Kramers’ relation

This result reflects the Kramers relation (Gardiner, 1985). A millisecond time of unbinding, i.e.. Tact 1 ms, corresponds in this case to a rupture force of 155 pN. For such a force the potential barrier AU is not abolished completely in fact, a residual barrier of 9 kcal/mol is left for the ligand to overcome. The AFM experiments with an unbinding time of 1 ms are apparently functioning in the thermally activated regime. 

From the Kronig-Kramer relation it immediately follows that A(e) = 0, as the function to be integrated is odd, and hence the resulting projected density of states becomes... 

The Kronig-Kramers relation is of fundamental importance for optics and for physics in general13). Here, these equations do not seem practical because of the integration of the wavelength from 0 to oo. However, these are very useful for calculating the molar ellipticity magnitude from the observed ORD curve 14). 

Intrinsic Viscosity. Dilute-solution viscometry of samples in toluene was carried out in a Cannon-Ubbelohde semimicrodilution viscometer (size 25) in a temperature-controlled bath (25.0 0.2 °C). At least three concentrations were measured, and the results were extrapolated to infinite dilution by using the Huggins and Kramers relations... 

Linear viscoelasticity theory predicts that one component of a complex viscoelastic function can be obtained from the other one by means of the Kronig-Kramers relations (10-12). For example, the substitution of G t) — Ge given by Eq. (6.8b) into Eq. (6.3) leads to the relationship... 

The beauty of the linear viscoelastic analysis lies in the fact that once a viscoelastic function is known, the rest of the functions can be determined. For example, if one measures the comphance function J t), the values of the components of the complex compliance function can in principle be determined from J(t) by using Fourier transforms [Eqs. (6.30)]. On the other hand, the components of the complex relaxation moduh can be obtained from those of / (co) by using Eq. (6.50). Even more, the real components of both the complex relaxation modulus and the complex compliance function can be determined from the respective imaginary components, and vice versa, by using the Kronig-Kramers relations. Moreover, the inverse of the Fourier transform of G (m) and/or G"(co) [/ (co) and/or /"(co)] allows the determination of the shear relaxation modulus (shear creep compliance). Finally, the convolution integrals of Eq. (5.57) allow the determination of J t) and G t) by an efficient method of numerical calculation outlined by Hopkins and Hamming (13). 

The Cotton effects can be either positive or negative. ORD and CD are related phenomena and data obtained by each method can be transformed to the other by means of the Kronig-Kramers relations [e.g., Ref. (18)]... 

The Kramers-restricted form of the Hamiltonian that was used in Cl theory is not suitable for Coupled Cluster theory because it mixes excitation and deexcitation operators. One possibility is to define another set of excitation operators that keep the Kramers pairing and use these in the exponential parametrization of the wavefunction. This would automatically give Kramers-restricted CC equations upon rederivation of the energy and amplitude equations. A more pedestrian but simpler alternative is to start from the spin-orbital formulation and inspect the relations that follow from the Kramers relation of the two-electron integrals. This method does also readily give the relations between the Kramers symmetry-related amplitudes. We will briefly discuss the basic steps in this approach, a detailed description of a possible algorithm is given in reference [47],... 

The final check of the Kramer relations was to decrease the sample thickness while keeping the strain amplitude constant. The sample volume was decreased proportionally. These results are presented in Figure 3. In Figure 3 the modulus values are reduced by the SE 30 modulus at the same frequency as measured on the RMS. 

The only remaining variable of interest in the Kramer relations is the sample aspect ratio (h /R). To test this the sample thickness was varied at constant strain amplitude. As we saw in Figure 3 there appeared to be a dependence of the calculated loss modulus on sample thickness. Some portion of the effect of sample thickness is not being corrected for by the Kramer (8) relations. 

If both components of G are known at a single frequency, both components of J can be very simply calculated by equations 27 to 30 of Chapter 1. On the other hand, if one component is known over the whole frequency range, the other can be obtained from it by mechanical analogs of the Kronig-Kramers relations. " ... 

The Kronig(47)-Kramers(48) relations arise mathematically from two considerations, the first being that fluid behavior is in the linear-response regime, and the second being that causality is satisfied. By linear response it is meant that the response of a fluid to a series of applied forces is the sum of the distinct responses that would have been created by applying separately each force in the series. The causality requirement mandates that the shear stress relaxation function G(t), which describes the shear stress required to maintain constant a shear strain initially imposed at t = 0, must satisfy G(t) = 0 for r < 0. The results of these considerations(47,48) are the Kronig-Kramers relations, which may be written... 

Furthermore, the actual G cff) and G foo) undoubtedly satisfied the Kronig-Kramers relations to within experimental error. If the ansatz is correct, the ansatz... 

Figure 13.39 Test of the Kronig-Kramers relations. G t) is calculated from Eqs. 13.16 and 13.17, using the temporal scaling ansatz functional forms as fitted to data from Colby, et a/. (15) to represent G and G". Solid and dashed lines represent, respectively, G t) from G and from G", for polybutadiene in phenyloctane at volume fractions (top to bottom) 1.00,0.49,0.28,0.14,0.062, and 0.027.

Sixth, the functional forms and experimental material-dependent parameters were shown in Subsection 13.5.3 to he consistent with the Rronig-Kramers relations. 

First, because the parameters give smooth functional forms for G a>) and G"(co) over a full range of frequencies, they may be used to confirm that the fitting functions satisfy the Kronig-Kramers relations. Indeed, calculations of G t) from G and separately from G" are in agreement, as required by these relations. 

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