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Ne diffraction from Cu(110)
B. Salanon
To cite this version:
B. Salanon. Ne diffraction from Cu(110). Journal de Physique, 1984, 45 (8), pp.1373-1379.
�10.1051/jphys:019840045080137300�. �jpa-00209875�
Ne diffraction from Cu(110)
B. Salanon
Centre d’Etudes Nucléaires de Saclay, Service de Physique des Atomes et des Surfaces, 91191 Gif sur Yvette Cedex, France
(Reçu le 24 novembre 1983, révisé le 16 mars 1984, accepté le 30 mars 1984)
Résumé. 2014 On présente les résultats expérimentaux de la rétrodiffusion d’atomes de néon d’énergie thermique (64 meV) sur la face (110) du cuivre. La diffraction est observée jusqu’au cinquième ordre, ainsi que le phénomène de résonance avec les états liés. On ajuste les niveaux d’énergie des états liés à l’aide du potentiel de Morse hybride déplacé. Les intensités diffractées extrapolées à zéro kelvin sont reproduites en utilisant le potentiel de Morse corrugué modifié (MCMP) qui est un potentiel non abrupt réaliste. On trouve que, dans sa partie répulsive, le potentiel est plus modulé que dans le cas de la diffraction de l’hélium par Cu(110). Ceci montre que les forces attractives influencent de manière importante la modulation du potentiel total.
Abstract 2014 Experimental results of diffraction of thermal energy (64 meV) Ne atoms from the (110) face of copper
are presented Diffraction is observed up to fifth order and strong selective adsorption features were measured.
The bound state energy levels are fitted using the shifted Morse hybrid potential. Diffraction intensities extrapolat-
ed to zero kelvin are fitted using the so called modified corrugated Morse potential (MCMP) which is a realistic
soft potential. The potential is found to be more corrugated and steeper in its repulsive part than in the Cu(110)/He
case. This shows that attractive forces play an important rôle in the corrugation of the potential.
Classification
Physics Abstracts
68.20 - 79.20R
Introduction
Low energy atom diffraction has already proven to be a useful tool for studying surfaces, but until recently
the use of a rather crude potential prevented from drawing much information from experimental data.
Indeed the commonly used potential was the so
called hard corrugated wall (HCW) that can be written
as :
R and z are respectively the coordinates parallel
and perpendicular to’ the surface, T is a shape func-
tion describing the corrugation of the potential.
In order to improve this situation some authors have
suggested some various soft potentials and solved exactly the scattering equations either in an iterative
Bom type procedure [1, 2] or in a close coupling
type one [3, 4]. Such potentials have been success- fully used in the Cu(110)/He case where it has been shown that a very satisfactory fit of the observed
intensities’ could be obtained [5-7]. Our model is
mainly the modified corrugated Morse potential
(MCMP) defined as follows
Vo(z) is the laterally averaged potential, it exhibits
a well the bound states of which can give rise to
resonances. D is the well depth, x is the range para- meter of the Morse potential. Such a Vo(z) is satis- factory except for the exact shape of its attractive part which has been theoretically shown to behave
like C3lZ3 for z large [8]. It has been shown that this behaviour does not affect the diffracted intensities except near resonances [9]. The G # 0 Fourier com-
ponents are chosen to be exponential functions of z
with a range parameter 3 x. This is in qualitative
agreement with several theoretical calculations [6, 10]. Moreover, with this form the matrix elements
required for the iteration procedure are analytically computable [7].
Such a potential, with conveniently chosen vG’s,
exhibits satisfactory properties, namely it is expo-
nentially repulsive and the corrugation amplitude
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019840045080137300
1374
of its isopotential lines increases with energy thanks to the choice of 3 x in V G(z).
Such potentials can give a good agreement for the
Cu(110)fHe case but also in the Cu(113)/He, Cu(115)/He, Cu(117)/He cases [11]. In this paper we want to show that the potential given by (1) can also successfully explain diffraction data for neon atoms
elastically scattered from Cu(110) with an incident
energy Ei = 63.8 meV. In part 1 we explain how the scattering equations are solved Part 2 is devoted to the study of resonances and to the determination of the parameters of VO(z). In part 3 we fit the experi- mentally observed diffracted intensities and tenta-
tively conclude about the Cu-Ne interaction.
1. Solation of the scattering equations, fitting para- mdem
In order to get the diffracted intensities, we solve the Lippmann-Schwinger equation for the T matrix.
The basic approach, dealt with in several papers [1, 2] is to divide the potential into two parts :
and
U and v are chosen so that the eigenstates of the
Hamiltonian Ho are known, and so that v is as small
as possible. We choose for U a Morse potential that
need not be VO(z), the zeroth order Fourier compo- nent of V(R, z). The T matrix equation is then writ-
Fig 1. - Typical behaviour of V(x, z) for a one dimensional
corrugation. The analytic expression is given by equation (1).
ten as :
with
Equation (3) can be solved by the Born series, nume- rically computed under the Neumann iterative proce- dure symbolized as :
The explicit expression of equation (4) is given in
reference [7] as well as the matrix elements of interest
Practically the fit of experimental data is achieved by varying x, D and the vG’s. As is shown hereafter,
x and D are approximately known when the bound states energy levels are known. The adjustment of
the scattered intensities is achieved mainly by varying
the vG’s, x and D being only slightly changed from
their values given by the bound states.
As the vG’s coefhcients are not easily related to the shape of the potential, we found it more convenient
to use as a parameter a shape function defined as the
isopotential surface for E = E,, the incident energy.
This isopotential surface is given by its Fourier coefficients. Moreover one can use as a starting point
the isopotential surface deduced from overlapped
atomic densities and from a density potential linear
relation [12]. This is of course not exact but turns
out to be useful as a first step in an adjustment pro- cedure.
2. Experiment, selective adsorption, determination of
Vo(z).
2.1 EXPERIMENT. 2013 With the apparatus which was
used for He, we have obtained a neon supersonic
beam (angular width 0.2°, velocity spread AV/V =
4 %. The beam was low frequency chopped, the detec-
tor being an ionization stagnation gauge whose ionic current is measured by a lock-in amplifier. The angular aperture of the detector seen from the sample was
0.50. The base pressure in the UHV chamber was
0.7 x 10-’° torr. The crystal surface was cleaned
in situ by cycles of Ar ion bombardment and anneal-
ing. The crystal was one used previously for He diffraction, it was checked with helium prior to neon
diffraction experiments. For experimental details
see reference [13]. Figure 2 shows a diffraction pattern for the incidence angle 0, = 50.50.
The experiment was performed for one incident
energy (Ei = 63.8 meV, k, = 24.6 A -1 ). The inci-
dence plane contained the 001 > direction. Various incident angles were used and the scattered intensi- ties were corrected for instrumental broadening.
Intensities we measured from 70 K up to 770 K and
extrapoled to 0 K.
Fig. 2. - Diffraction pattern for 0, = 50.50.
2.2 SELECTIVE ADSORPTION. - It is well known that
some information can be derived from the knowledge
of the bound state energy levels of V o (z). These
levels can be deduced from selective adsorption experiments. We recall thereafter the kinematical relation between the incident and diffracted wave vector and energy. The conservation of energy and of parallel momentum can be written as :
ko is the incident wave vector, Ko is the surface parallel component of ko, G is the surface reciprocal lattice
vector of interest, eG is the normal component of the final wave vector. When (k(;)2 is negative the G diffract-
ed channel is said to be closed or evanescent and cannot be seen as a diffracted peak. Nonetheless these closed channels contribute to the diffracted
amplitudes. They give rise to perturbations in the
open channel intensities when the initial state is
degenerate with a z-bound state coupled with it,
when one has h’12 m(k’G)’ = ei, where 8, is the energy of a bound state of Yo(z).
By analysing the intensity of the specular beam as a
function of the incident angle 8i (as shown in Fig. 3)
one can tentatively attribute G vectors and bound
state energy levels to the strongly localized extrema
that can be seen.
We observe strong minima for 8i = 83°, 78.7°, 76.0°, 74.00, 72.5°, 71.5°. These features can be attri-
Fig. 3. - Plot of specular intensity versus incident angle.
The verticals bars indicate the calculated angles of reso-
nance, in the symbol m m designates the index of the
n
reciprocal lattice vector, n being the vibrational quantum number. Vertical bars with a E indicate the emergence of the corresponding G diffracted beam. The discontinuity
of the intensity is due to the flashing of the crystal during
measurements.
buted to resonances with the G = (1, 0) vector. Three
other perturbations can be seen for 0; = 70°, 67.5°, 65.0°. The feature for 0, = 70.0° could be attributed either to a resonance with the G = (1, 0) vector,
or to a resonance with a G = (2, 0) vector on a deeper
level. The features for 0, = 65.00 and 0, = 67.5°
can be attributed unambigously to resonances with G = (2, 0) as the beam G = (1, 0) is open for 0,
68.3°.
Table I shows the energy levels deduced from the
resonances with G = (1, 0), figure 4 shows a plot
of E1/6 as a function of n, the number of the level. The linear dependence is typical of a 9-3 potential, an extrapolation can give two more levels. One gets E7 _ - 1.82 meV and so = - 10.8 meV, the cor- responding resonance angles are 0, = 70.6° for G =
(1, 0), 8 = 87 and OJ = 70.0° for G = (2, 0), 8 = so.
Table I. - Angles of observed resonances with G = (1, 0) and bound state energy levels deduced from the angles.
1376
Fig. 4. - Plot of £1/2 and £1/6 as functions of the vibrational quantum number n. E is the corresponding bound state
energy level. £1/2 is linear in the case of the Morse potential.
So there is a strong evidence that the feature seen
for Oi = 70.0° is a resonance with G = (2, 0) and
s = so = - 10.8 meV. Such a resonance with a =
- 10.8 meV and G = (1, 0) is kinematically unat-
tainable. Some more features of the resonance pat-
tern are also tentatively attributed : namely the
resonance condition for G = (1, 1) is fulfilled for
oi = 810 and B = E1, 0i = 77.40 for 8 = E2, 8 = 83
for Oi = 74.90. Around Oi = 77.40 a very visible dip
can be seen ; for Oi = 81° and 74.20 shoulders can be
seen. This indicates a coupling by the V11 Fourier component, i.e. the presence of an out of incidence
plane corrugation.
2.3 FIT OF THE BOUND STATE ENERGY LEVELS BY A MODEL POTENTIAL. - We use mainly the so called
9-3 potential, the Morse potential and the shifted Morse hybrid potential (SMH).
i) 9-3 potential.
A least square adjustment of the experimentally
observed energy levels gives :
D = 11.9 meV
ii) Morse potential.
As can be seen on table II the Morse potential
is clearly unable to fit all the energy levels and we
just give here the potential obtained by adjusting the
two deepest levels, so = - 10.8 meV, 81 = - 8.3 meV.
We get D = 12.15 meV and x= 1.23 A - 1.
Both these potentials show severe drawbacks.
The 9-3 potential, although it gives a good fit of the
energy levels, cannot be used to find the steepness of the exponentially repulsive part of the potential
used for diffraction. Moreover some authors have discarded that kind of model for theoretical reasons
[14].
The Morse potential does not fit correctly the
energy levels as its attractive part is not satisfactory.
So the value of x thus deduced is doubtful in the Cu/Ne
case. That is why we use a model potential which is qualitatively correct in both the attractive and repul-
sive regions [14]. This shifted Morse hybrid potential (SMH) is written as :
zo and z. are determined by the matching of vo(z)
and of dV’/dz for z = zp. The minimum of Vo(z) being - DO(I + A) = - D’.
Do, Xo and A are fixed by the three deepest energy levels experimentally observed C3 is determined in order to fit as well as possible the upper bound state energy levels. All the energy levels were obtained in the WKB approximation. We get xo = 2.04 A-1, Do = 6.41 meV, A = 0.93 i.e. D’ = 12.4 meV. The fit
of C3 is constrained by the matching conditions for
V o(z) which determine the range of acceptable values
Table II. - Various fits of the experimental bound state energy levels (EXP). 9-3 is the fit with the 9-3 potential (D = 11.9 meV, 6 = 2.86 A). M is the fit with a Morse potential (D = 12.15 meV, X = 1.23 A-1). SMH is
the fit with the shifted Morse hybrid potential (Do(1 + A) = 12.4 meV, Xo = 2.04 A -1). In the case of 9-3 and SMH, C3 is the coefficient of the attractive term.
of C3 (C3 > 1 100 meV A3). So we get a very satis-
factory fit of experimental data with C3 = 1.2 eV A3.
Table II shows the comparison between the experi-
mental values and the energy levels given by various
models. The value of C3 determined with the SMH
potential is higher than the prediction given by Vidali
and Cole (C3 = 450 meV A3 [15]). The ratio of static
polarizabilities a(Ne)/a(He) = 2 is lower than the ratio of C3 coefficients C3(Ne)/C3(He) = 4. But one
has D(Ne)/D(He) = 2. Our value of C3 is probably
model dependent but D is not, so more theoretical work on Van der Waals and repulsive forces is needed
in order to predict a reliable shape for the Cu/Ne potential.
3. Fit of the observed intensities.
3.1 REMARKS ABOUT THE 0 K EXTRAPOLATION. -
For Ei = 63.8 meV, on account of the high mass of
neon atoms, a substantial part of the incident beam is inelastically scattered So the signal given by the
detector is the superposition of the elastic and of the inelastic part of the scattered beams. The elastic part follows a Debye Waller type law but the inelastic part decreases when temperature decreases [16].
In many cases, the extrapolation to 0 K is made unprecise because of this superposition. This is the predominant source of errors for weak diffracted peaks. One can say that the precision for normalized
intensities lower than 0.10 is about 20 % ; it is improved
for intensities higher than 0.10 or 0.15 to about 10 %.
3.2 DETERMINATION OF D AND x. - In order to get
an adjustment of the observed diffracted intensities
we have to vary at least three parameters : x, D and vio (Eq. (1)). As the SMH potential is qualitatively
and quantitatively satisfactory we want to use the parameters of its repulsive part : xo, Do and A, to fix x and D. So we use x=xo and D=D’=Do(1 + A).
As can be seen on figure 5, the thus obtained Morse potential is different from the SMH potential only
in the attractive part This is unimportant as the exact shape of the attractive part is not very sensitive for the calculation of the diffracted intensities far from
resonances [9]. Of course the values of D and x can be varied to improve the fit but the initial values deduced from the adjustment of the SMH potential prove to be
near to the final choice. As the extrapolated intensities
are not very precise it is difficult to determine x with
a precision better than 10 %. In fact a satisfactory global agreement is obtained with x = 1.9 A-1 and D = 12.2 meV as can be seen on figure 6. x = 2 A -1 gives a slightly less good fit. Table III gives the cal-
culated and experimental intensities. We do not fit D which is determined by the bound state energy levels,
moreover the influence of its variation on the inten- sities can be compensated by a variation of corrugation
Fig. 5. - Comparison between the Morse potential used
for diffraction and the shifted More hybrid potential used
for the fit of bound state energy levels. Although very
different in their attractive part, these two potentials are
similar in their repulsive part
within a range of approximately 1 meV. We do not directly adjust the vG’s but rather the shape of the isopotential surface for E = Ei. The in-plane dif-
fracted intensities are normalized to unity. In spite
of the evidence of a slight out-of-plane corrugation
we use a one dimensional corrugation for in-plane
diffraction. Indeed the peak to peak ratios are not
much changed for in-plane beams when the out-of-
plane corrugation is weak. By using a cosine shape
for the isopotential surface we find a good agreement with data for a total amplitude Az = 0.23 A. The
values of VG’s are then vlo = 0.040, V20 = 0.008.
We have also checked that if the deepest level
s = - 10.8 meV is not used, thus giving D ~ 9 meV, then it becomes impossible to get a fit as good as
with D = 12 meV, while keeping x = 1.9 A -1.
1378
Table III. - Comparison of the observed intensities with theoretical results. The experimental intensities were
normalized to unity with respect to U, the observed in plane total intensity. Upper values are experimental, lower
ones are calculated using MCMP (D = 12.1 meV, X = 1.9 A-1, 4Z = 0.23 A).
Fig, 6. - Comparison of the observed diffracted intensities and of the results of diffraction calculations using the MCMP.
D = 12.1 meV,x= 1.9 A-’, Az = 0.23 A.
4. Conclusion
We have shown that our model potential is able to fit correctly the data within experimental errors. This
potential is the sum of a corrugated repulsive part
and of a non corrugated attractive part The corru- gation amplitude of the isopotential surface for E; = 63.8 meV is Az = 0.23 A. This can be compared
with the corrugation found for the Cu(110)/He case
where the corresponding amplitude is Az = 0.13 A [7].
The high corrugation in the case of Cu(110)/Ne as
well as the occurrence of clear selective adsorption
features show that the theoretical density potential proportionality relation cannot be used directly.
Indeed Puska et ale have calculated immersion
energies for He and Ne embedded in an electron gas.
They show density potential relations for both gases [ 17]. The proportionality constant is P whose
values are BHe = 270 eV(au)3, BNe. = 900 eV(au)3, showing that Ne atoms should see lower densities
than He and consequently be reflected by a lower corrugation. This does not seem to be the case. As
the above values of fl are obtained in a local theory including exchange and correlation effects it seems
that non local polarization forces should be included in order to understand the high value of Az. As Ne
is more polarizable than He, Van der Waals forces
can make the Ne atom penetrate more deeply in the potential and consequently experience more corrugat- ed electronic isodensities. However the high range parameter (X = 1.9 A)-1 for Ne compared with that
of He (X = 1 A)-1 remains difficult to understand in terms of a local density formalism, even when corrected
by non local attraction terms. More precise calcu-
lations of the total metal/Ne potential are needed Acknowledgments.
I would like to thank Drs. G. Armand and J. Lapu- joulade for their, help during this work and M. Lefort,
Y. Lejay and E. Maurel for their important contri-
butions in obtaining experimental data.
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