Normalized autocorrelation functions

Figure Bll.2.1 shows the normalized autocorrelation functions of various micelles loaded with octadecyl rhodamine B chloride (ODRB) at pH 7 (PBS buffer)3 . The differences in size of the micelles are clearly reflected by the differences in diffusion times td- The translational diffusion coefficients are reported in Table Bll.2.1, together with the hydrodynamic radii and the aggregation numbers.

When the excitation light is polarized and/or if the emitted fluorescence is detected through a polarizer, rotational motion of a fluorophore causes fluctuations in fluorescence intensity. We will consider only the case where the fluorescence decay, the rotational motion and the translational diffusion are well separated in time. In other words, the relevant parameters are such that tc rp, where is the lifetime of the singlet excited state, zc is the rotational correlation time (defined as l/6Dr where Dr is the rotational diffusion coefficient see Chapter 5, Section 5.6.1), and td is the diffusion time defined above. Then, the normalized autocorrelation function can be written as (Rigler et al., 1993)... 

Figure 4.6. Normalized autocorrelation function. Autocorrelation function for collagen single molecules. The autocorrelation function G(nAt) is normalized by dividing all points by the first experimental point G(l). The autocorrelation function decays to a value of the average squared intensity of scattered light divided by G(l). The average squared intensity is proportional to the weight average molecular weight, whereas the rate of decay is related to the translational diffusion coefficient. Reproduced from Silver, 1987.

Fytas et al. [18[ observed polymer dynamics associated with height fluctuations of PEO-PS copolymers attached to glass in toluene. Some of the measured normalized autocorrelation functions are shown in Fig. 8. The data was well represented by a single exponential function with a decay constant that exhibited minimal dependence on q. There is no calculation of the structure factor for such a system, but it is possible to rationalize the dependence of the decay constant on the 5/3 power of the chain density and the cube of the number of monomers in the PS segment. 

Another piece of information that we wanted to extract from our experiments was connected with the dynamic behavior of spatial variables. If we consider three successive particles in the chain and we denote by the distance of the middle one from the center of mass of the other two and by the distance between these two, we can compute the normalized autocorrelation function of these two variables. They are shown in Fig. 9 as can be immediately observed, they decay to zero on a time scale which is much greater than that of the velocity variable. Also, the center of mass decays faster than R . In the next section we shall argue that this suggests that the virtual potential characterizing the itinerant oscillator model has to be assumed to be fluctuating around a mean shape, which, moreover, will be shown to be nonlinear and softer than its harmonic approximation. 

In this chapter, the focus will be on how information can be extracted, utilizing the second category described earlier (Fig. 8.1b). In its general form, the normalized autocorrelation function of the detected fluorescence fluctuations will show a complex dependence on the reaction rates and the coeflicients of the translational diffusion, and cannot be expressed in an analytical form. Fortunately, for a rather broad range of molecular reactions the reaction-induced fluorescence fluctuations can be treated separately from those due to translational diffusion [19]. If diffusion is much slower than the chemical relaxation time(s) and/or the diffusion coeflicients of all fluorescent species are equal, then the time-dependent fluorescence correlation function can be separated into two factors. The first factor, Gd( ), depends on transport properties (diffusion or flow) and the second, R t), depends only on the reaction rate constants ... 

The photon autocorrelation function was analyzed by the method of cumulants (2.) 9 in which the logarithm of the normalized autocorrelation function, ci,t), is fit to a polynomial using... 

Figure 8. (a) Normalized autocorrelation function for E 2 in the Henon-Heiles system at E = % (solid line). FallofT with rate K to the statistical value (arrow) is shown as a dotted curve, (b) Average value of E 2 in the Henon-Heiles system at E = (solid curve) and of , (dashed curve). Exponential falloff with rate K to statistical value (arrow) with rate K is given by dotted curve. Initial conditions as described in test. (From Ref. 37b.)... 

Figure 6. A comparison of normalized autocorrelation functions determined by PCS for diluted samples (1 X I0-3 g/cm3) of silica S3 (0) and S4 (A).

Figure 4. Normalized autocorrelation functions of unmodified erythrocyte surface profile (Control), of erythrocyte, modified with zinc ions (Zn), of erythrocyte, modified with lead ions (Pb).

The value A/ [G(t)-B] / [AbT is defined as the normalized autocorrelation function g(x) The translational diffusion coefficient can be calculated from F and the scattering vector. 

For a system of polydispersed raacromolecules undergoing only translational diffusion the normalized autocorrelation function, g(x) is related to a sum of exponents which is expressed by a Fredholm integral equation... 

An uncomplicated solution technique that can be applied to equation (16) is the method of cumulants. In 1972 Koppel showed that the logarithm of the normalized autocorrelation function was identical to the cumulant generating function for the distribution of decay constantsl. The coefficients of the cumulant expansion can be related to the moments of the F(r) distribution. The Koppel equation can be expressed by... 

It is now a straightforward matter to compute the dielectric susceptibility Ae (cy). From Eq. 39 the normalized autocorrelation function of the total dipole moment Co(M t) reads ... 

Electric-Field Autocorrelation Function We consider the autocorrelation function of the electric field E,(t) of the light scattered by solutes. As we have seen in Section 2.4, is a complex quantity. We introduce another normalized autocorrelation function gi(r), which is defined as... 

Fig. 11.3 (a) SAXS profiles of iPP samples of Table 11.1 isothermally CTystallized from the melt at the indicated values of temperature (a) and corresponding normalized autocorrelation functions. SAXS data are recorded at Tc- Values of long spacing (L), and thickness of crystalline (4) and amorphous (k = L- 4) layers (c). They are evaluated from the autocorrelation function in (b) (full symbols) and also using the Bragg law for the calculation of L and successive multiplication of L times the crystallinity index (see the text) for 4 (open symbols)... 

Fig. 5.18 (A) Normalized autocorrelation fimction of the fluorescence fluctuations for (frran top to bottom) iV = 25, 50, 100 or 200, all with k At = k ffAt = 0.1. The autocorrelation functions are normalized relative to the square of the mean fluorescence amplitude. (B) Normalized autocorrelation function of the fluorescence fluctuations for N = 50, and k At = 0.1, k At = 0.1 (short dashes), k nAt = 0.02, k jfAt = 0.02 (long dashes), k At = 0.02, k ffAt = 0.1 (solid curve), or k At = 0.01, koffAt = 0.05 (dot-dashed curve). The results were averaged over 2 x 10 time steps...

Here rr is the mean square of the frequency fluctuations ([Pg.450]

Suppose we have a normalized autocorrelation function of a phase space function A(p, q) where, as before, A implicitly depends on time through p and q (McQuarrie, 1976 Hansen and McDonald, 1976) ... 

Examples of time correlation functions are shown in Figure 9.1, where the normalized autocorrelation functions for D, rj and k are displayed. The figure contains also the correlation function for the bulk viscosity, which is, however, not of concern in the... 

It is convenient to define a normalized autocorrelation function for the field, 0 (t), such that... 

In the frame of the Bloch/Wangsness/Redfield (BWR) relaxation theory [2, 17], the fluctuations of the spin Hamiltonians are described with the aid of (preferably normalized) autocorrelation functions of the type... 

Note that the spectral density defined in this way is independent of the subscript m for isotropic systems since it is based on the normalized autocorrelation functions given in Eq. 28. 

Fig. 10. Numerically evaluated, normalized autocorrelation function for the RRM for p=5 together with a fitted curve for a stretched exponential, Eq. 131, and an exponential curve. The stretched-exponential parameter is 5=0.66...

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