Three dimensional classical theory of rainbow scattering of atoms from surfaces

Chemical Physics 375 (2010) 337–347

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Three dimensional classical theory of rainbow scattering of atoms from surfaces Eli Pollak a,*, Salvador Miret-Artés b a b

Chemical Physics Department, Weizmann Institute of Science, 76100 Rehovoth, Israel Instituto de Física Fundamental, Consejo Superior de Investigaciones Científicas, Serrano 123, 28006 Madrid, Spain

a r t i c l e

i n f o

Article history: Available online 26 May 2010 Keywords: Surface scattering Rainbows Classical dynamics Angular distributions Three dimensional

a b s t r a c t In this work, we extend to three dimensions our previous stochastic classical theory on surface rainbow scattering. The stochastic phonon bath is modeled in terms of linear coupling of the phonon modes to the motion of the scattered particle. We take into account the three polarizations of the phonons. Closed formulae are derived for the angular and energy loss distributions. They are readily implemented when assuming that the vertical interaction with the surface is described by a Morse potential. The hard wall limit of the theory is derived and applied to some model corrugated potentials. We find that rainbow structure of the scattered angular distribution reflects the underlying symmetries of the surface. We also distinguish between ‘‘normal rainbows” and ‘‘super rainbows”. The latter occur when the two eigenvalues of the Hessian of the corrugation function vanish simultaneously. Ó 2010 Elsevier B.V. All rights reserved.

1. Introduction The scattering of atoms by surfaces continues to be of both theoretical [1,2] and experimental interest [3]. Recent reviews may be found in Refs. [4–6]. Much interest has focussed on quantum diffraction, which appears even for atoms as heavy as Ar [7,8]. However, most of the features measured for the scattering of heavy atoms are classical in nature [9–12]. It is thus of interest to further develop the classical mechanics theory of atom surface scattering. There are a number of major features which any theory should account for. One is the rainbow structure of the angular distribution. For in-plane scattering this leads to a typical double peaked distribution whose minimum lies around the specular scattering angle. A second feature is the broadening of the distribution which is due to the interaction of the projectile with the surface and bulk phonons. The third and fourth features have to do with the angular dependence of the energy transfer to the surface [13,14] and the angle dependent sticking probability, respectively. The classical theory of atom surface scattering has a long history. One may resort to molecular dynamics simulations [15,16] however, these do not necessarily provide an explanation or deep insight into the processes. Brako and Newns [17] developed a classical model in which the motion of the incident particle is coupled linearly to the phonon bath of the surface. They derived an analytical expression for the angular distribution, as well as for the energy and momentum transfer. Their theory served as the basis for many theoretical studies of the scattering, especially by Manson and coworkers [1,2]. However, Brako and Newns did not * Corresponding author. E-mail address: [email protected] (E. Pollak). 0301-0104/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.chemphys.2010.04.039

include in their theory any corrugation and so it could not account directly for the rainbow phenomenon. As a result they also ignored the coupling between the phonon bath and the horizontal motion of the atom along the surface. The classical rainbow effect in surface scattering was discovered by McClure [18]. The first three dimensional theoretical treatment was given by Garibaldi et al. [19]. It was later studied by Horn et al. [20] but they did not derive an explicit expression for the angular distribution which accounted for the structure. A model which provided insight into the effect of the corrugation is the ‘‘washboard model” of Tully [21] which was further extended in a more recent paper in Ref. [22]. However, as in the previous analysis the washboard model assumes an impulsive collision of the projectile with the surface. We have recently formulated a classical perturbation theory within the generalized Langevin equation formalism for the angular distribution which takes into account the continuous interaction of the projectile with the surface [23]. It also included the interaction of the projectile with the surface phonons through the motion along the horizontal direction. The theory was further improved by using a translationally invariant model Hamiltonian in Ref. [24] providing deeper insight into the intertwining between the energy loss and angular dependence of the scattering. Asymmetric rainbow scattering was then studied in Ref. [25]. However, all of these studies were restricted to the two dimensional theory of in-plane scattering. In this present paper we generalize the theory to three dimensional scattering. We include both vertical and horizontal coupling to the phonon baths and as in the case of inplane scattering the coupling to the horizontal phonon baths is translationally invariant and so presents a more realistic model for the collision dynamics.

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First we solve perturbatively for the motion in the horizontal directions, this gives the change in the horizontal momentum as effected by the collision. Then we obtain the change in the vertical momentum through energy conservation. The origin of time is chosen as the point at which the unperturbed trajectory, as derived from the vertical Hamiltonian

In Section 2 we present the theory in the limit that there is no coupling to the baths. We show how the perturbation theory reduces to known results derived previously by assuming hard wall scattering with a massive particle. We study some simple examples of surfaces with different symmetries (the egg carton shape, a chiral and an hexagonal surface) and show how these different symmetries show up in the full 3D rainbow scattering. These results suggest the possibility of using three dimensional rainbow scattering for characterization of surfaces and their corrugation as an alternative and complementary way to particle diffraction. The treatment proposed is analytical and readily implemented in a fitting procedure of experimental results. The theory is then generalized in Section 3 to include the interaction of the three projectile degrees of freedom with the surface. We end with a discussion and suggestions for new experiments aimed at elucidating the three dimensional rainbow scattering structures.

in the vertical direction reaches the turning point ðztp Þ so that at time t ¼ 0 the vertical velocity vanishes. We then need to consider the motion from the initial time t 0 when the atom is far from the surface until the time t 0 when it is again far from the surface. With this construction, the unperturbed trajectory in the vertical direction is an even function of time. The unperturbed motion in the horizontal directions is then given as:

2. Theory of 3D scattering with a frozen surface

xðtÞ ¼ x0 þ

We consider three dimensional scattering of a particle with mass M. The three degrees of freedom are the vertical distance z and the horizontal coordinates x and y. The interaction potential Vðx; y; zÞ is mainly a function of the instantaneous vertical distance of the atom from the surface. Typically the surface is corrugated so that this distance is modulated by the periodic corrugation function hðx; yÞ with periods lx and ly which correspond to the lattice lengths of the surface. Assuming weak corrugation one may approximate the three dimensional potential to be

ð2:1Þ

where V 0 means derivative with respect to the vertical coordinate and the Hamiltonian is

H¼

p2x þ p2y þ p2z þ Vðx; y; zÞ: 2M

ð2:2Þ

Throughout this paper, the vertical potential VðzÞ will be approximated as a Morse potential

VðzÞ ¼ V 0 ð1  expðazÞÞ2  V 0

ð2:3Þ

characterized by a binding energy V 0 of the incident particle to the surface and the stiffness parameter a. We assume that at the initial time, the particle is characterized by the initial momenta px0 ; py0 and pz0 which define the incident scattering angles:

0qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 p2x0 þ p2y0 A; h0 ¼ tan1 @ pz0 ! py0 1 /0 ¼ tan px0

ð2:7Þ

ð2:8Þ ð2:9Þ

From Hamilton’s equations of motion for the horizontal momenta we then readily find that

px ðt 0 Þ ¼ px0 

py ðt 0 Þ ¼ py0 

Z

t0

dtV 0 ðzt Þ

@hðxðtÞ; yðtÞÞ  px0 þ dpx ; @xðtÞ

ð2:10Þ

dtV 0 ðzt Þ

@hðxðtÞ; yðtÞÞ  py0 þ dpy : @yðtÞ

ð2:11Þ

t 0

Z

t0

t 0

Explicit expressions may then be obtained for the shifts in the momenta by inserting on the right hand sides of these equation the zeroth order solutions for the equations of motion. Specific examples are given below. The final vertical momentum ðpz þ dpz Þ is then obtained from energy conservation:

pz0 dpz ¼ px0 dpx þ py0 dpy :

ð2:12Þ

If one of the initial horizontal momenta is of the same order of magnitude as the respective momentum change, then one should include the relevant quadratic term on the right hand side. We assume that the absolute value of the vertical momentum is always large as compared to the absolute value of the vertical momentum change, if not, as in grazing collisions, the perturbation theory fails. The final scattering angles 0 6 h 6 p=2; p 6 / 6 p which depend on the initial conditions of the horizontal momenta are by definition

  pz0 þ dpz ðx0 ; y0 Þ p0

hðx0 ; y0 Þ ¼ cos1 ð2:4Þ

’ h0 þ

cosð/0 Þdpx ðx0 ; y0 Þ þ sinð/0 Þdpy ðx0 ; y0 Þ   ; pz 

ð2:13Þ

0

ð2:5Þ /ðx0 ; y0 Þ ¼ tan1

and the radial initial momentum is:

p20 ¼ 2ME0 ¼ p2x0 þ p2y0 þ p2z0 :

p2z þ VðzÞ 2M

px0 ðt þ t0 Þ; M py yðtÞ ¼ y0 þ 0 ðt þ t 0 Þ: M

2.1. The model Hamiltonian

Vðx; y; zÞ ¼ VðzÞ þ V 0 ðzÞhðx; yÞ;

Hz ¼

ð2:6Þ

We note that p=2 6 h0 6 0 due to the existence of the surface and the fact that by construction the initial vertical momentum is taken to be negative. The range of the azimuthal angle is p 6 /0 6 p. 2.2. Perturbation theory The derivation of an explicit expression for the angular distribution is predicated on the assumption of weak corrugation. This allows for a perturbative solution of Hamilton’s equations of motion.

’ /0 

! py0 þ dpy ðx0 ; y0 Þ px0 þ dpx ðx0 ; y0 Þ

cosð/0 Þdpy ðx0 ; y0 Þ  sinð/0 Þdpx ðx0 ; y0 Þ : p0 sinðh0 Þ

ð2:14Þ

These results provide an explicit expression for the three dimensional deflection functions and their dependence on the horizontal location of the initial trajectory. The final angular distribution is obtained by integration over the horizontal coordinates in the unit cell. These integrations are facilitated by changing the integration variables from the initial coordinates ðx0 ; y0 Þ to their values at the turning point ðxð0Þ; yð0ÞÞ. Since the unperturbed motion is that of a free particle, the

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Jacobian of this transformation is unity. Suppressing the time notation ðxð0Þ ! x; yð0Þ ! yÞ we then have that the expression for the final angular distribution is:

Z

Z

t0

dtV 0 ðzt Þf ðxðtÞ; yðtÞÞ ’ f ðx; yÞ

lx

This result can be simplified by introducing the following change of variables:

cosð/0 Þdpx ðx; yÞ þ sinð/0 Þdpy ðx; yÞ uðx; yÞ ¼ ; p0

1

dtV 0 ðzt Þ

1

t 0

¼

1 Pðh; /Þ ¼ dx lx ly 0   Z ly cosð/0 Þdpx ðx; yÞ þ sinð/0 Þdpy ðx; yÞ dyd h þ h0   p0 cosðh0 Þ 0   cosð/0 Þdpy ðx; yÞ  sinð/0 Þdpx ðx; yÞ  d /  /0 þ : ð2:15Þ p0 sinðh0 Þ

Z

4aV 0 f ðx; yÞ

X

tanðUÞ2 :

We thus find that:

dpx ðx; yÞ @hðx; yÞ ’2 p0 cosðh0 Þ @x

ð2:24Þ

and

dpy ðx; yÞ @hðx; yÞ : ’2 @y p0 cosðh0 Þ

ð2:25Þ

This implies that the Jacobian of Eq. (2.18) is proportional to the Jacobian of the first derivatives of the corrugation function:

ð2:16Þ

0

@ 2 hðx; yÞ @ 2 hðx; yÞ @ 2 hðx; yÞ Jðu; v ; x; yÞ ¼ 4 cos ðh0 Þ@  2 2 @x @y @x@y 2

v ðx; yÞ ¼

cosð/0 Þdpy ðx; yÞ  sinð/0 Þdpx ðx; yÞ : p0

@ðu; v Þ @ðx; yÞ   1 @dpx ðx; yÞ @dpy ðx; yÞ @dpx ðx; yÞ @dpy ðx; yÞ ¼ 2  @x @y @y @x p0

Jðu; v ; x; yÞ ¼

ð2:18Þ and the expression for the angular distribution formally simplifies to:

1 1 J ðcosðh0 Þðh þ h0 Þ;  sinðh0 Þð/  /0 Þ; x; yÞ: lx ly

!2 1 A:

ð2:17Þ

The Jacobian of this transformation is

Pðh; /Þ ¼

ð2:23Þ

ð2:26Þ The rainbow lines are then given by the condition that the Hessian of the corrugation vanishes – this is precisely the condition derived by Garibaldi et al. [19]. 2.3.2. The ‘‘standard” egg carton corrugation function Perhaps the simplest possible corrugation function takes the form

hðx; yÞ ¼ hx sin

    2px 2py þ hy sin : lx ly

ð2:27Þ

The horizontal momentum shifts are then seen to be given by:

ð2:19Þ

The rainbows are then the lines along which the Jacobian vanishes. 2.3. Examples

  dpx ðx; yÞ 2p x ;  K x cos p0 cosðh0 Þ lx   dpy ðx; yÞ 2p y ;  K y cos ly p0 cosðh0 Þ

ð2:28Þ ð2:29Þ

where the angular coefficients are Structural studies of surfaces with atomic and molecular beam diffraction have been widely developed by Rieder’s group [26,4]. An exhaustive analysis about clean surfaces and monolayers has been carried out. Here we extract some empirical information about corrugation functions in order to illustrate the 3D theory. 2.3.1. The hard wall limit It is of interest to show how the formal results of the previous subsection reduce to known results for hard wall potentials. Using the Morse potential, one obtains the hard wall limit by allowing the stiffness parameter a ! 1. The unperturbed trajectory of the Morse oscillator is known analytically:

expðazt Þ ¼ 

cosðUÞ 2

sin ðUÞ

½coshðXtÞ þ cosðUÞ

ð2:20Þ

with

X2 ¼

2

2a Ez M

ð2:21Þ

where Ez is the incident energy in the vertical direction and

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V0 cosðUÞ ¼  : Ez þ V 0

ð2:22Þ

For very large stiffness parameters we then have that the derivative of the potential vanishes for almost all times, except around t ¼ 0. This then allows us to approximate the time integrals as:

Kj ¼ 

2phj p0 cosðh0 Þlj

Z

1

dtV 0 ðzt Þ cosðxj tÞ;

j ¼ x; y

ð2:30Þ

1

and the frequencies are:

xj ¼

2ppj0 ; Mlj

j ¼ x; y:

ð2:31Þ

The expression for the Jacobian is then found to be:

Jðu; v ; x; yÞ ¼

    4p 2 2px 2py sin K x K y sin lx ly lx ly

ð2:32Þ

so that the angular distribution is given as:

Pðh; /Þ ¼

1

p2

1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 K x  ðcosð/0 Þðh þ h0 Þ þ sinð/0 Þ tanðh0 Þð/  /0 ÞÞ2

1  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 2 K y  ðsinð/0 Þðh þ h0 Þ  cosð/0 Þ tanðh0 Þð/  /0 ÞÞ2 ð2:33Þ One notes that the angular distribution will have rainbow peaks if either x ¼ 0; lx =2 or y ¼ 0; ly =2. In the ðx; yÞ plane, this creates four lines with a rectangular structure, reflecting the rectangular symmetry of the corrugation function. The four points ð0; 0Þ; ð0; ly =2Þ; ðlx =2; 0Þ; ðlx =2; ly =2Þ are special, since at these four points both eigenvalues of the Hessian matrix of the corrugation function vanish. One thus expects a more intense peak at these four points which may considered as super rainbow angles. This topological

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a

b

Fig. 1. An egg carton corrugation function and angular distribution. Panel a shows the square corrugation function, panel b the resulting angular distribution. Note the square symmetry of the corrugation function and its reflection in the angular distribution. The four major peaks are the super rainbows, as discussed in the text. The ‘‘standard” rainbows are reflected by the ridges as seen in the 3D angular distribution plot. The latter has been artificially broadened by a Gaussian function with a 1o angular width, for presentational purposes.

E. Pollak, S. Miret-Artés / Chemical Physics 375 (2010) 337–347

341

a

b

Fig. 2. A chiral corrugation function and angular distribution. Panel a shows the chiral corrugation function, panel b the resulting chiral angular distribution. The chirality is evidenced through the four peaks which have different heights. Other details are given in the text. The angular distribution has been artificially broadened as in Fig. 1.

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a

b

Fig. 3. An hexagonal corrugation function and angular distribution. Panel a shows the hexagonal corrugation function, panel b the resulting hexagonal distribution. In this case there are not any ‘‘super rainbows”, note though the steep ridges along the rainbow lines. The angular distribution has been artificially broadened as in Fig. 1.

analysis is related to catastrophe theory and chaotic scattering [6]. In the angular ðh; /Þ plane, these four points correspond to the two possibilities:

h þ h0 ¼ ðK x cosð/0 Þ þ K y sinð/0 ÞÞ; /  /0 ¼ 

K x sinð/0 Þ  K y cosð/0 Þ tanðh0 Þ

ð2:34Þ

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or to

h þ h0 ¼ ðK x cosð/0 Þ  K y sinð/0 ÞÞ; /  /0 ¼ 

K x sinð/0 Þ þ K y cosð/0 Þ : tanðh0 Þ

ð2:35Þ

2.3.3. Numerical examples One of the purposes of the present study was to understand to what extent one can use the full three dimensional angular distribution as a probe of the surface structure. For this purpose, we consider three different cases. The first one is a surface with a symmetric square unit cell. The second case is an asymmetric (chiral) surface and the third case is an hexagonal surface. In all cases we find that the angular distribution clearly reflects the underlying symmetry. In all cases we choose the incident scattering angles h0 ¼ 45 and /0 ¼ 45 . In panel a of Fig. 1 we plot a square corrugation function ðlx ¼ ly ¼ 1Þ obtained from Eq. (2.27), with hx =lx ¼ hy =ly ¼ 0:05. The resulting angular distribution with K x ¼ K y ¼ 10 (see Eq. (2.30)) is shown in panel b of the figure. As discussed above, the square structure of the corrugation is borne out in the angular distribution. The second example is scattering from a handed surface. This is effected by choosing the corrugation function to be:

       2px 4px 4px þ hx2 sin þ cos lx lx lx     2py 4py þ hy2 sin : þ hy1 sin ly ly

hðx; yÞ ¼ hx1 sin

This corrugation function is shown in Panel a of Fig. 2, using equal lattice lengths in the x and y directions and a weak asymmetry ðhi2 =hi1 ¼ 0:2; i ¼ x; yÞ. The change in the horizontal momenta in this case may be written as:

       dpx ðx; yÞ 2px 4px 4px þ K x2 cos  sin ;  K x1 cos p0 cosðh0 Þ lx lx lx ð2:36Þ     dpy ðx; yÞ 2p y 4p y þ K y2 cos :  K y1 cos ly ly p cosðh0 Þ

ð2:37Þ



Choosing K x1 ¼ K y1 ¼ 10 as in Fig. 1 and K i2 =K i1 ¼ 0:2; i ¼ x; y we find the angular distribution shown in panel b of Fig. 2. Note that the four rainbow peaks have different heights, this is the signature of the chiral structure of the underlying corrugation. As a final example, we consider a hexagonal surface. This is obtained from the corrugation function [4]

       2p x 2py 2pðx  yÞ þ cos þ cos : hðx; yÞ ¼ h cos l l l

ð2:38Þ

The hexagonal structure of the corrugation function is seen in panel a of Fig. 3. The resulting hexagonal angular distribution is shown in panel b of the figure. In summary, we have shown that the angular distribution of scattered atoms may be used as a probe of the underlying symmetries and structure of the surface. Thus far the computations have been exemplified for frozen surfaces, that is we have ignored any broadening induced by phonon interactions. The three dimensional theory including phonon interactions will be considered in the next section. 3. 3D theory of scattering from a thermal surface 3.1. Preliminaries We now assume that both the vertical and horizontal coordinates fluctuate due to interaction with the thermal phonon bath

of the surface. When the particle is bound to the surface, it diffuses freely along the horizontal direction. Its equation of motion in the horizontal direction is then the standard Langevin equation with a periodic potential. However, when the particle is far away from the surface, the interaction of the horizontal coordinate with the phonon bath vanishes. It is therefore reasonable to model the interaction of the horizontal motion with the phonon bath through coupling which is linear in the bath modes and is modulated by a space dependent function gðzÞ which vanishes when the vertical distance from the surface is large. This then leads us to assuming that the Hamiltonian governing the scattering event is:

H¼

p2x þ p2y þ p2z þ VðzÞ þ V 0 ðzÞhðx; yÞ 2M2 !2 3 pffiffiffiffiffi N M c 1X j 2 2 0 z 4p þ x xj  V ðzÞ 5 þ jz jz z 2 j¼1 x2jz 2 !2 3 pffiffiffiffiffi   N M c 1X l 2 p x j x 4p2 þ x2 xj  x g x ðzÞ 5 sin þ jx jx x 2 j¼1 lx x2jx 2p 2 !2 3 pffiffiffiffiffi   N cjy M ly 1X 2py 2 2 4 g y ðzÞ 5; pjy þ xjy xjy  sin þ 2 j¼1 ly x2j 2p

ð3:1Þ

y

where the phonon bath is characterized by the mass weighted momenta and coordinates pji ; xji ; j ¼ 1; . . . ; N; i ¼ x; y; z. The term coupling the horizontal motions to the respective phonon bath is periodic in the horizontal coordinates, this is necessary to assure the translational invariance of the model. When the particle is far from the surface it does not interact with the phonons, so that the bath Hamiltonian (in mass weighted coordinates and momenta) is defined to be:

HB ¼

1 2

  p2ji þ x2ji x2ji :

N X

ð3:2Þ

j¼1;i¼x;y;z

As is well known, for the linearly coupled harmonic baths the equations of motion in the continuum limit are Generalized Langevin Equations (GLE’s). Introducing the spectral densities

J i ð xÞ ¼

N c2ji pX dðx  xji Þ; i ¼ x; y; z 2 j¼1 xji

ð3:3Þ

and associated friction functions:

gi ðtÞ ¼

2

p

Z 0

1

dx

J i ðxÞ

x

cosðxtÞ;

i ¼ x; y; z

ð3:4Þ

the GLE for the horizontal directions takes the form (as may be readily seen by using the known forced harmonic oscillator solution for the bath variables and inserting it into the equations of motion for the system degrees of freedom)

  pffiffiffiffiffi 2pxt MF x ðtÞ cos g x ðzt Þ lx   @Vðxt ; yt ; zt Þ 2pxt g x ðzt Þ þ M cos ¼ M€xt þ @xt lx      Z t d lx 2 p xt 0 0 0 g dt gx ðt  t0 Þ sin ð z Þ ;  x t 0 lx dt 2p t0   pffiffiffiffiffi 2pyt g y ðzt Þ MF y ðtÞ cos ly   @Vðxt ; yt ; zt Þ 2pyt €t þ g y ðzt Þ þ M cos ¼ My @yt lx      Z t d ly 2pyt0 0 g y ðzt0 Þ : dt gy ðt  t0 Þ sin  0 ly dt 2p t0

ð3:5Þ

ð3:6Þ

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The GLE for the vertical motion is more complicated, but we will not need it explicitly in the theory developed below. We do note that the vertical motion is a function of the vertical friction function gz ðtÞ and noise F z ðtÞ. As with the frozen surface, we initiate trajectories at the time t0 . The projectile is initially sufficiently distant from the surface, such that at the vicinity of z0 all the coupling functions vanish and the motion is that of a free particle. The noise functions therefore depend only on the initial conditions of the respective phonon bath. We note that N X

F i ðtÞ ¼

  pj cji xji cos½xji ðt þ t0 Þ þ i sin½xji ðt þ t 0 Þ ;

xji

j¼1

i ¼ x; y; z ð3:7Þ

3.3. The energy losses to the phonon baths and the vertical momentum change The energy loss to the bath may be divided into two parts, an average energy loss and a fluctuational energy loss:

DEB ¼

i; j ¼ x; y; z;

ð3:8Þ

where b ¼ 1=ðkB TÞ is the inverse temperature and the averaging is over the thermal distribution associated with the classical bath Hamiltonian as given in Eq. (3.2). Typically the Debye frequency of the crystal is much larger than the frequency of motion of the projectile when bound to the surface. We may therefore safely assume that the friction functions are Ohmic:

gi ðtÞ ¼ 2gi dðtÞ; i ¼ x; y; z:

ð3:9Þ

ðhDEB ii þ dEBi Þ:

ð3:17Þ

i¼x;y;z

As readily seen from Ref. [27], after some manipulation, one finds that for Ohmic friction, the average energy loss to the bath due to the motion in the x direction is:

   2 d 2pxt g x ð zt Þ sin dt lx t 0   4pxðt0 Þ  Dx  cos Dx;1 ; lx 2

hDEB ix ¼ Mgx

and the noise functions obey the fluctuation dissipation relations:

M hF i ðt 1 ÞF j ðt 2 Þi ¼ dij gi ðt1  t2 Þ; b

X

lx 4p 2

Z

t0

dt



ð3:18Þ

where 2

2

Dx ¼ E0 sin ðhi0 Þ cos ð/i0 Þgx

Z

t0

dt

t0

g 2x ðzt Þ

 2 ! dg x ðzt Þ þ 2 dt xx 1

ð3:19Þ and 2

Dx;1 ¼ E0 sin ðhi0 Þ cos2 ð/i0 Þgx

Z

t0

dt t 0

3.2. The horizontal momentum changes

 cosð2xx tÞ g 2x ðzt Þ þ The perturbation theory used in the previous section is applied here also with respect to the coupling to the phonon bath, which is assumed to be weak. Denoting the momentum shifts in the absence of the baths with a 0 superscript, we readily find from the Langevin equations (3.5) and (3.6) that the horizontal momentum shift in the x direction may be written as:

px ðt 0 Þ ’ px0 þ dp0x þ Dpx;1 þ Dpx;2  px0 þ dpx

ð3:10Þ

and similarly for the horizontal y direction. Here, the friction induced momentum shift is (after an integration by parts):

Dpx;1 ¼ gx

px0 2

Z

t0 t 0

2

dtg x ðzt Þ

ð3:11Þ

and the noise induced momentum shift is !   N   N pffiffiffiffiffi 2px X 2 px X X js Dpx;2 ¼ M cos cjx X jc xjx ðt0 Þ þ sin cjx pjx ðt 0 Þ lx lx xjx j¼1 j¼1     2px 2px  cos Dpx;2c þ sin Dpx;2s lx lx

ð3:12Þ with

X jc ¼

Z

t0

t0

X js ¼ 

Z

dtg x ðzt Þ cosðxx tÞ cosðxjx tÞ;

ð3:13Þ

t0

t 0

dtg x ðzt Þ sinðxx tÞ sinðxjx tÞ

ð3:14Þ

and we used the notation:

xjx ðt 0 Þ ¼ xjx cos½xjx t0  þ

pjx

xjx

sin½xjx t 0 ;

pjx ðt 0 Þ ¼ xjx xjx sin½xjx t 0  þ pjx sin½xjx t 0 :

ð3:15Þ ð3:16Þ

Analogous expressions hold for the momentum changes in the horizontal y direction.

 2 ! dg x ðzt Þ : dt x2x 1

ð3:20Þ

The fluctuational energy loss in the horizontal direction is then found to be:

   pffiffiffiffiffi lx Z t0 d 2 p xt g x ðzt Þ F x ðtÞ sin dEBx ¼  M dt 2p t0 dt lx "  X N pffiffiffiffiffi lx 2px ¼ M sin cjx X jc pjx ðt0 Þ lx 2p j¼1 #   N 2p x X cjx X js xjx xjx ðt 0 Þ þ cos lx j¼1     2px 2px  sin dEBx s þ cos dEBx c : lx lx

ð3:21Þ

One also readily finds that the variance of the fluctuational energy loss is proportional to the average energy loss:

hdEBx i2 ¼

2 hDEB ix : b

ð3:22Þ

Similar results are found for the energy losses in the y direction, one must replace x with y everywhere in Eqs. (3.17)–(3.22) and in addition, the terms cos2 ð/i0 Þ in Eqs. (3.19) and (3.20) should be replaced 2 by sin ð/i0 Þ. We note that the averaged energy losses are function of the x and y coordinates. This dependence may lead to energy loss rainbows, as described elsewhere [28]. If however, the horizontal momenta are sufficiently large one may use the ‘‘high horizontal frequency limit” in which the term Dx;1 is negligible as compared with Dx . In this limit the energy losses are independent of the impact parameter and the energy loss rainbow phenomenon disappears. Similar relations hold also for the energy losses in the vertical direction. The average energy loss to the bath due to the vertical 1 motion is (note that the dimensions of gx and gy are time while 3 2 that of gz is time =mass ):

E. Pollak, S. Miret-Artés / Chemical Physics 375 (2010) 337–347

hDEB iz ¼ M gz

Z

!2 pz Dpz;1 dV 0 ðzt Þ  0 dt M

t0

dt

t0

ð3:23Þ

pffiffiffiffiffi Z dEBz ¼  M

t0

dV 0 ðzt Þ F z ðtÞ dt t0   N pffiffiffiffiffi X pz Dpz;2 Z js ¼ M cjz Z jc xjz ðt 0 Þ þ pjz ðt0 Þ   0 : x M jz j¼1 dt

ð3:24Þ

where

Z jc ¼ Z js ¼

Z Z

t0

dt

dV 0 ðztþt0 Þ cosðxjz tÞ ¼ 0; dt

ð3:25Þ

dt

dV 0 ðztþt0 Þ sinðxjz tÞ: dt

ð3:26Þ

t 0 t0

t 0

ignored. The corrugation dependent angular distribution then becomes:

Iðh; /; x; yÞ ¼

and the associated fluctuational energy loss is

The variance of the vertical fluctuational energy loss obeys the same relation as for the horizontal energy losses:

2 hdEBz i ¼ hDEB iz : b 2

! pffiffiffi bE2 cot2 ðh0 ÞY 2h bE0 jcotðh0 Þj pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp  0 dðY / Þ; hDEB iz phDEB iz

Iðh; /; x; yÞ ¼

ð3:27Þ

! pffiffiffiffiffiffiffiffi bE0 cosðh0 Þ bE0 cos2 ðh0 ÞY 2h pffiffiffiffiffiffiffiffiffiffiffi exp  Hhh pHhh

Y h ¼ h þ h0 

pz ðt 0 Þ ¼ pz0 þ dpz :

ð3:28Þ

pz0 dpz ¼ px0 dpx þ py0 dpy þ MDEB :

The angular deflection functions are obtained via perturbation theory as before, except that here one must also take into consideration the energy losses. Using Eq. (3.29) we then find for the scattering angle that the deflection function is: 0 1

Bpz þ dpz ðx0 ; y0 ÞC hðx0 ; y0 Þ ¼ cos1 @ q0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A p20  2MDEB   dpy MDEB 1 dp ’ h0 þ  sinðh Þ cosð/0 Þ x þ sinð/0 Þ 0 p0 p0 cosðh0 Þ p20  h0 þ Dh:

ð3:30Þ

The expression for the azimuthal deflection function is formally the same as in Eq. (2.14), except that now the horizontal momentum shifts include contributions from interaction with the baths, as given in Eq. (3.10). To obtain the angular distribution one must average over the phonon bath:

0

dx

ð3:34Þ

2

ð3:29Þ

3.4. The angular deflection functions and distribution

lx

 0  dpx þ Dpx;1 MhDEB i 1  sinðh Þ 0 cosðh0 Þ p0 p20

and the variance is [29]

Energy conservation then implies the relation

Z

ð3:33Þ

with

as:

1 lx ly

ð3:32Þ

where the corrugation dependent shifts Y h and Y / are the friction free limits of Eqs. (A.2) and (A.3). In this case, the angular ðhÞ distribution is broadened by the vertical energy loss to the surface, while the azimuthal ð/Þ angular distribution remains as it was in the absence of friction. It is also useful to consider the case of in-plane scattering. In this case, the initial momentum in the horizontal y direction vanishes and the final and initial azimuthal angles are set to 0. The measurement is limited to motion along the horizontal x coordinate. As a result the width Hh/ ¼ 0 and the normalized in-plane corrugation dependent angular scattering distribution reduces to the known result for in-plane scattering (Eq. (2.45) of Ref. [24]):

The final momentum in the vertical direction is written down

Pðh; /Þ ¼

345

Z 0

ly

dy

Z

1

N Y

1 j¼1;k¼x;y

dpkj dxkj expðbHB Þ ZB 2ph

 dðh þ h0  DhÞdð/  /0  D/Þ: Z lx Z ly 1 ¼ dx dyIðh; /; x; yÞ; lx ly 0 0

ð3:31Þ

where Z B is the bath partition function and D/ is a shorthand notation for the shift in the azimuthal angle. Carrying out the averaging over the bath is somewhat lengthy, the details are given in the Appendix, the expression for the corrugation dependent function Iðh; /; x; yÞ is given in Eq. (A.17). It may be the case that the horizontal dissipation is much weaker than the vertical dissipation and so it may be to lowest order

Hhh ¼

sin ðh0 ÞhDEB i E    Z 01 4px 2 cosð2xx tÞ : þ gx dtg x ðzt Þ cosð2h0 Þ þ cos lx 1

ð3:35Þ

4. Discussion We have extended our previous classical perturbation theory to include the full dimensionality of an atom that is scattered from a surface. The perturbative treatment recovers previously known results for rainbow scattering in three dimensions. We have also demonstrated that the three dimensional structure of the rainbow and its reflection of the surface corrugation should be considered as a complementary and alternative method to the standard particle diffraction for characterizing surfaces. Explicit formulae have been presented for the energy losses to the surface, thus providing also the 3D theory of energy loss. Energy loss distributions provide insight into the phonon dynamics as characterized by friction coefficients and fluctuations. Closed expressions are provided for the experimental observables, allowing implementation in a fitting procedure of experimental results and thus interpretation and prediction of additional properties. We noted that there is a difference between the ‘‘standard” rainbow scattering which reflects the vanishing of one of the eigenvalues of the Hessian of the corrugation function and super rainbow scattering which occurs when both eigenvalues of the Hessian disappear. In the latter case, the peaks are even further pronounced. These results present a challenge to experiment. Would it be possible to measure the complete angular distribution under a variety of conditions (surface temperature, angle of incidence, incident energy)? Such detailed measurements should lead to a much more accurate representation of the surface. In the least they would reveal the underlying symmetry of the corrugation. The geometrical characterization of clean surfaces is an active field. Many elementary processes on surfaces require knowledge of the surface morphology. Kinks and/or steps can be crucial to understanding chemical reactivity, phase changes, epitaxial growth, etc. Fourier series is the starting point for a morphological

346

E. Pollak, S. Miret-Artés / Chemical Physics 375 (2010) 337–347

study if the periodicity of the surface is kept over a large spatial domain. He diffraction studies are used for such characterizations. Their advantage with respect to scanning tunneling microscopy or atomic force microscopy is that they provide a global picture of the morphology. The preliminary cases studied in this paper, suggest that perhaps it is not necessary to use He and quantum diffraction, but that it may be replaced by measuring the rainbow structure of scattered heavy particles, where diffraction is not important. The Fourier structure of the resulting three dimensional rainbow structures remain a project for the future. This could be useful to experimentalists for the characterization of a variety of surfaces, including surfaces with kinks, ratchet-like potentials [30], single adsorbates [31], self assembled monolayers (SAMs) or surface liquids [32,33], uniaxial incommensurate monolayers [34], or surface diffusion [35]. Acknowledgments It is a pleasure to dedicate this paper to Professor Peter Hänggi upon his 60th birthday. Peter has been a friend, collaborator and teacher for many years. This present paper has its roots in a joint paper on the Kramers turnover problem [27]. We thank him not only for stimulating collaborations but also to his dedication to science and to his fellow colleagues and collaborators. We wish him many more years of fruitful work. We thank Dr. Bill Allison for stimulating discussion and for suggesting the study of chiral surfaces. We also thank Dr. Santanu Sengupta for many discussion on 3D scattering from surfaces and Dr. Jeremy Moix for his critical reading of this paper. We gratefully acknowledge support of this work by a grant of the Israel Science Foundation and the Albert Einstein Minerva Center for Theoretical Physics of the Weizmann Institute. S.M.A. would like to thank the Ministry of Science and Innovation of Spain for a project with reference FIS2007-62006.

DY / ¼

  Dpy;2 Dpx;2 1 :  sinð/0 Þ cosð/0 Þ sinðh0 Þ p0 p0

We then note that all the bath variables appear in the fluctuational terms DY h and DY / . Specifically:

pffiffiffiffiffi   N cosð/0 Þ M 2px X DY h ¼ cos cjx xjx ðt0 ÞHxj c cosðh0 Þ p0 lx j¼1 !   N c jx 2px X p ðt ÞH þ sin lx xjx jx 0 xj s j¼1 pffiffiffiffiffi N sinð/0 Þ M 2py X þ cosð Þ cj xj ðt 0 ÞHyj c cosðh0 Þ p0 ly j¼1 y y !   N c jy 2py X þ sin p ðt ÞH ly xjy jy 0 yj s j¼1 pffiffiffiffiffi   N M M tanðh0 Þ X Z js þ c Z x ðt Þ þ p ðt Þ ; 0 0 j jc j z z xjz jz p20 j¼1

Iðh; /; x; yÞ 

1

N Y

1 j¼1;k¼x;y;z

2

X js xjx ;

ðA:8Þ

X jc xjx ;

ðA:9Þ

Y js xjy ;

ðA:10Þ

Y jc xjx :

ðA:11Þ

2

sin ðh0 Þ

xx sin ðh0 Þ

xx 2

Hyj s ¼ Y js þ

sin ðh0 Þ

xx

Gaussian integration over the bath variables then gives the result

! 2vh v/ H/h ¼ exp    ; 4bE0 cos2 ðh0 Þ 4bE0 sin2 ðh0 Þ 4bE0 sinð2h0 Þ

v2/ H//

v2h Hhh

ðA:12Þ ðA:1Þ

where the variances are found to be: !   N 2  X N cjx 2 c2jx 2 2px X 2 2px Hhh ¼ cos2 ð/0 Þ cos2 H þ sin H lx lx x2j xj c x2j xj s j¼1 j¼1 x

x

!  X  X N N c2jy 2 c2jy 2 2 2py 2 2py H þ sin H þ sin ð/0 Þ cos ly ly x2j yj c x2j yj s j¼1 j¼1

Y h ¼ h þ h0 

Y / ¼ /  /0 þ

xx

hexpðiðvh DY h  v/ DY / ÞÞi

where the brackets denote thermal averaging over the three baths and we used the definitions:

 dp0 þ Dpx;1 1 þ sinð/0 Þ cosð/0 Þ x cosðh0 Þ p0 ! dp0y þ Dpy;1 MhDEB i  sinðh0 Þ ;  p0 p20

sin ðh0 Þ

Hxj c ¼ X jc þ

2

dpkj dxkj expðbHB Þ dðY h  DY h Þ ZB 2ph 

 dðY / þ DY / Þ Z 1 Z 1 1 d v dv expðivh Y h þ iv/ Y / Þ ¼ 4p2 1 h 1 /  hexpðiðvh DY h  v/ DY / ÞÞi;

ðA:7Þ

and we used the notation:

Hyj c ¼ Y jc þ

In this appendix we carry out the averaging over the harmonic bath variables, to obtain an expression for the angular distribution. We are interested in performing the integration

Z

ðA:6Þ

pffiffiffiffiffi   N M cosð/0 Þ 2p y X DY / ¼ cjy Y jc xjy ðt 0 Þ cos p0 sinðh0 Þ ly j¼1 !   N cjy 2py X Y p ðt Þ þ sin ly xjy js jy 0 j¼1 pffiffiffiffiffi   N M sinð/0 Þ 2p x X cjx X jc xjx ðt0 Þ cos  p0 sinðh0 Þ lx j¼1 !   N 2px X X js þ sin cjx pjx ðt 0 Þ lx x jx j¼1

Hxj s ¼ X js þ Appendix A. Derivation of an expression for the three dimensional angular distribution

ðA:5Þ

2

y

ðA:2Þ

! dp0y þ Dpy;1 dp0 þ Dpx;1 1 ;  sinð/0 Þ x cosð/0 Þ sinðh0 Þ p0 p0 ðA:3Þ

  Dpy;2 MdEB Dpx;2 1 DY h ¼ þ sinð/0 Þ  2 sinðh0 Þ ; cosð/0 Þ cosðh0 Þ p0 p0 p0

ðA:13Þ 2

H// ¼ sin ð/0 Þ cos2 2

!   N 2  X N c jx 2 c2jx 2 2px X 2 2px X þ sin X lx lx x2jx jc x2jx js j¼1 j¼1 2

þ cos ð/0 Þ cos

ðA:4Þ

y

2

sin ðh0 ÞhDEB iz ; þ E0



!  N c2  X N c2jy 2 2py X jy 2 2py 2 Y þ sin Y ; ly ly x2jy jc x2jy js j¼1 j¼1 ðA:14Þ

347

E. Pollak, S. Miret-Artés / Chemical Physics 375 (2010) 337–347

  N 2  X N cjx c2jx H/h 2px X 2 2 px Hxj c X jc þ sin H X ¼ cos2 2 sinð2/0 Þ lx lx xjx x2jx xj s js j¼1 j¼1 !  X  X N N c2jy c2jy 2 2py 2 2py  cos H Y þ sin H Y ly ly x2j yj c jc x2j yj s js j¼1 j¼1 y

2

Hhh ¼

y

ðA:15Þ

sin ðh0 ÞhDEB i E0    Z 1 4px 2 2 cosð2xx tÞ þ cos ð/0 Þgx dtg x ðzt Þ cosð2h0 Þ þ cos lx 1    Z 1 4py 2 2 cosð2xy tÞ ; dtg y ðzt Þ cosð2h0 Þ þ cos þ sin ð/0 Þgy ly 1 ðA:25Þ

and we used the identity: N  2hDE i c2jy  2 X B z Z jc þ Z 2js ¼ : 2 M x jy j¼1

ðA:16Þ

The Gaussian integration over the Fourier variables vh and v/ then gives the central result of this Appendix which demonstrates the corrugation dependent Gaussian broadening of the deflection functions: Iðh; /; x; yÞ ¼

bE0 j sinðh0 Þ cosðh0 Þ j qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p Hhh H//  H2/h h i1 0 2 bE0 H// cos2 ðh0 ÞY 2h þ Hhh sin ðh0 ÞY 2/  sinð2h0 ÞH/h Y h Y / A:  exp @ Hhh H//  H2/h

ðA:17Þ Explicit expressions for the angular width parameters can be obtained when the frictions are Ohmic. For this purpose we note the following useful identities: N X c2jx

x

2 jx

j¼1

N X c2jx

x2jx

j¼1

N X c2jx

xjx

j¼1

N X

Z

X 2jc ¼ gx

1

1

Z

X 2js ¼ gx

1

2 x

¼ gx x

ðA:18Þ

2

ðA:19Þ

dtg x ðzt Þ½1  cosð2xx tÞ;

X jc X js ¼ xx gx

c2jx X 2jc

2

dtg x ðzt Þ½1 þ cosð2xx tÞ;

1

Z

1

1

2

dtg x ðzt Þ;

ðA:20Þ

!  2 dg x ðzt Þ 2 dt½1 þ cosð2xx tÞ þ g x ðzt Þ ; dt x2x 1

Z

j¼1

1

1

ðA:21Þ N X

c2jx X 2js

2 x

¼ gx x

j¼1

Z

!  2 dg x ðzt Þ 2 dt½1  cosð2xx tÞ þ g x ðzt Þ ; dt x2x 1 1

1

ðA:22Þ cos2



 N 2  X N c jx 2 c2jx 2 2px X 2 2px H þ sin H c x lx lx x2jx j x2jx xj s j¼1 j¼1 2

¼

    4px 2 cosð2xx tÞ dtg x ðzt Þ 1 þ cos lx   1   Z 1

 4py 2 cos 2xy t dtg y ðzt Þ 1 þ cos þ gy cos2 ð/0 Þ ly 1 2

sin ðh0 Þ hDEB ix cos2 ð/0 Þ E0    Z 1 4p x 2 cosð2xx tÞ ; þ gx dtg x ðzt Þ cosð2h0 Þ þ cos lx 1

  N 2  X N c jx 2 c2jx 2 2px X 2 2px X þ sin X jc lx lx x2jx x2jx js j¼1 j¼1    Z 1 4p x 2 cosð2xx tÞ ¼ gx dtg x ðzt Þ 1 þ cos lx 1

ðA:23Þ

cos2

ðA:24Þ

and analogous expressions are found for the horizontal y direction. One then finds that:

H// ¼ gx sin ð/0 Þ

Z

1

ðA:26Þ and

    4px 2 cosð2xx tÞ dtg x ðzt Þ cos2 ðh0 Þ þ cos lx 1     Z 1 4py 2 cosð2xy tÞ : dtg y ðzt Þ cos2 ðh0 Þ þ cos  gy ly 1

H/h ¼ gx sinð2/0 Þ

Z

1

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