Fourier transform correlation function

In the Reverse Monte Carlo (RMC) method [5], the pair correlation function or the structure factor is calculated after each random move (Ssim(<]) or gsimfr)) and compared to the respective target function obtained from experimental diffraction data (Sexp(q) or gexp(r)). It is possible to calculate Ssm(q) with full periodicity from the atomic positions. This method is best in principle [10], but the computational cost is much greater than for any of the other available methods. It is also possible to obtain Ssm(q) by first calculating gsm(r) from the atomic positions and then Fourier transform this function and calculate Ssim(q). The disadvantage of this approach is that there is an additional computational cost associated with the Fourier transform of gsm(r) after each move. 

Here, I(co) is the Fourier transform of the above C(t) and AEq f is the adiabatic electronic energy difference (i.e., the energy difference between the v = 0 level in the final electronic state and the v = 0 level in the initial electronic state) for the electronic transition of interest. The above C(t) clearly contains Franck-Condon factors as well as time dependence exp(icOfvjvt + iAEi ft/h) that produces 5-function spikes at each electronic-vibrational transition frequency and rotational time dependence contained in the time correlation function quantity <5ir Eg ii,f(Re) Eg ii,f(Re,t)... 

For strongly structured microemulsions, g is negative, and the structure functions show a peak at nonzero wavevector q. As long as g < 2 /ca, inverse Fourier transform of S q) still reveals that the water-water correlation functions oscillate rather than decay monotonically. The lines in phase space where this oscillating behavior sets in are usually referred to as disorder lines, and those where the maximum of S q) moves away from zero as Lifshitz lines. ... 

In the case where x and y are the same, C (r) is called an autocorrelation function, if they are different, it is called a cross-correlation function. For an autocorrelation function, the initial value at t = to is 1, and it approaches 0 as t oo. How fast it approaches 0 is measured by the relaxation time. The Fourier transforms of such correlation functions are often related to experimentally observed spectra, the far infrared spectrum of a solvent, for example, is the Foiuier transform of the dipole autocorrelation function. ... 

Given a real-valued sequence (Jq, Ci,. ..,(T v-i, the the correlation function R t) and Fourier Transformation may be defined in both a continuous and discrete form ... 

It is a well known fact, called the Wiener-Khintchine Theorem [gardi85], that the correlation function and power spectrum are Fourier Transforms of one another ... 

The qualitative difference between low-density and high-density rotational relaxation is clearly reflected in the Fourier transform of the normalized angular momentum correlation function ... 

If the second term on the right-hand side of the equation is omitted, the latter is transformed into Eq. (2.76). As the omission is possible only for t < tj, Fourier transformation of the reduced equation holds for co-tj 1 only. Consequently, the equality (2.75) is of asymptotic character, and may not be utilized to find full g(co) or its Fourier-transform Kj(t) at any times. When it was nevertheless used in [117], the rotational correlation function turned out to be alternating in sign. The oscillatory behaviour of Kj(t) occured not only in a compressed gas, but also at normal pressure, when Kj(t) should vanish monotonically, if not exponentially. The origin of these non-physical oscillations is easily... 

According to Eq. (2.13), the spectra we are interested in are given by a Fourier transform of the orientational correlation functions of the corresponding order Similarly to what was done in Chapter 3, the correlation functions for linear and spherical molecules may be represented as a superposition of the partial (marginal) components... 

Given equilibrium quantum expectation values, we can calculate moments of the infra-red vibrational lineshape using a procedure originally outlined by Gordon.The infrared vibrational lineshape is given as the Fourier transform of the dipole moment correlation function ... 

The matrix obtained after the F Fourier transformation and rearrangement of the data set contains a number of spectra. If we look down the columns of these spectra parallel to h, we can see the variation of signal intensities with different evolution periods. Subdivision of the data matrix parallel to gives columns of data containing both the real and the imaginary parts of each spectrum. An equal number of zeros is now added and the data sets subjected to Fourier transformation along I,. This Fourier transformation may be either a Redfield transform, if the h data are acquired alternately (as on the Bruker instruments), or a complex Fourier transform, if the <2 data are collected as simultaneous A and B quadrature pairs (as on the Varian instruments). Window multiplication for may be with the same function as that employed for (e.g., in COSY), or it may be with a different function (e.g., in 2D /-resolved or heteronuclear-shift-correlation experiments). 

Two densities P(r) and p (r) are said to be indistinguishable if all their correlation functions and c are identical. As shown by Mermin and collaborators their Fourier transforms... 

The prerequisite for an experimental test of a molecular model by quasi-elastic neutron scattering is the calculation of the dynamic structure factors resulting from it. As outlined in Section 2 two different correlation functions may be determined by means of neutron scattering. In the case of coherent scattering, all partial waves emanating from different scattering centers are capable of interference the Fourier transform of the pair-correlation function is measured Eq. (4a). In contrast, incoherent scattering, where the interferences from partial waves of different scatterers are destructive, measures the self-correlation function [Eq. (4b)]. 

Equations (35) and (36) define the entanglement friction function in the generalized Rouse equation (34) which now can be solved by Fourier transformation, yielding the frequency-dependent correlators . In order to calculate the dynamic structure factor following Eq. (32), the time-dependent correlators are needed. 

In Equation (5), we can first notice (i) the factor 1/r6 which makes the spectral density very sensitive to the interatomic distance, and (ii) the dynamical part which is the Fourier transform of a correlation function involving the Legendre polynomial. We shall denote this Fourier transform by (co) (we shall dub this quantity "normalized spectral density"). For calculating the relevant longitudinal relaxation rate, one has to take into account the transition probabilities in the energy diagram of a two-spin system. In the expression below, the first term corresponds to the double quantum (DQ) transition, the second term to single quantum (IQ) transitions and the third term to the zero quantum (ZQ) transition. 

As seen from the above theoretical developments, accessing geometrical (and stereochemical) information implies at least an estimation of the dynamical part of the various relaxation parameters. The latter is represented by spectral densities which rest on the calculation of the Fourier transform of auto- or cross-correlation functions. These calculations require necessarily a model for describing molecular reorientation... 

From the observed correlation function the scattering pattern is obtained by Fourier transformation. As Eq. (2.31) is subjected to the Fourier transform, it will only act on the correlation function of the template because hu (a) is no function of r. With Eq. (2.29) we obtain the expected result... 

It is complete because of fiber symmetry. The 2D Fourier transform of this image is not related to the searched slice, but to a projection of the correlation function. In contrast, the sought-after slice in real space... 

Equations. For a ID two-phase structure Porod s law is easily deduced. Then the corresponding relations for 2D- and 3D-structures follow from the result. The ID structure is of practical relevance in the study of fibers [16,139], because it reflects size and correlation of domains in fiber direction . Therefore this basic relation is presented here. Let er be50 the direction of interest (e.g., the fiber direction), then the linear series expansion of the slice r7(r)]er of the corresponding correlation function is considered. After double derivation the ID Fourier transform converts the slice into a projection / Cr of the scattering intensity and Porod s law... 

The presented result is different from the radial correlation function y(r) = An js21 (s)(sm 2nrs)/2nrs) ds, which is computed from the isotropic scattering intensity by means of the three-dimensional Fourier transform. 

It is convenient to set Ap1 = 1, L = d - -dq = 1. Rounding errors are suppressed by replacing die intensity by 1/s2 (Porod s law) for big arguments (s > 8). A smooth phase transition zone (in all the example curves dz = 0.1) is considered by multiplication with exp (2nsdz/3)2 j. From this one-dimensional scattering intensity die correlation function is obtained by Fourier transformation. 

The Fourier transform of the correlation functions Cem(x) and Cabs(x) are the emission (Sem) and absorption (Sabs) spectra, respectively Sem = jdxCem(x)exp(-icox) the integration limits are (-00,-1-00),... 

---