Zeeman levels

The involvement of the matrix elements of the Zeeman operator in the interaction matrix 

The programming of the formulae needs a COMPLEX 16 arithmetic since the CF potential itself could be complex. Therefore, it is easy to implement the complex spherical transforms of the magnetic field, x and B, into the (complex) Zeeman matrix elements. With the magnetic field Eref aligned parallel to the principal rotational axis of an axial system, the Zeeman matrix stays real since then B](] = Bxef. Its counterpart for the perpendicular direction is also real, and this involves the following transforms x = - (l/V2)Bref and 

A more precise powder average is obtained by distributing the magnetic field over a number of grids (pi) lying in a sphere. Then 

The full list of parameters that enter the calculation of the magnetic energy levels is  

Racah parameters B and C for the given transition metal ion (Table 46)  

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Of the NMR-active nuclei around tluee-quarters have / > 1 so that the quadnipole interaction can affect their spectra. The quadnipole inter action can be significant relative to the Zeeman splitting. The splitting of the energy levels by the quadnipole interaction alone gives rise to pure nuclear quadnipole resonance (NQR) spectroscopy. This chapter will only deal with the case when the quadnipole interaction can be regarded as simply a perturbation of the Zeeman levels. 

The transition between levels coupled by the oscillating magnetic field B corresponds to the absorption of the energy required to reorient the electron magnetic moment in a magnetic field. EPR measurements are a study of the transitions between electronic Zeeman levels with A = 1 (the selection rule for EPR). 

Figure C 1.4.8. (a) An energy level diagram showing the shift of Zeeman levels as the atom moves away from the z = 0 axis. The atom encounters a restoring force in either direction from counteriDropagating light beams, (b) A typical optical arrangement for implementation of a magneto-optical trap.

The intensity of a non-degenerate n.m.r. transition between the nuclear Zeeman levels n and m is proportional to the difference in population of levels n and m as given by the Boltzmann distribution. This can be expressed by equation (35). 

Fig. 4 Relaxation pathways between quadrupole-perturbed Zeeman levels of / = 3/2 nuclear spin. Reprinted from [50]...

If the molecules are placed in a magnetic field, the different spatial orientations correspond to different Zeeman levels and the selective saturation explained above leads to a nonthermal population of the Zeeman niveaus. By exposing the sample simultaneously to an rf field with the proper frequency, magnetic dipole transitions between the Zeeman levels can occur and the thermal population can be restored. This will again increase the polarization of the fluorescence at most up to its presaturation value. The fluorescence polarization is thus used as detector for the rf transitions. 

In ions containing more than one unpaired spin the individual moments can interact with the local fields generated by the other electrons. When the ion is in a site of high symmetry (octahedral or tetrahedral), this spin-spin interaction adds a constant energy to all Zeeman levels which is not detected by transitions between the levels and hence does not appear in the ESR spectrum. For sites of lower symmetry, the electron distribution is polarized and the spin-spin interaction becomes dependent on A/. It depends on M rather than Ms because the spin-spin interaction detects only relative orientations of electron spins, not absolute orientations. 

A closely related phenomenon to that described is the observation of quantum beats167-160 in the radiative decay of a set of coherently excited states. Since transitions to the ground state can take place via two or more channels, interference effects may be observed. As an example, in Figure 23 we present the decay curve obtained by Dodd et al.180 for the decay of the coherently excited Zeeman levels of Cd. 

Upon rotation the average electron induced magnetic moment causes nuclear relaxation. This is called Curie relaxation (see Section 3.6). However, the full electron magnetic moment causes nuclear relaxation through dipolar and contact mechanisms by jumping over the various Zeeman levels. The electron jumps over the Zeeman levels with rates equal to the electronic R or R2 values and such jumps give also rise to nuclear relaxation. 

We should note that if g = ge, the contact shift is isotropic (independent of orientation). If g is different from ge and anisotropic (see Section 1.4), then the contact shift is also anisotropic. The anisotropy of the shift is due to the fact that (1) the energy spreading of the Zeeman levels is different for each orientation (see Fig. 1.16), and therefore the value of (Sz) will be orientation dependent and (2) the values of (5, A/s Sz S, Ms) of Eq. (1.31) are orientation dependent as the result of efficient spin-orbit coupling. On the contrary, the contact coupling constant A is a constant whose value does not depend on the molecular orientation. 

In the case of a Unear development of Zeeman levels it is as simple as... 

The fundamental principle governing the NMR technique centres on the induction of transitions between different nuclear Zeeman levels of a particular nucleus. To cause these transitions, a variable radiofrequency (RF), referred to as B, acts perpendicular to the applied magnetic field (B0), which is causing the nuclear alignments. When the frequency of the applied RF is identical to the precessional frequency (w0) of the nuclei being observed, a transition between nuclear spin states occurs. 

Fig. 2. Ground state Zeeman levels in an external magnetic field. The sum of the frequencies of the indicated transitions Aviz and Av i at the same magnetic field equals the zero field splitting Avhfs and their difference allows to determine the muon magnetic moment...

The nuclear spin Zeeman levels then have energies given by... 

Figure 8.6. Calculated Zeeman levels for D2 (J = 1, / = 1) in the magnetic field range 1800 to 2200 G. The six predicted A Mi = 1, AMj = 0 transitions are shown for a frequency of 1.3 MHz these predictions are to be compared with the experimental observations presented in figure 8.7.

A simple extension of the Stark analysis given above enables one to derive an expression for the intensities of the electric dipole transitions. The oscillating microwave electric field is applied perpendicular to the static magnetic field, so that the Zeeman levels experience a time-dependent perturbation, represented by the operator... 

Figure 9.28. Zeeman levels for N = 0 and 1 of NH 3E (v = 0) and the observed far-infrared laser magnetic resonance transitions [58]. These were recorded using four different FIR lines at 31.7615, 32.1466, 33.0822 and 33.1922 cm-1.

One of the classical NMR methods used to determine molecular correlation times is provided by spin-lattice relaxation experiments. The spin-lattice relaxation rate 1 /T is determined by transitions among the Zeeman levels. For a liquid, the expression for the spin-lattice relaxation rate [81] is... 

The situation with a pair of spins is illustrated in Fig. 1, where the static field B0 is along the z-axis. To first order and neglecting terms in the expansion of Eq. (2) which lead to a change in nuclear quantum numbers of 1 or 2 (i.e. Am — 1 and + 2 transitions), the effect of is to split the Zeeman levels into many closely spaced energy levels, thereby causing a distribution of resonant frequencies and consequently a broad line. Eq. (2) has been simplified by the van Vleck formula 2... 

In the triplet model the spin polarization is with respect to the internal molecular states, TjJ>, Ty>, and T > of the triplet and evolves with time according to the time-dependent Schrodinger equation into a spin polarization with respect to the electron spin Zeeman levels Ti>, Tq>, and T i> in an external magnetic field Bq. Consider a simple case of axially symmetric zero-field splitting (i.e., D y 0 and E = 0 D and E are the usual zero-field parameters). Tx>, [Ty>, and TZ> are the eigenstates of the zero-field interaction Hzfs, where Z is the major principal axis. The initial polarization arising from the population differences among these states can be expressed as... 

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