the mobility of positively 
 charged ions x injected into liquid he differs 
 markedly from that of ions in liquid he .
the temperature 
 and field dependence of the mobility suggests that a cation in he will drag with it some 50 he atoms , which , as a result 
 of electrostriction , form a solid - like layer . in subsequent work , it was found that the size of this  snowball
an alternate and in some way more accurate approach 
 to determine the number of atoms in a snowball is via gas - phase experiments ; 
 they can provide information about dissociation ( or evaporation or 
 separation ) energies defined as1where en is the total energy of henx .
dissociation energies ( or , more exactly , enthalpies ) may 
 be derived quantitatively from vant hoff plots by measuring 
 the partial pressure of henx in thermal equilibrium as a function of temperature .
unfortunately , these measurements are limited to very small 
 complexes . a simpler method to determine
the number of helium 
 atoms in a partial or complete solvation shell is to look for anomalies 
 in the abundance distribution of henx measured under nonequilibrium conditions ; abundance anomalies 
 often correlate with anomalies in dn . for example ,
kojima et al . injected ar or kr into a drift tube filled with helium gas at low density ( about 
 2 pa ) and low temperature ( 4.4 k ) ; the noble gas ( ng ) ions were found 
 to bind up to 14 helium atoms .
the abundance 
 distributions depended on experimental conditions ( helium pressure , 
 temperature , applied electric field ) , but for suitable choice of the 
 drift field , abrupt drops in the abundance were observed at he12ar and he12kr .
the same 
 ions were also formed prominently when helium droplets were doped 
 with ar or kr and ionized by electrons .
the agreement between these two different 
 experimental approaches stems from the fact that in both cases , the 
 observed ions are dissociation products of larger precursor ions . 

dissociation in the drift tube is induced by collisions of the ions 
 with the buffer gas , whereas in mass spectra of doped helium droplets , 
 dissociation is the result of the excess energy introduced upon ionization . 

the relation between the size dependence of dissociation energies dn and ion abundances in in experiments where cluster 
 ions undergo unimolecular dissociation has been explored by several 
 authors . for
complexes with very low heat capacities , local anomalies in in directly correlate with local 
 anomalies in dn .
experiments involving doped helium droplets have revealed 
 anomalies in the abundance of henx for many ions x , including alkali ions and their 
 dimers , other metal ions , halogen ions , h , h2 , and 
 h3 , other polyatomic 
 ions , and c60 , 
 c70 , and their aggregates . here , n = 12 is a frequently observed anomaly , 
 often interpreted to indicate formation of a solvation shell with 
 icosahedral symmetry , but the number of helium atoms 
 in the first solvation shell may be much larger .
for example , 1820 
 atoms are needed to complete the first solvation shell for mg , 20 for i2 , 60 for c60 , and 62 for 
 c70 . on the other hand
, much smaller anomalies may indicate closure of 
 subshells or formation of stable structures such as , at n = 2 , the formation of the linear [ he  na  na  he ] , followed by an anomaly at n = 6 when a 
 subshell is completed .
the richness of anomalies 
 observed for some metallic ions contrasts with the 
 small set of anomalies reported so far for ng s . here , n = 12 is the only anomaly observed for henar and henkr ( pronounced anomalies were reported for 
 he12kr2 and he12kr3 , but we will show that they were caused by xenon 
 impurities ) .
no anomaly was found for henne if doped helium droplets were ionized even though drift tube experiments indicated enhanced stability 
 of he13ne . in a 
 separate publication ,
we have solved 
 this discrepancy ; it arises from the difficulty to resolve nominally 
 isobaric ions .
in fact , the paucity of anomalies reported for henng can be traced to the challenge 
 of deducing ion abundances from mass spectra ; ( i ) for all ng s , 
 the mass number of the most abundant isotope ( ne , ar , kr , and xe ) is an integer 
 multiple of the he mass number , and ( ii ) the large number 
 of naturally occurring isotopes of kr ( 6 ) and xe ( 9 ) causes a multitude 
 of mass spectral coincidences for heavy ng s .
several ions , 
 differing in the number of helium atoms and/or the number of ng atoms , 
 will have the same nominal mass .
the resolving power of the 
 mass spectrometer employed in the present work exceeds that of instruments 
 used previously by an order of magnitude . as a result
, ions containing 
 different numbers of ng atoms can be resolved ; a large number of anomalies 
 are identified in the abundance distributions of henngx .
perhaps the most surprising ones 
 are anomalies in the henar series at n = 32 and 44 , in addition to the well - known 
 anomaly of he12ar .
it is tempting to assign 
 the anomalies to three solvation shells of ( approximately ) icosahedral 
 symmetry , with 20 atoms localized in the hollow sites of the he12ar icosahedron , thus forming a dodecahedron , 
 and an additional 12 atoms localized in the hollow sites of he32ar , thus forming an icosahedron ; the two outer 
 shells thus resemble the commensurate phase of helium on c60 , which offers 20 hexagonal and 12 pentagonal adsorption 
 sites .
the interpretation 
 draws on a recent path integral monte carlo study by galli et al . who concluded that three distinct solvation 
 shells of icosahedral symmetry arrange around na , but 
 no other alkali and alkali - earth ions were found to exhibit such a 
 high degree of order .
in fact , early experimental work suggested that 
 no more than two shells are part of the snowball , and many recent theoretical studies of ions solvated in 
 helium concluded that no structural order exists outside of the first 
 solvation shell .
even in the first layer , quantum 
 effects may destroy structural order and wash out shell effects .
we note that these theoretical studies included quantum 
 effects ( zero - point motion and the bosonic nature of helium ) to various 
 degrees , which explains some of the differences between them .
several other anomalies in distributions of henngx are reported and 
 interpreted in this work .
the solvation layer of henkrx appears to grow 
 by insertion of six - membered helium rings as the ionic core grows 
 from x = 1 to 2 to 3 , and helium  argon complexes 
 containing at least 2 ar atoms appear to be particularly stable if 
 they contain a total of 19 atoms .
striking differences are found between 
 the abundance distributions of henng for the heavy ng s ( kr , xe ) and henar ; a parallel to the different fragmentation dynamics 
 of these systems is tentatively drawn .
we also critically 
 evaluate previously published mass spectra and include recent results 
 for hennex .
so far , few quantum mechanical 
 treatments of henngx have been reported ; the nature of the ionic core , 
 charge delocalization , vibrational delocalization , and other quantum 
 mechanical effects pose significant challenges .
it is to be hoped 
 that results presented in this work will provide critical tests for 
 future theoretical work .
neutral 
 helium nanodroplets were produced by expanding helium ( purity 99.9999% ) 
 from a stagnation pressure of about 2 mpa through a 5 m nozzle , 
 cooled by a closed - cycle refrigerator ( sumitomo heavy industries ltd , 
 model rdk-415d ) to about 10 k , into vacuum .
the average number of 
 atoms per droplet formed in the expansion ranged from a few  
 10 to 10 ; these 
 droplets are superfluid with a temperature of 0.37 k. the resulting supersonic beam was skimmed by 
 a 0.8 mm conical skimmer , located 12 mm downstream from the nozzle . 
 the skimmed beam traversed a 20-cm - long pick - up region into which 
 either argon ( air liquide , purity 99.9999% ) , krypton ( 99.999% ) , or 
 xenon ( 99.999% ) was introduced at a partial pressure of a few tenths 
 of a millipascal .
after the pick - up region , the doped helium droplets 
 passed a region in which they were ionized by collisions with electrons 
 at energies ranging from 70 to 140 ev .
cations were accelerated to 40 ev into 
 the extraction region of a commercial time - of - flight mass spectrometer 
 equipped with a reflectron ( tofwerk ag , model htof ) ; its mass resolution 
 was m / m = 1/7000 .
the 
 ions were extracted at 90 into the field - free region of the 
 spectrometer by a pulsed extraction voltage . at the end of the field - free 
 region , they entered a two - stage reflectron that reflected them toward 
 a microchannel plate detector operated in single - ion counting mode . 

in the past , most attempts to identify 
 ng  helium complexes henngx of enhanced stability have been hampered 
 by the inability to resolve different ions with the same nominal mass . 

argon is the only other ng that is nearly monisotopic , 
 with ar at 99.600% and ar at 0.337% .
even 
 so , the nominal mass of he12ar , which appears 
 prominently in mass spectra , coincides with that of he22 and he2ar2 .
the exact 
 masses differ by 0.064 au ; a resolution of 1/1400 or better would 
 be needed to separate the ions .
the problem is aggravated in 
 experiments with krypton and xenon , which have 6 and 9 naturally occurring 
 isotopes , respectively . to make things worse , the masses of their 
 most abundant isotopes , kr and xe ( relative 
 abundance of 57 and 26.89% , respectively ) are integer multiples of 
 the helium mass . in past research
, the ions could not be separated from he21 and he33 , which are heavier by just 0.143 and 0.182 au , 
 respectively . in the present experiments , an instrument with resolution 

m / m  1/7000 is used , 
 and these ions are easily resolved .
this is demonstrated in figure 1 , which displays a mass spectrum of helium nanodroplets 
 doped with krypton .
the isotopes of kr are kr ( 0.3% ) , kr ( 2.3% ) , kr ( 11.6% ) , kr ( 11.5% ) , kr ( 57.0% ) , and kr ( 17.3% ) .
the hen series 
 forms the dominant peaks below 80 au and between 88 and 108 au .
peaks 
 due to isotopologues of henkr ( n  0 ) are marked by open dots ; those due 
 to the main isotope , henkr , are connected by lines .
an 
 abrupt drop in the ion yield between he12kr and he13kr is clearly 
 visible ; a weaker drop occurs at n = 14 . a mass spectrum 
 of helium nanodroplets doped with krypton .
isotopologues due to henkr and henkr2 ( n  0 ) 
 are marked by open circles and solid diamonds , respectively ; mass 
 peaks due to the most abundant isotopologues in each series , containing 
 one or two kr , are connected by lines .
experimental conditions : helium pressure 
 1.6 mpa , nozzle temperature 8.9 k , krypton pressure 0.2 mpa ( corrected 
 ion gauge value ) .
peaks due to isotopologues 
 of henkr2 are marked 
 by full diamonds ; those due to the main isotopologue , henkr2 , are connected 
 by lines .
the corresponding ion peaks coincide with isotopologues 
 of kr3 in nominal mass , but they are resolved .
isotopes due to henkr are marked by open dots ; they are fully resolved from isotopologues 
 of henkr2 , which 
 are marked by diamonds and connected by a line .
peaks due to pure hen ( not marked ) appear immediately to the right of henkr ( mass difference 0.143 au ) .
( a ) a section 
 of a mass spectrum of helium nanodroplets doped with krypton .
isotopologues 
 due to henkr and henkr2 are marked by open circles 
 and solid diamonds , respectively ; mass peaks due to henkr2 isotopologues are connected 
 by lines .
( b  d ) narrow bars : simulated mass spectra of henkr2 assuming unit 
 mass resolution .
panel b assumes equal contributions for all sizes n ; the wide bars represent the contribution of he12kr2 to the total spectrum .
mass peaks are 
 connected by lines in order to demonstrate the similarity with the 
 experimental pattern observed for henkr2 in panel a ( solid diamonds ) .
panel c assumes 
 that the abundance of he12kr2 is 
 enhanced by 50% relative to other sizes ; this extra contribution is 
 represented by wide bars .
panel d assumes that a 50% krxe contamination is present ; this extra contribution is represented 
 by wide bars .
a problem in measurements 
 of krypton- or xenon - doped helium droplets is crosstalk between ions 
 containing the same number x of ng atoms but different 
 numbers of helium atoms .
this is illustrated in figure 2b , which shows a simulated spectrum of henkr2 ( narrow bars connected by solid 
 lines ) .
the ion yield of the dominant isotopologue , henkr2 , was set to 1 for all 
 values of n. the contributions of he12kr2 are represented by the wide bars in figure 2b .
its isotopologues cover a wide mass range ; three 
 of them contribute to the signal at 212 au , namely , he12kr2 , he12krkr , and he12krkr .
their masses differ by less than 0.01 au 
 from that of he11kr2 , which provides the dominant contribution at 212 au ; these ions 
 can not be resolved .
in essence , the ion yield measured 
 by a mass spectrum represents the ion abundance of henkrx convoluted 
 with the isotopologues of krx ; therefore , 
 the ion abundance can be extracted from a mass spectrum by a deconvolution .
figures 35 display the ion abundances of henarx ( x  
 7 ) , henkrx ( x  3 ) , and henxex ( x  3 ) . as described in section 3a , the data are corrected for crosstalk from isotopologues . 

reproducible anomalies are marked and listed in table 1 together with previously reported anomalies , including recent 
 data for hennex . as discussed in section 3d , abrupt changes in the abundance
are expected 
 to correlate with abrupt changes in the dissociation ( or separation 
 or evaporation ) energies as defined in eq 1 . 

ion abundance of henarx with 1  x  7 ; reproducible anomalies are marked .
experimental conditions 
 were helium pressure 2 mpa ( 1.6 mpa ) , nozzle temperature 10 k ( 9.1 
 k ) , ar pressure ( corrected ion gauge reading ) 0.56 mpa ( 0.21 mpa ) , 
 and electron energy 70 ev ( 100 ev ) ; values in parentheses pertain 
 to henar .
ion abundance of henkrx ( 1  x  3 ) extracted 
 from mass spectra and corrected for the occurrence of isotopologues . 

experimental conditions were helium 
 pressure 1.6 mpa , nozzle temperature 8.9 k ( 9 k ) , kr pressure ( corrected 
 ion gauge reading ) 0.20 mpa , and electron energy 70 ev .
ion abundance of henxex ( 1  x  3 ) extracted from mass spectra and corrected for the occurrence 
 of isotopologues .
experimental 
 conditions were helium pressure 1.6 mpa , nozzle temperature 8.9 k , 
 xe pressure ( corrected ion gauge reading ) 0.17 mpa ( 0.24 mpa ) , and 
 electron energy 70 ev ; values in parentheses pertain to henxe3 .
data 
 for neon are from ref ( 32 ) . for each ng , 
 we have recorded several mass spectra , varying the initial size of 
 the helium droplets by varying the stagnation pressure and/or temperature 
 and the partial pressure of the ng in the pick - up region .
these conditions 
 did not significantly affect the anomalies in the ion abundance , although 
 it should be mentioned that droplet sizes remained within the somewhat 
 narrow range of 1010 . another parameter 
 that could potentially play a role is the time that elapses between 
 formation of the ions in the ionizer and the mass analysis , that is , 
 the time at which they are extracted into the time - of - flight mass 
 spectrometer . in the present work ,
this time amounts to about 10 s ; it could not be varied . for helium  argon complexes , the only magic number firmly 
 established so far was he12ar .
the anomaly 
 was observed when mass - selected atomic ions were injected into helium 
 gas at 4.4 k , and in mass spectra of helium 
 droplets doped with argon .
it is clearly reproduced in our data ( figure 3 ) . according to calculations by murrell et al .
the 
 authors applied the diatomics - in - molecules method in order to determine 
 the classical ground - state structures of henar up to n = 16 .
the ionic core 
 in henar is the trimeric , 
 symmetric ( dh symmetry ) [ he  ar  he ] , although charge 
 delocalization in this complex is very small , much smaller than that 
 in [ he  ne  he ] .
for hear , sun 
 et al . calculated that 99.6% of the charge 
 resides on the argon . in he12ar ,
the calculated dissociation energies of henar ( which do not include the 
 effects of zero - point motion ) are displayed in figure 6a .
completion of the ionic core at n = 2 
 and completion of the first solvation shell at n = 
 12 are accompanied by abrupt drops in dn , in agreement with abrupt drops in the ion abundance 
 ( figure 3 ) . according to murrell et al .
, two 
 more helium atoms can be squeezed into the first shell before helium 
helium 
 repulsion pushes additional atoms into a second shell . the drop in the calculated dissociation energies 
 at n = 14
( a ) dissociation 
 energies dn calculated 
 for henar ; the dashed line represents the local average of dn .
( b ) relative dissociation 
 energies of henar , that is , 
 the calculated values divided by the 
 local average .
( c  f ) relative experimental ion abundances of 
 henng for ar , kr , xe , and 
 ne ( from ref ( 32 ) ) . instead , the experiment reveals 
 strong anomalies at n = 32 and 44 ; they are outside 
 of the size range of previous experimental and theoretical investigations . 

we propose an appealing structural model for these magic numbers , 
 inspired by a path integral monte carlo study of henna by galli et al .
when applied to quantum systems of bosons , the method can provide 
 exact estimates for properties , within the statistical uncertainties 
 of the monte carlo calculation and the accuracy of the assumed interaction 
 potential . according to the calculations by murrell et al . , the distance between ar and the 
 two he atoms in the trimeric core is not much less than that between 
 ar and the other , neutral he atoms in he12ar . thus , he12ar has approximately icosahedral 
 symmetry .
if these faces are decorated by one he atom each , one obtains a dodecahedral 
 he32ar ( 20 vertices , 12 pentagonal faces ) . 
 if the faces of this complex are decorated by one he atom each , one 
 obtains the magic he44ar , with a total of three 
 closed solvation shells .
this description ignores the quantum nature 
 of helium , but it agrees with the findings for henna by galli et al . ,
namely , the existence of three distinct shells , angular order of 
 helium within each shell , and structural correlation between the shells . 

each layer provides a corrugated surface for the next layer , which 
 remains essentially rigid ; exchange of atoms within a shell and between 
 shells is small .
this type of growth of concentric shells of icosahedral 
 symmetry is not energetically favored for neutral clusters that interact 
 via a lennard - jones or similar potentials but is driven by the strong interaction of helium with the charge 
 at the center .
it is also interesting 
 to note that a commensurate layer of 32 helium atoms adsorbed on c60 ( which plays the role of he12ar ) features the same arrangement , namely , 20 atoms located 
 at the centers of the carbon hexagons plus 12 atoms located at the 
 centers of the carbon pentagons .
the rigidity of the helium layer is caused by the 
 strong corrugation of the c60 surface .
in this 
 case , though , the two shells ( containing 20 and 12 atoms , respectively ) 
 have nearly the same dissociation energies and radial distances from 
 the center ; they appear as just one shell in the experiment .
we also mention a study of c28 , which 
 has 12 pentagonal plus 4 hexagonal hollow sites .
it features 16 energetically 
 preferred sites , but another anomaly in the computed dissociation 
 energy occurs when 32 helium atoms form a single , ordered layer of 
 icosahedral symmetry .
the fact 
 that anomalies as large as n = 44 are observed is 
 remarkable because they suggest an unusually large rigid core surrounding 
 the ion .
most theoretical studies of cations solvated in helium concluded 
 that the solid - like structure does not extend beyond the first shell ; 
 atoms in the second shell are more delocalized and liquid - like .
gianturco 
 and co - workers employed the variational monte 
 carlo method and ab initio potential energy curves to optimize a trial 
 wave function ; in a second step , they applied the diffusion monte 
 carlo algorithm in order to obtain the energy and the geometric distributions 
 of the cluster .
they found a rigid solvation shell of icosahedral 
 symmetry for he12na , but delocalization in 
 the first shell increases if the number of helium atoms is smaller 
 or larger than 12 .
however , the microscopic 
 structure of the snowball is strongly dependent on the identity of 
 the ion ; shells around na were found to be the most distinct , 
 as judged from the radial density distributions and angular correlations . further theoretical
work will be needed to confirm 
 that the structure of the snowball around ar is , indeed , 
 similar to the one predicted for na .
first , anomalies 
 at n = 12 , 32 , and 44 were already reported for henag , plus another one at n = 10 .
a very different model , 
 including electronic excitations of the ion , had been invoked in the 
 interpretation ; no attempt was made to assign a structure .
second , 
 the ion abundance of henna has been reported by our group for n  27 ; 
 it displays distinct anomalies at n = 2 and 12 .
we have inspected the mass spectra once again , 
 but no reliable information can be drawn for complexes with n > 27 .
several anomalies are observed in figure 3 for henar2 , with the strongest ones at n = 17 
 and 20 .
several experimental and theoretical studies of argon cluster 
 ions have been reported ( see , e.g. , ref ( 53 ) for recent work ) , but we are not aware of any 
 experimental or theoretical work pertaining to henarx with x  2 except for a theoretical study of the linear [ ar  he  ar ] ion by sun et al .
this ion is 
 bound by only e = 0.063 ev , a factor of 20 less than the energy of ar2 .
thus , [ ar  he  ar ] would be metastable ; the most likely ionic core in henar2 is the argon dimer ion . 

we refrain from speculating about the structure of  magic  
 henarx ( x  2 ) .
however , one interesting 
 feature is apparent from figure 3 , namely , 
 the gradual decrease of the magic number n = 17 for x = 2 to n = 16 for x = 
 3 , n = 15 for x = 4 , and so on to n = 12 for x = 7 . in other words , the complex 
 with a total of n + x = 19 atoms 
 has enhanced stability .
in fact , the smallest magic number consistently 
 identified in experiments involving pure argon is ar19 ; it is usually assumed to have the structure of a double 
 icosahedron .
it is conceivable that the linear structure of ar3 and ar4 stabilizes the double icosahedron 
 for helium  argon complexes containing up to x = 4 argon atoms and 19  x helium atoms , 
 but it is not at all clear why the replacement of further helium atoms 
 with argon atoms should preserve the structure and high stability 
 of the double icosahedron , given the very different polarizabilities 
 of he and ar .
when krypton 
 ions are injected into low - pressure helium gas at 4.4 k , henkr containing up to 14 helium atoms form . 

the abundance of he12kr greatly exceeds that 
 of he13kr if a drift field is applied ; he12kr appears to be particularly 
 stable .
they reported an enhanced 
 ion yield for he10kr , but the resolution was 
 poor ; krypton isotopes were not resolved . in similar experiments performed 
 at higher resolution and involving single - photon ionization , kim et 
 al . reported enhanced ion yields for he12kr , he12kr2 , and he12kr3
we have confirmed the anomaly for he12kr but 
 could not record reliable data for complexes containing more than 
 one krypton atom .
the present data 
 ( figure 4 ) confirm the anomaly for he12kr , and they show another , weaker one for he14kr .
the henkr distribution is featureless for larger values of n , but a very broad maximum at around
possible reasons for this feature , 
 which is also observed for henxe but not henar , will be discussed 
 in section 3c .
the difference in ionization 
 energies between krypton ( ie = 14.00 ev ) and helium ( 24.59 ev ) is 
 even larger than that between argon ( 15.760 ev ) and helium .
hence , 
 one would expect negligible charge transfer between kr and he , and he12kr might have an approximately 
 icosahedral structure similar to that of he12ar ( we are not aware of any theoretical work pertaining to henkr , except for hekr ) . concerning the anomaly at he14kr , it is 
 worth noting that murrell et al
. computed 
 an abrupt drop in the dissociation energy of henar at n = 14 because two additional 
 helium atoms could be squeezed into the first solvation shell ( he12ar ) .
the anomaly predicted for n = 14 was not observed in the henar data ( figure 3 ) , but perhaps it shows 
 up in henkr thanks to the 
 larger size of kr ; the less compact solvation shell in 
 he12kr would more easily accommodate two additional 
 helium atoms .
the abundance of henkr2 reveals two weak anomalies at n = 7 and 12 plus stronger ones at 18 and 20 .
several very 
 weak anomalies ( at n = 10 , 12 , 16 ) are observed for 
 henkr3 plus stronger 
 ones at n = 24 and 26 .
these results are inconsistent 
 with work by kim et al . who reported strong local maxima for he12kr2 and he12kr3 .
in figure 4b of kim s work , the group of mass peaks assigned to 
 he12kr2 was about three times more 
 intense than that for he11kr2 ; in 
 our work , the enhancement is less than 10% .
we believe that the pronounced 
 anomalies in kim s work were caused by xenon impurities ( the 
 authors had doped helium droplets with either krypton or xenon ) .
the most abundant one is xe ( natural abundance 26.89% ) , while the most abundant isotope 
 of krypton is kr .
thus , the nominal mass of xe equals that of the main isotopologue of he12kr ; the exact masses differ by only 0.038 au . in order 
 to test our hypothesis that a krxe impurity was erroneously 
 interpreted as a magic he12kr2 , 
 and kr2xe as he12kr3
, we have simulated the corresponding mass spectra assuming 
 a resolution better than 1 au but insufficient to separate nominally 
 isobaric ions , in line with the experimental conditions in kim s 
 work .
a simulated spectrum of henkr2 with equal ion abundance for all values 
 of n is shown in figure 2b 
 ( narrow bars ) ; it has already been discussed in section 3a .
figure 2c shows a simulated 
 spectrum of henkr2 assuming a 50% higher abundance of he12kr2 ; the extra contribution is shown explicitly by the wide 
 bars .
figure 2d shows a simulated spectrum 
 of henkr2 with 
 no magic henkr2 but , instead , a 50% contamination with krxe ; the contribution 
 is shown explicitly by the wide bars .
the two simulated spectra in 
 figure 2c and d are quite similar ; both feature 
 a local maximum at 216 au .
the most telling difference between the 
 two spectra is the enhancement at masses 213 and 215 au due to the 
 abundant xe ( 26.44% ) and xe ( 21.18% ) isotopes . 

we conclude that the very strong 
 anomalies in the ion signal at around 216 and 300 au reported by kim 
 et al . were artifacts .
although their main 
 isotopes ( at nominal mass 132 au , with a mass difference of 0.039 
 au ) would be barely resolvable , the henkr series produces no peak at mass 129 , whereas xe is nearly as abundant as xe . in our spectra , 
 mass peaks at 129 au are below the 1% level of the peak at 132 au 
 ( he12kr ) , that is , a xenon impurity 
 can be clearly ruled out .
figure 4 reveals 
 an interesting pattern . the pair of dominant anomalies in the abundance 
 of henkrx shifts from ( 12,14 ) for x = 1 to ( 18,20 ) 
 for x = 2 to ( 24,26 ) for x = 3 , 
 each time a shift by 6 units . given the disparity in ionization energies 
 between krypton and helium and the large dissociation energies of 
 kr2 ( about 1.3 ev ) and kr3 ( 0.25 ev )
, the ionic core in henkrx will essentially be the krx ion , carrying close to 100% of the charge . 

an icosahedral arrangement of 12 he atoms in he12kr may be viewed as consisting of two five - atom rings plus two 
 he atoms at either end ( plus , perhaps , another two atoms ; see our 
 discussion of he14kr ) . upon solvation of kr2 , a third ring containing six helium atoms
would 
 be inserted between the five - atom rings and another ring containing 
 six helium atoms upon solvation of the linear kr3 .
kr is significantly larger than he ( the equilibrium distance of 
 kr2 equals 4.0  , while 
 the distance between adjacent atoms in condensed helium is 2.97  ) .
thus , it is reasonable to assume that the 
 additional helium rings surrounding kr2 or 
 kr3 would accommodate six rather than five 
 atoms .
figure 5 reveals a weak anomaly in the abundance distribution of henxe at n = 12 plus two 
 broad maxima at around n = 10 and 35 .
the abundance 
 distributions of henxe2 and henxe3 show only minor anomalies ; the two broad maxima are absent .
toennies and co - workers reported three broad maxima at n = 4 , 8 , and 15 in the henxe series , but the corresponding mass peaks were barely resolved .
janda and co - workers and neumark and co - workers reported 
 mass spectra of henxe and 
 henxe2 with unit 
 mass resolution but without the ability to distinguish between different 
 ions of the same nominal mass . they did not report any anomalies in 
 the distribution of henxe . 
 xenon has nine naturally occurring isotopes , but only two have an 
 odd atomic mass number , a = 129 and 131 .
thus , below 
 the mass of xe2 , the ion series henxe and henxe are free from contributions from 
 other helium  xenon isotopologues . the fact that no anomalies 
 were noticed in previous work is probably due to a combination of 
 low resolution and statistical fluctuations or contaminants . why is the anomaly 
 at he12xe so much weaker than that at he12ar and he12kr ?
the hexe bond length is approximately 3.4  , substantially larger than the he  he separation of 2.97  
 in condensed helium . in a cluster ,
the 
 bond length of hexe is probably > 3.4  because 
 of reduced charge delocalization . in a hypothetical icosahedral he12xe
, such a large radial hexe distance 
 would cause an even larger distance between adjacent peripheral helium 
 atoms .
in other words , helium atoms are too small to complete a solvation 
 shell at he12xe .
the ion abundances of 
 henxe2 and henxe3 can not be reliably 
 assessed unless one corrects for the various isotopologues .
the most 
 abundant isotope of xenon is xe ( abundance 26.89% ) ; 
 six other isotopes occur with significant abundances , namely , xe ( 1.92% ) , xe ( 26.44% ) , xe 
 ( 4.08% ) , xe ( 21.18% ) , xe ( 10.44% ) , and xe ( 8.87% ) .
the matrix method provides 
 a remedy , and as a result , we are able to identify weak anomalies 
 in the abundance distributions of henxe2 and henxe3 . the broad maxima at n  35 in the abundance distributions of henkr and henxe ( figures 4 and 5 ) contrast with the distributions of complexes containing 
 two or more heavy ng atoms . the features , which could also be characterized 
 as one broad minimum at around n = 18 , appear in 
 all our data for a variety of helium expansion conditions and partial 
 pressures in the pick - up cell , that is , a variety of droplet sizes 
 and ng cluster sizes .
, we tentatively draw a parallel to another striking difference 
 between clusters of argon and clusters of heavy ng s , namely , 
 the experimentally established propensity of ( bare ) argon clusters 
 to fragment upon ionization into mostly dimer ions , while clusters 
 of heavy ng s fragment overwhelmingly into monomer ions .
the difference is quite dramatic but not easily detected because 
 in most experiments , the distribution of neutral clusters prior to 
 ionization remains unknown ; an elaborate setup involving elastic scattering 
 between two supersonic beams is required to mass - select neutral clusters . in the experiments by buck and co - workers , bare ng clusters
were directly 
 ionized by electrons , whereas in our experiments ng clusters are embedded 
 in helium droplets and ionized indirectly by charge transfer from 
 he .
still , experiments by janda and co - workers suggest 
 that these different experimental approaches both show the same striking 
 difference in the fragmentation behavior between heavy ng clusters 
 and argon clusters
. however , we can not propose 
 a specific mechanism that would account for the different abundance 
 distributions between henng for heavy ng s and henar .
theoretical descriptions of the fragmentation dynamics of 
 small clusters of argon , krypton , or xenon have struggled for a long 
 time to explain the differences ; dimer ions were invariably found 
 to be the dominant fragment ions .
after all , 
 the binding energies of kr2 and xe2 are nearly as strong as that of ar2 . very recently ,
their 
 multiscale approach is complex ; it involves spin  orbit coupling 
 and different time scales to account for initial , nonadiabatic processes 
 as well as radiative and nonradiative transitions between electronically 
 excited states that occur on a microsecond time scale .
metastable 
 dissociation of excited ng dimer ions has been shown experimentally 
 to occur on this time scale , with profound differences between 
 light ( ne , ar ) and heavy ng s .
it remains to be seen if excited 
 states could possibly cause the peculiar broad features in the distributions 
 of henkr and henxe .
as discussed elsewhere , for clusters with small heat capacities 
 such as small henng ions , 
 the evaporative model suggests that the relative abundance is proportional 
 to the relative dissociation energy . here
, the 
 relative abundance is defined as the ratio of the observed abundance in and its local average in , and the relative 
 dissociation energy is defined as the ratio of dn and its local average dn , hence2the relation has to be 
 taken with some caution .
second , there is no unique definition of the local averages . for extended 
 data sets , a running average with gaussian weighting
a fit with a polynomial 
 is a better choice because points near the end of the data set ( e.g. , n = 1 or 2 ) would be lost otherwise . in the present work ,
we have fitted fourth - degree 
 polynomials . as an illustration , the dashed line in figure
6a is a fourth - degree polynomial fit to the theoretical 
 dissociation energies of henar for 1  n  
 16 ; the corresponding relative dissociation energies calculated from 
 eq 2 are shown in figure 6b .
the measured relative abundances are shown in figure 6c  e for ar , kr , and xe , respectively . also 
 shown ( figure 6f )
contrary to previous , conceptually 
 similar experiments our data reveal a strong anomaly at he13ne , similar to the one observed by kojima et al . in drift 
 tube experiments
however , calculated 
 dissociation energies of henne either feature anomalies at n = 10 and 12 or none at all .
the 
 relative abundances in figure 6 display more 
 clearly local anomalies that tend to be masked if they occur in regions 
 where in changes rapidly 
 with n. furthermore , they provide a better measure 
 of the strength of an anomaly .
he12ar is enhanced 
 by some 50% , he13ne by some 40% , while the 
 anomaly of he12xe measures only 10% .
we also 
 see that the calculated relative dissociation 
 energies of henar ( figure 6b ) do not track the relative abundance of these 
 ions ( figure 6c ) ; instead , they bear a resemblance 
 to the relative abundance of henkr .
our main conclusions from figure 6 are that ( i ) the computed dissociation energies of henar do not match the experimental results , 
 ( ii ) the anomalies in the relative abundance decrease in strength 
 from ar to kr to xe , and ( iii ) anomalies in the abundance of henne differ markedly from those 
 of ar , kr , and xe complexed with 
 helium .
the mass spectra also 
 show the occurrence of bare ng clusters ngx or , in other words , ngxhen complexes with n = 0 ( see , e.g. , the mass spectrum in figure 1 ) .
we do not show their ion abundance in figures 35 , but some remarks are in 
 order . first , bare cluster ions ngx ( x > 1 ) result from ionization - induced 
 fragmentation of helium droplets doped with ng atoms because neutral 
 ng clusters can form only within a helium droplet , and ngx will not be expelled from the droplet until the 
 droplet is ionized . the ion yield of ngx relative to that of ngxhen ( where n is 
 some small number > 0 ) increases systematically as the mass of the 
 ng increases . for neon ,
the ion yield of nex with x > 3 is less than that 
 of nexhen ( for small n ) . for argon ,
bare cluster ions 
 and mixed complexes have about the same yield , while for xenon , bare 
 cluster ions are an order of magnitude more abundant . in short , fragmentation 
 of doped helium droplets into bare ngx appears to increase as the mass of the dopant increases .
second , the trend described above for ngx mirrors a trend observed by janda and co - workers and neumark and co - workers for ng monomers ; those authors have discussed possible 
 reasons for this phenomenon .
we can not directly confirm their observations 
 because in our experimental setup , there is a possibility that atomic 
 ng results from leakage of the dopant gas from the pick - up 
 cell to the ionizer rather than from ionization - induced fragmentation 
 of doped helium droplets .
we note , however , that janda and co - workers reported a similar trend for ng dimer ions , 
 and here , our data clearly agree with theirs .
we have reported mass spectra of helium droplets 
 doped with argon , krypton , or xenon .
the resolution of the mass spectrometer 
 exceeds the resolution of instruments used in previous studies by 
 an order of magnitude .
this allows us to distinguish between ions 
 of nominally identical mass . as a result
, the abundance of the complete 
 ion series henngx can be established without interference from ions 
 containing a different number of ng atoms .
furthermore , a matrix method 
 is used to account for isotopologues that can not be resolved , such 
 as he10kr2 , he11kr2 , he12kr2 , and he12krkr .
a rich spectrum of anomalies is observed 
 in the abundance distributions ; several interesting patterns have 
 been interpreted .
perhaps the most striking is evidence for the existence 
 of three shells of helium atoms in a rigid icosahedral arrangement 
 that solvate ar .
this conclusion runs counter to that 
 of most theoretical studies of atomic cations in helium , which usually 
 conclude that the effects of electrostriction are not strong enough 
 to localize helium atoms outside of the first solvation shell .
however , 
 the theoretical treatment of charged complexes of ng s with 
 helium faces several challenges , including the nature of the ionic 
 core , the extent of charge delocalization , and the effect of quantum 
 effects including zero - point motion , its effect on the delocalization 
 of the charge , and the bosonic nature of he .
 helium 
 nanodroplets doped with argon , krypton , or xenon are ionized by electrons 
 and analyzed in a mass spectrometer . 
 henngx+ ions containing up to 
 seven noble gas ( ng ) atoms and dozens of helium atoms are identified ; 
 the high resolution of the mass spectrometer combined with advanced 
 data analysis make it possible to unscramble contributions from isotopologues 
 that have the same nominal mass but different numbers of helium or 
 ng atoms , such as the magic he2084kr2 + and the isobaric , nonmagic he4184kr+ . 
 anomalies in these ion abundances reveal particularly 
 stable ions ; several intriguing patterns emerge . 
 perhaps most astounding 
 are the results for henar+ , 
 which show evidence for three distinct , solid - like solvation shells 
 containing 12 , 20 , and 12 helium atoms . 
 this observation runs counter 
 to the common notion that only the first solvation shell is solid - like 
 but agrees with calculations by galli et al . for henna+ [ j. 
 phys . 
 chem . 
 a2011 , 115 , 730021568337 ] that reveal three shells of icosahedral 
 symmetry . 
 henarx+ ( 2  x  7 ) ions 
 appear to be especially stable if they contain a total of n + x = 19 atoms . 
 a sequence of anomalies 
 in the abundance distribution of henkrx+ suggests that rings of six helium 
 atoms are inserted into the solvation shell each time a krypton atom 
 is added to the ionic core , from kr+ to kr3 + . 
 previously reported strong anomalies at he12kr2 + and he12kr3 + [ kimj . 
 h. ; et al 
 . j. chem . 
 phys.2006 , 124 , 21430116774401 ] are attributed to a contamination . 
 only minor local anomalies appear 
 in the distributions of henxex+ ( x  3 ) . 
 the 
 distributions of henkr+ and 
 henxe+ show strikingly similar , 
 broad features that are absent from the distribution of henar+ ; differences are tentatively ascribed 
 to the very different fragmentation dynamics of these ions . 
Introduction
Experiment
Results and Discussion
Conclusions