Spectra from Experimental Functions

This is the most important application of approximation methods, since exact methods are of such limited use. 

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However, even when such a model is not being used, it is often useful to describe the relaxation modulus by use of Eq. 4.16 where the constants are inferred from experimental data by an empirical procedure. The resulting constants, T , G , are said to constitute a discrete relaxation spectrum. While these empirical parameters have no physical significance, in the limit of large N, they should, in principle, approach the underlying function G(f), which is a material property. Methods of determining the constants for a discrete spectrum from experimental data are described in Section 4.4. 

Distributions of relaxation or retardation times are useful and important both theoretically and practicably, because // can be calculated from /.. (and vice versa) and because from such distributions other types of viscoelastic properties can be calculated. For example, dynamic modulus data can be calculated from experimentally measured stress relaxation data via the resulting // spectrum, or H can be inverted to L, from which creep can be calculated. Alternatively, rather than going from one measured property function to the spectrum to a desired property function [e.g., Eft) — // In Schwarzl has presented a series of easy-to-use approximate equations, including estimated error limits, for converting from one property function to another (11). 

Interpretation of the resonance spectrum is quite tedious because both the size and refractive index varied during the experiment. Phase function data obtained during the experiment provide an estimate of the size, and this estimate was used to initiate computations of the resonance spectrum from Mie theory to match the experimental spectrum. Once the refractive index... 

As mentioned earlier, the sum of the squared error is found to be the most convenient measure of the error because much of the calculation may be done analytically. In the following sections the sum of the squared error will be formulated for each of the constraints, and the form of the equations in the unknown Fourier coefficients for each constraint will be determined. Values of both artificial and experimental data will then be substituted in these equations to determine these unknown Fourier spectral components of the extended spectrum. From these, the completely restored function may be determined. 

Some experiments only measure the power spectrum of correlation functions over very restricted frequency ranges. Hence, the correlation functions themselves cannot be reconstructed from the experimental data. This is the case in static transport coefficient measurements where only the power spectra of specific correlation functions at zero frequency are measured.12... 

Finally, consider the power spectra of the experimental approximate correlation functions which are displayed in Figures 24, 29, and 34. Note that each of these spectra has been normalized to unity at co = 0. Note also that the experimental spectrum from the angular momentum correlation function is much broader than the experimental velocity autocorrelation power spectra. The power spectra of the Gaussian II autocorrelation functions are in much better agreement with the experimental spectra at all frequencies than the power spectra of the other approximate autocorrelation functions. 

The fimction (A) is referred to as the distribution of relaxation times or the relaxation spectrum. In principle, once (A) is known, the result of any other type of mechanical experiment can be predicted. In practice (A) is determined from experimental data on E t). Since the distribution of relaxation times is so broad, it is more convenient to consider In A. Hence we introduce the function H(ln A), where the parenthesis denotes functional dependence, to replace (A) as... 

The spectrum from this two-dimensional model is shown in Fig. 1(b). The spectrum has been broadened by multiplying the autocorrelation function with Eq. (38) with a time constant of 35 fs. This is equivalent to con-voluting the peaks in the spectrum with a fairly broad Lorentzian with 40meV FWHM. The agreement with the experimental spectrum is very good proof that the central band is due to a breakdown of the Born-Oppenheimer approximation by vibronic coupling. 

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