Centre line intensity of a supersonic helium beam
Adrià Salvador Palau, Sabrina D. Eder, Truls Andersen, Anders Komar Ravn, Gianangelo Bracco, Bodil Holst
CCentre line intensity of a supersonic helium beam
Adrià Salvador Palau, Sabrina D. Eder, Truls Andersen, Anders Komár Ravn, Gianangelo Bracco,
2, 3 and Bodil Holst Department of Engineering, Institute for Manufacturing,University of Cambridge, Cambridge, CB3 0FS, UK Department of Physics and Technology, University of Bergen, Allégaten 55, 5007 Bergen, Norway CNR-IMEM, Department of Physics, University of Genova, V Dodecaneso 33, 16146 Genova, Italy (Dated: October 9, 2018)Supersonic helium beams are used in a wide range of applications, for example surface scatteringexperiments and, most recently, microscopy. The high ionization potential of neutral helium atomsmakes it difficult to build efficient detectors. Therefore, it is important to develop beam sourceswith a high centre line intensity. Several approaches for predicting the centre line intensity exist,with the so-called quitting surface model incorporating the largest amount of physical dependenciesin a single analytical equation. However, until now only a limited amount of experimental datahas been available. Here we present a comprehensive study where we compare the quitting surfacemodel with an extensive set of experimental data. In the quitting surface model the source isdescribed as a spherical surface from where the particles leave in a molecular flow determined byMaxwell-Boltzmann statistics. We use numerical solutions of the Boltzmann equation to determinethe properties of the expansion. The centre line intensity is then calculated using an analyticalintegral. This integral can be reduced to two cases, one which assumes a continuously expandingbeam until the skimmer aperture, and another which assumes a quitting surface placed before theaperture. We compare the two cases to experimental data with a nozzle diameter of 10 µ m , skimmerdiameters ranging from 4 µ m to 390 µ m , a source pressure range from 2 to 190 bar, and nozzle-skimmer distances between 17.3 mm and 5.3 mm. To further support the two analytical approaches,we have also performed equivalent ray tracing simulations. We conclude that the quitting surfacemodel predicts the centre line intensity of helium beams well for skimmers with a diameter largerthan µ m when using a continuously expanding beam until the skimmer aperture. For the case ofsmaller skimmers the trend is correct, but the absolute agreement not as good. We propose severalexplanations for this, and test the ones that can be implemented analytically. I. INTRODUCTION
The supersonic expansion of a gas into vacuum can beused to obtain a molecular beam with high centre lineintensities with narrow speed distributions [1–6]. Suchbeams are used in different applications, for example sur-face scattering experiments and atom beam microscopy[7–10]. Noble gas atoms are very hard to detect due totheir high ionization potential [2]. Therefore, precise pre-diction of the beam centre line intensity plays an impor-tant role in designing instruments and experiments witha sufficient signal to noise ratio.In a standard supersonic expansion source used in scat-tering experiments, a pressurised gas expands from asmall aperture called a nozzle into a vacuum. The ex-pansion is then collimated using an aperture placed atthe end of a conical structure that points towards thenozzle, forming a beam. This conical structure is com-monly known as a skimmer (see Fig. 1). The problem ofprecisely determining particle intensities after the skim-mer attains different levels of complexity depending onthe modified Knudsen number, Kn ∗ at the skimmer posi-tion, which determines the flow regime close to the skim-mer [11]. The modified Knudsen number was introducedby Bird [11] to describe the changes in the flow due to backscattering of atoms from the skimmer. Kn ∗ = Kn (cid:18) S (cid:107) (cid:19) − / ( η p − . (1)Where S (cid:107) is the parallel speed ratio, a measure of thevelocity spread of the beam defined in Sec. II B. η p isthe term leading the inverse power law of the repulsivecollision model. For a hard sphere gas η p → ∞ , andfor the Lennard-Jones potential η p = 13 [12]. Kn is theKnudsen number: Kn = λ r S = 1 r S σ √ n , (2)where λ is the mean free path of the gas particles and r S is the radius of the skimmer. n is the number den-sity at the skimmer and σ is the temperature dependentcollision cross section of the gas atoms. In this case, σ can be calculated either according to the stagnationtemperature, or according to the maximum between thestagnation temperature and the skimmer temperature.For the case of a cold source the collision velocity willbe dominated by the warmer skimmer. The need for themodified Knudsen number is justified by the change inthe mean free path due to backscattering of atoms fromthe skimmer; In eq. (2) λ is the mean free path forparticles unaffected by the skimmer presence.The Knudsen number is used to estimate the validity ofdifferent flow regimes. Navier-Stokes flow can be assumed a r X i v : . [ phy s i c s . a t m - c l u s ] O c t for Kn < . , and free molecular flow for Kn > [11]. Asthe gas moves away from the nozzle, the mean free pathof the particles increases and therefore the nature of theflow dynamics of the problem changes [11]. As explainedbefore, we use here the modified Knudsen number, butthe discussion of different flow regimes remains the same.The Knudsen number can only be assumed to be smallerthan 0.2 in the space very close to the expansion origin(the nozzle), and hence the Navier-Stokes equations can’tbe generally used to model the flow of the beam close to,and after, the skimmer. Here, Direct Simulation MonteCarlo methods (DSMC), or direct numerical integrationof the differential equation (under simplifying assump-tions of the physics of the system), can be used to solvethe Boltzmann equation [12, 13].At Kn ∗ (cid:46) , the centre line intensity of the beamis known to be strongly affected by interaction betweenthe beam and particles reflected from the skimmer [11].Considering the reflection of particles from the skim-mer wall makes solving the Boltzmann equation diffi-cult, as DSMC methods are often computationally heavy.Some work has been done regarding the effect of skim-mer geometries [11, 14–16]. However, much of this worklacks extensive validation due to the lack of experimentaldata. This, together with the complexity of some of theproposed approaches, has caused some authors to avoidskimmer attenuation by designing experiments where itis not present.Another relevant contribution to the beam centre lineintensity is the exponential decrease of intensity due tofree molecular scattering of the beam’s atoms with abackground gas in the vacuum chambers [14, 16]. Theimportance of this contribution will depend on the qual-ity of the pumping system in the experimental set-up andthe flux from the nozzle into the expansion chamber.Intensity calculations disregarding both the interac-tion between the beam and particles reflected from theskimmer, and collisions with background gas were pre-sented in a range of analytical models published in the1970’s and 1980’s, based on a Maxwellian velocity dis-tribution of the supersonic expansion [17–20]. Thesemodels coexist with simpler treatments, disregarding theMaxwellian nature of the beam’s velocity distribution(usually compensated by including a peak factor), for ex-ample [5, 13, 14, 21]. Others use Beijerinck and Verster1981 model that incorporates cluster formation and usesthe concept of a virtual source [8, 18, 22]. Analyticalmodels have the advantage of requiring only relativelysimple numerical solutions of the Boltzmann equationand of directly showing the dependencies with the dif-ferent variables in the system. Among the most prolificanalytical models are various adaptations of the quittingsurface model [20].In the quitting surface model, the spherical quittingsurface is assumed to be located at the distance from thenozzle at which the atoms reach molecular flow [20]. Theatoms then leave the quitting surface following straighttrajectories determined by Maxwell-Bolzmann statistics. The ellipsoidal Maxwellian velocity distribution over thesurface is given by three parameters: the most proba-ble velocity ¯ v along the parallel direction (correspondingto the radial direction from the centre of propagation),and the parallel and perpendicular temperatures, respec-tively T || and T ⊥ . These two temperatures are associatedwith the velocity spread of the beam in spherical coordi-nates [23], and in some models are reduced to a simplerdescription with only a radial temperature T || [20].There are two popular ways to estimate the positionof the quitting surface: i) calculating the terminal Machnumber using the continuum assumption and taking theposition of the quitting surface to be the distance fromthe nozzle where the terminal Mach number is close to be-ing reached (see for example [21, 24]), or ii) directly com-puting the expansion’s temperatures and observing thepoint where these temperatures de-couple. De-couplingis defined as the point where the perpendicular tempera-ture is much smaller than the parallel temperature. De-coupling is typically assumed at a distance where thetemperatures of the expansion fulfil T ⊥ /T || ≤ . , thusdetermining the position of the quitting surface. Alter-native cutoff values have also been proposed [13], pro-viding a certain degree of freedom to the choice of thequitting surface position. Typically, such temperaturesare calculated through a numerical solution of the Boltz-mann equation. Previous studies already used such anapproach to predict the velocity distribution and inten-sity in the beam expansion [13, 25–27]. Given that (ii) ismore general than (i), we use (ii) in this paper.The quitting surface position can either be placed be-fore the skimmer, at the skimmer or after the skimmer.If the quitting surface is taken to be before the skimmer,the parallel temperature T (cid:107) dominates. This means thatthe condition T ⊥ /T || ≤ . is reached close to the ex-pansion source, and that the perpendicular temperatureof the beam quickly approaches 0. If the quitting sur-face is calculated to be at or after the skimmer it meansthat T ⊥ tends to 0 slowly. In this case, the perpendiculartemperature T ⊥ is mostly used in the calculations, andthe expansion is assumed to stop at the skimmer, even inthe case that its calculation gives a position further awaythan the skimmer [20]. Regardless of where the expan-sion is assumed to stop, the centre line intensity is thencalculated by integrating over the section of the quittingsurface seen by the detector through the skimmer.In this paper, we present a dataset of centre line inten-sity measurements for a helium atom beam, using severaldifferent skimmer apertures and designs, source temper-atures, and skimmer to nozzle distances. We benchmarkthese intensity measurements with the quitting surfacemodel, and discuss its shortcomings. Additionally, wepresent a ray tracing simulation of the quitting surfacemodel. This is done using a modification of the ray trac-ing software known as McStas described in detail in [28].This paper contains a large number of variables, manyof which are used in several formulas. All formulas areintroduced with definitions as they appear in the text. Inaddition, to make it a bit easier for the reader to keep anoverview, we have included an Appendix E with a tablelisting all the variables with definitions. II. THEORETICAL FOUNDATIONA. The supersonic expansion
The expansion of gas through a small nozzle undergoestwo different physical regimes: an initial continuum flow,governed by the Navier Stokes equations, followed by amolecular flow regime. In a sonic nozzle (a Laval tubecut-off in the sonic plane), the total flux per unit time(from now on, centre line intensity) stemming from thenozzle is typically calculated using the isentropic nozzlemodel [18]. The sonic plane corresponds to the planewhere the Mach number M = v/c = 1 where v is theaverage velocity of the gas and c the local speed of sound[29]. The equation for the total intensity stemming froma nozzle then reads [18]: I = P k B T (cid:114) k B T m (cid:16) π d (cid:17) (cid:114) γγ + 1 (cid:18) γ + 1 (cid:19) / ( γ − , (3)where γ is the ratio of heat capacities ( / for Helium),and d N is the diameter of the nozzle. In theory, this di-ameter must be corrected with the size of the boundarylayer at the nozzle throat. However, this correction cantypically be neglected. k B is the Boltzmann constant, T and P are the flow stagnation temperature and pressureinside the nozzle, m is the mass of a gas particle. Inthe second flow regime, the expansion of the gas is calcu-lated using the Boltzmann equation, assuming the nozzleis a point source, and using the following collision inte-gral Ω( T eff ) (corresponding to the RHS of the Boltzmannequation, that gives the rate of change of molecules in aphase-space element caused by particles that have suf-fered a collision) [13, 25]. Ω( T eff ) = (cid:18) k B T eff πm (cid:19) (cid:90) ∞ Q (2) ( E ) ζ exp (cid:0) − ζ (cid:1) dζ. (4) ζ = (cid:114) Ek B T eff . (5)Where T eff is an effective average temperature intermedi-ate to the values of the parallel and perpendicular tem-peratures, Q (2) is the viscosity cross section and E isthe collision energy of two atoms in the centre-of-masssystem. For collisions between particles following Bose-Einstein statistics, the viscosity cross section can be writ-ten as follows [13, 30]: Q (2) ( E ) = 8 π (cid:126) mE (cid:88) l =0 , , ... ( l + 1)( l + 2)2 l + 3 sin ( η l +2 − η l ) , (6) where η l are the phase shifts for orbital angular momen-tum l , obtained solving the scattering of He atoms in thechosen two body potential.An ellipsoidal Maxwellian velocity distribution is as-sumed along the whole expansion [13]. The velocity dis-tribution of the atoms in the expansion, f ell , is defined inspherical coordinates by the two independent tempera-tures, T || and T ⊥ , and their two corresponding velocities v (cid:107) and v ⊥ as described in the introduction, f ell ( (cid:126)v ) = n (cid:18) m πk B T || (cid:19) (cid:18) m πk B T ⊥ (cid:19) · exp (cid:18) − m k B T || ( v || − ¯ v ) − m k B T ⊥ v ⊥ (cid:19) . (7)The numerical solution of the Boltzmann equation hasbeen implemented for the Lennard-Jones potential (LJ)[31], defined as follows: V LJ ( r LJ ) = 4 (cid:15) (cid:34)(cid:18) r m r LJ (cid:19) − (cid:18) r m r LJ (cid:19) (cid:35) , (8)where r LJ is the distance between any two interactingparticles. r m is the distance at which the potentialreaches its minimum, for the case of He correspondingto r m = 2 . Å, (cid:15) = 2 . meV [32]. A detailed de-scription of the potential and its implementation in theBoltzmann equation can be found in [13]. The simple LJpotential can be replaced by more sophisticated poten-tials, such as the Tang, Toennies and Yu (TTY) or HurlyMoldover (HM) potentials [33, 34]. However, results ofprevious calculations showed that this is only necessaryfor source temperatures below 80 K [13, 26, 35]. In thepresent study, the source temperature is higher than 80K and the LJ potential is adequate.The numerical solution of the Boltzmann equation inspherical approximation presented here provides the evo-lution of the gas velocity, and the temperatures T || and T ⊥ with respect to the distance from the nozzle. B. The quitting surface model
As mentioned in the introduction, the quitting surfacemodel assumes that the particles leave in molecular flowfrom a spherical surface of radius R F centred at the sonicpoint. The centre line intensity of the beam is calculatedby integrating over all the particles leaving from the quit-ting surface and arriving at the detector. In 1973, Sikoraseparated the quitting surface model in two approaches:one corresponding to what he called the quitting surfacemodel , and one which he called the ellipsoidal distribu-tion model . The first approach assumes a quitting sur-face placed before the skimmer and a Maxwellian velocitydistribution featuring only the radial component of thevelocity: v (cid:107) . The second approach, the ellipsoidal distri-bution model, assumes an ellipsoidal Maxwellian velocitydistribution featuring both v (cid:107) and v ⊥ , together with aquitting surface placed exactly at the skimmer. For therest of the paper we will refer to the two approaches asSikora’s quitting surface approach and Sikora’s ellipsoidaldistribution approach.Sikora’s ellipsoidal distribution approach was lateradapted by Bossel to be used for expansions stoppingbefore the skimmer. In other words, Sikora’s quittingsurface approach (assuming a quitting surface placed be-fore the skimmer) was adapted to incorporate ellipsoidaldistributions [19]. To avoid confusion, it is enough toconsider the position of the quitting surface itself: in thecase of Sikora’s ellipsoidal distribution approach, the ex-pansion is considered to stop at the skimmer. In thecase of Bossel’s approach, the expansion can be chosento stop at the skimmer or before it. Expansions stoppingafter the skimmer have thus far not been treated usingthe quitting surface model. An attempt of doing so ispresented in this paper (see Appendix A).Bossel’s approach is the most general approach de-scribed so far, as under the right assumptions it reducesto both approaches proposed by Sikora. Bossel’s ap-proach corresponds to integrating eq. (7) over the quit-ting surface area seen by the detector through the skim-mer: I D = τ I πa R L (cid:90) r D (cid:90) r S (cid:90) π g ( δ ) rρ cos β(cid:15) e − S (1 − (cid:15) cos θ ) D ( b ) dρdrdα, (9)where r D is the radius of the detector opening, r S is theradius of the skimmer (see Fig. 1), and a is the distancebetween the skimmer and the detector. r , β , θ , δ , α and ρ are geometrical parameters defined in Fig. 1. τ = T || T ⊥ is the fraction between parallel and perpendicular tem-peratures, which is used to simplify the integral through (cid:15) = (cid:0) ( τ sin θ + cos θ (cid:1) − / . g ( δ ) is the angular depen-dency of the supersonic expansion density at the quittingsurface, and L = (cid:82) π g ( δ ) sin δdδ corresponds to its inte-gral along the quitting surface. S = (cid:113) m ¯ v kT (cid:107) is the parallelspeed ratio at the quitting surface.Unfortunately, Bossel’s approach has no simple ana-lytical solutions and is often slow to compute over a widevariable space. For S i > Sikora showed that both hisellipsoidal distribution approach and quitting surface ap-proach can be approximated as [20]: I = I (cid:90) π d Φ2 π [ e − S i sin θ ] θ θ . (10)Here, Φ is the angle of rotation about the beam axis,and θ is the angle between the vector normal to thequitting surface and the vector connecting a given pointon the quitting surface with a point in the detector plane. θ and θ are the minimum and maximumangles that fulfil the condition that the line connecting apoint in the quitting surface and a point in the detector plane must cross the skimmer aperture. In the case ofSikora’s quitting surface approach, θ is defined from aspherical surface of radius R F , and S i = S (cid:107) = (cid:113) m ¯ v kT (cid:107)∞ isthe parallel speed ratio at the end of the expansion. Inthe case of Sikora’s ellipsoidal distribution approach, θ is defined from the skimmer aperture (the radius of thequitting surface is then the distance between the nozzleand the skimmer x S , R F = x S ), and S i = S ⊥ = (cid:113) m ¯ v kT ⊥ is the perpendicular speed ratio at the skimmer (see Fig.1 for a sketch featuring these geometrical terms). I is defined as the intensity arriving at the detector,assuming that there is no skimmer. This can be obtainedin two ways: I = I πr η D 1( x S + a ) . Using eq. (3) for I η D πr nv ∞ (cid:16) x S x S + a (cid:17) . Using density at skimmer.Here, η D is the efficiency of the detector incounts/partice. Sometimes, one might be interested toobtain the intensity per area. In order to do so, it suf-fices to divide I by πr .From eq. (10) it can be shown that for r S (cid:28) x S , r S (cid:28) a , ar S >> S i , and r D << a , the intensity arriving at thedetector reads [20]: I S = I (cid:40) − exp (cid:34) − S i (cid:18) r S ( R F + a ) R F ( R F − x S + a ) (cid:19) (cid:35) (cid:41) , (11) x S is the distance between the nozzle and the skimmer.This equation, with the assumption of S i = S (cid:107) , and theexpansion stopping before the skimmer is usually pre-ferred to using the perpendicular speed ratio, as measur-ing the parallel speed ratio of atoms is a well establishedtechnique [36]. The simplicity of the model has moti-vated its usage for example to optimize the intensity ofhelium microscopes [10, 37]. C. Scattering contributions
The atoms leaving the quitting surface do not travel ina perfect vacuum. Rather, they interact with the back-ground gas and the particles scattered from the chamberand skimmer walls. Such interactions can become signif-icant at high nozzle pressures. There have been variousapproaches for accounting for this, from DSMC simula-tions, to simpler numerical models based on assumptionson the scattering properties of the skimmer walls [14, 38].Analytical models for the skimmer contributions are sofar non-existent due to the difficulty of solving the Boltz-mann equation analytically in a typical nozzle-skimmergeometry. The method that has provided a better under-standing is the DSMC method (see, for example [11]).This method is not employed in this paper due to itscomplexity, but it can be assumed to be the preferablemethod when precise, localized predictions are desired.Here, we choose to only model the interaction with thebackground gas via free molecular scattering, as it canbe modelled by a simple exponential law [14, 16]: II S = exp (cid:0) − σ n B E x S − σ n B C a (cid:1) . (12) σ = r m / is the scattering cross section of the atomsin the Lennard-Jones potential. n B E and n B C are thebackground number densities in the expansion chamberand the subsequent chambers respectively, measured bya pressure gauge placed far away from the beam centreline. D. Overall trends
In this section we qualitatively describe importanttrends in the expected behaviour of the centre line in-tensities according to the theory presented above.1. For skimmers large enough, the exponential termin the equation for centre line intensity becomesnegligible, (eq. (11)). Thus, increasing the radiusof the skimmer further will not lead to an increasein the centre line intensity.2. Larger skimmers display a decrease in centre lineintensity at high pressure. This is due to thefact that a larger skimmer gives a smaller modifiedKnudsen number (eq. (1)) for a given pressure. Itis known that for smaller modified Knudsen num-bers in the so called transition regime, wide angledshock waves can form, which compromise the flowof the beam [11]. Note that the shock wave be-haviour is not modelled by the theory presentedabove.3. The closer the skimmer is to the quitting surface( ( R F − x S ) → ); the higher the centre line intensitywill be, as the denominator in the exponential ineq. (11) reaches its minimum. This effect is dueto the fact that a larger portion of the quittingsurface is captured and this gives a larger centreline intensity.4. Colder sources produce more intense beams be-cause the gas passing through the nozzle has ahigher density, which ends up influencing the centreline intensity equation (see eq. (3)).5. Numerical solutions of the Boltzmann equation asdescribed in Sec. II A predict an intensity dip atlow source pressures for small skimmers. This dipcannot be extracted from the equations in a simplemanner and will be discussed further in the maintext. E. The ray tracing simulation
As an independent test of eqs. (9) and (10), a ray trac-ing simulation of the quitting surface expansion was im-plemented. The simulation was performed using a mod-ification of the ray-trace software package known as Mc-Stas described in [28, 39, 40].In order to replicate the dynamics assumed during thederivation of eq. (9), a spherical source with ellipsoidalMaxwellian velocity distributions and an anisotropicnumber density was programmed. The McStas softwareworks with sources featuring uniform spatial ray proba-bility distributions that are later corrected for their realprobability weights determined by the physics of the sys-tem (in this case, the Maxwellian velocity distributionof the source, and the anisotropic number density). Thisposes a problem when simulating the quitting surface be-cause most of the rays yield probabilities that are toolow, bringing insufficient sampling at the detector. Toavoid this effect, we only computed the particles stem-ming from the surface of the quitting surface seen by thedetector through the skimmer (see Fig. 2). This reducesthe computation power needed for each experiment andtherefore allows for better statistics in the detector. y xz(x , y , z )
R RR ρ (x ,0, z )
D D x S x D R F δ P’P r S βθ P P’
DETECTORSKIMMER α ar } FIG. 1. Illustration of all variables used in the ellipsoidalquitting surface model. P is a point on the quitting surfacefrom which a particle leaves in a straight trajectory until P’ , apoint placed on the detector plane. The point on the quittingsurface is given by the set of Cartesian coordinates ( x, y, z ) ,which can be related to the polar coordinates r, α, ρ for in-tegration. x S is the distance from the nozzle to the skimmerand x D is the distance from the nozzle to the detector. There-fore a = x D − x S . The angles β and θ can also be expressedin terms of r , α and ρ . The simulation is performed as follows: first, a circu-lar target or focus of interest is set, which determines thearea of the detector, where the rays will hit. Then, thepoint P’ is generated randomly over the area of the de-tector. Subsequently, a point P over the quitting surfaceis randomly generated and its connecting vector (cid:126)r is com-puted. Only the points visible by the detector throughthe skimmer are allowed (see Fig. 2). Therefore a maxi- x S x D R F δ r S SKIMMER a m y d r DDETECTORNOZZLE x
P’P
FIG. 2. Diagram of the section of the quitting surface con-sidered in the ray tracing simulation, only the angle δ m seenby the detector through the skimmer contributes to the inten-sity at the detector. R F is the radius of the quitting surface, y is the distance between the axis of symmetry and the pro-jection of the maximum-angle ray on the quitting surface, r S is the skimmer radius and r D is the radius of the detec-tor. a is the distance between the skimmer and the detector, d is the distance from the skimmer to the point where themaximum-angle ray crosses the symmetry axis. x D is thedistance between the nozzle and the detector and x S is thedistance between the nozzle and the skimmer. x is the dis-tance from the point of emission of the maximum angle rayto the nozzle plane. mal angle δ m is set (see the derivation in Appendix C). δ m = arcsin yR F = arcsin dr S ( d + x S ) − (cid:113) d r R + R − ( d + x S ) R F ( dr S ) + R F (13)With d corresponding to the distance from the skimmerto the point where the maximum-angle ray crosses thesymmetry axis (see Fig. 2): d = ar S r D + r S . (14)Which means that the point P must be contained withinthe following angles: δ = (0 , δ m ) , φ = (0 , π ) . (15)In Cartesian coordinates, P is: P = R F (sin δ cos φ, sin δ sin φ, cos δ ) . (16)Following, a scalar velocity v is randomly generated be-tween two limiting values along the direction of the vector (cid:126)r . From its Cartesian components, the perpendicular andparallel velocities are obtained: v || = (cid:126)v · (cid:126)u r = v x sin δ cos φ + v y sin δ sin φ + v z cos δ,v ⊥ = (cid:126)v · (cid:126)u δ = v x cos δ cos φ + v y cos δ sin φ − v z sin δ,v ⊥ (cid:48) = (cid:126)v · (cid:126)u φ = − v x sin φ + v y cos φ. (17) A probability weight factor given by the Maxwellian ve-locity distribution of the beam is set for the ray travellingfrom P to P’ (see Figs. 2 and 1). The intensity recordedat the detector will be the sum of all probability weightfactors. Therefore, we can recover eq. (23) (AppendixB) in angular coordinates to infer the intensity contribu-tions: dI = I A D A S L f ell ( (cid:126)v ) g ( δ ) v d Ω dv. (18) A D = πr is the area of the detector. For the exper-iments presented here, this corresponds to the area ofthe pinhole placed in front of the detector (see Fig. 3), A S ≈ πy is the area of the section of the sphere fromwhich particles are simulated assuming r S (cid:28) R F (thecomputed section of the quitting surface is small enoughrelative to R F that its area approximates to the area ofa circle). L is defined as in eq. (29) (Appendix B) buttaking care to integrate only between 0 and δ m . d Ω is thesolid angle seen through the skimmer from the centre ofthe detector, this is approximately the same as the solidangle seen from P’ through the skimmer. This approxi-mation is true for detectors placed sufficiently far awayfrom the skimmer. III. EXPERIMENTAL SETUP FOR INTENSITYMEASUREMENTS
The setup used to obtain the experimental measure-ments presented in this paper is shown in Fig. 3. Allthe measurements have been obtained using the molecu-lar beam instrument at the University of Bergen, knownas MAGIE. This instrument is equipped with a home-built source which enables the skimmer and nozzle tobe positioned relative to each other with 50 nm preci-sion [5]. This is particularly important to ensure properalignment in centre line intensity experiments using smallskimmers. A detailed description of the system can befound in [41]. In contrast to most other helium atom scat-tering instruments with time-of-flight detection, MAGIEhas a movable detector arm, which allows us to mea-sure the straight through intensity of the beam with-out any sample. A centre line intensity measurementis performed by setting the initial pressure in the inletchannel and measuring the inlet channel temperature.For the experiments presented here, the beam source iseither "warm" (at ambient temperature) or "cold" (atroughly 125 K). The helium gas expands through a pin-hole aperture nozzle, µ m in diameter to a lower pres-sure chamber where it undergoes a supersonic expansion.We use a Pt-Ir electron microscope aperture as nozzle(purchased from Plano GmbH, A0301P) [5]. The expan-sion is then collimated by a skimmer placed . ± . ,or . ± . , or . ± . away from the nozzle.Figure 4 shows an example of the alignment procedure.The nozzle is moved across the skimmer opening in
50 nm steps in a 2D array and eventually moved to the positionof maximum intensity which is clearly visible. Note that adisplacement of just . leads to a noticeable changein intensity.Further downstream, at 973 mm from the nozzle, a µ m aperture is placed to further reduce the back-ground pressure and thus minimize the beam attenua-tion. Finally, at from the nozzle an ioniza-tion detector is set. The detector has an efficiency of η D = 2 . · − (provided by the manufacturer). Justin front of the detector another aperture is placed. Twodifferent apertures with diameters µ m and µ m respectively, were used in the experiments. This allowsus to measure the centre line intensity. A table with thediameter of the aperture for each intensity experiment isgiven in Appendix D.Five skimmers were used to collimate the beam, twomade of nickel, two made of glass and an additionalmetallic skimmer known as the Kurt skimmer. The nickelskimmers have apertures and µ m in diameter.They are produced by Beam Dynamics (model 2) andhave a streamlined profile [42] (see dimensions in Fig.5). The glass skimmers are home made using a Nar-ishige
PP-830 glass pulling machine, using Corning 8161Thin Wall capillaries with an outer diameter of 1.5 mmand an inner diameter of 1.1 mm. The glass skimmersare mounted on a Cu holder (see dimensions in Fig. 5).Their apertures are 18 and 4 µ m respectively, measuredusing an electron beam microscope. Stereo microscopemeasurements on the glass skimmers showed an outeropening angle of ≈ . ◦ for the first 200 µ m , followedby a more narrow section of ≈ . ◦ . The inner open-ing angle could not be determined, but due to the thinopening lip ( ≈ ), it is expected to be similar to theouter opening angle. This corresponds to what is knownas a slender skimmer. Slender skimmers are known toproduce better performance than wide angle skimmers,as long as the modified Knudsen number at the skimmeris kept large enough [11]. This condition is fulfilled in theexperiments presented here due to the large values of S || and the small skimmer openings.The Kurt skimmer is also home made. It is designedto be used with interchangeable apertures on 2 mm di-ameter discs. Two apertures are used in this study: 5and 100 µ m in diameter. The dimensions of the Kurtskimmer can be found in Fig. 5 (note the inverted coneshape before the aperture). The Kurt skimmer is madeof stainless steel type 1.4301. IV. RESULTS
Throughout Figs. 7-12 we use open circles for thenozzle-skimmer distance x S = 5.3 mm, triangles for x S =11.3 mm, and asterisks for x S = 17.3 mm. The labels areincluded in Fig. 7 only. Error bars are not included inthe plots because they are too small to show. A. Ray tracing benchmarking of the centre lineintensity integral
A spherical quitting surface is simulated using the el-lipsoidal quitting surface velocity distribution defined ineq. (7). The centre line intensity obtained through theray tracing simulation is then compared with eqs. (9) and(11) for different spans of the different variables presentin the equation. In all cases the result from the analyt-ical models lies within the statistical margin of error ofthe simulation (see Fig. 6). In the further sections ofthis paper we will just show the results from eqs. (9) and(11).
B. 120 µ m and 390 µ m skimmers In this section, the measured intensities for the largeskimmers from Beam Dynamics (see Fig. 5) (120 and 390 µ m diameters), are compared with the predictions fromeq. (12) for the two variations of the model described inSec. II A.
1. Warm source, T ≈ K The results for a warm source are shown in Figs. 7 and8. Fig. 7 shows the experimental results and eq. (12)with the expansion assumed to stop at the skimmer, and S i = S ⊥ . The experimental results are reproduced fairlywell over the whole range, but with a trend towards toohigh theoretical values for higher pressures. To obtain n B E → n B E ( P ) for eq. (12), we use a set of measuredbackground pressures in the expansion chamber. Fromobservation this dependency is linear, and the equationobtained is: n B E = 1 k B T ( m E · P + n E ) . (19) m E and n E are the linear fit coefficients from fittingthe measured background pressures P B with respect to P . Concretely, for this set of measurements m E =3 . · − P abar , n E = − . · − P a if P is given in bar and n B E in SI units (positive values of n B E are guaranteed bythe experimental pressure range, P ≥ bar). The num-ber density after the skimmer, n B C , was experimentallymeasured to be approximately 1/20 of n B E , eq. (19) wasused with the corresponding factor.Fig. 8 shows the values of eq. (12) for the 120 µ m and 390 µ m skimmers, where the expansion is assumedto stop before the skimmer (in this case for T ⊥ /T (cid:107) ≤ . ), and S i = S (cid:107) . At small source pressures there isgood agreement between experiments and simulations,but the dependency on the nozzle-skimmer distance islost. At high pressures the model becomes non-physicalbecause the point at which T ⊥ /T (cid:107) ≤ . is calculated tobe positioned after the skimmer. One must note thatthe decrease in centre line intensity at high pressures is ExpansionchamberP= e-8 -e-9 barNozzlechamberP= 1-201 barNozzle Ø10 µm Free molecular regimeP < e-10 bar Pinhole apertureØ=50 or 200 µmSkimmeraperturex =11.3±6 mm s R F Ionization detectorx = 2441 mm d Backgroundnoise reducingapertureØ=400 µmx = 973 mm A r S r D FIG. 3. Sketch of the experimental setup used for the centre line intensity measurements. A skimmer is used to select thesupersonic beam, followed by two apertures. Vacuum pumps are placed in each chamber to reduce interactions of reflectedparticles with the beam. R F is the radius of the quitting surface, from where the gas particles are assumed to leave followinga mollecular flow. not given by the model (eq. (11)) being un-physical, butinstead by S (cid:107) r /R → as P increases. If the expansionis assumed to always stop at the skimmer ( R F = x S )as in the case of Fig. 7, this condition does not holdany more and the predicted centre line intensity increasesmonotonically with P . In this case, eq. (12) is also used.The discrepancy at low pressures is discussed in Sec. V.
2. Cold source T ≈ K We present the measured intensities for a beam with asource temperature of ± K and we compare themwith the predictions from eq. (12). We obtain n B E → n B E ( P ) as in eq. (19): m E = 5 · − P abar , n E = 48 · − P a . In the case of cold sources, if one chooses todetermine the quitting surface position by the ratio oftemperatures T ⊥ /T (cid:107) ≤ . , the quitting surface is placedafter the skimmer already at quite low pressures. Thus,computing the eq. (12) for the case of S i = S (cid:107) and theexpansion stopping before the skimmer is only valid fora few measurement points. Therefore, we only presentthe results for the case of the expansion stopping at theskimmer and S i = S ⊥ . In general, the prediction powerof the model decreases for a cold source (see Fig. 9). C. Micro skimmers
The centre line intensity plots for micro skimmers showmarked dips in the intensity, especially for the cold sourcecases. Centre line intensity dips are also observed athigher pressures for a warm source (see Figs. 10 and11). The model predicts the dips for a cold source, butin both cases fails to fit the experimental data well. Thecentre line intensity measured for both skimmers is inthe same range, while the model predicts a more pro-nounced difference between the µ m skimmer and the µ m skimmer. D. The Kurt skimmer
To experimentally determine the importance of Kn ∗ -driven skimmer effects we use a skimmer designed in sucha way that such effects are expected to clearly domi-nate over the centre line intensity trends. This is thecase of the Kurt skimmer (see Sec. III), which due toits inverted-cone walls concentrates the reflecting parti-cles along the beam center line, leading to a low Kn ∗ (see eq. (1)). Comparing the Kurt skimmer intensitieswith the Beam dynamics skimmers, one sees that skim-mer effects are not clearly observed until about 40 bar,for nozzle-skimmer distances corresponding to x S > . mm (see Fig 12). This means that the discrepancies atlower pressures between eq. (12) and the micro-skimmer Centered U-axis position [mm] -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 N o r m a li z ed i n t en s i t y [ C oun t s / s ] U Y x s FIG. 4. Example of the alignment procedure, here done fora cold source at 60 bar and a 390 µ m diameter skimmer.The nozzle is moved relative to the skimmer in 50 nanometersteps for three values of x S . The optimum alignment positionof the nozzle relative to the skimmer is obtained by findingthe centre point of the maximum of intensity. The completeintensity plot of the beam is shown in the upper left corner. measurements cannot be explained by skimmer interac-tions only. In fact, the modified Knudsen number in thecase of micro-skimmers at 40 bar is expected to be largerthan in the case of the Kurt skimmer due to the /r S dependency (see eq. (1)).Note how skimmer interference in the case of the Kurtskimmer is not significant until the nozzle-skimmer dis-tance is set at . mm, (see Fig 12). A similar effect isseen, for a cold source, in the case of the 390 µ m BeamDynamics skimmer, where for x S = 5 . mm, skimmerinterference becomes evident (see Fig. 9). The same ef-fect is not clearly observed for the smaller, 120 µ m BeamDynamics skimmer. This can be seen as an experimentalconfirmation of the importance of the modified Knud-sen number, which predicts stronger skimmer effects forlarger skimmers.
E. Complete experimental data
In this section, we plot the complete dataset of mea-surements carried out during this study, with the excep-tion of measurements corresponding to the Kurt skim-mer, that are plotted separately. In order to preserve therelevant intensity magnitude, and thus make comparisonseasier the intensities plotted have been normalized to theradius of the aperture in front of the detector used toperform each measurement. Therefore, in this section,the intensities are given in counts / s · m . The centre lineintensity data for a warm source T ≈ K is shown in Fig. 13, and for a cold source T ≈ K in Fig. 14.Additionally, we plot the difference in centre line inten-sity per square meter between cold and warm sources foreach experiment (Figs. 15 and 16).From Fig. 15, one can observe that for large skimmerscold sources produce a higher centre line intensity thanwarm sources, especially for high source pressures. Thisis given by eq. (3) and by the larger speed ratios obtainedin cold beams. For the case of the 120 µ m skimmer, thisdifference reduces the further away the skimmer is placedfrom the nozzle due to the evolution of T ⊥ along the beamaxis.For the case of micro skimmers, cold sources are gener-ally less intense than warm sources, except for very largepressures. This is due to an intensity dip occurring forcold sources at low and medium pressures driven by theevolution of the beam’s perpendicular speed (see Discus-sion). The smaller the collimating skimmer is the largerthe influence of this dip on the measured centre line in-tensity. This is because larger skimmers collect particleswith a larger perpendicular temperature range. V. DISCUSSION
The analytical model based on Sikora’s ellipsoidal dis-tribution approach ( S i = S ⊥ , expansion stopped at theskimmer) predicts the centre line intensity of a heliumbeam generated by a source at ambient temperature withreasonable accuracy. However, the model has several lim-itations, each of which will be discussed in detail in thissection.1. Poor fit at high pressures : for most skimmers, themodel overshoots the measured intensities at highpressures ( P (cid:38) bar). This phenomenon is likelydue to a combination of two effects: skimmer inter-ference, and a continuing expansion of the beamafter the skimmer. By observing the data, we cansee that in the case of a warm source this over-shoot does not significantly vary when two skim-mers with the same design but different diameterare used (in this case, the Beam Dynamics skim-mers). This points towards the idea that skimmerinterference can’t be the main cause of the over-shoot, as the influence of the particles reflectedfrom the skimmer is expected to strongly dependon the skimmer radius. However, in the case of acold source, the overshoot is more significant for the µ m Beam Dynamics skimmer than its µ m equivalent. What is likely happening is that thehelium beam continues to expand significantly af-ter the skimmer following different dynamics thanbefore it, due to the removal of particles by theskimmer edges. According to the simulations of theexpansion performed in this study, this is particu-larly relevant for the case of a cold source, wherethe quitting surface is often predicted to be sev-eral centimetres after the skimmer. This renders0 (C)(B) Base: 27.9 mmHeight: 25.4 mm Rim: 2.5 mm (A)
Ø 120, 390 µm Ø 4, 18 µm Ø 5, 100 µm
FIG. 5. Drawings of the skimmers used for the centre line intensity measurements. (A) corresponds to the Beam Dynamicsskimmers, with diameters of 120 and 390 µ m , (B) to glass micro-skimmers mounted on copper, with diameters of 4 and 18 µ m and (C) corresponds to the Kurt skimmer, with inserted appertures of 5 and 100 µ m . Sikora’s treatment of a beam that expands due toits non vanishing T ⊥ at the skimmer un-physical asit assumes no further collisions after the skimmer.During the preparation of this paper, efforts wereundertaken to adapt Sikora’s model to a beam ex-panding after the skimmer using simple geometri-cal rules. This was motivated by the observationsmade by Doak et al, whom used micro-skimmers toperform focusing experiments and observed a devi-ation between expected and measure focal spot size.They suggested that this may have been due to thesupersonic expansion continuing after the beam haspassed through the skimmer aperture [43]. Thisadaptation can be found in Appendix A (Fig. 19)but did not produce very promising results. Atreatment using a DSMC simulation of the wholesystem is most likely a more accurate approach inorder to predict intensities at large pressure values.This approach is also much more complex than theanalytical models presented here.Another possible explanation of these discrepancieswould be the non-physical nature of a “hard" quit-ting surface. Replacing it with a “soft" treatmentmay yield interesting results. The centre line inten-sity would be calculated then by integrating over aseries of infinitesimally spaced successive quittingsurfaces.The higher overshoot at P (cid:38) bar for the smallerBeam Dynamics skimmer in the case of a coldsource occurs in all cases except one: x S = 5 . mm (see Fig. 9). In order to understand this pe-culiarity, one must re-visit the modified Knudsennumber. The case of x S = 5 . mm for a cold sourceand r S = 390 µ m , is the case expected to havethe lowest modified Knudsen number (largest r S and number density at the skimmer, see eq. (1)).Therefore, it is likely that this particular case is theonly one showing skimmer interference governed bythe interaction with reflected particles. 2. Low predictability of micro-skimmer intensities : onthe one hand, skimmer interference and skimmerclogging are known to be determined by the mod-ified Knudsen number Kn ∗ , which strongly de-pends on the skimmer diameter (eq. (1)). Micro-skimmers, are thus expected to show less inter-ference than their larger counter-parts under thesame conditions. This effect is clearly seen in Fig.9, where skimmer effects are present only for thelarger µ m skimmer.On the other hand, smaller skimmers sometimeshave very thin and long geometries, causing a pos-sible increase of pressure along the skimmer chan-nel. This effect is likely what causes the bad fitbetween the model predictions and the observedmicro-skimmer centre line intensities.Notwithstanding, it is important to note thatSikora’s ellipsoidal quitting surface model is able topredict the general trends of micro skimmer inten-sities. This includes the centre line intensity dip atlow pressure for small skimmers. This dip is drivenby the behaviour of the perpendicular speed ratioat low pressures, that is predicted by the simula-tion of the supersonic expansion to decrease firstand increase later (see Fig. 17).However, the experimental observability of this dipis actually determined by the radius of the skim-mer and the distance between the nozzle and theskimmer. If r S x S S ⊥ is small enough ( (cid:46) . ), then theterm (cid:20) − S ⊥ (cid:16) r S ( R F + a ) R F ( R F − x S + a ) (cid:17) (cid:21) in eq. (11) is smalltoo. This makes the exponential term in eq. (11)dominate, and the effect of the dip in S ⊥ can beclearly observed in the beam centre line intensity.This explains why this dip is only experimentallyobserved for the case of micro-skimmers.This good trend replication is particularly relevantfor purposes of optimization, where the value of1 var (normalized) I [ no r m a li z ed ] ar D r S FIG. 6. Plot of the ray tracing simulation (dashed lines) com-pared with eq. (9) and (11) (respectively, circles, crosses for S i = S (cid:107) and triangles for S i = S ⊥ , superposed). The greenline show the effect on the centre line intensity of varying thedistance between the skimmer and the detector, a . The blueand red lines show the intensity change when varying the ra-dius of the pinhole in front of the detector, r D , and the radiusof the skimmer, r S . The centre line intensity and the variablevalues have been normalized to 1 in order to show all depen-dences in a single plot. The calculations are done at a fixedskimmer position x S = 11 . mm (the centre position). a isvaried between 0.5 m and 2 m, r D is varied between 10 µ m and 100 µ m , and the radius of the skimmer, r S is varied be-tween 1 µ m and 10 µ m . While a variable is varied, the othersare kept fix at the maximum value of their span ( a = 2 m, r D = 100 µ m , r S = 10 µ m . The source temperature is 115 Kand the source pressure is 161 bar. Both the ray tracing sim-ulation and the centre line intensity model assume a quittingsurface placed just before the skimmer position ( R F = 11 . mm). interest is not so much the centre line intensity butthe combination of parameters maximizing it.3. Weak dependence on the nozzle-skimmer distanceof the S i = S (cid:107) variant : only when the expansion isallowed to stop at the skimmer and the perpendic-ular speed ratio is used, does the predicted centreline intensity significantly depend on the nozzle-skimmer distance, x S . This is expected, as in thiscase the thermal spread of the beam is caused bythe value of the perpendicular temperature at theskimmer T ⊥ , and this value varies strongly with x S . Despite S (cid:107) << S ⊥ causing a stronger expo-nential contribution in eq. (11), the variation on S ⊥ with the skimmer radius is much stronger thanthe fraction term in the exponential, making the S i = S (cid:107) variant actually less dependent on x S (as S (cid:107) remains constant). P [bar] I [ c oun t s / s ] FIG. 7. Plot of measured and predicted intensities for a warmsource (300 K), 120 µ m (pink) and 390 µ m (red) skimmers,and for three values of x S : 5.3 mm (circles), 11.3 mm (up-wards arrows) and 17.3 mm (asterisks). The intensities arecomputed assuming that the expansion stops at the skimmerwith S i = S ⊥ . Note that, for the larger skimmer, the centreline intensity becomes independent of the distance betweenthe skimmer and the nozzle, so that all the curves collapsein one simulated curve (in good agreement with what is ob-served experimentally). The difference in intensities betweenthe two skimmers is due to the fact that they were obtainedusing different pinholes in front of the detector (see AppendixD and Fig. 3) VI. CONCLUSION
We present a dataset of centre line intensity measure-ments for a supersonic helium beam and compare itto various intensity models. We show that these mod-els replicate the experimental data well for skimmerswith diameters and µ m . Particularly, we showthat Sikora’s ellipsoidal distribution approach, assuminga quitting surface placed at the skimmer position, withthe expansion dominated by the supersonic expansionperpendicular temperature T ⊥ fits the experimental databest.We present a ray tracing simulation approach, used tonumerically replicate the introduced centre line intensitymodels. We show that the ray tracing approach and ana-lytical models (Sikora’s and Bossel’s) follow very similardependencies with the different geometrical variables ofthe experiment.In the presented dataset, we observe Knudsen numberdependent skimmer interference for a µ m skimmer,and a specially designed µ m skimmer placed 5.3 mmaway from a cold source. We postulate that the rest ofthe discrepancies between the experimental data and themodel may be due to either backscattering interferencesat quasi-molecular flow regimes, or a continuation of thesupersonic expansion after the beam has passed throughthe skimmer. Another explanation may be that the as-2 P [bar] I [ c oun t s / s ] R >x
F S
FIG. 8. Plot of measured intensities for a warm source (300K), and 120 µ m (pink) and 390 µ m Beam Dynamics skimmers(red). The measured intensities are compared to eq. (12),with the expansion stopped before the skimmer and S i = S (cid:107) . Note how after the quitting surface has surpassed theskimmer, the model loses its predictability (light grey for x S =17 . , dark grey indicates the whole span for the differentvalues of x S ). The difference in intensities between the twoskimmers is due to the fact that they were obtained usingdifferent pinholes in front of the detector (see Appendix D). sumption of the quitting surface stopping abruptly at agiven distance is is too simple to adequately describe thephysics in this regime. ACKNOWLEDGMENTS
The work presented here was sponsored by the Euro-pean Union: Theme NMP.2012.1.4-3 Grant no. 309672,project NEMI (Neutral Microscopy). We thank YairSegev from the the Weizmann Institute of Science (Is-rael) for his very useful and detailed comments, especiallyfor his comments regarding theoretical background. Wethank Kurt Ansperger Design, Entwicklung und Bau vonPrototypen, Moserhofgasse 24 C / 1, 8010 Graz, Austriaand Department of Physics/Experimental Physics, Fine-mechanical Laboratory and Workshop, Karl-Franzens-University Graz, Universitatsplatz 5, 8010 Graz, Austriafor the design and production of the Kurt skimmer. Wethank Jon Roozenbeek for his useful edits.
APPENDIX A: ADAPTATION TO ANEXPANSION AFTER THE SKIMMER
An untreated case in literature is when collisional ex-pansion continues after the skimmer. A way to approachthis problem is to assume that the expansion is unaf-fected by this interaction and simply project the quitting P [bar] I [ c oun t s / s ] FIG. 9. Plot of measured and predicted intensities for a coldsource (125 K) and the Beam Dynamics skimmers: 120 µ m (pink) and 390 µ m (red). The intensities are computed us-ing eq. (12) and assuming that the expansion stops at theskimmer with S i = S ⊥ . The intensities are plotted for threevalues of x S : 5.3 mm (circles), 11.3 mm (upwards arrows) and17.3 mm (asterisks). Note how for P > bar and 390 µ m skimmer (red), in the case of x S = x S positions. All measurementswere taken with r D = 25 µ m (see Appendix D and Fig. 3) P [bar] I [ c oun t s / s ] FIG. 10. Plot of measured and predicted intensities for awarm source and the glass skimmers: 18 µ m (black) and 4 µ m (green). The intensities are computed using eq. (12) andassuming that the expansion stops at the skimmer with S i = S ⊥ . surface further ahead until its predicted radius R F (seeFig. 18).The centre line intensity must be calculated using eq.(12), with a → a (cid:48) , r S → r (cid:48) S , x S → x (cid:48) S : a (cid:48) = a − (cid:18) R F cos(arctan r S x S ) − x S (cid:19) (20)3 [bar]10 I [ c oun t s / s ] FIG. 11. Plot of measured and predicted intensities for acold source and the glass skimmers: 18 µ m (black) and 4 µ m (green). The intensities are computed assuming that the ex-pansion stops at the skimmer with S i = S ⊥ . P [bar] I [ c oun t s / s ] FIG. 12. Plot of measured and computed intensities for the µ m Kurt skimmer (black) and a 120 µ m (pink) Beamdynamics skimmer for a warm source. The intensities arecomputed assuming that the expansion stops at the skimmerwith S i = S ⊥ . Note how strong discrepancies are not ob-served except for the case of the µ m Kurt skimmer. Twodiscrepancy modes can be observed, a very significant one for x S = 5.3 mm and a less significant one for the rest of nozzle-skimmer distances. r (cid:48) S = R F sin (cid:18) arctan r S x S (cid:19) (21) x (cid:48) S = R F cos (cid:18) arctan r S x S (cid:19) (22) P [bar] I [ c oun t s / s m ] FIG. 13. Measured centre line intensities per area in counts/ s · m for a warm source, and for the following skimmer aper-tures: 120 µ m Beam Dynamics (blue), 390 µ m Beam Dy-namics (red), 18 µ m glass skimmer (black), and µ m glassskimmer (green). The circle, triangle, and asterisk markerscorrespond to the nozzle-skimmer distances, x S , of 5.3 mm,11.3 mm, and 17.3 mm respectively. P [bar] I [ c oun t s / s * m ] FIG. 14. Measured centre line intensities per area in counts/ s · m for a cold source, and for the following skimmer aper-tures: 120 µ m Beam Dynamics (blue), 390 µ m Beam Dy-namics (red), 18 µ m glass skimmer (black), and µ m glassskimmer (green). The round, triangle, and asterisk markerscorrespond to the nozzle-skimmer distances, x S , of 5.3 mm,11.3 mm, and 17.3 mm respectively. APPENDIX B: DERIVATION OF THE QSMODEL
The contribution to the number density by a differen-tial of the quitting surface dS placed at a point P to thepoint P (cid:48) is [19]: dN ( x D , , z D ) = n ( R F , δ, η ) f ell ( v, θ ) d v. (23)4 P [bar] -0.500.511.522.5 I C - I w [ c oun t s / s m ] FIG. 15. Measured differences between cold source and warmsource beam intensities per area in counts/ s · m for the fol-lowing skimmer apertures: 120 µ m Beam Dynamics (blue),390 µ m Beam Dynamics (red).The circle, triangle, and aster-isk markers correspond to the nozzle-skimmer distances, x S ,of 5.3 mm, 11.3 mm, and 17.3 mm respectively. The continu-ous line indicates that where experimental data was missing,data was extrapolated from the closest experimental points. P [bar] -1.5-1-0.500.511.52 I C - I w [ c oun t s / s m ] FIG. 16. Measured differences between cold source and warmsource beam intensities per area in counts/ s · m for the fol-lowing skimmer apertures: 18 µ m glass skimmer (black), and µ m glass skimmer (green). The round, triangle, and asteriskmarkers correspond to the nozzle-skimmer distances, x S , of5.3 mm, 11.3 mm, and 17.3 mm respectively. The continuousline indicates that where experimental data was missing, datawas extrapolated from the closest experimental points. In this equation, n ( R F , δ, η ) ≡ n ( R F ) g ( δ ) is the numberdensity at the quitting surface, that is allowed to dependon the angle δ to account for the fact that the nozzle is notactually point-like. f ell ( v, θ ) is the ellipsoidal Maxwelliandistribution defined in eq. (7). v is the modulus of thespeed vector and θ is the angle between the segment PP’ P [bar] S FIG. 17. Predicted value of S ⊥ for a cold source (125 K) ac-cording to the numerical calculation of the supersonic expan-sion presented in Sec. II A. The round, triangle, and asteriskmarkers correspond to the nozzle-skimmer distances, x S , of5.3 mm, 11.3 mm, and 17.3 mm respectively. x S x D R F r SSKIMMER a r
DDETECTORNOZZLE r’ S a’x’ S expansion cone S S FIG. 18. Diagram of the supersonic expansion for the case ofa radius of the quitting surface radius higher that the distancebetween the nozzle and the skimmer. The quitting surface isassumed to expand unaffected by the skimmer aperture, ex-cept by collimation. R F is the radius of the quitting surface, y is the distance between the axis of symmetry and the pro-jection of the maximum-angle ray on the quitting surface, r S is the skimmer radius and r D is the radius of the detec-tor. a is the distance between the skimmer and the detector, d is the distance from the skimmer to the point where themaximum-angle ray crosses the symmetry axis. x D is thedistance between the nozzle and the detector and x S is thedistance between the nozzle and the skimmer. and P (see Fig. 1). Following the derivation from [19],one obtains: N (P (cid:48) ) = τ n ( R F )2 πa (cid:90) r S (cid:90) π g ( δ ) r cos β · (cid:15) e − S (cid:107) (1 − (cid:15) cos θ ) D ( b ) drdα, (24)where S (cid:107) = U/c (cid:107) is the parallel speed ratio, (cid:15) = (cid:0) ( τ sin θ + cos θ (cid:1) − / , τ = T (cid:107) T ⊥ . The function D ( b ) is5 P [bar] I [ c oun t s / s ] FIG. 19. Plot of measured intensities for a warm source andthe Beam Dynamics skimmers: (300 K), and 120 µ m (pink)and 390 µ m Beam Dynamics skimmers (red). The measuredintensities are compared to eq. (12), with the expansionstopped after the skimmer and S i = S (cid:107) . defined as follows: D ( b ) ≡ √ π be − b + (cid:0) b + 1 (cid:1) [1 + erf ( b )] , b ≡ S (cid:107) (cid:15) cos θ (25)The angle β is shown in Fig. 1. N ( P (cid:48) ) corresponds tothe number density at a radial position from the axisof symmetry, to obtain the number density at a circulardetector we must integrate over the arriving differentialvolume: N total = ∆ x (cid:90) S N (P (cid:48) ) d S = 2 π ∆ x (cid:90) r D N ( x D , ρ ) ρdρ. (26)Imposing that the proportion of intensities must corre-spond to the proportion of number densities, we can ob-tain the expression for the centre line intensity arrivingat a circular detector: I D I = N total π (cid:82) R F R F − ∆ x (cid:82) π n ( r ) r g ( δ ) sin δdδdr . (27)We obtain: I D = τ I πa R L (cid:90) r D (cid:90) r S (cid:90) π g ( δ ) r · ρ cos β · (cid:15) e − S (cid:107) (1 − (cid:15) cos θ ) D ( b ) dρdrdα. (28)Where I is defined in eq. (3). L corresponds to theintegration of g ( δ ) along the half sphere (all the intensityemitted by the source is set to be contained in g ( δ ) ). L ≡ (cid:90) π g ( δ ) sin δdδ. (29). APPENDIX C: EQUATIONS FOR THE RAYTRACING CODE
Using trigonometry, it is possible to determine ex-actly the maximum possible δ m within a source-skimmer-detector geometry (see Fig. 2). δ m = arcsin yR F . (30)Now, we use the Pythagorean theorem to obtain y , theheight of the triangle containing the angle δ m , x is thebasis of the triangle as shown in Fig. 2. yd + ( x S − x ) = r S d , x = (cid:113) R − y . (31)Expanding eqs. (31) we obtain the following quadraticequation: (cid:18) ydr S − d − x S (cid:19) = R − y , (32)expanding in powers of y : y (cid:18) ( dr S ) + 1 (cid:19) + y (cid:18) − dr S ( d + x S ) (cid:19) +( d + x S ) − R = 0 (33)Which can be solved using the quadratic formula: y = 2 d ( d + x S ) r S ± (cid:113) d r R − d + x S ) + 4 R dr S ) + 2 = dr S ( d + x S ) ± (cid:113) d r R + R − ( d + x S ) ( dr S ) + 1 . (34)The distance d is also obtained using trigonometry (seeFig. 2). r S d = r D a − d → d = ar S r D + r S . (35)To determine whether to take the positive or negativesquare root in eq. 34, we can take the case x = R F (whichcorresponds to the case R F → ∞ ). In this case, fromtrigonometry it is easy to see that y = r S d ( d + x s − R F ) .Thus, the geometrically-sound case corresponds to thenegative square root. VII. APPENDIX D: r D - r S TABLE
TABLE I. Table showing the values for the skimmer radius r S ,and the radius of the pinhole placed in front of the detector r D , for the experiments presented in this paper. Skimmer diameter r S r D (warm) r D (cold)4 µ m µ m µ m µ m µ m µ m µ m µ m µ m µ m µ m not shown120 µ m µ m µ m µ m µ m µ m µ m µ m VIII. APPENDIX E: GLOSSARY OF SYMBOLS
Symbol
Description Kn ∗ Modified Knudsen numberKn Knudsen number S (cid:107) Parallel speed ratio η P Power law parameter on the collision model λ Mean free path of gas particles r S Skimmer radius n Number density of the gas at the skimmer σ Cross section of gas particles ¯ v Most probable velocity alongthe radial direction T (cid:107) Parallel temperature of the expansion T ⊥ Perpendicular temperature of the expansion v (cid:107) Parallel component of the velocity v ⊥ Perpendicular component of the velocityM Mach numberv Average velocity of the gasc Local speed of sound I Total intensity stemming from the nozzle T Stagnation temperature inside the nozzle P Stagnation pressure inside the nozzle k B Boltzmann constant γ Ratio of heat capacities d N Diameter of the nozzlem Mass of a gas particle Ω( T eff ) Collision integral in the Boltzmann equation T eff Effective average temperature of the gas Q Viscosity cross-sectionE Collision energy in the centre of mass system (cid:126)
Reduced Planck constant η l Phase shifts for orbital momentum lf ell Velocity distribution in the expansion V LJ Lennard-Jones potential r LJ Distance between two interacting particles r m Distance where V LJ is minimum (cid:15) Depth of the potential well in V LJ R F Radius of the quitting surface I D Centre-line intensity (ellipsoidal model) τ T (cid:107) /T ⊥ a Distance between the skimmerand the detector r D Radius of detector openingP Point on the quitting surfaceP’ Point on the detector (cid:126)r
Vector connecting P and P’r Distance from beam axis to where theskimmer plane intersects (cid:126)r x,y,z Cartesian coordinates β Angle between (cid:126)r and the xz plane θ Angle between P and (cid:126)rα
Angle between r (note, not \vec{r}) andthe xz plane ρ Distance between P’ and the detector centre g ( δ ) Angular dependency of the gas densityon the quitting surface L Integral of g ( δ ) over the quitting surface S i Speed ratio term in Sikora’s model I Sikora’s centre line intensity beforeapproximation Φ Angle of rotation about the beam axis T (cid:107)∞ Asymptotic value of the parallel temperature η D Efficiency of the detector in counts/part I Intensity arriving at the detector assumingno skimmer presence x S Distance between nozzle and skimmer I S Sikora’s centre line intensity assuming r S (cid:28) x S , r S (cid:28) a , a/r S >> S i , r D << an BE Background number density in theexpansion chamber n BC Background number density insubsequent chambers δ m Maximal angle on the quitting surfaced Distance from the skimmer to the point wherethe maximum-angle ray crosses the beam axis φ Azimuthal angle in spherical coordinates δ Polar angle in spherical coordinates( δ = 0 lays over x) A D Area of the detector A S Area of the skimmer x A Distance between the nozzle and thenoise-reducing aperture [1] R. Campargue, Rev. Sci. Instrum. , 111 (1964).[2] H. Pauly, Atom, Molecule, and Cluster Beams I , 1st ed.(Springer-Verlag, Berlin, 2000).[3] D. P. DePonte, S. D. Kevan, and F. S. Patton, Rev. Sci.Instrum. , 55107 (2006).[4] G. Scoles, B. D, and U. Buck, Atomic and MolecularBeam Methods , Vol. 1 (Oxford University Press, NewYork Oxford, 1988) p. 752.[5] S. D. Eder, B. Samelin, G. Bracco, K. Ansperger, andB. Holst, Rev. Sci. Instrum (2013).[6] U. Even, EPJ Tech. Instrum. , 17 (2015).[7] M. Koch, S. Rehbein, G. Schmahl, T. Reisinger,G. Bracco, W. E. Ernst, and B. Holst, J. Microsc. ,1 (2008).[8] A. Fahy, M. Barr, J. Martens, and P. Dastoor, Rev. Sci.Instrum , 023704 (2015).[9] S. D. Eder, T. Reisinger, M. M. Greve, G. Bracco, andB. Holst, New. J. Phys. , 73014 (2012). [10] A. Salvador Palau, G. Bracco, and B. Holst, Phys. Rev.A (2016).[11] G. A. Bird, Physics of Fluids , 1486 (1976).[12] G. a. Bird, Molecular Gas Dynamics and Direct Simula-tion of Gas Flows , 1st ed. (Oxford University Press, NewYork Oxford, 1994) p. 458.[13] T. Reisinger, G. Bracco, S. Rehbein, G. Schmahl, W. E.Ernst, and B. Holst, J. Phys. Chem A , 12620 (2007).[14] H. Hedgeland, A. P. Jardine, W. Allison, and J. Ellis,Rev. Sci. Instrum. , 123111 (2005).[15] J. Braun, P. K. Day, J. P. Toennies, G. Witte, and E. Ne-her, Rev. Sci. Instrum. , 3001 (1997).[16] M. J. Verheijen, H. C. W. Beijerinck, W. A. Renes, andN. F. Verster, Chem. Phys. , 63 (1984).[17] J. B. Anderson and J. B. Fenn, Physics of Fluids , 780(1965).[18] H. C. W. Beijerinck and N. F. Verster, Physica C ,327 (1981). [19] U. Bossel, Skimming of Molecular Beams from Diverg-ing Non-equilibrium Gas Jets , Deutsche Luft- und Raum-fahrt. Forschungsbericht (Deutsche Forschungs-und Ver-suchsanstalt für Luft-und Raumfahrt, 1974).[20] G. S. Sikora,
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