Boltzmann distribution, electron nuclear

Boltzmann distribution, electron nuclear dynamics (END), intramolecular electron transfer, 350-351... 

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. 

A major limitation for NMR spectroscopy is the intrinsically low sensitivity due to the rather unfavorable Boltzmann distribution for nuclear spins at thermal equilibrium. Thus, considerable effort in magnetic resonance spectroscopy is made towards sensitivity enhancement by hyperpolarization techniques, such as optical polarization, para-hydrogen-induced polarization enhancement, and dynamic nuclear polarization (DNP), a method which exploits the magnetization of unpaired electrons in stable radicals or transition metals to enhance nuclear polarization beyond the Boltzmann limit. In the chapter Dynamic Nuclear Hyperpolarization in Liquids, the fundamental theory for different polarization transfer... 

Transition State Theory [1,4] is the most frequently used theory to calculate rate constants for reactions in the gas phase. The two most basic assumptions of this theory are the separation of the electronic and nuclear motions (stemming from the Bom-Oppenheimer approximation [5]), and that the reactant internal states are in thermal equilibrium with each other (that is, the reactant molecules are distributed among their states in accordance with the Maxwell-Boltzmann distribution). In addition, the fundamental hypothesis [6] of the Transition State Theory is that the net rate of forward reaction at equilibrium is given by the flux of trajectories across a suitable phase space surface (rather a hypersurface) in the product direction. This surface divides reactants from products and it is called the dividing surface. Wigner [6] showed long time ago that for reactants in thermal equilibrium, the Transition State expression gives the exact... 

Nonequilibrium effects. In applying the various formalisms, a Boltzmann distribution over the vibrational energy levels of the initial state is assumed. The rate constant calculated on the basis of the equilibrium distribution, keq, is the maximum possible value of k. If the electron transfer is very rapid then the assumption of an equilibrium distribution over the energy levels is not valid, and it is more appropriate to treat the nuclear fluctuations in terms of a steady-state rather than an equilibrium formalism. Although a rigorous treatment of this problem has not yet appeared, intuitively it seems that since the slowest nuclear fluctuation will generally be a solvent orientational motion, ke will equal keq when vout keq and k will tend to vout when vout keq (a simple treatment gives l/kg - 1/ vout + 1/keq). These considerations are... 

To obtain hyperpolarizabilities of calibrational quality, a number of standards must be met. The wavefunctions used must be of the highest quality and include electronic correlation. The frequency dependence of the property must be taken into account from the start and not be simply treated as an ad hoc add-on quantity. Zero-point vibrational averaging coupled with consideration of the Maxwell-Boltzmann distribution of populations amongst the rotational states must also be included. The effects of the electric fields (static and dynamic) on nuclear motion must likewise be brought into play (the results given in this section include these effects, but exactly how will be left until Section 3.2.). All this is obviously a tall order and can (and has) only been achieved for the simplest of species He, H2, and D2. Comparison with dilute gas-phase dc-SHG experiments on H2 and D2 (with the helium theoretical values as the standard) shows the challenge to have been met. 

The mechanism is thus entered from the top, either by the left branch or the right branch, producing either the radical pair on the left or that on the right. Looking at an ensemble, the radical pairs initially all have the same electron spin multiplicity, and all nuclear spin states are populated according to the Boltzmann distribution (i.e. are populated practically equally). 

In all the experiments mentioned above, the Overhauser effect has been observed by irradiating the e.s.r. signal of the dissolved free radicals. However the essential conditions for production of an Overhauser effect are that the populations of the electron spin Zeeman levels should depart from their thermal equilibrium value and that, as the electron spins relax and attempt to restore the Boltzmann distribution among their levels, they should interact with the nuclear spins present in solution. 

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. 

Another technique for the study of reactions that is highly specific for radical processes is known as CIDNP, an abbreviation for chemically induced dynamic nuclear polarization." The instrumentation required for such studies is a normal NMR spectrometer. CIDNP is observed as a strong perturbation of the intensity of NMR signals in products formed in certain types of free radical reactions. CIDNP is observed when the normal population of nuclear spin states dictated by the Boltzmann distribution is disturbed by the presence of an unpaired electron. The intense magnetic moment associated with an electron causes a polarization of nuclear spin states, which is manifested by enhanced absorption or emission, or both, in the NMR spectrum of the diamagnetic product of a free radical reaction. The technique is less general than EPR spectroscopy because not all free radicals can be expected to exhibit the phenomenon. 

A critical point in the retrieving of the number of nuclear reactions in laser-solid experiments is that there is no control on the spectrum of the electrons accelerated in the interaction, as well as the acceleration mechanism is uncertain and difficult to fit in a predictable scheme. In most cases, the electron energy distribution is assumed to be Boltzmann-like and deconvolutions are performed starting from this assumption. 

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