Rate constant spectral density

Among the dynamical properties the ones most frequently studied are the lateral diffusion coefficient for water motion parallel to the interface, re-orientational motion near the interface, and the residence time of water molecules near the interface. Occasionally the single particle dynamics is further analyzed on the basis of the spectral densities of motion. Benjamin studied the dynamics of ion transfer across liquid/liquid interfaces and calculated the parameters of a kinetic model for these processes [10]. Reaction rate constants for electron transfer reactions were also derived for electron transfer reactions [11-19]. More recently, systematic studies were performed concerning water and ion transport through cylindrical pores [20-24] and water mobility in disordered polymers [25,26]. 

As we shall see, all relaxation rates are expressed as linear combinations of spectral densities. We shall retain the two relaxation mechanisms which are involved in the present study the dipolar interaction and the so-called chemical shift anisotropy (csa) which can be important for carbon-13 relaxation. We shall disregard all other mechanisms because it is very likely that they will not affect carbon-13 relaxation. Let us denote by 1 the inverse of Tt. Rt governs the recovery of the longitudinal component of polarization, Iz, and, of course, the usual nuclear magnetization which is simply the nuclear polarization times the gyromagnetic constant A. The relevant evolution equation is one of the famous Bloch equations,1 valid, in principle, for a single spin but which, in many cases, can be used as a first approximation. 

A common assumption in the relaxation theory is that the time-correlation function decays exponentially, with the above-mentioned correlation time as the time constant (this assumption can be rigorously derived for certain limiting situations (18)). The spectral density function is then Lorentzian and the nuclear spin relaxation rate of Eq. (7) becomes ... 

Spatial density profiles of atomic (and molecular) species can also be made via saturation fluorescence approaches. For a "2-level" atom, like Sr, a plot of 1/Bp vs 1/E (Bp is the fluorescence radiance, in J s 1m-2sn l, and Ex is the excitation spectral irradiance, in J s nT nm ) allows estimation of the quantum efficiency, Y of the fluorescence process (and thsu estimation of "radiationless" rate constants) and the total number density nj, of the species of interest by means of... 

Here, K% is a constant related to the NMR coupling constant,. S 2 is the spectral density associated with the second-rank orientational correlation function g2(f). If g2(f) is an exponential function, such as in rotational diffusion, Eq. (16) reduces to the famous Bloembergen-Purcell-Pound (BPP) expression for the spin-lattice relaxation rate [81,82]. 

For a heteronuclear two-spin system, we focus on terms in the spectral density at (t)h which do not induce transitions of the S spins, so the second term can be ignored to give an overall rate constant of (2 Wj + Wt + W2). On the other hand, for a homonuclear system, I = S, so the two terms can be combined and simplified to give a rate constant (2W1 + 2W2). Note that W0 does not appear, a consequence of the fact that for a homonuclear system — cos 0. 

Cf. the Fermi golden rule. Section 5.2.3.) The density of states pg (number of states per unit energy interval) is related to the spectral overlap J, and using the relations for (i given above the expressions derived by Fdrster (1951) and Dexter (1953) for the rate constant of energy transfer by the Coulomb and the exchange mechanism, respectively, may be written as... 

The size of the dipolar interaction depends primarily on the distance and orientation between the two dipoles, not on the correlation time. By contrast, the rate of change of the dipolar interactions depends on and hence is relevant to the efficiency of relaxation. The total amount of fluctuating fields is independent of Tc, although Tc determines the upper limit of the frequencies of the fields. The three curves in Figure A5-1 must enclose the same area, but their upper limits vary. In curve (a), molecular tumbling is very rapid and the spectral density is low. In curves (b) and (c), the upper limit of frequencies decreases with the lengthening of Tc, so the spectral density increases proportionately to maintain a constant area. 

The value of the spectral density, J(co), has a large effect on relaxation rate constants, so it is well worthwhile spending some time in understanding the form that this function takes. 

We have just seen that for a given frequency, spectral density is a maximum when tc is 1 /co0, so to have the most rapid relaxation the correlation time should be 1 /co0. This is illustrated in the plots below which show the relaxation rate constant, W, and the corresponding relaxation time (Tx = 1/W) plotted as a function of the correlation time. 

At very short correlation times (tc 1 /co0) there is some spectral density at the Larmor frequency, but not that much as the energy of the motion is spread over a very wide frequency range. As the correlation time increases the amount of spectral density at the Larmor frequency increases and so the relaxation rate constant increases, reaching a maximum when tc = l/Larmor frequency, and hence the rate constant, falls. 

The rate constant W2 corresponds to transitions which are at the sum of the Larmor frequencies of the two spins, (spectral density at this sum frequency which is relevant. In contrast, W0 corresponds to transitions at (spectral density at this difference frequency which is relevant. 

It turns out that the secular part depends on the spectral density at zero frequency, 7(0). We can see that this makes sense as this part of transverse relaxation requires no transitions, just a field to cause a local variation in the magnetic field. Looking at the result from section 8.5.2 we see that 7(0) = 2tc, and so as the correlation time gets longer and longer, so too does the relaxation rate constant. Thus large molecules in the slow motion limit are characterised by very rapid transverse relaxation this is in contrast to longitudinal relaxation is most rapid for a particular value of the correlation time. 

E8-7. What do you understand by the terms correlation time and spectral density Why are these quantities important in determining NMR relaxation rate constants ... 

Studies of single channels formed in lipid bilayers by Staphylococcus aureus alpha toxin showed that fluctuations in the open-channel current are pH-dependent (47). The phenomenon was attributed to conductance noise that arises from reversible ionization of residues in the channel-forming molecule. The pH-dependent spectral density of the noise, shown in Figure 6, is well described by a simple model based on a first-order ionization reaction that permits evaluation of the reaction parameters. This study demonstrates the use of noise analysis to measure the rate constants of rapid and reversible reactions that occur within the lumen of an ion channel. 

We performed a calculation of the relaxation rates using the phonon Green s functions of the perfect (CsCdBr3) and locally perturbed (impurity dimer centers in CsCdBr3 Pr ) crystal lattices obtained in Ref. [8]. The formation of a dimer leads to a strong perturbation of the crystal lattice (mass defects in the three adjacent Cd sites and large changes of force constants). As it has been shown in Ref. [8], the local spectral density of phonon states essentially redistributes and several localized modes appear near the boundary of the continuous phonon spectrum of the... 

If the spectral density function drops to zero at the frequency of a given transition as the result of an increase in the correlation time Tc, then the rate constant for the transition decreases. This transition rate constant reduction is important when the size of the molecule increases, when field strength increases, or when the solution viscosity increases (e.g., during a polymerization or as the result of cooling). 

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