Negative heat capacity in the critical region of nuclear fragmentation: an experimental evidence of the liquid-gas phase transition

We address the stability of multicharged finite systems driven by Coulomb forces beyond the Rayleigh instability limit. Our exploration of the nuclear dynamics of heavily charged Morse clusters enabled us to vary the range of the pair potential and of the fissibility parameter, which results in distinct fragmentation patterns and in the angular distributions of the fragments. The Rayleigh instability limit separates between nearly binary (or tertiary) spatially unisotropic fission and spatially isotropic Coulomb explosion into a large number of small, ionic fragments. Implications are addressed for a broad spectrum of dynamics in chemical physics, radiation physics of ultracold gases, and biophysics, involving the fission of clusters and droplets, the realization of Coulomb explosion of molecular clusters, the isotropic expansion of optical molasses, and the Coulomb instability of "isolated" proteins.

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Beyond the Rayleigh instability limit for multicharged finite systems: from fission to Coulomb explosion.

Multifragmentation MF results from 1A GeV Au on C have been compared with the Copenhagen statistical multifragmentation model ~SMM!. The complete charge, mass, and momentum reconstruction of the Au projectile was used to identify high momentum ejectiles leaving an excited remnant of mass A, charge Z, and excitation energy E* which subsequently multifragments. Measurement of the magnitude and multiplicity ~energy! dependence of the initial free volume and the breakup volume determines the variable volume parametrization of SMM. Very good agreement is obtained using SMM with the standard values of the SMM parameters. A large number of observables, including the fragment charge yield distributions, fragment multiplicity distributions, caloric curve, critical exponents, and the critical scaling function are explored in this comparison. The two stage structure of SMM is used to determine the effect of cooling of the primary hot fragments. Average fragment yields with Z>3 are essentially unaffected when the excitation energy is 170 the effective latent heat approaches zero. Thus for heavier systems this transition can be identified as a continuous thermal phase transition where a large nucleus breaks up into a number of smaller nuclei with only a minimal release of constituent nucleons. Z<2 particles are primarily emitted in the initial collision and after MF in the fragment deexcitation process.

Comparison of 1A GeV 197 Au+C data with thermodynamics: The nature of the phase transition in nuclear multifragmentation

We address unifying features of fragmentation channels driven by long-range Coulomb or pseudo-Coulomb forces in clusters, nuclei, droplets, and optical molasses. We studied the energetics, fragmentation patterns, and dynamics of multicharged (A+)n (n=55, 135, 321) clusters. In Morse clusters the variation of the range of the pair-potential induced changes in the cluster surface energy and in the fissibility parameter X=E(Coulomb)∕2E(surface). X was varied in the range of X=1–8 for short-range interactions and of X=0.1–1.0 for long-range interactions. Metastable cluster configurations were prepared by vertical ionization of the neutral clusters and by subsequent structural equilibration. The energetics of these metastable ionic clusters was described in terms of the liquid drop model, with the coefficients of the volume and surface energies depending linearly on the Morse band dissociation energy. Molecular-dynamics simulations established two distinct fragmentation patterns of multicharged clusters that i...

http://www.tau.ac.il/~jortner/Publications/Pub601-/691.pdf

Fragmentation channels of large multicharged clusters.

A geometric foundation thermo-statistics is presented with the only axiomatic assumption of Boltzmann's principleS(E, N, V) = klnW. This relates the entropy to the geometric area eS(E, N, V)/k of the manifold of constant energy in the (finite-N)-body phase space. From the principle, all thermodynamics and especially all phenomena of phase transitions and critical phenomena can unambiguously be identified for even small systems. The topology of the curvature matrix C(E, N) of S(E, N) determines regions of pure phases, regions of phase separation, and (multi-)critical points and lines. Phase transitions are linked to convex (upwards bending) intruders of S(E, N), where the canonical ensemble
defined by the Laplace transform to the intensive variables becomes multi-modal, non-local, (it mixes widely different conserved quantities). Here the one-to-one mapping of the Legendre transform gets lost. Within Boltzmann's principle, statistical mechanics becomes a geometric theory addressing the whole ensemble or the manifold of all points in phase space which are consistent with the few macroscopic conserved control parameters. This interpretation leads to a straight derivation of irreversibility and the second law of thermodynamics out of the time-reversible, microscopic, mechanical dynamics. It is the whole ensemble that spreads irreversibly over the accessible phase space not the single N-body trajectory. This is all possible without invoking the thermodynamic limit, extensivity, or concavity of S(E, N, V). Without the thermodynamic limit or at phase-transitions, the systems are usually not self-averaging, i.e.
do not have a single peaked distribution in phase space. The main obstacle against the second law, the conservation of the phase-space volume due to Liouville is overcome by realizing that a macroscopic theory such as thermodynamics cannot distinguish a fractal distribution in phase space from its closure.

Geometric foundation of thermo-statistics, phase transitions, second law of thermodynamics, but without thermodynamic limit

The volume W of the accessible N-body phase space and its dependence on the total energy is directly calculated. The famous Boltzmann relation S = k * ln(W) defines microcanonical thermodynamics (MT). We study how phase transitions appear in MT. Here we first develop the thermodynamics of microcanonical phase transitions of first and second order in systems which are thermodynamically stable in the sense of van Hove. We show how both kinds of phase transitions can unambiguously be identified in relatively small isolated systems of ∼ 100 atoms by the shape of the microcanonical caloric equation of state 〈 T(E/N) ⌨ and not so well by the coexistence of two spatially clearly separated phases. I.e. within microcanonical thermodynamics one does not need to go to the thermodynamic limit in order to identify phase transitions. In contrast to ordinary (canonical) thermodynamics of the bulk microcanonical thermodynamics (MT) gives an insight into the coexistence region. Here the form of the specific heat c(E/N) connects transitions of first and second order in a natural way. The essential three parameters which identify the transition to be of first order, the transition temperature T tr, the latent heat q lat, and the interphase surface entropy Δs ssurf can very well be determined in relatively small systems like clusters by MT. It turns out to be essential whether the cluster is studied canonically at constant temperature or microcanonically at constant energy. Especially the study of phase separations like solid and liquid or, as studied here, liquid and gas is very natural in the microcanonical ensemble, whereas phase separations become exponentially suppressed within the canonical description. The phase transition towards fragmentation is introduced. The general features of MT as applied to the fragmentation of atomic clusters are discussed. The similarities and differences to the boiling of macrosystems are pointed out.

Fragmentation phase transition in atomic clusters I

We discuss the role and the treatment of polarization effects in many-body systems of charged conducting clusters and apply this to the statistical fragmentation of Naclusters. We see a first order microcanonical phase transition in the fragmentation of Na 70 Z+ for Z = 0 to 8. We can distinguish two fragmentation phases, namely evaporation of large particles from a large residue and a complete decay into small fragments only. Charging the cluster shifts the transition to lower excitation energies and forces the transition to disappear for charges higher than Z = 8. At very high charges the fragmentation phase transition no longer occurs because the cluster Coulomb-explodes into small fragments even at excitation energy e* = 0.

Fragmentation phase transitions in atomic clusters III

Here we first develop the thermodynamics of microcanonical phase transitions of first and second order in systems which are thermodynamically stable in the sense of van Hove. We show how both kinds of phase transitions can unambiguously be identified in relatively small isolated systems of $\sim 100$ atoms by the shape of the microcanonical caloric equation of state $ $ I.e. within microcanonical thermodynamics one does not need to go to the thermodynamic limit in order to identify phase transitions. In contrast to ordinary (canonical) thermodynamics of the bulk microcanonical thermodynamics (MT) gives an insight into the coexistence region. The essential three parameters which identify the transition to be of first order, the transition temperature $T_{tr}$, the latent heat $q_{lat}$, and the interphase surface entropy $\Delta s_{surf}$ can very well be determined in relatively small systems like clusters by MT. The phase transition towards fragmentation is introduced. The general features of MT as applied to the fragmentation of atomic clusters are discussed. The similarities and differences to the boiling of macrosystems are pointed out.

Fragmentation Phase Transition in Atomic Clusters I --- Microcanonical Thermodynamics ---

We discuss the role and the treatment of polarization effects in many-body systems of charged conducting clusters and apply this to the statistical fragmentation of Na-clusters. We see a first order microcanonical phase transition in the fragmentation of $Na^{Z+}_{70}$ for Z=0 to 8. We can distinguish two fragmentation phases, namely evaporation of large particles from a large residue and a complete decay into small fragments only. Charging the cluster shifts the transition to lower excitation energies and forces the transition to disappear for charges higher than Z=8. At very high charges the fragmentation phase transition no longer occurs because the cluster Coulomb-explodes into small fragments even at excitation energy $\epsilon^* = 0$.

Fragmentation Phase Transition in Atomic Clusters II - Coulomb Explosion of Metal Clusters -

We present a statistical fragmentation study of doubly charged alkali (Li, Na, K) and antimony clusters The evaporation of one charged trimer is the most dominant decay channel (asymmetric fission) at low excitation energies For small sodium clusters this was quite early found in molecular dynamical calculations by Landman et al For doubly charged lithium clusters, we predict Li$_{9}^{+}$ to be the preferential dissociation channel As already seen experimentally a more symmetric fission is found for doubly charged antimony clusters This different behavior compared to the alkali metal clusters is in our model essentially due to a larger fissility of antimony This is checked by repeating the calculations for Na$_{52}^{++}$ with a bulk fissility parameter set artificially equal to the value of Sb

O. Schapiro

Papers

Fragmentation phase transition in atomic clusters I

Fragmentation phase transitions in atomic clusters III

Fragmentation Phase Transition in Atomic Clusters I --- Microcanonical Thermodynamics ---

Fragmentation Phase Transition in Atomic Clusters II - Coulomb Explosion of Metal Clusters -

Fragmentation Phase Transition in Atomic Clusters II -----Symmetry of Fission of Metal Clusters----