Spin levels

Application of an oscillating magnetic field at the resonance frequency induces transitions in both directions between the two levels of the spin system. The rate of the induced transitions depends on the MW power which is proportional to the square of oi = (the amplitude of the oscillating magnetic field) (see equation (bl.15.7)) and also depends on the number of spins in each level. Since the probabilities of upward ( P) a)) and downward ( a) p)) transitions are equal, resonance absorption can only be detected when there is a population difference between the two spin levels. This is the case at thennal equilibrium where there is a slight excess of spins in the energetically lower p)-state. The relative population of the two-level system in thennal equilibrium is given by the Boltzmaim distribution... 

Figure Bl.16.6. An example of CIDNP net effeet for a radieal pair with two hyperfme interaetions. Part A shows the spin levels and sehematie NMR speetnim for unpolarized prodnet. Part B shows the spin levels and sehematie NMR speetnim for polarized prodnet. Populations are indieated on eaeh level. Initial eonditions ...

Figure Bl.16.8. Example of CIDNP multiplet effect for a syimnetric radical pair with two hyperfme interactions on each radical. Part A is the radical pair. Part B shows the spin levels with relative Q values indicated on each level. Part C shows the spm levels with relative populations indicated by the thickness of each level and the schematic NMR spectrum of the recombination product.

Figure 4. Spin-orbit splitting in AT — 1 and 2 vibronic levels of the state of NCN. Solid lines connect the results of calculations thar employ ab initio computed potential curves [28], For comparison the results obtained by employing experimentally derived potential curves (dashed lines) [30,31] are also given. Full points represent energy differences between P — K — and P — K spin levels, and crosses are differences between P — K + I and P — K levels.

The electron spin resonance spectrum of a free radical or coordination complex with one unpaired electron is the simplest of all forms of spectroscopy. The degeneracy of the electron spin states characterized by the quantum number, ms = 1/2, is lifted by the application of a magnetic field, and transitions between the spin levels are induced by radiation of the appropriate frequency (Figure 1.1). If unpaired electrons in radicals were indistinguishable from free electrons, the only information content of an ESR spectrum would be the integrated intensity, proportional to the radical concentration. Fortunately, an unpaired electron interacts with its environment, and the details of ESR spectra depend on the nature of those interactions. The arrow in Figure 1.1 shows the transitions induced by 0.315 cm-1 radiation. 

The first term is characterized by a scalar, 7, and it is the dominant term. Be aware of a convention disagreement in the definition of this term instead of -27, some authors write -7, or 7, or 27, and a mistake in sign definition will turn the whole scheme of spin levels upside down (see below). The second and third term are induced by anisotropic spin-orbit coupling, and their weight is predicted to be of order Ag/ge and (Ag/ge)2, respectively (Moriya 1960), when Ag is the (anisotropic) deviation from the free electron -value. The D in the second term has nothing to do with the familiar axial zero-field splitting parameter D, but it is a vector parameter, and the x means take the cross product (or vector product) an alternative way of writing is the determinant form... 

Considering any of these paradigms, a minimal goal for toy models would be to manipulate the quantum dynamics of a small number of spin levels , and that requires a known and controlled composition of the wavefunction, sufficient isolation and a method for coherent manipulation. As illustrated in Figure 2.13, the first few magnetic states of the system are labelled and thus assigned qubit values. The rest of the spectrum is outside of the computational basis, so one needs to ensure that these levels are not populated during the coherent manipulation. 

Significant progress in signal enhancement methods for the central transition has been achieved by the implementation of double frequency sweeps (DFS) [62]. The basic idea of DFS, applicable for both static and MAS experiments, is to invert simultaneously the STs so that the populations of the outer spin levels are transferred to the CT energy levels before they are selectively excited (Fig. 4). 

Translational levels are too closely spaced to be considered quantized, while the very small differences between nuclear spin levels arise only when the molecules are subjected to a strong magnetic field. 

Figure 2. Electron spin levels in a magnetic field, showing the resonance conditions at H = Hq.

Figure 3. Spin levels for an electron interacting with the N atom (1=1) in the nitroxide radical. The three allowed transitions generate an ESR spectrum with hyperfine splitting, A.

An electron spin can relax by coupling with a neighboring electron spin. Therefore, when a paramagnetic metal ion interacts with a second paramagnetic metal ion, the electron relaxation rates of the two metal ions may be dramatically affected. If Si and S2 are the two spins coupled by a scalar interaction, new spin levels will be established due to the interaction, with total S varying in unitary steps from Si — S2I to Si + S2. The energies of these spin levels are given by )... 

Nuclear relaxation rates, iron-sulfur proteins, 47 267-268 Nuclear resonance boron hydrides and, 1 131-138 fluorescence, 6 438-445 Nuclear spin levels, 13 140-145 Magnetic properties of nuclei, 13 141-145 Nuclear testing... 

Electron spin resonance can provide detailed information about free radical (ions) in condensed media. Transitions between the electron spin levels are stimulated by radiation at frequencies satisfying the resonance condition ... 

The theory of CIDNP depends on the nuclear spin dependence of intersystem crossing in a radical (ion) pair, and the electron spin dependence of radical pair reaction rates. These principles cause a sorting of nuclear spin states into different products, resulting in characteristic nonequilibrium populations in the nuclear spin levels of geminate (in cage) reaction products, and complementary populations in free radical (escape) products. The effects are optimal for radical parrs with nanosecond lifetimes. 

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