and a large magnetic interaction, as depicted in Fig. 7.3 and represented by the Hamiltonian [3]... 

Much attention has been paid to Monte Carlo simulations of magnetic ordering, and its variation with temperature. Such models assume a particular form for the magnetic interactions, e.g. the Ising or Heisenberg Hamiltonian (see e.g. Binder... 

The interaction Hamiltonian that appears in Equation (5.37) can involve different types of interactions namely, multipolar (electric and/or magnetic) interactions and/or a quantum mechanical exchange interaction. The dominant interaction is strongly dependent on the separation between the donor and acceptor ions and on the nature of their wavefunctions. 

A more accurate description is obtained by including other additional terms in the Hamiltonian. The first group of these additional terms represents the mutual magnetic interactions which are provided by the Breit equation. The second group of additional terms are known as effective interactions and represent, to second order perturbation treatment, interaction with distant configurations . These weak interactions will not be considered here. 

The Breit-Pauli Hamiltonian with an external field contains all standard one- and two-electron contributions as well as the magnetic interaction of the electrons and their interactions with an external electromagnetic field. We may group the various contributions in the Breit-Pauli Hamiltonian according to one-and two-electron terms,... 

The nuclear spins give rise to additional terms in the Breit-Pauli Hamiltonian due to the interaction of the electrons with the magnetic moment of the nuclei and the electrostatic interaction with the electric quadrupole interaction of the nuclei. The magnetic interaction term of the spins with the nuclei is of the same type as the spin-spin interaction and following Abragam and Pryce (61) can be written as... 

The second moments of NMR signals in the observable spectral range can be calculated, provided that the local structure and the type of the internal magnetic interactions of the spin ensemble under study are known. According to the total Hamiltonian given in Eq. (1), it follows that... 

Now we can estimate the static magnetic field, where the electric field -induced spin-flips have a larger probability than the transitions due to the magnetic field of the EM wave. With the Hamiltonian of interaction of the spin... 

The first term in eq. (1) Ho represents the spherical part of a free ion Hamiltonian and can be omitted without lack of generality. F s are the Slater parameters and ff is the spin-orbit interaction constant /<- and A so are the angular parts of electrostatic and spin-orbit interactions, respectively. Two-body correction terms (including Trees correction) are described by the fourth, fifth and sixth terms, correspondingly, whereas three-particle interactions (for ions with three or more equivalent f electrons) are represented by the seventh term. Finally, magnetic interactions (spin-spin and spin-other orbit corrections) are described by the terms with operators m and p/. Matrix elements of all operators entering eq. (1) can be taken from the book by Nielsen and Koster (1963) or from the Argonne National Laboratory s web site (Hannah Crosswhite s datafiles) http //chemistry.anl.gov/downloads/index.html. In what follows, the Hamiltonian (1) without Hcf will be referred to as the free ion Hamiltonian. 

The fact that the magnetic interaction Hamiltonians are compound tensor operators can be exploited to derive more specific selection rules than the one given above. Furthermore, as we shall see later, the number of matrix elements between multiplet components that actually have to be computed can be considerably reduced by use of the Wigner-Eckart theorem. 

The operators so and ss are compound tensor operators of rank zero (scalars) composed of vector (first-rank tensor) operators and matrix (second-rank tensor) operators. We will make use of this tensorial structure when it comes to selection rules for the magnetic interaction Hamiltonians and symmetry relations between their matrix elements. Similar considerations apply to the molecular rotation and hyperfine splitting interaction... 

The Ising model assumes the magnetic interactions to be anisotropic. In fact, this phenomenon does not occur practically and the choice of an Ising Hamiltonian will be made on the basis of other factors such as the presence of a crystal field or a magnetic dipolar field both of which can polarize the spin in a certain direction of the crystal. The first case very often results from the existence of an orbital momentum7. ... 

The remaining important magnetic interactions to be considered are those which arise when a static magnetic field B is applied. The Zeeman interaction with a nuclear spin magnetic moment is represented by the Hamiltonian term... 

One can gain some insight into the nature of the Dirac wave equation and the spin angular momentum of the electron by considering the Foldy Wouthuysen transformation for a free particle. In the absence of electric and magnetic interactions, the Dirac Hamiltonian is... 

We now show how the many-electron Hamiltonian developed in the previous chapter may be extended to include magnetic interactions which arise from the presence of nuclear spin magnetic moments. Equation (3.140) represents the Hamiltonian for electron i in the presence of other electrons we present it again here ... 

It is possible to obtain the nuclear spin magnetic interaction terms by starting from the Breit equation. We recall that the Breit Hamiltonian describes the interaction of two electrons of spin 1 /2, each of which may be separately represented by a Dirac Hamiltonian ... 

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