Collision statistics of clusters: From Poisson model to Poisson mixtures

1 Collision statistics of clusters: From Poisson model to Poisson mixtures Sascha Vongehr, S haochun Tang ( 唐少春 唐少春 唐少春 唐少春 ) , and Xiangkang Meng ( 孟祥康 孟祥康 孟祥康 孟祥康 ) National Laborator y of Solid State Microstructure s, Department of Materials S cience and Engineering, Nanjing Universit y, Nanjing 210093, P.R . China Abstract: Cl usters traverse a gas and c ollide with gas particl es. T he gas particles ar e adsor bed and the clusters bec ome hosts. I f the clusters a re size selected , the n umber of guests will b e Poisso n distributed. W e review this b y showcasin g four labora tory procedures that al l rel y on the validity o f the Po isson mode l. The effects of a s tatistical distr ibution of the cl usters ’ sizes in a b eam of clusters ar e disc ussed. W e derive the average collis ion rates. Additio nally, we presen t Poisson mixture models that in volve also standard deviations. W e der ive the collisio n statistics for co mmon size d istributions o f hosts and also for some generalizatio ns thereof. T he mod els can be a pplied to large nob le gas clusters tra versing do ping gas. W hile outlining ho w to fit a ge neralized Poisso n to the statistics, we still find even these P oisson models to b e often insuf ficient. Keywords: collision statist ics, helium droplets, clusters, Poisson statistics PACC: 36.40.-c, 02.50.-r, 81.07.Nb 1. Introduction When clusters traverse gas, they co llide with the gas particles. For a clus ter of a giv en size, the random collisions are described by a Poisson pr ocess, but the size distribution o f clusters inside a 2 beam is usually broad. While fractionation le ads often to exponential distribu tions, random growth processes like phase change ag gregations give m ostly rise to log-norm al ones, be it in biology, econom ics or cluster physics [1,2,3] . Usually, in beams of clusters, the clusters’ num ber of atoms N has a standard deviation alm ost as large as the average size. This larg e initial uncertainty complicates the statist ics of the collision experiment. The ab solute differences of more than 100% between probabilitie s in simple and im proved models (figures 1 to 3) are m ostly not practically relevant. We draw from laboratory experience and only want to point out practically important corrections. Therefore, we review the simple Poisson m odel (section 2) by showcasing four procedures that rely on the validity of that m odel: Single molecule isolation spectroscopy (HENDI [4,5] ), the dependence of mass spectrometric peak s on changing doping g as density, and so on. With th e desire to be practically relevant, we first prov ide formulas for the influence of the host cluster dis tribution on the capture statistics. Firstly , we derive analytically exact expressions for the effects on av erage capture rates (section 3). These se ctions also introduce a necessarily strict notation. Sta tistical quantities like av erage and variance all exist for the probability distribution of the hosts and a lso for that of the guests. Then we introduce Poisson mixtures and derive exact expressions for the standard deviations a (section 4) from the Poisson mixtures’ general properties. Reporting a measurem ent via its average and standard deviation is the accep ted scientific standard. Why do we deal with the full com plexity of Poisson mixtures (section 5 to 7)? Our initial motivation was to model exotic collision and pick- up cross sections [6] due to electrical polarizability, sticking coefficients, etc. How ever, the m ixture models had already unexpected results when merely still considering simple cross sections. Som etimes, the shape of the actual distribution can mislead the researche r’s interpretations about physical p rocesses (especially a A downward s correctio n of an avera ge wit hout correctin g the deviatio n nece ssarily incre ases the hei ght of the maximu m. Actuall y, the maximum i s alwa ys lower (see black peaks, f igures 1 to 3). 3 section 5 and 7). Sometimes, the average and deviation shift only a little, but applicat ion of the Poisson model leads nevertheless to sig nificant errors (section 6). The treatment is only valid for larg e hosts picking up guests that do not prov ide too much energy (e.g. via condensation to a strong ly bound guest cluster). Espec ially when dealing with small host clusters, it is necessary to account for the evaporation r esulting from dissipation of impact and binding energy and more subtle effects on the final host cluste r size distribution resulting from preferential scattering of small clusters out of the be am. Such can only be done numerically, for example with a Mo nte Carlo modeling m ethod [7,8] , but it is difficult to infer conclusions valid under dif ferent conditions without running sim ulations every tim e again, especially if a detection method (lase r induced fluorescence (LI F), ionization, etc) is already nested inside the simulation. Ev aporation renders collisions history dep endent, i.e. the next collision depends on how m any there have been already . This violates the main assumption of a Poisson process. Hence, the widespread us e of so called effective cross sec tions while keeping the Poisson model cannot be a m athematically sound approach. Moreover, we show t hat the deviation needs to be decoupled from the average. There are m any statistical distributions that do so. It is our strategy to select the m ost promising ones by sticking to analytical models with wel l understood parameters for as long as possible. One eventually m ust give up analyticity and maybe also enter effective cross sections in o rder to account for evaporation, detector pa rameters etc. Nevertheless, the strate gy demands to first correctly account for the influence of the hosts ’ size distribution. It seems t o be the only effect that can be dealt w ith analytically. Moreover, this clearly distinguishes the sp ecific influence of the hosts’ size dis tribution on the shape of guest distributions. Beam depleti on by deflection and evapor ation are very important for sm all hosts, but with large hosts, the effects of dis tribution mixing are more im portant. In other words, we distill out the effects of the easily neglected large size “tail” of the host distributions. They turn out surprisingly important. 4 All probability distributions and prob ability density functions (PDF) that we used for calculations are listed in the appendix in order to facilit ate comparison. 2. The Poisson model and its practical use For clarity, we will present everything whi le having in m ind large noble gas clusters that capture collision partners efficiently and are not deflected much. In H elium nano-droplets for example, guest particles will condense v ery fast, and after k collisions, the host cluster contain s a guest cluster of size k . Generally, when a host travels through a length of gas, the probabi lity of colliding with k particles is k P . The probability not to collide is 0 k P e − = . It depends b on : k D σ = , where σ is the cross section and D the doping strength. : D nFL =  combines the gas related properties: parti cle number density n  , path length L , and F , which takes into account the velocity distrib ution c of the gas [9] . If collisions are independent of each other and the clusters are massive enough to col lide with several particles without be ing deflected, one can use 0 P to straightforwardly deduc e the Poisson distribution d 0 ! k k P P k k = . This distribution is equi- dispersed, i.e. the variance equals t he mean: ( ) 2 var : k k k k P k = − = ∑ (1) The Poisson distribution has therefore on ly one degree of freedom and is rather rigid. b “ : = ” defines a ne w symbol on it s left hand s ide, while “ : = ” defines a ne w one on t he right sid e. c ( ) ( ) 2 2 1 2 0 ˆ 2 ; x x t x F x e x e dt x v u π − − − − = + + = ∫ . The host cluster ’s speed is v and the mo st prob able speed ˆ u of the scatteri ng gas pa rticle is give n via 2 B ˆ 2 mu k T = . d The b ar over k d oes not extend over the p ower k . A bar always indicate s the avera ging over all k . 5 There are four methods of practical import ance in the laborato ry that all rely on the Poisson model bei ng valid. We will revis it them often later on in order to sh ow how the different host size distributio ns may have an impact on each of these methods: 1: Monome r isolation : No ble gas host clu ster beams are us ed for ultra cold isolation spectrosc opy. Typically, one i nvestigates isolat ed molecules via their LIF signal. The doping is chosen t o be so weak that the average number of picked up molec ules is belo w unity. With 0.3 k ≅ for example , only about 25% of hosts pick up a nything at all ( 0 k > ). For every ho st that picks up one guest, o nly 2 1 2 0.15 P P k = ≅ collide with two. The LIF is the n treated as if originating fr om isolated mol ecules only. Of a ll doped hosts, o nly ( ) ( ) 0 1 0 1 1 14% P P P − − − ≅ carry cluste rs instead of the de sired monomers. 2: Weak dop ing depende nce (WDD): When fe w collisions occur , the probability can be expressed as the W DD limit ( ) 0 lim 1 ! k k k P k k k → = − . A signal due to guests of size k is proporti onal to k P and theref ore a function of the respective avera ge to the k th power a t the origin ( 0 k = ). A signal’s li near rise (i.e. one to the 1 st power) accompanying a s light chan ge in the dopin g gas pressure i s often used to argue that the signal i s due to monomers 1 k = only. A qua dratic rise may ident ify a signal, such as a certain LIF freq uency, as originat ing from dimers (a cluster with two monomers ). One may al so observe small guest clusters and infer the involved cross sectio ns by employing 2 1 0 lim k P k k → = − and ( ) 2 3 2 0 lim 2 k P k k → = − . 3: Maxi mum when changi ng doping (MCD) : The deri vative ( ) var k k dP P k k dk = − shows tha t k P is maximal whe n the avera ge pick-up is k k = . This allow s deriving total cro ss sections [10] af ter finding th e doping strength that r esults in a maximum s ignal du e to hosts of si ze k . 4: Fixed doping ratios (F DR): Monomer isol ation tries to avoi d the total of all mult i-mer signal s because o ne may not be ab le to relate signal type to guest cluster size. Now t hink of signals with 6 known origin (guests of size k ) and of larger collision rates. Consider the mass abundanc e spectru m of guest clusters . Given one peak in the mass sp ectrum, the FDR ratio ( ) 1 k k P k k P − = predicts all other peaks. The hi ghest peak (the maximum around 1 / 2 1 / 2 k k P P − + = ) is found at 1 / 2 k k ≅ − . 3. Average collision rates The ho st clusters’ sizes N are statisti cally distributed. Since one observes a statistical ensemble of hosts in the beam, the e xpectation val ue of any observable Ψ is e 0 PDF dN ∞ Ψ = Ψ ∫ . A linear exp onential (EXP) wi th / 1 PDF N N N e − = has been observed for large c lusters gained from so-calle d supercritical ex pansions, e. g.: for Helium droplet s [11] He N . The EXP is equi- dispersed, i.e., the standard devia tion equals the mean N N ∆ = . I t the refore has only one degree of free dom. Whenever possib le, we derived all desirable equ ations for more general distributions dependent on the dispersion r atio, which is defined as: 2 2 : X X d X = ∆ (2) For instanc e, results for the EXP follow at once b y setting 1 N d = in formulas valid for the gamma ( Γ ) distrib ution. After sub-critical beam expansions , the condensed clus ters have log- e Notation: An average over all N use s angled br ackets, e.g. k P . Depende nce on discrete variab les is written withou t brackets, a s for P k . Depende nce on a contin uous variable uses rou nd brack ets, as in σ ( N ) . N is continuo us because the p resent work is about very large clusters. T here is no co nvenient discretizatio n of the host size d istribution s conside red here. Variables may serve as ide ntifying (rat her than c ounting) index, e.g. the standar d deviatio n 2 2 : N N N ∆ = − . 7 normal (L N) size distributions whose two degree s of freed om are given via the mean n and standar d deviation n ∆ of the logarit hm : ln n N = . The LN i s simply the well known normal distributio n in n -space. The followi ng formula is very useful, esp ecially when cl osed expressions are impossible and one needs to expand an ob servable Ψ in a power s eries: ( ) ( ) ( ) 2 ; for - distr. ; for LN- distr. exp 1 2 N N a N d a d a a n d N N a a − +  Γ Γ Γ  =    − ∆     (3) An importa nt example is a cluster’s geometrical cross section 2 2 / 3 S r N σ π = with the Wigner - Seitz radi us S r (e.g.: Helium 2.221 Å S r − = ) [12] . Since ( ) N k σ σ ∝ = , a shift of σ will shift t he expected average number of c ollisions or guests : k D κ σ = = . Thus, FDR an d the MCD both shift . In case of the EXP-distribution, t he observed a verage is onl y 90.3% as large as naivel y expected , because of ( ) ( ) 2 / 3 2 5 / 3 / 0.903 S r N σ π = Γ ≈ . This la rge correction holds for any expone ntial size distribution, r egardless of its width N ∆ . Cluster beams can usuall y be described via few N N ∆ ≅ ; the LN-dist ributed ones, to o. For Helium cluster s from contin uous nozzles it is experiment ally establis hed that they obey [13,14] 0.55 0.03 2.83 n N d ∆ = ± ⇒ ≅ (4) This lea ds to a correction of 2 ( / 3 ) e 96.7% n − ∆ = . This is not as impressive a s the 10% error for the EXP. Ho wever, it will be shown later how even this small deviatio n can lead to a di screpancy of almost 30% . This comes a bout because the new average is actually that of a differently shape d distributio n. The small shif t here does sadly not imply that the Poisson distribu tion can be truste d for LN-dist ributed hosts. Also, the corrections do not depend on t he doping. They ar e just as valid 8 in the limit of vanishing gas de nsity and should be taken into accou nt even when obser ving the WDD. 4. Poisson mixtures The just discussed average nu mber of collisi ons or guests k κ = is only th e first importa nt degree of f reedom of the gue st clusters’ d istribution k P , which is called a Poisso n mixture [15] . Poisson mixt ure models are wel l known in a ctuarial science to model total ins urance claim distributio ns. The Poisson mixture is said to be “mixed” by the hosts’ size distri bution. The size distributio n is called “mixing dist ribution”. Usin g { } 0,1 p ∈ for the normalization and the average of a distribution r espectively, un-mixed and mixed expre ssions are very similar , i.e., p p k k k P k = ∑ becomes p p k k k P κ = ∑ . The mixture ’s variance is ( ) 2 VA R : k k k P κ = − ∑ . A Poiss on mixture is al ways over-dispersed b ecause the Poi sson distributio n’s va r k = leads to the mixture ’s variance f being ( ) 2 VA R 1 k k d κ κ κ = + ∆ = + (5) This allows us to present corr ection formulas for deviation s even when lackin g a closed expression for the mixed distributions. The LN is convenie nt: Given an LN host size distri bution, any propo rtional variable a M N ∝ is also log-normal with ln M n a ∆ = ∆ and ( ) ln ln a M a n M N = + . Definin g : a k BN = leads thus straight to k ∆ and k d . For the f Changing t he sequence o f sum and i ntegratio n ( ... .. . = ∑ ∑ ) and sever al times ap plying ( ) 2 2 var ( ) me an( ) mean( ) x x x = − shows 2 VAR var k = + ∆ holds true for any mi xture model. 9 EXP, the distribution of k and thus k d can be derived with th e cumulative dis tribution functi on of the EXP. The results are ( ) ( ) 1 / 2 1 2 ln 4 ! ; for EXP-distr. 1 ; for LN- distr. exp a a k k a d π + −  Γ  = − +    ∆     (6) Even when lacking close d expressions for the mixtures, one can still derive so me further gener al statements by looking at the power expa nsions. Via expand ing ( ) 0 ! 1 ! i k k k i k i k P k e k i ∞ − + = = = − ∑ one may deri ve a general expr ession for Pois son mixtures (10) and investigat e the leading terms. The W DD has the sa me leading power after mixing (i.e. aft er averaging over t he hosts’ sizes) , but to i nfer total cross sections fr om signals of s mall guest clusters at weak dopi ng is not so easy anymore: T he proportionality fa ctors for 2 0 lim k P κ ≥ → and for the terms in 1 0 lim P κ → of higher tha n first order in κ depend on the size distributio n of the hosts: ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 ln ln ln 1 1 1 1 1 0 1 2 1 2 2 ... ; for EXP-distr. lim ; for L N- distr. ! 1 2 ... * k k k k k ak a a a k k k k k k P k e e e κ κ κ κ κ + + + + + → − ∆ ∆ + ∆    Γ − Γ Γ + Γ    =     − + −    (7) For the L N, this can be more intuitively expres sed with ( ) : 1 1 k d ε = + , because one can then write 3 2 1 0 lim 1 2 ... P κ κ ε κ ε κ →   = − + −   and 2 2 3 2 2 0 lim 1 2 ... 2 P κ κ ε ε κ ε κ →   = − + −   . The aver age of collisions κ may also b e varied by adju sting the avera ge host size N instead of the dopi ng gas pressure, but in practice this alters the host clusters’ beam expa nsion condition s, which cha nges the overall beam fl ux. Therefore, we let N untou ched and that is how the derivative k d P d κ shoul d be understoo d here. Using 1 a d d N d d B κ − = , one may drag 10 the derivati ve into the integra l PDF k P dN ∫ , where it c an only act on k P , because the PDF is independ ent of doping. It foll ows that the MCD is at ( ) 1 1 k k k P k P + = + (8) This con cludes the general part . Even simplif ying linear approximati ons (e.g.: a k BN = with 1 a = ) are gener ally not helpful. Obtaining a clos ed, let alone tract able expression for k P is possible on ly for certain mixin g distributions [11,16] . 5. Instructive example: linea r cross section with exponential (gamma) mixing A li near appro ximation a k BN = wit h 1 a = may be physically due to the cross section being (effecti vely) proporti onal to N . This leads to B N κ = . Since it is eas ily done an d needed l ater, the results a re here given f or the hosts’ sizes being Γ -distributed; the results for the EXP f ollow by s etting 1 N d = . The Poiss on mixt ure k P is the probability function of the Negati ve Binomial (NB) or “Pasc al” distribu tion (11) . For 1 N d = it is c alled “geometr ic” d istribution. The varianc e is ( ) VA R N N d d κ κ = + . Apart from the d ifference b etween va r and VAR , the derivati ve retains the form it h ad before the mixing: ( ) VAR k k d P P k d κ κ = − . Therefore, relative to κ instead of k , the MCD s urvives the mixi ng. The W DD becomes: ( ) ( ) ( ) ( ) 0 lim 1 ! N N k d k k k k N d P k d κ κ κ κ + → Γ = − + Γ (9) For clarity an d practical significance, the rest of this section ass umes an EXP distribution. The limit 0 lim k k P κ κ → = confirms t hat the WDD of 1 P is the s ame after the mixing (i. e. after averaging over h ost sizes) as it was before, at least to first order in κ . However, a ll k P with 11 1 k > are modifi ed: ( ) ( ) 0 lim 1 1 k k P k κ κ κ → = − + . Mono mer isolation is compromise d because for e very host that collide s only once, ( ) 0 1 1 1 P P P κ − − = pick up mult iple guests. Wit h 0.3 k κ = ≅ , now 23% of the doped hosts carry clusters inste ad of the desir ed monomers. Th e FDR becomes ( ) 0 1 1 k k P P P − = − . This impli es that a mass spectrum of the guest clusters looks now always as if the signals are syste matically suppresse d. Peaks proportional to k P monotonic ally decrease with k . In order t o illustrate the e ffects with an e xample, assume tha t the dopin g strength (gas den sity) and the a verage size of the hosts lead togeth er to on avera ge 10.5 k κ = = guests. T o compare the traditio nal with the improved model s, we calculat ed guest abundan ce spectra (figure 1). The Poisson model (grey) 10.5 10.5 ! k k k P e − = predicts the maximum at 10 k = . Taking the exponential host clust er size distribution in to account, the s pectrum (black) i s due to a geometric distribution 1 10.5 11.5 11.5 k k P       = . It looks as i f it is obtained from a violent fra ctionation of ver y large guest clusters. There is no maximum anymo re. Instea d, a strong right t ail with unex pectedly large guests a ppears. The effect is here especially obvio us because o f the simplified cross section and the EXP size distribution. However , this eff ect of the host’s size distribution mixing is always present to some extend. This generally leads to overestimation of the severity of fractionating detection processes or host dep letion due to binding energy relea se when guests meet in/on the host. 12 1 5 10 1 5 2 0 25 k 0.02 0.04 0.06 0.08 0.1 0.12 P k & X P k \ Figure 1 : Guest abunda nce sp ectra (= co llision stati stics) du e to t he Poi sson model (gra y) and d ue to a Poisso n mixture (blac k). A sh ape typical of gro wth pro cesse s turned into one typical for fr actionatio ns. This illu strative e xample assu mes a simp lified cross sec tion. 6. Log-normal host size dist ribution To mix a Poisson with an LN is difficult. As long as 0.6 n ∆ ≤ (compare equation (4)), th e Inverse Gau ssian (IG) distribu tion [17,18] trac es the LN extremely well . Conside ring the experimenta l accuracy, sub-criti cal cluster size di stributions c ould have been modeled with the IG all along. T he IG may also be a s well supported as the LN when co nsidering the unde rlying physical pr ocesses. We would like to dra w attention to the fact that the IG may be ver y useful for cluster ph ysics. Its Poisson mixture is a cl osed expression. We wa nt to fit the distrib ution of k D σ = . The cros s section σ is proporti onal to a N , so their distributio ns are also log-normal wi th ln ln n k a σ ∆ = ∆ = ∆ . Since 2 / 3 a ≅ , it follows f rom equatio n (4) that ln 0.37 k ∆ ≅ is small. Hence , one may with co nfidence expres s the ensemble of host clust ers via the IG’s ( ) , PDF k d κ while usin g equation (6), n amely 1 2 ln 2 exp ln k k κ   = ∆ +   , ln n k a ∆ = ∆ and ln ln k a n B = + . The collis ion probability can be integrated. T he probability of no collision is 0 exp k k P d d   = − Ω   with : 2 k d κ Ω = + . The general formula 13 (12) is now the probabilit y function of th e IG Poisson mixture distributi on (IGP). In the weak doping li mit, the following ex pression is val id for general k : ( ) ( ) ( ) ( ) 1 3 2 2 1 2 0 2 lim ! k k k k d k d d k k k k k k e P d d K d K k κ κ κ κ π − − − →   = + −   Ω   . E.g. for t he probability of only one c ollision one may wri te ( ) 3 3 2 1 0 1 lim 1 1 ... 2 k P d κ κ ε κ ε κ →     = − + − −       , which is the same as t he result for the LN up to 2 nd order i n κ (compare equ ation (7) and t hereafter). Thus, the main differen ce to the WDD bet ween Poisson model and LN mixed one is captured co rrectly by the IGP. Alre ady 2 2 0 lim P κ κ ε → = is twice the LN’s result. Th is could perh aps be measured to justify prefer ring the IG over the LN distribu tion. Only in t he limit k d = ∞ is the MCD s till at k κ = . Otherwis e it is at ( ) ( ) ( ) 3 2 1 2 k k d k k k k k d k d K d P P d k d K κ κ κ κ Ω − Ω −   Ω   = + −   Ω     . The proba bility of a single coll ision is therefore ( ) 1 1 k k d P P d d d κ κ κ κ = + − Ω Ω . For example , at 2 k d = , the maxi mum of 1,2 ,3 P is shifted by 6.7 , 10.8 and 1 3.7% away from 1 , 2 , 3 κ = respecti vely, requiring a h igher average doping to reach the maximum val ue. The shift upwards f rom k κ = worsens wit h k approach ing a limit of 27.9%. Guest cluster abundance spectra (figure 2) for h osts with the geometrical cross section of liquid drops, i.e. 2 / 3 a = , and a doping strength leadi ng to on avera ge 10 guests per host were calculate d: Uncorrected, the a verage value is assumed too high ( 10.34 k ≅ ) and t he Poisson model (whi te) predicts a max imum at 10 k = . The Poisson model wit h a corrected aver age (grey) of k κ = , here 10 κ = , predicts the peaks at 9 k = and 10 k = to be equ al. In the actual 14 spectru m due to the IGP (blac k), the pea k at 8 k = is the highes t. Also, a str ong right tail appea rs again. 1 5 10 15 20 25 k 0.02 0.04 0.06 0.08 0.1 0.12 P k & X P k \ Figure 2 : Guest abunda nce sp ectra due to the uncor rected Poisson model ( white), the mod el with corre cted average ( gray), a nd the P oisso n mixtur e (bla ck) that ta kes th e LN ho st size d istrib ution int o acc ount. Fitti ng the latter with a P oisso n curve would result i n large er rors. Mono mer isolation survives: With 0.3 0.29 k κ = ⇔ = , 2 1 0.159 P P ≅ is close to 2 1 0.15 P P = . Now 15% of the doped hosts carry clu sters instead of monomers . The c orrections are less s evere than with the EX P distribution, b ecause the LN is u nder-disperse d (equation (4) leads to 6.95 k d ≈ ). The s mallness of the power a cannot g be blamed . Hence, one may be tempted t o interpret the smal lness of corre ctions as reassurin g. The Poisson model seems only unable t o deal with EXP distrib uted host sizes , but this is decepti ve. What is routine ly done [19] is to fit the guests’ abundance sp ectrum with a Po isson distributi on in order to inf er the size of the g One may put a =1 agai n: In t he IGP -spectru m with κ =10 (Fig. 2) , 6 P would be the hi ghest, so the linear ap proxi mation, c ompare d to a = 2/3, onl y doubles t he shift fro m the u ncorre cted k = 10. With 1 a = and thus k κ = , monom er isolation at κ = 0. 3 would ha ve 17 .6% o f dop ed hosts car ry cluster s. 15 hosts. For the calculated case (f igure 2), such fitting underestimates the average pick-up by almost 20% , because a P oisson’s average a nd maximum are cl ose and the maximum peak went from 10 dow n to 8 k = during the mixin g with the host s’ size distrib ution. Calculating the average number N of atoms in th e hosts introduces the power 2 / 3 a = . This in turn worsens the matt er to only 3 / 2 ( 80%) 72% ≅ of the actual average. It is true tha t taking the LN i nto account re duces the average n umber of collisi ons κ to about 96.7%, but one should not conclude that utilizatio n of the Poisson model will only l ead to errors of about 4%. 7. Exponential host size distribution If the powe r 1 a < , the distrib ution of k will have a cusp around t he maximum even th ough N is EXP dis tributed; shifts will b e less severe t han those of section 5 . In order to calculate the Poisson mixt ure, we need to fit the PDF of k again. W e looked at many possibilities: Th e Power Inverse Gau ssian [20,21] (PIG) is at tractive beca use the LN belo ngs to it as a special lim itin g case. The PIG can f it the distrib ution of k for bot h types of size distri butions (!), but it h as no convenie nt Poisson mixture. T he Generalized Inverse Gaussian distribution (GIG) i ncludes many others as s pecial cases ( Γ , Hyperbolic, Re ciprocal Inverse Gaussian (RIG), …) and allows tractable Po isson mixture [22] . Of the whole GIG fa mily, Γ fits the PDF of k best, but, when sharing th e same k d and κ , the Γ is more p eaked h than the distr ibution of k . After fitt ing, one can use t he results for the Γ with li near approximation fr om above to c alculate an NB distri bution. However, t reating the L N taught us that a more peaked function can strongly undere stimate the h A differe nt fit ( e.g. a le ast sq uares fit) does not make the GI G fit well. More over, f unction al fits matc h average a nd d ispersion and ther eby prese rve tho se impor tan t quantitie s. 16 propertie s of interest here. Bei ng presented wit h these difficulties , we abandoned generality and only calculat ed for researchers in cluster science most important conditions. The exac t Poisson mixture ( equation (13)) f or the hosts ha ving an EXP size distribution and 2 / 3 a = leads t o the following results : The MCD is shifted downwards , i.e. it is reache d already at lower than expected gas pre ssures. The maxima of 1,2 ,3 ,..., 10 P are shifted by 3.0, 4 .6, 5.6, 6.2, 6.7, 7.0, 7.3, 7.6, 7.8 an d 7.9% away from 1 , 2 , 3 , ...,10 κ = to a lower avera ge doping. M onomer isolation is somewhat compro mised: With 0.3 0.27 k κ = ⇔ = and 2 1 0.178 P P ≅ , now 17% of t he doped hosts carry clusters instead of isolated monomers. Le t us again compare gues t cluster a bundance spectra . We assume that the experimental c onditions lead to on average 9.5 κ = guests (f igure 3): Uncorrecte d, the average is assumed too high ( 10.52 k ≅ ) and t he maximum is p redicted to be at 10 k = . A correcte d Poisson model still predicts the peak at 9 k = to be the highest. Taking the e xponential distribu tion fully into acc ount, the peak at 4 k = wins, a strong right tail with unexpect edly large clust ers appears, and the o verall shape suggest s fractionati on processes. The Po isson model fa ils because it fixes t he maximum to be c lose to the average. 1 5 1 0 15 2 0 25 k 0.02 0.04 0.06 0.08 0.1 0.12 P k an d X P k \ Figure 3 : Guest abunda nce sp ectra due to the Po isson mode l ( white), the model with cor rected average (gray), a nd the P oisson mixtur e (b lack) that takes the E XP h ost size distrib ution in to acc ount. 17 8. Generalized Poisson distribut ion To si mply facilitate a fitting of the shap e of the collision stati stics, one should b ear in mind that any mixing wi ll over-disperse an originally equ i-dispersed distribu tion. In other words , the deviation needs to be decouple d from the mean to allow one more d egree of fre edom. A well- studied alternative to the standard is the gene ralized Pois son [23] distributi on (GP). It has the mean ( ) 1 k u λ = − and ( ) 3 va r 1 u λ = − . It is over disper sed when 0 λ > and re duces to the Poisson at 0 λ = . Could a functional fit with a GP replace the Po isson mixture s? To find out , we rewrite t he GP as if it is already a mixture (14), i.e. we substitute GP k k P P → and pu t in the mixture’s a verage instead ( k κ → ). We fix λ in the n ew expression b y requiring also the variance ( ) 2 va r VAR 1 κ λ → = − to be correct, i.e. equal t o the Poisson mixt ure’s (5). This implies ( ) 2 1 1 k d λ κ − − = + . In orde r to prese nt an example (figure 4) resembling thos e above, doping an d average host si ze lead to on average 9.5 guests. The envelope of the guests’ spectr um due to an EXP-Poisson mixt ure (blac k lines) is not well repr oduced by the GP ( dashed black line). S uch fits may be sufficie nt with LN distribu ted host sizes (grey lines). 1 5 k 15 2 k 25 k 0.025 0.050 0.075 X P k \ & X P k \ GP Figure 4 : T he envelope s of gu est abunda nce sp ectra. The L N and E XP host size d istributi ons ar e take n into account (gr ey and black li nes r espectivel y). T he Poi sson mi xtures (so lid line s) are each fitte d with a generalized Po isson (da shed lines) . 18 9. Conclusion We cal culated the collision st atistics of size dis tributed host cluster s as far as possible analyticall y and without use of effective cross sections. The resultin g distributions allow a broader standard deviation, but there is a lar ge pool of probability dis tributions th at do so. In other wor ds, the restrictio n to employ only wel l understood parameters enabled us to select specific probability distributi ons. This mathemati cal rigor surp risingly disfavors the well known generalized Poisson model . Host size distri butions can be d ifferent from the most usual ones. Moreover, effective pick-up c ross sections may depe nd on a power a different fr om the geometri cal 2 / 3 a = size depen dence, for example whe n considering char ged particles, guest desorption [6] or cluste r surface layer c orrections [24] . There fore, we pro vided the Poisson mixt ure models by keepi ng expression s as general a s possible. Nevert heless, with the desire to stay practicall y relevant, common l aboratory metho ds were discussed and relativel y sim ple co rrection formulas for estimation of means and stan dard deviations ha ve also been provid ed. It was shown that even for LN distributed cluster sizes, knowi ng average number o f collision s and stan dard deviation can be insufficient. Taking the LN into account reduces the a verage number of collisions onl y to 96.7%. However , it is wrong to c onclude that usage of the standard Poisson model will only le ad to errors of maximally 4%. A simple example showed how the Poisson model can lead to an e rror of almost 3 0% instead. W e have also s hown that if the beam’s host clust ers have sizes that ar e EXP distribute d, the Poisson model will be oft en insufficient. Under the se conditions, monomer i solation will be compromised. An important result concerns the inter pretation of the distri butions shape: Even if the distributio n of the host cl usters does n ot change t he expectation val ues much, the shape of a mass ab undance spectrum can suggest a fractionati on of guests although no such fracti onation occurre d. We know of at least one other research gro up that wasted time and resourc es by trying to model t hese appar ent fractionations in vain. 19 The Poiss on model’s suc cess tempts on e to employ the gener alized one i n order to take mixing into accoun t, but neither d oes the underlying physi cs suppo rt this, nor do f unctional fits with generalized Poisson distribut ions happen to fit well: average and maximum are still bound too closely. The re is increasing in terest in embedd ing large structur es in noble cluster matrix es. This desires l arge host clusters. Corr espondingly lar ge cross sections render availa ble vacuum technol ogy insufficient. With the helium dropl et technique’s c luster sizes e ver growing [25] , the standar d Poisson model shoul d and can be improved analyti cally. Some necess ary models have been derived h ere and have be en already su ccessfully applied to account for pre -doping by residual b ackground gas in exp eriments involvin g large Helium dro plets. Moreo ver, we think tha t our conside rations will be helpful when incorporati ng effects like s ticking coeffi cients and beam depletion in a rigorous way on a theoretical le vel. 10. Appendix Probability Density Functions (variable o f interest is continuous) Gamma ( Γ ): ( ) ( ) 1 PDF N N N d Nd N N d N e Nd N − − = Γ Exponenti al (EXP): / 1 PDF N N N e − = Log-Normal ( LN): 2 1 1 PDF 2 exp[ ] 2 n n n n N π −  −  = ∆   ∆   2 exp 2 n N n   = + ∆   and 2 1 n N N e ∆ ∆ = − (or [ ] 2 ln 1 1 n N d ∆ = + ) transfo rm between : ln n N = and N -spaces. Inverse Gau ssian (IG): ( ) ( ) 2 1 , 2 PDF exp[ ] 2 N N d N N d N N N N d N N N π − = − 20 Probability Distributions (variable of interest is discrete) Poisson: 0 0 ! ; k k k P P k k P e − = = Generali zed Poisson (GP): 1 ( ) ! k u k k u u k P e k λ λ − − − + = General e xpression for P oisson mixture: ( ) 0 1 ! ! k i i k i k k P H k G i ∞ = = − ∑ (10) ( ) 1 a G + = Γ , ( ) ( ) ( ) 1 1 ai a k i H + + + = Γ Γ and ( ) ( ) ( ) 1 1 x x ax a k κ + + = Γ Γ in case of the EXP; 1 G = , 2 ln exp k H ki   = ∆   and ( ) ( ) 2 ln exp 1 2 x x k k x κ   = − ∆   with ln n k a ∆ = ∆ are valid for the LN bein g the mixing distribu tion. Negative Bi nomial (NB): ( ) ( ) ( ) ( ) 1 0 0 0 1 ; ! N N N N k d d d k k N N d P P P P d d k κ + Γ = − = +     Γ ( 11) IG Poisso n mixture i (IGP): ( ) ( ) ( ) ( ) 1 2 / 4 1 2 2 ! k k d k k d k k k e P d K k κ π + Ω − = Ω Ω (12) Computati onally, the IGP is b est gotten by re cursion [26] wit h ( ) 1 2 2 3 1 k k k k d P k P P k k κ κ − −   = − +   Ω −   . i ( ) ( ) ( ) 1 2 0 2 2 exp 4 c c c x K x u u x u du ∞ − − = +     ∫ is the mod ified B essel functio n of the second ki nd. Manipula ting t he IGP needs o ften co nsideri ng prop erties o f the Bessel functio n. For instan ce, 0 P follo ws from the general formula via ( ) ( ) 1 / 2 2 x x K e x π ± − = . 21 EXP-Poiss on mixture for liquid drop cro ss section ( 2 / 3 a = ): 2 3,3,1 , 2 5, 4 , 2 , 4 7 ,5, 4 ,5 5 / 3 5 / 3 5 / 3 1 1 T T T ! 2 k k P k κ κ κ         = − +       Γ Γ Γ         (13) 3 2 , , , , , 2 2 2 6 3 3 3 5 / 3 3 T : F [ , , 4 ] 3 k k k α β γ δ α β γ δ α κ     + +     +       − = Γ   Γ   contains t he generalized hypergeometri c function { } { } 2 2 F [ , , , , ] v w x y z . Generali zed Poisson written a s if it is alrea dy a Poisson mixtu re: ( ) ( ) ( ) 1 1 GP 1 1 ! k k k k P k e k κ λ λ κ λ λ λ κ − − − = − − + (14) 1 Villaric a M, Casey M J, Goodisman J a nd Chaiken J 1993 J. Chem. Phys. 98 4610 2 Wang C R, Hu ang R B, Liu Z Y and Zheng L S 199 4 Chem. Phys. Lett . 227 103 3 Limpert E , Stahel W A and Abb t M 2001 BioScience 51 341 4 Toennies J P and Vilesov A F 2004 Angew. Chem. Int. Ed. 43 2622 5 Stienkemeier F and Lehmann K K 2006 J. 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