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Cluster in Helium-Tropfen Fortgeschrittenen-Praktikum 3 Cluster in Helium-Tropfen Fortgeschrittenen-Praktikum 3 Introduction into Cluster physics [1] Clusters have been experimentally produced the first time in the middle of last century. Since 1980 the interest in cluster physics have grown rapidly because clusters are of special interest for catalysis, photography, physics and chemistry of aerosols and the growing of such complexes in space, and the structure of amorphous substances. As a general definition the expression cluster means an ensemble of particles (atoms or molecules) which are linked by certain kind of bond. Homogeneous clusters consist only of the same kind of atoms or molecules whereas mixed clusters consisting of different atoms are denoted as heterogeneous clusters. At this point the question arises which number of atoms or molecules is necessary to speak of clusters. Märk [2] defines as the minimum size 2 for a cluster, i.e. the dimer. Type of cluster Example Binding forces Mean binding energy Van-der Waals (rare gas)n, (H2)n Van der Waals forces 0.01-0.3 eV clusters (induced dipole interaction) Valence clusters Cn, S8,As4 Chemical forces (covalent 1-4 eV bond) Ionic clusters (NaCl)n Ionic bond (Coloumb force) 2-4 eV Hydrogen-bonded (H2O)n, (HF)n, Dipole-dipole 0.15-0.5 eV clusters (HNO3)n interaction Molecular clusters (I2)n, (organic molecules)n Like Van der Waals 0.3-1 eV with additionial covalent contribution Metallic clusters (alkali metal)n, Aln, Cun, Dispersion covalent metallic 0.5-3 eV Fen, Ptn Table 1. Classification of clusters by their binding properties and mean binding energies. Upper limit separating from the bulk state is assumed to be 106 particles. Clusters are generally characterized by their kind of bond between the cluster particles. Table 1 shows an overview of different kind of clusters. Helium clusters are bound via the Van der Waals Bond. [1] Haberland, Clusters of Atoms and Molecules [2] T. D. Märk, Lecture in Clusterphysics The Van der Waals bond The interaction between two polarizable molecules The interaction potential which is ascribed to the Van-der-Waals bond can be induced due to dipole interaction by atoms which are originally not polarized. If the distance between 2 atoms is small enough a fluctuations of the electron shell in one of the atoms is induced which leads to the formation of a dipole in this atom. This polarized atom vice versa leads to the induction of a dipole in the other atom. In sum a attractive dipole field between 2 atoms is formed, although it is a weak bond. In general the Van der Waals bond can be described by a Lennard-Jones potential (see Fig.1): Where ε is the depth of the potential well at the internuclear distance R=R0. The second part of the equation describes the induced dipole moment (interaction potential ~ R-6). The first term (~R-12) describes the steep repulsive part of the potential due to forbidden overlapping of occupied electron r − R shells. Often this term is replaced by a e 0 dependence like in a Morse potential which describe the repulsive part more accurate. Fig.1: Lennard Jones Potential (solid line) and the corresponding contributions (dotted lines) Three common ways exist to produce clusters: a) Gas aggregation sources: This is the oldest and easiest method for cluster production. Atoms or molecules are evaporated into a flow of rare gas atoms. The evaporated atoms are cooled in collision with the rare gas. When the atoms or molecules loose enough energy the cluster production is starting. b) Laser-ablation sources (surface sources, sputtering): Photon or heavy particle impact on a surface leads to the desorption of atoms or molecules. The released atoms or molecules are partially ionized and form plasma with a temperature of 104 K. Similar like in the gas aggregation sources the plasma is cooled by present rare gas and cluster formation is achieved c) Supersonic cluster sources: A gas under high pressure is expanded adiabatically through a small nozzle. It will be discussed the way to produce clusters and the theoretical background more detailed in following because the He cluster source is working on this principle. Supersonic expansion The principal scheme of an adiabatic expansion is shown in Fig. 2. A gas is introduced into a stagnation chamber under high pressure with low temperature. The parameters (pressure, temperature) are strongly depended on the used gas i.e. for lower binding energies higher pressure and lower temperature are needed (see below). The stagnation chamber has an orifice where the gas expands into the vacuum. The originally random velocity distributions of the particles in the stagnation chamber (which reveal a Maxwell Boltzmann distribution) are transformed into a very narrow energy distribution during the expansion. In general different nozzle forms are available (sonic, cylindrical, conical or Laval) it turned out that the nozzle with Laval geometry has the best nozzle shape for cluster production. In practical applications sonic geometry is most common because it is the easiest to make. Fig. 2: Schematic view of a supersonic expansion From the theoretical (thermo dynamical) point of view the expansion can be described as an adiabatic process where the heat stays constant. Therefore due to the conservation of energy the enthalpy in the source is: mv² H0 =H + (2) 2 The last term on the right side is the kinetic energy whereas the first term is the rest enthalpy H = cpT. Formula (2) can be written in terms of specific heat capacities at constant pressure (cp) and constant volume (cv) mv² cpT0 = cpT + (3) 2 Using some rearrangements of formula 3 one can obtain the following formula for the temperature of the cluster beam: (4) c v Where γ := p and M is the so called Mach-number M := which describes the ratio of the velocity of cv vs γkT the molecules and the local speed of sound v := . M increases during the expansion, because the m local speed of sound decreases. In Fig 3 the velocity distribution is shown for different Mach numbers. At Fig.3: Velocity distributions f(v) for different Mach numbers as a function of the normalized velocity v where α=(2kT/m)-0,5 M=0 (temperature T0) the broad distribution has the shape of an effusive beam whereas for higher Mach numbers the distribution became narrower. During the expansion not only the energy is conserved, also the entropy stays constant. Therefore the expansion is an adiabatic process where the Poison equation is valid: γ γ p ⎛ ρ ⎞ ⎛ T ⎞γ −1 = ⎜ ⎟ = ⎜ ⎟ (5) p0 ⎝ ρ0 ⎠ ⎝ T0 ⎠ This equation combines the stagnation values of pressure p0, density ρ0) and temperature with the corresponding local thermodynamic variables. From this equation and (4) one can obtain: 1 ρ ⎛ M 2 (γ −1) ⎞γ −1 ⎜1 ⎟ (6) = ⎜ + ⎟ ρ0 ⎝ 2 ⎠ γ p ⎛ M 2 (γ − 1) ⎞ γ −1 = ⎜1 + ⎟ (7) ⎜ ⎟ p0 ⎝ 2 ⎠ If the Mach number is known the local value of T, p, and ρ can be calculated by the equations (5-7) depending on the distance from the nozzle. During the expansion the value of pressure, temperature and density decrease very strongly. The Mach disk location xm can be calculated as a function of the nozzle diameter and pressure. x p m = 0.67 0 (8) d pb The Mach disk is depended from the intensity of the interaction between the flow and the background pressure pb. As can be observed from the equation for sufficiently low background pressure the Mach disk disappears. The upper limit for this point is reached when the free path of the background gas is comparable with the diameter of the Mach disk. At this point the Mach number is frozen by to a certain constant value. The region where the interaction of the flow with the background gas becomes negligible is called zone of silence (see Fig.4). The core of the expansion is isentropic in this region. Nevertheless for the real expansion one has to consider additional features of the nonideal supersonic jet. This can be seen in Fig.4. Due to an overexpansion very thin nonisentropic regions like the barrel shock at the sides and the Mach disk shock in forward direction are formed. In a descriptive explanation this can be explained that the supersonic flow needs information about the boundary conditions along the expansion. Fig. 4: Schematic diagram of a free supersonic jet structure This information is transported with the speed of sound but the particles in the expansion move faster. Thus leads to an overexpansion of the gas, but in order to get rid of these unknown boundary conditions the Barrels shock and the Mach disc shock with an M number below 1 are provided to change the flow of the expansion if the boundary condition is not satisfied. Nevertheless one has to consider following problems due to the presence of the background gas: a skimmer is placed after a certain distance to the nozzle. The skimmer prevents to reach cluster which are outside the beam line in the direction to the ion source. It turned out that the distance between nozzle and skimmer has a strong influence on cluster formation. If the distance is too large the beam is scattered by the background gas which leads to a decrease of the intensity of the beam. For too low distances an additional expansion starts in the skimmer causing turbulences which changes the properties of the initial cluster beam (velocity, mean cluster size, direction). Condensation-cluster formation During the expansion the temperature of the beam decreases successively. After a certain distance after the nozzle the atoms or molecule are nearly without interaction with the surrounding, i.e.
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