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Mean square displacement of xenon atoms in an epitaxial monolayer on (0001) graphite
Solid State Communications,VoI. 15, pp. 1585—1589, 1974.
Pergamon Press.
Printed in Great Britain
MEAN SQUARE DISPLACEMENT OF XENON ATOMS IN AN EPITAXIAL MONOLAYER ON (0001) GRAPHITE J.P. Coulomb, J. Suzanne, M. Bienfait and P. Masri Laboratoire de Croissance Cristalline Associé au C.N.R.S., Universite D’Aix-Marseile II, Centre de Luminy, 13288 Marseile Cedex 2, France (Received 22 March 1974; in revised form 6 May 1974 by E.F. Bertaut)
The thermal variation of the mean square displacement of xenon atoms adsorbed in epitaxy on (0001) graphite is measured by low energy electron diffraction in the 48—73°Ktemperature range. Lattice dynamics calculation is in good agreement with experimental results.
WHEN MEASURING the mean square displacement of surface atoms by low energy electron diffraction (LEED)1 one faces two types of difficulties in the interpretation of the results. First, one uses the kinematic scattering theory, although the dynamic effects occuring in LEED are far from being negligible. However from a simple example Jepsen et al.2 were able to estimate the accuracy of the kinematic determination of the surface Debye temperature to be 10—15 per cent. Secondly, very little information is available about the exact penetration depth, although one knows that the penetration of low energy electron concerns only the surface first layers. Therefore the experimental value of the mean square displacement is an average on the first surface layer atoms. This average approaches the surface displacement value at very low energy. Usually the (u2) of the surface are obtained at about 50 eV; however, one cannot estimate the accuracy of the determination, Recently, Theeten et al.3 proposed a method avoiding these two difficulties. They had direct access to the vibrations of the first atomic layer of the surface by minimizing the multi-diffraction effects. This method applies to a two-dimensional layer adsorbed in epitaxy on a crystal surface. The thermal behaviour of the superstructure spots is measured for a few incident beam energies fulfilling the following criterium. Dynamical effects due to the substrate have to be negligible, i.e. that its diffractedbeams have to be few
and to have low intensities. This means that we have to use a low energy incident beam giving rise to superstructure spots more intense than the ones coming from the substrate. We report here the results obtained in a particularly favorable case; xenon adsorbed on (0001) graphite. This system has several advantages: —
The first condensed atomic layer is in epitaxy
on graphite. It induces a ~~J3 X \/3 superstructure in LEED.4’ ~ There is an incident energy fulfilling the criterium of Theeten et al. ~ Xenon is-one of the few crystals having a kinematic LEED spectrum.6 The mean square displacement (u2) can be computed since a realistic lattice dynamics model has been previously achieved.7 Indeed tile atomic positions in epitaxy on the substrate as well as the force constants between atoms are known. —
—
—
EXPERIMENTAL DETERMINATION Previous publications8’9 provide the temperature and pressure ranges in which the first atomic layer adsorbed on (0001) graphite is stable. We fixed the pressure at about 1 X l08torr and changed the ternperature from 48 to 73°K.At the lowest temperature 1585
1586
MEAN SQUARE DISPLACEMENT OF XENON ATOMS
reached, the second layer can only appear in the 1ff’7 torr range. The crystal surface is studied by LEED. As soon as the first layer is condensed, the ‘~/3X ~J3superstructure spots appear~’5This means that xenon is in epitaxy according to the model shown in Fig. 1. Simultaneously, the intensity of the graphite lattice reflections strongly decreases. For instance the intensity of the specular spot is about five times as low. Strong absorption of electrons by xenon has already been noticed~9 It is due to important inelastic processes and is characterized by a very short extinction length (2.5—5.5 A) when the energy varies from 20 to 55 eV. The superstructure spots are visible for only three energies 35, 64, 95 eV; the one at 95 eV can hardly be distinguished from the background. In order to obtain reproducible measurements, we anneal several times the two-dimensional crystal. The crystal temperature is stabilized at 1/4°K. It is increased or decreased by steps of about 2°K from 48 to 730 K in a few minutes. A decrease of the sweeping rate does not modify the result. The intensity of the superstructure diffracted beam is measured by a photometer, which records the intensity variation versus temperature. The background is determined either in the neighbourhood of the spot, or at the place of the spot before xenon adsorption. The same values are obtained in both cases.
Vol. 15, No. 10
Where I(T’) and 1(T) dre the intensities at temperatures T’ and T respectively; Ku2 (T’)) and Ku2 (7)) are the corresponding mean square displacement of xenon atoms in the direction of the scattering vector; 2~is the angle between incident and diffracted beam; A the wavelength of the incident electrons. The incident electron energy is corrected for the inner potential and the contact potential difference between the xenon crystal and the electron gun filament. This correction amounts to a 10 eV shift to higher energies.6 Measurements are carried out at normal incidence. The value of Ku2) occuring in (1) corresponds to an angle of about 17° between the direction of vibration and the surface normal. The experiments have been repeated several times by increasing or decreasing the temperature on different two-dimensional xenon crystals. The intensity variation versus temperature is quite reversible during one experiment. The reproductibility a few days later is not so perfect, but it can be considered as quite satisfactory. Fig. 2 shows the vanation, during a few experiments, of Ku2(fl)—Ku2(48°K)) = ~(u2) vs temperature for the atomic layer of xenon in epitaxy on (0001) graphite. To the measurements carried out at 35 eV, we have added a determination at 64 eV. In this case the intensity of the 10 spots is close to the one of the superstructure, as can be seen on the photograph published in.5 The value of Ku2) is found to be close to that obtained at 35 eV. This shows that at 64 eV the dynamical effects coming from the substrate are also quite negligible.
At 35 eV the diffraction pattern merely exhibits the central hexagon of the ~/3 X ~~13superstructure. The spots resulting from the supçrposition of the beams diffracted by xenon and graphite have an almost zero intensity. In this case Theeten’s et al.3 criteriurn is fulfilled: the substrate backscatters no beams that might contribute to multidiffraction processes in the superstructure spots. Moreover we know that xenon6 induce negligible multiple scattering effects in LEED. Therefore we can analyse the thermal behaviour of the adsorbed xenon by means of the kinematic theory, in which case 10 the diffracted intensity follows the equation: log~~= 16~2cos2ø[~ 2(T’))] 2(fl)—~ 1(T) X
(1)
~ 4,26A~ FIG. 1. The xenon monolayer in epitaxy on (0001)
graphite.
Vol. 15, No. 10
ijZ~.
MEAN SQUARE DISPLACEMENT OF XENON ATOMS
~ /
/ 40
~,/
/
/
,
/
~
4/’
x
/0
20
4.,
1587
17 have proposed procedure is too elaborate, Masrithe et al. an expansion method to shorten analytical calculation. Schematically, one has u2> = KU2)E + (u)~+ Ku2) 4 +... (2) 2)E stands for the mean square displacement where Ku in the Einstein approximation (diagonal part of the D matrix) and the other terms are the 2’~’, 4th . order corrections. The model describing the vibrational properties 7 Theonposition of two-dimensional xenon crystal adsorbed graphite has been reported by Suzanne et aL of atoms is given in Fig. 1. We take into account only
~
, +
the central forces between first neighbours. These
/ , I
I
T°I(
FIG. 2. Mean square displacement variation of the xenon atoms in an epitaxial monolayer on (0001) graphite vs temperature with respect to10the atomic dis60 placement at 48°K. Two sets of measurements (+ and
~
,/‘
/
002
/
//
---
graphite.7
The calculation of the mean square displacement of atoms is now well known.16 It requires the knowledge of the dynamical matrix D of the crystal. Generally, this computation determines in an intermediate step, the eigeiivalues of the D matrix. Since this
~I’Thiiix.
/
~
u2
/ /
x,
I. 0,01
-— —
—
0
THEORETICAL DETERMINATION
U~iN
I
~nnard X) incident are curves carried Joneswere electrons. out potentials with Thenormal 35(~ directions eV with and IN of ; one vibration AN (0) see with text). are 64 ateV retical 17°and 12°from the calculated respectively. two kinds The of theoMie
valueOur of Ku2) of adsorbed xenon is definitely higher results call for some comments. First, the than the mean square displacement of graphite atoms.~~’12 The ratio Ku2 )Xe/~phjt~ is about 10. Thus xenon lies on a practically frozen graphite substrate, from the vibrational viewpoint. Secondly, adnon crystal or on a surface (11 1)~315 This shows sorbed xenon vibrates less than xenon atoms in a xethat attrative forces from graphite imply a partial hindrance of the vibrations of adsorbed xenon. This point was already emphasized in an experimental and theoretical determination of the vibrational entropy of the two dimensional xenon crystal adsorbed on
*2
__________________
-I
-
___________________________ I I~-ET1 w445 170 m552 170 lSBT 50
ii
I
100
FIG. 3. Mean square displacement of xenon atoms in a monolayer adsorbed in epitaxy on (0001) graphite vs temperature. The computation was carried out for two directions, the first, normal (1),two the kinds second, parallel (II) to the (0001) plan, by using (AN and IN) of Mie Lennard Jones potential. The values calculated by Allen et aL’3”4 for bulk and (111) surface atoms of xenon crystal are also drawn here.
1588
MEAN SQUARE DISPLACEMENT OF XENON ATOMS
forces can be derived from a Mie Lennard Jones potential.18 The D matrix worked out in the harmonic approximation will be published later. The computation of Ku2) was carried out for the directions perpendicular (1) and parallel (II) to the plan of the xenon layer. The convergence of equation (2) is very good for (ui) ((u~) 2/(u~)a 0.7%). It is a little slower for (u~)((u~)2/(u~)E 30% and (U~)4/(U~)E —
drawn each direction I orFig. lithe3 Ku2) computed Theforresults are reported where we have with two Mie Lennard Jones potentials. The first one called AN 18 acts between all neighbours, the second one called IN acts between first neighbours only. They are both phenomenological potentials and it cannot be claimed that either one has any obviously superior merit. One can see that for both potentials Ku~)is higher than(u~).This is not surprising because the vibrations of xenon atoms do not feel so strbngly the attraction of graphite in the directions lying in the layer as in the direction perpendicular to it.
Vol. 15, No~.10
Figure 3 allows to calculate Ku2) in a direction at 17°from the direction [0001]. The value of (u2(fl) —(u2(48°K)) = i~(u2)for this direction is reported in Fig. 2 in the case of IN and AN poientials. The theoretical curves agree with the measurements within the limit of experimental errors. From these results we can draw the following conclusions: (i) the mean square displacement of atoms in two-dimensional xenon crystal is appreciably lower 13’14 than those of the bulk or (ill) surface xenon atoms reported Fig. 3. This is easy to understand since the binding energy5’9 and the xenon-graphite force constants7 are greater than the correspond~ngparameters associated to the xenon—xenon pair. (ii) the model describing the average vibrational properties of exitaxial xenon on graphite is realistic. (iii) If one works under certain experimental conditions it is possible to use LEED for determining the mean square displacement of atoms in epitaxy on a substrate.
REFERENCES 1.
MAC RAE A.U., Surface Sci. 2, 522 (1964).
2.
JEPSEN D.W., MARCUS P.M. and JONA F., Surface Sci. 41, 223 (1974).
3.
THEETEN J.B., DOMANGE J.E. and HURAULT J.P., Solid State Commun. 13, 993 (1973).
4.
LANDER J.J. and MORRISON J.,Surface Sci. 6, 1(1967).
5.
SUZANNE J., COULOMB J.P and BfENFAIT M., Surface Sci. 40,414 (1973).
6.
IGNATIEV A. and RHODIN T.N.,Phys. Rev. B. 8, 893 (1973).
7.
SUZANNE J., MASRI P. and BIENFAIT M., Surface Sci. (in press).
8.
THOMY A. and DUVAL X.,J; Chim. Phys. 67, 1101(1970).
9.
SUZANNE J., COULOMB J.P. and BIENFAIT M., Surface Sci. (in press).
10.
International tables for X-ray Qystallography Vol. III p. 232 Kynoch Press, Birmingham. (1968).
11.
ALBINET G., BIBERIAN J.P. and BIENFALT M., Phys. Rev. B. 3, 2015 (1971).
12. 13.
BIBERIAN J.P., BIENFAIT M. and THEETEN J.B., Acta Cryst. A29, 221 (1973). ALLEN R~E.and de WETTE F.W.,Phys. Rev. 179, 873 (1969). Phys. Rev. 188, 1320 (1969).
14.
ALLEN R.E., de WETTE F.W. and RAHMAN A.,Phys. Rev. 179, 887 (1969).
15.
TONG S.Y., RHODIN T.N. and IGNATIEV A.,Phys. Rev. B8, 906 (1973).
16.
MARADUDIN A.A., MONTROLL E.W., WEISS G.H. and IPATOVA J.P., Theory of Lattice Dynamics in the Harmonic Approximation Academic Press New York (1971).
Vol. 15, No. 10
MEAN SQUARE DISPLACEMENT OF XENON ATOMS
17.
MASRI P. and DOBRZYNSKI L.,J. Phys. 32, 939 (1971); Surface Sc!. 32,623 (1972).
18.
HORTON G.K, Am. J. Phys. 36,93(1968).
On mesure par diffraction d’dlectrons lents la variation de l’amplitude quadratique moyenne des vibrations atomiques du xenon adsorbC en épitaxie sur le graphite (0001) dans la gamme de temperature 48 73°K. Un calcul de dynamique du réseau de cette grandeur est en bon accord avec les résultats expérimentaux.
a
1589
Pergamon Press.
Printed in Great Britain
MEAN SQUARE DISPLACEMENT OF XENON ATOMS IN AN EPITAXIAL MONOLAYER ON (0001) GRAPHITE J.P. Coulomb, J. Suzanne, M. Bienfait and P. Masri Laboratoire de Croissance Cristalline Associé au C.N.R.S., Universite D’Aix-Marseile II, Centre de Luminy, 13288 Marseile Cedex 2, France (Received 22 March 1974; in revised form 6 May 1974 by E.F. Bertaut)
The thermal variation of the mean square displacement of xenon atoms adsorbed in epitaxy on (0001) graphite is measured by low energy electron diffraction in the 48—73°Ktemperature range. Lattice dynamics calculation is in good agreement with experimental results.
WHEN MEASURING the mean square displacement of surface atoms by low energy electron diffraction (LEED)1 one faces two types of difficulties in the interpretation of the results. First, one uses the kinematic scattering theory, although the dynamic effects occuring in LEED are far from being negligible. However from a simple example Jepsen et al.2 were able to estimate the accuracy of the kinematic determination of the surface Debye temperature to be 10—15 per cent. Secondly, very little information is available about the exact penetration depth, although one knows that the penetration of low energy electron concerns only the surface first layers. Therefore the experimental value of the mean square displacement is an average on the first surface layer atoms. This average approaches the surface displacement value at very low energy. Usually the (u2) of the surface are obtained at about 50 eV; however, one cannot estimate the accuracy of the determination, Recently, Theeten et al.3 proposed a method avoiding these two difficulties. They had direct access to the vibrations of the first atomic layer of the surface by minimizing the multi-diffraction effects. This method applies to a two-dimensional layer adsorbed in epitaxy on a crystal surface. The thermal behaviour of the superstructure spots is measured for a few incident beam energies fulfilling the following criterium. Dynamical effects due to the substrate have to be negligible, i.e. that its diffractedbeams have to be few
and to have low intensities. This means that we have to use a low energy incident beam giving rise to superstructure spots more intense than the ones coming from the substrate. We report here the results obtained in a particularly favorable case; xenon adsorbed on (0001) graphite. This system has several advantages: —
The first condensed atomic layer is in epitaxy
on graphite. It induces a ~~J3 X \/3 superstructure in LEED.4’ ~ There is an incident energy fulfilling the criterium of Theeten et al. ~ Xenon is-one of the few crystals having a kinematic LEED spectrum.6 The mean square displacement (u2) can be computed since a realistic lattice dynamics model has been previously achieved.7 Indeed tile atomic positions in epitaxy on the substrate as well as the force constants between atoms are known. —
—
—
EXPERIMENTAL DETERMINATION Previous publications8’9 provide the temperature and pressure ranges in which the first atomic layer adsorbed on (0001) graphite is stable. We fixed the pressure at about 1 X l08torr and changed the ternperature from 48 to 73°K.At the lowest temperature 1585
1586
MEAN SQUARE DISPLACEMENT OF XENON ATOMS
reached, the second layer can only appear in the 1ff’7 torr range. The crystal surface is studied by LEED. As soon as the first layer is condensed, the ‘~/3X ~J3superstructure spots appear~’5This means that xenon is in epitaxy according to the model shown in Fig. 1. Simultaneously, the intensity of the graphite lattice reflections strongly decreases. For instance the intensity of the specular spot is about five times as low. Strong absorption of electrons by xenon has already been noticed~9 It is due to important inelastic processes and is characterized by a very short extinction length (2.5—5.5 A) when the energy varies from 20 to 55 eV. The superstructure spots are visible for only three energies 35, 64, 95 eV; the one at 95 eV can hardly be distinguished from the background. In order to obtain reproducible measurements, we anneal several times the two-dimensional crystal. The crystal temperature is stabilized at 1/4°K. It is increased or decreased by steps of about 2°K from 48 to 730 K in a few minutes. A decrease of the sweeping rate does not modify the result. The intensity of the superstructure diffracted beam is measured by a photometer, which records the intensity variation versus temperature. The background is determined either in the neighbourhood of the spot, or at the place of the spot before xenon adsorption. The same values are obtained in both cases.
Vol. 15, No. 10
Where I(T’) and 1(T) dre the intensities at temperatures T’ and T respectively; Ku2 (T’)) and Ku2 (7)) are the corresponding mean square displacement of xenon atoms in the direction of the scattering vector; 2~is the angle between incident and diffracted beam; A the wavelength of the incident electrons. The incident electron energy is corrected for the inner potential and the contact potential difference between the xenon crystal and the electron gun filament. This correction amounts to a 10 eV shift to higher energies.6 Measurements are carried out at normal incidence. The value of Ku2) occuring in (1) corresponds to an angle of about 17° between the direction of vibration and the surface normal. The experiments have been repeated several times by increasing or decreasing the temperature on different two-dimensional xenon crystals. The intensity variation versus temperature is quite reversible during one experiment. The reproductibility a few days later is not so perfect, but it can be considered as quite satisfactory. Fig. 2 shows the vanation, during a few experiments, of Ku2(fl)—Ku2(48°K)) = ~(u2) vs temperature for the atomic layer of xenon in epitaxy on (0001) graphite. To the measurements carried out at 35 eV, we have added a determination at 64 eV. In this case the intensity of the 10 spots is close to the one of the superstructure, as can be seen on the photograph published in.5 The value of Ku2) is found to be close to that obtained at 35 eV. This shows that at 64 eV the dynamical effects coming from the substrate are also quite negligible.
At 35 eV the diffraction pattern merely exhibits the central hexagon of the ~/3 X ~~13superstructure. The spots resulting from the supçrposition of the beams diffracted by xenon and graphite have an almost zero intensity. In this case Theeten’s et al.3 criteriurn is fulfilled: the substrate backscatters no beams that might contribute to multidiffraction processes in the superstructure spots. Moreover we know that xenon6 induce negligible multiple scattering effects in LEED. Therefore we can analyse the thermal behaviour of the adsorbed xenon by means of the kinematic theory, in which case 10 the diffracted intensity follows the equation: log~~= 16~2cos2ø[~ 2(T’))] 2(fl)—~ 1(T) X
(1)
~ 4,26A~ FIG. 1. The xenon monolayer in epitaxy on (0001)
graphite.
Vol. 15, No. 10
ijZ~.
MEAN SQUARE DISPLACEMENT OF XENON ATOMS
~ /
/ 40
~,/
/
/
,
/
~
4/’
x
/0
20
4.,
1587
17 have proposed procedure is too elaborate, Masrithe et al. an expansion method to shorten analytical calculation. Schematically, one has u2> = KU2)E + (u)~+ Ku2) 4 +... (2) 2)E stands for the mean square displacement where Ku in the Einstein approximation (diagonal part of the D matrix) and the other terms are the 2’~’, 4th . order corrections. The model describing the vibrational properties 7 Theonposition of two-dimensional xenon crystal adsorbed graphite has been reported by Suzanne et aL of atoms is given in Fig. 1. We take into account only
~
, +
the central forces between first neighbours. These
/ , I
I
T°I(
FIG. 2. Mean square displacement variation of the xenon atoms in an epitaxial monolayer on (0001) graphite vs temperature with respect to10the atomic dis60 placement at 48°K. Two sets of measurements (+ and
~
,/‘
/
002
/
//
---
graphite.7
The calculation of the mean square displacement of atoms is now well known.16 It requires the knowledge of the dynamical matrix D of the crystal. Generally, this computation determines in an intermediate step, the eigeiivalues of the D matrix. Since this
~I’Thiiix.
/
~
u2
/ /
x,
I. 0,01
-— —
—
0
THEORETICAL DETERMINATION
U~iN
I
~nnard X) incident are curves carried Joneswere electrons. out potentials with Thenormal 35(~ directions eV with and IN of ; one vibration AN (0) see with text). are 64 ateV retical 17°and 12°from the calculated respectively. two kinds The of theoMie
valueOur of Ku2) of adsorbed xenon is definitely higher results call for some comments. First, the than the mean square displacement of graphite atoms.~~’12 The ratio Ku2 )Xe/
*2
__________________
-I
-
___________________________ I I~-ET1 w445 170 m552 170 lSBT 50
ii
I
100
FIG. 3. Mean square displacement of xenon atoms in a monolayer adsorbed in epitaxy on (0001) graphite vs temperature. The computation was carried out for two directions, the first, normal (1),two the kinds second, parallel (II) to the (0001) plan, by using (AN and IN) of Mie Lennard Jones potential. The values calculated by Allen et aL’3”4 for bulk and (111) surface atoms of xenon crystal are also drawn here.
1588
MEAN SQUARE DISPLACEMENT OF XENON ATOMS
forces can be derived from a Mie Lennard Jones potential.18 The D matrix worked out in the harmonic approximation will be published later. The computation of Ku2) was carried out for the directions perpendicular (1) and parallel (II) to the plan of the xenon layer. The convergence of equation (2) is very good for (ui) ((u~) 2/(u~)a 0.7%). It is a little slower for (u~)((u~)2/(u~)E 30% and (U~)4/(U~)E —
drawn each direction I orFig. lithe3 Ku2) computed Theforresults are reported where we have with two Mie Lennard Jones potentials. The first one called AN 18 acts between all neighbours, the second one called IN acts between first neighbours only. They are both phenomenological potentials and it cannot be claimed that either one has any obviously superior merit. One can see that for both potentials Ku~)is higher than(u~).This is not surprising because the vibrations of xenon atoms do not feel so strbngly the attraction of graphite in the directions lying in the layer as in the direction perpendicular to it.
Vol. 15, No~.10
Figure 3 allows to calculate Ku2) in a direction at 17°from the direction [0001]. The value of (u2(fl) —(u2(48°K)) = i~(u2)for this direction is reported in Fig. 2 in the case of IN and AN poientials. The theoretical curves agree with the measurements within the limit of experimental errors. From these results we can draw the following conclusions: (i) the mean square displacement of atoms in two-dimensional xenon crystal is appreciably lower 13’14 than those of the bulk or (ill) surface xenon atoms reported Fig. 3. This is easy to understand since the binding energy5’9 and the xenon-graphite force constants7 are greater than the correspond~ngparameters associated to the xenon—xenon pair. (ii) the model describing the average vibrational properties of exitaxial xenon on graphite is realistic. (iii) If one works under certain experimental conditions it is possible to use LEED for determining the mean square displacement of atoms in epitaxy on a substrate.
REFERENCES 1.
MAC RAE A.U., Surface Sci. 2, 522 (1964).
2.
JEPSEN D.W., MARCUS P.M. and JONA F., Surface Sci. 41, 223 (1974).
3.
THEETEN J.B., DOMANGE J.E. and HURAULT J.P., Solid State Commun. 13, 993 (1973).
4.
LANDER J.J. and MORRISON J.,Surface Sci. 6, 1(1967).
5.
SUZANNE J., COULOMB J.P and BfENFAIT M., Surface Sci. 40,414 (1973).
6.
IGNATIEV A. and RHODIN T.N.,Phys. Rev. B. 8, 893 (1973).
7.
SUZANNE J., MASRI P. and BIENFAIT M., Surface Sci. (in press).
8.
THOMY A. and DUVAL X.,J; Chim. Phys. 67, 1101(1970).
9.
SUZANNE J., COULOMB J.P. and BIENFAIT M., Surface Sci. (in press).
10.
International tables for X-ray Qystallography Vol. III p. 232 Kynoch Press, Birmingham. (1968).
11.
ALBINET G., BIBERIAN J.P. and BIENFALT M., Phys. Rev. B. 3, 2015 (1971).
12. 13.
BIBERIAN J.P., BIENFAIT M. and THEETEN J.B., Acta Cryst. A29, 221 (1973). ALLEN R~E.and de WETTE F.W.,Phys. Rev. 179, 873 (1969). Phys. Rev. 188, 1320 (1969).
14.
ALLEN R.E., de WETTE F.W. and RAHMAN A.,Phys. Rev. 179, 887 (1969).
15.
TONG S.Y., RHODIN T.N. and IGNATIEV A.,Phys. Rev. B8, 906 (1973).
16.
MARADUDIN A.A., MONTROLL E.W., WEISS G.H. and IPATOVA J.P., Theory of Lattice Dynamics in the Harmonic Approximation Academic Press New York (1971).
Vol. 15, No. 10
MEAN SQUARE DISPLACEMENT OF XENON ATOMS
17.
MASRI P. and DOBRZYNSKI L.,J. Phys. 32, 939 (1971); Surface Sc!. 32,623 (1972).
18.
HORTON G.K, Am. J. Phys. 36,93(1968).
On mesure par diffraction d’dlectrons lents la variation de l’amplitude quadratique moyenne des vibrations atomiques du xenon adsorbC en épitaxie sur le graphite (0001) dans la gamme de temperature 48 73°K. Un calcul de dynamique du réseau de cette grandeur est en bon accord avec les résultats expérimentaux.
a
1589