Relaxed fitting

If the gradient vector is g and the Hessian matrix is H, then the displacements that would occur on optimization A, assuming the local energy surface is quadratic, will be given by  

Hence we could minimize the displacement vector with respect to the fitted parameters in place of the gradients. However, in many cases the quadratic approximation is not sufficient and in some cases the Hessian may not even be positive definite so we would have to include further tests to ensure that the fit is valid. 

There is also a second flaw in the conventional approach to fitting in that the curvature related properties are only strictly calculable directly from the second derivative matrix when the gradients are zero. Unless the fit to the structure is already perfect then trying to reproduce elastic and dielectric constants at the experimental structure is far from ideal. 

Both of the above difficulties can be resolved by performing a full optimization of the structure with a subsequent property calculation for each point during the fitting procedure. This method, which has become known as Velaxed fitting, thus yields the exact displacements and genuine physical properties (Gale 1996). 

Property Calcite Experimental Calculated Aragonite Experimental Calculated  

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Lavorel B., Millot G., Bonamy J., Robert D. Study of rotational relaxation fitting laws from calculation of SRS N2 Q-branch, Chem. Phys. 115, 69-78 (1987). 

The relaxation fits of the Mossbauer spectra of [Fe(HB(pz)3)2] yield [30] the temperature dependence of both the population of the iron(II) high-spin and low-spin states and the relaxation rate between these two states. The resulting population of the high-spin state has a striking resemblance to that of the magnetic moment shown in Fig. 1 and these populations provide clear support both for the spin-state crossover and for the difference in populations upon heating and cooling. 

Sub-glass relaxations fit this equation. Plotting cosh [straight lines from whose slopes (= mEJPi) the evolution of the parameter m with the frequency of the isochrones can be evaluated. The Fuoss-Kirkwood equation also allows determination of the relaxation strength of sub-glass absorptions. 

In the particular application to dielectric relaxation, fit) is the aftereffect function following the removal of a constant field [8]. The solution of Eq. (93) rendered in the frequency domain yields the Cole-Davidson equation [Eq. (10)] [28],... 

Fig. 133. Temperature dependenee of the power in the power-exponential relaxation fit to the ZF speetra of single-erystalline CePt2Snj. From Luke et al. (1997a).

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Figure 12.4 Arrhenius plots for sub-T (fitted to Arrhenius equation) and glass—rubber relaxation (fitted to VFTH equation).

Fig. XIV-2. Dielectric relaxation spectrum of a water-in-oil emulsion containing water in triglyceride with a salt concentration of 5 wt % at a temperamre of 25°C. The squares are experimental points and the lines are fits to Eq. XIV-4. (From Ref. 9.)...

Figure Bl.4.6. Left an experimental optieal THz pump-probe set-up using sub-pieoseeond THz pulse generation and deteetion by the eleetro-optie effeet. Right the applieation of sueh pulses to the relaxation of optieally exeited TBNC in toluene. The THz eleetrie field used for these experiments is shown in the upper-right inset. Tluee exponential deeay tenns, of order 2, 50 and 700 ps, are required to fit the observed temporal relaxation of the solvent [51].

Figure Bl.14.6. J -maps of a sandstone reservoir eore whieh was soaked in brine, (a), (b) and (e), (d) represent two different positions in the eore. For J -eontrast a saturation pulse train was applied before a standard spin-eeho imaging pulse sequenee. A full -relaxation reeovery eiirve for eaeh voxel was obtained by inerementing the delay between pulse train and imaging sequenee. M - ((a) and (e)) and r -maps ((b) and (d)) were ealeulated from stretehed exponentials whieh are fitted to the magnetization reeovery eurves. The maps show the layered stnieture of the sample. Presumably -relaxation varies spatially due to inliomogeneous size distribution as well as surfaee relaxivity of the pores. (From [21].)...

For example, if the molecular structure of one or both members of the RP is unknown, the hyperfine coupling constants and -factors can be measured from the spectrum and used to characterize them, in a fashion similar to steady-state EPR. Sometimes there is a marked difference in spin relaxation times between two radicals, and this can be measured by collecting the time dependence of the CIDEP signal and fitting it to a kinetic model using modified Bloch equations [64]. 

Figure B2.4.6. Results of an offset-saturation expermient for measuring the spin-spin relaxation time, T. In this experiment, the signal is irradiated at some offset from resonance until a steady state is achieved. The partially saturated z magnetization is then measured with a kH pulse. This figure shows a plot of the z magnetization as a fiinction of the offset of the saturating field from resonance. Circles represent measured data the line is a non-linear least-squares fit. The signal is nonnal when the saturation is far away, and dips to a minimum on resonance. The width of this dip gives T, independent of magnetic field inliomogeneity.

Figure B2.4.8. Relaxation of two of tlie exchanging methyl groups in the TEMPO derivative in figure B2.4.7. The dotted lines show the relaxation of the two methyl signals after a non-selective inversion pulse (a typical experunent). The heavy solid line shows the recovery after the selective inversion of one of the methyl signals. The inverted signal (circles) recovers more quickly, under the combined influence of relaxation and exchange with the non-inverted peak. The signal that was not inverted (squares) shows a characteristic transient. The lines represent a non-linear least-squares fit to the data.

Snap-Fit and Press-FitJoints. Snap-fit joints offer the advantage that the strength of the joint does not diminish with time because of creep. Press-fit joints are simple and inexpensive, but lose hoi ding power. Creep and stress relaxation reduce the effective interference, as do temperature variations, particularly with materials with different thermal expansions. 

The premise of the above analysis is the fact that it has treated the interfacial and bulk viscoelasticity equally (linearly viscoelastic experiencing similar time scales of relaxation). Falsafi et al. make an assumption that the adhesion energy G is constant in the course of loading experiments and its value corresponds to the thermodynamic work of adhesion W. By incorporating the time-dependent part of K t) into the left-hand side (LHS) of Eq. 61 and convoluting it with the evolution of the cube of the contact radius in the entire course of the contact, one can generate a set of [LHS(t), P(0J data. By applying the same procedure described for the elastic case, now the set of [LHS(t), / (Ol points can be fitted to the Eq. 61 for the best values of A"(I) and W. 

Stress Relaxation. Another important consequence of the viscoelastic nature of plastics is that if they are subjected to a particular strain and this strain is held constant it is found that as time progresses, the stress necessary to maintain this strain decreases. This is termed stress relaxation and is of vital importance in the design of gaskets, seals, springs and snap-fit assemblies. This subject will also be considered in greater detail in the next chapter. 

Usually, modified Newton-Raphson methods with relaxation are applied. Additional iteration loops are necessary for the determination of the dynamic pressure losses in ducts and duct fittings. 

FIG. 8 Log-log plot of the relaxation time r23, Eq. (25), vs chin length N, for four values of e. Full straight lines indicate power-law fits including the shortest chain length N — 6 the broken line indicates a fit where A = 16 is excluded. Effective exponents Zgff are quoted [13],... 

The measurement of correlation times in molten salts and ionic liquids has recently been reviewed [11] (for more recent references refer to Carper et al. [12]). We have measured the spin-lattice relaxation rates l/Tj and nuclear Overhauser factors p in temperature ranges in and outside the extreme narrowing region for the neat ionic liquid [BMIM][PFg], in order to observe the temperature dependence of the spectral density. Subsequently, the models for the description of the reorientation-al dynamics introduced in the theoretical section (Section 4.5.3) were fitted to the experimental relaxation data. The nuclei of the aliphatic chains can be assumed to relax only through the dipolar mechanism. This is in contrast to the aromatic nuclei, which can also relax to some extent through the chemical-shift anisotropy mechanism. The latter mechanism has to be taken into account to fit the models to the experimental relaxation data (cf [1] or [3] for more details). Preliminary results are shown in Figures 4.5-1 and 4.5-2, together with the curves for the fitted functions. 

Table 4.5-1 gives values for the fit parameters and the reorientational correlation times calculated from the dipolar relaxation rates. 

First, the stability of the fitted Llo structure relative to other crystal structure with the same composition can be studied. In the present case we calculated the cohesive energies of fully relaxed B2 and structure 40 compounds and found 4.41eV and 4.50 eV, respectively. These are both lower than the cohesive energy of the Llo structure. Structure B19 was also investigated but relaxation always transformed this structure into Llo. 

Many designs incorporate the phenomenon of stress-relaxation. For example, in many products, when plastics are assembled they are placed into a permanently deflected condition, as for instance press fits, bolted assemblies, and some plastic springs. In time, with the strain kept constant the stress level will decrease, from the same internal molecular movement that produces creep. This gradual decay in stress at a constant strain (stress-relaxation) becomes important in applications such as preloaded bolts and springs where there is concern for retaining the load. The amount of relaxation can be measured by applying a fixed strain to a sample and then measuring the load with time. 

This behavior is particularly relevant with push-fit assemblies and bolted joints that rely on maintaining their load under constant strain. Special design features or analysis may be required to counteract excessive stress-relaxation. 

The chromosomes of Escherichia coli and other bacteria are single, double-stranded DNA molecules with a total length of more than 1,000 pm. Relaxed DNA exists as a helical molecule, with one full turn of the helix occurring approximately every 10.4 base pairs. This molecule must undergo several folding and compaction steps to fit into an E. coli cell which is only 1-3 pm long. Despite this enormous compaction, bacterial DNA must be accessible for the bacterial enzymes that catalize DNA replication and transcription... 

Bacterial as well as eukaryotic chromosomes contain too much DNA to fit easily into a cell. Therefore, the DNA must be condensed (compacted) to fit into the cell or nucleus. This is accomplished by supercoiling the DNA into a highly condensed form. When relaxed circular DNA is twisted in the direction that the helix turns, the DNA becomes positively supercoiled, if it is twisted in the opposite direction, it is called negatively supercoiled. Bacterial DNA is normally found in a negatively supercoiled state. Supercoiling reactions are catalyzed by topoisomerases. 

All relaxation curves exhibited more than one phase at various degrees of conversion and at different temperatures. This clearly rules out the all-or-none mechanism (AON) although the AON model is able to fit easily to the measured equilibrium transition curve. However, a mechanism has been proposed which allows the existence of side... 

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