Dirac bubble potential for He–He and inadequacies in the continuum: Comparing an analytic model with elastic collision experiments
Michael Chrysos
Citation: J. Chem. Phys. 146, 024106 (2017); doi: 10.1063/1.4973612 View online: http://dx.doi.org/10.1063/1.4973612
View Table of Contents: http://aip.scitation.org/toc/jcp/146/2 Published by the American Institute of Physics
Dirac bubble potential for He–He and inadequacies in the continuum:
Comparing an analytic model with elastic collision experiments
Michael Chrysosa)
LUNAM Universit´e, Universit´e d’Angers, CNRS UMR 6200, Laboratoire MOLTECH-Anjou, 2 Bd Lavoisier, 49045 Angers, France
(Received 17 October 2016; accepted 21 December 2016; published online 11 January 2017) We focus on the long-pending issue of the inadequacy of the Dirac bubble potential model in the description of He–He interactions in the continuum [L. L. Lohr and S. M. Blinder, Int. J. Quantum Chem.53, 413 (1995)]. We attribute this failure to the lack of a potential wall to mimic the onset of the repulsive interaction at close range separations. This observation offers the explanation to why this excessively simple model proves incapable of quantitatively reproducing previous experimental findings of glory scattering in He–He, although being notorious for its capability of reproducing several distinctive features of the atomic and isotopic helium dimers and trimers [L. L. Lohr and S. M. Blinder, Int. J. Quantum Chem.90, 419 (2002)]. Here, we show that an infinitely high, energy- dependent potential wall of properly calculated thicknessrc(E) taken as a supplement to the Dirac bubble potential suffices for agreement with variable-energy elastic collision cross section experiments for4He–4He,3He–4He, and3He–3He [R. Feltgenet al., J. Chem. Phys.76, 2360 (1982)]. In the very low energy regime, consistency is found between the Dirac bubble potential (to which our extended model is shown to reduce) and cold collision experiments [J. C. Mesteret al., Phys. Rev. Lett.71, 1343 (1993)]; this consistency, which in this regime lends credence to the Dirac bubble potential, was never noticed by its authors. The revised model being still analytic is of high didactical value while expected to increase in predictive power relative to other appraisals.Published by AIP Publishing.
[http://dx.doi.org/10.1063/1.4973612]
I. INTRODUCTION
Quantum chemistry offers unrivalled resources to suit the needs of physicists for reliable models to address pairwise interactions and electric properties relevant to weakly inter- connected light atomic “clusters”: interaction potential,1–8 dipole polarizability,9–18hyperpolarizabilities,16–19and dipole moment.8,13,17,20–22The smallest “unit” of such clusters is col- lisional atomic pairs, their prototype being4He–4He and the related isotopologues3He–3He and3He–4He. Among the prin- cipal suppliers of spectroscopic data on such unbound atomic pairs, one can find mature areas of scientific inquiry, viz., collision-induced Raman scattering9,10,15,23–25 and absorp- tion8,21,26–29as well as the process of collision-induced hyper- Rayleigh scattering which is a younger area in this list.19 Not unrelated to this issue is the use of cold elastic scatter- ing30–33 as an observational means to assess the interaction potential of He–He. (An excellent overview, still timely today, of the related phenomena in low energy elastic scattering is found in Ref.34.) This process is formally close to collision- induced Raman scattering, even though the two processes only barely overlap in their content: in the latter, the partial waves enter the cross sections through polarizability matrix- elements, viz., ∼P
l(2l+1)( ¯α)2fi; in the former, they merely depend upon the interaction potential, viz.,∼P
l(2l+1) sin2δl, through the phase shiftsδl. The primary motivation behind the
a)Electronic address: michel.chrysos@univ-angers.fr
present article was our desire to provide, by means of a simple mathematical model rather than of a black-box set of data for the interaction He–He potential, a picture of the Raman pro- cess to answer some recent pending questions.25However, in view of a series of findings which appeared in the meantime (outlined below) choice was made to put that project on hold to give priority to another task: elastic scattering of atomic and isotopic He–He.
One of the most insightful and fascinating potential mod- els for pairwise interactions in helium consists of an attractive deltafunction in the shape of a spherical bubble of radiusr0 to mimic the interatomic potential.35–38At the origin of this model is the observation that4He–4He supports but a single bound state, a property that is highly evocative of a deltafunc- tion (which has the same property).35However, it is still very surprising that such an oversimplified model can reproduce distinctive features of the helium dimers35 and trimers.36–38 In fact, the Dirac bubble potential (Dbp) is renowned for its capability to reproduce, without the need for elaborate com- putations, a wealth of stability properties in helium clusters, viz., the binding energies of 4He3 (96.1 mK) and 4He23He (11.4 mK),36,37an Efimov state39for4He23He (Ref.38), and the property of the isotopic variants3He4He,3He2,4He3He2, and3He3to be unstable.35,37Even more remarkably, it repro- duces the unique bound state of4He2“with an amazing overlap of 0.99 942 with the exact value”35 and predicts correctly the average internuclear distance (52.6 Å) and delocalization (48 Å) for that state. As paradoxical as it may sound, most of the probability distribution in that giant helium dimer is in the
0021-9606/2017/146(2)/024106/9/$30.00 146, 024106-1 Published by AIP Publishing.
024106-2 Michael Chrysos J. Chem. Phys.146, 024106 (2017)
classically forbidden region of the interaction potential. This is corroborated by the puny amount of energy which is needed for bonding in a deltafunction and is in agreement with pio- neering observations made in ultra-low temperature helium, in the 1990s,40–43until when the question over whether such vastly delocalized distributions and weakly bonding atomic dimers are possible had been matter of debate. Ever since that time, the issue has been offering to the scientific com- munity unexpected confirmations of the existence of huge (62 Å) and highly fluctuating loose chemical bonds,42–44pro- viding clear evidence of the biggest little molecule in the world.45More recently, a quantum halo state was found in4He3
and4He23He, which suggests an elusive structure for helium trimers.46
Further to the success of the Dbp in describing distinc- tive features of small bound helium clusters, it has been claimed that this potential is also capable of giving “the correct trends”35in the energy dependence of the total cross section for 4He–4He, 3He–4He, and 3He–3He scattering “although these results do not agree quantitatively with experimental data30”.35The data to which this citation makes reference had been measured in variable collision energy experiments over a wide interval of relative velocities, ranging, for the case of
4He–4He, from 4000 to 95 m/s.30Nearly a decade after their appearance,30 an experiment conducted at collision energies from 1.35 to 0.5 K for4He–4He (viz., relative velocities from 105 to 65 m/s) showed that scattering in this energy region is almost pure s-wave and predicted it to be gigantic (integral cross sections ranging from 200 to 1000 Å2as collision energy is reduced) due to a possibly bound state near the continuum31 (at that time the stability of4He2dimer had not yet been fully confirmed40–43). The authors of Ref.31stressed the agreement of their measurements with predictions based on the He–He model potentials of Aziz1,31and underlined the fact that it is the s-wave contribution which was seen in their experiment because the s-wave grows at low energy.
In light of the above, one can thus rightfully ask the fol- lowing: To what extent is the Dbp model really capable of predictions in the continuum? And how can it be revised to better suit needs that make it useful in applications involv- ing unbound species? It is the scope of the present article to offer answers to those questions, and to seek the range of agreement and the main causes of the disagreement between the Dbp model and the elastic scattering experiments. The remainder of the article is organized as follows. In Sec.IIa simple extension (EDbp) to the existing model is developed to fix the problem of the inconsistency with experiment for unbound collisional species. To the best of our knowledge, such an extended bubble model has not been reported so far and could well find its place in textbooks as a very insight- ful model for bound or collisional pairs of helium. In Sec.III, results are shown along with critical comparisons with data measurements from early conducted backward scattering and cold collision experiments.30,31Sec.IVis the Synopsis.
Anticipating our conclusions, it turns out that the Dirac bubble potential for unbound helium pairs can be as good as it is for bound dimers and trimers provided that the boundary conditions were modified to account for the onset of a repulsive wall in the interatomic potential at close range separations.
II. THEORY
A. Scattering in the frame of the center of mass The collisional system He–He consists of two mov- ing atoms with position and velocity vectors~ri and~vi(=~mpii) (i= 1, 2), respectively, which experience an interaction poten- tial V(|~r1−~r2|). Therefore, in order to study the scattering process, appropriate account of the separation of the center- of-mass motion in the two-body system must be taken. In fact, starting from the two-particle Hamiltonian,
H= ~p21 2m1 + ~p22
2m2 +V(|~r1−~r2|), (1) there is a straightforward way (which is well known to students and thoroughly expressed in textbooks) to reach the expression
H= ~P2 2M +~p2
2µ+V(r), (2)
~rand~pbeing the vector position~r1−~r2and the momentum
1
2(~p1−~p2) of a fictitious particle with reduced mass µ, and
~Rand~Pthe vector position mM1~r1+mM2~r2and the momentum
~p1+~p2of the center of mass of the two nuclei (Mdesignates the total mass,m1+m2).
To make comparisons with backward scattering and cold- collision measurements,30,31the relative motion will be con- sidered below, viz., the motion of the fictitious particle with massµand relative velocityv= pµ(=~µk =q
2Eµ ),Ebeing the eigenenergy involved in
"
~p2 2µ+V(r)
#
ψE,l(r)=EψE,l(r). (3)
B. The EDbp for unbound collisional pairs
A Dbp supplemented with an extra region at close-range separations is what will be referred to hereafter as EDbp.
Specifically, choice was here made to divide the coordinate r into three parts: an outer part (III),r > r0, an intermedi- ate one (II),rc <r <r0, and an impenetrable inner part (I), r < rc, now introduced to allow for considerations that are similar to those met in the literature of hard sphere models in order to mimic the steep potential wall of He–He. Within this model, the wavefunctionψE,l(r) in the continuous part of the spectrum (E≥0) must satisfy, in the center of mass frame,
"
d2
dr2 −l(l+1) r2 +k2
#
ψE,l(r)= λ
r0δ(r−r0)ψE,l(r) (4) forr ≥rc. Forr≤rc,ψE,l(r)=0.
The presence of a new region in the potential makes it mandatory to reconsider the previous model since the solu- tions of Ref.35are no more valid. Accordingly, the continuum wavefunctions in regionsIIandIIIread
ψE,l(r)= r k
πAr[jl(kr) cosθl−yl(kr) sinθl], (5) ψE,l(r)=
r k
πr[jl(kr) cosδl−yl(kr) sinδl], (6) respectively, withAbeing a constant, andθlandδlphase shifts to be properly calculated below.47In these expressions,jl(yl)
designates regular (irregular) spherical Bessel functions,rthe radial coordinate,k(=q
E, =2µ~2) the wavevector magnitude, E(>0) the energy of the scattering state,µthe reduced mass of the collisional pair, andlthe orbital quantum number. In the regionI,ψE,l(r)=0.
Continuity of the wavefunction at the two interfaces and the property of properly introducing the discontinuity of the derivative ofψE,latr0provide three conditions (i.e., as many as the number of the unknown parameters). A fourth condition has been already introduced through factor
q k
π in Eqs.(5) and (6), to account for energy normalization, ψE,l
ψE0,l
=δ(E−E0).
Skipping the tedious calculations, we obtain cotθl=yl(krc)
jl(krc), (7)
A=jl(kr0) cosδl−yl(kr0) sinδl
jl(kr0) cosθl−yl(kr0) sinθl, (8) and
cotδl =λkr0ξlyl(kr0)−yl(krc)
λkr0ξljl(kr0)−jl(krc) (9) with
ξl=jl(kr0)yl(krc)−jl(krc)yl(kr0), (10) λ= 2πµα
~2r0;α(<0) is the strength of deltafunction.
A nontrivial step in this calculus was the use of the l-independent expressionjl(x)yl0(x)−jl0(x)yl(x)= x12, which is a useful property of the Wronskian,W =jl(x)yl0(x)−j0l(x)yl(x), of spherical Bessel functions48 (“0” stands for dxd). After a change of variabler = xk during that algebra, the repeatedly occurring quantityjl(kr)y0l(kr)−jl0(kr)yl(kr) (“0” being nowdrd) takes the simple value xk2 =kr12.
Cast in its most aesthetic form, the expression of Eq.(9) permits to obtain δl at once. It permits also to draw some remarks as to the physics of the process. For instance, if rc→0 (jl(krc) goes to δl,0; jl(krc) diverges), the correct limit, cotδl '
rc→0 yl(kr0)
jl(kr0) − 1
λkr0[jl(kr0)]2, is reached, which is the expression reported in Ref.35for the original bubble model.
In Figure1, the EDbp of He–He is illustrated as a function of separationr. Shown also in this figure is therc(E) profile in order for the EDbp model to optimally reproduce backward glory measurements (see below). Drawings (a) and (b) indi- cate the two interfering pathways which are involved in the scattering amplitudesf(θ) andf(π−θ) in an experiment with identical helium atoms in the center-of-mass frame.
C. The EDbp in the bonding part of the spectrum The approach developed above cannot be internally con- sistent without an appropriate treatment of the bound helium problem within the same model. This is because, owing to changes in the boundary conditions (by comparison with the standard Dbp), one should expect also to occur modifications inλ. This forces us to revisit the bound He–He problem, as shown hereafter (along with a discussion as to whether the same or a differentλvalue must be eventually considered).
FIG. 1. Cartoon illustrating the EDbp of He–He as a function of separationr.
The vertical arrow indicates an attractive deltafunction located atr0= 13.15 bohr. A repulsive wall with thicknessrcis shown at short distances (in blue).
The red curve shows howrc(E) must depend onEin order for the EDbp model to optimally reproduce backward glory measurements. Sketches (a) and (b) show the two interfering pathways which are involved in the scattering amplitudesf(θ) andf(π−θ) in an experiment with identical atoms in the center-of-mass frame.
In regionsIIandIII, the wavefunction of the ground bound level takes the forms
RII =Bsinh(κr)
κr +Ccosh(κr)
κr (11)
and
RIII =De−κr
κr , (12)
respectively; “R-form” notation (see footnote, Ref. 47) is used for this wavefunction. These expressions are sub- jected to the boundary conditionsCB=−tanh(κrc),Bsinh(κr0) +Ccosh(κr0)=De−κr0, andRIII0 (r0)−RII0(r0)=rλ0R(r0). These equations, along with normalization of the wavefunction, lead to the complete solution of the bound Schr¨odinger problem and to the determination of parametersB,C,D, and λ(κr0).
After some tedious algebra, the wavefunction reads R=
1−[2κ(r0−rc)+1]e−2κ(r0−rc) 4κ5r2
0
−12
×sinhκ(r<−rc) κr<
e−κ(r>−rc)
κr> , (13)
wherer<andr>are, respectively, the smaller and larger of r, r0. As forλ(κr0), its new expression is (x=κr0)
λ=− 2x 1−e−2x
F(x)+e2x
1+e2x , (14)
with
F= tanhx+tanh(κrc)
tanhx−tanh(κrc). (15) Obviously, withrc= 0, the expression of Eq.(14)reduces to
− 2x
1−e−2x, which is exactly the same expression as that of Eq.(7) in Ref.35.
We expect from an EDbp to conserve all the advantages of the Dbp. However, by pluggingκ =0.005 520 bohr1 and r0 = 13.15 bohr into Eqs.(14)and(15), new values forλ4–4 are obtained depending on the choice of rc, instead of the previousλ4–4= −1.0741 (the value ofκis from data for the
4He–4He binding energy; the value of r0 stems from maxi- mization of the overlap between the bound-state wavefunction of4He2 in the Dbp model and anab initiocomputation). For
024106-4 Michael Chrysos J. Chem. Phys.146, 024106 (2017)
instance, withrc = 3.2 bohr, viz., two atoms with interpene- trating van der Waals halos (see footnote Ref.49), Eq.(14) returnsλ4–4 =−1.3955. Furthermore, on the basis of the gen- eral expressionµλ =2π~α2r0(=constant), new values forλ3–4and λ3–3 are found by using the relationshipµλ4–4
4–4 = µλ3–43–4 = µλ3–33–3. It is bothersome that the use of a repulsive wall has an effect on the attractive well of the interatomic potential. Equally annoy- ing is the observation that a change of theλ4–4 value cannot avoid damaging the spectacular overlap35between the model and the exact bound-state wavefunction of4He2, as well as the fact that changes inλ3–4orλ3–3entail misleading conclu- sions about the binding properties of4He23He and4He3He2
trimers.37 In the following paragraph, we outline the way to counter these problems.
Figure2showsλ(κr0) as a function ofr0for a value ofκ taken from the binding energy of4He2(κ=0.005 520 bohr1).
In contradistinction to the Dbp (in whichλ(κr0) is a continuous and monotonous function ofr0(r0∈[0,∞)) with a minimum (r0,λ(κr0))=(0,−1)), in the EDbp the expression of Eq.(14) does not vary monotonously withr0(r0 ∈[rc,∞)) and is singu- lar atrc. Choice was made ofrc= 3.2 bohr as a typical value of this parameter to illustrate these properties (see footnote Ref.49).
D. The advantages of a functionrc(E)
In order to ensure that the predictions of the Dbp regard- ing the binding properties of helium dimers and trimers are left unchanged in the bonding part of the spectrum, we now assume thatrcis anE-dependent parameter in theE >0 part of the spectrum and zero otherwise. The need forrc(E) = 0, as long asE ≤0, is also dictated by theoretical considerations.
Specifically, by exploiting the isomorphism of Schr¨odinger’s equation for the boundl= 0 level with the free-particle partial wave Green’s functionsGl(r,r0,k), a general argument about the existence and the hierarchy of rotational bound levels in
FIG. 2. λ(κr0) (no units) as a function ofr0(in bohr) for a value ofκtaken from the binding energy of4He2(κ=0.005 520 bohr1). In the Dbp model (rc= 0),λ(κr0) is a continuous and monotonous function ofr0(r0∈[0,∞)) with a minimum (r0,λ(κr0))=(0,−1) (blue line curve). In the EDbp model (rc,0),λ(κr0) [Eq.(14)] does not vary monotonously withr0(r0∈[rc,∞)) and is singular atrc(red line curve). The discontinuityr0=rcis indicated by a vertical dashed line. The valuerc= 3.2 bohr (see footnote Ref.49) was used for the illustration.
He–He has been advanced and applied to the Dbp model (rc= 0) on the basis of some monotony property.50According to this argument, a boundl= 0 level cannot exist unlessλ<−1, a condition obviously satisfied by λ4−4(=−1.0741) but not byλ3–4(=µµ3–44–4λ4–4 =−0.9207) orλ3–3(=µµ3–34–4λ4–4=−0.8056).
This is a remarkably elegant way of reaching the conclusion that4He2 does support a bound state while3He4He or3He2
does not, in agreement with the observation and ab initio quantum mechanical data and computations for those three systems. Note in passing that the Green’s function, on which the aforementioned level-hierarchy argument is based, is for truly free particles, and nothing guarantees its validity in the case ofrc,0 (displaced origin). In fact, unlike with the func- tion λ(x) = − 2x
1−e−2x (x = κr0) of the Dbp model (rc = 0), which remains monotonous and continuous atx= 0, its min- imum beingλ(0)=−1 (see Ref.50), in the case of the EDbp (rc ,0),λsatisfies the more general expression of Eqs.(14) and (15), which is neither monotonous nor continuous at x=κrc.
It is now obvious that an appropriate use of EDbp makes it possible to encompass two advantages: on one hand, preserve the validity of the level-hierarchy argument for the function λ(κr0) by settingrc = 0 atE < 0 to ensure reliable predic- tions about the bonding properties of 4He2 and the absence of bonding in3He4He and3He2; on the other hand, keep our model sufficiently flexible so that the scattering features at E >0 are reproduced. As we will see below, the giant cross sections reported in Ref.31at low energy are reproduced par- ticularly well with the Dbp (rc= 0), offering a glimpse into the predictive power of Lohr and Blinder model35 for processes involving slowly scattered unbound He–He atoms. At a higher energy, instead, the need for an extended model (EDbp,rc,0) becomes instrumental.
III. RESULTS
A. Measurements, computations, and comparisons Comparisons between the two experiments for4He–4He are made in Figure3. The backward scattering measurements of Ref.30are shown plotted together with the s-wave mea- surements of Ref. 31(and their error bars) as a function of beam velocity, over the same range and on the same semilog- arithmic scale as in Ref.31. We remind that in the experiment of Ref.30the maximum value probed on the velocity axis was much larger (∼4000 m/s, not shown) than in the experiment of Ref.31but the minimum velocity was not as small. Notice the smooth transition from the data of Ref.30to those of Ref.31 as the collisions slow down. The solid-line curve in this figure is the effective cross section obtained from the HFD-B2(HE) potential.1,31 Only the short segment which is restricted to the portion of the velocity range probed in the experiment is shown, exactly as was done in Ref.31to highlight the impres- sive consistency of the measurements with the potential of Ref.1. To draw this segment, the theoretical cross section was used at first,1,31which was then converted to an effective cross section in order for the comparison to be meaningful; theoreti- cal (raw) cross sections always depend on relative velocity and are determined by the interaction potential. A detailed account of the effect of convolution with the velocity distribution
FIG. 3. 4He–4He effective integral scattering cross sections (in Å2) as a func- tion of beam velocity (in m/s). Shown are the s-wave measurements of Ref.
31and related error bars (unfilled black squares). The cross sections associ- ated with the HFD-B2(HE) potential of Aziz1(black curve) are shown after convolution with the velocity distribution of the thermalized target atoms. The data are shown plotted in the same range and on the same scale as in Ref.31;
see Fig. 3 (and Fig. 4), therein. Also shown are the effective cross sections measurements of the backward scattering experiment30(filled blue circles).
of thermalized target atoms, as well as other subtle points relevant to cross section conversions and their programing, is given as thesupplementary material; a more extended velocity range is covered therein.
Figure4shows integral cross sections for4He–4He scat- tering as a function of relative velocity. The figure should be viewed in parallel with Fig. 3. Raw cross sections (before convolution) taken from Ref. 31 and associated with the HFD-B2(HE) potential1 are shown plotted (blue solid-line curve) over the same range and on the same semilogarithmic
FIG. 4. 4He–4He integral scattering cross sections (in Å2) as a function of relative velocity (in m/s) in the center-of-mass frame. The EDbp cross sections (this work) are shown for a typical thickness (rc= 3.8 bohr) (unfilled magenta circles) and compared with the Dbp (rc= 0) ones (filled red squares). The cross sections associated with the HFD-B2(HE) potential of Aziz1are shown before (blue curve) and after convolution with the velocity distribution of the thermalized target atoms (black curve). Below 100 m/s, an impressive consistency between the response of the Dbp and the potential of Aziz is seen which deteriorates quickly into an extremely pronounced mismatch beyond 400 m/s (not shown).
scale as in Fig.3and Ref.31. Also shown is the segment of the previous figure (black solid-line curve) and the response of Dbp (λ=−1.0741;rc= 0) and EDbp (λ=−1.0741;rc= 3.8 bohr); the latter thickness is typical of a pairwise helium inter- action potential barrier (see below). Orbital angular momenta up tol= 30 in the partial-wave contributions to the total cross section were found to be enough for convergence whatever the velocity in the probed range (v <4000 m/s). There are several points to comment in this figure, in relation with Fig.3. In the slow velocity regime, the gigantic s-wave cross sections which had been observed31 and consistently calculated in the state of the art1,31are also reproduced with the Dbp model. This agreement, which becomes spectacular below 60 m/s, lends credence to the model of Lohr and Blinder also for weakly colliding unbound atoms (and not only for bound states as in Refs.35and37); interestingly, however, it was never noticed by these authors.35This agreement is all the more impressive since the oversimplified deltafunction model of Ref.35has already been shown to reproduce several distinctive features of dimers35and trimers.36–38In this context, it is worth noting some relevant statement pointed out in Ref.31about the obser- vation of “gigantic cross sections which increase with decreas- ing scattering energy, as expected for a potential with a possible bound state near the continuum.”31 Outside that regime, a pronounced declining trend is found for the Dbp model by comparison with the measured data.30This mismatch attains one order of magnitude at 400 m/s and quickly increases to sev- eral orders of magnitude asvis increased further (not shown).
Obviously, the need for an extended (EDbp) model becomes mandatory outside the region of weak collisions. We remind that outside that region, He–He scattering is far from being a pure s-wave because there is a multitude ofl-partial waves which make significant contributions to the cross section there.
As Mesteret al.have pointed out,31“the scattering is over 93%
s-wave” at 105 m/s, “increasing to 99.9%” as collisions further slow down. Notice the response of the EDbp, which, except for the lower velocities, is maintained at levels consistent with those of the measured data. The diving response of the EDbp model in the region of weak collisions is evidence of disal- lowed low-lpartial wave components in a region where those waves should be dominant. This stresses the need for a properly adjusted functionrc(E(v)) instead of an arbitrarily chosen fixed value.
Figure5shows integral scattering cross section calcula- tions for4He–4He along with measured data for that species over a much more extended velocity range than Figs.3and4.
The calculations were done in the EDbp model (λ=−1.0741, rc ,0) and are illustrated in the frame of the center of mass as a function of relative velocity. The results are shown (black crosses) for 51 values ofrcvarying from 5 to 0 bohr in steps of 0.1 bohr (in the descending order atv= 1000 m/s). A focus on an arbitrarily chosenrcvalue (rc= 3.8 bohr) helps to guide the reader’s eye across the data points (unfilled magenta circles).
The data of the backward glory scattering measurements were taken from Ref.30(Figure 10, therein) and are here depicted by blue symbols. The overall consistency between theory and experiment is judged remarkably good. Note, in particular, the shoulder which is displayed by the EDbp data for the higher values ofrcin the regionv∼200 m/s. This shoulder is similar
024106-6 Michael Chrysos J. Chem. Phys.146, 024106 (2017)
FIG. 5. 4He–4He integral scattering cross sections (in Å2) shown as a function of velocity (in m/s) over a much more extended range than in Figs.3and4.
The cross section displays glory oscillations. The calculations were done with the EDbp model (λ=−1.0741); the results (black crosses) are shown for 51 values ofrcvarying from 5 to 0 bohr in steps of 0.1 bohr (in the descending order atv= 1000 m/s). EDbp cross sections for a single typical thickness (rc= 3.8 bohr) are shown separately to guide the reader’s eye across the data points (unfilled magenta circles). The measured glory data30are shown with blue symbols. Also shown is the response of the Dbp (filled red squares).
Notice the shoulder displayed by the EDbp data for the higher values ofrcin the region 200–300 m/s and its similarity to a shoulder which is also present in the experimental cross sections.
to what is also present in the experimental cross sections. In the measured glory, this structure occurs aroundv '150 m/s and is the manifestation of a dip, smeared out to a plateau, due to s-wave phase shifts that go through zero, because of a can- cellation, in the real species, between attractive and repulsive forces.30 For comparison’s sake, cross sections of the Dbp model (rc = 0) are also shown (filled red squares). As v is increased, these cross sections (red squares) diverge by orders of magnitude (not shown) from the measured data.
The cross sections of the3He–4He species are illustrated in Fig.6. The same choice of colors and type of symbols as in Fig.5were used for all plots. The calculations (black crosses and unfilled magenta circles) were done for λ3–4(=µµ3–44–4λ4–4
=−0.9207). Also shown are the cross sections calculated with λ3–4 = −0.77 for the arbitrarily chosen value rc = 3.8 bohr (unfilled green circles). This value ofλ3–4had been suggested by Lohr and Blinder as another possibility for the strength of the deltafunction potential for the3He–4He species on the basis of considerations relative to trimers.37The wavy patterns seen in the EDbp cross sections is evidence of the need for a function rc(E(v)), instead of somercwhich is constant, so that the cross sections “surf” on (rather than produce) the “waves”; only with a properly tailoredrc(E) will the3He–4He cross sections vary monotonously as a function ofv(no glory). This monotony, which is characteristic of heteronuclear pairs, stems from can- cellations between maxima and minima in the cross section seriesσ3–4 =4πk2
P∞
l=0(2l+1) sin2 δl, as dictated by the even and odd parity quantum numberlwhich entersσ3–4on an equal footing. By contrast, in the case of homonuclear pairs, even and oddlenter the seriesσ4–4 = 8πk2
P∞
l=0,2,...(2l+1) sin2 δl and σ3–3=2πk2[P∞
l=0,2,...(2l+1) sin2δl+3P∞
l=1,3,...(2l+1) sin2δl]
FIG. 6. Same as in Fig.5but for3He–4He. The calculations (black crosses, unfilled magenta circles, filled red squares) were done forλ3–4(=µµ3–44–4λ4–4
=−0.9207). Also shown are EDbp cross sections forrc= 3.8 bohr calculated withλ3–4=−0.77 (unfilled green circles). Even if much less pronounced than the physical undulations of Fig.5, the presence of experimentally unobserved oscillations in the EDbp cross sections (heteronuclear species are not allowed to exhibit glory) offers evidence of the incapacity of a singlerc value to consistently reproduce the experiment. This observation stresses the need for arc(E(v)) function in order to provide a monotonic shape in the EDbp cross section that “surfs” on the oscillations.
with unequal weights (see, for instance, Ref. 51, Eqs. (13) and (14) therein; Ref.52, Eq. (9) therein).
Cross sections for 3He–3He scattering are displayed in Fig.7. Same colors and symbols as those chosen for the plots of4He–4He and3He–4He were used. Note, in particular, the deep minimum which is displayed by the higher-lying EDbp data points in the regionv ∼200 m/s and its resemblance to a dip observed near 300 m/s in the experimental cross sections.
This is a manifestation of the atomic Ramsauer-Townsend effect “caused by the passage of the dominantp-phase shift through zero, where attractive and repulsive forces cancel each other.”30
FIG. 7. Same as in Fig.5but for3He–3He. Notice the similarity between the deep minimum shown by the higher-lying EDbp data points in the region
∼200 m/s and what is observed near 300 m/s in the experimental cross sections.
Particular gratifying is the observation of glory oscillations with appropriate frequency and amplitude.
As a general remark, it is a distinctive feature of the phys- ical cross sections of He–He that the oscillations are very pronounced for the homonuclear pairs and completely disap- pear for the heteronuclear isotopologue, in spite of an identical interaction potential in all three cases; see Ref. 32(p. 416, Fig. 6.28). In this respect, note that, physically, the back- ward glory oscillations of4He–4He and3He–3He (Figs.5and 7, respectively) originate from the indistinguishability of the atoms via zero-angle interference between the primary beam particles and the particles which are backward-scattered by the repulsive part of the potential.30,32It is therefore not surprising that the Dbp model, which lacks repulsive part of the potential (rc= 0), produces such a poor response as the one shown by the red squares in Figs.5and7.
Going back to the EDbp calculations, notice the jerks which are observed inside the oscillating patterns of the mod- elled cross sections in Figs.5and7. These come from inter- ferences in the interval [rc,r0] situated between the wall and the well of the He–He potential in modelling it as an EDbp.
This explanation is corroborated by a series of other calcula- tions (not shown) with a potential consisting of a repulsive wall alone (λ=0, rc,0) or, reciprocally, with the Dbp (λ , 0, rc = 0). Both those calculations showed that the glory oscillations of 4He–4He and3He–3He were no longer jerky. Definitely, modelling simply (EDbp) means accept- ing these unphysical oscillations as a necessary evil. In the actual He–He pair (experiment, state-of-the-art potential), no such interferences ever occur for that there is no specific values r0 or rc determining where to locate the well or the wall of the potential, but only a smooth and continuous curve V(r) (HFD-B2(HE)1) going as a progressively steep- ening barrier at short r and as a shallow trough around the minimum.
B. Searching the “best”rc(E)
In this paragraph, we will see that there exist functions for the thickness profile which, once properly adjusted, can satisfactorily reproduce the measured cross sections, both in average magnitude and in pattern, including oscillation fre- quency and damping, over a wide range ofvvalues. It turns out that such functions rise first quickly and then very slowly with E, while satisfyingrc(E) →
E→00 to allow, at small collision ener- gies, the low-lpartial waves to penetrate into distances close to the unified atom limit. The near-saturating asymptotic trend ofrc(E) is consistent with previous observations and calcula- tions conducted in our group and endorses a view according to which at high collision energies (i.e., in the far high-frequency wing of the Raman spectrum) the van der Waals radii of the two atoms interpenetrate each other;23analogous findings have been reported also for other rare gases53(and their mix- tures54). The extent of this wall was found to increase only very slightly beyond the Ramsauer-Townsend region, being kept practically constant atrc'4 bohr. This value, which is smaller than the van der Waals diameter of atomic helium (5.4 bohr), offers evidence of two heavily interpenetrating helium halos, in agreement with the critical value (3.2 bohr) which was pinpointed in 2000 as the minimum He–He distance effectively probed spectroscopically.23
We chose first to representrcas some arctan to translate mathematically the very slow increase ofrcbeyond the Ram- sauer region, and then, we supplemented it with a Lorentzian function for the effects observed below 350 m/s in Figs.5–7.
A more refined calculation suggests rc(E)≈A[arctan(aE) +c2(E−d)b 2+1] (Ein meV,rcin bohr;A=a= 3,b= 2,c= 20, d = 0.2) as a compromise between simplicity and accuracy.
The shape ofE(rc) is shown schematically in Fig.1. Shown in Fig.8 are EDbp cross sections, for a homonuclear and a heteronuclear isotopic pair, which were obtained by running a Newton-Raphson inversion subroutine (curves). The measure- ments of Ref.30(symbols) were used to feed the subroutine.
The convergence is to within 0.1%. The resultingrcis illus- trated as a function ofEin the inset. The profile follows closely the functional form suggested above and illustrated in Fig.1.
Although rc(E) is not exactly the same for different combi- nations of helium isotopes, the degree of similarity between the rc(E) profiles is enough to suggest some universality to the model. The discrepancies between differingrc(E) profiles could well be due to the simplicity of the EDbp but could just as well be attributed to some physical effect in the corresponding He–He potentials for the three isotopologues. In this context we should note, however, that the material of Fig.8(inset) is mainly for information purposes as, strictly, this way to pro- ceed with data-inversion remains valid only for velocities that are sufficiently high to ensure that the raw (unconvolved) cross sections are practically indistinguishable from the effective (convolved) ones. In fact, for velocities below roughly 200 m/s, where the effects of the bath may be strong (see thesupplemen- tary material), a more appropriate way to carry out the analysis would be to remove those effects (de-convolve) before feeding the inversion subroutine (or, alternatively, to use quantum- mechanical raw cross sections, if available, as input). Such operations, which can be demanding, are expected to affect to some extent the height and width of the sharp peak seen in the inset of Fig.8. This task is part of an oncoming report.
FIG. 8. Integral cross sections (in Å2) as a function of velocity (in m/s) for a homonuclear (3He–3He) and a heteronuclear (3He–4He) isotopic pair. Shown are EDbp cross sections computed by means of an iterative “inverse scattering”
process (curves). The Newton-Raphson root-finding algorithm was used for the purpose. Also shown are the experimental data30that we aim to approach (symbols). Convergence to within 0.1% was achieved. Convergedrcvalues are shown plotted (in bohr) as a function ofE(in meV) in inset.
024106-8 Michael Chrysos J. Chem. Phys.146, 024106 (2017)
Before ending this section, we should make it clear that the present work is not intended for the purpose of determin- ing the repulsive wallVrep(r) in the interaction potential. Past works have adequately addressed that issue30,55,56 and have proved to be of high value for the current state of the art. Nor is it the purpose of this paper to determinerc(E) by a more complicated function and more parameters. Offering evidence that some simple form ofrc(E) (such as the one given above) is qualitatively enough for describing the principal features of scattering cross sections30is what we content ourselves with.
In this context, it is worth to clarify thatrc(E) is not related, directly or indirectly, to the classical inner turning pointrin of the particles’ relative orbit; and for that reason, it cannot be used as a device to determine, as a function of energy, the repulsive wallVrep(r) in the He–He potential. Instead of solv- ing the problemEVrep(rin) = 0, the interest in usingrc(E) lies in its significance as a phenomenological parameter:rc(E) describes the extent of a hypotheticalinfinitewall which would produce, at energyE, the same effect asVrep(r).
IV. SYNOPSIS
Further to the advantageous use of elastic scattering (glory oscillations, orbiting resonances, shadow scattering, etc.) as a tool for new physics (low-energy reactive collisions, etc.),33 its use as a long-proven device for determining interatomic potential parameters for van der Waals “clusters” may still hold resonance today.30–32,34This is all the more true given the tremendous performance of quantum-chemistry methods designed to crosscheck a large variety of thermophysical prop- erties. However, in spite of the advanced level of nowadays’
literature concerning potential models, there are still questions left open even for the simplest of the van der Waals “clusters”:
He–He. In particular, the problem of whether the Dirac bub- ble potential (Dbp)35,37is capable of consistently reproducing He–He scattering cross sections has not been adequately explored, although it has to some extent been addressed in the past.35This is an intriguing issue since the Dirac bubble potential enjoys high predictive power in modelling the stabil- ity properties of helium dimers35and trimers,36–38including several of their distinctive features (binding energies, bond lengths, Efimov states, etc.). Is the performance of Dbp model equally good for collision processes? It was the purpose of the present work to answer that question. In this paper, we have worked out a more general Dirac bubble model for He–He in which the boundary conditions are changed as a function ofE to mimic the onset of the repulsive forces at close range sepa- rations: as long as low collision energies,E, are concerned, the extended model is reduced to standard Dbp, in conformity with theoretical arguments and pioneering experiments conducted in the pure s-wave scattering regime;31in contrast, at higher E, a repulsive wall with a thicknessrcslowly approaching the van der Waals diameter of helium atom needs to be considered in order to reach agreement with a series of milestone back- ward glory scattering measurements30for atomic and isotopic He–He species.
We expect that the extended bubble model will be help- ful for engineering purposes (molecule helium microsolvation, spectroscopy on helium nanodroplets, etc.), but also from a
more fundamental standpoint as a means to quantify through elegant and informative Bessel algebra the role of the repulsive part of the interaction in scattering cross sections. The inter- est of such a model will, among other things, be appreciated in Raman processes as a tool to “separate” (as analytically as possible) the contributions23 of interaction polarizability, ¯α,
∆α, from those of the interaction potential. In this context, this model could be effective for answering recently raised questions;25pending problems of this nature were the initial motivation for this article. Finally, it would be interesting to study in a future work how thermophysical property values7 from this model potential compared with the state of the art at various temperatures. Work is in progress in that direction.
SUPPLEMENTARY MATERIAL
Seesupplementary materialfor an account of the effect of convolution with the velocity distribution of thermalized target atoms and for some points relevant to cross section conversions and their programing.
ACKNOWLEDGMENTS
This work benefited from stimulating discussions with Florent Rachet.
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