Comparison with Observed Data

Assuming the elastic coefficients for the acoustic branch are the same as for the optical branch, we can now compute the (Ul and coj values from the lattice constants using Equation 16.20 and compare them against experimental values in Table 16.1. 

The elastic coefficients Cn and C44 are obtained from velocity of soimd measurements, and the values in Table 16.1 were taken from Kiffel, Introduction to Solid State Physics, 2nd edition. The values for wx(act), taken from Kiffel, Solid State Physics, 7th edition, were obtained by observing the onset of strong reflectance in the infrared. In Qiapter 24, it is shown that ionic materials reflect strongly as well as absorb at frequencies slightly above their transverse resonant frequency. The values for ML(act) were obtained from the ratio of the static to optical dielectric constants using the Lyddane, Sachs, Teller relation (see Chapter 23) and were also taken from Kittel, Solid State Physics, 7th edition. All frequencies are in units of 10 /s. 

We see that the values for cot and col predicted by the model are in general agreement but are somewhat below the observed values. Considering the simplifying assumptions made in the model, i.e., using only nearest neighbor interactions, perhaps this is all that can be hoped for. 

One of the more useful predictions from the model is that the absorption peak from Equation 16.20 is proportional to the square root of the lattice stiffness divided by the reduced mass of the ion pairs. In the design of glasses and other optical components that must operate in the far-infrared, one wants to operate at frequencies well above the absorption at ca = 2/3//a. Therefore, one looks for materials with heavy ion pairs and low soimd velocities. Single crystalline NaCl is used for infrared windows out to about 15 pm while KBr is good out to about 25 pm. KRS-5, (thallium bromoiodide) is used for infrared applications out to 35 pm. The development of heavy metal glasses is discussed further in Chapters 14 and 24. 

For continuous media, the propagation velocities for longitudinal and transverse waves is given by v-l = fCuJp and Vj = sJCu/p and the dispersion relation is ct) = vjc. For a monatomic chain of atoms, the dispersion relation is given by Equation 16.7, (o = sin(fai/2). The ratio of /3/m can be related to the elastic stiffness coefficient 

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Although the statistical mechanical theories such as those described above yield exact analytic expressions for various quantities characterizing the conformation of an interrupted helix, those expressions are so complicated that it is of both theoretical and practical value to simplify them, with the imposition of suitable restrictions on parameters, to forms that are amenable to straightforward computations and also, hopefully, to direct comparisons with observed data. Various attempts have been made, and they are summarized in Poland-Scheraga s book (10). Though not available at the time this book was published, the approximations worked out by Okita et al. (13) are of great practical use for their wide applicability and simplicity. Their method is described below in some detail, because it has been consistently used in our statistical-thermodynamic analyses of helix-coil transition phenomena. 

Comparison with observed data is also favorable. After adding estimates for the electron correlation (from cluster calculations) and for nuclear motion effects, a heat of adsorption is predicted which is at the lower edge of observed data. ... 

Here Tq are coordinates in a reference volume Vq and r = potential energy of Ar crystals has been computed [288] as well as lattice constants, thermal expansion coefficients, and isotope effects in other Lennard-Jones solids. In Fig. 4 we show the kinetic and potential energy of an Ar crystal in the canonical ensemble versus temperature for different values of P we note that in the classical hmit (P = 1) the low temperature specific heat does not decrease to zero however, with increasing P values the quantum limit is approached. In Fig. 5 the isotope effect on the lattice constant (at / = 0) in a Lennard-Jones system with parameters suitable for Ne atoms is presented, and a comparison with experimental data is made. Please note that in a classical system no isotope effect can be observed, x "" and the deviations between simulations and experiments are mainly caused by non-optimized potential parameters. 

The first-principles calculation of NIS spectra has several important aspects. First of all, they greatly assist the assignment of NIS spectra. Secondly, the elucidation of the vibrational frequencies and normal mode compositions by means of quantum chemical calculations allows for the interpretation of the observed NIS patterns in terms of geometric and electronic structure and consequently provide a means of critically testing proposals for species of unknown structure. The first-principles calculation also provides an unambiguous way to perform consistent quantitative parameterization of experimental NIS data. Finally, there is another methodological aspect concerning the accuracy of the quantum chemically calculated force fields. Such calculations typically use only the experimental frequencies as reference values. However, apart from the frequencies, NIS probes the shapes of the normal modes for which the iron composition factors are a direct quantitative measure. Thus, by comparison with experimental data, one can assess the quality of the calculated normal mode compositions. 

Models have been formulated to enable the simulation of the concentration vs. radial distance profile as it develops with time, from which the time-dependent concentration vs. distance, d, profile, observed at the probe, can be extracted for comparison with experimental data. Models based on Eqs. (29) and (30) give similar results for conditions encountered practically. 

Although the existence or absence of a particular process can often be determined from observed data, an assessment of how well an algorithm represents the process is often difficult to make due to observation errors, natural variations in field data, and lack of sufficient data on individual component processes. In such circumstances, model validity must be inferred or possibly based on comparisons with laboratory data obtained under controlled conditions. Often laboratory data provide the basis for developing an algorithm since field data are so much more difficult and expensive to collect and interpret. Examples of system representation errors and their analysis were presented at the Pellston workshop (6 ). 

Comparisons between observed data and model predictions must be made on a consistent basis, i.e., apples with apples and oranges with oranges. Since models provide a continuous timeseries, any type of statistic can be produced such as daily maximums, minimums, averages, medians, etc. However, observed data are usually collected on infrequent intervals so only certain statistics can be reliably estimated. Validation of aquatic chemical fate and transport models is often performed by comparing both simulated and observed concentration values and total chemical loadings obtained from multiplying the flow and the concentration values. Whereas the model supplies flow and concentration values in each time step, the calculated observed loads are usually based on values interpolated between actual flow and sample measurements. The frequency of sample collection will affect the validity of the resulting calculated load. Thus, the model user needs to be aware of how observed chemical loads are calculated in order to assess the veracity of the values. 

The limitation in all of these flash experiments is that only broad featureless UV/vis bands are observed and hence assignment has to rely on comparison with matrix data and/or kinetic consistency. How much more informative vibrational spectroscopy would be There is good reason to be optimistic as in the recent work of Schaffner (8), where, incidentally, it is shown how important a role is played by traces of H2O in the detailed mechanism of the photochemistry of Cr(C0)6 ... 

Comparison with observations Soil and vegetation are only represented as single layer (topsoil) surfaces in the MPI-MCTM, hence their contamination is expressed as a mass per surface area. Soil burdens were converted into concentrations by dividing them by soil dry bulk density and a fixed soil depth of 10 cm. The average DDT concentration in soil between 40 °N and 60°N was compared to measured soil and sediment concentrations from Northern North America and Great Britain [Dimond and Owen (1996), Meijer et al (2001), and others compiled by Schenker et al (2008a)]. For intercomparison reasons only relative soil concentrations are compared to observational data. Each set of observations was normalised to its 1990 value. 

The first isolable alkenetitanium complex, the bis(pentamethylcyclopentadienyl)-titanium—ethylene complex 5, was prepared by Bercaw et al. by reduction of bis(penta-methylcyclopentadienyl)titanium dichloride in toluene with sodium amalgam under an atmosphere of ethylene (ca. 700 Torr) or from ( (n-C5Mc5)2Ti 2(fJ-N2)2 by treatment with ethylene [42], X-ray crystal structure analyses of 5 and of the ethylenebis(aryloxy)trimethyl-phosphanyltitanium complex 6 [53] revealed that the coordination of ethylene causes a substantial increase in the carbon—carbon double bond length from 1.337(2) A in free ethylene to 1.438(5) A and 1.425(3) A, respectively. Considerable bending of the hydrogen atoms out of the plane of the ethylene molecule is also observed. By comparison with structural data for other ethylene complexes and three-membered heterocyclic compounds, the structures of 5 and 6 would appear to be intermediate along the continuum between a Ti(11)-ethylene (4A) and a Ti(IV)-metallacyclopropane (4B) (Scheme 11.1) as... 

Recent developments and prospects of these methods have been discussed in a chapter by Schneider et al. (2001). It was underlined that these methods are widely applied for the characterization of crystalline materials (phase identification, quantitative analysis, determination of structure imperfections, crystal structure determination and analysis of 3D microstructural properties). Phase identification was traditionally based on a comparison of observed data with interplanar spacings and relative intensities (d and T) listed for crystalline materials. More recent search-match procedures, based on digitized patterns, and Powder Diffraction File (International Centre for Diffraction Data, USA.) containing powder data for hundreds of thousands substances may result in a fast efficient qualitative analysis. The determination of the amounts of different phases present in a multi-component sample (quantitative analysis) is based on the so-called Rietveld method. Procedures for pattern indexing, structure solution and refinement of structure model are based on the same method. 

FIG. 6. The corrugation observed while imaging Al(l 11) is a strong function of tip-sample separation as well as the electronic structure of the tip. Theoretical results for an s and d/ tip state are shown in comparison with experimental data (Vtias = —50 mV). (From Ref. 39.)... 

It is important to note that the conclusion drawn from the observed data is based on a comparison with virtual data that might have been collected in other identical experiments but were never really observed. In fact, a judgement is made on the data rather than directly on the model or hypothesis. No consideration is given to the plausibility of the original hypothesis or specific alternatives. It is an erroneous assumption that the p value is a measure of the validity of the null hypothesis. As noted, p merely makes a statement about the data on the assumption that the hypothesis is valid. 

Validation of models is desired but can be difficult to achieve. Models are empirically validated by examining how output data (predictions) compare with observed data (such comparisons, of course, must be conducted on data sets that have not been used to create or specify the model). However, model validations conducted in this manner are difficult given limitations on data sources. As an alternative approach, model credibility can be assessed by a careful examination of the subcomponents of the model and inputs. One should ask the question Does the selection of input variables and the way they are processed make sense Also, confidence in the model may be augmented by peer reviews and the opinion of the scientific community. Common faults and shortcomings are... 

For comparison with these data, the types of salt effects that have been observed for model protein and polypeptide systems that achieve true thermodynamic equilibrium in solution can be summarized into three classes ... 

Of the above assumptions, only the last can be verified directly by comparison with exptl data. However, since that comparison will be correct at the end of detonation and since the other assumptions are made consistent with the observed Gurney velocity at the end of casing expansion, he suggests that the casing motions in between will also be fairly well represented... 

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