Fragmentation phase transition in atomic clusters .4. Liquid-gas transition in finite metal clusters and in the bulk

Within the micro-canonical ensemble it is well possible to identify phase-transitions in small systems. The consequences for the understanding of phase transitions in general are discussed by studying three realistic examples. We present micro-canonical calculations of the fragmentation phase transition in Na-, K-, and Fe- clusters of N = 200 to 3000 atoms at a constant pressure of 1 atm. The transition is clearly of first order with a back-bending micro-canonical caloric curve T-P(E, V (E, P)) = {partial derivative S(E, V(E, P))/partial derivative E\(P)}(-1). From the Maxwell construction of beta(P)(E/N, P) = 1/T-P one can simultaneously determine the transition temperature T-tr, the specific latent heat q(lat), and the specific entropy-loss Delta s(surf) linked to the creation of intra-phase surfaces. T(tr)Delta s(surf)*N/(4 pi r(ws)(2)N(eff)(2/3)) = gamma gives the surface tension gamma. Here 4 pi r(ws)(2)N(eff)(2/3) = Sigma N-i * 4 pi r(ws)(2)m(i)(2/3) is the combined surface area of all fragments with a mass m(i) greater than or equal to 2 and multiplicity N-i. All these characteristic parameters are for similar to 1000 atoms similar to their experimentally known bulk values. This finding shows clearly that within micro-canonical thermodynamics phase transitions can unambiguously be determined without invoking the thermodynamic limit. However, one has carefully to distinguish observables which are defined for each phase-space point, like the values of the conserved quantities, from thermodynamic quantities like temperature, pressure, chemical potential, and also the concept of pure phases, which refer to the volume of the energy shell of the N-body phase-space and thus do not refer to a single phase-space point. At the same time we present here the first successful microscopic calculation of the surface tension in liquid sodium, potassium, and iron at a constant pressure of 1 atm.