Instrumentation autocorrelation functions

After normalization to the asymptotic baseline, g2(r) decays from two to unity if measured with a perfect instrument. A real instrument always suffers from some loss of coherence, and for a monodisperse solution of ideal, non-interacting solute molecules the intensity autocorrelation function g2(r) takes the form... 

The Siegert relation is valid except in the case of scattering volumes with a very small number of scatterers or when the motion of the scatterers is limited. We ignore the exceptions, which are rare in common uses of DLS, and consider only autocorrelations of the type shown in Equation (104). As mentioned above, modern DLS instruments use computer-controlled correlators to calculate the intensity autocorrelation function automatically and to obtain the results in terms of the function gi(s,/rf) therefore we only need to concern ourselves here with the interpretation of gi(s,td). 

A time response function of the apparatus can be measured by upconversion of the excitation beam. The width of such measured instrument response function is 280fs (FWHM). Comparing this result with the width of the autocorrelation function of the dye laser 110fs we observe 170fs broadening of the instrument response function due to group velocity... 

Autocorrelation functions were measured with a Brookhaven Instruments 128 channel multi-time full correlator. Other details of the spectrometer have been described previously (7). 

The traditional method of using DLS to determine the intensity autocorrelation function is to detect the scattered light at a fixed angle with a photomultiplier tube and then process this signal with a hard-wired electronic correlator." " Commercial instruments with hardware and software optimized for this purpose are available from manufacturers such as Brookhaven Instraments, Malvern Instruments, and others. Here we will utilize... 

The particles must scatter independently otherwise the diffusion coefficient, and particle size, cannot be determined unambiguously from the decay rate of the autocorrelation function. The net effects of multiple scattering are that the instrument factor B/A decreases, and the autocorrelation factor decays faster, leading to too low an estimate for particle size. Thus, multiple scattering limits the application of the technique to low concentration dispersions (< 0.01% by volume), although techniques have been developed to overcome this condition. 

Quasi-elastic light scattering (QELS) experiments were conducted on both neutralized and unneutralized sols, in a Brookhaven model BI-90 particle sizer. This instrument measured the autocorrelation function, C(t), and fit this function to... 

The autocorrelation function can be calculated in real time using a hardware correlator or in software after the collection of a photon count trace using a multichannel scalar card. Details of the instrumentation will be discussed in Chapter 3. Formally, the un-normalized autocorrelation of time series data is given as [6] ... 

Figures 2 and 3 show how the appearance of the residuals and autocorrelation function for a pulsed excitation experiment typically depend on the appropriateness of the fitting function. In panel A of both figures, L shows an actual excitation pulse profile (more properly, the instrument response function) that was generated by an argon ion laser that pumped a dye laser circulating rhodamine 6G, the tuned output of which was frequency-doubled to 295 nm (nanometer = 10" m = 10 A) by passage through a p-barium borate crystal (cf.

Fig. 15.10 Fluorescence decays for a polythiophene derivative in toluene solution at 293 K and in thin film. The dashed lines in the decays are the pulse instrumental response functions in solution (obtained with a Ludox solution) and in the solid state (obtained with a blank sapphire disc inside the Horiba-Jobin-Yvon integrating sphere). Autocorrelation functions (AC.), weighted residuals and Chi square values (x ) are also present as insets. Reproduced with permission from Ref. [49], Copyright 2007, the American Chemical Society...

Fig. 15.11 Fluorescence decays showing monoexponential fits of the reference eompounds (obtained for the calibration of the ps time-resolution apparatus) a 2,2 5, 2" 5",2" -quaterthi-ophene in methylcyclohexane = 425 nm) and b p-terphenyl in cyclohexane (Aex = 296 nm). For better judgment of the quality of the fits, autocorrelation functions (AC.), weighted residuals (W.R.) and values are also presented as Insets. The shorter pulse is the instrumental response...

In eq. (1) A is a, in principle constant, background signal and B is an instrumental factor. Note that eq. (1) applies only to scattered fields with Gaussian statistics an hypothesis which is not always fulfilled experimentally. Especially for particles larger than roughly 0.5 to 1 im additional time delay dependent factors can be distinguished in eq. (1) In a second step the time decay of the field autocorrelation function is related to the particles Brownian motion. Thereby it is assumed that the particles scatter independently. In particular for monodisperse samples gi(0,x) is an exponentially decaying function ... 

The analysis involved deconvolution by iterative reconvolution, background subtraction, and optional correction for shift of the instrument response function. Statistical tests included chi-square, the Durbin-Watson test, the covariance matrix, a runs test, and the autocorrelation function [6]. An alternative form of data analysis involves distributions of lifetimes rather than a series of exponentials. Differentiation of systems obeying a decay law made up of three discrete components from systems where there exists a continuous distribution of lifetimes, or a distribution plus one or more discrete components, is a nontrivial analytical problem. Methods involving the minimization of the chi-square parameter are commonly used, but recently the maximum entropy method (MEM) has gained popularity [7]. Inherent in the MEM method is the theoretical lack of bias and the potential for recovering the coefficients of an exponential series with fixed lifetimes which are free of correlation effects and artificial oscillations. Recent work has compared the MEM with a new version of the exponential series method (ESM) which allows use of the same size probe function as the MEM and found that the two methods gave comparable results [8]. 

A suitable mathematical algorithm must be used to invert Equation 8.11 to obtain the set of coefficients,, constituting the intensity-weighted PSD, or PSDj, from the raw data , Typically, two kinds of algorithms are employed in DLS-based instruments. The simplest is the method of cumulants, based on a least-squares fit of a polynomial (quadratic or cubic) in i5t to the reduced autocorrelation function, Y(St),... 

Clearly, the model cannot be estimated by ordinary least squares, since there is an autocorrelated disturbance and a lagged dependent variable. The parameters can be estimated consistently, but inefficiently by linear instrumental variables. The inefficiency arises from the fact that the parameters are overidentified. The linear estimator estimates seven functions of the five underlying parameters. One possibility is a GMM estimator. Let v, = g, -(y+< >)g,-i + (y< >)g, 2. Then, a GMM estimator can be defined in terms of, say, a set of moment equations of the fonn E[v,w,] = 0, where w, is current and lagged values of x and z. A minimum distance estimator could then be used for estimation. 

Re F stands for the real part of the function F. The function fcs(r) models a slowly varying background, which is usually present in all of the measurements. The constant background term B is measured by the autocorrelator using special time bins with extra delay. / () is the intensity of the local oscillator (may represent scattering due to the interface itself) the term 2/S0// 0 indicates the relative amount of particle-scattered light and reference scattered photons and should not exceed 0.1 for heterodyne detection. The quantity / is an instrumental constant, a value around 0.5 indicating a reasonably optimized system for homodyne detection. 

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