Many-Body effects and resonances in universal quantum sticking of cold atoms to surfaces

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4Many-Body e?ects and resonances in universal quantum sticking

of cold atoms to surfaces

Eric.R.Bittner ?

Department of Chemistry and Biochemistry,

The University of Texas at Austin,

Austin,Texas 78712-1167

John C.Light ?

The James Franck Institute and the Department of Chemistry,

The University of Chicago,

Chicago,Illinois 60637

(October 31,1994)

The role of shape resonances and many-body e?ects on universal quantum sticking of ultra cold atoms onto solid surfaces is examined analytically and computationally using an exactly solvable representation of the Dyson equation.We derive the self-energy renormalization of the the transition amplitude between an ultra cold scattering atom and the bound states on the surface in order to elucidate the role of virtual phonon exchanges in the limiting behavior of the sticking probability.We

demonstrate that,to ?rst order in the interactions for ?nite ranged atom-surface potentials,virtual

phonons can only rescale the strength of the atom-surface coupling and do not rescale the range of the coupling.Thus,universal sticking behaviour at ultra-low energies is to be expected for all

?nite ranged potentials.We demonstrate that the onset of the universal sticking behavior depends greatly on the position of the shape resonance of the renormalized

potential and for su?ciently low

energy shape

resonances,deviations from the universal s (E )∝√?email:https://www.wendangku.net/doc/fe18816694.html, ?email:https://www.wendangku.net/doc/fe18816694.html,

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quantum and classical analysis yield identical results.Recently Carraro and Cole[7]applied Brenig’s equations to study H sticking to liquid He?lm and obtained remarkable agreement with the experimental results[11].

As far as we know,the most comprehensive examination of the quantum sticking problem is that of Cloughtery and Kohn who developed an analytic model for quantum sticking.[8]Working up from a surface composed of N discrete lattice atoms to the continuum,N→∞,they demonstrated that within their one dimensional model,while polarization e?ects due to virtual phonon exchanges may lead to an increase in the sticking coe?cient,eventually these e?ects will be lost as pure re?ection becomes more important at low energy.

Experimental evidence for quantum sticking has come from a variety of di?erent sources.Most notably in the desorption of positronium(Ps)from Al surfaces[12]and the sticking of spin polarized hydrogen atoms on liquid4He ?lms[10,11].Slow Ps atoms,composed of an electron and a positron,are almost ideal atoms for studying quantum sticking.At experimentally accessible scattering energies of7.5meV,the thermal de Broglie wavelength of Ps is on the order of100?A and is much larger than the range of the potential well.Measurements of the thermal desorption rates of Ps from clean Al(111)along with detailed balance arguments indicate that the Al surface is a“blackbody”for Ps emission and hence the system fails to exhibit perfect re?ection of ultraslow Ps atoms.For the case of H on He?lms,the interactions are considerably weaker and there are a number of examples of experiments in which the sticking data extrapolates to s(0)=0[13,14,15,16,17,18,19,20,21].Surprisingly enough,recent data for H sticking onto thin He?lms,reported a gradual increase in the sticking probability as the temperature decreased between10 mK and.1mK[10,11].This prompted Hijmans,Walraven,and Shlyapnokov[22]to propose that the atom-surface potential might be a?ected by very long-range van der Waals forces(with relativistic retardation)from the underlying substrate.Indeed,calculations by Hijmans,Walraven,and Shlyapnokov[22]and by Carraro and Cole[7]indicate

√

that substrate e?ects could account for this trend and predict that the

T behavior was observed.

In this paper we wish to examine two issues which we believe have not been adequately addressed,and yet are necessary in order to properly interpret the experimental results mentioned above.The?rst issue is the role of many-body e?ects on the limiting behavior,and the second is the role of low energy shape resonances.In the next section,we examine the many-body contributions by solving formally the Lippmann-Schwinger equation for the scattering wavefunction in which we include the self-energy due to the virtual phonon transitions between the scattering wave and the inelastic channels.Since our theory incorporates the dynamical evolution of the surface directly into the?nal equations of motion,it is essentially exact(for single phonon exchanges)and non-perturbative. Using approximate forms of the low energy scattering wavefunction,which we show to become exact as k→0,we derive the renormalization of the sticking probability due to the the many-body interactions for realistic atom-surface interactions.We then demonstrate that polarization due to many-body e?ects serves only to rescale the strength of the interactions and does not rescale the e?ective range.We next examine the e?ect of low lying shape resonances and demonstrate the e?ect of increased penetration of the scattering wave into the potential region near these resonances. Finally,in the last section we apply the methods which we developed to examine the low energy limiting behavior of H sticking to thin and thick4He?lms.

II.THEORY

The starting point of our theory is the Lippmann-Schwinger(or Dyson)equation for the scattering wavefunction in which we include explicitly the self-energy or polarization due to virtual phonon exchanges.The self-energy re?ects the many-body nature of the problem and we shall assume that only single exchange processes are important.Close coupled forms of this equation have been used extensively by Stiles,et.al[23,24]and Jackson[25,26,27,28,29,30,31]to study inelastic molecule-surface scattering and Whaley and Bennett have used the formalism to study atom scattering from disordered surfaces[32].We write the elastically scattered component of the wavefunction as

φo(E)=φbare(E)+G bare(E)V self(E,T)φo(E),(1) whereφbare(E)and G bare are the bare wavefunction and Green’s functions in the absence of surface vibrations and V self(E,T)is the self energy operator

1

V self(E,T)=

Our notation is such that superscripted+and?refer to whether or not a virtual phonon was initially created or annihilated.Also,throughout this paper we shall de?ne H o as the static(zero-phonon)Hamiltonian and V ph q(z) as the force between the scattering atom and a phonon with wavevector q.We have also assumed that the bath is harmonic.

Closure is accomplished in the usual way by iteratively substitutingφo back into the Lippmann-Schwinger equation and analytically performing the summation over single phonon exchange diagrams.

φo=φbare+G bare V selfφbare+···

=(1?G bare V self)?1φbare(3) Before applying our theory to a physically realistic system,it is important to explore the e?ects induced by the additional polarization due to virtual phonon exchanges.Let us consider the“golden rule”transition rate between a very low energy scattering state and a single bound state,ψB,and attempt to relate the transition probabilities predicted using a dressed wave to those predicted from using the bare wavefunction.In short,we need to consider the matrix elements

ψB|V ph|φo = ψB| 1?V ph G o(E)V ph(G+(E)+G?(E)) ?1V ph|φbare ,(4) where G o(E)is a solution of

(H o?E)G o(z,z′)=δ(z?z′)(5) subject to the boundary condition G o(?∞,z)=0and G o(z,z′)→0as z→∞.

To evaluate this matrix element we need to make a series of approximations regarding the functional form of φbare.The bare scattering wave,φbare,is an solution of(H o?E)φbare=0subject to the boundary conditions:φbare(?∞)=0andφbare(z)→sin(kz)+tan(δk)cos(kz)as z→+∞.We can also de?ne the real regular and irregular functions,ψr,i(z),as solutions of

(H o?E)ψr,i(z;E)=0(6) subject to the boundary conditions

ψr(z,k)z→∞

?→sin(kz+δ)

?→cos(kz+δ).

ψi(z,k)z→∞

(7) Using these functions and the Wronskien relation,we can write G o(z,z′;k)as

1

G o(z,z′;k)=?

2mE is the scattering momentum.

For very low energies and temperatures,virtual transitions are predominantly to the bound state and we can obtain the single phonon exchange propagator from the bound(asymptotically closed)wavefunctions,weighted by the appropriate phonon densities of states,which in the continuum limit can be taken as

G+(z,z′)=ψB(z)ψB(z′) d2qω(q)(E B?ω(q)+k2)(n q(T)+1)

=ψB(z)ψB(z′) ωD0dωρ(ω)cos2(q(ω)a/2)

(n(ω)+1) ω2D?ω2

F(k)=2k2+E

B?ω

ω2D ω2D?(k2+E B)2

E in the limit of k→0for?nite ranged atom-phonon interactions.

What we have neglected in this evaluation is the possible contribution from shape and threshold resonances.In

particular,a threshold resonance or a shape resonance nearly at the threshold may produce an additional pole close

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enough to the real energy axis to change the limiting behavior of F(k).For the moment,let us consider the case that there are no threshold resonances and that we are well below the last shape resonance of the static potential surface. In this case,Σ(0)is a constant between1and-1and Eq.21produces only a rescaling correction to the DWBA result,

√

i.e.there is no change in the e?ective range of the potential and the s→

E behavior of the sticking coe?cient is recovered.

III.STICKING OF ATOMIC HYDROGEN ON LIQUID HELIUM FILMS

The interactions between H and liquid He are known rather well.In fact,variational estimates of the binding energy of H on liquid3He and4He date back to the late70’s[19,13].At long range,the potential can be obtained from summing over all pairwise van der Waals interactions between the H and the He atoms.At large distances (over200?A),retardation e?ects need to be included in order to provide the proper cut o?in the attractive forces.At shorter ranges,the potential can obtained from empirical?ts to atomic beam scattering data.[19].To lowest order, the e?ective H–liquid He potential can be constructed by integrating over H-He pairwise potentials and assuming that the liquid He is a semi–in?nite and incompressible?uid extending from z=?∞to z=0and with a sharp density pro?le,ρ(R,z)=ρoΘ(z+u(R)).

V(z)=ρo d2R′ +∞?∞dz′v pair(r)Θ(z′+R′),(23) where r=

of previous treatments[33].A number of forms for the e?ective potential have been proposed ranging from Morse potentials used by Zimmerman and Berlinski[13]to much more realistic forms which include the1/z3van der Waals contributions both from the He?lm and the underlying substrate.The e?ective potential used in our study is similar to that used by Carraro and Cole[7]and by others[34].We included both He?lm and substrate van der Waals polarization contributions as well as a relativistic cut o?factor which limits the long range behavior of the potential at distances greater thanλ=200?A.

V o=?(z)?C3 z3(z+d)3 γ(z)?C s1

?2

2πρo

ˉh q

S1/2 q

q3,(27)

Mρ

whereαis the He surface tension(α=0.27K?A?2)[35],and the“gravitational”acceleration is given by,g=C s/d4. As with He scattering from metal surfaces,we make the ansatz that the complicated dependence upon q in the atom-ripplon interaction can be factored into two terms,one which includes all information regarding momentum transfer and one which depends upon only the core interactions at the surface.The form which we use is

V ph q= 2Mωρ2β

k z n| φn|V ph q|φo |2ψ′′(E/ˉh?ωq)/(1?e?ˉhωqβ)(29) whereψ′′(ω)is the imaginary part of the ripplon susceptibility,which can be approximated using a Debye model,

3π

ψ′′(ω)=

Figure1shows the variation of s(T)for a30?A He?lm and bulk He as the H scattering energy is reduced below10 mK.Superimposed are the experimental data for the same temperature regime.Given the uncertainties in both the experimental data and in the sensitivity of our calculation to variations in the potential parameters,the agreement is somewhat satisfactory.For scattering from bulk He?lms,the substrate polarization is attenuated by the thickness of the?lm.Although the binding energy of H on bulk He is somewhat less than that for thin?lms,the shape resonances are far enough into the continuum that penetration into the potential region is not enhanced below1mK and quantum re?ection begins to dominate.For thin?lms,the situation is just the opposite.The additional polarization due to the underlying substrate dominates the long ranged behavior of the potential and the density of the shape resonances near threshold is increased.This gives rise to the observed increase in the sticking as the?lm thickness is decreased. Below about0.1mK,which is at the end of the most recent available sticking data,barring the presence of a lower lying shape resonance,the resonance enhancement e?ect is diminished and quantum re?ection should be observed. In the next?gure,(Figure2),we show the changes in s(T)as the He?lm thickness is increased from thin?lm to bulk at a constant scattering energy of.35mK.Again,although our results di?er from the experimental values by perhaps a factor of two or three on the average,the agreement between our calculations and the experimental data is on the whole quite good,especially given the apparent sensitivity of s(T)to the long ranged part of the potential and the lack of an accurate measurement of the substrate interactions.What is striking about our calculations is the bump in our computed s(T)which appears when the?lm thickness is just below50?A.This is due to the appearance of a threshold resonance,which eventually changes to a second bound state when the?lm thickness is around15-17?A.In Table1we show the variation in the binding energy as the?lm thickness is decreased as computed by diagonalizing the static surface Hamiltonian.Quantitative comparison to the the experimental data much below this region is di?cult due to other substrate e?ects which might come into play,such as roughening,corrugation e?ects, and inelastic coupling to the substrate phonon modes.In fact,the experiments were done using either sintered silver or epoxy as a substrate.For the sintered silver case,for coverages above30?A,the pores in the sinter should be ?lled[11];however,for coverages below that the pores may be only partially?lled.In any case,the experimental data also indicates an abrupt change in the sticking behavior which occurs when the He?lm thickness is below20to30?A. As a?nal demonstration of the role of shape resonances in quantum sticking,we consider the case of H sticking to“pseudo”liquid He where we have replaced the realistic e?ective potential used above with a square well.By adjusting the width of this well,λ,we can very easily examine(analytically,if we want),the enhancements to the sticking due to low lying resonances.In Fig.3we plot s(T)vsλat constant scattering energy well into the quantum re?ection regime(E=0.01meV).For the narrowest well(λ=10?A),the potential barely supports a single bound state with a binding energy of0.031558K.Asλis increased to50?A,more bound states are added to the well.This a?ects the sticking in a profound way.At each peak in this?gure,an additional bound state is added to the well.As λincreases further,the binding energy of the new state increases and sticking decreases.

Lastly,in Fig.4we consider a rather exceptional case in which the e?ective range of the atom-phonon interaction was chosen to be longer than the e?ective range of the attractive part of the potential.Here,we set1/β=20?A and vary both the range of the square well,λ,and the scattering energy.Two features are immediatly appearent. First is that the universal sticking behaviour is observed in all cases indicating that for?nite ranged atom-phonon interactions,universal sticking is to be expected.Secondly,is the dramatic enhancement of the sticking due to the low lying resonances.

IV.DISCUSSION

We have presented a theory for the sticking of ultra-cold atoms from surfaces,and we have presented evidence for what we believe to be the proper limiting behavior of s(T)for non-Coulombic potentials.Although our model calculations are limited to low scattering energy where only transitions to bound states are allowed,we believe that the limiting behavior shown here is in fact correct,agreeing with the predicted behaviors.Our calculations indicate that s(T)can have non-vanishing limiting behavior only in the presence of extremely low lying(threshold)resonances. We demonstrate that many body e?ects do persist into the quantum sticking regime;however,their net e?ect at very low energy is to rescale the strength of the potential but not to rescale the e?ective range.Since sticking is most sensitive to the range of the potential,the limiting low E behavior becomes independent of the strength of the interaction and the coupling to the inelastic channels.At very low scattering energy,there is a competition between polarization e?ects due to the virtual phonon exchanges and quantum mechanical re?ection by the long ranged part of the potential.In the absence of shape and threshold resonances,quantum re?ection is expected to win out and the √

universal s(E)∝

retardation e?ects and substrate contributions to the potential become very important and can dramatically change the sticking behavior.These subtle contributions are very di?cult to quantify experimentally due in part to the technical di?culties of working in the submillikelvin regime[10].However,given the sensitivity of the sticking to very subtle details of the long range part of the potential,these low energy experiments o?er unique insight to substrate e?ects and relativistic retardation e?ects that would otherwise be unmeasurable.

ACKNOWLEDGMENTS

We wish to thank Professor Steven J.Sibener and Dr.Carl Williams for for many discussions and suggestions this work.This work was supported by the Materials Research Center of the University of Chicago under the NSF grant DMR-88-19860.

APPENDIX A:ZERO ENERGY W A VEFUNCTIONS FOR1/Z3POTENTIALS

The essence of the theory of quantum sticking at very low scattering energies boils down to the fact that near the classical turning point,the scattering wavefunction vanishes linearly with momentum,k.Thus,any matrix element coupling this low energy scattering state to a bound state will also vanish.A demonstration of this can be seen by solving the following Schr¨o dinger equation,written in dimensionalless units.

ψ′′k+(g2/z3+k2)ψk=0(A1) on z≥1and subject to the the boundary condition,ψk(1)=0.Exact analytic solutions for this exist only for k=0 and the result is[5]

ψ0(z)=A

√

(z).The asymptotic expansion ofψ0is obtained by taking x→0and using the known asymptotic expansions for the Bessel and Neumann functions

J1(x)=x

16

+···(A3)

N1(x)=1

x

+

x

4

π).(A5)

Putting things together,one obtains

ψ0(z)=?Ag tanφo+1γ2g2 +z

A=?πgk+O[k5/2ln k].(A8) Similarly the phase shift can be estimated.The important observation is that the amplitude of the wavefunction inside the interaction region,as de?ned by z

TABLE I.E?ective Potential Parameters and Computed Binding Energies for H on Bulk4He and on thin4He?lms(d= 30?A).The reported value of E o for bulk4He is1.0K(See Ref.[11].)

3.00.20

4.8-0.981589-0.855165 2.90.20 4.8-1.059139-0.931325 2.80.19 4.8-1.155515-1.025170 2.80.20 4.8-1.139079-1.009945 2.80.21 4.8-1.122955-0.994995 2.70.20 4.8-1.221373-1.090980 2.60.20 4.8-1.305987-1.174390