Diffraction of Neutral Atoms and Molecules from Crystalline Surfaces

ADVANCES IN ELECTRONICS AND ELECTRON PHYSICS. VOL . 60

Diffraction of Neutral Atoms and Molecules from Crystalline Surfaces G . BOATO

AND

P. CANTINI

Istituto di Scienze Fisiche and GNSM-CNR Universita di Genova Genoa. Italy

I . Introduction ............................................................ I1. General Outline ......................................................... A . Brief Review of Experimental Techniques .............................. B . Elastic Diffraction ................................................... C . Inelastic Diffraction .................................................. D . Rotational Diffraction ................................................ 111. Quantum Theory of Atom-Surface Scattering .............................. A . Static Potentials and Elastic Diffraction Theories ........................ B. Vibrating Surfaces and Inelastic Scattering ............................. IV . Structural Information from Elastic Diffraction ............................. A . Simply Corrugated Surfaces........................................... B. Reconstructed and Periodically Deformed Surfaces ...................... C . Adsorbate-Covered Surfaces .......................................... V . Information on the Surface Potential Well ................................. A . General ............................................................. B . The He-Graphite System............................................. VI . Information on Surface Lattice Dynamics.................................. A . Debye-Waller Factor ................................................ B. Structures in the Angular Distribution .................................. C . Energy-Loss Measurements ........................................... VII . Conclusions ............................................................ References .............................................................

95 98 98 100 106 109 111 112 121 129 130 132 135 138 138 139 146 146 148 152 155 156

I . INTRODUCTION The scattering of neutral atomic beams. particularly helium. at thermal energies from crystalline surfaces has become a very practical and useful tool for studying both the surface structure and lattice vibrations. The diffraction of thermal atoms differs from low-energy electron diffraction (LEED) and other surface-scattering techniques essentially by the fact that in the first case the very top layer of the surface is explored. The 95

Copyright @ 1983 by Academic Press. Inc. All rights of rcproduction.in any form reserved. ISBN 0.12-01466&6

G. BOAT0 AND P. CANTINI

96

Fig. 7. Die gestrichelten Kurvenstiicke zeigen den EinfluB einer kleinen Verdrehung (- 20) des Kristalls in seiner Ebene.

\, ,, , \\

..I,'

,,'

_,

FIG.1 . Reproduction of some of Frisch and Stem (1933) illustrations of selective adsorption experiments: (a) diffraction figure; (b) part of the apparatus; (c) experimental geometry.

reason for this is the impenetrability of atoms, or, in more physical terms, the repulsion originating from the Pauli exclusion principle. The first observation of atom diffraction from surfaces is due to Estermann and Stem (1930), who had the particular objective of demonstrating the wave nature of atomic particles. However, Stem and collaborators

ATOM DIFFRACTION FROM SURFACES

97

carried out in Hamburg a series of very ingenious experiments that were the prelude to the present atom-surface diffraction method. They studied different ionic crystal surfaces, built a double-crystal diffractomer (Estermann et al., 1931), and observed the phenomenon of selective adsorption, later explained by Lennard-Jones and Devonshire (1937). For historical record and respectful memory, some of the results obtained by Frisch and Stem (1933) on selective adsorption are shown in Fig. l a (a photostatic reproduction of one of Stem’s figures). Similarly, Figs. l b and l c show the apparatus and the geometry used in this outstanding set of experiments. Unfortunately in 1933 Stem was forced to leave Germany and never resumed work in this field. About 35 years elapsed before significant progress was accomplished. This happened at the end of the 1960s because of two main reasons (1) the development of the supersonic beams and (2) the availability of controlled surfaces by the use of ultrahigh vacuum. Further motivatim was given by the theoretical paper of Cabrera et al. (1970), who first used a nonperturbative approach to the theory of atom diffraction. Interest in the gas-surface interaction was also aroused by the growth of rarefied-gas dynamics, in connection with space aeronautical problems. Important experiments were carried out by Smith et al. (1969), Fisher et al. (1969), and Subbarao and Miller (1969). At the beginning of the 1970s, the first detailed diffraction experiments were carried out with the use of nearly monochromatic supersonic beams; among these the experiments of Williams (1971a,b) are worth mentioning, since they first showed fine structures in the diffraction pattern and, from the technical point of view, made use of cryogenic vacuum. The atom diffraction technique was then ripe for a gradual development. The first meeting where the subject was thoroughly discussed and a new look taken at the problem was the LVIII Course of the E. Fermi International School (Varenna) held in 1973 (proceedings published by Editrice Compositori, Bologna, Italy, 1974). Subsequently, rapid progress involving an increasing number of workers was made. The purpose of the present review is to give an overview of the subject as it stands now, to discuss the main points in both theory and experiments, and to indicate possible future trends. This review does not pretend to be complete: we regret that some important work may not be sufficiently discussed or even mentioned. In return, we hope to give the reader a critical discussion of the principal information one can obtain from atom diffraction experiments. Review articles on the subject that appeared and the topics covered by these articles are mentioned below; these topics are not given wide space in this review. Cole and Frank1 (1978) and Armand and Lapujoulade (1979) wrote brief general reviews, where details on techniques and

98

G. BOAT0 AND P. CANTINI

theories may be found. Hoinkes (1980) treated thoroughly the gassurface interaction potential as determined by atom diffraction experiments. Engel and Rieder (1982) wrote a very detailed review on structural studies of surfaces, where much information and a complete bibliography can be found. Finally, Cardillo (1981a) gave a general survey of recent experimental work on all types of gas-surface interactions studied by means of molecular beam techniques. The present article starts with a qualitative treatment of the experimental methodology and a simplified explaination of the physical phenomena involved in atom diffraction (Section 11). A review of the theoretical methods useful for understanding and interpreting the experimental data is then given (Section 111). In Section IV a concise account of the structural information obtained by elastic diffraction is given, and in Section V the problem of determining the atom-surface potential from experiment is treated. The information obtained on surface lattice dynamics is discussed in Section VI, and finally, the conclusions and perspectives are reported in Section VII. 11. GENERAL OUTLINE

A . Brief Review of Experimental Techniques The apparatuses presently used in different laboratories for the study of molecular beam diffraction do not differ much in their main constituents. The production of neutral beams is based on the supersonic expansion from a high-pressure region (5-100 atm) through a small nozzle (50-5 pm) into a vacuum, the source being held at temperatures ranging from 10 to about lo00 K. The central part of the beam is skimmed and angularly defined by means of suitable collimators. As a result, one disposes of very narrow beams of neutral atoms and molecules, which are intense (- 5 x 1019 particles/steradians sec) and nearly monochromatic (velocity spread, Av/v = 1-5%). Details of molecular beam techniques and of scattering apparatuses for surface studies can be found by the reader in review articles by Cole and Frank1 (1978) and Engel and Rieder (1982), and they are very briefly discussed in this article. Special sources must be used for the production of beams of radicals or excited molecules. An important source of this kind is that of atomic hydrogen, which was extensively used for diffraction studies by the Erlangen group (Hoinkes el al., 1972a) and further developed in Genoa. The effusive beam is produced by a radiofrequency discharge and velocity selected by a magnetic hexapole (Barcellona et al.,

ATOM DIFFRACTION FROM SURFACES

99

1973) or by a rotating slotted disk (Finzel er al., 1975). With the source at room temperature, the velocity spread can reach 10% and the intensity can be 10l6molecules/steradians sec. The detection of molecular beams has also reached a high degree of sophistication. Although stagnation detectors (like that used by Stern) are still in use, electron bombardment followed by mass analysis of the produced ions is the most widespread method of detecting neutrals. A sensitivity of times the incident beam intensity has been reached. This technique is easily combined with time-of-flight measurements. Another detection method which proved to be effective and equally sensitive is based on the measurement of the energy delivered to a bolometric device-usually a low-temperature doped-semiconductor crystal-during the accomodation of the neutral particles on the sensitive surface. Special detectors must be used for radicals or metastable particles. In order to improve the signal-to-noise ratio, chopped beams and lock-in techniques are widely used. The scattering region is enclosed in a ultrahigh vacuum chamber under dynamical conditions. The vacuum is mantained with conventional techniques. Of great advantage has proved the use of cryogenic vacuum, particularly for the study of surfaces in the low-temperature region. As shown below, the use of cold surfaces may be determinant both for structural studies (the elastic peaks are attenuated by the presence of a temperature-dependent Debye- Waller factor) and for inelastic phenomena, when one-phonon processes are the main object of investigation. The crystal surface is cleaned or prepared and subsequently characterized in situ by the standard techniques used in modern surface studies [cleavage, heating, ion bombardment, annealing, etc., for surface preparation; LEED, Auger electron spectroscopy (AES), secondary-ion mass spectrometry (SIMS), etc., for surface cleanliness, analysis, and characterization]. The intensity and sharpness of diffraction peaks are very sensitive to the quality of the surface, so that atomic scattering itself can be advantageously used to obtain a check of the surface perfection. The surface is mounted on more or less elaborate crystal holders, possibly rotatable in all spatial directions. In fact, precision measurements need accurate determination of both the incident angles (polar and azimuthal) of the beam with the surface net and the outward scattering angle. Collimation problems are therefore important and usually delicate. For these and other experimental problems the reader is again referred to the above-cited review articles. We may conclude by saying that even if atom-surface scattering techniques are sophisticated and only partially commercialized, they are not particularly difficult to be installed and utilized.

100

G . BOAT0 AND P. CANTINI

B . Elastic Diffraction When a beam of particles having a mean wave vector k,, (and an energy E, = h2ki/2m) impinges on a stationary periodic surface (lattice vibrations are for the moment discarded), a discrete set of diffraction peaks is backscattered from the surface under the condition that the wavelength h = 2rr/ko is comparable with the surface lattice parameter a. Under these circumstances, there is no energy exchange between the particles and the surface. Because of the surface periodicity, the particle-surface potential can be expanded in a two-dimensional ( 2 D ) Fourier series V&) exp(iG R)

V(r) =

(1)

C

where r(R, z) defines the position of the particle, R is the projection of r on the surface, and the z direction is assumed to be the outward normal to the surface; G(m, n) represents a surface reciprocal lattice vector defined by G = ma* + nb* (2) where m and n are integers, and a* and b* are related to the surface unit vectors a and b by a . a* = b . b* = 2 r (3) a . b* = a * . b = 0 (4) The condition for having non-zero-order diffraction peaks is that some of the V , terms are different from zero. Otherwise, only the specular peak (G = 0) survives. The kinematic conditions for having elastic diffraction are

+

(5)

&=K,+G

(6)

ki = kg = k& (conservation of energy) and

(conservation of parallel momentum). Here kC(& + G, kGz) is the final wave vector of the particle, labeled by a reciprocal vector G. Expression (6) is the Bragg condition for 2 D diffraction. What we have stated until now is equally valid for neutral particles and low-energy electrons. Both techniques are in fact able to give quite similar information on the surface-lattice symmetry. However, low-energy electrons (Eo = 2-500 eV) explore a few atomic surface layers, whereas neutral atoms (E, = 4-200 meV) probe only the uppermost Dart of the surface and are nondestructive in character. The two techniques are therefore complementary for structural information.

ATOM DIFFRACTION FROM SURFACES 00

. o.p . .

ii i o

ii

i 2

'0,

0:

02.

1 ;

1p

:1

2 1.

2T.

2p

2 ;

2.2

101

' L i F (001)

22

20

I

FIG.2. Diffraction pattern of He scattered from LiF(001).

A typical experimental diffraction pattern is shown in Fig. 2 for the system He-LiF(001). In this example the incident beam (E,, = 63 meV, A = 0.58 A) was normal to the surface and the crystal was at 10 K. Only the diffraction peaks in an octant are shown; the others can be derived by the square symmetry of the surface unit cell. The peak widths are determined by the angular resolution and the velocity distribution of the beam. The diffraction-peak intensities are related to the atom-surface potential V(r). It is then important to give a qualitative description of this interaction potential. The discussion may remain specialized to systems showing effective diffraction, namely, to the case of light atoms (A large) impinging on chemically inactive periodic surfaces. The appropriate physisorption potentials are not exactly known, but they exhibit some common features, which are of great importance in understanding what kind of information on the surface structure a diffraction experiment can give. As an example, we consider the potential of the system He-graphite, one of the most accurately known. The methods used to derive it are extensively discussed in Section V,B,3. The potential as a function of z for a few positions R in the surface unit cell is shown in Fig. 3. It is seen that at short distances, very strong repulsion is present, originating from the mutual overlap of outer electron wave functions of the incident particle with those of surface atoms. The z dependence is nearly exponential and has very short range. On the other hand, at large

102

G. BOAT0 AND P. CANTINI

-201 1.0

I

I

2.0

I

1

3.0

I

1

4.0

I

I

5.0

z 181

FIG.3. Gas-surface potential of He-graphite.

distances, attraction due to London dispersion forces is prominent. According to the general treatment of Lifshitz (1956), the asymptotic form of the potential for large z is given by

V(z) = -cz-3

(7)

with C related to the dynamical polarizability of the atom and to the dielectric function of the solid (see Hoinkes, 1980). At medium distances, around the equilibrium position, there is the well region, where the potential is usually less accurately known. Figure 3 shows that for z values just above the equilibrium distance the incident atom sees an attractive potential which does not depend on R and gives only a minor contribution to the diffraction figure. In contrast, the potential at short distances is laterally dependent and periodic; of a similar nature is the region near the minimum. The potential region at short distances is the most important for diffraction. The effects of the periodically lateral dependence can also be represented by the Fourier components V,(z) of the interaction potential [see Eq. (l)]. In the long-range

ATOM DIFFRACTION FROM SURFACES

103

attractive region and for G # 0, the Vc's are in fact completely negligible, but they strongly increase when z becomes lower than the equilibrium distance (for example, see Fig. 22 later). A very interesting quantity is the zero Fourier component, given by

V(R, z) d2R

V,(z) = d - l / U.C.

where d is the area of unit cell (u.c.); Vo represents the laterally averaged gas-surface potential and can be used as a first approximation to V(r) when the other Fourier components are small. The description of V(r) through its Fourier components is fully exploited in the next sections. The above-illustrated trend of the gas-surface potential is common to all systems of interest for atom diffraction. The modulation of the repulsive potential may be more or less strong, and in a parallel manner the well depth may change with R (the two effects are described by the number and strength of sizable Fourier components), but two features remain: the repulsive potential is very steep and modulated, whereas the attractive region is long range and nonperiodic. The average well depth D , namely that associated with Vo(z),can range from 5 to 20 meV for He and rise to about 100 meV for H and H,, but its effect can be viewed, at least in a first approximation, as an acceleration of the particle before a hard collision against the steep repulsive periodic potential takes place. From the above analysis, we can understand the physical reasons for the commonly adopted approximation of representing the diffractive potential with a corrugated hard-wall model (see Section III,A,2). In all cases the observed diffraction pattern, namely, the intensity distribution among diffracted beams, essentially reflects the periodic shape (or corrugation) of the repulsive wall. The corrugation can be visualized by constructing the equipotential surfaces at the incident energies (25- 100 meV); in other words, the periodic repulsive wall is the locus of classical turning points of the particle scattered by the potential V(r). As a first example of the above discussion, we return to Fig. 2, showing the case of He-LiF. The diffracted peak intensities display a peculiar trend: their envelope as a function of the scattering angle Of has a maximum around the second-order peaks, i.e., at Of = 35-40'. This effect is called quantum surface rainbow, by analogy with the classical surface rainbow observed when rays are scattered by a periodic reflecting surface. The classical surface rainbow for atom scattering was discussed by McClure (1972); the theory of quantum rainbow is expounded in Section 111,A,2. A second example is shown in Fig. 4. Here the diffraction pattern is more complicated, due to the complex structure of layered compounds

104

G . BOAT0 AND P. CANTINI

io!I

He/2 H -Ta Se 2

............ ........... 0

0

.

.

o

$0.

o

* *

01

00’

2c

5c

.-A0

017

Reciorocal lattice

I .......’ -5OO -40°

..,....: ... -30° -

I

30O

FIG.4. Diffraction pattern of 2H-TaSeZ. A superstructure caused by charge-density waves is present.

such as 2H-TaSe2. The intense integer-order peaks are the result of the corrugation of the top layer of selenium atoms (having trigonal symmetry owing to the metal-atom layer below). The small fractional order peaks are due to a superstructure brought about by the presence of chargedensity waves (see Section IV,C), which slightly deform the top layer with a periodicity three times larger than the principal one. In other words, a (3 x 3) commensurate superstructure, stable below 90 K, is detected (P. Cantini and R. Colella, unpublished results). The previous discussion is a first qualitative approach to understand the diffracted intensities. There is, however, another effect of the attractive well that is relevant and, under special circumstances, can become dominant and eventually upset the above picture. This effect was observed by Frisch and Stern (1933) in the first experiments on atom diffraction and was explained by Lennard-Jones and Devonshire (1937) as the result of resonant transitions into the bound states of the attractive well. It was named “selective adsorption” by the latter authors, and this evocative name is still widely used today. In brief, under given incident conditions, the particle can make a resonant transition into a state bound to the surface insofar as the z direction is concerned, while traveling with in-

ATOM DIFFRACTION FROM SURFACES

105

creased kinetic energy parallel to it. The kinematic condition controlling a bound-state resonance (BSR)is k2Gz - k2 - [K,+ G(m, n)I2 = 2mej/h2 (9) where ej(< 0) is one of the energy level of the average potential V,(z) and (h2/2m)(& + G)2represents the parallel kinetic energy. From Eq. (9) it is seen that a BSR is labeled by three indices: ( m ,n), which define G, and j . The kinematics of a BSR process is illustrated graphically in Fig. 5 , using the K plane. The total energy hzki/2m is positive, so the particle lives temporarily in this metastable state; it ultimately leaves the surface via an elastic transition in one of the diffraction channels or it will suffer an inelastic process. As a result, the distribution of intensity among diffraction peaks can be, at resonance, significantly different than out of resonance. Various aspects of the selective adsorption and its importance for the experimental determination of the shape of the potential in the region of the well are discussed at length in later sections. We want only to mention here how the selective adsorption structures are experimentally observed. One usually works at constant energy E, = h2kg/2m,whereas one of the angles (0, , 4,) that the incident wave vector ko makes with the surface net is continuously changed. When the condiction (9) is met, an anomaly may be observed in both the specular and diffracted beams; the resonant structure has the form of a minimum (more often), a maximum, or a

Example: j =1

.' FIG.5. Kinematics of a BSR in the K plane.

106

L

0.2

Of 2

I 10

01

a (0)

(b)

FIG.6 . Bound-state resonance structures in the specular intensity plotted (a) as a function of +o at constant Ba, and (b) as a function of Bo at constant + o . [From Frankl et a / . (1978).]

more complex shape. This is shown for the system He-LiF(001) in Fig. 6a and b where either the polar O0 or the azimuthal angle 4owas changed. The angular location of minima and maxima satisfies Eq. (9) with excellent agreement. Rewriting Eq. (9) in the f7rm

[KO

+ G(m, n)]'

=

(2m/h2)(Eo-

~ j )

(10)

one deduces that the values of & (the projection of ko on the surface) leading to each bound state ej (
Owing to thermal motions, during a scattering process surface-atom displacements take place so that a static gas-surface potential is unrealis-

107

ATOM DIFFRACTION FROM SURFACES Ky

(i-') ' ..

'\.* '\

\

--

*

\

... . \ \\-

\

\

\\

-

.€0

\

01-threshold

-A;,

-.El '\

..

- €2.. -::.. :

..

... ..

. .. .. .. .. €3: .:. . - .. . < :. .. .. K,

(i-' )

I

I

I

4

5

6

FIG.7. Location of observed BSRs in the K plane: k,, NaF(001) [From Meyers and Frank1 (1975).]

=

6.10 k

l ; He scattered from

tic. In fact, lattice vibrations are present not only at finite temperature but even at 0 K by virtue of the so-called zero-point motions. The result is that a true potential must be time dependent and that in real scattering experiments, exchange of energy between particle and surface is always possible, in the form of created and annihilated phonons. At surface temperatures T, lower than a characteristic temperature, the inelastic efusually identified as the surface Debye temperature 8,, fects are sufficiently weak to leave almost unaffected the picture given previously in Section I1,B. Some intensity is subtracted from the elastic peaks and scattered in a wide solid angle, but no change of the intensity ratios among diffraction peak intensities is observed. This fact is operatively described by the introduction of a temperature-dependent corrective factor, smaller than unity, which multiplies the intensity of diffraction peaks. The experimentally measured diffraction probabilities can then written as ( PG)expt = ( PG)el exp( - W G ) (1 1) where (PdeXpt is the ratio of the observed peak intensity to the incident

108

G . BOAT0 AND P. CANTINI

beam intensity, whereas (P&, is the same quantity to be expected for a purely elastic process (static potential). The quantity exp(- 2W,) is called the Debye-Waller factor by analogy with what is found in X-ray, neutron, or electron diffraction; however, in these cases, Eq. (11) is easily justified and W, is found to be the thermal average W,

=

:((Ak

*

u)~)

(12)

- k, and u is the thermal atom diswhere Ak is the scattering vector placement. In molecular beam scattering, Eq. (12) is only approximately valid and even unapplicable, as we discuss in detail in Sections III,B,l and VI,A. The existing experiments essentially show that the thermal attenuation of the diffracted beams is approximately exponential in character at temperi.e., P , exp(A/T,). atures around 8,, Under the condition that 0 < T, << 8,,the intensity subtracted to elastic peaks appears as tails to these peaks and as a weak background between them (see Figs. 2 and 4). This inelastically scattered intensity contains in a concealed way a wealth of information on the elementary inelastic processes occurring during scattering. For low-mass particles and low surface temperatures, exp( - 2 W,) can be of the order of 0.1 and only a few phonons are involved at each inelastic collision. More precisely, it can be shown that when W, is of the order of unity, one-phonon processes are dominant: in this case, for a coherent process, the conservations of energy and of parallel momentum require that

Ef - E,

=

(hZ/2m)(kf2- kg) = k f w ,

AK = Kf - & = G

k

Q

(13)

(14)

where hugand hQ are the energy and parallel momentum of the annihilated (+) or created (-) phonon, and the subscript f refers to final states. Among all possible involved phonons, surface phonons are making a large contribution, as is discussed later. Assuming that the conditions for the occurrence of one-phonon processes are met, the experimental method for detecting them is based on a combination of both angular distribution and energy measurements. As pointed out by Williams (1971b), it must be realized that creation and annihilation processes can be spatially separated at sufficiently large incident angles. For in-plane scattering this is illustrated in Fig. 8. Different attempts to detect surface phonons and their dispersion relation by purely angular distribution gave satisfactory results, as is discussed in Section V1,B. However, in order to obtain sufficiently accurate information, an energy analysis must be carried out: this question is treated in Section V1,C.

ATOM DIFFRACTION FROM SURFACES

109

Reciprocal S u r f a c e Array

FIG.8. In-plane inelastic events involving phonons of parallel momentum hQ (Ewald representation).

D . Rotational Diffraction

When molecules are used as impinging particles another kind of surface diffraction process is possible, namely, that involving both conservation of parallel momentum and energy exchange between translational and internal degrees of freedom of .he incident molecule. In the real world, the only detectable processes of this kind are rotational transitions of very light molecules in the act of interzcting with a surface. In fact, no case is presently known of surface diffraction coupled with rotational transitions other than that occurring with H, and its isotopic species; on the other hand, vibrational transitions of hydrogen molecules involve too high energies in comparison with those of thermal beams used in practical experiments. We give here a short account of this interesting phenomenon but the subject is not further treated in this article. The reason for this choice is that “rotational diffraction” has not yet been studied in depth, and, what matters most, no satisfactory theory of the diffracted intensities and resonance phenomena is yet available. For the interested reader, an extended bibliography is reported in this section. Logan (1969) was the first to predict on theoretical grounds the existence of rotationally inelastic diffraction for H, scattering. The phenomenon was experimentally observed by Boato et al. (1974) and independently by Grant Rowe and Ehrlich (1975a), using LiF(001) and Mg0(001), respectively. The phenomenon was further studied in different systems by various authors (Grant Rowe et al., 1975; Grant Rowe and Ehrlich,

110

G. BOAT0 AND P. CANTINI

'i,,UL 00 -450

t

t m z W

c

Z T Oool0

- 30°

00

30"

60"

30'

60'

01

H*-LIF(001)

W

a I

3

It 1

1

O

2

w

Jo

I-

c

a

0

m

-

- 30°

0f

FIG.9. Elastic and rotationally inelastic diffraction in H,-LiF(001). [From Boato et al. (1976b).]

1975b; Boato et af ., 1976b,c; 1977; Cantini et a / ., 1977; Lapujoulade et al., 1981). The kinematic conditions for rotationally inelastic diffraction are given by the conservation of parallel momentum

Kf=K,+G and by the conservation of energy

A2kf/2m = Azk;/2m

+ AE,,,

(154 (13))

where AE,,, = kBT,,,(Jo - Jf)(Jo+ Jf + l), Trotbeing the rotational temperature and J the rotational quantum number. An example of rotational diffraction is given in Fig. 9, where the scattered intensity is shown for the system H,-LiF(001). Here Jf - Jo = 2 2 and only the rotational transitions 0 + 2 and 2 + 0 forpara-H, and 3 + 1 for ortho-H, are detected as additional diffraction peaks. Conditions (15a,b) are sufficient to determine the angular location of rotational peaks for all isotopic species of Hz. They lead also to a focusing, resulting from extremal effects in Of as a function of k,, (Grant Rowe et al., 1975). It is due to this focusing that the latter authors were able to detect rotational peaks, even using an effusive source with large velocity spread. In connection with the experimental results we would like to mention

ATOM DIFFRACTION FROM SURFACES

111

two main features. On the one hand, in systems like H,-LiF, H, , D,, HD-MgO, and H,-NiO, the total diffracted intensity associated with rotational peaks is comparable to the elastic intensity, which gives a considerable importance to rotational transitions in gas-surface interactions. On the other hand, when metal surfaces are considered, the ratio of the rotationally inelastic intensity to the elastic one is much smaller, i.e., of the order of 10+ (HD is an exception, owing to mass asymmetry). The effect may be attributed to a presence of a “softer” repulsive potential in metal surfaces. The diffraction probabilities observed in H, scattering for both elastic and rotational peaks are only partially understood and a reliable theory of all phenomena involved is still missing, as mentioned above. There are inherent difficulties in this theoretical problem; however several qualitative and significant aspects of the phenomenon have been clarified using different approaches (Wolken, 1975; Goodman and Liu, 1975; Garibaldi et al., 1976; Gerber et al., 1980; Adams, 1980). 111. QUANTUM THEORYOF ATOM-SURFACESCATTERING

The rigorous quantum scattering theory of a nonpenetrating particle interacting with a crystalline surface is a complex problem featuring all aspects of many-body theories. From the beginning, two main points should be kept in mind: (1) The interaction of a light projectile (such as H or He) with the surface does not affect appreciably the surface statics (equilibrium position of the atoms) and dynamics (lattice vibrations at the impact location) during the collision (2) Owing to simmetry breaking, the surface layer finds itself in static and dynamic conditions which may be appreciably different from the bulk

These facts allow separation of the whole theoretical problem into two parts, i.e., the determination of surface structure and lattice dynamics, and the solution of the scattering problem. However, a problem still remains, namely, the choice of a proper atom-surface potential V = V(r; r l ,

...

, rN)

(16)

where r and rl , which are functions of time, denote the coordinates of the gas atom and of the lth lattice atom respectively. Attempts were made to assume V to be well represented by the sum of pair interactions V(r)

=

C vl(r 1

112

G. BOAT0 AND P. CANTINI

This choice is approximately valid for a few systems, such as He on heavy noble-gas surfaces, where the three-body contributions are relatively small and vl can be substituted by the well-known atom-atom potential. Until now, however, a detailed check of the validity of pairwise summation was obtained only for static potentials used to describe elastic scattering; an example is given by Carlos and Cole (1979), who studied the He -graphite system. The most significant attempts to formulate applicable scattering theories were carried out by using “model potentials,” which were chosen ad hoc in order to theoretically describe processes of experimental relevance. The model potentials may be separated into two classes: (1) static potentials used for elastic diffraction, and (2) dynamic potentials used for inelastic scattering. A . Static Potentials and Elastic Diffraction Theories

Many static “model potentials” have been proposed or used to discuss a great many experimental results. For an extensive and critical review we refer to the paper by Hoinkes (1980), who also gives semiempirical rules to derive them. We concentrate here on three different families of model potentials to outline three different and productive theoretical approaches to elastic scattering. 1 . Pairwise Summation and Coupled Differential-Equations Method

As discussed in Section II,B the interaction potential is composed of a short-range repulsive and long-range attractive part which both originate from pair interactions. For several solids, the potential is commonly approximated by a sum of static pair potentials vl(r - rl) between the gas atom and each atom of the solid in its equilibrium position. Several forms for 2rl have been proposed, as well as different ways to practically perform the summation (see Steele, 1974). In any case, the gas-solid potential must take into account the surface periodicity and can be expressed by the Fourier series (1). The Fourier components VG(z)=

d-l

V ( R , z ) exp( - iG R ) d 2 R

(18)

with G # 0, are generally much smaller than the laterally averaged zero component Vo given by Eq. (8). For many observed systems only few Fourier components are relevant; generally, they are of repulsive character.

ATOM DIFFRACTION FROM SURFACES

113

The Schrodinger equation for a particle of mass m and wave vector k,, in a potential V(r) is

By the Bloch theorem in two dimensions, the wave function +(r) can be expressed by =

C +G(z)exp[i(& G

+ GI Rl

(20)

Substituting in Eq. (19), one obtains a set of coupled differential equations - (2m/h2) C VG--dz)+G’(z)= 0 [(d2/dz2) + k&zl+G(~) G’

(21)

where k&, is given by Eq. (5). Far from the surface, the potential is zero, hence (d2/dz2 + k&)+G(Z) = 0

(22)

At sufficiently large distances, the solutions of Eq. (21) are therefore plane waves for all G’s giving k:, > 0 (open diffraction channels, identified hereafter by F vectors), while they are evanescent waves for all G’s giving k2,, < 0 (closed channels, identified either by N vectors, if leading to possible BSRs, or by E vectors, if k2,, < -2mD/h2, D being the well depth). The asymptotic form of the solutions for z -+ may then be written +F(z) = exp(- ikozz)6F,0 + AF exP(ikF,z)

(23)

for open channels, and

+dz)

= ANexp(- K ~ Z )

(244

for closed channels, K being (-k;)l’*. If the AF are known, from Eqs. (23) the diffraction probabilities PF(peak intensities) can be derived as PF =

(kFz/kOZ)IAFJ2

(25)

The sum of diffraction probabilities over all open channels must obey the condition 2, PF = 1 (unitarity condition for elastic scattering). The problem is now to solve the infinite set of coupled differential equations (21)-with the boundary conditions given by Eqs. (23) and (24)-in order to find the amplitudes AF . Approximate solutions were ob-

114

G . BOAT0 AND P. CANTINI

tained by Wolken (1973) and Tsuchida (1%9, 1974,1975),and in more detail by Chow and Thompson (1976b) and Chow (1977a,b), using different model potentials. The work of Chow and Thompson (1976b) is of particular importance because it showed for the first time several of the bound-state effects that were experimentally studied afterward. These authors obtained a satisfactory approximate solution of Eqs. (21) by including a finite basis set of G vectors, namely, the lower order open channels F and some of the closed channels N leading to possible BSRs. The unitarity of the approximate solution guaranteed the satisfactory choice of the basis set. The numerical results for He-LiF(Wl), obtained by using a Yukawa-6 pairwise potential, showed the appearance of several sharp resonant structures having the shape of both maxima and minima in the specular and other diffraction-peak intensities as a function of the azimuthal angle $I~.These results cleared the field from the prejudice that only experimental minima could be observed, owing to inelastic processes occurring during the residence time of the atom in the metastable state. A second important result by Chow and Thompson concerns the validity of the free-atom approximation, expressed by k;,

=

g

-

(K,,

+ N)2 = 2mei/h2

(26)

[compare with Eq. (9)]. This approximation is acceptable when Eo >> ej , which is usually the case for thermal atom collisions. The free-atom approximation is no longer valid for really adsorbed atoms 0 > Eo > - D , or-a more interesting case for diffraction-when two different resonances (ej ; N) and (ej, ; N’) happen to occur at the same incident conditions. Around this “crossing region” a better approximation is

(ki,

-

2 m ~ ~ / h ~ ) (-k 2rnq/h2) i,~ = I(xjJVN-,Ixj,)1’

(27)

(see also Chow, 1977b),where is the eigenfunction corresponding to ej, associated with the laterally averaged potential V,, . Equation (27) shows that the observed splitting of two near BSR anomalies is proportional to the (N - N’) Fourier component of the total potential (this is a bandstructure effect). Returning to the general problem, the solution of the infinite set of coupled equations (21) is often a formidable task, even with modern computers. On the other hand, a “brute force” solution of this type may cause lack of insight into the physical problem. For these reasons, and in trying to interpret the experimental data, other approximate theoretical methods were used, examples of which are the unitarized distorted-wave Born treatment by Cabrera et al. (1970) and the semiclassical methods. To this last class belong the theories given by Doll (1974), Berry (1975), Miller (1975), and McCann and Celli (1976). A more successful method is that based on a hard-wall corrugated model, which is discussed in the next.

115

ATOM DIFFRACTION FROM SURFACES

Close-coupling calculations, with their accurate results, still remain a possible check, together with experimental data, of the validity of approximate theories in the simplest cases.

2. The Corrugated Hard- Wall Potential The corrugated hard-wall model, also known as hard corrugated surface (HCS) model, has been widely used to interpret experimental data. A HCS model corresponds to a potential r

where {(R) is known as corrugation function; {(R) has 2D periodicity, i.e., it may be expressed as =

c, { c exP(iG

R)

*

G

(29)

The HCS potential is a first but realistic approximation for He interacting with a surface, for the reasons qualitatively explained in Section I1,B. A quantum theory using this potential was first used in an early paper by Boato et al. (1973) in order to interpret the diffraction of He and Ne scattered from LiF(001). The theory was fully developed by Garibaldi et al. (1975). It was found very useful to understand the peculiar shape of the diffraction pattern, known as quantum surface rainbow. Historically, the quantum theory of surface rainbow is akin to the problem of diffraction of sound waves having wave length A from a undulated reflecting mirror of period a A , studied by Lord Rayleigh (1896). In the HCS model the asymptotic form of the wave function for z + co is written

-

+(r) = exp(- iko r)

+

F

AF exp(ik,

r)

(30)

valid for z > max {(R). Rayleigh assumed that the far-field solution expressed by Eq. (30) is strictly valid all the way to the surface. With this Rayleigh hypothesis and imposing the boundary condition $(r) = 0 at the surface we get -exp[ - iko,<(R)] =

C AF exp[iF F

*

R

Multiplying both sides of this identity by exp{-i[G and integrating over the unit cell, one obtains

A8 =

BG,FAF F

(3 1)

i- &,{(R)] *

R

+ kG,{(R)]} (32)

116

G . BOAT0 AND P. CANTINI

where the coefficients A$ = BG,F

-&?-I

= &?-’ ]u,c,

exp[-i(kol exp[+

i(kFz

-

+ k,,){(R) - iG kG,)((R)

R] d2 R

+ i(F - G)

*

R] d 2 R

(334 (33b)

can be evaluated from the assumed ((R). The infinite system of linear equations (32) may be approximately solved, yielding the scattering amplitudes A F [and the diffraction probabilities PFfrom Eq. (25)]. It follows from Eq. (33b) that B G , G = 1 for all G’s; if, in addition, for all relevant channels (kFz - kGz){(R)is very small and can be neglected, the approximation A F = A: applies. This is commonly known as the eikonal approximation first used in atom scattering by Garibaldi et al. (1975) to explain the quantum rainbow. The eikonal approximation can be quite satisfactory for a qualitative interpretation of experimental data, particularly when many F vectors are involved. However, Chow and Thompson (1976a) and other authors warned against the danger that the nondiagonal elements B G , F may not be negligible under physically relevant circumstances. Their approach was to truncate the sum in Eq. (32) and retain enough terms to obtain convergent results. Another way to obtain the scattering amplitudes from Eq. (3 l), probably the most useful one, is that proposed by Garcia (1976) and known as the GR method. Briefly, it simplifies to writing Eq. (31) in the form

with

MG,R= exp[iG R + i(kGz + k,,)((R)] (35) Equation (34) must be satisfied at every point R of the unit cell. The GR method assumes a finite number n of points Rj, withj = 1, . . . , n, distributed in the unit cell, whereas the summation is truncated after n terms. Equation (34) can now be regarded as a system of n linear equations, which is solved for the A G ’ s . The computational applicability of the GR method was shown by Garcia (1977). The Rayleigh hypothesis gives a convergent solution of Eqs. (32) and (34) when the surface corrugation does not exceed a certain amplitude. A more general solution of the quantum mechanical HCS problem was consequently searched for (Masel et al., 1975; Goodman, 1977; Toigo et al., 1977; Garcia and Cabrera, 1978; Armand and Manson, 1978). The starting point is the Lippman-Schwinger equation which gives for a HCS

117

ATOM DIFFRACTION FROM SURFACES

$(r) = exp(-zk,,

+ ikGzlZ

- r) + -

d2R’f(R’) G

{exp[i(&

+ G)

(R - R‘) (36)

((Rf)l}/k3z

The wave function $(r), with the asymptotic behavior expressed by Eq. (30), yields the following scattering amplitudes:

The density of sources f(R) necessary to calculate AF from Eq. (37) is obtained by the appropriate boundary condition $[R, z = ((R)] = 0, giving - exp[-

ik~,S(R)l = G

[exp(iG * R)/k,l

LC.

f(R’) exp[-i(&

+ G) *

+ ik,lS(R) - C(R’>lId’R’

R’ (38)

Garcia and Cabrera (1978) proposed a numerical method to obtain f(R) from Eq. (38). Model calculations based on this procedure for 1D corrugations of different shape and amplitude seem to prove that the method is applicable to any kind of cumgation, no restriction concerning the corrugation amplitude or discontinuities in ( ( x ) or d((x)/dx being necessary (Armand et al., 1979). The solution of the problem, however, requires handling of very large matrices. We can conclude by saying that the HCS is a model potential suited to describe diffraction patterns obtained with He. In this description the real missing part is the existence of BSRs; the introduction of a potential well in front of the hard wall improves the calculations and explains the presence of BSRs, as is shown next. Other details on the use of the HCS model can be found in the review article by Engel and Rieder (1982).

3 . Corrugated Surface with a Well As discussed in Section III,A, 1, the close-coupling calculations of Chow and Thompson (1976b) proved that the observed BSR features could be well reproduced by using a purely elastic theory, except for a Debye- Waller correction. Several calculations carried out by using simple but realistic atom-surface potentials showed remarkable agreement with the experiments (Harvie and Weare, 1978; Wolfe et al., 1978; Garcia et al., 1979b). An elastic theory capable of reproducing in detail BSR effects without large computational efforts and giving physical in-

118

G. BOAT0 AND P. CANTINI

sight into the problem is that proposed by Celli et al. (1979). For brevity, we limit ourselves to the discussion of this theory. The calculation is based on the use of a model potential of the type V(r)

=

VR[z - C(R)1 + V,(Z)

(39)

where V, represents a short-range repulsion, [(R)being the effective corrugation function, and V, a long-range attraction. More precisely, V, and V, are separated by two planes, at z = zo - 6 and z = zo, with 6 > 0, such that in the region between them vR[z -

5(R>1 =

for z > zo - 6

(404

VA(z) = - D

for z < zo

(40b)

where D is the well depth and 6 can be arbitrarily small. In the calculations reported on p. 121, VR is often taken as a hard corrugated wall, whereas V, can be chosen as a zr3 term [see Eq. (7)]; in this case, the resulting well has a flat bottom. This model potential is shown in Fig. 10, with the observation that the indicated AD is set equal to zero. Returning to potential (39), the wave function can be found separately for VR - D and V, and matched at z = zo. In the region zo - 6 < z < zo , the wave function must have the form

{B&exp[i(&

+(r) =

+ G)

R + kh,~]

G

+ BE exp[i(K,, + G) R - k&,z]} (41) where k&%= ki - (K,, + G)2 + 2mD/h2. Far from the surface, +(r + m) = exp(zKo R - kozz) + C, AG exp[i(& + G ) R + kGg] _G_

(42) If it is now assumed that the scattering problem for V, - D has been solved (e.g., with the methods of Section 111,A,2), then for z < zo, the amplitudes B& of the diffracted waves are related to the incident amplitudes BE by B& = S(G, G’)B,, (43) G’

where S(G, G’) are known scattering-amplitude coefficients. Assuming that the scattering problem has been solved also for V,(Z > z,)-a simple problem since V, shows no periodicity-one then obtains for G # 0, AG = TGB&,

BG

=

RGB&

(44)

where RG and TG are the reflection and transmission coefficients, respectively, for incidence from the left (toward positive z), with kGzlTG12 + kbzlRGIZ= kbz

(45)

ATOM DIFFRACTION FROM SURFACES

I

T

rn, k o

I

I

I

' '

!

"R

I I

4

119

Al I+ I I

2,-

s

20

FIG.10. HCS potential with a flat-bottom well; V , and/or D are modulated by the corrugation function. The corrugation amplitudes are represented by A( and AD, respectiveli.

For G = 0 , one has

where R; and T; are coefficients for incidence from the right, where [TAP = kOz/k&in agreement with Eq. (45). Equations (46) include the incoming wave. Combining Eqs. (44)and (46), one obtains

B&

= S(G, 0)T;

+ 2 S(G, G')R&B&t

(47)

G'

According to Celli et al. (1979), G' in Eq. (47) can be restricted in most cases to the set of the N reciprocal lattice vectors for which 2rnD/A2 > kk: > 0 (diffraction into the well, see Section III,A, 1). As a consequence, Eq. (47) becomes a matrix equation of small rank for the unknown quantities B&. Finally, the diffraction intensities PFcan be obtained in the form of PF

=

(k,,/koz)JA2,1 = (kkZ/khz)

/C S(F, N)B&RN/G+ S(F, O ) l z N

(48)

as a combination of the elastic amplitudes S(F, N) and S(F, 0) for diffrac-

120

G . BOAT0 AND P. CANTINI

tion from the potential V , - D and of the reflection coefficients R, from the attractive well. A simple way to characterize the reflection from the well uses the phase shift BN as a function of the perpendicular energy EN = h2kir/2m;BN is related to the reflection coefficient RN by RN = exp(isN).Following Celli et al. (1979) and, in more detail, Hutchison (1980), the phase shift BN = 277j(EN) can be obtained by interpolation with the use of the relationship EN(j)

=

-D[1 - c(j + 8)y

(49)

which is a simplified version of the equation for the bound-state energy levels ej obtained by Matteraet al. (1980a);C and a are parameters related to the width, steepness, and asymmetry of the potential well. The above theory was used by Celli et af. (1979) to discuss in physical terms the selective adsorption process: a simple result is obtained when only one N vector is involved (isolated BSR). In this case Eq. (48) can be written

which contains a typically resonant term with a width

r = 2(ds/dE)-yi

-

p ( ~N)I) ,

(51)

In Eq. (50), S(F, 0) is the nonresonant amplitude corresponding to AF in Eq. (31). Equation (51), together with Eq. (49), shows that the resonance width goes to zero as the bound-state level approaches the threshold ( E ~+ 0). For intermediate levels the width increases, passes through a maximum, and reduces again for the deepest levels (for which 1 - IS1 decreases). The line shape of an isolated resonance can be written as a function of x =2 ( ~ E j ) / r in the form x + i

= 1 +

by

+ bz + 2b1 - 2bZx 1

+ x2

(52)

Forb, = 0, Eq. (52) becomes 1 + b,(b, + 2)/(1 + x2),which represents a Lorentzian line shape and is a maximum for b, > 0 or b, < -2, and a minimum for 0 > b, > -2. On the other hand, following this theory, the line shape is a maximum-minimum (i.e., non-Lorentzian) when b, # 0. The formulation given here is valid for any surface symmetry. In addition, when two N vectors are involved (BSR crossing region, see Section III,A,l), Celli and co-workers are able to give a simple explanation of resonance splitting and symmetry effects. In conclusion, the

ATOM DIFFRACTION FROM SURFACES

121

theory of Celli et al. (1979) gives a satisfactory physical picture of the intensities, widths, and line shapes observed in BSR structures. Whereas this and other elastic theories (see beginning of this section) give a good qualitative explanation of resonance effects, a detailed agreement with experiment was obtained only for He scattered from LiF(OO1) (Harvie and Weare, 1978; Garcia et al., 1979a). Similar calculations carried out for He-graphite (Garciaet al., 1980) showed only a partial agreement with experimental data; for example, the resonant structures are predicted to be narrower and more intense. Quantitative discrepancies in the BSR line shapes with respect to the prediction of elastic theories are observed in several experiments. These discrepancies have to be ascribed to inelastic effect, that are certainly effective in resonant transitions, as initially suggested by Lennard-Jones and Devonshire (1937). The use of an optical potential (Chow and Thompson, 1979; Garcia et al., 1980; Wolfe and Weare, 1980) can phenomenologically account for some of these deviations. Alternatively, Hutchison (1980) and Greiner et al. (1980) proposed a variant of the elastic theory by Celli et al. (1979), by multiplying each scattering coefficient S(G, G ’ ) by exp[- W(G, G ’ ) ] , 2W(G, G ’ ) being the appropriate Debye- Waller coefficient. This prescription works surprising well for explaining the experimental data at low incident energy or near-grazing incidence. The importance of the connection between BSRs and inelastic scattering is further discussed in Section III,B,3.

B . Vibrating Surfaces and Inelastic Scattering If one wants to go beyond the static potential approximation, only very simplified model potentials can be adopted to solve the theorical quantum problem. In quantum scattering theories, the relevant information is contained in the transition T matrix, which has to be evaluated between states compounded of particle states labeled by the initial wave vector k, and final wave vector k (hereafter written as subscripts), and of crystal states labeled by phonon occupation numbers a, p. A T-matrix element is then where Tk+ko is an operator in the space of crystal states. Following Manson and Celli (1971), the differential scattering probability for an atom to go in the solid angle dfl, losing energy A E = E, - E , is

122

G. BOAT0 AND P. CANTINI

where L is a quantization length. To obtain the probability p a , Gibbs' thermal equilibrium distribution may be used. As suggested by Levi (1979), the transformation introduced by Van Hove (1954) can be applied by using a Fourier representation of the 6 function, to obtain the correlation function expression

where the time evolution of the T matrix is driven by the free Hamiltonian of the solid. Equations (54) and (55) are exact and are the starting point for several approximate inelastic scattering theories. We now discuss three different theoretical problems. The first is the form of the Debye-Waller factor; the second is the derivation of the inelastic scattering probability; the last is a theoretical treatment of inelastic BSRs. 1 . Debye-Wuller (D-W) Factor

In order to derive the elastic diffraction probability, we must specialize the general correlation function in Eq. (55) to the case AE = 0. The exponential factor vanishes and the integral is dominated by long times. Asymptotically, a factorization of the correlation function can be applied (Levi and Suhl, 1979), thus

The next step is a direct computation of the average (Tkcko),a problem which presumes the knowledge of both the lattice dynamics and the time-dependent atom-surface potential. Levi and Suhl (1979) proved that for the interaction of fast light particles with heavy atom substrates, a D-W factor completely identical to that used in X-ray, electron, and neutron scattering is obtained, namely, Eqs. (1 1) and (12) are valid. The result was found to be equally valid for both completely correlated and completely uncorrelated (Einstein oscillators) surface-atom displacements. For slow atoms, however (i.e., when the collision time is comparable with the characteristic time of lattice vibrations), the application of the standard theory is not straightforward. Several corrections to the stand) proposed, depending on the mass, size, and ard exponent ((Ak * u ) ~were velocity of the incoming particle and on the crystal structure and dynamics (Beeby, 1971; Hoinkes et al., 1972b; Armand et ul., 1977). All these corrections can be justified by a more general formulation based on Eq. (56) and/or making use of semiclassical approaches (Levi and Suhl, 1979; Meyer, 1981). A detailed description of the semiclassical approximation is

ATOM DIFFRACTION FROM SURFACES

123

beyond the scope of the present work. We only give some of the conclusions which are relevant for interpreting the experiments, as derived from Levi and Suhl(l979). These authors find that a D-W factor exp(- 2W) is still valid, with the exponent given by 2W(k, k,,) = h-’

2 [[ F,(t)F,(t’):

(um(t)u,(t’))dt dt’

m,n

(57)

where F,(t) is the force by which the mth crystal atom in its equilibrium position acts upon the incident particle, whereas u,(t) is the corresponding displacement. The forces F(t) in the integral are evaluated along the trajectory of the particle in the field of the static lattice (u, = u, = 0) from the incoming k,, to the outgoing k state. Using the stationary properties of lattice vibrations and introducing the tensors A,,,(T) = h-’ Bm,n(T)

I

F,(t)F,(t

+ 7 ) dt

= (um(O)un(~))

(58b)

the formally simple and general formula for the D-W exponent 2W(k, 16) =

IAmn(T):Brnn(T) d7 m,n

(59)

is obtained. This formula is able to qualitatively describe the size and time-of-interaction effects, observed and/or proposed by different authors. Leaving the discussion of experimental results to Section VI,A, we limit ourselves to a few important consequences contained in Eq. (59)

(I) Ordinary D-W factor. As expected, for fast collisions and in case the well is neglected, Eq. (59) reduces to the elementary formula ((Ak * u)’) . (2) Beeby effect. In the presence of a nonnegligible well and fast collision, the formula given above (1) can only be used for the repulsive part of A,,,. For the part of A,,, involving attractive forces, B,,, in Eq. (59) has to be evaluated not at T = 0, as in (1) above, but at considerably longer times, when all displacement correlations ultimately vanish. The result is that in the elementary formula Ak ought to be replaced by Ak‘, with the vertical energy effectively increased by the well depth so that K2 = k”, + 2mD/h2, i.e., the particle is accelerated in the well before the “hard” collision. This correction was first proposed by Beeby (1971) in a somewhat different context. ( 3 ) Levi effect. When the overall collision time is large compared to the period of typical surface-lattice vibrations, the D-W factor is substantially increased. This effect was proposed by Levi (1975) and extensively treated in the paper by Levi and Suhl (1979).

124

G. BOAT0 AND P. CANTINI

(4) Armand effect. In atom-surface collisions, the assumption of single atom-atom interactions is not justified, owing to the finite size of the incident atom. A rough correction was first suggested by Hoinkes et al. (1972b). A more rigorous theory was proposed by Armandet al. (1977), who took into account correlations between displacements of neighboring surface atoms. The correction is important at short phonon wavelengths and leads to a reduction of Wand to an increase of the diffracted intensity. Equation (59) is inclusive of the Armand effect, as shown by Levi and Suhl (1979). 2. Inelastic Scattering Probability Quantum theories of inelastic scattering have been proposed in considerable number during the last 10 years. All theories contain some approximation. The close-coupling formalism proposed by Wolken (1974) was used by Lin and Wolken (1976) to calculate energy transfers in the He-Ag( 111) scattering. The renormalized distorted-wave Born approximation was initially employed by different authors (Manson and Celli, 1971; Goodman and Tan, 1973; Benedek and Seriani, 1974; Lagos and Birnstein, 1975,a,b). The approximate method more often applied in recent years has been the impulsive approximation (Beeby, 1972a,b, 1973; Weare, 1974; Adams and Miller, 1979), particularly when connected with a time-dependent HCS (Benedek and Garcia, 1979; Armand and Manson, 1979; Marvin and Toigo, 1979; Levi, 1979). To give a summary description of the inelastic formalism proposed by Levi (1979) for the vibrating HCS, we introduce the eikonal approximation in the 7'-matrix elements of Eq. ( 5 3 , i.e., Tktko

-ih

m

I

exp[iq(R, t ) ] d2R

where q(R, t) is the semiclassical phase, the integral being extended over the impact parameters R. The phase can be split into three parts q(R, t ) = qi(R) + r/z(R) + qAR, t )

(61)

where ql(R) = A K . R is the phase difference for a reflecting plane; q2(R) = (kO2+ k,){(R), where 5 is the corrugation function of the static surface; and q3(R, t) is the fluctuating phase, related to the surface lattice vibrations. By assuming a hard vibrating surface, one finds d R , t ) = (koz + kA @XR*,t ) (62) where 65 is the lifting of the surface at (R*, t), R* being given by R <(R)K,Jk,, . In turn, 6[(R*, t) may be written in terms of local displace-

+

ATOM DIFFRACTION FROM SURFACES

125

ment as 6<(R*, t ) = 6(R*) * u(R, t )

(63) The inelastic scattering probability for a vibrating HCS then becomes, if the Armand effect is neglected,

d3P/(dEd2a) klkozl 81S12exp( - 2W) (64) where S is an amplitude akin to that occurring in elastic scattering [see Eqs. (33a) and (SO)]. More explicitly,

where AK is the parallel momentum transfer, no more restricted to be a reciprocal lattice vector G. A first result obtained from Eq. (64) is then that all effects occurring in elastic scattering (like the rainbow), also occur in the inelastic scattering; in particular, in the neighborhood of intense diffraction peaks, strong inelastic scattering is expected. Similar results were obtained by Armand and Manson (1979) and by Benedek and Garcia (1979). The remaining factor 5,containing the effects of lattice dynamics, may be usefully expanded as

where 8,describes, in the approximation of an harmonic solid, the contribution to the scattering event brought about by the simultaneous exchange of 1 phonons. At this point, we can limit ourselves to one-phonon scattering, occurring when 2W [Eq. (57)] is small; in these circumstances the kinematic conditions (13) and (14) apply. Following Levi (1979) the final expression for the inelastic scattering probability is given by Eq. (64) with E substituted by the first term of the sum (66)

where M is the lattice-atom mass and A E is the energy loss Eo - E. Equation (67) contains on the one hand the Bose-Einstein population factor and on the other the surface-projected phonon density

where e,(:) are the surface components of the polarization vector of the phonons belonging to branch v and having parallel momentum Q (see Chen et al., 1972, 1977). The surface-projected phonon density contains

126

G . BOAT0 AND P. CANTINI

contributions from both the continuum spectrum associated with bulk bands and, with a larger weight, the discrete surface modes. In conclusion, the eikonal approximation applied to the vibrating HCS model, as expressed by Eq. (64),contains the most relevant features of the inelastic scattering probability. The dependence on the static potential is contained in ISP, whereas Eqs. (67) and (68) show that atom scattering is sensitive to low-frequency modes and in particular to Rayleigh phonons. More refined theories include multiple-phonon processes, size effects, interaction time of incident particles, etc. Particular mention should be made of the paper by Marvin and Toigo (1979), who took into account the effect of the attractive well. However, they did not consider explicitly the presence of BSRs. A description of how this last phenomenon can be treated in inelastic scattering is given next.

3 . Inelastic Scattering and Bound-State Resonances Structures in the inelastic background of the scattered intensity as a function of er were repeatedly observed in different atom- surface experiments (Williams, 1971a; Boato et al., 1976a; Frank et al., 1977). An interpretation of these structures as inelastic BSRs was early given by Cantini et al. (1976). A deeper study and a theoretical description of the processes involved in inelastic BSRs were presented by Cantini and Tatarek (1981). The proposed formalism can be considered an extension of the paper by Celli et al. (1979) [see Section III,A,3]. The starting potential is a vibrating repulsive wall plus a time-independent attractive well, that is,

m, 111 + V*(4

(69) The formal extension of Eq. (47) to the inelastic one-phonon processes gives b&

=

V(r,

1) = VR[Z -

s(G

k

Q , 0)T;

+ 2 [s(G k Q , G’)&,b& G‘

+ S(G * Q , G’ * Q)&,*Q~&,Q]

*

(70)

where S(G k Q , G’ Q ) and s(G k Q , G ’ ) are the elastic and onephonon inelastic scattering amplitudes, respectively; they refer to the interaction of the particle with the time-dependent repulsive wall described by V , - D . Kinematic conditions (13) and (14) must be fulfilled for each intermediate state. For any final scattered state, represented by f(Ko + F k Q , Eo k hw,), Eq. (70) gives a matrix equation for b+ which can be solved in complete analogy with the elastic problem, and from it the inelastic probability can

ATOM DIFFRACTION FROM SURFACES

127

be derived. As in Section III,A,3, a simple result is obtained when only one N vector is involved, whereas two different energy levels ej and ejt are supposed to take part in the process. In this case the inelastic scattering probability to go in a final state is given by

d3P/dEd 2 0 = (d3P0/dE d 2 0 ) ) 1+ A ,

+ A, + A,[*

(71)

where d3Po/dEdzO is the probability for a single interaction with V , - D , given by Eq. (64). Equation (71) contains three resonant contributions A,, A, , and A,, whose kinematics is illustrated in Fig. 1 1 ;A, corresponds to an elastic res, followed by a phonon stimulated desorption. It onant transition in ( E ~ N), is given by

akin to the second term in parenthesis of Eq. (50). In Eq. (72), eNis the energy associated with the motion perpendicular to the surface for an elastic transition into the N channel; i.e., 2mEN/fi2 = kiz = kg - (I?, + N)2. The resonant condition = ej selects the incident wave vector lC,,(ko,Bo , +o). The process corresponding to A, is schematically illustrated with process (2) in Fig. 11. The term A, corresponds to an inelastic resonant transition in ( E ~ , ,N), say a transition in a bound state assisted by one phonon, followed by an “elastic emission” of the particle. It is given by A , = - - iC - iS(f, n)R,s(n, O)/s(f, 0)(1 - IS@, n)l) 2(€, - Ej,)/rn +i x’ + i

(73)

Now E , is the perpendicular energy after the inelastic transition in the N channel;i.e.,2rne,/fi2 = (k2, ? 2rno,/h) - (& + N k Q),. Theresonance , at every incident lC,, a different family of phonons condition E , = E ~ selects wN,j,(Q). Such an inelastic resonant process is indicated by process (3) in Fig. 11. The last term A3 has the form of a double resonance A, = -B’/(X+ i)(x’

+ i)

(74)

where X and x’ are defined by Eqs. (72) and (73), and

Of course, A, is different from zero only when both the direct elastic resonant transitions are allowed by the initial lC,, ; A, gives then, at selected incident conditions, an (ej, N) and the inelastic, from (ej, N) to ( q ,N),

128

G. BOAT0 AND P. CANTINI

FIG. 11. Perpendicular energy changes for a direct (1) and three resonant (2), (3), and (4) inelastic processes associated with phonon creation: (1)

K,=&+F-Q

(2)

Kf = (&

(3)

Kf = (&

+ N) + (F - N - Q) + N - Q) + (F - N)

(4)

Kf = (&

+ N) - Q + (F - N)

extra resonant contribution for the same selected phonon family wNj,(Q) as in process (3) in Fig. 11. The resonant effect described by A, is shown as process (4) in Fig. 11. Since the process indicated by process (4) occurs at the same “selected” incident ko as process (2), it can manifest itself as a spectacular change in the resonance line shape, as it is shown by experimental data reported in Section V1,B. We point out finally that the resonance amplitudes B, C , and B‘ do not contain effects due to statistics and to the phonon spectrum. This can be shown immediately by using the eikonal approximation for the vibrating HCS, with the result that

Idf,0)l2 lS(f, 0)l’El o(

(76)

where El is given by Eq. (67). As a consequence the ratio of two s (always contained in Al ,A 2 , and A3) is equal to the ratio of the corresponding S, given in turn by Eq. (65).

ATOM DIFFRACTION FROM SURFACES

129

IV. STRUCTURAL INFORMATION FROM ELASTICDIFFRACTION The best available atom diffraction probe for structural investigations is helium. The reasons for this are manifold. Not only He is light, but it has small polarizability, thus giving rise to a shallow potential well (and few bound states) with most surfaces. The weak He-He interaction prevents condensation in the gas phase and therefore allows the production of very intense beams by supersonic expansion at quite low energies. The size of He atoms is not negligible, but is small enough to enable one to explore the profile of outermost electron density of most surfaces. The accuracy with which the profile is reproduced depends on the He hard-sphere radius and on the corrugation function. On the other hand, the reliability of the profile determination by He scattering has received a strong theoretical support from the recent work by Esbjerg and Ngirskov (1980). They approximate the potential V(r), through which He interacts with the surface-electron-density profile n(r), by replacing the nonuniform electron distribution in which the atom is embedded with an homogeneous electron gas having density ff = n(ro), r,, being the location of the center of He atom. Moreoever, they prove that the energy needed to embed He in a homogeneous electron medium is well approximated by

v,

=

pii

(77)

with p = lo5 meV A3 in the relevant energy range. Of course, the resulting V(r) is always repulsive in this approximation. For all mentioned reasons, whenever structural investigations are carried out, He is largely preferable to H and H2 (having larger well depth and often being reactive) and to Ne (having larger atomic mass and higher polarizability). The entirely nondestructive character of He must also be emphasized, in contrast to electron, ion, and X-ray probes. In this section we give a short account of the work accomplished to arrive at structure determination. A much more detailed survey can be found by the reader in the review article by Engel and Rieder (1982). We divide structural investigations into three main classes depending on the surface complexity; (1) simply corrugated surfaces, (2) reconstructed and periodically deformed surfaces, and (3) adsorbate-covered surfaces. A few typical examples for each of these classes are given. In most cases, the HCS model has been employed to derive the static corrugation function [(R)from the diffraction-peak probabilities. Equations (25) and (33a) and the eikonal approximation (AF = A$) have been used for this purpose. A choice of the experimental scattering conditions

130

G . BOAT0 AND P. CANTINI

may be useful in order to minimize the effect of both BSRs and multiple scattering. This aim has been reached by using sufficiently large incoming energies and small incident angles. Some times corrections were applied for the D-W factors by making use of expressions based on Eq. (11) and for the potential well by replacing k,, in Eq. (33a) with k& [see Eq. (41)]. A . Simply Corrugated Surfaces

The first example is the (001) face of LiF, mentioned previously in Section I1,B. From a detailed study of the diffraction pattern under different incident conditions (Boato et al., 1976a), the corrugation function can be found by using the HCS model with the eikonal approximation (Garibaldi et al., 1975). Neglecting all but the first-order terms of the Fourier expansion (29), which now reads 2TX

+ cos-

+ 251, [cos--(X277 2T + cos-((x a

-

a

+ Y)

Y)

1+

*

.

*

one finds = 0.0752 f 0.0009 A, corresponding to a peak-to-peak corrugation of 0.602 A. With a more detailed analysis and using the GR method, Garcia (1976) found that the best-fit values for the first two = 0.0767 0.0007 A and = Fourier components of [(X, Y) were 0.0042 2 0.0007 A. The introduction of a well with 5-meV depth causes to decrease to about 0.072 A. The peak-to-peak corrugation of about 0.6 A is in qualitative agreement with the difference in ionic radii of Fand Li+. Further measurements, a more complete analysis of the data, and an improved theoretical description of the ionic and electronic distribution at the surface, should eventually allow information to be obtained on the relaxation of this and other ionic surfaces. The second example of a simply corrugated surface is the basal plane of graphite, a lamellar solid. No relaxation effects are expected at the surface, owing to the weak Van der Waals forces binding the layers. A diffraction pattern by He scattering is shown in Fig. 12 together with a sketch of the graphite surface geometry. From the whole set of measurements (Boato et a f . , 1979b) of this hexagonal surface one derives = 0.023 f 0.002 A, corresponding to a peak-to-peak corrugation of 0.21 A. The higher order Fourier components are negligible. The graphite surface appears to be less corrugated than that of ionic crystals.

*

cl1

ATOM DIFFRACTION FROM SURFACES 0.04

,0.03

r

I

Surface

131

unit cell

-

> k 20.02-

(I)

Y

I-

f

SCATTERING A N G L E

9,

ef

FIG. 12. Diffraction pattern of He scattered from the basal plane (OOO1) of graphite: =

00;

eo = 100.

A third example is given by simple metal surfaces. A close-packed metal surface is supposed to appear very flat to an incoming He atom. In fact, as first pointed out by Smolukowski, conduction electrons have the tendency of filling out the space between ion cores and therefore to smooth out the surface corrugation. The extent to which this smoothing occurs can be inferred from the diffraction experiments. These have shown that for hexagonal compact surfaces such as Ag( 111) (Boato et al., 1976c; Home and Miller, 1977),the corrugation is indeed very small, as it appears from the almost complete absence of diffraction (see Fig. 13). From the weak intensity of the first-order diffraction peaks-the only equal to (2.5 ? 0.2) x A can be inones observed-a value of fered, corresponding to a peak-to-peak corrugation of -0.022 A, a factor of 10 less than graphite. For less packed surfaces a larger corrugation is expected. For instance, in Ni(ll0) close-packed rows of atoms are present, separate by throughs a d wide, d being the Ni-Ni distance. Perpendicularly to the rows, a peak-to-peak corrugation of 0.05 & 0.01 A was found by Rieder and Engel (1979), whereas the corrugation is almost absent along the rows. In order to find corrugations comparable to graphite or ionic crystals, one must choose such loosely packed metal surfaces as W(112) (Tendulkar and Stickney, 1971) or the stepped-surface Cu(ll7) (Lapujoulade and Lejay, 1977) (see Fig. 14), which are strongly corrugated only in one direction. In all these cases the parallel troughs are so

G. BOAT0 AND P. CANTINI

132

Surface

0.15

unit

cell

,

z 0

>

0.10

(I]

z

w

I-

z

a

; a05

a = 2.88

w

Il-

'.,.\.

!!

a

0 v)

I

00

I

I

..'

I

100

SCATTERING

.....'.

...

I 200

ANGLE

I

i

10 :.

.. I

'.... I

30' 8f

FIG.13. Diffraction pattern of He scattered from Ag(lll), do= 0".

wide that conduction electrons are unable to fill them out completely. It is interesting to note that the profile across the steps reported in Fig. 14 was derived on the basis of Eq. (37), since the eikonal approximation is too rough for treating such a case (see Garcia et al., 1979b). B . Reconstructed and Periodically Deformed Surfaces

The atom diffraction technique has proved to be of great advantage for understanding the structural features of complex surfaces, i.e., surfaces with large unit cell or complicated corrugation or both. By giving informa-

r

0.03

FIG. 14. Diffraction pattern of He scattered from Cu(117). [From Lapujoulade and Lejay (1977).]

ATOM DIFFRACTION FROM SURFACES

133

tion on the topmost surface layer, He diffraction is able to complement the results obtained by LEED, ion scattering, and surface-extended X-ray adsorption fine-structure (SEXAFS) techniques. The principal problem area for which rapid progress has been made is that of the reconstruction of semiconductor surfaces, through work carried out at Bell Telephone Laboratories by Cardillo and co-workers. It is well known that the geometrical configuration of these reconstructed surfaces is very complicated and in several cases not yet understood. Surface reconstruction takes place to saturate the strongly directional dangling bonds produced when the surface is created. Three surfaces were thoroughly studied, namely Si(OOl), GaAs(1 lo), and Si( 111). Apart from the experiment on Si(OO1) 2 x 1 (Cardillo and Becker, 1980) which confirmed this surface to be reconstructed in the form of dimer arrays but showed the presence of a lack of ordering, very interesting results were obtained for GaAs( 110) 1 x 1 and Si( 111) 7 X 7. The reconstructed surface of GaAs(ll0) is considered to be experimentally well understood. The unreconstructed surface is composed of parallel ridges and troughs of gallium arsenic bonded chains; the surface reconstructs by simply tilting the Ga-As bond by about 27". Thus the resulting surface is strongly corrugated across the troughs, but only slightly along them. The already established structure was fully confirmed by He scattering (Cardillo er al., 1981). Diffraction patterns obtained at different incident angles O0 and in the two opposite directions across the throughs (indicated by +o = 180" and +o = 0") are shown in Fig. 15. The asymmetry in the two azimuths is due to the tilting of Ga-As bonds during surface reconstruction. Simple evaluations of the two main corrugations by means of both the HCS model (eikonal approximation) and the locations of the classical rainbow angles (as deduced from the diffraction peak envelopes) give average values for cl, = 0.54 and tO1= 0.14 A, respectively. The corrugation parameters are in satisfactory agreement with close-coupling calculations carried out by Laughlin (1982) using a realistic soft-wall model potential. Moreover, sophisticated theoretical techniques were used by Hamann (1981) in order to calculate the electron charge density at the surface down to very low values. By using Eq. (77), Hamann derives that the classical turning points of He at 20 meV occur along the a.u. The two principal corrugations electron-density contour at 3 x are thus evaluated to be about 0.44 and 0.15 A, respectively, in remarkable agreement with experiment. In conclusion, the data by Cardillo and co-workers on GaAs(ll0) have given greater confidence both in the experiments and in the simple potentials (such as the HCS) used to interpret them. Contrary to GaAs( 110) 1 x 1, the Si(ll1) 7 x 7 face is a well-known

134

G . BOAT0 AND P. CANTINI 12 ~ 1 0 - ~

(a)

FIG. 15. Diffraction pattern of He scattered from reconstructed GaAs(ll0): (a) = 0.98 A; (b) I$,, = o", A = 0.98 A. [From Cardillo et al. (19811.1

b0 = 18o", A

puzzle in surface reconstruction. The surface was studied by Cardillo and Becker (1979) who measured the He diffraction pattern at different incident angle and energies. The 7 x 7 reconstruction was confirmed to be present up to the top layer. A multitude of diffraction peaks over a wide range of scattering angles was observed; their intensities are of comparable magnitude and change rapidly with the incident angle. The whole trend is difficult to disentangle; the only clear result is that the surface is strongly corrugated. However, Cardillo (1981b) was able to overcome this difficulty by making accurate measurements of the specular intensity as a function of the incident angle. The results are reported in Fig. 16; they show an oscillatory structure with at least one definite period. The observed trend may be interpreted in terms of interference of waves reflected from terraces displaced 3.2 A along the surface normal. This finding, together with other knowledge on the Si(ll1) 7 X 7 surface, enabled Cardillo to propose a simple model for the 7 x 7 reconstruction: the unit cell is essentially made of two triangular terraced regions displaced one from the other by two silicon layers (see lower part of Fig. 16). Although Cardillo's results cannot be considered conclusive, they are likely to lead soon to the resolution of the 7 x 7 puzzle. To conclude this section, we report briefly on some work carried out on another kind of complex surface, namely, that of a layered material de-

ATOM DIFFRACTION FROM SURFACES

135

12 10 0

-; v)

8-

64-

2-

20'

I

30'

I

40°

1

50°

I

I

€Do 70°

80 80°

90°

FIG.16. The specular intensity as a function of Bo for He-Si(ll1) 7 x 7. The proposed model for reconstruction is sketched ( h = 0.57 A, [01]* azimuth; terrace spacing: .1 3.28 A, j 2.95 A). [From Cardillo (1981b).]

formed by the presence of charge-density waves. The 3 x 3 commensurate superstructure of 2H-TaSe2 is an example and it has already been shown by the diffraction pattern in Fig. 4.Another example is that studied by Cantini er al. (1980) who measured the perpendicular deformation due to charge-density waves in the x superstructure of 1T-TaS,. Large superstructure peaks were observed. By using the HCS model, the overall peak-to-peak deformation was estimated to be quite large at T = 80 K, namely, about 0.4 A.

m

C . Adsorbate-Covered Surfaces

The study of the properties of surfaces covered by an ordered monolayer of adsorbed atoms or molecules has become a subject of wide interest because of fundamental aspects (2D phase transitions, dynamical

136

G . BOAT0 AND P. CANTINI

( i1 -r 5)

2.6

1 5 (*-I-)

2 6

4 = f 1.3'

,100

00

100

200

30° 40°

50°

60°

Qf

FIG. 17. Diffraction pattern of He scattered from Ni(ll0) Bo = 25", Ts = 105 K. [From Rieder and Engel (1980).]

+ H (2 x

6): h

=

0.63 A,

properties of monolayers, structural investigations, etc.) and interest in applications (physical and chemical adsorption, surface catalysis, etc.). The use of atom diffraction techniques is also very promising in this field, since it gives direct information not only on the monolayer structure, but also on the outer electron distribution of adatoms at the surface. A number of structural studies have been carried out, among which we like to mention the papers by Rieder and Engel (1979, 1980) on H-covered Ni(llO), by Lapujoulade et al. (1980b) on 0-covered Cu(llO), and by Ellis et al. (1981) on Xe-covered graphite (this last system was studied by H-atom diffraction). For some of these studies, details can again be found in the review article by Engel and Rieder (1982). We concentrate on one selected structure, i.e., the 2 x 6 phase of H adsorbed on Ni(llO), occurring for a coverage of 0.8. We first emphasize the He-atom diffraction is superior to LEED when investigating H-

LEI

138

G . BOAT0 AND P. CANTINI

surface is displayed in 3D in Fig. 18a and shows paired zig-zag hydrogen chains parallel to the bare-surface troughs. The proposed structure is sketched in Fig. 18b. The lower height of 0.18 A for the twofold sites (dashed circles) compared with the height of 0.25 A for the three fold sites (solid circles) seems to be in agreement with the expected H-Ni bond lengths. This final study is surely a most effective example to show the capability of the atom diffraction technique for understanding surface structures. V . INFORMATION ON

THE

SURFACE POTENTIAL WELL

A. General

As briefly described in Section II,B, the study of BSR structures in the elastically diffracted intensity is very effective for obtaining, through the levels ej ,information on Vo(z), Eq. (8). In addition, from the measurement of energy splittings occurring at the crossings of different BSRs the matrix are determined, as pointed out in Section III,A,l in elements (~jlVN-Nt1~) the discussion of Chow and Thompson’s theoretical findings. Therefore, the measured splittings can give information on the Fourier components V,(z) of the potential in the region of the well. The power of this experimental method, when applicable, lies in its simplicity, since the overall information on the potential is obtained by the use of kinematic relations, i.e., by measuring the angles at which resonant structures appear for any given k,, . Suitable systems and favorable experimental conditions for accurate measurements of the levels ej and especially for the determination of the matrix elements should be sought. First, V , must be much larger than any other V , , otherwise the free-atom approximation [Eq. (26)] is no longer valid over wide regions of the K space. Second, only a few Fourier components (one or two) and a limited number of levels ej must be present, otherwise a large number of crossings occur; under these circumstances, splitting make the measurements confused and intricate. On the other hand, the relevant Fourier components must not be too weak, otherwise BSRs may not be observed, as in the case of He-metal systems. A last requirement is to work at low incident energies and large incident angles, where the resonant structures are usually best resolved. Systems carefully investigated for the BSR levels were H and D on (001) alkali halide surfaces (LiF, NaF, KCl) (Finzel et al., 1975; Franker al., 1977); H and D on graphite (Ghio et al., 1980); 3He and 4He on LiF and NaF (Derry er al., 1978); 4He on NiO(001) (Cantini et al., 1979); 3He

ATOM DIFFRACTION FROM SURFACES

139

and 4He on graphite (0001) (Derry et al., 1979, 1980; Boato et al., 1979b); studies of energy levels were also carried out for H2 and D2diffracted from LiF(OOl), Ni0(001), and graphite. Several of these studies allowed more or less accurate determination of the form of Vo(z),but only a few of them gave information on the Fourier components. For a more complete review of the experimental results we refer to the previously mentioned article by Hoinkes (1980). We should like to add a promising piece of information: from BSRs associated to rotational peaks in the scattering of HD from Pt(11l), it has been possible for the first time to measure the energy levels ej of the interaction potential between a simple molecule and a close-packed metal surface (Cowin et a / . , 1981). In Section V,B we discuss in detail only the results relative to the He-graphite system, which is the most accurately studied until now.

B . The He-Graphite System 1 . Determination of Energy Levels

The selective adsorption in the elastic scattering of He from the basal plane of graphite was systematically studied by Derry et al. (1979, 1980) and by Boato et al. (1978, 1979a,b). The study of Derry et a / . (1979) concentrated on a very accurate determination of the energy levels for both 3He and 4He, by absolute measurement of the incident angles. The specular intensity was explored with an incident energy E, = 17.3 meV (corresponding to ko = 5.76 A-l) in a region of the K space where the freeatom approximation is clearly valid. In Fig. 19 a typical polar scan is presented. Precise measurements with 4He were also carried out in similar conditions by Boato et al. (1979b), with some uncertainty in the absolute measurements of O0; on account of this, e4 was set to coincide with the value by Deny et al. (1979). The final results obtained by the two experimental groups are shown in Table I. A remarkable agreement between the two sets of measurements is present: the statistical error for the deepest level E, is less than 1%, which shows the capability of the experimental method. 2. Determination of Energy Splittings One reason for the high accuracy secured for the energy values ej in He-graphite is the presence of only one sizable nonzero Fourier component in the interaction potential. This was predicted by simple pairwise evaluation of the potential and was confirmed by the measurements of Boato et al. (1979a,b), who carefully studied the crossing of possible reso-

G. BOAT0 AND P. CANTINI

140

I 40 '

1

5L

300

I

I

500

,

I

60'

I

Oo

FIG. 19. Accurate absolute measurements of angle of incidence Oo at which BSR minima appear in 4He-graphite: +o = 17", Eo = 17.3 meV. [From Derry et al. (1979).]

nances. The specular intensity was measured at an incident energy Eo = 22 meV (ko = 6.49 k l ) by exploring azimuthal angles between +o = 0" [(lo) azimuth] and +o = 30" [(ll) azimuth], and polar angles O0 up to about 80". As an example, typical polar scans for 30" < O0 < 46", obtained at different +o's are reported in Fig. 20, where several crossing of the ej (10) and ej (01) BSRs are observed. The resonances are split TABLE I ENERGYLEVELS~j A N D MATRIXELEMENTS (xjlvl~lxp)OF 4HE-GRAPHITE

~

0 1

2 3

4

-12.06 -6.36

-2.85 -1.01 -0.17

~~

-11.98 -6.33 -2.85 -0.99 -0.17

0.28 0.19 0.12 0.09 0.03

0.18 0.16 0.10 -

0.12 0.11 -

0.08 -

Uncertainties are *O. 1 meV. Uncertainties are kO.06 meV for cj, and from kO.01 to k0.02 for matrix elements; E* is set to be equal to that given in the first column. Boato et al. (1979b). See footnote b for errors.

141

D

I

I

I

1

I

I

1

I

40'

00 FIG.20. Polar scans of specular intensity at different azimuths I # I ~in He-graphite. Several BSR crossings are observed. [From Boato et al. (1979b).]

by the first Fourier component V,, of the periodic potential (in the case of Fig. 20, N,, - No, = G,i and GI, = Glo). Figure 21 shows how the splitting associated with the crossing of two selected resonances appears in the O 0 , plane. The reported experimental points correspond to the minima of the observed resonant structures. The solid lines refer to the free-atom approximation, whereas the dashed lines represent the best fit obtained by using Eq. (27), withj = 1, N = (lo), and N' = (01). The matrix elements (>(JIVlolG)obtained with this method by Boato et ul. (1979b) are reported in Table I together with their evaluated errors. A few matrix elements were determined also by Derry et af. (1980) at very low incident energy (E, = 4.7 meV); they agree with the values of Table I.

+,

142

G. BOAT0 AND P. CANTINI

I

I

43"

I

I

I

44O

I

1

45O

00

FIG.21. A detailed view of the splitting of two BSRs. [From Boato et al. (1979b).]

3. The He -Graphite Potential

The very accurate values of energy levels ej and matrix elements (xi[Vlolxjr) for He-graphite were used to propose reliable interaction potentials. This can be done in different ways, either by employing atomsurface model potentials or by looking for sums of appropriate pairwise potentials. A rough model potential, but one still well suited for describing the experimental results, is a modified HCS potential with a well (see Section III,A,3). The hard wall is assumed to have the shape function

((R) = 2{,,[cos(2r/a)X

+ cos(2r/a) Y + cos(2r/a)( Y

-

X)]

(80)

with = - 0.023 A as experimentally determined by Boato et al. (1978). The attractive well is chosen in a shape convenient for scattering calculations as proposed by Garcia et af. (1979a,b), namely, V*(z)

=

-D,

z < P

(814

V*(Z)

=

- m a + P)Aa + z)?,

z

P

(8 1b)

2

ATOM DIFFRACTION FROM SURFACES

143

With the parameters D = 14.4 meV, (Y = 1.33 A, and p = 0.93 A, the experimental ej are well reproduced (Garciaet al., 1980); moreover the C, coefficient for the zP3dependence at large distances [see Eq. (7)] turns out to be 166 Hi3 meV, close to the theoretical values 173 A3meV of Bruch and Watanabe (1977) and 186 A3 meV of Vidali et al. (1979). However, the matrix elements are not in agreement with the experiments. Hutchison and Celli (1980) suggested that this disagreement depends on the assumption of a R-independent flat bottom. By assuming a periodic variation of the well depth AD(R) = 2D,[~0~(2rr/a)X + cos(2r/a) Y

+ COS(~T/U)( Y - X)] (82)

in the range 0 < z < p (see Fig. lo), Hutchison and Celli showed that experimental matrix elements can be well reproduced if D, is chosen equal to -0.25 meV. The relevance of the HCS with a modulated well depth rests on separating the role of the different potential parameters: represents the hard-wall corrugation function; the well depth D and width p are related to the BSR levels, whereas the bottom variation amplitude D, determines the splitting of the deepest levels. A different model potential well suited to explain the experimental results is the Tommasini potential

V~(Z) = D{[1

+ A(z

- ~ , ) / p ] --~2[1 ~

+ A(z

- ~,)/p]-’}

(83)

discussed extensively by Mattera et al. (1980a). In Eq. (83), D is the well depth, A a reciprocal range parameter, p a variable exponent such that - 1 Il/p I1; A essentially describes the width of the potential, while p is related to asymmetry; z, denotes the position of the minimum and is a parameter not sensitive to scattering. The three-parameter Tommasini potential has the advantage of containing as special cases such widely used potentials as the ( 2 4 n) Lennard-Jones potential (X = p/z,), and the Morse potential Cp + 03). Its energy levels have approximately the form given by Eq. (49). The best-fit parameters needed to describe the ej levels by Derry et al. (1979) are D = 16.1 meV, A = 1.485 A-l, and p = 4.4. The flexibility of the Tommasini potential was proved by using it in a large number of systems; it does not give the correct zP3 asymptotic behavior, but the observable BSR levels approaching the continuum are likely to be nonsensitive to the long-range zP3 term. Starting from model potential (83), a parametrization of the first (and only relevant) Fourier component for He-graphite was proposed by Boato et al. (1979b) in the form V d z ) = -PloD[1

+ X(Z - z e > l / ~ ) - ~ ~

(84)

144

G. BOAT0 AND P. CANTINI

t

FIG.22. Zero-order (V,) and first-order (Vl,,) Fourier components of the He-graphite potential.

The measured matrix elements are well represented by using Eq. (84) with a = 3 and plo = 0.019; the choice a = 3 gives a much better fit to the experimental results than the more conventional choice a! = 2, used by several theoreticians in conjunction with a Morse potential having a modulated repulsive term. However both a! = 2 and a = 3 give too large a corrugation-compared to the HCS model-within the experimental energy range from 20 to 65 meV. Figure 22 shows V,(z) and VlO(z) determined by Boato et al. (1979b). The whole potential V(R, z) is that represented in Fig. 3. A third approach to the He-graphite potential is based on the assumption that V(r) can be expressed as the sum of He-C pair interactions. A potential of this type was first proposed by Steele (1973, 1974) and further critically discussed by Carlos and Cole (1978, 1979, 1980a). The calculation assumes that the potential is a sum of contributions arising from different graphite layers, each labeled by the integer n , in the form

ATOM DIFFRACTION FROM SURFACES

145

In Eqs. (85a,b), z, = z + nd ( d = 3.37 A) and 4 is a lattice vector in the basal plane. Further, j = 1, 2, 6, and 6, being the positions of the two carbon atoms in the unit mesh. By Fourier transforming Eq. (85b) one obtains the contribution to Vc(z) originating from the nth plane as

V,, = Pcd-' /u,c, exp( - iG * R)u(z,, R) d2R with

PC = exp(-

iG 6,)

+ exp( - iG

6,)

(87)

Starting from different He-C pair potentials u(r), Carlos and Cole (1978, 1979, 1980a) calculated the He-graphite potential and, from it, the energy levels ej and the matrix elements (xjlVlo(xj,).A first attempt to fit the experimental data was made by assuming a variety of central potentials u(lr - rll). With these potentials the energy levels were reproduced by an appropriate choice of parameters. In each case, however, the matrix elements computed with isotropic forms of u were smaller by 20-70% than the experimental values. Only by using anisotropic pair potentials was good agreement obtained with the experimental matrix elements (Carlos and Cole, 1979, 1980a). The necessity of such a choice was explained in terms of the anisotropy of dielectric function and charge density in graphite. The resulting Fourier components Vo(z) and Vlo(z), are very close to those shown in Fig. 22 in the region of the well. High-order Fourier components are again found to be negligible. However, the potential determined by Carlos and Cole also shows a high-energy corrugation amplitude which is larger than that experimentally determined. Two concluding remarks should be made. First, there is no unique potential that explains the experimental data. However, the physical arguments put forward by Carlos and Cole render the potential of Figs. 3 and 22 highly reliable. Second, excellent agreement is found between atomic beam scattering data and accurate measurements of thermodynamic properties of He submonolayers physisorbed on grafoil (Elgin and Goodstein, 1974; Elgin et al., 1978). The binding energy and the low-coverage specific heat deduced from these measurements (see also Silva-Moreira et al., 1980) are in excellent agreement with those evaluated from bandstructure calculations making use of the matrix elements derived from atom diffraction (Carlos and Cole, 1980b). We should like to emphasize that the He-graphite potential used prior to atom-scattering experiments was significantly in error; it overestimated the well depth by about 20%, also giving it a somewhat different R dependence. [For a critical review of this subject, see Cole et al. (1981).]

146

G. BOAT0 AND P. CANTINI

VI. INFORMATION

ON

SURFACE LATTICE DYNAMICS

Whereas bulk-phonon disperson relations are currently determined from inelastic neutron diffraction, little is known experimentally about the dispersion relations of surface phonons. Although such knowledge is an important prerequisite for a complete understanding of solid surfaces and their dynamical properties, standard techniques such as EELS or Raman and Brillouin scattering are not very effective in studying surface phonons. Because of their strong interaction with the surface and their thermal energy, atomic beams provide a unique tool for investigating lowfrequency surface phonons. The required angular and energy resolutions have become available only in the last few years; very recent experiments have finally shown that the success of atomic beams to perform highresolution surface-phonon spectroscopy is now guaranteed. The older efforts to obtain information on surface lattice dynamics were devoted either to “integral” studies of the inelastic atom scattering, through the D-W factor, or to “differential” studies, where structures in the inelastic angular distribution were uniquely related to the energy ha, and to the parallel wave vector Q of the exchanged phonons. Both old and quite recent experimental results will be discussed in the following sections. A . Debye- Wuller Factor

As discussed in Sections II,C and III,B,l the D-W factor exp(- 2W), operatively described by Eq. (ll), contains information on the mean square displacement (2) of the surface atoms, which in turn depends on the phonon spectrum. It was shown that due to the potential well, to the finite size of the atom probe, and to the nonnegligible interaction time, appropriate corrections must be introduced in the ordinary expression of 2 W . Unfortunately, no experimental evidence exists to demonstrate without doubt the validity limits of each proposed correction. Hereafter, we shall give a short description of the results and their possible explanation. In many experiments the D- W factor was measured from the thermal attenuation of the specular beam, made of He, H, or D. In order to apply the conventional formula, now expressed by 2W

=

((Ak * u)’)

=

2k&(u2,)

(88)

at least two corrections should be considered for reasons discussed in Section III,B,l, namely, the Beeby correction (kgz is substituted by kh2, =

I47

ATOM DIFFRACTION FROM SURFACES

k;, + 2rnD/h2, D being the well depth) and the size-effect correction. The validity of the Beeby correction is now generally accepted. It was first verified by Hoinkes et a/. (1972b, 1973) for the system H- LiF. The results are shown in Fig. 23, where the effect of taking into account a realistic well depth is clearly proved, The same authors applied also a size-effect correction, but did not use the better justified expression later proposed by Armand. By using the Debye model, (u:) can be expressed by

where M is the mass of surface atoms, eSis the surface Debye temperais the Debye function. Hoinkes and co-workers applied to ture, and their data the high-temperature limit of Eq. (89); that is,

+,,

(uz) = 3hTS/MkB@

(90)

A value for eSof 415 ? 42 K was found, compared with the bulk value 8 = 730 K . The ratio eS/0agrees well with the theoretical predictions (Allen et a/., 1969; Chen et al., 1972), based upon the weaker forces experienced by the surface atoms of the solid. An extensively studied surface was NaF(001) (Wilsch et al., 1974; Krishnaswamy et a/., 1978) using H, D, and He probes. The surface

TS . ( c o s 2 0 0

+&J

-

0

400K 8M)K

1200K 1600K

FIG.23. The D-W attenuation in scattering of H from LiF(001). Only the introduction of a well depth D = 18 meV makes the experimental points align on the same straight line: D = 17.8 meV (a), 0 meV (b), 71 meV (c); Ts = 725 K ( O ) , 475 K ( x ) , 225 K (0). [From Hoinkes et a / . (1973).1

148

G . BOAT0 AND P. CANTINI

Debye temperatures obtained in the three different experiments (370,416, and 425 K, respectively, without the size-effect correction), compare rather well with each other if account is taken of the experimental errors; eS/8is again in agreement with theoretical predictions. Another accurate study of the D-W factor was carried out by Lapujoulade et al. (1980a) for the system He-Cu(001); in this case also the conventional theory appears to work quite well when both well-depth and size-effect corrections (Armand et al., 1977) are taken into account. In spite of this apparent success, several doubts were thrown upon the D-W elementary expression given by Eq. (88). The cited paper by Krishnaswamy et al. (1978) showed that with or without the Beeby correction, Eq. (88) is not able to explain the angular dependence over the entire range 0" < O0 < 90". Much stronger doubts were earlier aroused by the large elastic diffraction probabilities found for Ne scattered by LiF(001) (Boato et al., 1976a). Other discrepancies were observed in the thermal attenuation measured in Cu(OO1) by using beams of both He (Mason and Williams, 1978) and Ne (Lapujoulade et al., 1980a). Some of these anomalies are qualitatively explained by Levi and Suhl (1979) and by Meyer (1981), but several features are not well understood. As expected, it seems certain that the more one departs from the fast-collision approximation, the stronger are the anomalies. But, as a matter of fact, a complete systematic experimental study of the D- W factor in atom scattering is yet to be accomplished. B . Structures in the Angular Distribution

In a number of cases, structures in the angular distribution of the scattered intensity are observed. In particularly favorable cases and under the condition that one-phonon events are dominant, it is possible to relate univocally the kinematic parameters ( O o , +o; Of, +f) of the structures to the energy fiwq and parallel momentum hQ of the exchanged phonons (see Section 1I.C). One kind of such experiments is based on the existence of a spatially forbidden region for inelastically scattered particles. At the boundary of this region particles associated with Rayleigh phonons should concentrate. The method was suggested by Williams (1971b) and an interpretation of the results for both LiF (Williams, 1971b) and NaF (Mason and Williams, 1974) in terms of surface-phonon dispersion relations was given. However, Williams' analysis was incorrect since he assumed Q to be perpendicular to the plane of incidence, and, in fact, the results for NaF were unsatisfactory. A reinterpretation of Williams' results was recently given by Avila and Lagos (1981),who observed, following a remark

ATOM DIFFRACTION FROM SURFACES Q,

[k’]

\

\

(a)

\

\

I

F

’1

0.5 L)

\

I

1.0

[1013rod sec-l]

NaF(001)

149

. /’

/ W W R

b

< 100 > < 110 >

FIG.24. Data of Mason and Williams (1974), as analyzed by Avilaand Lagos (1981) (a). The Rayleigh-phonondispersionrelation (b)is obtained: G = A,(00); 0, (1i)annihilation; 0 , (11)creation; +, (01) annihilation; 0,(Of) creation.

by Benedek (1979, that the kinematic relations (13) and (14) applied to Rayleigh waves identify for each elastic peak F a cone in k space having its vertex in kF. This cone corresponds to the allowed region for onephonon inelastic events in the real space. A “frontier of allowed zone” thus exists for out-of-plane scattering, which is experimentally observable as a sudden cutoff in the inelastic intensity (this fact was already noted by Williams). Avila and Lagos pointed out that for Rayleigh phonons, the intensity should have a singularity at the frontier of allowed zone. The angles corresponding to such singularity determine both kw, and Q of the involved phonons. The data of Mason and Williams (1974) on NaF are reported in Fig. 24 together with the proposed interpretation. The Q values now obtained from the experiment tend to align along the ( 110) direction (dashed line in Fig. 24a), whereas the w , Q plot (Fig. 24b) gives the surface-phonon dispersion relation. This compares well with the expected linear part of the Rayleigh-phonon dispersion relation (dashed straight line). A similar analysis can be made for LiF results.

150

G . BOAT0 AND P. CANTINI

A different kind of experiment is based on the study of inelastic BSR structures (see Section 111,B73). An inelastic BSR, characterized by (ej, N), selects at each incident angle and energy a well-identified family of phonons. Their frequencies oNJare given by the kinematic condition

k$

?

2rn~,,~(Q)/h - (K,,+ N

f Q)2 =

2rnej/A2

(91)

On the other hand, when a scanning of the angular distribution is taken, each final scattering angle 0, corresponds to inelastic processes which select a family of phonons of(Q) given for in-plane scattering by the parabolic equation.

kg

f 2rnof(Q)/A -

(I?,

+ F f Q)”sin2

0, = 0

(92)

Since the family of phonons yielding a resonant contribution [Eq. (91)] and that selected by 8, [Eq. (92)] in general do not coincide, the location (do, 0,) of a sharp resonant structure in the tail of the diffraction peak fixes through conditions (91) and (92) both the energy Awq and the parallel momentum AQ of the particular phonon involved. Thus, this procedure is also capable of yielding a dispersion curve without energy analysis of the scattered particle. The existence of inelastic BSR structures was first demonstrated by Cantini et al. (1976, 1977), who also showed that a rough dispersion relation compatible with Rayleigh phonons could be determined for He- LiF. More recently, the inelastic BSRs were observed and carefully studied in the system He-graphite (Cantini and Tatarek, 1981). In Fig. 25, typical examples of inelastic BSRs measured with high sensitivity are shown. They were observed in the tail of the specular peak near the scattering angle e, = 57”. These structures correspond to the transition into the BSR labeled eO(10-01) after a phonon has been exchanged with the surface. As the incident angle is increased the line-shape changes from a minimum to a maximum (around O0 = 51”) and again to a minimum. This is due to the intervention of the elastic BSR labeled el( 10-01). A double resonance takes place, as described in Section III,B,3 [process (4) of Fig. 111. Further, Cantini and Tatarek (1981) and, in more detail, Boato et al. (1982) showed how a systematic study of inelastic BSRs at different incident angles do and final angles 0, is able to furnish a dispersion relation for the graphite surface phonons. The dispersion relation is indistinguishable, within sizable experimental errors, from the TA, branch of the bulk graphite, relative to phonons traveling along the hexagonal plane. Figure 26 shows a different type of inelastic BSR structure which appears in the scattered intensity, namely, that corresponding to the transition in a bound state after the creation of one phonon, without a change of G. The process, called “specular inelastic” selective adsorption, was

151

ATOM DIFFRACTION FROM SURFACES

. . . .

00

52.45O

. . ..

52.20°

. .

. .

52.03'

.. ..

51.86' 51.70°

. . .. . .

51.53' 51.36' 51.20' 51.03O 50.86O 50.70"

I

55"

I

Of

I

60'

FIG.25. Inelastic BSR structures observed in the tail of specular peak in He-graphite: k, = 9.07 A, & = 30". [From Cantini and Tatarek (1981).]

experimentally studied by Cantini and Tatarek (1982). The resonant process appears as a sequence of maxima, each associated to a level ej of Vo(z),which become particularly strong at low surface temperatures Ts . The intensity of each inelastic structure of Fig. 26 can be understood from Eq. (73). Applying the approximation given by Eq. (76), one obtains C == Si&nSoo/Sio(l -

I ~ o o l )2: Soo/(1 -

ISOol)

(93)

where the S terms are elastic amplitudes. As the corrugation of graphite is small, lSoolapproaches unity and ICI >> 1 (strong maximum). In addition, the temperature dependence is weakly influenced by Bose statistics (phonon creation is in fact involved), but strongly affected by the D-W factor; the result is that as the temperature increases, the inelastic maxima reduce their height in the same way as the elastic peaks. Once again, the angular position of the observed maxima can be used to obtain the phonon dispersion relation.

152

G . BOAT0 AND P. CANTINI r

-

10

€2

€1

€0

FIG. 26. Specularly inelastic BSR structures in He-graphite: ko = 11.05 00 = 74", l#Jo = 0".

kl,

Finally, Brusdeylins et al. (1981b) studied inelastic BSRs in the systems He-LiF and de-NaF, in connection with measurements of energy losses by the time-of-flight (TOF) method (see Section V1,C). They observed a large number of structures in the inelastic background between diffraction peaks, most of them maxima, but some minima, which they attributed to resonance transitions followed by the exchange of one phonon [selective desorption, or process (2) in the scheme presented in Fig. 113. Brusdeylins and co-workers prove that one-phonon exchange takes place, by directly measuring the energy of the outgoing particles. The large number of structures seen by them is due both to the high monocromaticity of the incident beam and to the presence in He-alkali halides of a larger corrugation amplitude. The observed resonant events and line shapes can be understood in the framework of the theory by Cantini and Tatarek (1981). C . Energy-Loss Measurements

Energy analysis of the scattered beam is a prerequisite for a valuable surface-phonon spectroscopy by atom scattering. We limit our attention to one-phonon inelastic scattering of He from LiF and mention an early attempt to reach this goal made by Fisher and Bledsoe (1972), who observed some structure in the TOF spectrum. Their energy resolution (AEIE = 20%) was largely insufficient to give a clear interpretation of the data. Later, Horne and Miller (1978) used better resolution to obtain evidence for a single Rayleigh mode, similar results were reported by Feuerbacher et al. (1980).

153

ATOM DIFFRACTION FROM SURFACES

It was the experiment by Brusdeylins et al. (1980, 1981a) which showed that the crucial point was the angular and energy resolution of the apparatus. Using a supersonic expansion from 200 atm at T = 80 K through a 5-pm-diam. nozzle, these authors were able to get a beam having an angular spread [full-width at half-maximum (FWHM)] of 0.30" and an energy spread (FWHM) of less than 2%. Long flight distances were used, and very sharp TOF spectra were obtained, with an energy resolution of about 0.6 meV. A typical TOF spectrum obtained by Brusdeylins et al. (1981a) for in-plane scattering is shown in Fig. 27, at Ts = 300 K. Six peaks are well resolved. Peak 3 is the remnant of elastic scattering [the (fi) diffraction peak is near in angle]; peak 2 is due to a weak but widespread tail in the incident-velocity distribution. The other peaks are related to one-phonon events. Three of them, namely, 1,4, and 6 are due to Rayleigh phonons, whereas peak 5 is very likely a structure in the bulk-phonon contribution at low frequencies. This can be understood more easily by looking at Fig. 28, which is a plot of the measured frequency w versus parallel momentum transfer AK in the extended Brillouin-zone representation. The points correspond to events detected in a large number of TOF spectra taken at different values of Oo . The solid lines show the theoretically expected Rayleigh-phonon dispersion curve (Chen et al., 1977; Benedek, 1976), whereas the dashed line is the parabola 0

= (A2k;/2m){[(AK/k, cos 0,)

+ tan OOl2 -

1)

(94)

derived from the kinematic conditions (13) and (14), if account is taken of the fact that in the experimental apparatus of Brusdeylins and coworkers, the difference Of - Oo was fixed and equal to 90".Equation (94) is then the locus of possible in-plane phonons events (w. A m , at the incidence condition ( k o , 6,). The location of the six peaks detected at ko = 6.06 k1 and Oo = 64.2" in the TOF spectrum of Fig. 27 is indicated in Fig. 28. The results can be summarized by saying that for the first time, the full dispersion curve for Rayleigh phonons was measured, and a high accuracy was attained. The curve agrees with the theoretical expectations, with an exception made near the Brillouin-zone boundary; the lower frequency observed is likely due to changes in interatomic forces (polarizability, relaxation, etc.) not taken into account in the surface-dynamics calculations. Bulk-phonon events (as peak 5 in Fig. 27) are shown to participate in the inelastic scattering, but the dominance of Rayleigh phonons, particularly at low frequencies [see Eqs. (67) and (68)], is clearly demonstrated.

154

G. BOAT0 AND P. CANTINI

1.5 Time

2.0 of

Flight

2.5 [msec]

FIG.27. A high-resolution TOF spectrum for the system He-LiF(Wl):( 100) azimuth. k, = 6.06 k*, Oo = 64.2". [From Brusdeylins et al. (1981a).]

The measurements by Brusdeylins and co-workers fulfill the longawaited expectation of the specialists in atom scattering and, more important, open a fruitful field for future developments. A second important result in energy analysis is given by the experi-

FIG. 28. A plot of measured inelastic TOF peaks in the o,AK diagram. The whole Rayleigh-phonon dispersion curve appears. [From Brusdeylins et al. (1981al.l

ATOM DIFFRACTION FROM SURFACES

155

ments of Mason and Williams (1981), who used a LiF(001) surface as the energy analyzer (this technique is similar to the triple-axis spectrometer used in inelastic neutron diffraction). They applied their method to the Cu(OO1) face and to an ordered monolayer of Xe on the same surface at low temperatures. They observed both surface and bulk phonons and were able to measure a part of the Rayleigh-phonon dispersion curve for Cu(001). Of greater interest are the measurements carried out on the Xe overlayer: Mason and Williams find that the layer is ordered, as expected from previous LEED measurements. More important, crystal analyzer scans at fixed B0 were made; the data correspond to energy losses or gains of 2.5 and 5.0 meV, showing that a single frequency is involved. This frequency should correspond to the vibration of Xe atoms, perpendicular to the surface; the dispersion relation appears to be flat, resembling that of an Einstein oscillator. To end this section we should like to mention that interesting energychange measurements by TOF analysis were made in the scattering of Ne from LiF(001) (Semerad and Horl, 1982; Mattera et al., 1980b). Here the time spectra show only broad structures, which appear to be related not only to multiphonon effects, as expected from qualitative arguments (large incident mass), but also to one-phonon events, thus confirming the large D-W factor previously reported (Boato et al., 1976a). VII. CONCLUSIONS The present review shows how the atom-surface scattering method is rapidly advancing along different pathways, with respect to both experimental and theoretical techniques. We may distinguish between two main lines of development, namely, elastic diffraction and inelastic scattering. The first line has already reached a highly satisfactory level from the experimental point of view. Elastic He diffraction has become a commonly used technique for determining structures of the topmost surface layer, thus giving independent or complementary information compared to other standard techniques such as LEED or ion scattering. In connection with this, the examples reported on semiconductor reconstructed surfaces and H-covered metal surfaces should be convincing about the capabilities of the method. A systematic study of physisorbed monolayers and other surfaces largely affected by destructive techniques such as LEED might be a subject for future investigation. Further, we have repeatedly emphasized that the study of elastic BSR structures is a subject of great interest. Since the experimental technique is quite simple and accurate, this method is a powerful one for the deter-

156

G. BOAT0 AND P. CANTINI

mination of atom-surface potentials in the region of the well. The extension of the method to state-selected beams of H, , D, , and HD might clarify the dependence of the gas-surface potential on the rotational state of the molecule. Along with experiments, we have described how simple elastic theories, capable of explaining the experimental results, were developed. Although the HCS model was proved to be sufficiently flexible for describing most diffraction patterns, further theoretical effort should be made toward the study of more realistic “soft” potentials. The second line of research, i.e., inelastic scattering, is just now in full development, particularly since the experimentally needed energy resolution has only recently been reached. A first problem in inelastic scattering is the proper use of the D-W factor exp(-2W). In a few cases the conventional formula 2W = ((Ak * u ) ~ )modified , by appropriate corrections, was proved to be able to furnish reliable information on the mean square surface displacement ( u z ) (or on the surface Debye temperature @). However, the behavior of 2W becomes more intricate when multiple scattering is involved or heavier (and slower) incident atoms are used. A deeper study of the D- W attenuation, both by systematic experiments and by more refined theory, is still necessary. On the other hand, recent measurements have definitely shown the usefulness of atomic beam energy spectroscopy in giving direct information on the spectrum of surface phonons. The TOF technique seems now capable of determining the phonon-dispersion relation for several bare and adsorbate-covered surfaces. The theory of inelastic scattering needs further refinement, particularly for metal surfaces, where the energy transfer from incident atoms to lattice ions is still a problem owing to the presence of free electrons. Anyway, the theoretical approach should go beyond the vibrating HCS model; moreover, the effect of the surface-projected phonon density [Eq. (68)] on the inelastic scattering probabilities should be further clarified. With these refinements, the He atom ought to become a very useful probe for the investigation of surface dynamics. ACKNOWLEDGMENT The authors are greatly indebted to all members of the Genoa group on gas-surface scattering for useful discussions and constant cooperation.

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