Dependence of structure factor and correlation energy on the width of electron wires

The structure factor and correlation energy of a quantum wire of thickness $b\ll a_B$ are studied in random phase approximation and for the less investigated region $r_s<1$. Using the single-loop approximation, analytical expressions of the structure factor have been obtained. The exact expressions for the exchange energy are also derived for a cylindrical and harmonic wire. The correlation energy ${ϵ}_c$ is found to be represented by $\epsilon_c (b,r_s)= \frac{\alpha(r_s)}{b} + \beta(r_s); ln(b) + \eta(r_s)$,forsmall$b$ and high densities. For a pragmatic width of the wire, the correlation energy is in agreement with the quantum Monte Carlo simulation data.