Boltzmann distribution of spins

The derivation is also subject to the following conditions and assumption 1) NMR is determined at low RF power. 2) A Boltzmann distribution of spin states obtains so that diagonal elements of the density matrix (e, see below) are given by equation 7,... 

Changes in temperature of the sample alter the Boltzmann distribution of spins in the various energy levels in proportion to 1/T, as indicated in Eq. 2.20. For example, a reduction in temperature from 27°C (300 K) to — 60°C causes an increase in signal intensity of 40%. 

Resonance absorption involves a net gain in magnetic energy by the sample, which is only possible when a greater proportion of nuclei are in the lower spin state or states. The mechanisms whereby a Boltzmann distribution of spins tends to be, or is established or maintained, are the relaxation processes, and they are very important phenomena for three reasons. 

The Ti relaxation time is the time constant for magnetisation to recover on the z axis (Figure 7.3b), restoring the equilibrium Boltzmann distribution of spins. This relaxation process occurs by energy loss to the surrounding lattice (any nearby molecules) and the efficiency of the process is determined by molecular reorientations occurring near the Larmor frequency (e.g. molecular rotations and internal motions). T, is measured with the inversion-recovery (IR) or saturation-recovery (SR) pulse sequence (see Section 7.7.2). 

Relaxation refers to all processes which regenerate the Boltzmann distribution of nuclear spins on their precession states and the resulting equilibrium magnetisation along the static magnetic field. Relaxation also destroys the transverse magnetisation arising from phase coherenee of nuelear spins built up upon NMR excitation. 

Methods of disturbing the Boltzmann distribution of nuclear spin states were known long before the phenomenon of CIDNP was recognized. All of these involve multiple resonance techniques (e.g. INDOR, the Nuclear Overhauser Effect) and all depend on spin-lattice relaxation processes for the development of polarization. The effect is referred to as dynamic nuclear polarization (DNP) (for a review, see Hausser and Stehlik, 1968). The observed changes in the intensity of lines in the n.m.r. spectrum are small, however, reflecting the small changes induced in the Boltzmann distribution. 

The origin of postulate (iii) lies in the electron-nuclear hyperfine interaction. If the energy separation between the T and S states of the radical pair is of the same order of magnitude as then the hyperfine interaction can represent a driving force for T-S mixing and this depends on the nuclear spin state. Only a relatively small preference for one spin-state compared with the other is necessary in the T-S mixing process in order to overcome the Boltzmann polarization (1 in 10 ). The effect is to make n.m.r. spectroscopy a much more sensitive technique in systems displaying CIDNP than in systems where only Boltzmann distributions of nuclear spin states obtain. More detailed consideration of postulate (iii) is deferred until Section II,D. 

By reference to Fig. 18 and by assuming a Boltzmann distribution of electron spins among the three states, the temperature dependence of the signal intensity for Tj gave AEqj = 61 cm ( = 175 cal mol ) and J was calculated to be -f-16cm ... 

Fig. 9. This Larmor spectrum was measured in the analysis trap by resonant excitation (at 104 GHz) of the transition between the two spin states (spin up and down) of the bound electron. The asymmetric line shape of the resonance curve is due to the strong magnetic inhomogeneity in the analysis trap in combination with the thermal Boltzmann distribution of the ion s axial oscillation amplitude...

There is another reason why the magnitude of 7) is important. Suppose we have a Boltzmann distribution of nuclei precessing in a magnetic field, and we irradiate the collection with photons of precisely the correct frequency (and energy) to cause transitions (spin flips) between the lower (m = + ) level and the upper (m = -i) states. Because there is initially such a small difference between the populations of the two states, it will not be long before the populations are equalized through the absorption of the photons This, of course, means the spin system has become saturated and no further net absorption is possible. However, if we turn off the source of rf radiation, the system can relax back to the Boltzmann distribution (at a rate controlled by Ti) and absorption can... 

Perhaps the most convincing proof of scalar interaction is afforded by the observation of contact shifts. These arise from the Boltzmann distribution of electron spins in the magnetic field, which tend to align with a small excess of spins antiparallel to the field, resulting in a net magnetic field at a nucleus I. Using the Curie Law, this shift is given by—... 

Signal enhancement due to NOE is an example of cross-polarization, in which a polarization of the spin states in one type of nucleus causes a polarization of the spin states in another nucleus. Cross-polarization will be explained in Section 4.6. In the current example (proton-decoupled spectra), when the hydrogens in the molecule are irradiated, they become saturated and attain a distribution of spins very different from their equilibrium (Boltzmann) state. There are more spins than normal in the excited state. Due to the interaction of spin dipoles, the spins of the carbon nuclei sense the spin imbalance of the hydrogen nuclei and begin to adjust themselves to a new equilibrium state that has more spins in the lower state. This increase of population in the lower spin state of carbon increases the intensity of the NMR signal. 

Optical electron spin polarisation (OEP) is the term used to describe a non-Boltzmann distribution of the populations of the three zero-field or Zeeman components of an optically-excited triplet state. This non-thermal equilibrium can be a stationary or a non-stationary state. The optical excitation, that is e.g. the UV excitation, must be neither narrow-band nor polarised, and at low temperatures, OEP is the normal case for most triplet states in organic tt-electron systems. The OEP is... 

A well-known and important phenomenon in the area of nuclear-spin resonance (NMR) in gases, liquids, or solid samples is dynamic nuclear-spin polarisation (DNP) (see e.g. [M6]). This term refers to deviations of the nuclear magnetisation from its thermal-equilibrium value, thus a deviation from the Boltzmann distribution of the populations of the nuclear Zeeman terms, which is produced by optical pumping (Kastler [31]), by the Overhauser effect [32], or by the effet solide or solid-state effect [33]. In all these cases, the primary effect is a disturbance of the Boltzmann distribution in the electronic-spin system. In the Overhauser effect and the effet solide, this disturbance is caused for example by saturation of an ESR transition. Owing to the hyperfine coupling, a nuclear polarisation then results from coupled nuclear-electronic spin relaxation processes, whereby the polarisation of the electronic spins is transferred to the nuclear spins. 

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