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We calculate the effective macroscopic dielectric constant $\varepsilon_a$ ofa periodic array of spherical nanocrystals (NCs) with dielectric constant$\varepsilon$ immersed in the medium with dielectric constant $\varepsilon_m\ll \varepsilon$. For an array of NCs with the diameter $d$ and the distance$D$ between their centers, which are separated by the small distance $s=D-d \lld$ or touch each other by small facets with radius $\rho\ll d$ what isequivalent to $s < 0$, $|s| \ll d$ we derive two analytical asymptotics of thefunction $\varepsilon_a(s)$ in the limit $\varepsilon/\varepsilon_m \gg 1$.Using the scaling hypothesis we interpolate between them near $s=0$ to obtainnew approximated function $\varepsilon_a(s)$ for $\varepsilon/\varepsilon_m \gg1$. It agrees with existing numerical calculations for$\varepsilon/\varepsilon_m =30$, while the standard mean-field Maxwell-Garnettand Bruggeman approximations fail to describe percolation-like behavior of$\varepsilon_a(s)$ near $s = 0$. We also show that in this case the chargingenergy $E_c$ of a single NC in an array of touching NCs has a non-trivialrelationship to $\varepsilon_a $, namely $E_c = \alpha e^2/\varepsilon_a d$,where $\alpha$ varies from 1.59 to 1.95 depending on the studiedthree-dimensional lattices. Our approximation for $\varepsilon_a(s)$ can beused instead of mean field Maxwell-Garnett and Bruggeman approximations todescribe percolation like transitions near $s=0$ for other materialcharacteristics of NC arrays, such as conductivity.

Dielectric Constant and Charging Energy in Array of Touching Nanocrystals