Magnetic field Hamiltonian

In addition, there could be a mechanical or electromagnetic interaction of a system with an external entity which may do work on an otherwise isolated system. Such a contact with a work source can be represented by the Hamiltonian U p, q, x) where x is the coordinate (for example, the position of a piston in a box containing a gas, or the magnetic moment if an external magnetic field is present, or the electric dipole moment in the presence of an external electric field) describing the interaction between the system and the external work source. Then the force, canonically conjugate to x, which the system exerts on the outside world is... 

We now come back to the important example of two spin 1/2 nuclei with the dipole-dipole interaction discussed above. In simple physical tenns, we can say that one of the spins senses a fluctuating local magnetic field originatmg from the other one. In tenns of the Hamiltonian of equation B 1.13.8. the stochastic fiinction of time F l t) is proportional to Y2 (9,( ))/rjo, where Y, is an / = 2 spherical hannonic and r. is the... 

While all contributions to the spin Hamiltonian so far involve the electron spin and cause first-order energy shifts or splittings in the FPR spectmm, there are also tenns that involve only nuclear spms. Aside from their importance for the calculation of FNDOR spectra, these tenns may influence the FPR spectnim significantly in situations where the high-field approximation breaks down and second-order effects become important. The first of these interactions is the coupling of the nuclear spin to the external magnetic field, called the... 

Optically detected magnetic resonance (ODMR) has yielded valuable information about dynamics of long-lived pholoexcitations of conjugated polymers. The technique relies upon the paramagnetic interaction of excitations with an applied magnetic field. For a particle with non-zero spin, placed in a magnetic field, the Hamiltonian is ... 

The Hamiltonian function for an electron in a constant magnetic field of strength H parallel to the s axis is1... 

In the Hamiltonian conventionally used for derivations of molecular magnetic properties, the applied fields are represented by electromagnetic vector and scalar potentials [1,20] and if desired, canonical transformations are invoked to change the magnetic gauge origin and/or to introduce electric and magnetic fields explicitly into the Hamiltonian, see e.g. refs. [1,20,21]. Here we take as our point of departure the multipolar Hamiltonian derived in ref. [22] without recourse to vector and scalar potentials. 

A nucleus in a state with spin quantum number 7 > 0 will interact with a magnetic field by means of its magnetic dipole moment p. This magnetic dipole interaction or nuclear Zeeman effect may be described by the Hamiltonian... 

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

However, when it comes to the simulation of NFS spectra fi om a polycrystalline paramagnetic system exposed to a magnetic field, it turns out that this is not a straightforward task, especially if no information is available from conventional Mossbauer studies. Our eyes are much better adjusted to energy-domain spectra and much less to their Fourier transform therefore, a first guess of spin-Hamiltonian and hyperfine-interaction parameters is facilitated by recording conventional Mossbauer spectra. 

If we replace the z-component of the classical angular momentum in equation (6.87) by its quantum-mechanical operator, then the Hamiltonian operator Hb for the hydrogen-like atom in a magnetic field B becomes... 

The basic physics governing the measurement of the EDM in all types of electrically neutral systems is almost the same as discussed in this section. If the system under consideration has a magnetic moment p and is exposed to a magnetic field B, then the interaction Hamiltonian can be written... 

The spin-orbit Hamiltonian (HB0) requires some explanation. The energy of interaction between the magnetic moment M and the magnetic field caused by the orbital motion of an electron can be derived as(134)... 

In Equation (6) ge is the electronic g tensor, yn is the nuclear g factor (dimensionless), fln is the nuclear magneton in erg/G (or J/T), In is the nuclear spin angular momentum operator, An is the electron-nuclear hyperfine tensor in Hz, and Qn (non-zero for fn > 1) is the quadrupole interaction tensor in Hz. The first two terms in the Hamiltonian are the electron and nuclear Zeeman interactions, respectively the third term is the electron-nuclear hyperfine interaction and the last term is the nuclear quadrupole interaction. For the usual systems with an odd number of unpaired electrons, the transition moment is finite only for a magnetic dipole moment operator oriented perpendicular to the static magnetic field direction. In an ESR resonator in which the sample is placed, the microwave magnetic field must be therefore perpendicular to the external static magnetic field. The selection rules for the electron spin transitions are given in Equation (7)... 

When an electron is placed in a magnetic field, the degeneracy of the electron spin energy levels is lifted as shown in Figure 1.1 and as described by the spin 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... 

When we include the Zeeman interaction term, gpBB-S, in the spin Hamiltonian a complication arises. We have been accustomed to evaluating the dot product by simply taking the direction of the magnetic field to define the z-axis (the axis of quantization). When we have a strong dipolar interaction, the... 

With the magnetic field oriented along the x-axis, the Hamiltonian is ... 

For Cr(m) complexes, D is relatively small (comparable to the X-band microwave quantum, 0.317 cm-1) and all three fine structure lines are observable. This is not always the case. Consider high-spin Fe(m) in an axial ligand field with D > > hv0, E = 0. With the same Hamiltonian as above and the magnetic field along the z-axis, the energies are ... 

For a pseudo-axial ligand field the appropriate perturbation Hamiltonian, H, is, in the absence of a magnetic field, (c.f. Section 2)... 

---