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 ph ase-transitions in small systems. The consequences for the understandi ng of phase transitions in general are discussed by studying three rea listic examples. We present micro-canonical calculations of the fragme ntation phase transition in Na-, K-, and Fe- clusters of N = 200 to 30 00 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 ca n simultaneously determine the transition temperature T-tr, the specif ic latent heat q(lat), and the specific entropy-loss Delta s(surf) lin ked to the creation of intra-phase surfaces. T(tr)Delta s(surf)N/(4 p i 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 co mbined surface area of all fragments with a mass m(i) greater than or equal to 2 and multiplicity N-i. All these characteristic parameters a re for similar to 1000 atoms similar to their experimentally known bul k values. This finding shows clearly that within micro-canonical therm odynamics phase transitions can unambiguously be determined without in voking the thermodynamic limit. However, one has carefully to distingu ish observables which are defined for each phase-space point, like the values of the conserved quantities, from thermodynamic quantities lik e temperature, pressure, chemical potential, and also the concept of p ure phases, which refer to the volume of the energy shell of the N-bod y phase-space and thus do not refer to a single phase-space point. At the same time we present here the first successful microscopic calcula tion of the surface tension in liquid sodium, potassium, and iron at a constant pressure of 1 atm.