Critical temperature for first-order phase transitions in confined systems

Abstract

We consider the Euclidean $D$-dimensional $-\lambda |\varphi {|}^4+\eta |\varphi {|}^6$ ($\lambda ,\eta >0$) model with $d$ ($d\le D$) compactified dimensions. Introducing temperature by means of the Ginzburg--Landau prescription in the mass term of the Hamiltonian, this model can be interpreted as describing a first-order phase transition for a system in a region of the $D$-dimensional space, limited by $d$ pairs of parallel planes, orthogonal to the coordinates axis $x_1,x_2,...,x_d$. The planes in each pair are separated by distances $L_1,L_2,...,L_d$. We obtain an expression for the transition temperature as a function of the size of the system, , $i=1,2,...,d$. For D=3 we particularize this formula, taking $L_1=L_2=...=L_d=L$ for the physically interesting cases $d=1$ (a film), $d=2$ (an infinitely long wire having a square cross-section), and for $d=3$ (a cube). For completeness, the corresponding formulas for second-order transitions are also presented. Comparison with experimental data for superconducting films and wires shows qualitative agreement with our theoretical expressions

REVTEX, 11 pages, 3 figures; to appear in Eur. Phys. Journal B