## 3 Results and Discussion
For the lowest concentration, sample S5 (c = 4 \(\times\) 10\({}^{-4}\)%), we managed to successfully fit the observed \(\tau\)(T) behavior using the Neel-Brown equation \(\tau\)(T) = \(\tau_{0}\) exp\(\left(\frac{E_{\rm B}}{k_{\rm B}T}\right)\) with parameters \(\tau_{0}\) = 5.6 \(\times\) 10\({}^{-10}\) s and E\({}_{\rm B}\)/k\({}_{\rm B}\) = 366 K. This is significant because it indicates that the Fe\({}_{3}\)O\({}_{4}\) nanoparticle ensemble in S5 is diluted enough, so each particle relaxes individually across an energy barrier, E\({}_{\rm B}\), that only depends on the bulk and surface magnetic anisotropy. The key value that supports this conclusion is the vertical axis intercept of the ln\(\tau\) vs. 1/T dependence, ln\(\tau_{0}\), which in this case yields a \(\tau_{0}\) between 10\({}^{-9}\) and 10\({}^{-11}\) s. Values of the inverse attempt frequency (\(\tau_{0}\) = 1/2\(\pi\)f\({}_{0}\)) within this range are a known signature of non-interacting nanoparticle ensembles where each superspin relaxes independently [20,27]. In addition, we note that E\({}_{\rm B}\)/k\({}_{\rm B}\) has the same value in all five samples because they were prepared by progressively diluting the same nanopowder. Figure 5 shows the temperature dependence of the relaxation time measured on samples S1 (upright triangles), S2 (diamonds), S3 (inverted triangles), S4 (circles), and S5 (squares), using \(\chi^{\prime\prime}\) vs. T\(\parallel_{t}\) data, such as the ones in Figure 4a. The main feature here is  
Figure 4: (**a**) Temperature dependence of the out-of-phase susceptibility, \(\chi^{\prime\prime}\) vs. T, measured on the S3 (c = 1.7 \(\times\) 10\({}^{-3}\)% \(\
u\)/\(\
u\)) sample at different frequencies: 100 Hz (filled squares), 300 Hz (open diamonds), 1000 Hz (filled diamonds), 3000 Hz (empty circles), and 10,000 Hz (filled circles). (**b**) Temperature dependence of the relaxation time, \(\tau\), (solid symbols) obtained from ac-susceptibility data and best fit of the DBS model equation \(\tau\)(T) = \(\tau_{\rm r}\) exp \(\left(\frac{E_{\rm B}+E_{\rm A}}{k_{\rm B}T}\right)\) (solid line). that the slope of the ln\(\tau\) vs. 1/T plots increases with the increase in c, indicating that the superspin needs to overcome a higher energy barrier in denser samples. The solid lines are best fits of Equation (2), which allow the energy barrier to superspin reversal \(\rm E_{B}+E_{ad}\) to be determined. As discussed above, we found \(\rm E_{ad}=0\) K and \(\rm E_{B}/\rm k_{B}=366\) K for the S5 sample. For the other four samples, the best fits yield additional energy barriers that increase with the increase in concentration, from \(\rm E_{ad}/\rm k_{B}=138\) K for S4 (c = 8.3 \(\times\) 10\({}^{-4}\)%) to \(\rm E_{ad}/\rm k_{B}=745\) K for S1 (c = 4 \(\times\) 10\({}^{-2}\)%). These results are summarized in the inset table and shown by the solid symbols in Figure 6. The solid line here is a guide to the eye showing that the additional barrier to superspin reversal, \(\rm E_{ad}/\rm k_{B}\), increases linearly with the common logarithm of the concentration \(\rm log_{10}\)C. We note that upon the increase in the volume concentration, the additional energy barrier reaches values comparable to its \(\rm E_{B}/\rm k_{B}\) counterpart at c\(\sim\)5 \(\times\) 10\({}^{-3}\)%. For sample S1, we found \(\rm E_{ad}/\rm k_{B}>2\rm E_{B}/\rm k_{B}\), which clearly demonstrates the strong effect of the interparticle interactions even at low nanoparticle volume concentrations, below 10\({}^{-1}\)%. At the microscopic level, the observed dependence of the barrier to magnetization reversal on concentration indicates that, up to c = 4 \(\times\) 10\({}^{-4}\)%, the nanoparticles' magnetic moments (the superspins) only need to overcome \(\rm E_{B}\) (i.e., the barrier resulting from the bulk and surface magneto-crystalline anisotropy) in order to flip along an easy magnetization axis via thermal activation. This is the same as the case of an isolated magnetic nanoparticle, as described by the Neel-Brown model.