Determination of hydrogen cluster velocities and comparison with numerical calculations
Alexander Täschner, Esperanza Köhler, Hans-Werner Ortjohann, Alfons Khoukaz
aa r X i v : . [ phy s i c s . a t m - c l u s ] J a n Determination of hydrogen cluster velocities and comparison withnumerical calculations
A. T¨aschner, a) E. K¨ohler, H.-W. Ortjohann, and A. Khoukaz Institut f¨ur Kernphysik, Westf¨alische Wilhelms-Universit¨at M¨unster, D-48149 M¨unster,Germany (Dated: 30 August 2018)
The use of powerful hydrogen cluster jet targets in storage ring experiments led to the need of precise dataon the mean cluster velocity as function of the stagnation temperature and pressure for the determinationof the volume density of the target beams. For this purpose a large data set of hydrogen cluster velocitydistributions and mean velocities was measured at a high density hydrogen cluster jet target using atrumpet shaped nozzle. The measurements have been performed at pressures above and below thecritical pressure and for a broad range of temperatures relevant for target operation, e.g., at storagering experiments. The used experimental method is described which allows for the velocity measurementof single clusters using a time-of-flight technique. Since this method is rather time-consuming and thesemeasurements are typically interfering negatively with storage ring experiments, a method for a precisecalculation of these mean velocities was needed. For this, the determined mean cluster velocities arecompared with model calculations based on an isentropic one-dimensional van der Waals gas. Basedon the obtained data and the presented numerical calculations, a new method has been developed whichallows to predict the mean cluster velocities with an accuracy of about 5%. For this two cut-off parametersdefining positions inside the nozzle are introduced, which can be determined for a given nozzle by onlytwo velocity measurements.PACS numbers: 47.40.Ki, 05.70.Ce, 36.40.-cKeywords: hydrogen; molecular clusters; velocity; time-of-flight method; Laval nozzle
I. INTRODUCTION
Cluster beams have been studied in the last centuryextensively with respect to their physical and chemicalproperties and even today the interest in technologicalapplications is increasing rapidly . Prominent exam-ples are the use for cluster beam deposition, cluster im-pact lithography, and the application as target beamsin, e.g., storage ring experiments which has started onlyin the last few decades. For applications in hadronphysics experiments or in experiments with high intenselaser beams, the most important advantage is that theyprovide a pure target material inside a vacuum chamberwith densities in the range between gas beams and thesolid state targets. Cluster beams consisting of particleswith sizes from the nanometer to the micrometer scalepropagate through vacuum with almost no increase oftheir angular spread, so that it is possible to providea spatially well defined interaction zone for, e.g., aparticle beam in a storage ring or a laser beam.In hadron physics experiments at a storage ring thecluster beams are typically produced by expansion ofgaseous materials in Laval type nozzles. Althoughsuch targets can be operated in principal with all kindof gaseous materials ranging from hydrogen to, e.g.,xenon, the use of hydrogen is of special interest as ef-fective proton target for the investigation of elementaryreactions. Examples for experimental facilities usingsuch hydrogen cluster beams as internal targets at astorage ring are the ANKE experiment and the formerCOSY-11 experiment, both situated at the COSY accelerator of the Forschungszentrum J¨ulich. For the a) Author to whom correspondence should be addressed. Elec-tronic mail: [email protected] established internal target experiments mainly the den-sity and the purity of the cluster beams were important,but for the design of new experimental facilities whichcan be operated with event rates increased by one ortwo orders of magnitude, the precise knowledge of mi-croscopic properties, like velocity or mass distributions,or time structure, became of high importance. Thisinformation is especially important for the simulationof the interaction between intense pulsed laser beamsand cluster beams. An example for a future experi-ment at a storage ring is the 4 π detector PANDA atthe planned accelerator centrum FAIR in Darmstadt(Germany). For this experiment a cluster jet targethas been developed at the University of M¨unster wherethe number of target atoms per unit area is above10 atoms/cm at a distance of 2 . µ m. During the expansion ofthe fluid through the nozzle into a first vacuum chamberclusters are produced. Directly behind the nozzle theshape of the jet beam, consisting of both clusters and agas beam, is determined by the shape of the divergentpart of the production nozzle. In order to prepare a welldefined cluster beam for experiments and to suppressthe disturbing residual gas background from the gasbeam, a set of two skimmers is used to separate differ-ential pumping stages. The second skimmer, which isdenoted in the following as collimator, determines thefinal shape and size of the cluster beam at all furthervacuum stages, and especially at the interaction pointwith the beam of the storage ring in the scattering de Lavalnozzle(15 K – 40 K) skimmergas + clusters clusterjetH a r –13b a r Ø (cid:1)
20 µm collimator
FIG. 1. Schematic view of the principle of operation of acluster jet source which produces clusters from hydrogengas in front of the Laval nozzle. chamber. For more details see Ref. 6. A schematicsketch of this setup operated with gaseous hydrogen isshown in Fig. 1.For an optimized use as internal target in storage ringexperiments, e.g., for hadron physics experiments, it isessential to determine and to adjust the target thickness n T , the number of target atoms per unit area. With theknowledge of the thickness the luminosity L = f N C n T of the internal experiment can be calculated (see, e.g.,Ref. 8), where f is the revolution frequency and N C the number of circulating particles in the storage ring.Given the cross section σ of a specific reaction betweentarget and storage ring beam particles, the mean rate˙ N R = σ L of these reactions can be determined. Thisdetermination is especially important for adjusting thetarget thickness to achieve a desired reaction rate andto estimate the rate of background events. Using aCartesian coordinate system, where the cluster beampropagates along the z axis and the stored beam alongthe x axis and assuming that the transverse width ofthe stored beam is negligible compared to the size ofthe cluster beam, it is possible to calculate the targetthickness n T in units of number of atoms per squarecentimeter at a specific distance z behind the nozzle di-rectly from the volume density distribution ρ ( x, y, z ) : n T = N A M a Z + ∞−∞ ρ ( x, y, z ) d x , (1)where M a is the molar mass of the gas atoms and N A the Avogadro constant. In case of the cluster jet targetthe thickness can be measured by inserting movablerods into the cluster jet. At the cluster target prototypefor the PANDA experiment such rods are mounted in ascattering chamber located approximately two metersbehind the nozzle (Fig. 2). Here the rod diameter of d = 1 mm is small compared to the size of the clus-ter jet which is typically in the order of about 10 mm.Clusters colliding with these rods are stopped and leadby evaporation to an increase of the vacuum pressurein this chamber. In Fig. 3 a typical measurement ofthe vacuum pressure is presented, where the pressureincrease is plotted as a function of the rod position.With such kind of measurements the size as well as theposition of the cluster jet within the vacuum stage canbe determined easily. Moreover, assuming a specificvolume density distribution ρ ( x, y, z ) this pressureprofile can be described by the following equation : p ( x ) = p b + u R TS M x − x + d/ Z x − x − d/ d x ′ + ∞ Z −∞ d y ′ ρ ( x ′ , y ′ , z ) . (2) clusterjet x y z FIG. 2. Scattering chamber with moveable rods used tomeasure the target thickness.
In this equation p b is the background pressure, x thecenter position of the cluster jet, R the universal gasconstant, M the molar mass of the gas, S the knownpumping speed of the used pumping system, and u the mean velocity of the clusters. Note that thevelocity u depends on the stagnation condition, i.e., thetemperature T and the pressure p , as well as on thenozzle geometry. However, for constant numbers for p and T the velocity u is constant. Therefore, in order tocalculate the target thickness in a first step the targetdensity distribution ρ ( x ′ , y ′ , z ) has to be adjusted todescribe the relative shape of the measured pressureprofile. For an absolute target density determinationthe mean velocity u has to be known. In order to min-imize the uncertainty of the extracted volume density,the mean velocity has to be measured with an accuracywhich does not exceed the uncertainty of the pressureand of the pumping speed. At the presented setup bothuncertainties amount to about 10%.Since the cluster beams from the described clus-ter target sources are optimized for highest volumedensities, the used nozzles differ significantly both inshape and size from the ones commonly used by groupsspecialized in the investigation of velocity and massdistributions, e.g., work on hydrogen clusters reported v a c uu m p r e ss u r e / ( − m ba r) position of rod / mm FIG. 3. Example of a pressure profile measured in thescattering chamber. in Ref. 9. In the cited work a pin hole nozzle with aminimum diameter of 5 µ m was used, whereas for theproduction of the cluster beams described here trumpedshaped nozzle geometries with minimum diameters ofabout 20 µ m are used. Both factors can significantlychange the velocity distribution of the clusters in thegenerated beam and therefore new systematic studiesof these cluster target beams had to be performed.The mean velocity of the hydrogen clusters producedwith a similar trumped shaped nozzle with a throatdiameter of 37 µ m was already measured at the targetfor the E835 experiment at FERMILAB and it wasfound, that it can be described adequately by the max-imum local velocity u max of a perfect gas accelerated inan isentropic expansion through a convergent-divergentnozzle : u max = r κκ − R T M . (3)In this equation T is the gas temperature at the inletof the nozzle and κ is the adiabatic index. Thesemeasurements were done at a pressure below 8 barand temperatures between 15 K and 40 K where thehydrogen is in the gas phase before entering the nozzle.However, later optimization studies on hydrogen clus-ter targets showed that a significant performanceincrease is possible, i.e., an increase of the achievablemaximum target thickness by orders of magnitude. Oneprerequisite for this is that the target is operated withhydrogen being in the liquid phase before entering thenozzle. Since it is known from previous measurements,e.g., Ref. 9, that the phase transition from the gasphase into the liquid phase has a significant impacton the cluster velocity, precise measurements on thevelocity distributions and mean velocities as well as acomparison with the situation obtained with hydrogenbeing in the gas phase before entering the nozzle werestrongly needed. Based on this data verifications andoptimizations of calculations will be possible.For this reason a dedicated time-of-flight systemwas designed and installed at the M¨unster cluster jettarget. Detailed studies on the velocity distributions asfunction of the operating parameters were performedwhich is presented in the first part of this work.Although the time-of-flight system enabled for aprecise determination of the velocity distributions andwith this for the calculation of the mean velocities, themeasurement time of several hours makes it impracticalto use this system regularly for the volume density de-termination described above. This is especially true foroptimization studies where the temperature and pres-sure at the nozzle inlet is changed very often. Thereforeit was essential to be able to calculate the mean clustervelocity as function of the stagnation conditions in thetypical operation region with a precision, as discussedbefore, below 10%. Many groups, e.g., Ref. 9, Ref. 13,and Ref. 14 have measured mean velocities of clustersand extracted fluid properties like the temperatureof the cluster at the end of the expansion based ondifferent equations of state. Ref. 15, for example, usedan equation of state to produce a theoretical predictionfor the mean velocity by assuming a constant finaltemperature. The values predicted with this methoddeviate from the measurements presented in the same work by about 20%–40% for the data points measuredat temperatures below the boiling point. Therefore anew method had to be developed to allow for moreaccurate predictions at the discussed stagnation condi-tions. In the second part of this work such a method ispresented, introducing two parameters which had to bedetermined only once by a fit to the measured data. Inthe operation region of the investigated cluster sourcethis techniques provides precise predictions with anaverage absolute deviation of only about 5% comparedto the measured mean cluster velocities. This appliesboth to the regions of liquid and of gaseous hydrogenin front of the nozzle. II. EXPERIMENTAL SETUP
In Fig. 4 the schematic view of the utilized time-of-flight setup is shown. Clusters produced in the clustersource are ionized by a pulsed electron gun mounted ata short distance behind the collimator. This electrongun is operated in a pulsed mode with a repetitionrate of about 20 Hz and a pulse width of approximately20 µ s. The current of the electron beam is reduced insuch a way that for each pulse no more than a singleionized cluster is registered. The ionized clusters itselfare detected by a Channeltron after a flight path of4 . ± .
02 m. Due to this long distance and pulsewidths in the microsecond time scale, the observedtime-of-flight times being in the range between 4 msand 26 ms can be obtained with high resolution. Thestart and stop pulses are detected by a timer systembased on a MC9S08QG8 micro controller by FreescaleSemiconductor and the time difference is send to acomputer. A detailed description of the used softwarefor the micro controller can be found in Ref. 7.In order to extract time-of-flight information withhigh resolution the complete setup has to be calibratedwith respect to possible timing offsets introduced bythe pulsed electron gun device and the cluster detectionsystem. For this purpose the calibration source forions with known kinetic energy is used (Fig. 5). Thesource consists of two coaxial cylinders. The outercylinder is electrically grounded while the inner cylinderis connected to a voltage source providing a potential ofup to 4 kV. The calibration source is mounted in sucha way, that the cylinder axis is perpendicular to theaxis of the incoming cluster beam, so that the clusterscan enter through a hole with a diameter of 10 mm.In the inner cylinder the clusters are stopped by a no zz l e sk i mm e r c o lli m a t o r e - pulsede - -gun PCcluster jett Start t Stop t TOF
FIG. 4. Schematic view of the time-of-flight setup for thevelocity determination of single clusters. clusterjet e-ionsgroundedpos . voltageinsulator
FIG. 5. Cross section of the calibration source used toproduce hydrogen ions with known kinetic energies. wedge shaped plate, evaporate and are converted intohydrogen gas. This gas is ionized by the pulsed electronbeam entering along the axis of the two cylinders. Thepotential difference between the cylinders accelerate theproduced ions to a known kinetic energy while theyare extracted through a 2 mm hole along the clusterbeam axis. The ions leaving this unit drift towardsthe detection system which consists of an array of agrounded entrance orifice followed by a Channeltron.Here the Channeltron input is set on a negative po-tential of 2 . ∼ µ s is a direct measure of the timing offset causedby the electronics. The three other peaks correspond todifferent hydrogen ions, namely H + , H +2 , and H +3 . Thetime offset and the length of the flight path between the g H + H H d N / d t / µ s − time−of−flight / µs100 V data FIG. 6. Example of a time-of-flight distribution using thecalibration source. electron gun and the Channeltron could be extracted bymeasuring the mean time-of-flight for the different ionsas function of the acceleration voltage. For these mea-surements the electron gun was operated at a repetitionrate of about 25 kHz and a pulse duration of about 2 µ s.With this calibration setup a time resolution of about3 µ s was reached which is predominantly given by thepulse duration of the electron gun. III. VELOCITY DISTRIBUTIONS OF CLUSTERS
Using the presented time-of-flight setup the veloc-ity distributions of hydrogen clusters produced in theM¨unster cluster jet target setup were measured usinga nozzle with a minimum diameter of 28 µ m. For thesemeasurements the pulse duration of the electron gunwas increased to 20 µ s, so that the time resolutionincreases to about 21 µ s which is still very precisecompared to the measured standard deviation of thetypical velocity destributions of the clusters of severalhundred to thousand microseconds.In Fig. 7 and 8 the measured distributions of thecluster velocity are shown. In the first figure a constantpressure of 8 bar was used in front of the nozzle andfor the second figure 14 bar were applied. In bothfigures the distributions at different fluid temperaturesfrom 20 K up to 50 K are displayed. The distributions
0 200 400 600 800 1000 1200 t e m pe r a t u r e / K r e l a t i v e f r equen cy velocity / (m/s) 250 350 450 550 650 t e m pe r a t u r e / K r e l a t i v e f r equen cy velocity / (m/s) FIG. 7. Distributions of the cluster velocities as functionof the inlet temperature of the fluid at a constant pressurebefore the nozzle of 8 bar. The inlayed graph shows a zoominto a small temperature range.
0 200 400 600 800 1000 1200 t e m pe r a t u r e / K r e l a t i v e f r equen cy velocity / (m/s) 450 550 650 750 850 t e m pe r a t u r e / K r e l a t i v e f r equen cy velocity / (m/s) 250 350 450 550 650 t e m pe r a t u r e / K r e l a t i v e f r equen cy velocity / (m/s) FIG. 8. Distributions of the cluster velocities as functionof the inlet temperature of the fluid at a constant pressurebefore the nozzle of 14 bar. The inlayed graphs show a zoominto two small temperature ranges. are scaled in such a way that the total area of eachspectrum is the same in the respective figure.In case of Fig. 8 where the data at 14 bar are shownthe distributions above the boiling point of around34 K are relatively sharp with a standard deviationof about 10 m/s and have a negative skew. At thephase transition between gas and liquid a double peakstructure is visible with a narrow peak at higher meanvelocity on top of a broad peak with a lower meanvelocity. The inlayed graph shows the developmentof this structure in the temperature range between33 K and 34 K. At a temperature of around 33 . . . . IV. MEAN CLUSTER VELOCITIES
As mentioned before, the precise knowledge of themean cluster velocity u is needed for the estimationof the volume density of the cluster beam. Since it isnot feasible to measure this quantity for each possiblestagnation condition within the operation region of thecluster target, a method is needed to predict thesevalues with an accuracy below about 10%. In contrastto other publications, e.g., Ref. 9, the main focus lieshere on the description of the mean cluster velocity ofthe complete velocity distribution which in our case canbe directly calculated from the measured velocities ofthe single clusters. We therefore will not quote themean velocity of the two peaks observed in the phasetransition region separately. In our case, where thesetwo peaks completely overlap there is also no modelindependent way to extract this information. A. Experimental results
To summarize the above findings three examples forthe measured time-of-flight distributions of hydrogenclusters are presented in Fig. 9. The distributionswere measured at a constant hydrogen pressure of 8 barat the nozzle inlet, but at different temperatures of20 K, 29 . ∼ µ s which increases up to 1800 µ s at 20 K. Nearthe boiling temperature of 30 K a double peak structure d N / d t / ( µ s ) − d N / d t / ( µ s ) − T = 50 K T = 29.8 K T = 20 K d N . / d t / ( µ s ) − time−of−flight / µs FIG. 9. Distribution of the time-of-flight of hydrogenclusters produced at three different hydrogen temperaturesat the nozzle inlet and with the same inlet pressure of 8 bar. v e l o c i t y / ( m / s ) temperature / Kdata, 8 bar isobarperfect gas FIG. 10. Mean velocity of hydrogen clusters as function ofthe temperature at the nozzle inlet and with the same inletpressure of 8 bar. The solid line shows the maximum gasvelocity of perfect gas according to Eq. (3). was observed with a small peak at higher velocityon top of a broad peak with a lower mean velocityindicating the different cluster production mechanismfrom the two coexisting phases. This dependance ofthe production process on the hydrogen phase state isalso visible in Fig. 10 where the mean cluster velocityis plotted as function of the hydrogen temperature atthe inlet of the nozzle at a constant inlet pressure of8 bar.Above the boiling point the data can be describedadequately by the maximum gas velocity of a perfectgas (Eq. (3)), which agrees well with the observationspresented in Ref. 10 taken at lower inlet pressure ofbelow 5 bar. However, the data below the boiling pointdeviate from the calculated ones by up to a factor ofthree, which is in agreement with the results presentedin Ref. 9.A collection of measured mean velocities for differentisobars is presented in Fig. 11. Since the boiling pointshifts towards higher temperatures for higher stagna-tion pressures the transition between high velocity tolow velocity shifts accordingly. For comparison, in thisgraph the data presented in Ref. 9 is also displayed. Inthe temperature region where two peaks are observed,only the dominating peak (Peak 4 in Ref. 9) is shownfor better comparison, since we discuss here only themean velocity of the clusters. As mentioned before, ourdata are in good agreement with the results of Ref. 9although a pin hole nozzle with a minimum diameterof 5 µ m was used there. A more detailed discussion ofthis data is given in Sec. IV B.In Ref. 15 it is shown that the cluster velocity canbe predicted by calculating the fluid velocity basedon isentropic expansion from the stagnation temper-ature down to the temperature of the triple point.However, the calculations presented there deviate fromthe measured data in the region below the boilingpoint by about 20%–40%, which is too large for theapplication discussed here. The data presented inRef. 15 and 9 indicate that the temperature at the endof the expansion, the so called terminal temperature, isdependent on the stagnation conditions and is found tobe always above the triple point temperature. Basedon this knowledge and the fact that the used nozzle in v e l o c i t y / ( m / s ) temperature / K8 bar12 bar14 bar17 bar20 bar, Knuth 1995perfect gas FIG. 11. Measured mean hydrogen cluster velocities asfunction of the stagnation temperature for different constantstagnation pressures. For comparison the measurementpresented in Ref. 9 obtained with a pinhole nozzle and astagnation pressure of 20 bar is shown. The solid line iscalculated from Eq. (3) assuming a perfect gas. our case is comparably long, one can expect that theterminal temperature is already reached inside the noz-zle. Therefore, it is essential to calculate the velocityof the fluid inside the nozzle as function of the distancefrom the nozzle throat. As will be discussed below itis possible to introduce two new parameters for thesecalculations which can be fixed by only two velocitymeasurements. This enables the precise prediction ofthe cluster velocities u ( p , T ).Since above the boiling point the data are alreadydescribed well using the simple assumption of a perfectgas, the method for calculating the local velocity insidethe nozzle is explained first using this simple model.In a next step the calculations will be done with anequation of state which can describe a fluid with agaseous and a liquid phase. B. Model calculations
In order to calculate the position dependent proper-ties of the hydrogen fluid inside the nozzle, a stationaryquasi-one-dimensional model is used. The details ofthese calculations, which are based on the dimensionsof the used Laval nozzle shown in Fig. 12, are presentedin Appendix A.The applied method can be used with any equationof state. The simplest model is the perfect gas with thefollowing equation of state : p = ρ R s T , (4)where p is the pressure and R s = R/M is the specificgas constant. In contrast to the ideal gas, which hasthe same equation of state, the specific heat at constantpressure or volume is constant in the case of the perfectgas. In Fig. 13 the calculated local velocity is shownfor a perfect gas as function of the position inside thenozzle. The two curves correspond to two differentstagnation temperatures at the nozzle inlets, namely,25 K and 50 K. For both curves a stagnation pressure of10 bar was assumed. It is obvious that a few millimeters
181 8 R ⌀
30 90 ° ⌀ .
028 7 ° ⌀ FIG. 12. Cross section of the used Laval nozzle manufac-tured in the CERN workshop from copper. behind the nozzle throat the velocity is almost constantand at its maximum value. The limit of the localvelocity u max is reached if the ratio A ( T z ) /A ∗ betweenthe local area and the area of the throat approachesinfinity and can be expressed by Eq. (3). The twohorizontal dashed lines in Fig. 13 indicate the valueof this maximum velocity u max for the two stagnationtemperatures.Since in case of the perfect gas the local velocityis already almost constant a few millimeters behindthe nozzle throat, the maximum velocity u max can beused as a first order estimate for the mean clustervelocity. In Fig. 11 a corresponding model calculationis compared to measurements with different stagnationpressures. Above a certain temperature, which changesdepending on the stagnation pressure, the model agreeswell with the measured data, but for lower temper-atures the measured velocities are up to a factor ofthree lower than the model predictions. Comparing = 10 bar, T = 50 Kp = 10 bar, T = 25 K v e l o c i t y o f t he ga s / ( m / s ) position in nozzle / mm 0 100 200 300 400 500 600 700 800 900−40 −20 0 20 40 60 80 100 p = ba r , T = K p = ba r , T = K v e l o c i t y o f t he ga s / ( m / s ) position in nozzle / µm FIG. 13. Local velocity of the perfect gas as function of theposition inside the CERN nozzle for two different stagnationconditions. The two dashed horizontal lines correspond tothe maximum gas velocity. v apo r p r e ss u r e c u r v e m e l t i ng c u r v e s ub li m a t i on c u r v e Münster typecluster jet targetsE835PROMICE/WASA p r e ss u r e / ba r temperature / K FIG. 14. Phase diagram of hydrogen with the triple point(TP), the critical point (CP), the vapor pressure curve,the melting curve, and the sublimation curve based onRef. 18 and Ref. 19. The typical operation region of theconventional cluster jet targets of the PROMICE/WASA and the E835 experiments are indicated by a dashed lineand for the M¨unster type target with a dashed dotted line.The point where the highest target thickness was obtainedwith the M¨unster type target is indicated by a cross. these specific temperatures with the phase diagram ofhydrogen shown in Fig. 14 it can be seen, that forpressures below the critical pressure these temperaturescan be identified as the pressure dependent boilingtemperatures of normal hydrogen.As mentioned above, the perfect gas equation ofstate was only used to explain the method used forcalculating the local velocity inside the nozzle. Thesimplest model which describes a fluid with both agaseous and a liquid phase is the van der Waals gaswith the following equation of state: p = R s Tv − b ′ − a ′ v , (5)with the specific volume v = 1 /ρ and the two constants a ′ and b ′ which can be calculated from the pressureand temperature at the critical point of the used gas(see, e.g., Ref. 21). In case of hydrogen a criticaltemperature of T c = 33 .
19 K and a critical pressureof p c = 13 .
15 bar were used, which were taken fromRef. 18. The detailed description of the method usedto calculate the local properties for the van der Waalsgas is given in Appendix B.In Fig. 15 the calculated local velocity is presentedas function of the position inside the nozzle for twodifferent combinations of stagnation pressure and tem-perature. The dashed lines show the solutions for theperfect gas and the solid line the solutions for the vander Waals equation of state. It is clearly visible that thevalue of the local velocity does not saturate for the vander Waals model in contrast to the values calculatedfor the perfect gas. Since it was observed by othergroups, e.g., Ref. 15 and Ref. 9, that the temperature atthe end of the expansion path of the cluster formationis strongly dependent on the stagnation conditions,this terminal temperature cannot be used to predictprecisely the mean velocity of the clusters. Instead,we chose here the position inside the nozzle as newparameter to produce such a prediction. This choice v an de r W aa l s ga s p = 10 bar, T = 50 Kperfect gas v an de r W aa l s ga s p = 10 bar, T = 25 K v e l o c i t y o f t he ga s / ( m / s ) position in nozzle / mm FIG. 15. Local velocity of the fluid based on the van derWaals model (solid lines) and the perfect gas (dashed line)as function of the position inside the CERN nozzle for twodifferent stagnation conditions. can be motivated by the production process of clusterswhich are formed by condensation from a gas. Inthis case it is obvious that at a certain point insidethe nozzle the mean number of collisions between theclusters and the surrounding molecules is so low andthe mass of the clusters so high that further collisionsdo not change the mean velocity anymore.In Fig. 16–18 the measured mean cluster velocitiesfor different isobars at 8 bar, 14 bar, and 17 bar arecompared to calculated local velocities at three differentpositions of 0 . . . z l ≈ . z g ≈ . u vdW ( p , T , z ) is calculated. The mean cluster v e l o c i t y / ( m / s ) temperature / Kmean cluster velocityvan der Waals gas, z = 0.5 mmvan der Waals gas, z = 1.0 mmvan der Waals gas, z = 2.0 mmperfect gas FIG. 16. Mean cluster velocity as function of the stagnationtemperature for an isobar at 8 bar. The solid line iscalculated assuming a perfect gas whereas the other linesrepresent the local velocity at three different positions of0 . v e l o c i t y / ( m / s ) temperature / Kmean cluster velocityvan der Waals gas, z = 0.5 mmvan der Waals gas, z = 1.0 mmvan der Waals gas, z = 2.0 mmperfect gas FIG. 17. Mean cluster velocity as function of the stagnationtemperature for an isobar at 14 bar. The solid line iscalculated assuming a perfect gas whereas the other linesrepresent the local velocity at three different positions of0 . velocity can then be predicted using the followingequation: u C = ( u vdW ( p , T , z l ) for T < T tr ( p ) u vdW ( p , T , z g ) for T ≥ T tr ( p ) . (6)The cluster production mechanism differs depending onthe phase state of the fluid in front of the nozzle. Incase of a gas at the inlet the clusters are formed bycondensation and in case of a liquid from breakup andevaporation. Therefore, it is plausible that for the twomechanisms, having a completely different expansionpath as indicated in Fig. 27, one has to allow fortwo different values for the position z . This is donein the above equation by the two parameters z l and z g . For pressures p below the critical pressure of p c = 13 .
15 bar, the specific temperature T tr ( p ), whichis used to switch between the two velocity regimes,is the pressure dependent boiling temperature. Forpressures above this point the phase transition betweengas and liquid is continuous, so that there is no explicit v e l o c i t y / ( m / s ) temperature / Kmean cluster velocityvan der Waals gas, z = 0.5 mmvan der Waals gas, z = 1.0 mmvan der Waals gas, z = 2.0 mmperfect gas FIG. 18. Mean cluster velocity as function of the stagnationtemperature for an isobar at 17 bar. The solid line iscalculated assuming a perfect gas whereas the other linesrepresent the local velocity at three different positions of0 .
20 40 60 80 100 120 140 160 0 5 10 15 20 25 30 35 40 45 50 hea t c apa c i t y a t c on s t an t p r e ss u r e / ( J / m o l K ) temperature / K 8 bar17 bar 0 5 10 15 20 25 0 10 20 30 40 50P c p r e ss u r e / ba r temperature / Kboiling temperaturetransition temperature FIG. 19. Calculated heat capacity c p as function of thetemperature based on the van der Waals model for twodifferent isobars at 8 bar and at 17 bar. In the inlay figurethe transition temperature defined as the position of themaximal heat capacity is displayed together with the vaporpressure curve. boiling temperature. Nevertheless, for the method de-scribed here such a well defined temperature is needed.Obviously there are different choices possible, however,an approach described in Ref. 22 is well suited sincethe transition temperatures produced by this method,displayed in the inlay of Fig. 19, are an direct extensionof the vapor pressure curve to higher pressures. Thistransition temperature is defined as the temperaturewith the maximal heat capacity at constant pressure c p = ( ∂h/∂T ) p . In Fig. 19 the temperature dependenceof this heat capacity is displayed for one isobar belowand for one above the critical pressure. For pressuresbelow the critical pressure the boiling temperature isdirectly visible as a discontinuity of the heat capacity,whereas in case of pressures above the critical pressurethe heat capacity exhibits a clear maximum. Theposition parameters z l and z g are adjusted in such away that the deviation between the predicted veloc-ity u C and the measured mean cluster velocities areminimized. In case of the studied cluster jet targetthe best fit values of these position parameters were z l = 0 . ± .
014 mm and z g = 1 . ± .
20 mm. Withthese values the described method produces very pre-cise predictions for the observed mean cluster velocitieswith mean absolute deviation of about 5.1% betweenmeasured and predicted velocities. Since the precisionof this prediction is better than the above mentionedrequired precision of 10%, the use of more sophisticatedequations of state, which were used in the work of othergroups, e.g., Ref. 11, was not required. Furthermore,the excellent agreement between the measured data andthe predictions suggest that the influence of the transi-tion to the clustered phase on the used thermodynamicparameters, e.g., entropy, are fully represented by thechoice of the two position parameters.In Fig. 20 the measured data for different isobars arecompared to the values calculated using Eq. (6) showingthe good agreement for the different data sets. Asmentioned above the terminal temperature or densitycannot be used as a parameter for reaching the requiredprecision. In Fig. 21 the local temperatures inside thenozzle, calculated at the same positions as used for
200 400 600 800 1000 20 25 30 35 40 45 50 8 bar isobar v e l o c i t y / ( m / s ) temperature / Kdatafit 200 400 600 800 1000 14 bar isobardatafit 200 400 600 800 1000 17 bar isobardatafit FIG. 20. Comparison between the measured mean clustervelocities at different isobars with the prediction made usingEq. (6). the local velocities shown in Fig. 20, are presented.It is obvious that these terminal temperatures are notconstant and, therefore, cannot be used as parametersfor a velocity prediction here. The values for theterminal temperature range between 14 K and 18 K forstagnation temperatures below the transition tempera-ture and around 11 K above this temperature. This isin good agreement with the values presented in Ref. 9.In Fig. 22 the terminal densities are shown which werecalculated in the same way as the terminal tempera-tures. It is obvious that also this parameter cannotbe used to make sufficiently precise predictions since itchanges over an order of magnitude in the stagnationtemperature region below the transition temperature.In summary, with the proposed position parameters z l and z g a high predictive power of the model calculationis reached. In order to further investigate this obser-vation detailed studies with different nozzle geometriesare planned in the future.The relevance of the nozzle geometry itself might beillustrated by the comparison of the presented modelcalculations with results from Ref. 9 obtained with apinhole nozzle with a minimum diameter of 5 µ m. Sincethe exact geometry of this pinhole nozzle is not known,in Fig. 23 the measured data is compared to the values
10 11 12 13 14 15 16 17 18 20 25 30 35 40 45 50 t e r m i na l t e m pe r a t u r e / K stagnation temperature / K 8 bar isobar14 bar isobar17 bar isobar FIG. 21. Local temperature at different isobars calculatedat the same positions as the local velocities shown in Fig. 20. t e r m i na l den s i t y / ( m o l / l ) stagnation temperature / K 8 bar isobar14 bar isobar17 bar isobar FIG. 22. Local density at different isobars calculated at thesame positions as the local velocities shown in Fig. 20. calculated using Eq. (6) based on the fitted values forthe position parameters presented above. Obviouslythe velocities measured using a pinhole nozzle differsignificantly, i.e., up to 50%, from the calculated ones.This indicates the relevance of the nozzle geometry, e.g.,the length and the shape of the exit trumpet, on themean cluster velocity.
V. VOLUME FLOW THROUGH THE NOZZLE
Using the method described above, not only the localproperties but also global properties like the mass flowcan be calculated from the critical properties using thefollowing formula : ˙ m = ρ ∗ u ∗ A ∗ . (7)In Fig. 24 the measured volume flow towards the nozzleis shown as function of the temperature in front of thenozzle using a pressure of 17 bar. The volume flow wasmeasured directly in the gas supply line using a com-mercial mass flow meter. It is clearly visible that thecalculations based on the van der Waals model describe v e l o c i t y / ( m / s ) temperature / K20 bar, Knuth 1995fit to MCT data FIG. 23. Comparison of the data presented in Ref. 9,obtained with a pinhole nozzle and a stagnation pressureof 20 bar, and the prediction based on Eq. (6) using the fitparameters obtained from the data measured with the Lavalnozzle presented in Fig. 12. v o l u m e f l o w / ( l / m i n ) temperature / K datavan der Waals gasperfect gas FIG. 24. Volume flow through the nozzle as function of theinlet temperature at a constant inlet pressure of 17 bar. the data very well especially above a temperature ofaround 38 K whereas the calculations for the perfect gasfail to explain the data. The largest deviations betweenthe perfect gas model and the data are visible belowthe transition temperature of about 35 K. Contrary, aqualitatively good description is reached by the van derWaals although it is also found that the data at lowertemperatures are not fully described.A similar discrepancy between calculations basedon the van der Waals model and experimental databelow the transition temperature was also observed byother groups which studied the flow of water vapor. Adescription is given for example in Ref. 23 where thisobservation was explained as a local deviation from thethermal equilibrium caused by the finite evaporationrate of the liquid. Models using rate equations canbe used principally to calculate the flow in such acase, however, the achieved precision of the discussedmodel is already sufficient for the desired investigationspresented in this work.
VI. SUMMARY
For a precise measurement of the velocity of singlehydrogen clusters produced in the source of a highdensity cluster jet target a time-of-flight setup usinga pulsed electron gun was built up. A rich data samplefor mean hydrogen cluster velocities and velocity distri-butions are provided for different stagnation conditionsboth above and below the critical pressure.The mean values of the obtained cluster velocitydistributions were compared with model calculationsbased on both the equation of state of the perfect gasand of the van der Waals gas. It was found that a pre-cise prediction of the measured data is possible by thevan der Waals model if two cut-off position parametersare introduced for which the local velocities inside thenozzle are calculated. By adjusting these two parame-ters to the measured data, a precise prediction of themean cluster velocities is possible. In principle, thesetwo positions can be fixed by measuring one velocity ata temperature above and one below the boiling point.The average absolute deviation between the predictedvelocities and the measured mean cluster velocities are1found to be only about 5%. Therefore, the approachpresented in this work provides an excellent tool topredict the mean cluster velocities in a regime especiallyrelevant for high intense cluster jet beams. For aspecific nozzle, an essential parameter required, e.g.,for the determination of the absolute target thicknessvia the scanning rod method, can be provided withoutfurther measurements. In order to further investigatethe observed excellent predictive power of the positionparameters, a detailed study of the dependence of theseparameters on the nozzle geometry is planned.
ACKNOWLEDGMENTS
The authors would like to thank H. Orth for thevery inspiring and helpful discussions and H. Baumeis-ter and W. Hassenmeier for their support duringthe design of the target device. We are gratefulto M. Macri and J. Ritman for providing powerfulvacuum pumps. The work provided by the teamsof our mechanical and electronic workshops is verymuch appreciated and we thank them for the excel-lent manufacturing of the various components. Theresearch project was supported by BMBF (06MS253Iand 06MS9149I/05P09MMFP8), GSI F&E pro-gram (MSKHOU1012), EU/FP6 HADRONPHYSICS(506078), EU/FP7 HADRONPHYSICS2 (227431), andEU/FP7 HADRONPHYSICS3 (283286).
Appendix A: Calculation of local properties inside thenozzle
In order to calculate the position dependent prop-erties of the hydrogen fluid inside the nozzle, a sta-tionary quasi-one-dimensional model is used. Here thefluid properties are assumed to vary only along the z -axis, i.e., the symmetry axis of the nozzle, while theproperties are considered to be constant in a planeperpendicular to the jet beam axis. This model impliesthat the flow is inviscid and without wall friction,external forces, and heat transfer with the walls. Usingthese assumptions the flow has to be isentropic andthe following relation between the local cross sectionarea A and velocity u can be derived :d AA = (cid:0) Ma − (cid:1) d uu , (A1)where Ma = u/a is the Mach number, which is theratio between the local velocity u and the local speedof sound a . For this relation three different cases canbe discussed: • Ma <
1: The flow is called subsonic. An increaseof the local velocity ( du >
0) is directly correlatedwith a decrease of the local area ( dA < • Ma >
1: The flow is called supersonic. Anincrease of the local velocity ( du >
0) is directlycorrelated with an increase of the local area ( dA > • Ma = 1: The local velocity u ∗ is equal to thelocal speed of sound a ∗ . In this case the localarea is either at its maximum or minimum. Forpractical purposes only the case of minimal areais relevant.Based on these considerations the flow through theused Laval nozzle is assumed to be subsonic in frontof the nozzle throat ( z <
0) and supersonic behindit ( z > Ma ( z = 0) = 1). Inthe following text all properties at the position, wherethe Mach number equals unity, the so called criticalproperties, are marked with an asterisk (*). Thisassumption leads directly to the knowledge of the size ofthe critical area A ∗ which has to be equal to the areaof the nozzle throat A t = π r t , where r t is the innerradius of the nozzle at its throat. In case of the quasi-one-dimensional model the energy conservation can beexpressed by the following equation : h + u h + u , (A2)where h , are the specific enthalpies at two positionsinside the nozzle. In case of the nozzle flow the velocitybefore the nozzle is considered to be zero ( u = 0)so that the local velocity can be calculated from thefollowing formula: u ( z ) = p h − h ( z )) , (A3)where h = h ( T , ρ ) is the specific enthalpy before thenozzle and h ( z ) = h ( T z , ρ z ) is the local specific enthalpyat a position z inside the nozzle. Here T z denotesthe local temperature and ρ z the local density at thisposition. The stagnation density ρ can be calculatedfrom the stagnation pressure p and the stagnationtemperature T before the nozzle if the equation of stateof the fluid is known. Using this equation of state thespecific enthalpy h ( T, ρ ) and the specific entropy s ( T, ρ )can be calculated. Since the flow is considered to beisentropic the density ρ can be calculated directly fromthe temperature T by solving the equation s ( T, ρ ) = s ( T , ρ ) . (A4)Therefore, Eq. (A3) is only dependent on the localtemperature T z .In order to calculate the local velocity the followingsteps have to be considered:1. Calculate the radius r z = r ( z ) of the nozzle atdesired position z and from this the area A z = π r z at this position.2. Calculate the ratio A z /A t of the local area A z and the area of the nozzle throat.3. Search for the temperature T z which satisfies thefollowing equation: A z A t = A ( T z ) A ∗ . (A5)This temperature is the local temperature at thedesired position.24. Calculate the local velocity u z = u ( T z ).For the calculation of the local radius an analyticformula was derived from the dimensions of the usedLaval nozzle which is shown in Fig. 12. The searchfor the temperature which satisfies Eq. (A5) is mostcomplex. It is done by a C which uses the Brent Method for the rootfinding. The limits of the temperature interval aredependent on the flow type at the desired position.In front of the nozzle throat the flow is subsonic, sothat the temperature has to be between the stagnationtemperature T in front of the nozzle and the criticaltemperature T ∗ ( T ≥ T z ≥ T ∗ ). Behind the nozzlethroat the flow is supersonic and the temperature mustbe lower than the critical temperature T ∗ but largerthan zero. In these numerical calculations a minimumtemperature T min > : ρ u A = ρ u A , (A6)where ρ , are the local densities at the two positions1 and 2, u , the local velocities and A , the localareas. From this the ratio A ( T z ) /A ∗ from Eq. (A5)can directly be derived: A ( T z ) A ∗ = ρ ∗ u ∗ ρ ( T z ) u ( T z ) . (A7)In this equation ρ ∗ is the critical density and u ∗ thecritical velocity. The local density ρ ( T z ) and velocity u ( T z ) are calculated by solving Eq. (A4) and afterwardsusing Eq. (A3). Appendix B: Van der Waals equation of state
In case of the perfect gas equation of state (Eq. (4)) acertain density or specific volume can be calculated foreach temperature and pressure value. This is not thecase for the van der Waals equation of state (Eq. (5)).Depending on the temperature up to three differentvalues for the density lead to the same pressure value.This can be seen in Fig. 25 where the pressure is shownas function of the molar volume for different temper-atures. Furthermore, it is obvious that for certaintemperatures this equation of state exhibits a behaviorwhich contradicts physics laws: it can lead to negativevalues for the pressure as well as for certain ranges ofthe molar volume where the pressure increases when thevolume is increased. Both can be cured using the wellknown Maxwell construction shown in Fig. 26. Here thevan der Waals equation of state is replaced between thepoints P and P by a constant pressure, which is thevapor pressure at the given temperature. There aremultiple methods to find the temperature dependentpoints P and P . The most common one mentioned intext books is to estimate a pressure between P and P that leads to equally large areas S and S . Althoughthis method is optimal to teach the concept of theMaxwell construction it has many disadvantages when −20 0 20 40 60 80 0 0.05 0.1 0.15 0.2 0.25P c van der Waals gas p r e ss u r e / ba r molar volume / (l/mol) T = 70 K T = 50 K T = T c = 33.19 K T = 30 K T = 25 K 0 0.05 0.1 0.15 0.2 0.25 −20 0 20 40 60 80perfect gas p r e ss u r e / ba r molar volume / (l/mol) T = 70 K T = 50 K T = 33.19 K T = 30 K T = 25 K FIG. 25. Volume dependency of the pressure for selectedisotherms for the van der Waals model of normal hydrogen(left) and the perfect gas (right). used in a numeric implementation. For this purposeit is much more effective to use an equivalent methodusing the specific Gibbs free energy g and solving thefollowing system of equations (see, e.g., Ref. 21): g ( T, v ) = g ( T, v ) (B1) p ( T, v ) = p ( T, v ) . (B2)Since Eq. (5) diverges at v = b ′ the numeric solutionof this equation is very challenging. In this work theCONLES algorithm was used to find the solution ofthis constrained system of equations. Initial values werecalculated based on the approximative equations givenin Ref. 27. For the calculation of the specific Gibbs freeenergy and the specific entropy the equations given inRef. 28 were used which provide an excellent approachto the calculation of these quantities based on a givenequation of state and a chosen value for the specificheat capacity at constant pressure c p ( T ) of the idealhydrogen gas. Since the specific heat capacity has to becalculated rather often, the data, specified in Ref. 29 as −5 0 5 10 15 20 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4P c p r e ss u r e / ba r molar volumen / (l/mol) S S P P FIG. 26. Schematic usage of the Maxwell construction usedto determine the vapor pressure in the coexistence regionbetween the liquid and the gas phase. Para hydrogen Normal hydrogen l T l / K u l T l / K u l T l were chosenempirically and only the parameters u l were fitted to thedata given in Ref. 29. a summation, was fitted by the following formula, whichwas inspired by the work presented in Ref. 28 and 30: c p R s = 2 . N e X l =1 u l (cid:18) T l T (cid:19) exp( T l /T )(1 − exp( T l /T )) , (B3)with the parameters u l and T l given in Table I.In Fig. 27 the density of normal hydrogen is presentedas function of the temperature for different isobarsand two selected isentropes. The boundary of thecoexistence region, where the isobars are simple verticallines, is indicated by the thick solid line. It is obviousthat both isentropes cross this boundary so that thecalculation of the fluid properties has to be done alsoin the coexistence region. For these calculations a newparameter called quality x is useful, which is the ratioof the mass of the vapor phase m g and the total mass m = m g + m l of the vapor and the liquid phase (see,e.g., Ref. 31): x = m g m g + m l . (B4)This definition leads to the following equations, whichconnect the value of a certain quantity q with the twovalues of the quantity at the liquid q l and the vapor −6 −5 −4 −3 −2 −1
0 5 10 15 20 25 30 35 40 45 5020 bar10 bar4 bar1 bar0.1 bar10 mbar1 mbar0.1 mbar0.01 mbarCisentrope: p = 10 bar, T = 25 Kisentrope: p = 10 bar, T = 50 K den s i t y / ( k g / m ) temperature / K FIG. 27. Density of normal hydrogen as function of the tem-perature for different isobars and two selected isentropes.The boundary of the coexistance region is indicated by thethick solid line. q g side of the boundary of the coexistance region (see,e.g., Ref. 31): v = x v g + (1 − x ) v l , (B5)1 ρ = x ρ g + (1 − x ) 1 ρ l , (B6) h = x h g + (1 − x ) h l , (B7) s = x s g + (1 − x ) s l , (B8)where v is the specific volume, ρ the density, h thespecific enthalpy, and s the specific entropy. In orderto calculate the specific entropy s ( T, ρ ) the followingscheme is used: • For T ≥ T c : Calculate s ( T, ρ ) directly from theequation of state. • For
T < T c : Calculate the density of the liquidphase ρ l ( T ) and of the vapor phase ρ g ( T ). – For ρ ≤ ρ g or ρ ≥ ρ l : Calculate s ( T, ρ )directly from the equation of state. – For ρ g < ρ < ρ l :1. Calculate the specific entropy of theliquid phase s l = s ( T, ρ l ) and the vaporphase s g = s ( T, ρ g ).2. Calculate the quality x = x ( ρ, ρ g , ρ l ).3. Calculate the specific entropy s = x s g +(1 − x ) s l .In order to calculate the position dependent quanti-ties inside of the nozzle the velocity of sound a is neededto find the critical velocity u ∗ . In the coexistance regionthe following equation from Ref. 32 is used:1 ρ a = αρ g a + 1 − αρ l a , (B9)where a l,g = a ( T, ρ l,g ) is the velocity of sound of theliquid and the vapor phase calculated from the equationof state and α is the void fraction, defined as the ratioof the volume V g of the vapor phase to the total volume V l + V g : α = V g V l + V g . (B10) H. Pauly,
Atom, Molecule, and Cluster Beams II - ClusterBeams, Fast and Slow Beams, Accessory Equipment and Ap-plications (Springer Berlin Heidelberg, 2000). S. Barsov, U. Bechstedt, W. Bothe, N. Bongers, G. Borchert,W. Borgs, W. Br¨autigam, M. B¨uscher, W. Cassing, V. Cherny-shev, B. Chiladze, J. Dietrich, M. Drochner, S. Dymov, W. Er-ven, R. Esser, A. Franzen, Y. Golubeva, D. Gotta, T. Grande,D. Grzonka, A. Hardt, M. Hartmann, V. Hejny, L. vonHorn, L. Jarczyk, H. Junghans, A. Kacharava, B. Kamys,A. Khoukaz, T. Kirchner, F. Klehr, W. Klein, H. R. Koch, V. I.Komarov, L. Kondratyuk, V. Koptev, S. Kopyto, R. Krause,P. Kravtsov, V. Kruglov, P. Kulessa, A. Kulikov, N. Lang,N. Langenhagen, A. Lepges, J. Ley, R. Maier, S. Martin,G. Macharashvili, S. Merzliakov, K. Meyer, S. Mikirtychi-ants, H. M¨uller, P. Munhofen, A. Mussgiller, M. Nekipelov,V. Nelyubin, M. Nioradze, H. Ohm, A. Petrus, D. Prasuhn,B. Prietzschk, H. J. Probst, K. Pysz, F. Rathmann, B. Ri-marzig, Z. Rudy, R. Santo, H. Paetz gen. Schieck, R. Schle-ichert, A. Schneider, C. Schneider, H. Schneider, U. Schwarz,H. Seyfarth, A. Sibirtsev, U. Sieling, K. Sistemich, A. Selikov,H. Stechemesser, H. J. Stein, A. Strzalkowski, K.-H. Wat-zlawik, P. W¨ustner, S. Yashenko, B. Zalikhanov, N. Zhuravlev,K. Zwoll, I. Zychor, O. W. B. Schult, and H. Str¨oher, Nucl.Instrum. Methods A , 364 (2001). S. Brauksiepe, D. Grzonka, K. Kilian, W. Oelert, E. Roder-burg, M. Rook, T. Sefzick, P. Turek, M. Wolke, U. Bechstedt,J. Dietrich, R. Maier, S. Martin, D. Prasuhn, A. Schnase,H. Schneider, H. Stockhorst, R. T¨olle, M. Karnadi, R. Nellen,K. Watzlawik, K. Diart, H. Gutschmidt, M. Jochmann,M. K¨ohler, R. Reinartz, P. W¨ustner, K. Zwoll, F. Klehr,H. Stechemesser, H. Dombrowski, W. Hamsink, A. Khoukaz,T. Lister, C. Quentmeier, R. Santo, G. Schepers, L. Jarczyk,A. Kozela, J. Majewski, A. Misiak, P. Moskal, J. Smyrski,M. Sokolowski, A. Strzalkowski, J. Balewski, A. Budzanowski,S. Bowes, A. Hardt, C. Goodman, U. Seddik, and M. Zi-olkowski, Nucl. Instrum. Methods A , 397 (1996). R. Maier, Nucl. Instrum. Methods A , 1 (1997). PANDA Collaboration, “Strong interaction studies with an-tiprotons”, Technical Progress Report (FAIR GmbH, 2005). A. T¨aschner, E. K¨ohler, H.-W. Ortjohann, and A. Khoukaz,Nucl. Instrum. Methods A , 22 (2011), arXiv:1108.2653[physics.ins-det]. A. T¨aschner,
Entwicklung und Untersuchung von Cluster-Jet-Targets h¨ochster Dichte , Doctoral Thesis, Westf¨alischeWilhems-Universit¨at, M¨unster (2013). F. Hinterberger,
Physik der Teilchenbeschleuniger und Io-nenoptik (Springer Berlin Heidelberg, 2008). E. Knuth, F. Schunemann, and J. P. Toennies, J. Chem. Phys. , 6258 (1995). D. Allspach, A. Hahn, C. Kendziora, S. Pordes, G. Boero,G. Garzoglio, M. Macri, M. Marinelli, M. Pallavicini, andE. Robutti, Nucl. Instrum. Methods A , 195 (1998). W. Christen, K. Rademann, and U. Even,J. Phys. Chem. A , 11189 (2010),http://pubs.acs.org/doi/pdf/10.1021/jp102855m. A. T¨aschner, S. General, J. Otte, T. Rausmann, andA. Khoukaz, AIP Conf.Proc. , 85 (2007). H. Buchenau, E. L. Knuth, J. Northby, J. P. Toennies, andC. Winkler, J. Chem. Phys. , 6875 (1990). W. Christen, J. Chem. Phys. , 024202 (2013). J. Harms, J. P. Toennies, and E. L. Knuth, J. Chem. Phys. , 3348 (1997). E. D. K¨ohler,
Das M¨unsteraner Cluster-Jet Target MCT2,ein Prototyp f¨ur das PANDA-Experiment, & die Analyse derEigenschaften des Clusterstrahls , Diploma Thesis, Westf¨alischeWilhems-Universit¨at, M¨unster (2010). J. D. Anderson,
Modern Compressible Flow: With HistoricalPerspective , international 2 revised ed. (McGraw-Hill Educa-tion (ISE Editions), 1990). R. D. McCarty, J. Hord, and H. M. Rode, “Selected prop-erties of hydrogen (engineering design data),” Monograph 168(National Bureau of Standards, Washington, DC, 1981). J. W. Leachman, R. T. Jacobsen, S. G. Penoncello, and E. W.Lemmon, J. Phys. Chem. Ref. Data , 721 (2009). C. Ekstr¨om, Nucl. Instrum. Methods A , 1 (1995), proceed-ings of the 17th World Conference of the International NuclearTarget Development Society. W. Nolting,
Grundkurs Theoretische Physik, 4, Spezielle Rel-ativit¨atstheorie, Thermodynamic (Vieweg, 1997). B. I. Sedunov, Journal of Thermodynamics , 1 (2011). D. W. Sallet, Heat and Mass Transfer , 315 (1991). W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P.Flannery,
Numerical Recipes 3rd Edition: The Art of ScientificComputing , 3rd ed. (Cambridge University Press, 2007). M. Shacham, Internat. J. Numer. Methods Engrg. , 1455(1986). M. Berberan-Santos, E. Bodunov, and L. Pogliani, J. Math.Chem. , 1437 (2008). B. A. Younglove and M. O. McLinden, J. Phys. Chem. Ref.Data , 731 (1994). H. W. Wooley, R. B. Scott, and F. G. Brickwedde, J. Res.Natl. Bur. Stand. , 379 (1948). E. W. Lemmon and R. T. Jacobsen, J. Phys. Chem. Ref. Data , 69 (2005). H. D. Baehr,
Thermodynamik. Grundlagen und technischeAnwendungen (Springer-Verlag GmbH, 2005). VDI-Gesellschaft Verfahrenstechnik und Chemieingenieurwe-sen, ed.,