Rare gas scattering from molten metals examined with classical scattering theory

Transcription

1 Rare gas scattering from molten metals examined with classical scattering theory André Muis and J. R. Manson Department of Physics and Astronomy, Clemson University, Clemson, South Carolina Received 2 November 1996; accepted 20 March 1997 Classical limit expressions of the differential reflection coefficient for atoms scattering from a surface are compared with recent experiments for the scattering of monoenergetic beams of Ne, Ar, and Xe with incident energies in the ev range from the molten metal surfaces Ga, In, and Bi. We find that single collision events usually make the greatest contribution to the backscattered intensity, double collision events make a significant but smaller contribution, and scattering of atoms that are completely trapped and subsequently thermally desorbed has a small probability. In the case of large mass incident projectiles and small mass target atoms we find some evidence for collective mass effects of the target. An analysis of the temperature dependence of the intensities shows that these surfaces act neither as a smooth continuous repulsive barrier nor as a collection of discrete scattering centers; rather they exhibit a behavior in between these two extremes American Institute of Physics. S I. INTRODUCTION The scattering of rare gas atoms from solid surfaces, especially He scattering, is a highly developed experimental method for obtaining information on surface structure and dynamics. 1 4 Atom scattering from liquid surfaces is much less developed, but certainly has the potential of providing similar surface-specific information on structure and dynamics. Hurlbut and Beck were the first to carry out scattering experiments with atomic and molecular projectiles on liquid metals. In 1959 they reported seeing diffuse, cosine-like distributions from low-energy N 2 and Ar scattering from liquid indium and gallium films. 5 McCaffery and co-workers found internal excitations in I 2 with scattering experiments of I 2 on liquid gallium. 6 An extremely interesting and novel series of experiments has been recently carried out by Ronk et al. 7 in which monoenergetic beams of Ne, Ar, and Xe were scattered from molten Ga, In, and Bi, and the scattered atoms were detected with both angular and energy resolution using time-of-flight TOF methods. These experiments used a monoenergetic jet source for the atomic beam in a fixed-angle geometry in which the incident beam, detector, and normal to the surface were all in the same plane sagital plane and the incident and detector directions were both at an angle of 55 with respect to the surface normal. They investigated how the scattered intensity and energy transfer depended on the identity of the incident and surface atoms, the incident energy of the scattered atoms, and the temperature of the molten metals. In this paper we provide a fundamental theoretical basis for describing the results of Ronk et al., which is based on semiclassical and classical limits of general atom-surface scattering theory, developed to include multiple collisions with the surface. We find that when the theory includes both single and multiple collisions very good agreement is obtained with the experimental data. Most of the backscattered intensity is due to single collisions between incident atoms and surface atoms, and the single collision contribution produces an asymmetric TOF peak shape with a width that is very nearly the same as the experimental observations. Double collisions typically make an important yet smaller contribution to the total intensity, and give significant intensity to the low-energy side of the scattered intensity. The good comparison of the multiple scattering theory with the low-energy part of the intensity indicates that the fraction of the incident beam that is trapped and subsequently thermally desorbed in equilibrium with the surface temperature is quite small. In some situations, particularly when the incident projectile has a mass relatively large compared to that of the surface atoms, collective effects of the surface were indicated by the fact that a larger effective mass for the surface was necessary in order for the theory to agree with the experimental data. The behavior of the classical limit expressions for atom-surface scattering varies considerably depending on the strength of the surface corrugation, 8 with welldefined limiting expressions for the cases of a flat continuous surface and for a surface of discrete atomic cores. An analysis of the temperature dependence of the observed scattered intensities indicates that these liquid metal surfaces are neither a flat continuum nor a collection of discrete scattering centers, but rather a surface with an intermediate corrugation roughness. The remainder of this paper is organized as follows. In Sec. II, which is the next section, we develop the theory. An extensive comparison with the experimental data is presented in Sec. III, and a discussion of the results and concluding remarks are given in Sec. IV. II. THEORY In the classical limit, the single-collision scattering of an atomic-like projectile from a surface at temperature T S,in which the collision time is short compared to a vibrational period, is completely solved in terms of closed-form analytic expressions. We express these here in terms of the differential reflection coefficient dr (1) /d f de f, which gives the J. Chem. Phys. 107 (5), 1 August /97/107(5)/1655/9/$ American Institute of Physics 1655

2 1656 A. Muis and J. R. Manson: Rare gas scattering from molten metals fraction of particles scattered into the small solid angle d f and into the small energy interval de f centered at the final energy E f. Single collision scattering of an atomic projectile from a collection of discrete scattering centers is expressed as the following: dr 1 m2 1/2 p f d f de f p fi iz k B T S E 0 exp E fe i E 0 2 4k B T S E 0, 1 where p q is the momentum of a particle in state q which we will often write as p q (P q,p qz ) with components parallel P q and perpendicular p qz to the surface, fi 2 is the form factor of the scattering center, m is the projectile mass, M c is the crystal atom mass, T S is the surface temperature, and E 0 (p f p i ) 2 /2M c is the binary collision recoil energy. Although Eq. 1 appears Gaussian-type in energy transfer E f E i, the momentum-dependent prefactors and the energy dependence of E 0 can give rise to highly asymmetric peak shapes. For example, the intensity exists only for E f 0 and vanishes when E f 0. A second example arises under conditions in which E f E i for which the Gaussian-type function goes to a decreasing exponential form in E f similar to the exponential in the Maxwell Boltzmann distribution. 12 However, in spite of the fact that the differential reflection coefficient has an exponential decay in this limit, it is not an equilibrium distribution because Eq. 1 describes single collision processes only. A Maxwellian thermal distribution would arise after the atoms become trapped for a substantial time during which they make many collisions with the surface and are eventually desorbed after coming into equilibrium with the substrate. There is a somewhat different classical expression for the single collision scattering from a surface that can be considered as a flat continuous barrier: dr 1 d f de f 2 m2 v 2 R p f p iz S fi u.c. 3/2 k B T S E 0 exp E fe i E 0 2 2v 2 R P 2 4k B T S E 0, 2 where P is the parallel momentum exchange P P f P i, S u.c. is the area of a surface unit cell, and v R is a weighted average of sound velocities parallel to the surface 13 and the term in the Gaussian-type exponent involving v R and P arises from scattering from vibrational correlations at the surface. Typically, v R is expected to be of the order of the Rayleigh velocity of the solid. 13 Equation 1 applies to high-energy inelastic neutron scattering 9 as well as to ion scattering, 16 while Eq. 2 has been shown to describe inelastic He-surface scattering at energies above 0.1 ev and for temperatures above room temperature. 17 In this latter case the flat continuous surface is ascribed to the locus of classical turning points due to the Pauli exchange repulsion, with the low-density surface electron distribution extending outward from the surface. The two cases of classical scattering in Eqs. 1 and 2 can be regarded as the extreme limits of a flat uncorrugated surface 2, and that of a highly corrugated surface 1, in which the classical turning point shrinks to a small discrete surface around the cores of the target atoms. Between the two extreme limits one can obtain a continuous range of functional dependence, determined by the strength of the surface corrugation roughness, which bridges the behavior of Eqs. 1 and 2. 8 For incident conditions in which the differential reflection coefficient is nearly Gaussian in shape high incident energies and small mass ratios, m/m c 1, the characteristics of the scattered intensity are straightforward to describe. In this case the calculated peak position of Eq. 1 is very nearly given by the zero of the argument of the exponential, E f E i E 0. This is the classical recoil expression describing the situation in which the impulse momentum is deposited in a single stationary crystal atom. This can be expressed in terms of the total scattering angle f i, where the incident angle i and the final angle f are measured from the surface normal, in the Baule form as E f f ()E i, where f 12 sin 2 cos The same result holds for Eq. 2 when the term in P 2 in the exponent is small, which is usually the case when V R 1000 m/s. The intensity at the point of most probable energy transfer is dictated by the prefactor envelope function and its value goes, for the discrete case of Eq. 1, as 1 I MAX k B T S E 0 1/2, 4 and for the continuum case of Eq. 2 goes as 1 I MAX k B T S E 0 3/2. 5 The characteristic power law exponent provides a clear signature difference between the two extreme limits. The width of the energy distribution, as expressed in terms of the mean square deviation from the most probable energy, is given to a good approximation by 18 E 2 2gE i k B T S, 6 where g TA g 1 cos /f, 7 2 and where g TA 1 f 2f cos, 8 is the value taken by g() in the trajectory approximation TA. The trajectory approximation is obtained upon assuming that the recoil energy shift E 0 appearing in the numerator of the argument of the exponential in Eq. 1 is constant, in which case E 2 2 E 0 k B T S 2 gta ()E i k B T S. For large scattering angles such as in the surface backscattering

3 A. Muis and J. R. Manson: Rare gas scattering from molten metals 1657 experiments discussed here, the narrowing of the intensity peak by the energy dependence of E 0 can be quite significant and the TA may be a poor approximation. 19 Similar arguments hold for the peak widths derived from Eq. 2 if v R is small. If v R is significant then Eq. 2 gives a peak that is further narrowed by the Gaussian-type term in the parallel momentum transfer P. Another approximation that has often been used in surface scattering is the hard cubes model, which is based on the assumption that parallel momentum is conserved in the collision process, i.e., P P f P i The hard cubes condition is obtained from Eq. 2 in the limit v R. As has been noted previously, the hard cubes model is of limited usefulness in atom-surface scattering under conditions in which the energy transfer can be a substantial fraction of the incident energy. 21 When the energy transfer is large the momentum exchanges are also large in both the perpendicular and parallel directions, as is borne out in the explicit comparisons with experiment presented below. The single scattering equation 1, because it basically describes binary collisions with initially vibrating scattering centers in a thermal bath at temperature T S, has no adjustable parameters. Similarly, Eq. 2 depends on the single parameter v R, which, in principle, can be calculated if the phonon spectral density is known. 13 It is now of interest to consider multiple collisions involving successive hits on different target atoms. Multiple scattering is most readily described as a convolution of successive single-particle collisions. Through single plus double scattering terms, the multiple collision differential reflection coefficient is expressed explicitly as dr M p f,p i dr 1 0 p f,p i d f de f d f de f N n1 0 deq n d q dr 1 n p f,p q dr 1 0 p q,p i, d f de f d q de q where the summation runs over the set of target atoms with which second collisions can occur, n is the solid angle subtended by the classical cross section of the nth atom, and the single scattering differential reflection coefficients are normalized to unity. Higher-order multiple scattering terms can readily be added onto Eq. 9 as multiple convolutions of the single scattering differential reflection coefficient. For example, the next term in Eq. 9, the triple scattering, would be the double convolution of three such terms: dr 3 p f,p i d f de f N n1 N n1 0 deq dr 1 n p f,p q d q n d f de f 0 deq n d q 9 dr 1 p n q,p q dr 1 0 p q,p i. 10 d q de q d q de q The multiple convolution form of the higher-order scattering contributions in Eqs. 9 and 10 would imply that the temperature and energy dependence of these terms could under some initial conditions be different from that of the single scattering contribution. However, under conditions in which the single scattering intensities are very nearly Gaussian in shape, the functional form of the temperature dependence of the intensity for the multiple scattering will be approximately the same as for the single scattering peaks, as given in Eqs. 4 or 5. The width of the multiple scattering peaks will also have approximately the same temperature dependence as in 6, because of the property that the square width of two convoluted Gaussian functions is the sum of the squares of the two component widths. This will hold true to a good approximation for both forms of the single scattering intensity shown in Eqs. 1 and 2. For the calculations that are compared with experimental data in Sec. III below, we have used Eq. 1 for the single scattering, Eq. 9 for calculating the double scattering contributions, and have ignored triple- and higher-order scattering terms. Since the solid angles subtended by the target atoms involved in the second collisions are usually small, we assume that the angular dependence of the intermediate differential reflection coefficients can be ignored, and the angular integration is replaced by a multiplicative factor of n. In the cases at hand, we have checked the results against the full three-dimensional integral, and the comparisons are quite favorable. Regardless of this simplifying approximation, the multiple scattering model depends on only a single physical parameter, the classical cross section of the target atom, which we usually express in terms of /4, the fractional solid angle subtended by the cross section at the nearest neighbor distance. Also, we take the form factor fi 2 to be a constant, which is appropriate for classical hard sphere scattering. In order to simulate the double scattering trajectories in the liquid metals examined here, we have assumed that the initial scattering center is surrounded by six similar surface atoms in a symmetrical hexagonal pattern in the surface plane, and additionally we average over all azimuthal orientations of the six-atom configuration. The mean interparticle distances do not need to be specified since the only parameter needed is the solid angle subtended by the cross section of the second atoms, n. Thus, the differential reflection coefficient that we use for the comparisons in Sec. III below is dr M p f,p i dr 1 0 p f,p i d f de f d f de f 6 n1 0 deq n d q dr 1 n p f,p q dr 1 0 p q,p i d f de f d f de, 11 f where the angular brackets signify the average over azimuthal angles of the surface atoms. For actual numerical comparison with the data, two additional modifications must be made to Eq. 11. First, the data were reported in time-of-flight t, and not final energy

4 1658 A. Muis and J. R. Manson: Rare gas scattering from molten metals TABLE I. Table of properties of the rare gases and metals Refs. 22,23. For the metals, D is the mean nearest distance in the liquid, T M is the melting temperature, D is the Debye temperature Ref. 24, and min /4 is the fractional solid angle subtended by a metal atom of atomic radius d at the mean interparticle spacing D calculated according to d 2 /4D 2. max /4 is similar to min /4, but calculated including the finite size of the projectile gas atom d G according to (d d G ) 2 /4D 2 using for d G the covalent radius of Ar. Mass amu Atomic radius Å Covalent radius Å D Å min / 4 max / 4 T M K D K Ne Ar Xe Ga In Bi FIG. 1. The time-of-flight spectrum for 0.44 ev Ar scattering from liquid In at T S 436 K. The experimental points are shown as circles. The vertical lines indicate the elastic position IB, the single scattering S, and the double forward scattering DF positions as calculated from binary scattering using Eq. 3. The theoretical curves are as follows: i short-dashed, single collisions only; ii long-dashed, only the double collisions averaged over all nearest neighbors using a value of /40.044; and iii dashdotted, single plus double collisions. The dotted curve is a flux-corrected Maxwell Boltzmann distribution. E f. 7 To obtain the differential reflection coefficient for TOF we must multiply Eq. 11 by the Jacobian de f /dt md 2 /t 3 where D is the TOF flight distance between the surface and the detector that is D 28.8 cm for these experiments. Second, the experiment uses a density detector that is velocity dependent. Thus, in order to compare directly with the detected count rate, Eq. 11 must also be divided by the final velocity, which is equivalent to multiplying by a factor of the flight time t. III. COMPARISON WITH EXPERIMENT In this section we analyze the extensive and carefully measured data of Ronk et al. for rare gases scattered by liquid metal surfaces using the theory developed in Sec. II. For all measurements the incident beam and the detector direction were in the plane of the surface normal. The incident beam and detector direction each made an angle of 55 with the normal, which gives a total scattering angle between the incident and final momentum vectors of 70. In Fig. 1 we show the data and calculated results for a monoenergetic beam of Ar with an energy of 42 kj/mol 0.44 ev incident on a surface of molten In at a temperature of 436 K The measured intensity as a function time of flight is shown as data points, and only a reduced number of measured points are shown. The data exhibits a smooth, broad, Gaussianlike peak with a long tail at large TOF times low energies and a short tail at small times high energies. Three vertical lines are drawn through the data. The vertical marked IB is the TOF time for the specularly reflected incident beam, i.e., for elastically scattered particles that have neither lost nor gained energy. The vertical marked S is the expected position for energy loss produced by binary scattering from a single stationary surface atom as given by the Baule expression of Eq. 3. In order to indicate the effects of multiple scattering, the vertical marked DF is the expected TOF position for in-plane double forward scattering off of two initially stationary surface atoms. This double forward scattering trajectory consists of an initial collision scattering the incident projectile parallel to the surface while still in the scattering plane, and subsequent scattering off of a second surface atom into the detector. The double forward scattering energy is calculated by double application of the Baule expression of Eq. 3, and it appears at higher energy than the single scattering because each of the two scattering events has a small scattering angle of 35, whereas the single scattering peak has a total scattering angle twice as big. The curves show the calculations, with the short dashed curve giving the results for single scattering, as calculated from Eq. 1. The long dashed curve is the calculated contribution arising from the double scattering with a hexagonal ring of six surface atoms positioned about the central scattering center and averaged over all azimuthal angles in the plane of the surface, as described above in connection with Eq. 11. The sum of both single and multiple scattering contributions is the dash dotted curve. Clearly the majority of the incident Ar is scattered with an energy loss, although a small fraction gains energy from the thermally vibrating surface, as can be seen from a comparison of the data with the IB line. The calculated most probable energy for single scattering is at a slightly higher energy than that predicted by the simple binary formula of Eq. 3, while the double scattering peaks at an energy very near to the simple double forward prediction. The total calculated single plus double scattering intensity matches the data very well and even predicts the very long time tail at low energies. The relative importance of the double to single scattering contribution is governed by the parameter /4, which was chosen to be in this case. This value compares very favorably with the simplest estimate given by the square of the ratio of the In covalent radius to the average interparticle spacing of liquid In see Table I given by (1.44/3.14) 2 /

5 A. Muis and J. R. Manson: Rare gas scattering from molten metals 1659 FIG. 2. A series of time-of-flight spectra for a 0.95 ev incident Ar beam on the three liquid metals Ga and In at T S 436 K and Bi at T S 553 K. The experimental points are shown as circles Ref. 7. The theoretical curves are as follow: i short-dashed, single collisions only; ii long-dashed, only double collisions averaged over all nearest neighbors; and iii dash dotted, single plus double collisions. For Ar/Ga scattering an effective target mass of 1.75 Ga masses was used. In all cases there is negligible evidence for appreciable capture and subsequent thermal desorption, as can be seen from the Maxwellian desorption curve dotted curve for the temperature T S, which peaks at much lower energies. FIG. 3. A TOF spectrum for Ne with incident energy E i 0.66 ev scattering from liquid In at T S 436 K. The experimental data and theoretical curves are shown in the same manner as in Fig. 1, except for the additional vertical line DB that shows the TOF position of the in-plane double backward scattering One might expect that a measurable fraction of the incident Ar atoms would penetrate into the liquid or be trapped at the surface long enough to be thermalized and subsequently leave the surface with an equilibrium distribution. The dotted line in Fig. 1 shows a Maxwell Boltzmann distribution flux corrected at the temperature of the surface. The Maxwellian curve peaks at very large times in a region where only a few percent of the particles are being scattered, but even at these low energies the combined single plus double scattering theory describes the intensity quite well. Thus, there is no indication of substantial trapping or capture of the Ar. It is interesting to note that the dominant contribution to the calculated intensity at large times comes almost entirely from the multiple scattering. The trajectories contributing to the low-energy tail are primarily double backward scattering events, i.e., those trajectories consisting of two successive large angle back-scattering events, as previously suggested. 7 Presented in Fig. 2 are a series of additional TOF spectra for Ar, now at an incident energy of 92 kj/mol 0.95 ev, scattering from all three molten metals for which measurements were made: Ga and In both at T S 436 K and Bi at T S 553 K. The experimental data and theoretical calculations are shown in the same manner as in Fig. 1. In all three cases the agreement of the multiple scattering calculation with the data is very good. In order to obtain agreement with the measured intensity in the long tail at large TOF, a somewhat stronger amount of multiple scattering had to be assumed which results in larger values of /4 than the value obtained for Fig. 1. However, except for the case of Bi, the chosen values of /4 are in near agreement with the maximum geometrical value max /4 obtained from the combined covalent radii of Ar and the surface atom as shown in Table I. The comparison with the Maxwellian curves shows that there is again very little indication of a significant amount of thermal diffuse Ar emission. For the case of Ar/Ga the initial calculation using the atomic mass of Ga for the surface produced a spectrum that was peaked at an energy loss that was larger than measured. This is evident from Fig. 5 below, which shows the most probable energy transfer as a function of incident energy. Since Ga is the lightest and smallest in size of the three metals studied, this was interpreted in terms of an effective mass in which the incoming Ar collides with more than one Ga atom. 7 The agreement for Ar/Ga shown in Fig. 2 was obtained using an effective mass that is 1.75 times the Ga mass. This larger effective mass is consistent with the proposal that the incident Ar collides with more than one Ga atom in a typical collision. 7 For the case of Ar scattering from Bi, the results of the calculations indicate that multiple collisions are more important than for the two other metals. Figure 3 shows a TOF spectrum for Ne scattering from liquid In at T S 436 K. The experimental points are shown as circles and the theoretical curves are the same as in Fig. 1. In this case we also are able to show with a vertical line marked DB the position of the double backward scattering trajectory that consists of two successive in-plane backscattering events, each with a scattering angle of 125. In this case the calculated total single plus double intensity peaks at the same position as the experiment, although the overall distribution is slightly narrower than the measured spectrum on the high-energy side. The long tail of intensity

6 1660 A. Muis and J. R. Manson: Rare gas scattering from molten metals FIG. 4. A TOF spectrum for Xe with incident energy E i 2.5 ev scattering from In at T S 436 K. /40.18 and the effective mass is M e 2.00M In. The experimental points and theoretical calculations are shown in the same manner as in Fig. 1, except that the Maxwell Boltzmann intensity has been reduced by a multiplicative factor of 0.2. The solid line shows the single scattering contribution calculated without a larger effective mass for the In and similarly for the DF position. on the low-energy side large TOF times is well explained by the double scattering contribution. The value of / is not significantly different from max / calculated from Table I. In particular, the absence of significant experimental intensity in the vicinity of the maximum in the Maxwellian distribution indicates that the thermal scattering is negligible. The case of Xe incident with energy 240 kj/mol 2.5 ev is shown in Fig. 4. The experimental data and theoretical calculations are again shown as in Fig. 1. However, this is a case in which the Xe projectile mass isotope averaged of 131 amu is larger than the In target mass of 115 amu. Thus, it is not possible to show the single scattering position S because a single binary collision with a stationary In atom cannot scatter Xe into the detector under these incident conditions. However, because of the initial thermal motion of the surface atoms, it is possible to scatter some Xe into the detector, and this contribution is shown in the solid line. Clearly this peaks at energies too small to agree with the measured TOF spectrum, and the addition of double scattering contributions, although giving intensity predominantly at higher energies, is not sufficient to bring the calculations into agreement with experiment. As for the case of Ar/Ga, this is interpreted as due to collective mass effects. The dash dotted curve in Fig. 4 is the total single plus double scattering intensity calculated with an effective mass of twice the atomic mass of In. Although the double scattering contribution is larger than the single scattering, indicating that multiple scattering is more important in this system, the peak position of the total calculated intensity now agrees well with the experimental data and the calculated curve is only slightly narrower than the measured width. Once again, there is no substantial indication for thermal diffuse scattering as FIG. 5. Position of the most probable energy transfer as a function of incident energy for Ar scattering from the three liquid metals: Ga and In at T S 436 K and Bi at T S 553 K. The theoretical calculations are shown by the curves as labeled. the Maxwellian curve peaks at a TOF above 800 s, where the measured and calculated intensity is negligible. A measure of the energy exchanged is shown in Fig. 5 which compares the measured and calculated most probable energy transfers as a function of incident energy for Ar scattering from all three liquid metals. What is plotted in Fig. 5 is E E ie MP, 12 E i E i where E MP is the most probable energy or maximum of the peak in the scattered intensity as a function of energy i.e., the peak in the differential reflection coefficient of Eq. 11. The experimental data are shown as points and the theoretical calculations, using the actual atomic masses of the surface atoms, are shown in the curves as marked. A value of /40.15 was used for all three metals, and this value is consistent with those obtained by the direct comparison with the TOF spectra at higher energy as in Fig. 2. The calculated results are weakly dependent on /4, and, in fact, calculations carried out wiith the parameter-free single scattering expression of Eq. 1 give very similar results. As is evident from Figs. 1 and 2 experiment and theory are in good agreement for Ar/Bi, and for Ar/In the discrepancy in the peak positions is small, as is evident from Figs. 1 and 2. For Ar scattering from the light mass and small size Ga, the most probable energy loss is measured to be less than what is calculated using the Ga atomic mass. As discussed in connection with Fig. 2, this has been interpreted as a collective mass effect, 7 and our calculations from Eq. 11 can be made to agree with the experimental data if we use an effective mass for the Ga of about 1.75 times its atomic mass. Figure 6 is similar to Fig. 5, except it is for the case of Ne atoms scattering from the three liquid metals. Clearly, as suggested by Fig. 3, the most probable energy loss measured for Ne/In agrees nicely with the theoretical predictions. For the case of Ne/Ga the measured most probable energy transfer is smaller than the theoretical predictions obtained using

7 A. Muis and J. R. Manson: Rare gas scattering from molten metals 1661 FIG. 6. Position of the most probable energy transfer as function of incident energy for Ne scattering from the three liquid metals: Ga and In at T S 436 K and Bi at T S 553 K. The experimental data points and theoretical calculations are as shown in Fig. 5. the actual atomic mass of Ga. This can again be interpreted in terms of scattering from an effective mass that is approximately 1.4 times as large as the Ga atomic mass. The case of Ne/Bi is more difficult to understand because the theoretically predicted most probable energy transfer is smaller than the measured values. If this is interpreted in terms of an effective mass model, it would require a smaller mass for Bi, which is approximately 60% of its atomic mass. The anomalous results for the Ne/Bi system may also be interpreted in terms of possible impurities on the surface. A small coverage of undetected light mass impurities, such as bismuth oxide, would lead to a shift in the TOF spectrum toward smaller final energies, together with a broadening of the peak. A repetition of the Ne/Bi experiments in the future will hopefully resolve this issue. The temperature dependence of the scattering data is considered in Fig. 7. Since the most characteristic signature in the temperature dependence of the theory of Eqs. 1 and 2 is the decreasing intensity with increasing temperature at the most probable energy, this is what is shown in Fig. 7 for Ar scattering from In and Ga. For the case of Ar/Ga the scattered intensity at the most probable energy transfer was measured from just above the melting point at T S 303 K to T S 460 K, while for Ar on In the measurements were made over a somewhat shorter range, from above the melting point at T S 430 to 525 K. Also shown in Fig. 7 with the short dashed and the dash dot ted curves are the respective T S 1/2 and T S 3/2 behaviors predicted by the discrete and continuum models of Eqs. 1 and 2. In both the case of Ar/Ga and Ar/In the intensity decreases with increasing T S, but does not agree with either the T S 1/2 or the T S 3/2 limiting cases. Instead the data seems to decrease in reasonable agreement with a T S 1 behavior. This discrepancy between theory and experiment would seem to indicate that the liquid metal target as seen by the rare gas projectiles is neither a collection of discrete scattering centers, nor is it the opposite extreme of a flat vibrating surface. Rather, the situation seems to be something intermediate in which the surface barrier felt by the rare gas projectiles is corrugated, but not corrugated so strongly as to appear to be a collection of discrete particles. Recent theoretical calculations have shown that the two expressions of Eqs. 1 and 2 can be regarded as limiting cases of strong and weak surface corrugation, respectively, and the temperature dependence varies smoothly between these two extreme limits as a function of the strength of the surface corrugation. 8 Thus, the temperature behavior of the measured data in Fig. 7 is consistent with the interpretation that the surface is neither flat nor discrete but, instead, has an interaction potential energy that is represented by a corrugated repulsive barrier. In passing, it should be noted that the temperature dependence data of Fig. 7 was originally reported in the form of the integrated intensity under the TOF curves, which was shown to fit well to a 1/T S 1/2 function. 7 If the differential reflection coefficient is a nearly Gaussian function in the final energy, as is the case for Eqs. 1 and 2 under the present experimental conditions, an integration over final energy at a fixed final angular position will reduce the prefactor by a factor of (k B T S E 0 ) 1/2. This implies that a most probable intensity that varies as T S 1 is consistent with the total integrated intensity varying as T S 1/2. IV. DISCUSSION AND CONCLUSIONS FIG. 7. Temperature dependence of the intensity at the most probable energy for Ar with incident energy E i 0.44 ev scattering from liquid Ga and In. The T 1/2 and T 3/2 curves are the two temperature dependences predicted by Eq. 1 and Eq. 2. The experimental points disagree with these two predictions and appear to follow more closely a 1/T dependence. In this work we have shown that the essential features of the scattering of rare gases with translational energies in the ev range from molten metals are explained with a classical scattering theory that includes multiple collisions with the surface atoms. The scattered intensity observed when a wellcollimated monoenergetic incident beam strikes the surface is a single broad scattered peak in the TOF energy-resolved spectrum. The maximum of this peak is at lower energy than the incident beam and there is a long tail at lower energies and a shorter tail at higher energies extending to energies significantly larger than the incident energy.

8 1662 A. Muis and J. R. Manson: Rare gas scattering from molten metals Perhaps the most interesting result is that, for cases in which the rare gas projectile is less massive than the liquid metal atom the position, width, shape, and asymmetry of the observed scattered peak is reproduced reasonably well by the simplest version of the classical theory, which is for a single binary scattering with a single initially vibrating liquid atom, a theory that contains no free parameters. With the addition of multiple scattering events the tails of the TOF spectra at large and small energies are well explained by the theory. For cases in which the projectile atom is more massive than the target atom, a single binary scattering event with an initially stationary metal atom cannot backscatter at scattering angles larger than the critical angle given by Eq. 3. Thus, for more massive rare gases such as Xe, we find the expected result that multiple scattering plays a much more important role in the observed backscattering intensity. The importance of multiple scattering events is evident in Xe scattering from In, and also in the scattering of Ar and Ne from Ga, which is the least massive of the target metals investigated. In all three of these combinations, the initial experimental investigations indicated a collective scattering process with the target atoms in which the incident projectile was scattering simultaneously with more than one target atom. 7 This collective effect is accounted for in the present theoretical analysis by assuming a larger effective target atom mass in order to obtain good agreement with the data. This larger mass was no greater than twice the atomic mass of the metal atoms. The theoretical scattering intensities of Eqs. 1 and 2 predict well-defined signatures for the temperature dependence of the scattering spectrum. The most evident signatures are that the widths of the scattered peaks have a square root dependence on T S, and the peak maximum the intensity at the most probable energy transfer decreases with a characteristic inverse power law on T S that depends on the corrugation strength of the surface barrier potential. The experimental measurements showed widths of the TOF spectra that increased with surface temperature in agreement with the T S prediction, but the range of temperatures over which the measurements were made was not sufficiently large to give a good test of this dependence. However, the range of temperatures was sufficient to check the power law dependence of the intensity at the most probable energy exchange, and this is shown above in Fig. 7. Previous work on He scattering from the Cu001 surface at temperatures above 300 K and energies above 0.1 ev produced broad classical intensity peaks whose T S dependence agreed with the T S 3/2 behavior predicted by a flat continuum surface of Eq ,25 On the other hand, Na ion scattering at energies of several hundred ev appears to give a scattering behavior that is consistent with the T 1/2 behavior expected from a target of discrete scattering centers, 16,19 and similar behavior is expected for other weakly interacting systems, such as in neutron scattering. 9 The experimental behavior for Ar scattering from both liquid In and Ga is consistent with neither of the two predicted T S dependencies. As seen in Fig. 7 the measurements are more consistent with a T 1 behavior. Such an intermediate temperature dependence indicates that these liquid surfaces do not present a flat repulsive barrier potential to the incoming rare gas but are strongly corrugated. However, this corrugation is not so strong that the surface appears as a collection of discrete scattering centers. 8 A very remarkable feature of the observed experimental TOF intensities is that there is very little intensity scattered at the thermal energies, where one would expect to observe diffuse scattering of trapped and equilibrated rare gas atoms. In fact, as shown in Figs. 1 4, in almost all cases what little intensity is observed in the neighborhood of the peak in the Maxwell Boltzmann distribution is well explained by the present multiphonon theory. Thus the present data indicates that at these temperatures and incident energies there is virtually no thermal diffuse scattering intensity. This interpretation is consistent with the above arguments about the nature of the interaction potential derived from the temperature dependence of the data, i.e., the surface potential apparent to the incident rare gases is a corrugated repulsive barrier that does not permit appreciable penetration and subsequent trapping of the projectiles. In conclusion, we have shown that binary collisions with the vibrating surface atoms, together with successive multiple collisions, can do a very good job of explaining the nature of the scattered intensity spectra for the recent measurements of rare gases striking liquid metal surfaces. The success of these initial calculations would indicate that more extensive measurements over a larger range of incident angles and energies, as well as over a larger range of surface temperatures, will lead to more detailed theoretical comparisons which will reveal much more sophisticated information about the nature of the interaction potential. For example, the relation between the temperature dependence of the intensity and the surface corrugation can be explored by enhancing the incident momentum normal to the surface, either by increasing the incident energy or by looking at smaller incident and detection angles. On the other hand, experiments at glancing incident angles, in which the momentum perpendicular to the surface is small, should be very sensitive to the Van der Waals attractive potential and the effects of the adsorption well in the potential. ACKNOWLEDGMENTS We would like to thank G. Nathanson and W. Ronk for very helpful discussions during the course of this work. This work was supported by the National Science Foundation under Grant NO. DMR J. P. Toennies, Springer Series in Surface Sciences, edited by W. Kress and F. W. de Wette Springer, Heidelberg, 1991, Vol. 21, p J. A. Barker and C. T. Rettner, J. Chem. Phys. 97, J. A. Barker and D. J. Auerbach, Surf. Sci. Rep. 4, E. Hulpke, in Helium Atom Scattering, Springer Series in Surface Sciences Springer-Verlag, Berlin, 1992, Vol F. C. Hurlbut and D. E. Beck, U. C. Eng. Proj. Report No. HE , University of California, A. J. Kenyon, A. J. McCaffery, C. M. Quintella, and M. D. Zidan, Chem. Phys. Lett. 190, ; J. Chem. Soc. Faraday Trans. 89, W. R. Ronk, D. V. Kowalski, M. Manning, and G. M. Nathanson, J. Chem. Phys. 104, J. R. Manson unpublished.

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