Relationship between blocking temperature and strength of interparticle interaction in magnetic nanoparticle systems

In magnetic nanoparticle systems, the variation of the blocking temperature $T_B$ with the measuring frequency $f_m$ is often used to determine the strength of the interparticle interactions (IPI) through a parameter $\Phi$ or the Vogel-Fulcher temperature $T_o$. Presence of IPI is inferred if $T_o$$>$0 and $\Phi=$$\Delta$$T_B$$/$$[T_B\Delta$$log_{10}$$f_m]<$0.13 where $\Delta$ signifies changes in $T_B$ and $f_m$. Here it is shown that these two parameters are related by the Eq. $\Phi=$$\Phi_o$$\approx$ 0.11 to 0.15 is a constant of the system depending on the magnitudes of measuring frequency and $\Phi=$$\Phi_o$$[1-T_o/T_B]$ where the attempt frequency $f_o$ of the N\'eel relaxation. Experimental verification of this relationship is also presented using data on a variety of nanoparticle systems.