sticking coefficient at ultralow energy: quantum reflection

Pergamon Prt,grr\< I” Surface Scwwe. Vol 57. NC, I, ,‘,I (,I-‘J3. IYYX

0 1YYX Elsevier Science Ltd All n&h& reserved. Printed ID Great Br~lil~n

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STICKING COEFFICIENT AT ULTRALOW ENERGY: QUANTUM REFLECTION

CARLO CARRARO* AND MILTON W. COLE**

‘Dt-partmznt of Chcmicnl Enginrering, University of California, Berkeley, California 94720 “*Department of Physics, Pennsylvania State University, University Park, Pennsylvania 16802

Abstract

Some sixty years ago, Lennard- Jones and Devonshire predicted that the low energy behavior of the sticking coefficient is s oc fi, which vanishes as the energy approaches zero. This behavior is called guantvm reflection. The prediction differs from the classical limit s = 1, because the wave function of the impinging particle is reflected at large distance from the surface. The present article explains why and when the prediction is valid, when it is not valid, and the nature of the experimental confirmation to date. Other related predictions are discussed.

1. Introduction

2. Theory

Contents

A. Model atom-surface Hamiltonian

l3. Rigorous results in Born approximation. I. Short range potentials

C. Rigorous results in Born approximation. II. Long range potentials

D. Rigorous results in Born approximation. III. Resonances near zero energy

E. Nonperturbative many-body effects: exact threshold behavior of sticking

coefficient

3. Experiment

A. Review of early experiments

62

64

64

69

72

73

74

79

79

61

62 C. Carraro and M. W. Cole

B. Hydrogen atom scattering from helium films

C. Hydrogen atom scattering from liquid helium

4. Summary and conclusions

Acknowledgements

Appendix

A. Elastic potential

B. Coupling to ripplons

C. Effect of substrate

References

Acronyms

DWBA Distorted wave Born approximation

WKB Wentzel Kramers Brillouin

80

83

86

87

87

87

89

90

90

1. Introduction

Understanding the nature of inelastic collisions between slow atoms and cold surfaces is

of vital importance in controlling the kinetics of thermal equilibrium, which is of paramount

interest in atomic and low-temperature physics. It is well-known that in the very low energy

regime, many physical phenomena exhibit behavior which is qualitatively different from

that seen at higher energy. Energy exchange at solid or liquid surfaces is no exception

to this rule. Recent progress in atom trapping and cooling has made the extreme low

temperature (microkelvin and even nanokelvin) regime experimentally accessible, and has

allowed the determination, for the first time, of the ultra-low energy behavior of the sticking

coefficient. Such behavior has been long anticipated; for reasons which will become clear,

it is called quantum reflection. Here, we review the theory of sticking at ultra-low energy,

and of the experimental evidence of quantum reflection. We shall see that the problem

involves both general physical principles, such as impedance mismatch, as well as quite

elusive and esoteric behavior, such as retardation of van der Waals forces.

Quantum Reflection 63

The basic physics of ultra-low energy sticking is straightforward. An atom incident at

low energy has a wave length A that is long compared to the characteristic distance scale

of the gas-surface interaction V(F), which is an extreme example of impedance mismatch.

The general theory of waves yields the expectation that in this limit the wave function

is likely to be reflected long before the atom arrives at the attractive well. Hence, the

sticking coefficient s falls to zero; the specific prediction [l] is that s is proportional to the

amplitude of the impinging wave within close proximity to the surface, which leads to the

dependence

SCXdE. 0.1)

This is the statement of quantum reflection. Such quantum reflection behavior differs

dramatically from the the classical expectation that s ought to approach unity at low E

for the case of a very cold surface. The latter belief is intuitively obvious, because the

approach to the surface of an infinitely slow particle takes infinite time, during which the

particle exerts a nonzero force on the surface. Thus, there will ensue some excitatation of

the surface, resulting in energy loss; the final energy of the particle will then be less than

zero, meaning that the particle will be trapped at the surface, and hence, s = 1.

Despite its simplicity, the prediction of quantum reflection had to await heroic efforts of

atomic physicists for its definitive test. The reason is that gas-surface interactions possess

very long range, which make it easy for an impinging particle to adjust its de Broglie wave

length smoothly as it travels towards the the surface. In this way, penetration into the

adsorption well can be achieved with low reflection. In fact, the smoothest of all surface

potentials, the Coulomb (or image) potential of charged particles, has been proven not

to quantum reflect at all. In general, quantum reflection is expected to be important

below an energy threshold which, for neutral particles interacting through the van der

Waals potential, is extremely low; so low, in fact, as to be reachable only with great

experimental effort. However, once the quantum reflection regime has been reached, its

onset provides interesting information concerning the very long-range decay of the surface

potential, including the retardation regime of van der Waals forces.


The need to attain exceedingly low temperatures, in order to observe quantum reflec-

tion, was first recognized by Goodman [2] and then again by Brenig [3]. Nonetheless, this

recognition does appear to have eluded some researchers and, from time to time, the finite

values of the sticking coefficient observed in some experiments at low energy (but not low

enough!) have been mistakenly construed as evidence that quantum reflection is somehow

violated (41. It has been claimed that exceptions to quantum reflection can occur when

the coupling between incident particle and surface is large [5]. However, those claims have

been shown to be in error [6,7].

The review is organized as follows. Section 2 deals with the theory of low energy

surface scattering, with particular emphasis on the threshold behavior of the sticking co-

efficient. In $3, we specialize to the physical system where quantum reflection has been

observed, namely, atomic hydrogen reflecting from liquid helium surfaces and helium films.

Concluding remarks and an outlook on future work are contained in $4.

2. Theory

A. Model Atom-Surface Hamiltonian

Low-energy scattering of an atom from a condensed medium is a difficult quantum

many-body problem. Throughout this review, we shall assume without proof that

i) because the target is a condensed medium (liquid or solid), its wave function decays

exponentially at infinity (at least in one direction);

ii) the Hamiltonian of the condensed medium has been diagonalized. The ground state

and the excitation spectrum are known;

iii) elastic scattering from the condensed medium is described by a one-body static po-

tential. From (i) and (iii) it follows that the asymptotic scattering states of the atom

are free-particle states. (The case where the one-body potential is of the Coulomb type

deserves special treatment. See below.)

iv) the atom can undergo inelastic collisions with the condensed medium. These processes

(which are responsible for sticking) are caused by interactions whose range is comparable

to, or shorter than, that of the static (elastic) potential.


That (i) is true is self-evident. Assumption (ii) is made for notational convenience;

most general conclusions about the threshold behavior of the sticking coefficient can be

drawn without a specific knowledge of the surface excitation spectrum [see ref. [8], which

also contains a proof of (iii)]. Note that (iii) does not imply, nor does it require, that the

one-body elastic potential can be defined uniquely. Finally, assumption (iv) is justified on

physical grounds, since the interactions that give rise to elastic and inelastic scattering at

low energies originate from the same physical mechanism (e.g., the electronic polarizability

of the medium).

A model Hamiltonian of the atom-surface system can be split up into a free part, which

we take to describe an atom in an external potential and a set of harmonic oscillators

and an interaction part, which couples the oscillators to the particle linearly

(2.2) a ’ /

The free Hamiltonian HO is obviously separable. The interaction part VI allows for

energy to be transferred between the atom and the condensed medium, through creation

or annihilation of elementary excitations.

This model Hamiltonian is relevant to many physical systems. In the case of atom

scattering from liquid helium, the excitation spectrum includes bulk excitations, such as

phonons and rotons, as well as quantized surface capillary waves, the ripplons. In the case

of scattering from solid surfaces, the modes include bulk and surface phonons [9]. Charged

particles scattering from solids can also couple to plasma modes. In all these examples,

the bulk excitation spectrum is labeled by a three-dimensional momentum vector @‘, while

surface modes have wave vectors restricted to a two dimensional plane.

Other geometries, besides the planar one, can be important. Most notable is the

spherical symmetry which would be appropriate, e.g., for the case of atom scattering from

a liquid cluster, electron scattering on a closed-shell atom, or nucleon scattering from a

closed-shell nucleus. In this review, we focus on scattering from planar, infinite surfaces.


A schematic depiction of the sticking process appears in Fig. 1. It is worth mentioning

that many of the physical phenomena we discuss, especially the threshold behavior at low

energy, have a direct counterpart in s-wave scattering from finite systems, like clusters or

atomic nuclei.

Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..__..........................

Parallel wave vector

. . . .

. . . . . . . . . . . . .

.

. . . . . . ;;$

. . . . . . . s

Fig. 1. Schematic depiction of the sticking process. An incident atom, whose total energy and surface-parallel wave vector are indicated by the open circle, undergoes a transition to a bound state, indicated by the closed circle. The shaded region is the domain of scattering states, bounded from below by the free particle parabola. The bound state parabola assumes that the particle has an effective mass equal to the free mass. The angular frequency and wave vector shown must correspond to the dispersion relation of a mode of the surface. This relation is indicated by the curve consisting of x’s.

To make a concrete connection to the case of ultra-low energy hydrogen scattering from

liquid helium, which will be the main focus of !j3, we also assume that the primary inelastic

channel is the excitation of (quantized) surface modes. At this stage, this assumption is

made purely for notational convenience, and does not affect the conclusions derived about


the threshold behavior of the sticking coefficient, which hold in general for couplings of

any form, under assumptions (i)-(iv). The relative importance of scattering from bulk

vs. surface modes should be gauged on a case-by-case basis. For H atoms [lo] and He

atoms [ll] incident on liquid helium surfaces, it is established that surface excitations

are the dominant mode. Thus, neglecting direct coupling of the incident particle to bulk

modes, the harmonic oscillator operators a$, a< represent quantized fluctuations of the

surface profile v(J), where g, $ are now two-dimensional vectors. The surface normal

mode quantization is carried out explicitly in the Appendix for the case of ripplons. The

interaction Hamiltonian can be written by recognizing that the interaction originates from

the same fluctuations of the medium and atom as those, which give rise to the static

interaction:

(2.3)

where 7~ is discussed in Appendix B and

v,‘m = J d2R e’+%(~~),

with

(2.4)

v(r) = &$w . .?=r

(2.5)

Here, the static surface potential has been taken to depend on z only, which is appropriate

for an uncorrugated surface such as the surface of a liquid. Note that (4) to (6) are

consistent with assumption (iv).

Several important parameters can be extracted from the Hamiltonian. Their values

provide a rationalization for the trends found in real calculations. Important energy scales

are the kinetic energy of the incident atom, the binding energy of the lowest bound state

of the static potential, and the average energy of surface exitations. Important length

scales are the mean square surface displacement, the range of a surface bound state, and

(when they can be defined) the effective range and scattering length of the potential. An

important dimensionless coupling constant can be defined as the ratio of the surface root

mean square displacement to the range of the bound state wave function, ao:

A=imiy

a0 (2.6)


Qualitatively, X is a measure of the overlap of surface fluctuations and the final state of a

surface bound state. At very low incident energy, the square of this ratio is essentially equal

to the ratio of the mean energy transferred in a single collision to the average energy of a

surface mode. Hence, a small value of X (weak coupling regime) indicates that dominant

scattering events are those in which at most one surface mode is excited. Furthermore,

in the limit, where the incident atom energy is much smaller than the binding energy,

inelastic reflection will be negligible: almost all inelastic processes will lead to sticking.

(As is intuitively manifested in Fig. 1, insofar as low incident energy means few available

inelastic scattering states.)

One might also interpret the ratio X2 as the deviation of the Debye-Waller factor from

unity or, equivalently, as a measure of the elastic reflection probability:

&1=1-X2.

$?rom this estimate of l&l, and from the unitarity condition

(2.7)

(2.8)

one obtains an approximate upper limit for the sticking coefficient

s I x2. (2.9)

In the case of weak coupling (X < l), this upper limit is qualitatively different from

the naive classical prediction that s -+ 1 at low incident energy. As pointed out by

Brenig [7], this is a consequence of properly treating the substrate quantum mechanically,

thereby accounting for the fact that energy exchange between particle and surface occurs

in quantized bits. Equation (2.7) does not treat the incident atom quantum mechanically,

however. Therefore, it fails at sufficiently low incident energy. Its range of validity can

be estimated in the WKB approximation, where the wave vector of the elastic potential is

given by

k(z) = j/k2 + 2mV(z)/P. (2.10)

Quantum Reflection

For the WKB approximation to be valid, one must have

69

(2.11)

which means that the potential must vary slowly over the (WKB) wave length of the in-

cident particle. This condition fails to be met below a sufficiently small incident energy,

EC, except for potentials that decay more slowly than l/z2 (see s2.C). Thus, (2.7) be-

comes qu&atively inaccurate below EC, where a full quantum mechanical treatment of

the sticking process is needed. The value of EC below which semiclassical theory fails can

be surprisingly small for physically relevant potentials [2,3]. For example, for the retarded

van der Waals potential between a Cs atom and the surface of liquid helium, it is estimated

that EC NN lo-‘OK [12]. H ence, the WKB approximation does an excellent job describ

ing the sticking process (and the absence of quantum reflection) in most experimental

situations.

B. Rigorous Results in Born Approximation. I. Short-Range Potentials

We begin by studying the sticking coefficient in Born approximation, whose pertur-

bative nature is accurate in some practical situations, where the atom-surface interaction

is weak (A2 << 1).

Consider an incident atom beam impinging on a surface, which bounds a medium

occupying the half space I < 0. We assume for the time being that the surface is at zero

temperature, and that the incident beam is monoenergetic. Let Ei = h2(ii1 + k,)2/2m =

El1 + E, be the incident beam energy. The sticking coefficient is defined as the sum of

transition probabilities from a continuum state at incident energy Ei, normalized to unit

incident flux, into each of N bound states, In >. In Born approximation, this reads as


where the distorted wave < zlk, > solves the Schrijdinger equation

2 2

-$-&+V(z)-Ez >

< zlk, >= 0,

and In > is a bound state.

Effective range theory can be used to prove that

s(E) 0: fi

(2.13)

(2.14)

at low energy, if both of the following assumptions are valid:

i) the potential is of (sufficiently) short range, so that one can find a finite value of z, say

~0, such that kzo < 1 and for all I > zc, I*V(z)) 5 k2; and

ii) the elastic potential has a finite scattering length or, equivalently, the low-energy phase

shift vanishes linearly with k

6(k) oc k, k --) 0.

The proof is as follows. The distorted wave obeys the Schrodinger equation

d+‘(z) + [u(z) + k2]$(z) = 0

u(z) = - $%(z).

Now write

u(z) = uo(.z) +w(z)

with

so that

Q,(Z) = u(z) - k2, z < zo

uo(z) = 0, z > zo

Us 5 k2 for all z.

(2.15)

(2.16)

(2.17)

(2.18)

(2.19)

(2.20)

Consider the zeroth-order wave function @c(z), which obeys the Schrodinger equation

&‘(z) + @o(z) + k2)40(4 = 0. (2.21)

The solution is


$0(z) = sin(kz + 60(k)), I > ~0 (2.22)

$O(%) = aC(%) + bS(%), 22 < zo where C(Z) and S(Z) are linearly independent solutions of the Schriidinger equation for the

original potential u(.z) at zero energy, and therefore, they do not depend on k. Now, the

matching conditions at E = .rc imply that

a,bcx k. (2.23)

Defining $(z) = tic(z) +$1(-z), then @I(Z) = O(k2) by virtue of (2.20). It follows that the

amplitude of the wave function is of order k for z < ~0. But, since .rc is also the range

of the static potential and of the inelastic coupling, the matrix elements < nlVilkz > too

must be proportional to k, so that s(E) K fi, which is indeed the statement of quantum

reflection in the Born approximation.

The vanishing of the matrix element as k -+ 0 was just proved under the assump-

tions of sufficiently short-range potential and finite scattering length. In the next two

sections, we show that quantum reflection can indeed be violated, if either of these two

assumptions is abandoned. Provided they hold true, the solution to the scattering problem

essentially requires that we match a wave of finite amplitude and infinitesimal wave vector

(the asymptotic scattering state) with one of finite wave vector (the scattering state in the

potential well), which therefore will necessarily have infinitesimal amplitude.

It is important to realize that the incident particle is reflected outside of the matching

distance ~0, and not at the surface. In other words, it is the attractive tail of the surface

potential, not the repulsive wall, that causes quantum reflection. Quantum reflection is

present even in the absence of a repulsive wall. Interestingly, a purely attractive surface

potential can be realized in practice; in such case, there is a continuum of bound states

into which the sticking atom may fall. One such potential is the potential felt by a he-

lium atom scattering at low energy from the surface of liquid helium. The helium atom

will be quantum reflected in the limit of zero incident energy. Note that the latter situ-

ation, which has been studied experimentally [13], exemplifies the fact that the quantum

reflectionphenomenon pertains even to the case of a continuous bound state spectrum.


C. Rigorous Results in Born Approximation. II. Long-Range Potentials

Long-range potentials, falling off as negative powers of the distance, are very inter-

esting from a theoretical viewpoint because the proof of quantum reflection given in the

previous section may not apply here, since assumption (i) of 52.B may fail. Therefore, we

consider now elastic potentials of the type

V(z) = -cJc, a > a, (2.24)

where a is a small distance cutoff at the surface. Examples of long-range potentials are

the van der Waals dispersion potential (n = 3, crossing over to n = 4 in the retarded

regime) and the Coulomb potential (n = 1). These long-range potentials lack a character-

istic length scale, and one might wonder whether their smooth approach to infinity allows

smooth matching of the asymptotic scattering wave function with the wave function inside

the potential well, without quantum reflection. More precisely, we ask whether the scat-

tering state can adjust its wave length smoothly from the large asymptotic value of k-l

to the value of l/J- near the surface. If this is possible, then the WKB

approximation will hold true,

Wz) I I --yjg- e k2(4, 2 > a. (2.25)

and the sticking coefficient will be nonzero even at very low energy.

Following Ref. [3], we adopt dimensionless units where lengths are measured in units

of the lower cutoff a, and energies in units of h2/(2ma2). Inequality (2.25) thus reads

For this to be satisfied for all a > 1, we ask that it be satisfied when the 1.h.s. attains its

maximum, i.e., for a k-dependent value

z*(k) = k-2/” [ $;;);I ? (2.27)

For n < 2, inequality (2.26) is always satisfied. Smooth matching is possible, and there

is no quantum reflection, irrespective of the strength of the potential, as is demonstrated by

ezact calculations for the one-dimensional Coulomb potential, n = 1, described in Ref. [7].


The case n = 2 is marginal [14]. For c2 > 1, inequality (2.26) is satisfied for all k

values and all z > 1. Thus, if the coupling constant is strong enough, there is no quantum

reflection for n = 2, as for the Coulomb potential.

For n > 2, inequality (2.26) ceases to be satisfied for I = .a* when the incident wave

vector is smaller than a characteristic value

k c

= 21/(“--2) &iq(, + 1)(“+1)/(“-2) 33”/2(n-2),&2(n-2)&h-2) .

(2.28)

Thus, as the incident energy is lowered towards zero, quantum reflection will always set

in, although this energy threshold may be extremely small. (This conclusion is valid even

though the effective range expansion fails for these long range potentials: both scattering

length and effective range have logarithmic singularities at low energy.) The sticking

coefficient is thus predicted to vanish as a for these potentials. This prediction has been

confirmed by experiment [15].

D. Rigorous Results in Born Approximation. III. Resonances near Zero Energy

In this section, we examine the possibility that the elastic potential has infinite scat-

tering length, despite its short range. Consider for simplicity a hard-wall-plus-square-well

potential: V(Z) = 00, r<O

V(Z) = -90, 0 < z < a (2.29)

w(z) = 0, z > a,

where ‘uc is a positive constant. The scattering state at energy E = A2k2/2m outside the

well has the form

$J> = sin(kz + 4(k)), (2.30)

while inside the well,

$< = A(k) sin(Ka), (2.31)

whereK = dV k + 2mvo/ti and A has been taken to be real. Matching at a = a determines

the amplitude inside the well

A(k)2 = k2

K2 cos2 (Ka) + k2 sin2(Ka). (2.32)


It normally implies A(k) 0: k for k -+ 0 (quantum reflection). However, an exception to

this rule occurs when

cos(Ka) = 0. (2.33)

If this is the case, jA(k = 0) 1 = 1 and there is a shape resonance at zero energy. Under this

circumstance, the wave function inside the well reaches the edge of the well with exactly

zero slope, which allows it to match the infinite wave length of the zero energy scattering

state without loss of amplitude, i.e., without quantum reflection. However, as we shall

see in the next section, many-body effects restore quantum reflection, and the sticking

coefficient does vanish at zero energy, although its behavior near zero energy will be highly

nonmonotonic, a characteristic mark of resonance scattering. Note that an infinitesimally

deeper potential would support an additional bound state.

The circumstance of a shape resonance near zero energy is quite fortuitous, since it

requires the depth and range of the potential to satisfy the condition

a (2.34)

with n an integer. Thus, the mechanism of resonant scattering has been usually overlooked

in discussions of quantum reflection, until Hijmans et al. [16] and Carraro and Cole [17]

invoked it to explain the experimental measurements of the sticking coefficient of atomic

hydrogen on liquid helium films [ES], to be discussed in 53.

E. Nonperturbative Many-Body Effects: Exact Threshold Behavior of the

Sticking Coefficient

The previous sections have dealt with exact results for the low energy behavior of

the distorted wave function, from which one can compute the sticking coefficient in the

distorted wave Born approximation (DWBA). In this section, we ask whether similar

rigorous statements can be made about the true sticking coefficient and not, just its first

order perturbative expression. As we shall see shortly, this question is answered in the

affirmative.


The DWBA to the sticking coefficient suffers from two principal shortcomings. First,

it is the first order term of a perturbative expansion in the parameter X2 defined in (2.6),

and thus it assumes that the coupling term in the Hamiltonian is small. Furthermore,

it allows only a single surface excitation to partake in the trapping process. One can

imagine situations where these assumptions break down, and hence, one has to resort

to nonperturbative methods to compute the sticking probability. Secondly, the DWBA

violates unitarity; the sticking coefficient computed in DWBA could exceed unity, which

is of course an absurd result.

Due in part to these shortcomings of the DWBA, it was conjectured in the litera-

ture [5] that the prediction of universal quantum reflection in ultra-low energy surface

scattering may fail. It was argued that strong coupling to the surface excitations might

destroy quantum coherence of the incident wave packet. Numerical “evidence” in support

of this conjecture was claimed in simulations of surface scattering of a charged particle

in the strong-coupling regime [19]. Unfortunately, those claims overlooked the fact that

the Coulomb potential, strong or otherwise, does not quantum reflect, even in the Born

approximation [2,7].

Although the many-body scattering problem cannot be solved exactly in general, we

now show that the threshold value of the sticking coefficient is known exactly:

$no s(E) = 0, -+

(except in the case of very long ranged potentials studied in 92.C). Indeed, one can rely

on the observation that in the limit of zero incident energy (and of a surface at T = 0),

there is no inelastic rejlection channel available to the incident particle. Contrary to some

previous studies, which claimed that sticking exhausts unitarity in this case [20], we will

prove the opposite conclusion, that elastic reflection, and not sticking, exhausts unitarity

under this circumstance. Furthermore, it will be clear that many-body effects actually

conspire to make the sticking coefficient vanish at zero energy even in the presence of a

low-lying resonance, in which case the Born approximation would erroneously predict a

violation of quantum reflection.


To prove the limit (2.35), we use the exact (closed form) solution of the many body

scattering problem in a restricted Hilbert space where inelastic reflection events are not

allowed. Such solution was worked out long ago by Wigner and Eisenbud [21]. A slight

generalization of their method is given in Ref. [22]. Closed-form solutions of the many-

body problem can be found by applying different restrictions to the Hilbert space. One

celebrated approach, that of Tamm [23] and Dancoff [24], actually allows for inelastic

scattering in the continuum, provided a restriction is placed on the number of surface exci-

tations (one quantum per mode). Clougherty and Kohn [14] have carried out a calculation

of the sticking coefficient ii. la Tamm-Dancoff. The results confirms that the threshold

behavior is given correctly by (2.35) also in the case when inelastic reflection events are

allowed. Because in practice these events are unimportant, we neglect them altogether in

the following analysis.

. . . +-- ..*... +

+-f!z?4+,=4 Fig. 2. Diagrammatic representation of (2.36). The broken line propagates

the particle in the bound state, while the solid line propagates it in the continuum. The wavy line represents the ripplon propagator.

In regards to the problem of low energy atom-surface scattering, the solution of the

many-body problem amounts to resumming the diagrammatic expansion of Fig. 2 for the

Green’s function of the atom. Knowledge of the atom’s Green’s function then allows us

to reconstruct the atomic wave scattering amplitude, which obeys a Lippmann-Schwinger

equation:

pc, >*= (k, > +A Jqc dq’q’D(q’, (Ez - Eo - Eql)/h) 0

J

O” dk’ Ik: >< k:lv,:lo > ww

x + 0 = E, -E;+iO < OlV,:lk, >*,


where D(q,w) is the propagator of surface excitations. Here, we assume that the static

potential supports only one bound state, ]$~c >. Such a physically important situation

pertains to the case of atomic hydrogen sticking on the surface of liquid helium, which will

be studied in detail in $3. However, the results can be easily generalized to the case of

many bound states.

Equation (2.36) is suitable for numerical inversion. It is often assumed that the force

is either independent of q [3], or that it is a smooth function of q in the neighborhood of

a wave vector qo, defined as the root of Es = FL+,, + Eq,, [17]. Then a simple analytical

solution is readily available for the transition amplitude into the bound state:

and hence, for the sticking coefficient:

4%) = -$$ < olv;Jk, >* I~~“. (2.38)

(2.37)

where E, - E. - Eq

h ’ (2.39)

x” = Im(l,,) (2.40)

is the imaginary part of the generalized susceptibility of the substrate, and

Ik = J O” dk’ 1 < OlVJk > I2

0 7 k2 - (k’)2 + iv ’ (2.41)

The physical meaning (2.36) is that the scattering state of the impinging atom is

not only distorted by the presence of the static interaction Vi(z), but is also allowed to

make virtual transitions into the bound state by interacting with a virtual cloud of surface

excitations.

The sticking coefhcient given by (2.38) fulfills the unitarity condition s(E) 5 1. Note

that the discussion of the threshold behavior, presented in 52.B in the context of the Born

Approximation, continues to hold true when the perturbation series is resummed to all

orders, which is a direct consequence of the absence of inelastic reflection at zero incident


energy. In practice, the energy of the incident beam can never be made exactly to vanish.

However, the relative importance of inelastic reflection and sticking is easily estimated. In

the case of a hydrogen atom beam sticking on the surface of liquid helium, the estimate

relies simply on the available ripplon phase space and on the fact that V,I(z) ‘v W(z)/&

in a low energy process. With EO denoting the binding energy, the inelastic reflection

coefficient is suppressed by a power of (E/Eo)~ with respect to the sticking coefficient, an

extremely small number under usual experimental conditions.

Consider now the implications of (2.38) for the case when the potential has a resonance

near zero energy. The transition amplitude squared is given by

where y is a k-independent constant. Thus,

Ik=-r 2 &+%&) (

(2.42)

(2.43)

and the sticking coefficient is given by

4rn-yfk s(E) = - [h2(b + myx’/A2)2 + (m2-y2/A2)(xtt)2] - 2mx”yk + h2k2 *

(2.44)

Note that the effect of surface polarization, embodied in x’, results simply in the shift of

the position of the resonance from b = 0 to b = -myx’/li2. However, we see that for

k --) 0, the sticking coefficient now vanishes linearly with k as

in contrast to the naive Born approximation (cf., 32.C).

Note that the effect of dissipation, embodied in the imaginary part of the substrate

susceptibility, x”, is to prevent the resonance width from vanishing near zero energy [25],

which is a manifestation of the distruction of quantum coherence caused by inelastic scat-

tering. Here, quantum coherence is responsible for the presence of a resonance, that is,

for the build up of quantum mechanical amplitude inside the well, which would help to

bypass quantum reflection. Thus, dissipative effects that broaden the resonance promote

quantum reflection, rather than prevent it, contrary to what is sometimes stated in the

literature [5].

Quantum Reflection

3. Experiment

A. Review of Early Experiments

79

Definitive experimental tests of the theory of sticking at ultralow energies have been

carried out using hydrogen atom beams as a probe and liquid 4He (LHe) as the substrate.

Conceptually, the H/LHe system is very well suited to test the theory. This is because (a)

there is essentially no problem of surface imperfection due to the high purity of the liquid,

(b) both the surface and bulk modes are rather well known, (c) the pertinent interatomic

interactions are relatively well known, (d) the extreme quantum regime can be most easily

attained for the lightest of all atomic probes, hydrogen. However, one should also not lose

sight of the fact that the problem takes on a practical importance in the application to

spin-aligned H atom confinement experiments. Such experiments have been performed for

some time in the pursuit of Bose-Einstein condensation [26,27]. In those experiments, LHe

films are used because H does not dissolve in LHe, and the He is inert so surface adsorption

and spin flipping are limited, resulting in very slow recombination of H to Hz. Because

experiments often involve thermal hydrogen beams, it is useful to observe [28] that the low

energy behavior

s(E) = sl&‘* (3.1)

implies a corresponding low temperature law

J;; s(T) = -pT’/*. (3.2)

The sticking coefficient also enters (through detailed balance) the problem of desorption

from liquid He.

Among the dynamical properties of the LHe/gaseous H interface, the sticking coeffi-

cient plays a central role, since sticking is the dominant inelastic channel at low energies,

as we saw in $2. Hence, transport properties can be expressed in terms of the sticking

probability. For example, the Kapitza conductance between gaseous atomic hydrogen and

the liquid helium surface, defined as the ratio of the heat flux to the temperature difference

between gas (Tg) and surface (Ts), is given by [29]

GK(T',T,) = %~~o(Tgb(Tg,Z), (3.3)


where vo(Z”,) is the surface collision rate and the accommodation coefficient is given by

(3.4)

where r(T, T,) = Taln[s(T,T,)]/dlnT, and the explicit dependence on the surface tem-

perature has been noted throughout. Equation (3.4) is valid when the sticking channel

dominates [30].

Experimentally, the first determination of the sticking coefficient of atomic H on both

4He and 3He surfaces was obtained by Jochemsen et al. [31]. They found s N 0.046 at

T N 200 mK for L4He and s z 0.016 at T II 100 mK for L3He. The accommodation

coefficient was observed by Helffrich et al. [32] to decrease with temperature in the range

0.18-0.4 K. By the end of the 1980’s, the record low temperature determination of s was

firmly in the hands of the Amsterdam group [33]. Their experiment determined the flow

rate of spin-polarized H in a capillary tube, whose walls were wetted by liquid helium.

They observed an enhanced flow rate, compared to ordinary Knudsen flow, which they

related to a vanishing sticking probability. When plotted as a function of incident energy

(see Fig. 4 below), the Amsterdam data does indeed suggest a vanishing s(E) as E -+ 0;

however, the sticking coefficient seems to decrease proportionally to E rather than E’i2.

This is an indication that the threshold for quantum reflection is not yet reached even

at temperatures as low as 100 mK. Experiments in the microkelvin regime had to await

progress in evaporative cooling [34,35]. Using this technique, the MIT group has been

able to probe the sticking coefficient at these unprecedented low energies in a series of

experiments, where s(E) was first measured for H on He films [18], and subsequently on

the bulk liquid [15].

B. Hydrogen Atom Scattering from Helium Films

The experimental cell used in the first measurements of sticking in the submillikelvin

regime consisted of a cylindrical tube 4.4 cm in diameter and 65 cm tall. The walls of the

tube were covered with a saturated 4He film. The film was kept at a temperature of 50

mK during the experiments. In the upper part of the cell, H atoms, produced in an RF


discharge source, were confined by static magnetic fields. A quadrupole field effected radial

confinement, while two solenoids confined the atoms longitudinally. The energy barrier EC

at the bottom of the trap was adjusted either to evaporatively cool the confined atoms

(by leaking out the hotter ones) or to measure their temperature (by dumping the entire

contents of the trap). The atoms that escaped from the trap eventually stuck to the walls,

where they recombined to molecular hydrogen. The heat of recombination was measured

by a bolometer.

Starting with atoms of known energy distribution, the radial confining field could be

lowered suddenly to let the atoms interact with the helium-covered walls. It should be

noted that, because of the experimental design, the H atoms predominantly interacted

with the upper portion of the cell walls. Because the cell was over two feet tall, only a

thin helium film is expected to be present on the walls, even if bulk liquid existed at the

bottom of the cell [36]. After specified time intervals, the confining field was restored, and

the energy distribution measured again. Comparison with the original distribution yielded

the decay rate of the population in the trap. The population decayed because of sticking

events. (Spin relaxation, which would also lead to population decay, was determined to

take place only after sticking occurred.) The population decay rate at energy E, T(E)-~,

was used to extract the sticking coefficient s(E) through the relation

T(E)-1 = 7) (3.5)

with D the cell diameter, M the atomic mass, and n a geometric factor, determined by

Monte Carlo simulations of atomic trajectories in the cell. For energies above 1 mK,

a different method was used to determine s(E). It involved letting a pulse of atoms

escape from the barrier at the bottom of the trap, and detecting the heat of adsorption

bolometrically. This method involved a significant uncertainty in the factor 7, and resulted

in larger errors.

The sticking coefficient, measured over a range of energies 100pK 5 E 5 20mK [18],

is plotted in Fig. 3. The MIT results clearly show that s(E) rises as the energy is lowered

below 1 mK. This behavior is incompatible with the the theory of sticking on bulk liquid


helium [37]. It was soon realized [16,17] that quantitative agreement between theory and

experiment is recovered by taking proper account of the finite depth of the He liquid, and

thus of the presence of a substrate. Hijmans et al. [16] have carried out calculations of the

sticking on thin He films in the DWBA. Carraro and Cole [17] have addressed the effect of

the substrate through a fully nonperturbative calculation, following the method described

in 52.E.

1

10”

10-2

10-3 10-6 10-5 10’4 10-a 10-z 10-t 1

E WI

Fig. 3. Sticking coefficient of H atoms on He films. Circles and crosses:

experimental data from ref. [18]. Solid line: nonperturbative theory; dot-dashed

line: distorded wave Born approximation; dashed line: predicted sticking coeffi-

cient on bulk helium (theoretical results from ref. [17]). See text for details.

Figure 3 shows that both the effect of the substrate and nonperturbative corrections

are quite significant. The dashed line is the result of a calculation of the sticking coefficient

on bulk films, using the potential VII of Appendix A with p = 0.587 A-i. This potential

supports a single bound state. The dot-dashed line instead is calculated with potential

VIII of Appendix C, for a film of thickness d = 50 A on a substrate with C, = 5000 KA3


(this is a reasonable value to assume, lacking better characterization of the experimen-

tal conditions; for reference, graphite and copper substrates would give C. = 4400 and

5500 K A3 respectively [38]). There is now qualitative agreement between experiment

and calculation. Quantitative agreement is obtained when higher order corrections to the

DWBA are included, as in (2.38). This is shown by the solid curve. Further DWBA calcu-

lations, where the film thickness was taken as a variable, showed that effect of the substrate

on s(E) is felt for films as thick as 15011. This is because the helium-hydrogen interaction

is extremely weak, since He is the least polarizable material. Thus, any substrate below

a He film enhances the wall attraction. For sufficiently thin films, a second bound state

appears, implying that the sticking coefficient on He films is dominated by the presence of

a low-lying resonance.

C. Hydrogen Atom Scattering from Liquid Helium

In order to clarify the relation to theory for infinite bulk LHe (and implicitly to avoid

the resonance near zero energy that shows up for thin helium films), a new experiment

was designed by the MIT group, where the interaction between H atoms and helium

occurred near the bottom of the cell. Thus, in the presence of saturated liquid, this

experiment was able to measure the sticking coefficient on the bulk liquid, and also to

perform measurements as a function of film thickness in subsaturated conditions.

The measured sticking probability on bulk helium is shown in Fig. 4. These measure-

ments provide the first clear experimental verification of the onset of quantum reflection

in atom-surface scattering. Also shown in Fig. 4 are the predictions of Hijmans et al. [16]

and of Carraro and Cole [17]. There is excellent agreement between the latter theory and

experiment.

The discrepancy between the calculation of Hijmans et al. [16] and that of Carraro

and Cole [17] can be traced to a difference in the form of the adsorption potential at short

distances (in the case of sticking on bulk liquid, nonperturbative effects accounted for in

ref. [17] are negligible). Basically, the potential used by Carraro and Cole switches off

the short range (many body) effects at a distance of about 25A, which is 25% larger than


that used by Hijmans et al., and larger than we would have guessed prior to the sticking

coefficient measurements on thin films [18]. In our opinion, this fact points to the existence

of interesting, and not understood, many body effects affecting the interaction of H atoms

with He films at distances that are somewhat larger than the known width of the helium

surface [39].

0.1

?= x

0.01

0.1 1 10 100 1000

T [mKl

Fig. 4. Sticking coefficient of H atoms on liquid He (Reprinted from ref. [15]

with permission). Circles represent the ultra-low temperature MIT data [15],

while triangles are data from the earlier capillary flow experiment [33]. The solid

lines are the prediction of ref. [17] assuming binding energies of 1.1 and 1.0 K.

The broken line is the prediction of ref. [16].

Finally, Fig. 5 shows how s(E) varies as the 4He pressure is increased towards sat-

uration, i.e., for increasing film thickness. Although precise values of film thickness were

not determined in the experiment, the observed trend in the data is consistent with the

predictions of ref. [17].

We should stress that both the theory of Hijmans et al. [16] and that of Carraro and

Cole [I71 make a very important prediction regarding the effect of relativistic retardation

of the van der Waals potential. (This prediction concerns the form of the potential at

very large distances, of order hundreds of A; hence, it is unaffected by small discrepancies

in the potential at short length scales.) The prediction is the the sticking coefficient in


the microkelvin regime is extremely sensitive to the form of the potential at those very

large distances at which retardation is expected to be important. Omitting the effect of

retardation of the van der Waals potential leads to a prediction of the sticking coefficient

which is at least three times larger than experimentally observed, and cannot be reconciled

with the data by invoking uncertainties in binding energy, ripplon dispersion, substrate

polarizability, and the like. Indeed, the earlier measurements of s(E) on thin films [18]

are in reasonable agreement with a calculation of Goldman for bulk liquid, which omitted

retardation. Thus, the experiments done at different pressure, showing the continuous

crossover between thin film and bulk regimes, should be regarded as a striking confirmation

of the retardation of van der Waals forces.

1

g 0.1

0.01 0 5 10 15 20 25

Film Thickness [nm]

Fig. 5. Sticking coefficient of H atoms at 300 mK on He films of various thicknesses (Reprinted from ref. [15] with permission).

C. Carraro and M. W. Cole

4. Summary and Conclusions

The phenomenon of quantum reflection has been demonstrated experimentally in the

case of ultra slow H atoms incident on liquid 4He. It would seem warranted to further

explore the idea by studying other cases. A few examples of such systems will indicate

that other problems are worth exploring. The case of H sticking to cold 3He surfaces

seems intriguing because of the role of ripplons on a fermion superfluid; to our knowledge,

these modes have not been explored to date. The case of 3He sticking to Cs is also worth

investigating because Cs is a solid, the number of bound states is small, and the recently

calculated potential [40] has relevance to the problem of wetting transitions. Equally inter-

esting, at least, would be the study of the low energy behavior of the sticking coefficient for

a charged particle, in which case the quantum reflection concept does not apply. We hope

that experimentalists are not deterred from such a study by the difficulties of achieving

low energy beams of charged particles.

The significance of the observation of quantum reflection and of the remarkable agree-

ment between experiment and theory is worth commenting on. At first sight, one might

say‘that a good agreement is not surprising in the case of H sticking to bulk liquid 4He,

since the interaction is weak and standard perturbation theory is valid and has never been

questioned. However, it should be noted that theory preceded the experiments: the the-

oretical curve of Fig. 4 is not a fit to the data! Rather, it is a prediction based on the

results of the thin film experiments, whose analysis requires a theory of the strong coupling

regime. We used a physically sensible one-body potential, whose parameters are either

known (i.e., the van der Waals dispersion coefficients, the H immersion energy in the liq-

uid, and, within a tight range, the binding energy) or were fixed by comparison to the thin

film data. This allowed us to predict the behavior of s(E) on thick films and bulk. The

fact that the prediction was confirmed by experiments attests to the validity of the theory

of sticking in both weak and strong coupling regimes.

One may well ask what other significant physical questions have been answered or

remain to be answered by such tour de force experiments. In our opinion, a key issue is

the retardation of surface forces. That subject, part of the Casimir problem, is fundamental


to our understanding of both the quantum vacuum and the expansion of the universe [41].

Yet there exist few, if any, definitive tests of the concept. Low energy sticking was shown

to be incompatible with unretarded surface potentials; further experiments would be useful

for a definite quantitative validation of Casimir forces.

Acknowledgements

This research has been supported by the National Science Foundation. We would like

to thank W. Brenig, F. Goodman and T. Greytak for many stimulating conversations on

this subject.

A. The Elastic Potential

Appendix

The interaction between atomic He and H is known rather accurately. At long range,

the functional form of the potential follows from first principles. The leading term is the

dipole-dipole van der Waals interaction

?&&v(r) = -c&F (~4.1)

At distances so large that retardation effects become important (typically a few hundred

A), the potential falls off as -C7/r7. The dispersion coefficient depends on the polariz-

abilities of He and H; its value is Cs = 19500 K A6 [42]. At distances of several A and

below, the potential is obtained by fitting atomic beam phase shifts to phenomenological

functional forms. A first approximation then treats the He liquid as a semiinfinite contin-

uum extending from z = --oo to t = 0, where it is bounded by a sharp interface. This

results in a total potential V(z) on the H atom :

where V(r) is the He-H pair potential, p = 0.0218 AP3 is the number density of bulk He,

and R = (2, y). The van der Waals tail of the He-H potential determines the long range

behavior of V(z) [43]:

V(z) -+ -c3/z3; C3 = $KT6 = 223 KA3. (A.3)


Retardation has been shown in many cases to reduce this potential by a function well

approximated by [44]

(A-4)

with a characteristic length X which is typically 200 A. This is the correct form of the

potential when .z is large enough that the atomic structure of the liquid is not resolved. At

shorter distances, the interaction becomes a complicated many-body problem. However,

the observation that H scattering from the surface is predominantly elastic justifies a

description in terms of an effective one-body potential. This is also supported by the

many-body calculations of Mantz and Edwards [45], which provide the best variational

estimate to date of the binding energy.

To write the elastic potential in a simple analytic form, we start from the parameter-

ization of Goldman [29]:

VI(z) = po/(e BE + 1) - C&(Z6 + 2,“). (-4.5)

we fix ~0 = 37 K, Cc = 911 K A3, zc = 3.8 A as in ref. (291, but keep p as an adjustable

parameter, which eventually determines the binding energy. The latter has been measured

to be in the range 0.9 K 5 Eg 5 1.15 K [46,47], about 50% higher than the variational

estimate [45]. These values of EB most likely imply a single bound state for VI. As it

stands, VI does include van der Waals dispersion, but the value of the coefficient CG used

by Goldman is some four times higher than the known dispersion Ca. To correct for this,

and include the effect of retardation, we write

ti&) = %(')f(') -c3(z6+e;) (1 - f(4)-&) (-4.6)

f(z) = l/(1 + exp(z - V/a), i.e., we turn off the Goldman dispersion with a fermi-type function at a conservatively

large distance b = 25 A; a = 4 A.

Note that any effective one-body interaction can be thought of as generated by an

effective two-body He-H potential [29]:

V(T) = 1 82

--V(%) 27rrp 8.z2 z=T'

C-4.7)

B. Coupling to Ripplons


The He surface was treated as a flat boundary in (A.2). A realistic description must

take into account oscillations of the height ~(2) of the surface about the equilibrium

position. Quantization of the normal modes of such capillary-gravity waves, or ripplons

[48], allows the surface displacement to be written as

The free ripplon hamiltonian is

while the ripplon propagator is given by

D(q,w) = ftq 1

Mp w2 - (wq - iv)2 ebc - wq),

(4

(A.lO)

where n --* O+. Here Q = 0.27 KAe2 is the He surface tension, M is the mass of a He atom,

and we neglect the effect of gravity. We assume an upper cutoff for the ripplon spectrum

at qc = 1 A-‘, corresponding to LC x 12 K [49].

The presence of ripplons affects the He substrate-H atom interaction, which is now

obtained from (A.2) by replacing the upper limit of the z’ integration with ~(2). Ex-

panding in powers of IL, the zeroth order term yields the elastic interaction V(z); the linear

term in the ripplon operators gives the H atom-ripplon coupling

(A.ll)

and V(T) is related to V(z) through (A.7).

This coupling describes processes in which a H atom emits or absorbs a ripplon;

momentum parallel to the unperturbed interface is conserved.

90

C. Effect of Substrate

C. Carraro and M. W. Cole

If the He is actually a film of thickness d on a substrate with dispersion coefficient

C,, the results we have derived above must be modified to account for the presence of the

substrate. The latter affects the physics in two ways: it changes the elastic potential V(z)

and the ripplon frequency spectrum wq.

For d of order a few tens of A or larger, only the long range behavior of V(z) is

changed. Putting (’ = z + d, we replace (A.7) by

VIII(Z) = vi(t)f(z) - c3 (z6~zg) - f) (1 - fb>)r(4 - +o. (A.12)

Thus the substrate can considerablyenhance the long range attraction, provided C, >>

C’s, which is always the case, since He provides the weakest van der Waals force of any

condensed medium.

The dispersion relation for the capillary-gravity waves on the surface of a film is [48]

wi = (gq + zq3)tanhqd, MP

(A.13)

where the acceleration in the substrate potential is g = Cs/d4 (neglecting retardation). It

turns out that gravity effects are important only for d 5 10 A. For much thicker films the

correction is negligible. The same is true of the hyperbolic tangent term, which can be put

equal to unity in the relevant range of qd.

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sticking coefficient at ultralow energy: quantum reflection

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