Thermal equilibrium spin energies

Figure 2 shows the thermal equilibrium spin population for a three-spin system. Generally there is no degeneracy present. When an appropriate polarizing agent is used, the energy levels IIV > and IV > or IVI > and IIII > become degenerated (Fig. 2a). Irradiation of EPR transition and CE transitions leads to positive (Fig. 2b) or negative enhancement (Fig. 2c) of the nuclear polarization.

Figure 4-6 illustrates the relaxational eontribution to the motion. Figure 4-6A shows moment vectors for a spin system in the absenee of the rf field (Hi = 0) the magnetization eomponents are = Mq, = 0, My = 0, beeause in the xy plane the magnetization eomponents caneel. In the presenee of the rf field at the resonanee frequency the spin system absorbs energy, increasing the angle between Ho and M and perturbing the thermal equilibrium so that and My components are induced and M < Mo (Fig. 4-6B). With the passage of time (comparable to the relaxation times Tj and Tj), relaxation back to the equilibrium configuration takes place, so M. increases toward Mo, whereas nd My decrease toward zero as a consequence of the gradual loss of coherence of the moment vectors.

It may be concluded that during the contact time in the competing process for the energy in the various spin systems, the carbon atoms are trying to reach thermal equilibrium with the proton polarization, which is in itself decreasing with a time constant, (Tig, H). In fact the protons undergo spin diffusion and can be treated together, whereas the carbon atoms behave individually. Therefore one implication is that we can also expect to obtain a C-13 spin polarization proportional to the proton polarization. 

When nuclei with spin are placed in a magnetic field, they distribute themselves between two Zeeman energy states. At thermal equilibrium the number (N) of nuclei in the upper (a) and lower (j8) states are related by the Boltzmann equation (1) where AE=E — Ep is the energy difference between the states. In a magnetic field (Hq), E = yhHo and... 

Clearly, if a situation were achieved such that exceeded Np, the excess energy could be absorbed by the rf field and this would appear as an emission signal in the n.m.r. spectrum. On the other hand, if Np could be made to exceed by more than the Boltzmann factor, then enhanced absorption would be observed. N.m.r. spectra showing such effects are referred to as polarized spectra because they arise from polarization of nuclear spins. The effects are transient because, once the perturbing influence which gives rise to the non-Boltzmann distribution (and which can be either physical or chemical) ceases, the thermal equilibrium distribution of nuclear spin states is re-established within a few seconds. 

Figure 12.3 outlines the essential features of the PASADENA/PHIP concept for a two-spin system. If the symmetry of the p-H2 protons is broken, the reaction product exhibits a PHIP spectrum (Fig. 12.3, lower). If the reaction is carried out within the high magnetic field of the NMR spectrometer, the PHIP spectrum of the product consists of an alternating sequence of enhanced absorption and emission lines of equal intensity. This is also true for an AB spin system due to a compensating balance between the individual transition probabilities and the population rates of the corresponding energy levels under PHIP conditions. The NMR spectrum after the product has achieved thermal equilibrium exhibits intensities much lower than that of the intermediate PHIP spectrum. 

At high temperatures, a nanoparticle is in a superparamagnetic state with thermal equilibrium properties as described in the previous section. At low temperatures, the magnetic moment is blocked in one potential well with a small probability to overcome the energy barrier, while at intermediate temperatures, where the relaxation time of a spin is comparable to the observation time, dynamical properties can be observed, including magnetic relaxation and a frequency-dependent ac susceptibility. 

The Monte Carlo (MC) method can be used to efficiently calculate thermal equilibrium properhes (see Fig. 3.2). However, since it is an energy-barrier-based method, it will fail to generate dynamic features such as the precession of the spins, and will be able to generate the dynamic magnetizahon in the overdamped limit (X —> oo) only if an appropriate algorithm is used [35]. 

At each temperature the equilibrium spin glass state is considered to consist of a ground state plus thermally activated droplet excitations of various sizes. A droplet is a low-energy cluster of spins with a volume if and a fractal surface area L. The typical droplet free-energy scales as... 

Figure 4 Schematic representation of the populations of the nuclear spin energy levels of a quadrupolar nucleus with spin 5/2 (such as Mg) under a strong magnetic field and a perturbative quadrupole coupling showing (A) populations at thermal equilibrium, (B) populations after complete saturation of the satellite transitions, and (C) populations after complete inversion of the satellite transitions, following the order first, inversion of STl and ST4 and then inversion of ST2 and ST3. The numbers at left of each level (named pj in the text) are proportional to the population of that level, with —hVl/ 2k T= 0. ...

The results obtained from thermal spin equilibria indicate that AS = 1 transitions are adiabatic. The rates, therefore, depend on the coordination sphere reorganization energy, or the Franck-Condon factors. Radiationless deactivation processes are exothermic. Consequently, they can proceed more rapidly than thermally activated spin-equilibria reactions, that is, in less than nanoseconds in solution at room temperature. Evidence for this includes the observation that few transition metal complexes luminesce under these conditions. Other evidence is the very success of the photoperturbation method for studying thermal spin equilibria intersystem crossing to the ground state of the other spin isomer must be more rapid than the spin equilibrium relaxation in order for the spin equilibrium to be perturbed. 

In a real system, there is not just one isolated nucleus. In reality, there are many nuclei, and all them can occupy a particular spin state. After a certain time after the application of the external magnetic field, the spin system will reach the state of thermal equilibrium with a thermostat. This means that we should consider an ensemble of spins consequently, applying the canonical ensemble methodology, it is easy to calculate the ratio of the populations of the two spin energy states [45] ... 

---