Fragmentation phase transition in atomic clusters I
TL;DR: In this article, the authors studied the thermodynamics of microcanonical phase transitions of first and second order in systems which are thermodynamically stable in the sense of van Hove and showed how both kinds of phase transitions can unambiguously be identified in relatively small isolated systems of ∼ 100 atoms.
Abstract: 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.
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TL;DR: In this paper, an experimental indication of negative heat capacity in excited nuclear systems is inferred from the event by event study of energy fluctuations in Au quasi-projectile sources formed in Au+Au collisions at 35 A.MeV.
208 citations
Purdue University1, University of Catania2, Lawrence Berkeley National Laboratory3, University of California, Davis4, Texas A&M University5, Sungkyunkwan University6, Brookhaven National Laboratory7, Kent State University8, Saint Mary's College of California9, Heidelberg University10, GSI Helmholtz Centre for Heavy Ion Research11, Ohio State University12, Uppsala University13, University of California, Los Angeles14, Massachusetts Institute of Technology15, United States Naval Research Laboratory16
TL;DR: In this paper, the Copenhagen statistical multifragmentation model (SMM) was used to identify high momentum ejectiles leaving an excited remnant of mass A, charge Z, and excitation energy E* which subsequently multifragments.
Abstract: 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.
73 citations
TL;DR: In this article, a geometric foundation for thermodynamics is presented with the only axiomatic assumption of Boltzmann's principle, which 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.
Abstract: 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.
35 citations
TL;DR: The microcanonical ensemble as mentioned in this paper describes the equilibrium statistics of extensive as well of non-extensive Hamiltonian systems, and it can address nuclei and astrophysical objects as well.
Abstract: Equilibrium statistics of Hamiltonian systems is correctly described by the microcanonical ensemble. Classically this is the manifold of all points in the N-body phase space with the given total energy. Due to Boltzmann's principle, eS=tr(δ(E-H)), its geometrical size is related to the entropy S(E,N,...). This definition does not invoke any information theory, no thermodynamic limit, no extensivity, and no homogeneity assumption, as are needed in conventional (canonical) thermo-statistics. Therefore, it describes the equilibrium statistics of extensive as well of non-extensive systems. Due to this fact it is the fundamental definition of any classical equilibrium statistics. It can address nuclei and astrophysical objects as well. All kind of phase transitions can be distinguished sharply and uniquely for even small systems. It is further shown that the second law is a natural consequence of the statistical nature of thermodynamics which describes all systems with the same -- redundant -- set of few control parameters simultaneously. It has nothing to do with the thermodynamic limit. It even works in systems which are by far than any thermodynamic "limit".
30 citations
TL;DR: In this article, the authors 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.
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 βP (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
ws
2
N
eff
2/3
) = γ gives the surface tension γ. Here 4πr
ws
2
N
eff
2/3
= ΣN
i*4πr
ws
2
m
i
2/3
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.
28 citations
References
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Book•
01 Jan 1973TL;DR: CRC handbook of chemistry and physics, CRC Handbook of Chemistry and Physics, CRC handbook as discussed by the authors, CRC Handbook for Chemistry and Physiology, CRC Handbook for Physics,
Abstract: CRC handbook of chemistry and physics , CRC handbook of chemistry and physics , کتابخانه مرکزی دانشگاه علوم پزشکی تهران
52,268 citations
TL;DR: In the data for the 63 elements, trends that occur simultaneously in both the columns and the rows of the periodic table are shown to be useful in predicting correct values and also for identifying questionable data.
Abstract: A new compilation, based on a literature search for the period 1969–1976, is made of experimental data on the work function. For these 44 elements, preferred values are selected on the basis of valid experimental conditions. Older values, which are widely accepted, are given for 19 other elements on which there is no recent literature, and are so identified. In the data for the 63 elements, trends that occur simultaneously in both the columns and the rows of the periodic table are shown to be useful in predicting correct values and also for identifying questionable data. Several illustrative examples are given, including verifications of predictions published in 1950.
3,569 citations
TL;DR: In this article, the authors survey the hierarchy of theoretical approximations leading to the jellium model, including various extensions, including local density approximation to exchange and correlation effects, which greatly simplifies self-consistent calculations of the electronic structure.
Abstract: The jellium model of simple metal clusters has enjoyed remarkable empirical success, leading to many theoretical questions. In this review, we first survey the hierarchy of theoretical approximations leading to the model. We then describe the jellium model in detail, including various extensions. One important and useful approximation is the local-density approximation to exchange and correlation effects, which greatly simplifies self-consistent calculations of the electronic structure. Another valuable tool is the semiclassical approximation to the single-particle density matrix, which gives a theoretical framework to connect the properties of large clusters with the bulk and macroscopic surface properties. The physical properties discussed in this review are the ground-state binding energies, the ionization potentials, and the dipole polarizabilities. We also treat the collective electronic excitations from the point of view of the cluster response, including some useful sum rules.
1,357 citations
Book•
01 Dec 1963
1,237 citations