J. Phys. Chem. 1996, 100, 1475-1482
1475
Energy Analysis and Model Calculations of Collisional Acceleration in Seeded Molecular Beams. Application to Xe/He and C60/He B. Tsipinyuk, A. Budrevich, and E. Kolodney* Department of Chemistry, Technion-Israel Institute of Technology, Haifa 32000, Israel ReceiVed: June 22, 1995; In Final Form: October 4, 1995X
A simple theoretical model for the aerodynamic acceleration of heavy particles in seeded molecular beams in the hyperthermal energy range 1-60 eV is presented. The model gives a microscopically detailed picture of the collisional acceleration dynamics in terms of the number of collisions at a given distance downstream from the nozzle aperture. The basic model applies to an ideal isentropic free jet center line expansion with hard spheres collisions but is also modified to include capillary nozzle effects. The model predictions are compared with experimental measurements of terminal kinetic energies of Xe accelerated in He in the constant flux mode up to 11.5 eV and C60 accelerated in He in the constant flux mode up to 56 eV and in the constant temperature mode up to 41 eV. Good agreement between calculation and experiment is obtained. The applicability of the hard sphere approximation for describing the acceleration dynamics of superhot C60 in He shows that C60 can be treated here as a “quasi-atom” and that the possible coupling between vibrational relaxation and the velocity slip effect can be ignored in this case. Overall, we are presenting new experimental and theoretical tools for the generation, energy analysis, and modeling of the collisional acceleration of large particles in the hyperthermal energy range of 10-100 eV.
1. Introduction Aerodynamical acceleration in seeded supersonic molecular beams provides a unique method for obtaining neutral ground state particles (atoms, molecules, and clusters) with hyperthermal energies.1-6 These particles usually serve as colliders for a variety of scattering experiments (gas-gas, gas-surface, and gas-liquid) in the 1-20 eV energy range.7-12 The highest kinetic energy is achieved by acceleration of a highly diluted heavy particle in a light carrier gas (usually He or H2) under extreme stagnation conditions, namely, very high source pressure P0 and temperature T0. The usual ways by which one can manipulate the terminal kinetic energy of the heavy particle are (a) changing P0, thus controlling the velocity slip between the heavy particle and the light carrier gas atom, (b) changing the nozzle temperature T0, and (c) controlling the dilution ratio. As a result, the two common working modes with a seeded beam are the constant flux mode with variable nozzle temperature (P0/xT0 ) const) and the variable flux mode (changing P0) with constant nozzle temperature. The hyperthermal kinetic energy range is well established. It includes acceleration of Xe up to 37 eV14,15 and scattering experiments with Xe up to 30 eV,9-13 Hg atoms up to 12 eV,16,17 diatomics like I2 up to 12 eV,18 and a variety of large and heavy polyatomics in the range 1-20 eV.19-22 Very recently we reported on the first beam generation23 and beam scattering experiments24 with molecular species at the new energy range 20-70 eV, as demonstrated by C60 acceleration in He up to 56 eV and in H2 up to 73 eV. C60 was serving as a model for a highly vibrationally excited, very large molecule or cluster, accelerated under standard expansion conditions (P0D ) 10-20 Torr‚cm with D as the nozzle diameter) to terminal kinetic energies that are still far below the zero-slip isentropic expansion limit. In this study our goal is to present a microscopically detailed, simple theoretical modeling for collisional acceleration in seeded beams and apply it to the experimental measurements of C60/He beams. X
Abstract published in AdVance ACS Abstracts, December 1, 1995.
0022-3654/96/20100-1475$12.00/0
Analysis of noncontinuum, nonequilibrium kinetic effects (relaxations) between beam species is not a simple one and requires the solution of the Boltzmann equation for the ellipsoidal Maxwellian velocity distribution of each species.25 One of the most important effects for the molecular beam researcher and the one treated here is the velocity slip. Existing theoretical methods for predicting the velocity slip effect in binary mixtures and the terminal kinetic energy of the heavy particle are based on solving the Boltzman equation using either the approximated moments method15,25-27 or direct Monte-Carlo simulation.28,29 The scattering potential usually assumed for the moments method is the Lennard-Jones 6-12 potential while the Monte-Carlo simulation was carried out within the hard sphere approximation. Both methods include all the relevant conservation laws such as for number density, momentum, stress tensor, and heat flux, but the microscopic collision dynamics cannot be visualized. The basic working assumption is that the center line flux of an axis-symmetric free jet can be treated as a radially expanding spherically symmetric flow source. A clear introduction to the properties and analysis of continuum and noncontinuum free jet molecular beams is given by Miller.30 After solving the Boltzmann equation, one usually finds that the relative velocity slip can be described as a function of a single dimensionless correlation parameter named the velocity slip parameter (VSP). Good agreement with experimental results of inert gas atom acceleration was usually obtained only for a velocity slip smaller than 10% and a VSP > 10.15,25-28 Agreement is rather poor for a larger slip and a VSP < 5-10. Obviously there is a need for a simple microscopic collisional approach for modeling the acceleration dynamics in terms of the number of collisions at a given downstream distance. This model should also give a correct prediction for the terminal kinetic energy in the problematic region of large velocity slip (20-80% lower than the isentropic limit). This region is especially relevant for very heavy particles like large molecules or clusters at high nozzle temperature (up to 2000 K). The model developed and applied in this article relates to the three prerequisites presented above. The microscopic © 1996 American Chemical Society
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Figure 1. Schematic drawing of the seeded molecular beam apparatus: (1) two-stage nozzle oven where Toven ) 950-1000 K (for C60) and nozzle temperature is T0 ) 1130-1930 K; (2) skimmers chamber; (3) movable beam chopper; (4) UHV detection chamber; (5) mass spectrometer equipped with a homemade 90° cylindrical energy analyzer; (6) on-line view port; (7) telescopic micropyrometer situated on the beam axis; (8) stagnation pressure (P0) gauge; (9) precision needle valve. P1, P2, P3, and P4 are pumping stages.
collisional approach provides a detailed description of the acceleration dynamics in terms of the number of collisions with the light carrier gas and the distance from the nozzle. The simple model can easily be implemented using elementary mathematical software packages on a laboratory computer and has good predictive power for heavy particle acceleration within the large slip region. Please note that the model has no adjustable parameters. In section 3 a description of the basic model is given, and in sections 4 and 5 the model is applied to the acceleration of Xe and C60 in He, respectively. The acceleration and cooling of vibrationally hot C60 represent an intriguing situation of a possible coupling between vibrational relaxation and the velocity slip effect. This unique situation could be highly relevant to acceleration of very hot large clusters expanded from high-temperature nozzles. We discuss this possibility in light of our preliminary results on vibrational cooling of C60 in He beams and show that good agreement is obtained between our model (based on the hard spheres approximation) and the experimental results. This agreement provides evidence that C60 could be treated as a “quasi-atom” in the seeded beam experiments reported here. 2. Experimental Section The molecular beam apparatus used for the generation of the high-temperature Xe and C60 seeded beams and for measurements of the hyperthermal kinetic energies is shown schematically in Figure 1. Briefly, a seeded beam is generated by nozzle expansion with a stagnation temperature T0 and a stagnation pressure P0. The beam is traveling along three differential pumping stages to the ionizer region of a crossed beam quadruple mass spectrometer (Balzers, QMG-421, 1-1024 amu) equipped with a 90° cylindrical energy analyzer and mounted in an ultrahigh-vacuum (UHV) chamber with a base pressure of 5 × 10-10 Torr. The beam is skimmed twice, chopped for time-of-flight (TOF) experiments or modulated for phasesensitive detection, and collimated before entering the UHV chamber. Kinetic energy measurements were usually carried out in the constant flux mode (P0/xT0 ) const). The flux is stabilized to within (2% by forcing the He carrier gas into the nozzle region from a high-pressure gas cylinder (150 atm) through a flow-controlling precision needle valve. Good stabilization is assumed as long as the stagnation pressure is much lower than the cylinder pressure and the nozzle diameter does not change during the experiment. The latter may be a
problem when working with a high-temperature refractory metal nozzle due to trace amounts of oxygen in the carrier gas or a reactive seed molecule. For all our experiments, over a period of more than 500 h per nozzle, the ratio P0/xT0 remains constant to within (2% (for a given initial P0). The resulting pressure rise (P ∝ xT) can thus serve as an accurate thermometer for the nozzle temperature T0, as was demonstrated before.31 The two unique components in our molecular beam apparatus are (a) the beam source, which is a two-stage, all-ceramic differentially heated nozzle oven enabling the generation of superheated C60 vapors up to 2100 K (1935 K reported in this experiment), and (b) the electrostatic energy analyzer used for the hyperthermal kinetic energy (Ek) measurements. Both were described before,23,32 and only a few essential details will be given here. The nozzle first stage is the evaporation oven, temperature controlled to better than 1° and typically maintained in the region 950-990 K (0.1-0.2 Torr C60 vapor pressure). Pure C60 (99.9%) was used (MER). The C60 molecules are seeded in He at pressures of 200-4500 Torr and expanded out of a capillary nozzle with orifice diameter 100-200 µm and capillary length 0.4-0.7 mm. Nozzle temperatures were measured by the TOF technique with He and Ar, both at the supersonic (neat gas expansion) and effusive mode and with Xe at the effusive mode. TOF spectra were usually taken at the counting mode with a multichannel scaler/averager. An alternative method used was microoptical pyrometry of the nozzle front face along the beam axis. The overall accuracy of the temperature measurements is (15 K with reproducibility of (10 K. All the C60 and Xe seeded beam energy measurements were carried out in a continuous beam mode using the energy analyzer with a resolution of ∆E/E ) 0.07. This method has several advantages over the TOF method for the hyperthermal energy range 1-100 eV.23 Overall, the energy analyzer method has better a signal to noise ratio by a factor of 102 to 103 and improved resolution (up to a factor of 10) as compared with the TOF method for hyperthermal beams. These advantages may be critical in low-level signal experiments like beamsurface scattering,24 where one usually has to make a painful compromise between signal intensity and resolution (both angular and energetic). The TOF method for hyperthermal energies was used for only the early stage comparative assessment of the energy analyzer method. Since the analyzer is working in a mode of variable pass energy, the experimentally
Collisional Acceleration in Seeded Molecular Beams
Figure 2. Kinetic energy distributions N(Ek) of C60 seeded in helium for different backing pressures P0 and nozzle temperatures T0: (a) P0 ) 300 Torr, T0 ) 1705 K; (b) P0 ) 600 Torr, T0 ) 1705 K; (c) P0 ) 1000 Torr, T0 ) 1705 K; (d) P0 ) 1800 Torr, T0 ) 1705 K; (e) P0 ) 2800 Torr, T0 ) 1870 K; (f) P0 ) 4160 Torr, T0 ) 1935 K.
J. Phys. Chem., Vol. 100, No. 5, 1996 1477 carrier gas atoms with mass mc. We define Vc(x) as the mean velocity of a carrier gas atom located on the beam centerline at a distance x from the nozzle orifice, and Vs(x) as the heavy atom velocity at the same distance. The heavy atom is accelerated via sequential collisions with the carrier gas atoms with Vc(x) > Vs(x). The present model is based on three simplifying assumptions: (1) Collisions are treated within the hard sphere approximation for the interaction potential. (2) In a coordinate system moving with velocity Vs relative to the laboratory coordinate system the energy transferred from carrier to seed atom is an average over all impact parameters. (3) The heavy atom velocity directions, both before and after collision, are the same as the carrier gas atom velocity direction before collision. In the moving system (with velocity Vs) the carrier gas atom velocity before collision is Vc - Vs and the seed atom velocity after collision is V′s - Vs, where V′s is the seed atom velocity (after collision) in the laboratory system. From assumptions 1 and 2 it follows that ms(V′s - Vs)2 ) 〈Λ〉mc(Vc - Vs)2 with 〈Λ〉 ) 2mcms/(mc + ms)2.33 One can write now a simple relation between the seed atom velocity V′s after collision (average value) and the velocities Vs and Vc before collision, which is given by
V′s ) Vs +
x2mc (V - Vs) ms + mc c
(1)
The probability dw for a collision between a carrier gas atom and the heavy atom during the travel time over a distance dx downstream along the beam centerline is determined by the equation Figure 3. Relative widths (fwhm) ∆Ek/Ek of C60 energy distributions in He as a function of the stagnation pressure P0 for constant nozzle temperature T0 ) 1705 K. The experimental points are denoted by the empty triangles.
measured ion signal has to be transformed as described before23 and results in N(Ek) distributions as presented for seeded C60 in Figure 2. Peak energies could be measured to within (0.2 eV, with good reproducibility. An effusive beam measurement served for calibrating the energy analyzer and extracting its resolution. As shown in Figure 2, the measured kinetic energy distributions of C60 seeded in He are remarkably narrow for the given expansion conditions (P0D values of 10-20 Torr cm). The relative energy widths ∆Ek/Ek (∆Ek taken as full width at half maximum) are presented in Figure 3 as a function of the stagnation pressure. Above ∼800 Torr of He, the relative width of the observed hyperthermal beam distribution is ∆Ek/Ek ) 0.14, which result in ∆Ek/Ek ≈ 0.11 after Gaussian deconvolution of the analyzer transfer width. For all the experiments reported in this paper He was used as the carrier gas. Much higher energies could be obtained by seeding in hydrogen, and indeed we have measured a maximum energy of 73 eV for the C60-hydrogen beam with a nozzle temperature of 1400 K and a 3200 Torr stagnation pressure. However, systematic and reproducible measurements with hydrogen were not possible at high energies due to repeated nozzle-clogging problems. Furthermore, since hydrogen is a molecular carrier gas that is still rotationally relaxing during expansion, hydrogen-seeded beams are not suitable for the hard sphere model presented here. 3. Basic Model Our goal is to provide a simple collisional model for the acceleration of a heavy atom (the seed atom) with mass ms within a radially expanding beam (straight streamlines) of light
dx dw ) σ‚N(x)[Vc(x) - Vs(x)] Vs(x)
(2)
where σ is the hard sphere cross section for collisions between carrier and seed atoms and N(x) is the x-dependent carrier gas density along the jet centerline. Let us assume that the i-th collision of the heavy atom with a carrier gas atom took place at a distance xi from the source and Vs,i is the heavy atom velocity after the i-th collision. The probability W(xi,x) of a collision between a carrier gas atom and the heavy atom along the way from xi to x should obey the kinetic equation dW(xi,x) ) (1 - W(xi,x)) dw. Using expression 2 for dw, we now obtain
[
]
x dx W(xi,x) ) 1 - exp -∫x σN(x)[Vc(x) - Vs,i] i Vs,i
(2′)
Next we will assume that the (i + 1)-th collision will take place (on the average) at the distance xi+1, defined by the condition
∫xx
dx σN(x)[Vc(x) - Vs,i] ) 1 Vs,i
i+1
i
(3)
That is, the value ∆xi ) xi+1 - xi is essentially the mean free path between the i-th and (i + 1)-th collisions. If we now introduce a dimensionless distance y ) x/D, where D is the nozzle diameter, and a dimensionless density n(y) ) N(y)/n0 with n0 ) P0/KBT0 and KB the Boltzmann constant, then eqs 3 and 1 can be rewritten in the form
P0D yi+1 σ ∫ n(y)[Vc(y) - Vs,i]Vdys,i ) 1 KBT0 yi Vs,i+1 ) Vs,i +
x2mc [V (y ) - Vs,i] ms + mc c i+1
(4)
(5)
The coupled eqs 4 and 5 can now be solved with respect to yi+1 and Vs,i+1 if yi and Vs,i and the functions n(y) and Vc(y) are
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Figure 4. Measured (open circles) and calculated kinetic energies of Xe atoms seeded (0.2%) in He as a function of nozzle temperature T0 under constant flux conditions (P0/xT0 ) 24.0 Torr K-1/2). The effective nozzle diameter D is 168 µm. The straight solid line is the isentropic limit (zero-slip), the squares stand for VSP calculations (see eqs 9 and 10), and the triangles and diamonds are the calculated values using the present model with two different values of hard sphere cross sections. The crossed symbols at the highest T0 point relate to a different nozzle with an effective diameter of 136 µm.
given. The initial conditions for finding y1 and Vs,1 are y0 ) 0 and Vs,0 ) (2KBT0/ms)1/2. For the n(y) and Vc(y) functions one can use the ideal isentropic expansion expressions30 (with specific heat ratio γ ) 5/3)
Vc(y) ) M(y)‚V∞[3 + M2(y)]-1/2
(6)
n(y) ) 33/2[3 + M2(y)]-3/2
(7)
where M(y) is the y-dependence of the Mach number along the beam centerline:30,34
M(y) )
{
y2/3[3.232 - 0.7563y-1 + 0.3937y-2 - 0.0729y-3] 1 + 3.337y2 - 1.541y3
(8) for y > 0.5 for 0 < y < 1.0
V∞ ) (5KBT0/〈m〉)1/2 is the isentropic terminal velocity, and 〈m〉 is the average molecular weight 〈m〉 ) ∑cjmj with cj and mj as the j-th gas component fraction and the molecular weight, respectively. 4. Acceleration of Xenon in HeliumsExperiment and Calculations In order to demonstrate the method presented before, we will first apply it to the simplest case of acceleration of a heavy atom (Xe) in a helium beam with a dilution ratio of 2 × 10-3. Figure 4 shows the kinetic energies of the accelerated Xe atoms for several nozzle temperatures (circles). The measurements were carried out in the constant flux mode (P0/xT0 ) 24.0
Figure 5. Calculated collisional acceleration dynamics for Xe atoms seeded (0.2%) in He under constant flux conditions (P0/xT0 ) 24.0 Torr K-1/2). The relative distance yi, the free path ∆i ) yi - yi-1 (left ordinate), and the kinetic energy Ei are presented as a function of the collision number i. The solid curves and the dash-dot curve (∆i) are for hard sphere cross section σ ) 36.3 Å2, and the dashed curves are for hard sphere cross section σ ) 100 Å2. The two vertical dotted lines stand for the maximum number of collisions (“freezing” of the acceleration process). The effective nozzle diameter D is 168 µm.
Torr K-1/2) with a constant dilution ratio. Since we are working with a capillary nozzle, we have used an effective (calculated) diameter derived from measurements of total gas throughput in the source chamber in the limit of very large stagnation pressure. At this limit the nozzle capillary effect is negligible (see also eq 13 and Figure 9 for a pressure range where Peff = P0). The effective nozzle diameter for Xe acceleration experiments is D ) 168 µm. The straight solid line in Figure 4 describes the isentropic expansion limit (zero slip) E∞(T0) ) 5/ K T m /〈m〉. Since we do not measure the actual gas velocity, 2 B 0 s it is more appropriate to use the term of an energy slip defined as the difference between the measured terminal kinetic energy of the heavy species and the zero-slip isentropic limit (dilution ratio corrected). The calculated energy values using eqs 4-8 are given in Figure 4 by the diamonds. Figure 5 presents some of the collisional acceleration dynamics according to our simple microscopic approach (solid lines). The initial stagnation conditions are T0 ) 1465 K and P0 ) 920 Torr. The left ordinate represents the relative distance yi (scaled in D units) where the i-th He-Xe collision occurs and the free path ∆i ) yi - yi-1. The kinetic energy Ei of the Xe atom after the i-th collision is presented on the right ordinate. For this calculation we have used the hard sphere cross section σ ) π(RXe + RHe)2, where RXe and RHe are van der Waals atomic radii for Xe and He atoms, respectively.35 As can be seen from Figure 5, the free path ∆i between two sequential collisions rises rather smoothly and slowly from ∼10-3 for the first collision to ∼10-1 for i ≈ 70 and then steeply increases up to ∆80 ) 57 for the 80th collision. The free path for the 81st collision is already more than the experimental setup dimensions. The implication of this behavior is that at the relative distance y80 ) 69.3 there is “freezing” with regard to energy exchanging collisions between carrier and seed particles. Please note that collisional processes eventually lead every nonequilibrium relaxation effect
Collisional Acceleration in Seeded Molecular Beams
J. Phys. Chem., Vol. 100, No. 5, 1996 1479
in the molecular beam to a “freezing” or termination point. The average number of accelerating collisions between He atoms and Xe is 80, and the average terminal energy of the Xe atom is E80 ) 6.3 eV. This value is 2 eV less than the measured one for the same T0 and P0 conditions (see Figure 4). Figure 4 also presents Xe atom energies calculated by the method of moments of the Boltzmann equation (squares) using the relation30
(Vc - Vs)∞ ) 0.5(VSP)-1.07 V∞
(9)
where VSP is the so-called velocity slip parameter, which for Xe(0.2%)/He is given by
P0D
VSP ) (1.87 × 103)
T4/3 0
(10)
(P0 in Torr, D in cm, and T0 in K). It should be noted that the expression (eq 10) for VSP is based on linearization of the collision integral using an attractive C6/r6 potential.30 This approach is appropriate for low-energy cross sections, as is usually the case in low-temperature binary free jet expansions, but tends to be gradually inadequate as the source temperature increases. This general trend is reflected in the calculated results as compared with the experimental measurements. As follows from Figure 4, Xe atom kinetic energies calculated using eqs 9 and 10 for large T0 values and correspondingly large P0 ∝ xT0 (small VSP) values are further from agreement with the experimental results as compared with those calculated using the method presented here. However, for small T0 and P0 (large VSP) values, the calculation by the moments method is in better agreement with experiment (the 300 K point). At this stage we would like to discuss some additional aspects of our collisional acceleration model and consider the sensitivity of our calculations with regard to different parameters. (i) The calculated terminal energy of the heavy atom depends rather weakly on the fraction Λ of the carrier gas atom energy transferred to the heavy atom per collision in a coordinate system in which this atom was at rest before the collision. In a previous section it was mentioned that this part was equal to the average value in the hard sphere approximation 〈Λ〉 ) 2mcms/(mc + ms)2, namely, half of the maximum energy transfer (in the abovementioned coordinate system). The calculation shows that, even for Λ ) 1.5〈Λ〉, the energy of the seed Xe atom increases by only ∼3% (∼0.2 eV for the example presented in Figure 5). This behavior results from a compensating effect between the value of Λ and the total number of accelerating collisions. An increase in Λ leads to a corresponding increase in the free path before the next collision and therefore a decrease of the total number of collisions. This decrease compensates almost completely for the increase in Λ, and only a weak increase of the heavy atom terminal energy is observed. (ii) A basic assumption that simplified our former calculations was the energy independent value of the hard spheres cross section σ between carrier and seed particles. A more realistic approach is to take into account the dependence of the radii RXe and RHe determining σ on the relative velocity of the colliding atoms. Therefore, σ in eq 4 should be a function of the difference ∆V ) Vc(y) - Vs,i and consequently be included under the integral sign. It should be remembered that the hard sphere cross section is defined as πR20 with R0 as the distance of closest approach for the HeXe pair (see assumption 1 in section 3). The function σ(∆V) itself with σ as the hard sphere
cross section can be obtained from high-energy beam scattering experiments or extracted from the exponentially repulsive part of the interatomic potential.36 For the velocity range ∆V < 1.0 × 105 cm/s, effective for acceleration under our experimental conditions and for our model calculations, R0 varies in the range 3.0-3.5 Å.36 In the former calculation we have used R0 ) 3.4 Å (σ ) 36.3 Å2), which is near the zero crossing value (3.565 Å) of the potential energy curve.36 Making use of the explicit form of σ(∆V) is therefore expected to have a negligible effect on the acceleration dynamics. The absolute total elastic scattering cross sections as given for example in refs 36 and 37 in the form of nearly straight lines on a log-log plot of σ(∆V) are not appropriate for our model, since they reflect a large contribution of low-angle scattering, where energy exchange is negligible. The increase in cross section with decreasing relative velocity in this case is due to long-range attractive forces, while the acceleration dynamics is probing the short range, exponentially repulsive part of the potential energy curve. Energy-exchanging collisions are characterized by a low impact parameter, and the large total elastic cross sections (up to ∼103 Å2) contributed by the longrange region of the HeXe interatomic potential are therefore not effective for acceleration and should not be used within the frame of our model. However, in order to demonstrate the possible effect of increased cross section (e.g., due to attractive forces), we have repeated the calculation with an extremely large σ value of 100 Å2. This value corresponds to a 0.5 meV attractive potential energy (about 20% of the HeXe well depth36 and smaller than 1% of the average amount of energy transferred per single collision). The resulting new energy values are given in Figure 4 by the triangles, and the dynamical details of the calculation are demonstrated in Figure 5 for the case of T0 ) 1465 K and P0 ) 920 Torr. As shown in Figure 5 by the dashed lines, the terminal energy of the Xe atom increases from 6.2 eV for σ ) 36.3 Å2 to 7.5 eV for σ ) 100 Å2. Close inspection of Figure 5 shows that using the σ ) 100 Å2 value results in a substantial increase of the total number of collisions before the freezing point is reached from 81 collisions up to 128. However, the kinetic energies Ei exceed those calculated for σ ) 36.3 Å2 only during the last 13 collisions. (iii) The model calculations show that the analytical form of the density function n(y) has an important effect regarding the calculated terminal value of the heavy atom energy. The specific form of n(y) as given by eqs 7 and 8 is partially responsible for the relatively low calculated energy values. A density function with a weaker y dependence will probably yield a calculated energy value closer to the measured one. The physical basis for a different form of n(y) can be associated with nozzle capillary effects regarding beam intensity. Section 5 deals in a more detailed way with these effects as related to C60 acceleration in He. 5. Acceleration of C60 in HeliumsExperiment and Calculations The model developed in this paper and applied to the collisional acceleration of Xe atoms in He carrier flux is suitable in principle only for treating kinetic energy and momentum transfer in atom-atom collisions. For polyatomic species one has to consider also the possible involvement of internal degrees of freedom. The unique and intriguing situation regarding the acceleration of superhot large molecules or clusters relates to the high vibrational energy content that is roughly of the same order as the final kinetic energy attained in the expansion. In our experiment C60 with an initial vibrational energy as high as 21 eV is accelerated up to ∼60 eV of kinetic energy.
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Figure 6. Calculated collisional acceleration dynamics for C60 molecules (0.1 Torr) seeded in He under constant flux conditions (P0/xT0 ) 62.5 Torr K-1/2). The relative distance yi (left ordinate) and the kinetic energy Ei (right ordinate) are presented as a function of the collision number i. The two vertical dashed lines stand for the maximum number of collisions (“freezing” of the acceleration process). The effective nozzle diameter D is 103 µm.
Preliminary vibrational thermometry measurements of C60 cooling in He-seeded beams indicate that hot C60 may lose up to 6-8 eV out of the initial 20 eV during expansion.38 The basic question here is whether part of this energy, pumped away from molecular vibrations, can show up as C60 kinetic energy or maybe most or all of it is dissipated as heating of the carrier gas. We believe that the more complicated situation of coupling between vibrational cooling and acceleration in superhot polyatomic species is mainly relevant for large molecules or clusters characterized by floppy vibrational modes and a low degree of symmetry. However, for the spherically symmetric C60 with high structural rigidity, one can expect “quasi-atom” behavior, at least regarding the microscopic acceleration mechanism. Indeed, as will be shown later, the reasonably good agreement obtained between predictions of our simple acceleration model and the experimental results confirms this assumption. We will therefore assume for the purpose of calculation that the C60 molecule can be treated as a structureless quasi-atom with a mass of 720 amu. The cross section σ for He-C60 collisions is taken as σ ) (π/4)[dC60 + dHe]2, where dC60 ) 10.0 Å is the outer diameter of the C60 molecule39 and dHe ) 2.44 Å is the van der Waals diameter of the helium atom. Please note that the dependence of this hard sphere cross section on the relative velocity of the colliding particles should be significantly less pronounced than for atom-atom collisions, since dC60, which is the most important factor in σ, can be realistically considered as a constant. The collisional acceleration dynamics of C60 molecules (PC60 ) 0.1 Torr) seeding in He carrier gas is calculated according to eqs 4-8 and is illustrated in Figure 6. The calculation describes the gradual acceleration dynamics for two pairs of T0, P0 values, under constant flux conditions (P0/xT0 ) 62.5 Torr K-1/2) and an effective nozzle diameter of 103 µm. These are the experimental conditions for the measured C60 kinetic energies as presented in Figure 7. Small variations in PC60 in
Tsipinyuk et al.
Figure 7. Measured and calculated kinetic energies of C60 molecules (0.1 Torr) accelerated in He as a function of nozzle temperature T0 under constant flux conditions (P0/xT0 ) 62.5 Torr K-1/2). Shown are experimental points (triangles) and calculated energies (dashed line). The effective nozzle diameter D is 103 µm. The straight solid line is the isentropic limit (zero slip).
the range 0.1-0.2 Torr have negligible effect on the measured kinetic energies due to the very high dilution ratio. Close inspection of the simulation in Figure 6 shows that for the initial conditions T0 ) 1130 K, P0 ) 2100 Torr, the freezing point is achieved after 483 collisions at a distance of 129 nozzle diameters downstream (y483 ) 129), and the terminal energy obtained is 28.6 eV. However, for the pair T0 ) 1795 K, P0 ) 2650 Torr a freezing is achieved already after 435 collisions (y435 ) 107), but the terminal energy is 43.0 eV. This conclusion that the total number of collisions effective for acceleration (until the freezing point is reached) decreases with T0 but the terminal kinetic energy increases is a characteristic of the constant flux conditions as given by P0/xT0 ) const and can be arrived at on the basis of qualitative elementary arguments.23 It is also interesting to note that already after two nozzle diameters (T0 ) 1795, P0 ) 2650, y393 ) 2) the kinetic energy is reaching a value of 35.4 eV, which is 82% of the terminal value. This behavior is a general characteristic of all the acceleration processes presented here and results from the behavior of yi, which for C60-He rises steeply only during the last 50 collisions, whose contribution to the terminal kinetic energy is relatively small. This behavior is also in line with the rapid rise of the carrier gas mean velocity Vc(y) as given by eqs 6 and 8. Figure 7 presents the experimental measurements (triangles) under constant He flux conditions (P0/xT0 ) 62.5 Torr K-1/2) and variable nozzle temperature in the region 1130-1930 K, for D ) 103 µm. The solid thin line stands for the isentropic limit (dilution ratio corrected), and the calculated values, lower by 7-8% than the measured ones, are given by the dashed line. Although the agreement obtained is remarkable, considering the simple model applied and the various assumptions made, we believe that for this situation as well as for the previous one of
Collisional Acceleration in Seeded Molecular Beams Xe acceleration the difference between the calculated values and the measured ones can be related to the capillary nozzle effects. The associated effects of expansion from a capillary nozzle, as compared with an ideal orifice nozzle, were already discussed in the literature both theoretically40 and experimentally.41 Our basic model is based on the formulas for ideal isentropic expansion along the free-jet center line. The nozzle was assumed to be a good approximation for an ideal orifice with a short subsonic converging section. The gradual area decrease results in a gradual increase of the Mach number inside the nozzle up to the planar sonic (M ) 1) surface at the exit aperture.40 Although the calculated energies are in reasonably good agreement with the measured values (using an effective nozzle diameter D), it is a clear experimental observation that capillary nozzles with a length to diameter ratio of 3-5 gave the best gas dynamic performance in terms of center line beam flux and terminal energies (up to a 10% increase). Flow acceleration inside the constant diameter capillary is not derived from gradual narrowing, as for the ideal converging nozzle, but is related to pressure gradients that develop toward the exit aperture.40 The flow is not isentropic although it can be treated as adiabatic, and the Mach number changes along the capillary radius. Pitot tube experiments40 showed that the sonic surface is curved upstream and is penetrating inside the capillary up to one nozzle diameter. Namely, acceleration starts already inside the capillary. On the one hand, a certain density decrease is expected at the nozzle aperture,40 but at the same time an increase of up to a factor of two (as measured for H2 and Ar) is observed41 for the center line beam intensity. It is therefore concluded that the beam flux divergence from a capillary source is smaller as compared with the divergence from an orifice source, namely, n(y) is expected to decrease more slowly with y. To the best of our knowledge, no analytical form of n(y) for capillary source is reported in the literature, but in order to take this effect into account, we have constructed a n(y) function that qualitatively describes the two effects discussed above.42 We have repeated the terminal kinetic energy calculation for C60 in He using eqs 4-6 and eq 8 with the new n(y) function and obtained approximately a 10% increase in terminal energies, in agreement with experiment. However, due to the somewhat arbitrary nature of the new density function, this result should be considered only as a qualitative one. Figure 8 presents the experimental results (squares) for acceleration of C60 in helium under constant nozzle temperature T0 ) 1705 K and variable stagnation pressure P0. The calculated values are given by the dotted line, and the isentropic limit (dilution ratio corrected) is given by the upper solid line. As can be seen from the figure, at higher pressures of P0 g 1500 Torr, the calculated energies are lower by roughly 5-7% than the measured ones. With a pressure decrease, in the region of 1200 g P0 g 900 Torr, the calculated and measured curves nearly converge, and for lower pressures (P0 e 900 Torr), the calculated energies even exceed the measured ones. As discussed before, under low stagnation pressure conditions a substantial pressure gradient is built up inside the capillary nozzle such that the effective pressure (Peff) at the nozzle aperture becomes lower than the P0 value measured by the stagnation pressure gauge. Some experimental support for this behavior is given by Figure 3, which shows a dramatic decrease of the relative energy distribution width ∆Ek/Ek as a function of P0, in the P0 ) 650-800 Torr range. The effective pressure at the nozzle aperture can be calculated if the total throughput through the nozzle source (into the source chamber) is measured. The value of Peff is obtained by equalizing the gas flux from the nozzle1 and the total throughput:
J. Phys. Chem., Vol. 100, No. 5, 1996 1481
Figure 8. Measured and calculated kinetic energies of C60 molecules accelerated in He as a function of He stagnation pressure (P0) for constant nozzle temperature T0 ) 1705 K. Shown are experimental points (squares), calculated energies using P0 (dotted line), and calculated energies using Peff (dashed line). The thin solid line (upper) is the isentropic limit (zero slip).
πD2Peff(P0)
A
x〈m〉KBT0
Pthr(P0) KBTthr
) C(Pthr)‚
(11)
where A ) xγ[2/(γ + 1)]0.5(γ+1)/(γ-1) with γ ) 5/3; Pthr(P0) and Tthr are the corresponding pressure and temperature at the point where the total throughput is measured, and C(Pthr) is the conductance for pumping from the point where the throughput was measured. The values of Pthr(P0) were experimentally measured, and the conductance C(Pthr) was calculated according to standard fromulas.43 The He effective pressure Peff at the nozzle exit plane calculated by eq 13 is presented in Figure 9 as a function of P0. The results of the calculation using eqs 4 and 5, in which the values of P0 were replaced by Peff as given in Figure 9, are presented in Figure 8 by the dashed line. The agreement with the experimental results is much better now for the lower pressure region but has slightly degraded for higher pressures. Summarizing this section, we find that the agreement demonstrated in this study between experimental and calculated energies for C60 accelerated in He beams is quite remarkable. This agreement can be viewed as supporting the initial assumptions that C60 can be modeled as a hard sphere (within our experimental conditions) and that the coupling between vibrational relaxation and acceleration for C60-He is small. It is possible that vibrational cooling is mainly due to soft collisions (e.g., grazing collisions) with negligible contribution to kinetic energy, while accelerating collisions are mostly head-on. 6. Summary In this work we have presented experimental results and theoretical modeling of energy analysis and collisional acceleration dynamics of heavy species in seeded supersonic molecular beams in the hyperthermal energy range of 1-60 eV. The model was developed using a microscopic collisional approach for predicting the heavy species terminal energies in binary mixtures far away from the zero-slip isentropic limit. It employs only independent physical quantities without any adjustable parameters. The basic model describes the acceleration dynamics in an ideal isentropic free-jet expansion within the hard spheres approximation for the interaction potential between
1482 J. Phys. Chem., Vol. 100, No. 5, 1996
Figure 9. He effective pressure Peff (Torr) at the nozzle exit plane calculated by eq 13 as a function of the He stagnation pressure P0 (Torr). The calculation is based upon total throughput measurements. The solid line is a best fit to the calculated values.
collision partners. The model was applied to measured terminal energies of Xe accelerated in He beams up to 11.5 eV and C60 accelerated in He beams up to 56 eV. Good agreement between the basic model calculation and experiment is obtained and is further improved by modifications related to capillary nozzle effects. The remarkable agreement demonstrated between the model predictions and the C60 terminal energies for both the constant flux mode and the constant temperature mode is attributed to the applicability of the basic assumptions of hard sphere behavior of C60 in our experiments and the relatively weak coupling between the vibrational relaxation of the superhot C60 molecule and the collisional acceleration dynamics. Acknowledgment. This research was supported by the Israel Science Foundation (ISF) administered by the Israel Academy of Sciences and Humanities and in part by the fund for the promotion of research at Technion. References and Notes (1) Scoles, G., Ed. Atomic and Molecular Beam Methods; Oxford University Press: Oxford, 1988. (2) Pauly, H.; Toennies, J. P. Methods Exp. Phys. A 1968, 7, 227. (3) Anderson, J. B.; Andres, R. P.; Fenn, J. B. AdV. Chem. Phys. 1965, 10, 275. (4) Fluendy, M. A. D.; Lawley, K. P. Chemical Applications of Molecular Beam Scattering; Wiley: New York, 1973. (5) Milne, T. A.; Greene, F. T. AdV. High Temp. Chem. 1969, 2, 107. (6) Kolodney, E.; Amirav, A. Chem. Phys. 1983, 82, 269. (7) Amirav, A. Comments At. Mol. Phys. 1990, 24, 187. (8) Barker, J. A.; Auerbach, D. J. Surf. Sci. Rep. 1985, 4, 1. (9) Buck, U.; Mattera, L.; Past, A.; Haaks, D. Chem. Phys. Lett. 1979, 62, 562. (10) (a) Sheen, S. H.; Dimoplow, G.; Parks, E. K.; Wexler, S. J. Chem. Phys. 1978, 68, 4950. (b) Amirav, A.; Cardillo, M. J.; Trevor, P. L.; Lim, C.; Tully, J. C. J. Chem. Phys. 1987, 87, 1796. (11) Saecker, M. E.; Govoni, S. T.; Kowalski, D. V.; King, M. E.; Nathanson, Science 1991, 252, 1421.
Tsipinyuk et al. (12) Rettner, C. T.; Barker, J. A.; Bethune, D. S. Phys. ReV. Lett. 1991, 67, 2183. (13) Danon, A.; Vardi, A.; Amirav, A. J. Chem. Phys. 1990, 93, 7506. (14) Campargue, R.; Lebehot, A. Rarefied Gas Dynamics; Proceedings of the 9th International Symposium; 1974; p CII-1. (15) Campargue, R. J. Phys. Chem. 1984, 88, 4466. (16) Danon, A.; Vardi, A.; Amirav, A. Phys. ReV. Lett. 1990, 65, 2038. (17) Kolodney, E.; Amirav, A.; Elber, R.; Gerber, R. B. Chem. Phys. Lett. 1985, 113, 303. (18) Kolodney, E.; Amirav, A.; Elber, R.; Gerber, R. B. Chem. Phys. Lett. 1984, 111, 366. (19) Danon, A.; Kolodney, E.; Amirav, A. Surf. Sci. 1988, 193, 132. (20) Kolodney, E.; Powers, P. S.; Hodgson, L.; Reisler, H.; Wittig, C. J. Chem. Phys. 1991, 94, 2330. (21) Danon, A.; Amirav, A. J. Phys. Chem. 1989, 93, 5549. (22) Dagan, S.; Danon, A.; Amirav, A. Int. J. Mass Spectrom Ion Processes 1992, 113, 157. (23) Budrevich, A.; Tsipinyuk, B.; Kolodney, E. Chem. Phys. Lett. 1995, 234, 253. (24) Budrevich, A.; Tsipinyuk, B.; Kolodney, E. Submitted to Phys. ReV. Lett. (25) Miller, D. R.; Andres, R. P. Rarefied Gas Dynamics; Proceedings of the 9th International Symposium; 1969; Vol. II, p 1385. (26) Schwartz, M.; Andres, R. P. Rarefied Gas Dynamics; Proceedings of the 10th International Symposium; 1977; Vol. 51, Part I, Progress in Astronautics and Aeronautics, p 135. (27) Takahashi, N.; Moriya, T.; Teshima, K. Rarefied Gas Dynamics; Proceedings of the 13th International Symposium; 1985; p 939. (28) Cattolica, R. J.; Gallagher, R. J.; Anderson, J. B.; Talbot, L. AIAA J. 1979, 17, 344. (29) Chatwani, A.; Fiebig, M. Rarefied Gas Dynamics; Proceedings of the 12th International Symposium; 1988; Vol. II, p 785. (30) Miller, D. R. In Atomic and Molecular Beam Methods; Scoles, G., Ed.; Chapter 2. (31) Danon, A.; Amirav, A. ReV. Sci. Instrum. 1987, 58, 1724. (32) Kolodney, E.; Tsipinyuk, B.; Budrevich, A. J. Chem. Phys. 1994, 100, 8542. (33) See for example: Thompson, M. W. Defects and Radiation Damage in Metals; Cambridge University Press: Cambridge, 1969. (34) Ashkenas, H.; Sherman, F. S. Rarefied Gas Dynamics; Proceedings of the 4th International Symposium; 1964; Vol. II, p 84. (35) (a) Cook, G. A. Argon, Helium, and Rare Gases; Wiley Interscience: New York, 1961; Vol. I. (b) Samsonov, G. V. Properties of the Elements; Metallurgy: Moscow, 1976; Vol. I. (36) Danielson, L. J.; Keil, M. J. Chem. Phys. 1988, 88, 851. (37) Pirani, F.; Vacchiocattiri, F. J. Chem. Phys. 1977, 56, 372. (38) The value of 6-8 eV is achieved only under extreme stagnation conditions (P0, T0). Under usual expansion conditions (T0 ) 1000-1800 K, P0 ) 300-2000 Torr), vibrational energy losses are 3-6 eV. The vibrational thermometry method used is based on measurements of the massspectrometric electron energy dependent C+ 58 appearance curve for C60 at different temperatures (T0 ) 1190-1875 K), corresponding to an average vibrational energy content of 10-20 eV. (Kolodney, E.; Tsipinyuk, B.; Budrevich, A. J. Chem. Phys. 1995, 102, 9263.) This method was recently applied by us for studies of the vibrational relaxation of superhot C60 in supersonic beams and the vibrational excitation of C60 scattered from surfaces. (Tsipinyuk, B.; Budrevich, A.; Kolodney, E. In Preparation). (39) Dresselhaus, M. S.; Dresselhaus, G.; Ekland, P. C. J. Mater. Res. 1993, 8, 2054. (40) Murphy, H. R.; Miller, D. R. J. Phys. Chem. 1984, 88, 4474. (41) Campargue, R. Rarefied Gas Dynamics; Proceedings of the 4th International Symposium; 1964; Vol. II, p 279. (42) The empirical density function used is of the type n(y) ) A/(y + 1)2. The value of A ) 0.584 was chosen in such a way that the two n(y) functions (the new one and eqs 7 and 8) will have the same value at y ) 1. When plotting both functions, one can see that indeed the new n(y) qualitatively describes the two major effects associated with capillary nozzle expansion, namely a slightly lower density as compared with orifice nozzle up to one nozzle diameter (y ) 1) and a factor of two higher density downstream. (43) O’Hanlon, J. F. A User’s Guide to Vacuum Technology; John Wiley & Sons: New York, 1989.
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