1. Introduction

Dynamic Light Scattering (DLS) is probably the most used optical technique for sizing nano-particles dispersed in a fluid [1,2]. It is commonly used in physics, biophysics and chemistry for studying systems that need to be investigated in situ and, possibly, in real time. DLS is based on the determination of the diffusion coefficient of particles freely moving in a fluid, a task that can be tackled by measuring the auto- or cross-correlation function of the light intensity scattered by the sample. Modern multi-tau digital correlators can perform this job excellently, by recovering correlation functions over an extremely wide range of lag-times, from nanoseconds to seconds or larger [3]. However, unless the measurements are repeated many times, they do not provide any estimation of the error bars and covariance coefficients matrix associated to the measure.

The problem of characterizing the statistics of DLS data was tackled by several authors in the ’70s, soon after the advent of the technique [4–6], but it was only in the late ’80s, early ’90s that, thanks to the pioneering work of K. Schätzel on digital multi-tau correlators, was fully mastered. In his seminal paper published in 1990 [7, 8], he provided a numerical expression for the full covariance matrix (and error bars) of the auto-correlation function estimator that is recovered in DLS under known given experimental conditions, such as the measuring time, the average count rate and the coherence factor β. Provided that the so called triangular averaging is properly taken into account, his formula was worked out without imposing any assumption on the sampling times, a requirement that is crucial when using multi-tau correlators where the sampling times Δt might be comparable or even larger than the correlation time τ_c.

In the same paper [7], Schätzel provided also two analytical expressions for the covariance matrix and the variance for the specific (but quite common) case of a Lorentzian spectrum, where the correlation is characterized by a single exponential decay function. However, these formulas do not include explicitly the effects of the triangular averaging and, as pointed out very clearly in his paper, if for example the variance formula is used as is, it gives highly inaccurate estimates of the error bars at points where Δt ≳ τ_c.

In this work we have revisited these formulas and worked out two new exact analytical expressions in which the effects of the triangular averaging are corrected to all orders. By the use of extensive computer simulations and experimental tests carried out on dilute dispersions of calibrated latex spheres, we have shown that the new formula for the variance works quite accurately for any value of the ratio Γ = Δt/τ_c and can be applied well beyond the specific case of a single exponential decay auto-correlation function. As a matter of fact, it can be extended also to bell-shaped polydisperse systems and works quite well also with cross-correlation functions. As to the covariance formula, within the limits of its applicability (discussed in Sect. 3), our new formula works quite well for any value of Γ, whereas the original Schätzel formula is accurate only when Γ ≪ 1.

2. The Schätzel formulas

Dynamic Light Scattering is based on the measurement of the normalized Intensity-Intensity auto-correlation function

According to Schätzel [11], the best estimate of g₂ that can be attained by using a digital correlator which works by acquiring data at a sampling time Δt for a finite measuring time T, is given by

K. Schätzel computed [7,8] the covariance matrix Cov_g2(τ_k, τ_l) associated to channels τ_k and τ_l of Eq. (2) in terms of the normalized field correlation function χ(τ_k) that, under common working conditions of a (homodyne) DLS experiment (many independent diffusing scatterers in the scattering volume), is described by a random Gaussian process and therefore can be linked to g₂(τ_k) by the Siegert relation [12]

In the same article, Schätzel worked out also the analytical expressions for Cov_g2(τ_k, τ_l) and the variance ${\sigma }_{g_2}^2\left({\tau }_k\right)$for the specific case of a Lorentzian spectrum. Following his (simpler) notation [χ(τ_k)→ χ_k, Cov_g2 (τ_k, τ_l) → Cov_g2 (k, l), ${\sigma }_{g_2}^2\left({\tau }_k\right)\to {\sigma }_{g_2}^2\left(k\right)$], we write the single exponential decay that characterizes the Lorentzian spectrum as

The diagonal terms of this matrix give the variance

As mentioned above, Eqs. (6) and (7) were worked out under the assumption that the sampling time is much smaller that the correlation time, which corresponds to have Δt ≪ τ_c or Γ ≪ 1. No explicit expression was reported by Schätzel when this assumption is not fulfilled. In this work we have solved this problem and, by introducing the triangular averaging to all orders, we have worked out two exact analytical formulas that generalize Eqs. (6) and (7) for any value of Γ. Their expressions (see Appendix B for demonstration) read:

Notice that Eq. (9) cannot be derived from Eq. (8) simply by the substitution k = l (see Appendix B). Notice also that, the factor 1/M appearing in Eqs. (6) and (7) has been replaced in Eqs. (8) and (9) with the factor 1/(M − k), which takes into account the effective number of points used for computing Eq. (2). This correction becomes significant when the maximum lag-time is comparable with the measuring time. Finally, it is straightforward to show that, under the limit Γ ≪ 1, Eqs. (8) and (9) tend to Eqs. (6) and (7), respectively.

A comparison between the covariances given by Eqs. (6) and (8) is reported in Fig. 1.

 

Fig. 1 Comparison between Eqs. (6) and (8). (a) and (b): behaviors of Cov_g2 (k, l) given by our Eq. (8) as a function of k for fixed Γ = 0.1 and variable l (a) and for variable Γ and fixed l = 5 (b). All the curves were computed by setting M = 10⁵, β = 1, and 〈n〉 = 1. The peaks correspond to the variances (k = l). (c): relative residuals between (6) and (8) for the curves of (a); (d): relative residuals between Eqs. (6) and (8) for the curves of (b).

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Figure 1(a) shows the behavior of Cov_g2(k, l) given by our Eq. (8) as a function of k for a fixed Γ = 0.1 and different l varying in the range 5 – 100, whereas Fig. 1(b) shows the behavior for different Γ varying in the range 0.05 – 0.4 and a fixed l = 5. The various curves were computed by setting, M = 10⁵, β = 1, and 〈n〉 = 1. The peaks appearing in all the curves of Figs. 1(a) and 1(b) correspond to the variances (k = l). The relative residuals between Eqs. (6) and (8) [(Eq. (6)–Eq. (8))/Eq. (8)] are reported in Figs. 1(c) and 1(d). Figure 1(c) shows that, when gamma is constant (Γ = 0.1), the highest (positive) deviations occur around the variance peak (k = l), whereas the lowest (negative) deviations occur for k ≫ l. In any case, except for k = l, they are always confined within ∼ ±1%. Conversely, Fig. 1(d) shows that, upon increasing Γ, the deviations between Eqs. (6) and (8) become more and more pronounced, up to ∼ +20% (k = l) and ∼ −10% (k ≫ l) for the curve with a Γ = 0.4.

The comparison between the variances given by Eqs. (7) and (9) is reported in Fig. 2 where the behaviors of ${\sigma }_{g_2}^2(k)$ given by our (open symbols) and the Schätzel formula (solid symbols) are reported as a function of k for two values of Γ (0.05 and 0.8) with M = 10⁵ (Fig. 2(a)) and M = 2 × 10³ (Fig. 2(b)). The other two parameters were fixed to β = 1 and 〈n〉 = 1.

 

Fig. 2 Comparison between Eqs. (7) and (9). (a) and (b): behaviors of ${\sigma }_{g_2}^2(k)$ given by Eqs. (7) (open symbols) and (9) (solid small circles) as a function of k for two values of Γ = 0.05 (red circles) and Γ = 0.8 (blue squares) with different statistical accuracies, namely M = 10⁵ (a) and M = 2 × 10³ (b). All the four curves were computed by setting β = 1 and 〈n〉 = 1. (c): relative residuals between Eqs. (6) and (8) for the curves of (a); (d): relative residuals between Eqs. (6) and (8) for the curves of (b).

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Figure 2(a) shows that, as soon as Γ ∼ 1, the differences between the two equations become striking, with relative residuals that, for Γ = 0.8 are as large as ∼ +30–40% (Fig. 2(c)). Conversely, when Γ ≪ 1, as expected, the two equations tend to become identical, with relative residuals that, for Γ = 0.05 are smaller than ∼ +0.1% within the entire k-range (not visible in Fig. 2(c)). Figure 2(b) reports an example of the differences between Eqs. (7) and (9) when the measuring time is not much larger than the maximum lag time, that is when the condition M ≫ k is not fulfilled (M = 2 × 10³ and k_max = 1000). Here the effects of the k–dependent factor 1/(M − k) appearing in Eq. (9) become dominant as k approaches M, whereas is k–independent for Eq. (7). Thus, even for Γ = 0.05, the relative residual between the two equations are as large as ∼ 60% (Fig. 2(d)).

3. Numerical simulations

We carried out several numerical simulations aimed at testing the correctness of our Eqs. (8) and (9) and their comparison with the corresponding Schätzel’s Eqs. (6) and (7). In order to generate synthetic DLS data, we followed the procedure described in [9,10] and the corresponding correlograms, i.e. the correlation function estimators given by Eq. (2), were computed for a finite measuring time T = M Δt at a given average count rate so that 〈n〉 = 〈c.r.〉 Δt. A multi-tau scheme [13,14], in which the lag times are grouped in stages of linear correlators whose integration times Δt_s increase as a pseudo-geometrical progression, was adopted. Such a scheme reads

The use of the multi-tau scheme given by Eq. (11) makes the application of the covariance Eqs. (8) and (9) rather troublesome. Indeed, two correlogram points g₂(τ_k) and g₂(τ_l) whose lag-times belong to different stages are acquired with different sampling times, so that they are subjected to different triangular averaging effects. In this case, Eqs. (8) and (9) cannot be used and a different approach, similar to the one described in [15], must be applied. An example of the statistical analysis of covariance data limited to channels τ_k and τ_l that belong to the same stage is reported in Appendix C.

In this work we focused our study on the variance formula. To this aim, we obtained accurate estimates of 〈g₂(τ_k)〉 and ${\sigma }_{g_2}^2({\tau }_k)$ by accumulating statistics over N independent batches (i.e. repetitions of the simulation, each one providing a different estimator [g₂(τ_k)]_i) so that

The numerical values given by Eq. (13) were compared with the theoretical expressions given by the Schätzel formula (Eq. (7)) and our formula (Eq. (9)), which both depend on the four parameters M_k, 〈n〉_k, Γ_k and β. Notice that, being the simulation carried out for a multi-tau correlator, the first three parameters depend on the lag time τ_k: M_k = T/Δt_k, 〈n〉_k = 〈c.r〉. Δt_k, and Γ_k = Δt_k/τ_c where τ_c is the filed-field decay time and Δt_k the sampling time associated to τ_k. Whereas M_k and 〈n〉_k are known from the simulation settings (or from the experiment), Γ_k and β can be estimated by fitting 〈g₂(τ_k)〉 to a single exponential decay function

3.1. Auto-correlation, monodisperse case (single exponential decay)

DLS data corresponding to a monodisperse sample characterized by a single exponential decay field correlation function (Eq. (5)) with τ_c = 10⁻⁴s were generated for a single coherence area (β = 1), at an average count rate 〈c.r.〉 = 10⁵ Hz for a measuring time T = 1s. The sampling time was Δt₀ = 10⁻⁷s, the multi-tau scheme was p = 28, m = 4, S = 10, so that that the maximum lag-time was τ_max = 0.74 s. N = 10⁴ independent batches were generated for good statistics.

In Fig. 3(a) we report, as a function of τ_k, the behaviors of 〈g₂(τ_k)〉 (red circles) and g₂(τ_k) (green triangles, representing an example of the correlation function recovered within a single batch) together with the corresponding single exponential decay fits (Eq. (14), black curves). From the two fits we recovered τ_c = (1.0002 × 10⁻⁴ ± 2 × 10⁻⁸)s and β = 1.001 ± 0.001 for 〈g₂(τ_k)〉 and τ_c = (1.006 × 10⁻⁴ ± 1 × 10⁻⁶)s and β = 0.99 ± 0.01 for g₂(τ_k), in excellent agreement with the expected values. The accuracy of the fits is also evidenced by the relative [(data-fit)/fit] residuals plots (Fig. 3(b)), where the residuals corresponding to 〈g₂(τ_k)〉 have been zoomed by a factor 100. Notice that the g₂(τ_k) residuals are non systematic over the entire τ_k range, with r.m.s. fluctuations that decrease significantly from the smaller to the larger τ_k values (approximately from ∼ 5 × 10⁻² to ∼ ×10⁻³). A similar trend is shown by the 〈g₂(τ_k)〉 residuals that, except for some discontinuities occurring in correspondence of the various changes of stage (associated to the multi-tau scheme expressed in Eq. (11)) are on average smaller by a factor 100 than the single batch residuals, as expected by the fact that 〈g₂(τ_k)〉 has been obtained by averaging N = 10⁴ independent batches.

 

Fig. 3 Results of the simulation test described in Sect.3.1. (a): behaviors of the average 〈g₂(τ_k)〉 (red circles) and single batch g₂(τ_k) (green triangles) auto-correlation functions obtained by computer simulations for the case of a monodisperse sample with a single exponential decay time τ_c = 10⁻⁴ s. The simulation was carried out by setting β = 1, 〈c.r.〉 = 10⁵ Hz, T = 1 s and N = 10⁴. The black curves represent single exponential decay fits (Eq. (14)). For the sake of clarity data and fit corresponding to 〈g₂(τ_k)〉 have been shifted upwards by 0.5. (b): relative residuals between simulated data and fit; the residuals corresponding to 〈g₂(τ_k)〉 have been zoomed by a factor 100. (c): comparison between the standard deviation σ_g2(τ_k) estimated by using Eq. (13) and the predictions of the original Schätzel (dotted line) and our (solid line) formulas. (d): relative residuals between σ_g2(τ_k) and our formula (red circles). In this panel the blue squares represent the residuals when the correlation analysis is carried out by using a different multi-tau scheme with p = 16, m = 2, S = 20.

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By making use of Eq. (13) we estimated quite accurately σ_g2(τ_k), whose behavior as a function of τ_k is reported in Fig. 3(c) and compared with the standard deviations given by the Schätzel formula (Eq. (7), dashed line) and our formula (Eq. (9), solid line). As evident, the original Schätzel formula is able to reproduce the numerical simulations only for τ_k ≲ τ_c, which corresponds to a stage with a sampling time Δt = 1.024 × 10⁻⁴ s and therefore Γ ∼ 1. Conversely, our formula works quite accurately over the entire range, as shown in the relative [(simulation-theory)/theory] residuals plot of Fig. 3(d) (red circles). In this figure we have also reported the residuals that are obtained when the analysis is carried out by using a different multi-tau scheme, namely the one commonly used in commercial hardware correlators where p = 16, m = 2, S = 20 (blue squares). Notice that our formula is capable to reproduce also the upturn exhibited by the standard deviation in correspondence of the last stage, where the lag-times become comparable with the measuring time. As mentioned at the end of the last section, in this region the assumption (M − k) ≫ 1, is no longer valid and this is probably the reason why the relative residuals tend to become progressively more and more systematic.

3.2. Auto-correlation, polydisperse case (Log-Normal distributions, σ_τc /〈τ_c〉 = 50% and 100%)

In this second test we investigated the possibility of exploiting Eqs. (9) for describing the variance of a polydisperse sample characterized by a given distribution N(τ_c) of decay times. In this case

We carried out several tests by using Log-Normal distributions with the same average value 〈τ_c〉 = 10⁻⁴ s and increasing relative standard deviations up to a maximum value of σ_τc/〈τ_c〉 = 100%. For each distribution we generated synthetic DLS data under the same conditions of Sect.3.1, (β = 1, 〈c.r.〉 = 10⁵ Hz, T = 1s, N = 10⁴) and computed the corresponding correlograms by using the same settings (Δt₀ = 10⁻⁷s, p = 28, m = 4, S = 10, so that τ_max = 0.74s). The average and a single batch correlograms (red and green symbols) were fitted to Eq. (14) (black curves) following the same procedure outlined for Fig. 3 and the results are shown in Fig. 4 for the case of a σ_τc/〈τ_c〉 = 50% distribution. From the two fits we recovered (τ_c)_eqw = (8.86×10⁻⁵ ±3.2×10⁻⁷)s and β = 0.981±0.002 for 〈g₂(τ_k)〉 and (τ_c)_eqw = (8.90×10⁻⁵ ±9.1×10⁻⁷)s and β = 0.997±0.006 for g₂(τ_k), in clear disagreement with the expected values β = 1 and 〈τ_c〉 = 10⁻⁴ s. Consistently, the residual plots exhibit systematic deviations that are somewhat masked by noise for g₂(τ_k), but clearly visible for 〈g₂(τ_k)〉.

 

Fig. 4 Results of the simulation test described in Sect.3.2. (a): behaviors of the average 〈g₂(τ_k)〉 (red circles) and single batch g₂(τ_k) (green triangles) auto-correlation functions obtained by computer simulations for the case of a polydisperse sample characterized by a Log-normal distribution of decay times with average value 〈τ_c〉 = 10⁻⁴s and relative standard deviation σ_τc/〈τ_c〉 = 50%. The simulation was carried out by setting β = 1, 〈c.r.〉 = 10⁵ Hz, T = 1 s and N = 10⁴. The black curves represent single exponential decay fits (Eq. (14)). For the sake of clarity data and fit corresponding to 〈g₂(τ_k)〉 have been shifted upwards by 0.5. (b): relative residuals between simulated data and fit. (c): comparison between the standard deviation σ_g2(τ_k) estimated by using Eq. (13) and the predictions of the original Schätzel (dotted line) and our (solid line) formulas. (d): relative residuals between σ_g2(τ_k) and our formula (red circles). In this panel the blue squares represent the residuals when a σ_τc /〈τ_c〉 = 100% polydispersity was considered.

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The behavior of σ_g2(τ_k) as a function of τ_k (computed by using the values (τ_c)_eqw = 8.86 × 10⁻⁵ s and β = 0.981) is reported in Fig. 4(c) and compared with the standard deviation given by the Schätzel formula (Eq. (7), dashed line) and our formula (Eq. (9), solid line).

As for Fig. 3(c), the original Schätzel formula is able to reproduce the numerical simulations only for τ_k ≲ τ_c, whereas our formula works quite accurately over the entire range. The corresponding residual plot (Fig. 4(d), red circles), shows that the relative residuals are somewhat systematic, but their amplitudes are always smaller than ≲ ±5%. In this figure, we have also reported the residuals (blue squares) obtained in the case of a larger polydispersity, when σ_τc/〈τ_c〉 = 100%, where they increase up to a maximum of ∼ 10%.

We can therefore conclude that our Eq. (9) works reasonably well also in the case of moderately large (∼ 50 – 100%) bell-shaped polydisperse samples.

3.3. Cross-correlation, monodisperse case (single exponential decay)

The third test was aimed at ascertaining the correct functioning of our formula in the case of a cross-correlation function. As known, when in a DLS experiment lag-times in the range of microseconds or smaller are to be measured with high accuracy, the use of cross-correlation (instead of auto-correlation) of the same optical signal is highly recommended. This is realized by splitting the collected scattered light into two intensity signals, and detecting each signal with different independent single photon counting units. In this way by cross-correlating the two pulse streams, optical correlations are maintained, whereas any defect associated to the detectors (afterpulse, dead time, dark count) that typically show up in the sub-microsecond range are filtered out.

We first generated an analog Intensity signal corresponding to a monodisperse sample characterized by a single exponential decay field correlation function (Eq. (5)) with τ_c = 10⁻⁴s, within a single coherence area (β = 1) for a measuring time T = 1s. Then, by applying two independent Poisson filters (affected by 6% afterpulse defects) to this signal, two pulse streams at different average count rates 〈c.r.〉_A and 〈c.r.〉_B were generated and cross-correlated. The simulation was carried out by using the same correlogram settings of Sects.3.1 and 3.2, fixing 〈c.r.〉_A = 10⁵ Hz and let 〈c.r.〉_B varying between were 10⁵ and 5 × 10⁴ Hz.

The comparison between simulations and theory was carried out by using for the parameter 〈n〉, the geometric mean between 〈n〉_A and 〈n〉_B, i.e.

 

Fig. 5 Results of the simulation test described in Sect.3.3. (a): behaviors of the average auto- and cross-correlation functions 〈g₂(τ_k)〉_A−A (blue squares), 〈g₂(τ_k)〉_B−B (green triangles), 〈g₂(τ_k)〉_A−B (red circles) and an example of a single batch cross-correlation function g₂(τ_k)_A−B (orange lozenges) obtained by computer simulations for the case of a monodisperse sample with a single exponential decay time τ_c = 10⁻⁴ s. The simulation was carried out by setting β = 1, 〈c.r. 〉_A = 2〈c.r.〉_B = 10⁵ Hz, T = 1 s and N = 10⁴. The black curves represent single exponential decay fits (Eq. (14)). For the sake of clarity data and fits of the various curves have been shifted upwards by increasing steps of 0.5. (b): relative residuals between cross-correlation simulated data and fit; the residuals corresponding to 〈g₂(τ_k)〉_A−B have been zoomed by a factor 100. (c): comparison between the standard deviation σ_g2(τ_k) associated to the cross-correlation function and the predictions of the original Schätzel (dotted line) and our (solid line) formulas. (d): relative residuals between σ_g2(τ_k) and our formula (red circles). In this panel we have also reported the residuals obtained for the two auto-correlation functions (green triangles and blue blue squares) and the ones obtained for the cross-correlation when 〈c.r.〉_A = 〈c.r.〉_B = 10⁵ Hz (black dots).

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As for the previous tests, the behavior of σ_g2(τ_k) as a function of τ_k shows that our formula (Fig. 5(c), solid line) works quite accurately over the entire range. The corresponding residual plot (Fig. 5(d), red circles), shows that the relative residuals are slightly systematic, with deviations always smaller than ≲ ±5%. In this figure, we have also reported the residuals obtained for the two auto-correlation functions (green triangles and blue squares), which are similar to each other (B − B residuals are higher than A − A residuals because of the smaller count rate) and slightly larger than the cross-correlation residuals. Finally, we report the residuals obtained for the cross-correlation when 〈c.r.〉_A = 〈c.r.〉_B = 10⁵ Hz (black dots). In this case, the residuals are even smaller and similar to the ones obtained for the in Sect.3.1.

We can therefore conclude that our Eq. (9) works reasonably well also in the case of single exponential decay cross-correlation function, provided that the average count rate to be used in these equations is given by the geometrical mean of the average count rates of the two channels. Furthermore, it is shown that the optimal working conditions occur when the two channels are balanced, i.e. when 〈c.r.〉_A ∼ 〈c.r.〉_B.

4. Experimental results

The applicability of our formulas to real data was tested by using an home made DLS apparatus [13], in which the sample, a dilute dispersion of latex spheres contained in a standard 10mm square cuvette, was shined with the beam of a frequency doubled ∼ 100mW Nd:YAG laser (Coherent, mod. Compass 315M-100) operating at λ = 532nm. The light scattered at 90° was collected with a mono-mode fiber (OZ Optics mod. LPC-01-532-3.5/125) whose output was coupled to a commercial cross-correlator detector (ALV Langen, mod. ALV/SO-SIPD) followed by a 65ns home made dead time circuit. The particles were calibrated polystyrene spheres from Thermo-Fisher Scientific Inc. (catalog n.3040A) with a DLS-certified average diameter d = 41 ± 3nm (polydispersity not provided), dispersed in Milli-Q filtered water at room temperature (T = 22±0.1°C). The auto- and cross-correlation functions were computed by using the software correlator described in [13]. The average count rates of the two channels were 〈c.r.〉_A ∼ 1.5 × 10⁴ Hz and 〈c.r.〉_B ∼ 1.0 × 10⁴ Hz, so that their geometrical mean (see Eq. (16)) was 〈c.r.〉 ∼ 1.2 × 10⁴ Hz. The measuring time was T = 0.7s, the gate time Δt₀ = 2.5 × 10⁻⁷s and the multi-tau scheme p = 28, m = 4, S = 9, so that the maximum lag-time was τ_max = 0.44 s. N = 10⁴ independent batches were generated for good statistics.

In Fig. 6(a) we report, as a function of τ_k, the behaviors of the average cross-correlation functions 〈g₂(τ_k)〉 (red circles) and an example of a single batch cross-correlation function g₂(τ_k) (blue lozenges) together with their corresponding single exponential fits (Eq. (14)). The corresponding relative residuals are reported in Fig. 6(b). Whereas the residuals associated to g₂(τ_k) appear to be non systematic, the one associate to 〈g₂(τ_k)〉 (zoomed by a factor 50) exhibits some systematic trend, probably due to sample polydispersity. Indeed, by fitting the 〈g₂(τ_k)〉 data with the first two cumulants, we obtain residuals (orange triangles) that are much smaller (zoomed by a factor 100 in the figure) and non systematic. The final z-averaged particle diameter recovered by using the Stokes-Einstein relation was 〈d〉_z = 44.3 ± 0.1nm with a polydispersity of 〈σ〉_z = 8.2 ± 0.4nm, in fairly good agreement with the expected certified value.

 

Fig. 6 Results of the experimental test described in Sect.4. (a): behaviors of the average cross-correlation function 〈g₂(τ_k)〉 (red circles) and an example of a single batch cross-correlation function g₂(τ_k)_B (blue lozenges) obtained by DLS data on a aqueous dispersion of d = 41 ±3nm latex spheres whose scattered light was collected at 90° with a mono-mode fiber and coupled to two photo-multipliers whose average count rates were 〈c.r.〉_A ∼ 1.5×10⁴ Hz and 〈c.r.〉_B ∼ 1.0 × 10⁴ Hz. The measuring time was T = 0.7 s and N = 10⁴ independent batch were analyzed. The black curves represent single exponential decay fits (Eq. (14)). For the sake of clarity data and fit corresponding to 〈g₂(τ_k)〉 have been shifted upwards by 1. (b): relative residuals between simulated data and fit; the residuals corresponding to 〈g₂(τ_k)〉 have been zoomed by a factor 50. The orange symbols are the non systematic residuals obtained when 〈g₂(τ_k)〉 is fitted with the with the first two cumulants. (c): comparison between the standard deviation σ_g2(τ_k) estimated by using Eq. (13) and the predictions of the original Schätzel (dotted line) and our (solid line) formulas. (d): relative residuals between σ_g2(τ_k) and our formula for the data of (c) (red circles) and when the measurements are taken at a higher count rate (green squares).

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The behavior of σ_g2(τ_k) as a function of τ_k is reported in Fig. 6(c). As for the simulation test, the figure shows that our formula (Fig. 6(c), solid line) works quite accurately over the entire range. The corresponding residual plot (Fig. 6(d), red circles), shows a non-systematic trend always smaller than ≲ 2 – 3%. In this figure, we have also reported the residuals obtained when the measurements are taken at higher count rates, namely 〈c.r.〉_A = ∼ 1.1 × 10⁵ Hz and 〈c.r.〉_B ∼ 7.5×10⁴ Hz (green squares) corresponding to a geometrical mean 〈c.r.〉 ∼ 9.1×10⁴ Hz. In this case the residuals are slightly systematic (∼ −5%) at low lag-times, probably due to the 65ns dead time circuit, whose effects become visible at high count rates. Simulations carried by using detectors with and without dead time defects confirm this hypothesis (data not shown).

5. Conclusions

In conclusion, we have worked out two exact analytical expressions that describe the covariance matrix Cov_g2(τ_k, τ_l) and the variance ${\sigma }_{g_2}^2\left({\tau }_k\right)$ associated to estimator g₂(τ_k) of the normalized auto-correlation function that is recovered in a Dynamic Light Scattering experiment when the sample is characterized by a single exponential decay correlation function. The new formulas (Eqs. (8) and (9)) depend on four dimensionless parameters that are either known or can be recovered from the experiment, namely: the coherence factor β given by the zero intercept of g₂(τ_k) (subtracted by 1), the number of sampling times M = T/Δt where T is the measuring time and Δt the sampling time, the reduced decay time Γ = Δt/τ_c where τ_c is the (field) decay time, and the average number of counts 〈n〉 = 〈c.r.〉 Δt, where 〈c.r.〉 is the average count rate.

Equations (8) and (9) generalize the original Schätzel formulas (Eq. (6) and (7)), where the effects of the triangular averaging due to the finite sampling time, although thoughtfully discussed by Schätzel, were non explicitly reported. In the new formulas we have also introduced a small refinement not present in the original Schätzel formulas, that allowed us to obtain accurate estimates of Cov_g2(τ_k, τ_l) and ${\sigma }_{g_2}^2\left({\tau }_k\right)$ also in the cases when the maximum lag-time is comparable with the measuring time T.

The correctness and accuracies of the new formulas were tested by numerous computer simulations. However, since the use of the covariance formula must be restricted to channels to that are acquired with the same sampling time Δt and therefore is of rather limited interest for multi-tau correlators, we focussed our tests on the use of the variance formula (Eq. (9)), which is the one more commonly used in the data fitting procedures. These simulations allowed us to investigate the possibility to extend the use of Eq. (9) beyond the particular case of a single exponential decay correlation function. Indeed, we showed numerically that it works accurately (≲ 5 – 10%) also in the case of: (i) auto-correlation functions characterized by a (bell-shaped) distribution of decay times with polydispersities up to ∼ 50 – 100% and, (ii) cross-correlation functions between two channels, characterized by the same single exponential decay time, provided that the average count rate is computed as the geometrical mean of the average count rates of the two channels. Finally, we checked that the new formula for the variance is also able to reproduce accurately the error bars obtained by averaging experimental data taken on dilute dispersions of calibrated polystyrene particles.

As to the covariance matrix, we showed that, within the restriction of having the two channels belonging to the same stage, our new covariance formula (Eq. (8)) works quite well for any value of Γ, whereas the original Schätzel formula is accurate only when Γ ≪ 1. An example related to the case of a monodisperse system is reported in appendix C.

As a final comment, it is interesting to compare our results with the statistical analysis adopted in Fluorescence Correlation Spectroscopy (FCS). In this case, one of the most useful formulas is the one found in 1974 by D.E. Koppel [16], who derived a simple analytical expression for the standard deviation of the fluorescence correlation function. His formula was worked out, among the others, under the assumptions of Gaussian statistics (implying a high number of fluorescent particles and consequently a small zero-lag time intercept) and a sampling time much smaller than the correlation time. Under these two conditions (β ≪ 1 and Γ ≪ 1) Koppel formula, rewritten as in [17, 18], is fairly similar to both our and Schätzel formulas. Interestingly, under these conditions, these three formulas are able to describe not only DLS and FCS data, but also data coming from optical techniques based on multiply scattered light, such as Diffusive Correlation Spectroscopy [18] and Diffusive Wave spectroscopy [19]. Finally, it should be pointed out that, when Γ ≳ 1 (regardless of β), the Koppel’s formula is similar only to Schätzel formula [see Fig. 3(c)] and fails to describe the experimental error bars [17]. A correct estimate of the FCS error bars for any Γ condition was provided by Saffarian and Elson in 2003 [20]. A comparison between our results and this work is left to future work.

In conclusion, we believe that our new variance formula would turn out to be a fairly useful tool for the wide community of scientists working in the field of DLS. Nowadays, this technique is ubiquitously based on the use of multi-tau correlators, where, if not properly taken into account, the triangular averaging would introduce huge errors in the estimates of the uncertainties to be associated to the measured correlation function.

Appendix

A Triangular averaging

Equation (1) is based on the assumption that the time resolution used for the measurement of the scattered intensity is infinite. However, in a real experiment, the detected signal is integrated on a finite sampling time Δt. As a consequence, the measured intensity is given by

(17)$$\widehat{I}\left(t\right)=\frac{1}{\mathrm{\Delta }t}{\int }_{t-\mathrm{\Delta }t/2}^{t+\mathrm{\Delta }t/2}I\left(t^{\prime }\right)\text{d}{t}^{\prime }=\frac{1}{\mathrm{\Delta }t}\left[I\otimes R_{\mathrm{\Delta }t}\right]\left(t\right)$$

where ⊗ denotes the convolution product and R_a(x) is the rectangular function defined as R_a(x) = 1 for |x/a| ≤ 0.5 and zero elsewhere. By inserting Eq. (17) into Eq. (1), it is straightforward to show that

(18)$${\widehat{g}}_2\left(\tau \right)=\frac{1}{\mathrm{\Delta }t}\left[g_2\otimes {\mathrm{\Lambda }}_{\mathrm{\Delta }t}\right]\left(\tau \right)$$

where Λ_a(x) is the triangular function defined as Λ_a(x) = 1 − |x/a| for |x/a| ≤ 0.5 and zero elsewhere. Eq. (18) is called triangular averaging because ĝ₂ is an average version of g₂, with the weighing function being given by the triangle Λ_Δt.

As long as the Δt ≪ τ, the triangular averaging introduces negligible distortions in ĝ₂ (accuracy is better than 10⁻³ for Δt < τ/7, see [14]), but, as pointed out by Schätzel in [7]), it might introduce significative distortions in the estimate of the covariance matrix that is associated to correlation function estimator g₂(τ_k) given by Eq. (2).

K. Schätzel was able to express Cov_g2(τ_k, τ_l) and ${\sigma }_{g_2}^2\left({\tau }_k\right)$ as a sum of several numerical series involving only the field correlation function χ(τ_k) (from now on indicated as χ_k). As already mentioned, the original covariance formula (Eq. (39) of [7]) and variance formula (Eq. (41) of [7]) are related to an ideal correlation function where no triangular averaging has been taken into account. When this correction is properly introduced, the various terms χ_k appearing in these two equations must be replaced with ${\widehat{\chi }}_k=\sqrt{{\left|{\widehat{\chi }}_k\right|}^2}$, where

(19)$${\left|{\widehat{\chi }}_k\right|}^2=\frac{1}{\mathrm{\Delta }\tau }\left({\left|{\chi }_k\right|}^2\otimes {\mathrm{\Lambda }}_{\mathrm{\Delta }t}\right)\left(\tau \right)=\frac{1}{{\left(\mathrm{\Delta }t\right)}^2}{\int }_{(k-1)\mathrm{\Delta }t}^{(k+1)\mathrm{\Delta }t}{\left|\chi \left(t\right)\right|}^2\left(\mathrm{\Delta }t-\left|t-k\mathrm{\Delta }t\right|\right)\text{d}t.$$

However, as pointed out by Schätzel in [7], only the terms with k = 0 and k = ±1 are responsible for non-negligible corrections, whereas all the remaining ones with |k| > 1 can be ignored and set to |χ̂_k|² = |χ_k|². The first two terms read

(20)$${\left|{\widehat{\chi }}_0\right|}^2=\frac{1}{{\left(\mathrm{\Delta }t\right)}^2}{\int }_{-\mathrm{\Delta }t}^{\mathrm{\Delta }t}{\left|\chi \left(t\right)\right|}^2\left(\mathrm{\Delta }t-\left|t\right|\right)\text{d}t$$

(21)$${\left|{\widehat{\chi }}_{\pm 1}\right|}^2=\frac{1}{{\left(\mathrm{\Delta }t\right)}^2}{\int }_0^{2\mathrm{\Delta }t}\left|\chi {\left(t\right)}^2\right|\left(\mathrm{\Delta }t-\left|t-\mathrm{\Delta }t\right|\right)\text{d}t$$

which correspond to Eqs. (58) and (59) of [7].

In conclusion, from a practical point of view, the general procedure for estimating the covariance and variance associated to a measured correlation function is: (i) measure ĝ₂(τ_k) with good statistics, so that all the terms |χ_k|², the intercept β and the average number of counts 〈n〉 can be recovered accurately from the data; (ii) replace |χ₀|² and |χ_±1|² with the respective |χ̂₀|² and |χ̂_±1|² by using Eqs. (20) and (21); (iii) use Eqs. (39) and (41) of [7] by stopping the infinite series when convergence is attained.

B. Demonstration of Eqs. (8) and (9)

In the case of a Lorentzian spectrum all the χ̂_k terms can be computed analytically. By inserting Eq. (5) into Eq. (19) we get for k = 0

(22)$${\widehat{\chi }}_0=\sqrt{\frac{1}{{\left(\mathrm{\Delta }t\right)}^2}{\int }_{-\mathrm{\Delta }t}^{\mathrm{\Delta }t}{e}^{-2\mathrm{\Gamma }\left|t\right|\mathrm{\Delta }t}\left(\mathrm{\Delta }t-\left|t\right|\right)\text{d}t}=\sqrt{\frac{2\mathrm{\Gamma }-1+e^{-2\mathrm{\Gamma }}}{2{\mathrm{\Gamma }}^2},}$$

whereas, for k ≠ 0

(23)$${\widehat{\chi }}_k=\sqrt{\frac{1}{{\left(\mathrm{\Delta }t\right)}^2}{\int }_{\left(k-1\right)\mathrm{\Delta }t}^{\left(k+1\right)\mathrm{\Delta }t}{e}^{-2\mathrm{\Gamma }\left|t\right|/\mathrm{\Delta }t}\left(\mathrm{\Delta }t-\left|t-k\mathrm{\Delta }t\right|\right)\text{d}t=}\frac{\text{sinh}\left(\mathrm{\Gamma }\right)}{\mathrm{\Gamma }}{e}^{-\mathrm{\Gamma }k}=\frac{\text{sinh}(\mathrm{\Gamma })}{\mathrm{\Gamma }}\chi k.$$

Correspondingly, the correction of the correlation function ĝ₂ is

(24)$${\widehat{g}}_2\left({\tau }_k\right)=1+\beta \text{exp}\left(-2\mathrm{\Gamma }k\right)\frac{{\text{sinh}}^2\left(\mathrm{\Gamma }\right)}{{\mathrm{\Gamma }}^2}.$$

By replacing the field correlation function χ_k with its specific corrective terms χ̂_k we can work out analytically all the 8 sums appearing in Eq. (39) of [7] as

(25)$$\begin{array}{l}\sum _{i=-\infty }^{+\infty }{\left|{\widehat{\chi }}_i\right|}^2{\left|{\widehat{\chi }}_{i+k-l}\right|}^2=\frac{{\text{sinh}}^4\left(\mathrm{\Gamma }\right)}{{\mathrm{\Gamma }}^4}{e}^{-2\mathrm{\Gamma }\left(k-l\right)}\left[k–l+\text{coth}\left(2\mathrm{\Gamma }\right)\right]+2\frac{{\text{sinh}}^2\left(\mathrm{\Gamma }\right)}{{\mathrm{\Gamma }}^2}\left({\widehat{\chi }}_0^2-\frac{{\text{sinh}}^2\left(\mathrm{\Gamma }\right)}{{\mathrm{\Gamma }}^2}\right)e^{-2\mathrm{\Gamma }\left(k-l\right)}\\ \sum _{i=-\infty }^{+\infty }{\left|{\widehat{\chi }}_i\right|}^2{\left|{\widehat{\chi }}_{i+k+l}\right|}^2=\frac{{\text{sinh}}^4\left(\mathrm{\Gamma }\right)}{{\mathrm{\Gamma }}^4}{e}^{-2\mathrm{\Gamma }\left(k+l\right)}\left[k+l+\text{coth}\left(2\mathrm{\Gamma }\right)\right]+2\frac{{\text{sinh}}^2\left(\mathrm{\Gamma }\right)}{{\mathrm{\Gamma }}^2}\left({\widehat{\chi }}_0^2-\frac{{\text{sinh}}^2\left(\mathrm{\Gamma }\right)}{{\mathrm{\Gamma }}^2}\right)e^{-2\mathrm{\Gamma }\left(k+1\right)}\\ \text{Re}\left({\widehat{\chi }}_k{\widehat{\chi }}_l^{*}\sum _{i=-\infty }^{+\infty }{\widehat{\chi }}_i{\widehat{\chi }}_{i+k-l}^{*}\right)=\frac{{\text{sinh}}^4\left(\mathrm{\Gamma }\right)}{{\mathrm{\Gamma }}^4}{e}^{-2\mathrm{\Gamma }k}\left[k-l+\text{coth}\left(\mathrm{\Gamma }\right)\right]+2\frac{{\text{sinh}}^3\left(\mathrm{\Gamma }\right)}{{\mathrm{\Gamma }}^3}\left({\widehat{\chi }}_0-\frac{\text{sinh}\left(\mathrm{\Gamma }\right)}{\mathrm{\Gamma }}\right)e^{-2\mathrm{\Gamma }k}\\ \text{Re}\left({\widehat{\chi }}_k{\widehat{\chi }}_l^{*}\sum _{i=-\infty }^{+\infty }{\widehat{\chi }}_i{\widehat{\chi }}_{i+k+l}^{*}\right)=\frac{{\text{sinh}}^4\left(\mathrm{\Gamma }\right)}{{\mathrm{\Gamma }}^4}{e}^{-2\mathrm{\Gamma }(k+l)}\left[k+l+\text{coth}\left(\mathrm{\Gamma }\right)\right]+2\frac{{\text{sinh}}^3\left(\mathrm{\Gamma }\right)}{{\mathrm{\Gamma }}^3}\left({\widehat{\chi }}_0-\frac{\text{sinh}\left(\mathrm{\Gamma }\right)}{\mathrm{\Gamma }}\right)e^{-2\mathrm{\Gamma }(k+l)}\\ \text{Re}\left(\sum _{i=-\infty }^{+\infty }{\widehat{\chi }}_i{\widehat{\chi }}_{i+k}^{*}{\widehat{\chi }}_{i+l}^{*}{\widehat{\chi }}_{i+k+l}\right)=\frac{{\text{sinh}}^4\left(\mathrm{\Gamma }\right)}{{\mathrm{\Gamma }}^4}{e}^{-2\mathrm{\Gamma }k}\left[k-l+\text{coth}\left(\mathrm{\Gamma }\right)+e^{-2\mathrm{\Gamma }l}(\text{coth}(2\mathrm{\Gamma })-\text{coth}(\mathrm{\Gamma }))\right]+\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+2\frac{{\text{sinh}}^3(\mathrm{\Gamma })}{{\mathrm{\Gamma }}^3}\left({\widehat{\chi }}_0-\frac{\text{sinh}(\mathrm{\Gamma })}{\mathrm{\Gamma }}\right)\left(1+e^{-2\mathrm{\Gamma }l}\right)e^{-2\mathrm{\Gamma }k}\\ {\left|{\widehat{\chi }}_k\right|}^2{\left|{\widehat{\chi }}_l\right|}^2\sum _{i=-\infty }^{+\infty }{\left|{\widehat{\chi }}_i\right|}^2=\frac{{\text{sinh}}^6(\mathrm{\Gamma })}{{\mathrm{\Gamma }}^6}{e}^{-2\mathrm{\Gamma }(k+l)}\text{coth}(\mathrm{\Gamma })+\frac{{\text{sinh}}^4(\mathrm{\Gamma })}{{\mathrm{\Gamma }}^4}\left({\widehat{\chi }}_0^2-\frac{{\text{sinh}}^2(\mathrm{\Gamma })}{{\mathrm{\Gamma }}^2}\right)e^{-2\mathrm{\Gamma }(k+l)}\\ {\left|{\widehat{\chi }}_l\right|}^2\text{Re}\left({\widehat{\chi }}_k\sum _{i=-\infty }^{+\infty }{\widehat{\chi }}_i{\widehat{\chi }}_{i+k}^{*}\right)=\frac{{\text{sinh}}^5(\mathrm{\Gamma })}{{\mathrm{\Gamma }}^5}{e}^{-2\mathrm{\Gamma }(k+l)}\left[k+\text{coth}(\mathrm{\Gamma })\right]+2\frac{{\text{sinh}}^4(\mathrm{\Gamma })}{{\mathrm{\Gamma }}^4}\left({\widehat{\chi }}_0-\frac{\text{sinh}(\mathrm{\Gamma })}{\mathrm{\Gamma }}\right)e^{-2\mathrm{\Gamma }(k+l)}\\ {\left|{\widehat{\chi }}_k\right|}^2\text{Re}\left({\widehat{\chi }}_l\sum _{i=-\infty }^{+\infty }{\widehat{\chi }}_i{\widehat{\chi }}_{i+l}^{*}\right)=\frac{{\text{sinh}}^5(\mathrm{\Gamma })}{{\mathrm{\Gamma }}^5}{e}^{-2\mathrm{\Gamma }(k+l)}\left[l+\text{coth}(\mathrm{\Gamma })\right]+2\frac{{\text{sinh}}^4(\mathrm{\Gamma })}{{\mathrm{\Gamma }}^4}\left({\widehat{\chi }}_0-\frac{\text{sinh}(\mathrm{\Gamma })}{\mathrm{\Gamma }}\right)e^{-2\mathrm{\Gamma }(k+l)}\end{array}$$

By substituting these 8 terms into Eq. (39) of [7], Eq. (8) of this article can be worked out.

Similarly, for the variance, we can express analytically the 7 sums appearing in in Eq. (41) of [7] as

(26)$$\begin{array}{l}\sum _{i=-\infty }^{+\infty }{\left({\left|{\widehat{\chi }}_i\right|}^2\right)}^2=\frac{{\text{sinh}}^4\left(\mathrm{\Gamma }\right)}{{\mathrm{\Gamma }}^4}\text{coth}(2\mathrm{\Gamma })+{\left({\widehat{\chi }}_0^2\right)}^2-\frac{{\text{sinh}}^4\left(\mathrm{\Gamma }\right)}{{\mathrm{\Gamma }}^4}\\ \sum _{i=-\infty }^{+\infty }{\left|{\widehat{\chi }}_i\right|}^2{\left|{\widehat{\chi }}_{i+2k}\right|}^2=\frac{{\text{sinh}}^4\left(\mathrm{\Gamma }\right)}{{\mathrm{\Gamma }}^4}{e}^{-4\mathrm{\Gamma }k}\left[2k+\text{coth}(2\mathrm{\Gamma })\right]+2\frac{{\text{sinh}}^2(\mathrm{\Gamma })}{{\mathrm{\Gamma }}^2}\left({\widehat{\chi }}_0^2-\frac{{\text{sinh}}^2\left(\mathrm{\Gamma }\right)}{{\mathrm{\Gamma }}^2}\right)e^{-4\mathrm{\Gamma }k}\\ {\left|{\widehat{\chi }}_k\right|}^2\sum _{i=-\infty }^{+\infty }{\left|{\widehat{\chi }}_i\right|}^2=\frac{{\text{sinh}}^4\left(\mathrm{\Gamma }\right)}{{\mathrm{\Gamma }}^4}{e}^{-2\mathrm{\Gamma }k}\text{coth}(2\mathrm{\Gamma })+\frac{{\text{sinh}}^2\left(\mathrm{\Gamma }\right)}{{\mathrm{\Gamma }}^2}\left({\widehat{\chi }}_0^2-\frac{{\text{sinh}}^2\left(\mathrm{\Gamma }\right)}{{\mathrm{\Gamma }}^2}\right)e^{-2\mathrm{\Gamma }k}\\ \text{Re}\left({\widehat{\chi }}_k^2\sum _{i=-\infty }^{+\infty }{\widehat{\chi }}_i{\widehat{\chi }}_{i+2k}^{*}\right)=\frac{{\text{sinh}}^4\left(\mathrm{\Gamma }\right)}{{\mathrm{\Gamma }}^4}{e}^{-4\mathrm{\Gamma }k}\left[2k+\text{coth}\left(\mathrm{\Gamma }\right)\right]+2\frac{{\text{sinh}}^3\left(\mathrm{\Gamma }\right)}{{\mathrm{\Gamma }}^3}\left({\widehat{\chi }}_0-\frac{\text{sinh}\left(\mathrm{\Gamma }\right)}{\mathrm{\Gamma }}\right)e^{-4\mathrm{\Gamma }k}\\ \text{Re}\left({\widehat{\chi }}_k^2\sum _{i=-\infty }^{+\infty }{\widehat{\chi }}_i{\widehat{\chi }}_{i+k}^{*2}{\widehat{\chi }}_{i+2k}\right)=\frac{{\text{sinh}}^4\left(\mathrm{\Gamma }\right)}{{\mathrm{\Gamma }}^4}{e}^{-4\mathrm{\Gamma }k}\left[\text{coth}\left(2\mathrm{\Gamma }\right)-\text{coth}(\mathrm{\Gamma })\right]+\frac{{\text{sinh}}^4\left(\mathrm{\Gamma }\right)}{{\mathrm{\Gamma }}^4}{e}^{-2\mathrm{\Gamma }k}\text{coth}(\mathrm{\Gamma })+\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+\frac{{\text{sinh}}^2(\mathrm{\Gamma })}{{\mathrm{\Gamma }}^2}{e}^{-2\mathrm{\Gamma }k}\left({\widehat{\chi }}_0^2-\frac{{\text{sinh}}^2(\mathrm{\Gamma })}{{\mathrm{\Gamma }}^2}\right)\left(1+2\frac{\text{sinh}(\mathrm{\Gamma })}{\mathrm{\Gamma }}{e}^{-2\mathrm{\Gamma }k}\right)\\ {\left({\left|{\widehat{\chi }}_k\right|}^2\right)}^2\sum _{i=-\infty }^{+\infty }{\left|{\widehat{\chi }}_i\right|}^2=\frac{{\text{sinh}}^6(\mathrm{\Gamma })}{{\mathrm{\Gamma }}^6}{e}^{-4\mathrm{\Gamma }k}\text{coth}(\mathrm{\Gamma })+\frac{{\text{sinh}}^4(\mathrm{\Gamma })}{{\mathrm{\Gamma }}^4}\left({\widehat{\chi }}_0^2-\frac{{\text{sinh}}^2(\mathrm{\Gamma })}{{\mathrm{\Gamma }}^2}\right)e^{-4\mathrm{\Gamma }k}\\ {\left|{\widehat{\chi }}_k\right|}^2\text{Re}\left({\widehat{\chi }}_k\sum _{i=-\infty }^{+\infty }{\widehat{\chi }}_i{\widehat{\chi }}_{i+k}^{*}\right)=\frac{{\text{sinh}}^5(\mathrm{\Gamma })}{{\mathrm{\Gamma }}^5}{e}^{-4\mathrm{\Gamma }k}\left[k+\text{coth}(\mathrm{\Gamma })\right]+2\frac{{\text{sinh}}^4(\mathrm{\Gamma })}{{\mathrm{\Gamma }}^4}\left({\widehat{\chi }}_0-\frac{\text{sinh}(\mathrm{\Gamma })}{\mathrm{\Gamma }}\right)e^{-4\mathrm{\Gamma }k}\end{array}$$

and derive Eq. (9) of this article.

Notice that Eq. (9) has been derived starting from the general expression given by Eq. (41) of [7]. Indeed, it cannot be derived by simply substituting k = l in the covariance, Eq. (8), because this substitution would lead to a wrong count of the term |χ̂₀| appearing in the variance. For example, in the first sum appearing in the general expression of the variance

(27)$$\sum _{i=-\infty }^{+\infty }{\left({\left|{\widehat{\chi }}_i\right|}^2\right)}^2=\sum _{i\ne 0}{\left({\left|{\widehat{\chi }}_i\right|}^2\right)}^2+{\left({\left|{\widehat{\chi }}_0\right|}^2\right)}^2$$

the index i and it encounters χ̂₀ only for i = 0. For the covariance, instead, since the first term is

(28)$$\sum _{i=-\infty }^{+\infty }{\left|{\widehat{\chi }}_i\right|}^2{\left|{\widehat{\chi }}_{i+k-l}\right|}^2=\sum _{\begin{array}{l}i\ne 0\\ i\ne l-k\end{array}}{\left|{\widehat{\chi }}_i\right|}^2{\left|{\widehat{\chi }}_{i+k-l}\right|}^2+{\left|{\widehat{\chi }}_{l-k}\right|}^2{\left|{\widehat{\chi }}_0\right|}^2+{\left|{\widehat{\chi }}_0\right|}^2{\left|{\widehat{\chi }}_{k-l}\right|}^2$$

the index i encounters χ̂₀ twice, for i = 0 and i = l − k. These two indices are the same in the variance, but if we calculate it by simply substitution of l = k in the analytical expression of the covariance, we automatically consider a double contribution of χ̂₀ and, consequently, an incorrect value of the variance.

C. Numerical simulations (covariance)

As pointed out in the main text, Eqs. (8) and (9) that describe the covariance matrix can be applied only when the two correlogram points g₂(τ_k) and g₂(τ_l) are acquired with the same sampling time Δt. Thus, in a multi-tau correlator, only channels that belong to the same stage (see Eq. (11)) can be described by such equations. With this restriction in mind, we carried out several statistical analysis similar to the ones reported in the main text where we used N independent batches for obtaining accurate estimates of Cov_g2(τ_k, τ_l) as

(29)$${\text{Cov}}_{g_2}({\tau }_k,{\tau }_l)=\frac{1}{N}\sum _{i=1}^N{\left[g_2({\tau }_k)\right]}_i{\left[g_2({\tau }_l)\right]}_i-\left[\frac{1}{N}\sum _{i=1}^N{\left[g_2({\tau }_k)\right]}_i\right]\left[\frac{1}{N}\sum _{i=1}^N{\left[g_2({\tau }_l)\right]}_i\right]$$

that were then compared with Eqs. (8) and (9).

Figure 7 reports such a study for the same case of Sect.3.1, i.e. a monodisperse sample characterized by a single exponential decay auto-correlation function with τ_c = 10⁻⁴ s. All the simulation settings were identical to the ones reported in Sect.3.1, i.e. β = 1, 〈c.r.〉 = 10⁵ Hz, T = 1s, with the exception of N = 3 × 10⁵. Also for the correlograms we used the same settings, i.e. Δt₀ = 10⁻⁷s, p = 28, m = 4, S = 10, so that τ_max = 0.74s.

 

Fig. 7 Comparison between Eqs. (6) and (8). (a): comparison between the behaviors of Cov_g2(τ_k, τ_l) as a function of τ_k of the auto-correlation function obtained by computer simulations (open symbols) and the predictions of the original Schätzel (dotted line) and our (solid line) formulas for the case of a monodisperse sample with a single exponential decay correlation function with τ_c = 10⁻⁴ s. Different symbols refer to channels acquired with different sampling times indicated on the bottom. The covariances are evaluated at fixed lag-times corresponding to l = 16 for all the shown stages. The peaks correspond to variance (k = l). The measuring time is T = 1 s and the average count rate is 〈c.r.〉 = 10⁵ Hz. (b): absolute residuals between simulations and our formula (open symbols) and the Schätzel formula (solid symbols). The values of Γ corresponding to each stage are reported on top of the panel.

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In Fig. 7(a) we compare the behavior of Cov_g2(τ_k, τ_l) as a function of τ_k (open symbols) with the predictions of the Schätzel formula (dashed line) and our formula (solid line). Different colors refer to channels acquired with different sampling times indicated on the bottom of the figure. The covariances are evaluated at various fixed lag-times τ_l, all of them given by the l = 16-th channel of the corresponding linear correlator. The peaks correspond to variance (k = l). For the sake of clarity and because their behaviors can be extrapolated by the next neighbors, the first one and the last three stages have not been shown.

The comparison between simulations and the two theories can be appreciated in Fig. 7(b), where we report the absolute residuals between simulations and our formula (open symbols) and the Schätzel formula (solid symbols). Notice that, for the Schätzel formula, the points corresponding to the variances of the last three stages are out of scale. The values of Γ associated to each stage are reported on top of the panel. As one can notice, whereas the residuals corresponding to our formula are always small regardless of Γ, for the Schätzel formula they became increasingly larger as Γ approaches and become larger than unity. Thus, our formula appears to be much more accurate than the original Schätzel formula.

Funding

Fondazione Cariplo, Grant n. 2016-0648, Project: Romancing the stone: size controlled HYdroxyaPATItes for sustainable Agriculture (HYPATIA).

Acknowledgments

We thank Daniele Redoglio for the support given during the acquisition of the experimental data and Mattia Rocco for the loan of some of the electronic instrumentation used in this work.

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