in the integrand the integral over $x$ converges fast. However the integral over $\zeta$ running from 0 to $\infty$ is not convenient for numerical evaluation. This problem can be solved by substituting $\zeta=xt\omega_c$. Now $t$ will run from 0 to 1, and in terms of $x$ and $t$ numerical calculation of the integral in (\[Lif\]) becomes convenient: $$\label{E_conv}
    E_{pp}^{flat}(d)=\frac{\hbar c}{32\pi^2d^3}\sum\limits_{\mu}
    \int\limits_0^1
    dt\int\limits_0^{\infty}dxx^2\ln\left(1-R_{\mu}e^{-x}\right).$$ The roughness correction can be calculated as follows. The force between a sphere and a plate $F_{sp}$ is related with the interaction energy per unit area between two plates $E_{pp}$ by the relation: $$\label{F-E}
    F_{sp}(d)=2\pi R E_{pp}(d),$$ where $d$ is the minimal distance between bodies and $R$ is the sphere radius. Relation (\[F-E\]) holds true in the limit $R\gg d$ that is the case for our experiment. Roughness gives contribution to the energy $E_{pp}$, which can be presented as $$\label{Efull}
    E_{pp}=E_{pp}^{flat}+\delta E_{pp}^{rough},$$ where the first term corresponds to the interaction energy between plates.

The second term in Eq. (\[Efull\]) is responsible for the roughness correction. It can be presented in the form [@Mia05] $$\label{corr}
   \delta E_{pp}^{rough}=\int\frac{d^2k}{(2\pi)^2}G(k,d)\sigma(k).$$ Here $\sigma(k)$ is the roughness spectrum and $G(k,d)$ is the response function derived in [@Mia05] both are functions of the wave number 