Fragmentation phase transition in atomic clusters IV

Abstract

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 backbending micro-canonical caloric curve T _P (E, V (E,P)) = {∂S(E, V (E,P))/∂E _P}⁻¹. From the Maxwell construction of β_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 Δs _surf linked to the creation of intra-phase surfaces. T _trΔs _surf*N/(4πr ²_ws N ^2/3_eff ) = γ gives the surface tension γ. Here 4πr ²_ws N ^2/3_eff = ΣN _i*4πr ²_ws m ^2/3_i is the combined surface area of all fragments with a mass m _i ≥ 2 and multiplicity N _i. All these characteristic parameters are for ∼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.