Zero frequency component

Presented in this manner, the analysis may proceed similarly to the treatment obtained from the Fourier analysis. C is the zero frequency component of the fit and A and B may be treated as the real and imaginary parts of the complex number. 

The diffusion constant D is determined by the zero-frequency component of this spectral density of the velocity correlation function, by the relation... 

There are many experiments which determine only specific frequency components of the power spectra. For example, a measurement of the diffusion coefficient yields the zero frequency component of the power spectrum of the velocity autocorrelation function. Likewise, all other static coefficients are related to autocorrelation functions through the zero frequency component of the corresponding power spectra. On the other hand, measurements or relaxation times of molecular internal degrees of freedom provide information about finite frequency components of power spectra. For example, vibrational and nuclear spin relaxation times yield finite frequency components of power spectra which in the former case is the vibrational resonance frequency,28,29 and in the latter case is the Larmour precessional frequency.8 Experiments which probe a range of frequencies contribute much more to our understanding of the dynamics and structure of the liquid state than those which probe single frequency components. 

As revealed from Eqs. (1) and (2), or their candid forms (4) and (5), the longitudinal relaxation is determined by the spectral densities in the order of o>h toc, whereas the transverse relaxation involves the contribution from the zero frequency component Jo(0). In the case of solid matter, tc is generally very long. Hence, the transverse relaxation is predominantly determined by the zero frequency component Jo(0). In Eq. (5), for example, the zero frequency term (the first term) dominates the other terms that are reciprocally proportional to Tc for co2x2 1. Tic increases as xc increases (i.e. as the material under consideration becomes solider), whereas T2c decreases infinitely as xc increases. For example, Tic is generally in an order of several tens several hundreds of seconds for the crystalline component and in an order of a few tenths of a second for rubbery components of polymers. On the other hand, T2c is of an order of a few tens of microseconds for the crystalline or glassy component and a few milliseconds for the rubbery component of polymers. In this work, Tic and T2c are used for characterizing different components in crystalline polymers. 

What has been presented here underscores the fact that the elastic scattering is the Fourier transform of the time-independent component of the intermediate scattering function. Naturally, the Fourier transform of a constant function produces a 5-function at > = 0, which is the elastic scattering, but this is not the same as the zero frequency component of the scattered intensity. If the time correlation function has a component that exhibits some time decay or relaxation, then the integral of the time dependant part of... 

Equation (6.14) associates the zero frequency component of the velocity time correlation function with the long-time diffusive dynamics. We will later find (see Section 6.5.4) that the high frequency part of the same Fourier transform, Eq. (6.15), is related to the short-time dynamics of the same system as expressed by its spectrum of instantaneous normal modes. 

Note that since the diffusion constant is the zero-frequency component of the Fourier transform of the velocity correlation functions [65], Eq. 

FID. The FID is then governed by the zero frequency component of the fluctuations in the local field. This is the case with rubbers and polymer melts. 

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