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 clusters

(rare gas)n, (H2)n Van der Waals forces (induced dipole interaction)

0.01-0.3 eV

Valence clusters Cn, S8,As4 Chemical forces (covalent bond)

1-4 eV

Ionic clusters (NaCl)n Ionic bond (Coloumb force) 2-4 eV

Hydrogen-bonded clusters

(H2O)n, (HF)n, (HNO3)n

Dipole-dipole interaction

0.15-0.5 eV

Molecular clusters (I2)n, (organic molecules)n Like Van der Waals with additionial

covalent contribution

0.3-1 eV

Metallic clusters (alkali metal)n, Aln, Cun, Fen, Ptn

Dispersion covalent metallic 0.5-3 eV

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

shells. Often this term is replaced by a 0Rr

e−

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:

H0 =H + 2

²mv (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)

cpT0 = cpT + 2

²mv (3)

Using some rearrangements of formula 3 one can obtain the following formula for the temperature of the cluster beam:

(4)

Where γ := v

p

cc

and M is the so called Mach-number M := sv

v which describes the ratio of the velocity of

the molecules and the local speed of sound v := mkTγ . M increases during the expansion, because the

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:

1

000

−⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛=

γγγ

ρρ

TT

pp (5)

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:

11

2

0 2)1(1

−

⎟⎟⎠

⎞⎜⎜⎝

⎛ −+=

γγρρ M (6)

12

0 2)1(1

−

⎟⎟⎠

⎞⎜⎜⎝

⎛ −+=

γγ

γMpp (7)

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.

b

mpp

dx 067.0= (8)

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. the relative energy to the other particles which are coexpanded can be in the meV region. At this point the temperature of the beam is below the binding energy and cluster formation starts with the production of a dimer:

A + A + A → A2+ A (9)

The formation of a dimer occurs in a three body collision due to momentum and energy conservation. The formation of a dimer is the starting point for the formation of larger clusters in a row of collisions with other atoms. The cluster temperature increase clearly during this growing process but this will compensated with evaporation of monomers and collisions with other particles. Cluster growing can be explained with the classical nucleation theory in a simple way which can be described as the gas-fluid phase transition (see Fig. 5 which shows the phase diagram for the gas-fluid transition). The gas phase region is separated from the fluid region by the vapour pressure line.

Fig.5: Phase diagram showing the vapour pressure curve separating the gas phase and fluid phase

In the phase diagram shown in Fig. 5 the supersonic expansion follows the adiabatic line which cross in point A the vapour pressure curve (boundary between gas phase and liquid phase). Cluster formation starts in the supersaturated region (point B), where the vapour pressure is high enough to allow the condensation. At this point the expansion leaves the adiabatic line due to the release of condensation heat and goes to point C, where the equilibrium state between gas and liquid exists.

Experimentally the mean size of a cluster beam distribution is dependent of three parameters. The following scaling law for H2 clusters was reported:

4.20

5.10

TDpN ∝ (10)

Where p0 is the pressure and T0 the temperature in the stagnation chamber. D is the reduced diameter of

the nozzle D = θtan

d (with d the diameter of the nozzle and θ the cone half angle of the nozzle).

Therefore the largest influence to increase the mean size of the cluster beam is the lowering of the temperature. The other two parameters are strongly dependent of the vacuum system. A change of the nozzle diameter is limited by the pump system which is used in the cluster chamber. If the pressure in the cluster chamber is too high, the clusters are destroyed by collisions with the background gas.

The rare gas helium

Helium is the lightest rare gas (mass: 4 amu) and has 2 stable isotopes, 3He (0.000137%) and 4He (99.999863%). Helium has several special properties which can be explained with the phase diagram (see Fig. 6). This figure reveals that for very low temperatures in the boundary region T→ 0 helium does not become solid at standard pressure. The minimum pressure to achieve a transition into the solid state is 25bar. The explanation for this behaviour is that due to the weak Van der Waals bond and the low masses of He atoms. Because of this properties the kinetic and potential energy has the same order of magnitude independent of the temperature. In addition the liquid state can be divided into two region which are separated by the λ-line at the temperature of Tλ=2.173 K. Above this temperature He is a standard liquid whereas below this temperature helium reveals a supraliquid state. This suprafluid liquid has a very low viscosity which allows a frictionless stream of helium through capillary and thinnest channels. In addition it reveals indefinitely high heat conductivity.

Depending on the starting point of an expansion, i.e. the initial temperature and pressure, different formation mechanism of cluster occurs and different cluster size distributions can be achieved see Fig.6.

Fig.6. Pressure-temperature phase diagram for 4He

In this temperature range and at the stagnation pressure of 20 bar the cluster formation in an isentropic expansion can be categorized into three regimes:

Regime I: In this region with high rather temperature for a helium cluster beam expansion the isentropes are nearly linear and can be described with the Poisson equation. The isentropes cross the phase line from the gas phase to the liquid phase. Clusters are formed by condensation from the gas phase. The cluster beam contains atoms with maximum cluster size of 12,000 atoms.

Regime II: In this intermediate region between gas and liquid phase the isentrope pass near or at the critical point which is at a temperature of Tc=5.2 K and pressure pc=2.27 bar. In this region the transition between gas phase condensation and liquid expansion takes place.

Regime III: An expansion in this region below the critical point has a deviation from the ideal gas behaviour and the isentropes are bend downwards. The phase line of the gas-liquid transition are crossed from the liquid phase, therefore already liquid helium passes the nozzle. In such expansion very large clusters are formed (mean size 106 atoms) which are cooled by evaporation of helium atoms.

The Table 2 shows the boiling points at 1 bar for different rare gases and for hydrogen. Helium has the lowest boiling point of all gases, therefore two things has to be considered. First low temperatures of the helium beam in the region below 15 K have to be achieved and second it is very important to work with clean gases. All contaminations of the helium gas freeze at these low temperature because they have higher boiling points. This causes problems with clogging of the nozzle.

Rare gas xenon krypton Argon neon hydrogen helium

boiling point (K) 165.1 120 87 27 20 4.2

Table 2. Overview of boiling points for rare gases and hydrogen

Description of the droplet source

Fig. 7 shows a schematic view of the cryostat with the mounted droplet source. Liquid helium flows through a vacuum isolated tube to the cold head, where it expands (and evaporates) through a nozzle approximately 1 mm in diameter. The helium gas flows back on the outside of the tube. The cold head is made entirely of copper to provide a high heat conductivity and therefore a uniform temperature. The helium gas for the droplet formation is precooled before it enters the cold head. A small reservoir ensures the temperature of the helium equals that of the cold head. To lower the impact of the heat radiation from the vacuum chamber at room temperature a cooling shield is mounted around the cold head.

Fig.7 Schematic view of the helium droplet source

The used nozzle is a commercial Pt-aperture for electron microscopy. The hole is 5µm +/- 1µm with a capillary length of 2.5µm. It resists pressures up to 35 bar. For the measuring and controlling the temperature of the cold head a temperature sensor and a heating wire are mounted. A temperature controller is used to adjust the heating current. The recommended temperature range for the silicon diode is from 1 K up to 450 K. The silicon diode is delivered in 4-wire-technique which enhance the accuracy of the temperature value because the resistance of the connection wire is compensated. In addition a change of the connection resistance caused due to the cooling is also taken into account using 4-wire technique.

Aspects of the ionization of Van der Waals clusters

As a consequence of the closed shell atomic configuration, neutral vdW clusters have a very small binding energy. The atoms are held together by a weak induced dipole-dipole interactions. The valence electrons remain localized around the atoms and not yet local dipole moments are present. Upon ionization, dramatic changes occur in the electronic structure. The positive charge (hole) tends to delocalize due to interatomic hopping, with the consequence of weakening of the van der Waals bonds and at the same time of inducing polarization in the surrounding neutral atoms. These two effects are competitive, since the larger the hole delocalization, the smaller the energy gain due to polarization. The interplay between the kinetic energy of the hole and the interatomic Coulomb energy clearly plays a central role in the charge distribution within the ionized clusters and in the size dependence of the ionization potential Ip(n) of van der Waals clusters. The kind of binding changes from an induced dipole-induced dipole in the neutral cluster to a much stronger charge-dipole interaction in the ionized cluster. Clusters change their geometry after ionization because of the influence of the charge. The ionization process, including a change in cluster geometry upon ionization, causes a release of excess energy into the vibrational degrees of freedom initiating severe fragmentation in the case of van der Waals clusters. This leads to mass distributions of ionic clusters which deviate significantly from the precursor size distributions of the neutral clusters and to the formation of particular strong and weak peaks in the respective mass spectra. These “magic numbers” indicate energetically favorable structures of the (charged) clusters.

In the case of helium it can be proposed that ionized n-atom rare gas clusters consist of a positively charged subcluster of 2 atoms surrounded by n-2 neutral atoms. The physical picture underlying this model is that after the hole reaches delocalization over 2 atoms (n>2), it becomes energetically less favorable to delocalize the hole also over one of the remaining n-2 atoms than to polarize neutral atoms around the ionized subcluster of 2 atoms.

Electron impact ionization

Electron impact ionization is a very powerful tool for studies of molecules and clusters. One of the first electron impact ionization studies has been carried out by Dempster [3] and further notable work was performed by Nier (i.e. Nier type ion source) [4]. Although sometimes electron impact is said to be not so accurate like photoionization studies the information which can be deduced by electron ionization experiments is very important. Especially in atmospheric chemistry and in fusion plasmas electron impact ionization plays a decisive roll. Additionally electron impact ionization is used for technical applications like pressure gauges or coatings. Here the main information we are interested in is the appearance energy. The ionization energy is theoretically defined as the energy required to remove an electron from the neutral particle (atom, molecule or cluster) in its ground state to form the ion also in its ground state [5]. The term “appearance energy” is sometimes applied in the case of the formation of a fragment ion from a

molecule, that is to say, when ionization and dissociation occur. Then by analogy, the appearance energy of a fragment ion could be theoretically defined as the energy required to produce the ion and its accompanying neutral fragment, with respect that all entities involved are in their ground state. In actuality ionization- and appearance energies measured in electron impact work do not necessarily correspond to these definitions because of the production of excited states of the particles in the exit channel. So in our case the measured appearance energy is the minimum energy of the bombarding electrons at which the formation of molecular ions can be detected. With a highly precise apparatus the measured values for the ionization energies can be very close to the values given by the theoretical definition. Using the present apparatus the accuracy of the measured values lies within a range from 100 meV to 1eV depending on the target particle and the corresponding signal intensity.

Using an electron as a projectile and a simple diatomic molecule as a target the following processes can occur [6]:

M + e → M+ + es + ee single ionization

→ M2+ + es + 2ee double ionization

→ Mn+ + es + nee multiple ionization

→ MK+ + es + ee K shell (inner) ionization

→ M** + es → M+ + es + ee autoionization

→ M+* + es + ee → A+ + B + es + ee metastable fragmentation

→ M2+ + es + 2ee autoionisation

→ M+ + es + ee + hν radiative ionization

→ A+ + B + es + ee dissoziative ionization

→ A+ + B + es ion pair formation

Where es is the scattered electron and ee are the ejected electrons. Other reactions are possible, but these effects, e.g. vibrational or rotational excitation, are not topic of this work. Most of these ionization reactions can be classified as direct ionization processes where the ejected and scattered electrons leave the ion within 10-16s. In addition to these processes there exists an alternate ionization channel where the electrons are ejected one by one. This autoionization process can be described as a two-step reaction. First a neutral particle is raised into a highly excited state which can exist for a finite time. After this time radiationless transition into the continuum occurs. This can only happen if the time within which this process happens is smaller than the time for predissociation. Very similar to the single autoionization a multiply charged ion can be formed by a two step autoionization process. In a first step a singly charged ion is produced by ejection of an electron from an inner shell. This internally ionized molecule is then transformed into a multiply charged ion by a series of radiationless transition (Auger effect). For the activation of such a process it is necessary to use much higher energies than for single ionization reactions. In case of the N2 molecule one would need about 420eV electrons to eject an innershell electron and start this Auger process [7].

[3] A.J. Dempster, Phys. Rev., 11:316 (1918) [4] A.O. Nier, Rev. Sci. Instrum., 18:398 (1947)

[5] F.H. Field, J.L. Franklin, Electron Impact Phenomena, Academic Press Inc., New York, (1957) [6] T.D. Märk, G.H. Dunn (Eds.), Electron Impact Ionization, Springer, Vienna, (1985) [7] N.A. Cherepkov, S.K. Semenov, Y. Hikosaka, K. Ito, S. Motoki, A. Yagishita, Phys. Rev. Lett. 84:250 (2000)

Determination of appearance energies

The determination of reliable ionization and appearance energies is of crucial importance to the understanding of high energy chemical processes. After many years of research the accurate determination and interpretation of threshold energies at which a molecule is ionized after electron impact still remains difficult [6, 8, 9]. Reasons of this are a number of technical obstacles and additionally the complicated physical situation of a quantum mechanical many-body system. Even if it is possible to achieve acceptable experimental conditions there are numerous possibilities that the situation changes and/or deteriorates during the experiment [10].

The nature of the underlying many-body process close to the ionization threshold was successfully described nearly 50 years ago by Wannier [11]. Originally, Wannier divided the problem into three different radial zones: The reaction zone (between 0 and 1 Bohr), where quantum mechanics has to be applied, the Coulomb zone (up to 1µm), where the escape of the electrons is influenced by Coulomb forces, and the outermost zone (up to ∞) where all the particles essentially move freely. By using this three zone model in addition to a “simple” hydrogen atom as a target for the electron impact theory Wannier was able to predict that the ionization cross section σ near the threshold should follow a power law (Wannier law) in the threshold region:

149100

21)(

),( 41

2

−−

=

≡−

ZZZ

p

µ

εαεµεσµ

(11)

Here ε is the energy above the ionization energy and Z is the charge of the ion after ionization. It should be noted that for Z=1, the coefficient µ=2.75 and the Wannier exponent p=1.127. Twenty years later, Rau [12] and Peterkop [13] were able to extend the quantum mechanical treatment into the Coulomb region, but they essentially confirmed Wannier’s law. Addtionally, three high resolution experiments during the 1980s supported the Wanniers exponential law [14, 15, 16].

In this experiment we are only interested in the appearance energy, so one can modify equation (11) to a simple power law for the fit procedure. In short the cross section σ(E) is fitted over an energy range which incorporates the threshold region:

σ(E ) = b , if the incident electron energy E < E0 (12)

= b + c (E-E0)p , if the incident electron energy E > E0 (13)

The fit then involves four parameters; b the background signal, E0 the appearance energy, a scaling constant c and p an exponential factor which can be compared with the Wannier factor in the case of targets without more than one ionic state in the threshold region. In addition two one can use a weighting factor to increase the importance of small signal values near the threshold:

nw

+=

11 (14)

This parameter w is only dependent of the number of counts n per energy bin. Fig. 8 shows a measurement of appearance energies including the fitted curve produced by the described fitting procedure.

Fig. 8. Typical experimental result for ionization energy measurements and execute the corresponding fit.

The main two factors affecting the results of the given fit procedure are the energy resolution of the incident electron beam and the stability of the electron current within the scanned energy range. When knowing the energy resolution the determination procedure of ionization energies can be extended in consideration of this value, i.e. involving a deconvolution procedure of the raw data using the roughly gaussian shape of the electron energy resolution. Table 3 gives a listing of different ionization potentials of atoms and molecules [17].

Target IP (eV) [17]

Xe 12.12987

Ar 15.759±0.001

Kr 13.999±0.001

N2 15.581±0.008

O2 12.0697±0.0002

N2O 12.889±0.004

Table 3: Ionization energies for positive ions of some rare gases and molecules derived from spectroscopic and photoionization results.

[8] D.L. Hildenbrand Int. J. Mass Spectrom., 197, (2000), 237 [9] D. Muigg, G. Denifl, A. Stamatovic, O. Echt, T.D. Märk, Chem. Phys., 20, (1998), 409 [10] T. Fiegele, G. Hanel, I. Torres, M. Lezius, T.D. Märk, J. Phys. B., 33, (2000), 4263 [11] G.H. Wannier, Phys. Ref., 90, (1953), 817 [12] A.R. Rau, Phys. Rev. A, 4, (1971), 207 [13] R. Peterkop, J. Phys. B. 4, (1971), 513 [14] J.B. Donahue, P.A. Gam, M.V. Hynes, R.W. Hamm, C.A. Frost, H.C. Bryant, K.B. Butterfield, D.A. Clark,

W.W. Smith, Phys. Rev. Lett., 48, (1982), 1538

[15] H. Kossmann, V. Schmidt, T. Andersen, Phys. Rev. Lett., 60, (1989), 1266 [16] P. Schlemmer, T. Rösel, J. Jung, H. Erhard, Phys. Rev. Lett., 63, (1989), 252 [17] NIST database, http://webbook.nist.gov

Aufgaben

Das Praktikum wird an der Helium-Cluster-Anlage durchgeführt. Ein Ziel ist es, in Heliumtropfen eingebettete Argoncluster zu erzeugen und ihre Größenverteilung mit einem Flugzeit-Massenspektrometer zu messen. Dazu werden die Cluster durch Stöße mit Elektronen ionisiert. Die Größenverteilung der Cluster wird auf das Vorhandensein geometrischer Schalen hin untersucht. Im zweiten Teil des Versuchs nutzt man die einstellbare Stoßenergie der Elektronen, um das Ionisationspotential der Argon- (Magnesium-) und Heliumcluster zu bestimmen. Das Ergebnis wird mit einem Experiment an einzelnen Argon- (Magnesium-) und Heliumatomen verglichen und auf die Natur des Ionisationsprozesses geschlossen.