Lowest mode

If we suppress the mode dependence of and consider only its lowest mode value ( then the intrinsic viscosity is given by... 

Fig. 5.1 Mode number dependence of the relaxation times Tj and T2 (solid lines). The dashed-dotted line shows the relaxation time ip in the Rouse model (Eq. 3.12). The horizontal dashed line displays the value of r. The dashed and the dotted lines represent the relaxation time when the influence of the chain stiffness is considered mode description of the chain statistics iq (dashed, Eq. 5.11) and bending force model tp (dotted, Eq. 5.7). The behaviour of the relaxation time used in the phenomenological description is also shown for the lowest modes (see text). (Reprinted with permission from [217]. Copyright 1999 American Institute of Physics)...

Figure 7 Application of the dimer method to a two-dimensional test problem. Three different starting points are generated in the reactant region by taking extrema along a high temperature dynamical trajectory. From each one of these, the dimer isjirst translated only in the direction of the lowest mode, but once the dimer is out of the convex region a full optimization of the effective force is carried oat at each step (thus the kink in two of the paths). Each one of the three starting p>oints leads to a different saddle point in this case.

The correlation soon becomes dominated by the relaxation of the lowest mode ... 

In order to show that this procedure leads to acceptable results, reference is briefly made to the normal coordinate transformation mentioned at the end of Section 2.2. By this transformation the set of coordinates of junction points is transformed into a set of normal coordinates. These coordinates describe the normal modes of motion of the model chain. It can be proved that the lowest modes, in which large parts of the chain move simultaneously, are virtually uninfluenced by the chosen length of the subchains. This statement remains valid even when the subchains are chosen so short that their end-to-end distances no longer display a Gaussian distribution in a stationary system [cf. a proof given in the appendix of a paper by Ham (75)]. As a consequence, the first (longest or terminal) relaxation time and some of the following relaxation times will be quite insensitive for the details of the chain... 

By choosing a level shift in the range Xk < i < kk+i we take a step which initially at least increases the function along the k lowest modes and reduces it along all higher modes. Therefore, if at each step we select a level shift in this range we may eventually expect to enter the local region of the k th excited state. 

Using the same method as for the first excited electronic state, we select a level shift in the region Xi < p < X2. This procedure may indeed lead to a transition state but in this way we always increase the function along the lowest mode. However, if we wish to increase it along a higher mode this can only be accomplished in a somewhat unsatisfactory manner by coordinate scaling. Nevertheless, this method has been used by several authors with considerable success.14 The problem of several first-order saddle points does not arise in electronic structure calculations since there is only one first excited state.15... 

Hence, in the diagonal representation the gradient and Hessian are identical except for opposite sign in the lowest mode.19 Therefore, a first-order saddle point of the function coincides with a minimum of the image and we may determine the transition state by minimizing the image function. 

In RSO minimizations we minimize Eq. (6.19) within the trust region. If instead we wish to maximize the lowest mode and minimize the others we may use the model... 

Certain simplifications that allow the dynamic response to be reconciled with equivalent static loadings are examined. In earthquake loading the dominant effects are found to occur in the lowest mode for which no cross sectional distortion takes place. In wind loading the dynamic response is spread over several modes. The maximum dynamic tensile stresses at the windward base of the tower can be estimated using simple gust effect factors. 20 refs, cited. 

The validity of this relation for a range of n-alkanes means no more than that their Raman LA lowest mode (accordion mode) has a frequency inversely proportional to the number of carbon atoms in the chain and this is true only for the crystalline material). 

Fig. 4. Nonlinear conductance oscillations at low field from a 6 /mi junction, (a) shows the oscillations as a function of both B and V. (A smoothed background has been subtracted to emphasize the oscillations.) The brightest (and darkest) lines, corresponding to tunneling between the lowest modes, break the V-B plain into regions I, II, and III. Additional positively-sloped bright and dark lines in II arise from other ID channels in the wires and are disregarded in our theoretical analysis. Also present is a slow modulation of the strength of the oscillations along the abscissa, (b) Absolute value of the peak of the Fourier transform of the conductance at a fixed V in region II as a function of V. Its slow modulation as a function of V is easily discerned.

Alternatively, when process (3) is slower than (4) or (5), but faster than (1) or (2), A will again relax by the route (3) followed by (4) or (5), but now (3) will be rate determining. This will give a linear variation of 1// A with x. B will relax independently, and more rapidly, via (4) and (3), with linear dependence of 1// B on x. There will thus be a double relaxation phenomenon with two relaxation times, PA involving only the vibrational heat capacity of A, and / B only that of B, both showing linear concentration dependence. This mechanism is analogous to the relaxation behaviour discussed in Section 3.1 for pure polyatomic gases, which show double dispersion because vibration-vibration transfer between modes is slower than vibration-translation transfer from the lowest mode. 

RATIOS OF REDUCED RELAXATION TIMES FOR LOWEST MODES FOR DEUTERIDES AND HYDRIDES... 

The general conclusions probably are independent of the details. The two higher-frequency peaks, derived from (o and can be reasonably well described in a local mode approximation, but the lowest mode, derived from (Oy, is so mixed with the acoustical modes that it should be treated together with the acoustical modes in a more complete calculation. 

Figure 6. The power spectrum density of the lowest mode energy for the FPU p model. The initial value of each mode energy is fixed at E = 0.5. The system size is given as N = 8,16,32,64,128 from left to right, respectively. The line F(f) =A/ 1 + (f/fs) 1 is obtained by fitting the power spectrum density in the frequency between two J. marks. The shoulder at/ =fs is indicated by a " mark. [Reprinted with permission from Jpn. J. Appl. Phys. 35 2387-2393 (1996). Copyright 1996 by The Institute of Pure and Applied Physics.]...

Figure 14.5 shows residuals extracted by removing the three lowest modes of the wooden bar strikes of Figure 14.4. The increased high frequency energy is evident in the hard stick, as well as some extra high frequency modes. These modes could be cross-modes (nonlongitudinal) on the bar, or modes of the stick itself Experimentation by striking the stick on a nonresonant hard surface (marble stone) and recording the sound showed that the two modes at about 17 k and 20 k were indeed modes of the plastic stick itself...

As noted previously in this chapter, Eqs. (9.13)-(9.16) indicate that the line shape of the G t) curve is a universal function of the normalized molecular weight M/Me in the region covered by the pB t) and pc(t) processes. The relaxation times tb and tc can also be normalized with respect to the relaxation time of the lowest mode of the pA t) process, KM /6,... 

Fig. 14.21 Comparison of the H t) distributions of samples A (middle figure), B (bottom one) and C (top one) at individual glass transition points or AT = 0. In each figure, the relaxation times of the Rouse-Mooney normal modes (for samples A and B) or the Rouse normal modes (for sample C) are indicated (upper +). In the entangled cases (samples A and B) the relaxation times of the individual dominant (lowest) modes in the jUB(t) and iac t) processes are also indicated (lower +). The vertical dotted...

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