Zeeman spin levels

For a Hamiltonian of the form in Eq. (10), it is clear that matrix elements of the form (j H(t) k) in the equation for transition rate (Eq. (8)) are only non-zero if the Zeeman spin levels j and k differ in magnetic quantum number, m, by M, where M is the index in Eq. (10). The non-zero matrix elements (j H(t) k) then depend only on the corresponding A2m and the matrix elements of the T2m operators, which are just numbers. Thus, the integral in Eq. (8) for the transition rate can be rewritten as a sum of integrals43 labelled J Mco0) where... 

One such analysis in the review period involves the characterization of the rotation of the methyl groups in pyridoxine (vitamin B6).49 The temperature dependencies of the 1H spin-lattice relaxation time T, and Tld (the relaxation time constant characterizing the relaxation of dipolar order, a population distribution over the Zeeman spin levels, which corresponds to a density operator component T20, he. I z - Ii. I2, to equilibrium) at three different applied field strengths and for a variety of temperatures were determined, yielding the curves in Fig. 30. The only motion that could affect... 

Figure 5. Energy level diagram for a spin 5/2 nucleus showing the effect of the first-order quadrupolar interaction on the Zeeman energy levels. The (m V2 m = -V2) transition (shown in bold) is independent of the quadrupolar interaction to...

This equation means that the spin populates the two Zeeman energy levels according to the Boltzmann distribution. 

Earlier in this chapter when dealing with the nuclear Zeeman interaction we calculated the behaviour of the nuclear spin levels of H2 ignoring the effects of nuclear shielding. We now return to this question in more detail. We have shown that the Zeeman interaction for a nucleus of spin I should be written in the form... 

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... 

Transitions between Zeeman energy levels, and hence nuclear magnetic relaxation, are caused by fluctuations (time variations) in the local interactions at the nucleus that can cause transitions. The transition rates between these energy levels that cause nuclear spin relaxation depend on two factors ... 

The terms are the energy of the each of the spin levels of the system before application of a magnetic field. The terms and E the first- and second-order Zeeman terms, respectively, quantify the response of the system to the applied magnetic field to the first and second orders. The exponential terms represent the relative population of each of these levels as a function of temperature according to Boltzmann statistics. The derivation of and Tor complex systems using perturbation theory is beyond the scope of this introductory chapter however, for illustration, the equation is applied to three cases to demonstrate its general use. 

We take for our example a system of two-spin-levels separated by a Zeeman-splitting of hco0. In the matrix description of these two levels, the off-diagonal spin-lattice coupling is given by (3.15) or (3.16) with the relevant matrix being... 

ESR studies as applied to micellar systems rely on the sensitivity of a free radical probe to its microenvironment. Molecular species with a free electron posses labeled intrinsic angular momentum (spins), which in an external magnetic field undergoes Zeeman splitting. For a system with 5 = 1/2, two Zeeman energy levels are possible whose energy gap (AE) is given by ... 

Fig. 2. Zeeman energy levels for electron spin 5 and nuclear spin / (a) scalar coupling, dominant relaxation transition S / (b) dipolar coupling, dominant relaxation transition S-1+. ...

Originally the three-spin effect was postulated to explain the positive enhancements observed for certain fluorine nuclei in solvents also containing protons.The possibility arose that the fluorine nuclei were being influenced by the proton polarization as well as by the direct coupling to the electron spins. The basic idea behind the three-spin effect can be seen by considering the transitions among the Zeeman energy levels shown in Fig. 20. Since the electron—proton interaction is of the dipolar kind, the predominant coupled transition will be an... 

Note that one could have transitions of a nucleus up one Zeeman level and another nucleus down one Zeeman level to maintain the magnetization while altering the dipolar order. Because the dipolar reservoir is usually isolated from the Zee-man reservoir, a system can be prepared so that its Zeeman spin temperature is different from its dipolar temperature... 

Figure 2. Effect of the quadrupole interaction on the Zeeman energy levels of an 1 = 1 nucleus with axial symmetry. I is the total spin, m its component, and 6 is the angle between the applied magnetic field and the principal (z) axis of the quadrupole splitting tensor.

Figure 5. Energy level diagram for a spin 5/2 nucleus showing the effect of the first-order quadrupolar interaction on the Zeeman energy levels.

The barrier to rotation of CH3 groups, V 5.0 kJ/mol, was first estimated from a correlation between barrier contributions of other groups and the minimum internuclear distance for various organic compounds (using d(Ge-C) = 1.98 A) [3]. Very low values based on H NMR relaxation measurements (2.7 kJ/mol) [34] and experimental and calculated entropies (3.1 kJ/ mol) [57] cannot be correct see also [90, p. 19]. Consistent values for solid Ge(CH3)4 have been obtained from far-infrared studies [52], neutron scattering [80,90,91], and proton Zeeman spin-lattice relaxation experiments [94] (barrier height V and torsional ground level Eq in... 

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