Relaxation moment problem

We have not yet specified if the operator to be handled is Hermitian (real eigenvalues) or whether it is a relaxation operator (eigenvalues either real or in the lower half of the complex plane). Uie moment problem related to a Hermitian operator is addressed as the classical moment problem, while by relaxation moment problem we mean the treatment of relaxation operators. 

The moment problem has been almost exclusively studied in the literature having (implicitly) in mind Hermitian operators (classical moment problem). With the progress of the modem projective methods of statistical mechanics and the description of relaxation phenomena via effective non-Hermitian Hamiltonians or Liouvillians, it is important to consider the moment problem also in its generalized form. In this section we consider some specific aspects of the classical moment problem, and in Section V.C we focus on peculiar aspects of the relaxation moment problem. 

It is not possible to extend right away the results of the classical moment problem to the relaxation moment problem. However, our survey of Section V.B has been done in such a way that it is possible to select which relations maintain their validity in the relaxation moment problem and which are to be disregarded. Thus little remains to be said except for a few comments. 

Before closing this section, we wish briefly to comment on how to deal with the relaxation moment problem in this case the usual definition of moments,... 

The first illustration is provided by ferroelectrics belonging to the family of pyridinium salts. Complex interplay between the contributions of van der Waals, Coulomb, dipolar and hydrogen-bonding interactions are expected because of the hybrid nature of the compound. The majority of reported NMR experiments are proton second-moment and relaxation studies on polycrystalline samples. The most sophisticated NMR methods with regard to resolution, symmetry and time-scale interpretations applied to the historical problem of assigning a pure order disorder or displacive mechanism to a ferroelectric phase transition will provide the second example with the study of squaric acids and perovskites compounds like BaTi03. 

Measurements of relaxation times fall broadly into two classes, those which monitor the populations of some chosen states, and those which measure in some way the impedance of the system to the propagation of a thermal disturbance many laser experiments fall into the first class, whereas ultrasonic dispersion or shock-tube measurements fall into the second. Although artefacts can occur if unsuitable population v. time profiles are used [76.P3], there is, in general, no real difficulty in using equation (2.14) to obtain the vibrational relaxation rate we need not discuss this point further at the moment. Problems may well arise, though, in the determination of rotational relaxation rates in this way, as I will show. 

As we have seen above, the temporal evolution of Mg is due to the longitudinal relaxation The temporal evolution of Mj (or My), which is described as transverse relaxation, is fundamentally different. It corresponds to the loss of phase coherence between the individual magnetic moments. Transverse relaxation can be described as an entropy relaxation in the sense that the final state (0, 0, Mq) is characterized by a higher entropy than the initial state (M, My, M ). As in the case of the temporal evolution of Mg, the analytical form of the My f(t) function is not predictable without careful analysis. This problem will also be discussed later. 

It is well known that in bulk crystals there are inversions of relative stability between the HCP and the FCC structure as a fxmction of the d band filling which follow from the equality of the first four moments (po - ps) of the total density of states in both structures. A similar behaviour is also expected in the present problem since the total densities of states of two adislands with the same shape and number of atoms, but adsorbed in different geometries, have again the same po, pi, P2/ P3 when the renormalization of atomic levels and the relaxation are neglected. This behaviour is still found when the latter effects are taken into account as shown in Fig. 5 where our results are summarized. 

All of the heteroatoms possess at least one naturally occurring isotope with a magnetic moment (Table 15). The nuclei 14N, 170 and 33S also possess an electric quadrupole moment which interacts with the electric field gradient at the nucleus, providing a very efficient mechanism for relaxing the nuclear spin. The consequence of this facilitation of relaxation is a broadening of the NMR signals so that line widths may be 50-1000 Hz or even wider. To some extent this problem is offset by the more extensive chemical shifts that are observed. The low natural abundances and/or sensitivities have necessitated the use of accumulation techniques for all of these heteroatoms. The relative availability of 170 and 15N enriched... 

Below we (1) disregard the chemical reaction running in the c-phase, (2) consider the temperature TB of the surface constant, and (3) consider the relaxation time of processes running in the gas to be quite small compared to the relaxation time (time of variation) of the distribution of heat in the c-phase. Restricting ourselves to consideration of intervals of time which are significant compared to the relaxation time of processes in the gas, we shall consider that the state of the nearest layer to the gas, in which the chemical reaction is concentrated, corresponds at each moment to the heat distribution in the c-phase. After establishing this correspondence the problem reduces to consideration of the comparatively slow variation of the heat distribution in the c-phase. 

However, the NMR properties of solid-phase methane are very complex, due to subtle effects associated with the permutation symmetry of the nuclear spin set and molecular rotational tunnelling.55 Nuclear spin states ltotai = 0 (irred. repr. E), 1 (T) and 2 (A) are observed. The situation is made more complicated since, as the solids are cooled and the individual molecules go from rotation to oscillation, several crystal phases become available, and slow transitions between them take place. Much work has been done in the last century on this problem, including use of deuterated versions of methane for example see Refs. 56-59. Much detail has emerged from NMR lineshape analysis and relaxation time measurements, and kinetic studies. For example, the second moment of the 13C resonance is found to be caused by intermolecular proton-carbon spin-spin interaction.60 Thus proton inequivalence within the methane molecules is created. 

Although at first glance 2D correlation spectroscopy of C with quadrupolar nuclei seems unfeasible due to relaxation problems, there are two metal nuclei where, owing to the relatively small quadrupolar moment, such a correlation has been performed. Gunther and co-workers were the first... 

Nuclei with spin 7 1 display another problem arising from the electric quadrupole moment which interacts with the electric field gradient created by the surrounding electrons. This property leads to a decrease of the relaxation times and, accordingly, an increase of the linewidths. This situation theoretically limits the use of NMR to nuclei with highly symmetric environments. The cluster effect , as suggested by Rehder [8], sometimes leads to important exceptions to this rule, as shown below. 

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