- Email: [email protected]
Investigation of surface diffusion of Hf on W(100) by the density fluctuation method
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ELSEVIER
Surface Science 304 (1994) 59-64
Investigation of surface diffusion of Hf on W( 100) by the density fluctuation method .J.Bqben
*, W. Gubernator
I~sfit~te of ~~~men~ul Physics, Wrocfaw U~iversi~, pf. Maxa Borna 9, PL SO-205 Wrocfuw, Poland
(Received 10 August 1993; accepted for publication 10 October 1993)
Abstract Surface diffusion of hafnium deposited in ultrahigh vacuum onto a microcrystal of tungsten was investigated using the field emission method of density fluctuations. Surface diffusion coefficients were determined by analysis of the experimental time autocorrelation functions measured in the temperature range 370-470 K. Values of the activation energy for the surface diffusion of Hf on W, 0.53 and 0.54 eV were obtained for two series of measurements performed under similar conditions of deposition of Hf submonolayers.
1. Introduction There are two kinds of experimental methods of examining surface diffusion; monitoring changes in a concentration profile of adsorbed particles (i.e. laser-induced desorption) and examining an adlayer in thermal equilibrium (i.e. density ~uctuations). These methods often give quite different values for the diffusion parameters which is not fully understood by now. Monte Carlo experiments were applied to compare the results of both the profile evolution and equilibrium techniques [1,21. The review by Gomer [3] presents results about diffusion on metals and points out the unsolved problems. The experimental methods are still being improved and are
* Corresponding
author.
expected to give the coinciding results in the future. The adsorption system was previously investigated with respect to its field electron emission [4,5] and nucleation properties [5]. The results show that Hf submonolayers in thermal equilibrium with the substrate produce characteristic bright spots in the field emission patterns situated on the (100) planes of tungsten. In this investigation the fluctuations of the field emission current from these spots were examined using the density fluctuation method [6-101. The time autocorrelation functions were measured experimentally and analyzed to obtain the parameters of the surface diffusion of Hf on W. The improved procedure of calculating diffusion coefficients from the experimental autocorrelation function introduced recently by Song and Gomer [ll] was applied.
0039~6028/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved
SSDZ 0039-6028(93)E0847-N
J. BCben, W Gubernator /Surface
60
2. Experimental
The measurements were carried out in a sealed-off, glass field-emission microscope (FEM). The tungsten field emitter was spot-welded to a support loop of a W-wire. Hafnium was evaporated in situ onto the W-tip from a source made of a Hf-wire heated by electric current. Prior to sealing-off, the microscope was thoroughly baked and degassed. The residual gas pressure of the FEM tube was monitored using a Bayard-Alpert gauge and was in the lo-’ Pa range. In order to perform measurements of the field emission current fluctuations, the FE microscope was modified by the addition of a probe hole plate and an electron collector. The FEM image of a W-tip could be observed on a phosphor screen, and the field-electron beam was deflected by a magnetic field using an external magnet. The combination of probe hole and deflection enabled us to probe any surface region of the emission from the W-microcrystal, which had an area of approximately 1 X lo-r2 cm2. A schematic diagram of the experimental arrangement is shown in Fig. 1. Prior to the deposition of Hf the W-substrate surface was cleaned by flashing the emitter over 2300 K. Hafnium was evaporated onto the tungsten substrate held at room temperature for 15 min in each series of measurements. Hf aggregates were thermally equilibrated by heating the emitter at 450-470 K, i.e. about 500 K below the temperature where considerable self-diffusion of
Science 304 (I 994) 59-64
the underlying tungsten occurs [12]. The exact concentration of the deposited atoms is not known. A rough estimate could be made from visual observation of the FE patterns indicating less than a monolayer of Hf. The probed area gave the characteristic bright spots of the (loo} plane of the W-tip. The temperature of the W-tip was controlled by means of resistive heating of the support loop from a stabilized power supply, and was estimated using the resistance-current dependence. The accuracy is believed to be within 25 K. The temperature during the autocorrelation measurements was altered in the range 370-470 K, (i.e. slightly above 0.1 of the melting point of the tungsten substrate or below 0.2T,, of hafnium).
3. Determination
of the diffusion
coefficients
Determination of the diffusion coefficient by the fluctuation method consists of comparison of the experimental and theoretical autocorrelation functions. The general formula for the theoretical autocorrelation function is [7,10]
Xexp( - ‘r1LAz’2), where R(t) is the autocorrelation function, JXZis the area of the probe region, and D is the diffusion coefficient. For a circular probe region the above integral takes the form [lo] R(t)
-
= 1 - e-2az[ Z0(2cy2) - Z,(2a2)],
R(O)
FREAMFIJFIER
H
Fig. 1. Experimental
SELECllM
AMPLIFIER
setup.
where a2 = to/t = ri/4Dt, r,, is the radius of the probe region, and I,, I, are the modified Bessel functions. The finite resolution of the FEM makes that the fluctuations of the field emission current do not exactly reflect the surface density fluctuations of the adsorbate in the probe region. A correction for this effect was given by Gomer and
J. Bqben, WVGubernator /Surface
Auerbach [13]. It consists of replacing R(t/t,) by R(t/t, + 2(A/r,>2> where A = 1.3 X 10-4p(r,/ka~1/2)1/2 cm; p = 1.5 is a compression factor, rt is the emitter radius, (Y= 0.8-0.9 is an image correction, 4 is the emitter work function and k = 3.5 is a field-voltage proportionality factor. The resolution correction calculated for this experiment is 2(A/r,)* = 0.33. In a recent paper of Song and Gomer [ill the way of examining the faster decay regime of the autocorrelation function is presented. The authors have introduced corrections in the autocorrelation shape to compensate the cutting-off low frequencies. The low-frequency cutoff results from the bandwidth of the amplifying filter. In order to incorporate the correction in the autocorrelation function R(t) the spectral density S(w) multiplied by a window function U(w) (a rectangular one was applied) has to be back transformed to obtain the corrected function R’(t). Because the Fourier transform of a product is the convolution of the Fourier transformed functions we can write I’
=
&/~a~(~)U(m) eiwt dw
=
+/1”
~(t-t’)U(t’)
7-r
dt'
m
-
=R(t)
quency cutoff is about 1.2 for to = 5 ms and about 1.05 for to = 1 ms and has been introduced to the calculation of the diffusion coefficient. To determine the diffusion coefficient D the resolution corrected theoretical curves are fitted to the experimental autocorrelation functions A(t) by using the least-mean-squares procedure and to is determined:
2 iz; (F$$ - 1 A(ti)
= minimum,
where A(ti) is an experimental autocorrelation function. to is related to the diffusion coefficient D (to = ri/4Dt>.
2 sin wOt’
dt’
t’ -
Fig. 2. Comparison of R’(t) and R(t) for t,, = 5 ms.
A(O)
1 m =- 2= /_ R( l - t’) 27r6( t’) cc i
61
Science 304 (1994) 59-64
a/:qt-t’) slnjot’ df’. cc
A frequency cutoff below w. gives a faster decay for a small time interval and oscillations for the long time limit. Fig. 2 shows the differences between R’(t) numerically calculated and the resolution corrected Z?(t) for w. = 27~ (which corresponds to a cutoff frequency of 1 Hz according to the amplifying filter used). The solid curve is the resolution corrected autocorrelation function R(t) and the dashed curve represents R’(t). The time scale correction resulting from the fre-
The fitting procedure was applied to the first 50 points only, which is one half of the number of measured values, to reduce the influences of the low-frequency cutoff.
4. Results and discussion The experimental results consist of two series of autocorrelation functions, shown in Fig. 3, measured for similar adsorbed Hf submonolayers and under the conditions described in detail in Section 2. Fig. 4 shows the experimental autocorrelation curves for t < 10 ms and the fitted, resolution
J. &berr, W Gubernaror /Surface Science 304 f1994).59-64
Fig. 3a,b. Two series of normalized autocorrelation deposition.
functions A(t). Both series were measured under similar conditions of Hf
corrected theoretical curves. Only for 370 K a good fit was obtained. The experimenta decay for the other temperatures is for longer times slower than the theoretical one.
A similar observation was reported by Song and Gomer [ill. We believe that this effect cannot be due to boundary conditions because these would modify the autocorrelation function in such
2.0
fbf -
I
1 Delay
Time
[ms]
Fig. 4a,b. The autocorrelation functions Aft) for t < 10 ms (the first 50 measured points). Crosses are experimental points. Solid lines are fitted, resolution-corrected autocorrelation functions. Curves are shifted on the Y-axis by 0.2 each for better viewing.
J. BTben, W. Gubernator /Surface
a way that they would decay more rapidly. The influence of the boundary was investigated by Gesley and Swanson [IO] and they found that a reflecting boundary will reduce the values of the spectral density function only for low frequencies. In its long time limit the autocorrelation function should decay as t-’ [7]. The slower decay may result from no-longer valid assumptions in the hydrodynamic description. This occurs when the mutual interaction of adatoms is sufficiently strong. It was shown by Mazenko et al. [7] that for a system undergoing a second-order phase transition the autocorrelation function decays as t-‘/l5 for the transition temperature. Going from the hydrodynamic region to the transition temperature the autocorrelation should “cross-over” from t-” to t-1/15. Very slow decays of the cross-correlation functions were observed for potassium on tungsten. The order-disorder transition temperature obtained [14] is in accordance with that calculated from the LEED spots intensity measurements related to the ~(2 X 2) surface structure [15]. Decays of the autocorrelation functions slower than t-’ may result from reasonabIy strong mutual interaction of the adatoms, causing that the adsorption system is outside the purely hydrodynamic region. The adsorbed layer may consist of
10 -*
Science 304 (1994159-64
63
a mixture of a structured and a dilute gas. The transition to the dilute gas phase may be of the second-order type. Temperature dependences of the diffusion COefficients D are presented in Fig. 5. Squares and diamonds correspond to different experimental series. Assuming that D = D, exp(E/k,T), the activation energy E and diffusion constant D, were calculated. The values obtained are 0.53 and 0.54 eV, and 2.7 X 10e4 and 3.3 X lob4 cm2/s, respectively.
5. Conclusion The experimental results presented here were analyzed in terms of the hydrodynamic model of adsorbate density fluctuations. A correction to the autocorrelation functions resulting from the finite resolution of the FEM was included. The diffusion coefficients calculated versus l/T in the Arrhenius-like plot enabled us to find the activation energy for surface diffusion of Hf on W(100). Some deviations in the decay of the experimental autocorrelation functions were observed with respect to that theoretically predicted. This may indicate that the Hf layer does not fully satisfy the assumptions of the hydrodynamic model of adsorbate density fluctuations.
_I
6. A~~owiedgements
nnnnn E = OQQQ.0 E = 10
0.53 0.54
The authors express gratitude to Professor Ch. Kleint for enabling them to carry out the fluctuation measurements in Leipzig University as well as for many valuable discussions. We also would like to thank Professor R. Medewski for stimulating discussions. The help of Dr. S. Surma in elaboration of the results is acknowledged. This study was supported by the KE3N Committee, Warsaw under Project No. 201259101, which is gratefully ac~owledged.
eV eV
-‘I.~{ 2.0
2.2
2.4
2.6
2.8
1000/T [K-l] Fig. 5. Temperature dependence of the diffusion coefficients. Squares and diamonds correspond to different series.
7. References [l] M.C. Tringides, J. Chem. Phys. 92 (1990) 2077. [2] C. Uebing and R. Gomer, J. Chem. Phys. 95 (1991) 7626.
64
J. Bqben, FK Gubernator / Surface Science 304 (I994j 59-64
[3] R. Gomer, Rep. Prog. Phys. 53 (1990) 917. [4] R. Mqclewski, Acta Univ. Wratislav. 455 (1983) 119. [S] Z. Szczudko, K. Sendecka, W. Gubernator and A. Ciszewski, to be published. [6] R. Gomer, Surf. Sci. 38 (1973) 373. [?] G. Mazenko, J.R. Banavar and R. Gomer, Surf. Sci. 107 (1981) 459. [8] D.R. Bowman, R. Gomer, K. Muttalib and M. Tringides, Surf. Sci. 138 (1984) 581.
[9] G.W. Timm and A. van der Ziei, Physica 32 (3966) 1333. [lo] M.A. Gesley and L.W. Swanson, Phys. Rev. B 32 (1985) 7703. [ll] Y. Song and R. Gomer, Surf. Sci. 290 (1993) 1. [lZ] D.W. Basset, Proc. R. Sot. A 286 (1965) 191. [13] R. Gomer and A. Auerbach, Surf. Sci. 167 (1986) 493. [14] J. Beben, Solid State Phenom. 12 (1990) 1’7. [IS] M.S. Gupalo, Fiz. Tverd. Tela. 22 (1980) 2311.
...i
.,
.... .,.(...... ..... :i,(.,,, ,,, ..~.....)...,,
..1....
..,..
,.:.:(_
,...,....:.
::
““‘..::“.~.~v.. ..(.../............,,,, ~~~~~,~~:~:~s:~:~i~:~::~..........-... ” ” ‘%‘. . Ic’: “‘.‘.....V
.
..y..
.,.,.,..
h”‘.:.:.:
:‘.‘+:.:.I . . . . . ,...... ..1...
.\...... :.:.: . .. . . . .. . . :.:.: .. . . . . . . .. . . . . . . . . . . . . . . .. .. . . . . ..~... .
.
.
.
”
‘surface ...
.
.
”
.
.
.
. .
.
.
.
...‘. . . . . . . ..:,:: ~,:::
scienc&
.:.. ..,._ ...... “‘:‘~Y.;.“~ ..%I _(_ __ ._ _,,,_,, “-. “--l’;:~:: .‘A. .a.. ..:,: ..,‘ .,_,,,, .,:.,, ,,>>.y,.:.: ,___ ,__ ... “‘Y~=‘“+.: .....n ..~.:... .,.._ _“.‘::.:.:.:.:..:.:.:.:~,::~~ ,,,.,,; _._ “‘.“‘?“.,...:......: .,........ >...‘., “,.,~,:‘,,., “‘,~~.:.::.~.:.::~:::::.:::;: :.:,:,:.; .,.:.,., :,:.
ELSEVIER
Surface Science 304 (1994) 59-64
Investigation of surface diffusion of Hf on W( 100) by the density fluctuation method .J.Bqben
*, W. Gubernator
I~sfit~te of ~~~men~ul Physics, Wrocfaw U~iversi~, pf. Maxa Borna 9, PL SO-205 Wrocfuw, Poland
(Received 10 August 1993; accepted for publication 10 October 1993)
Abstract Surface diffusion of hafnium deposited in ultrahigh vacuum onto a microcrystal of tungsten was investigated using the field emission method of density fluctuations. Surface diffusion coefficients were determined by analysis of the experimental time autocorrelation functions measured in the temperature range 370-470 K. Values of the activation energy for the surface diffusion of Hf on W, 0.53 and 0.54 eV were obtained for two series of measurements performed under similar conditions of deposition of Hf submonolayers.
1. Introduction There are two kinds of experimental methods of examining surface diffusion; monitoring changes in a concentration profile of adsorbed particles (i.e. laser-induced desorption) and examining an adlayer in thermal equilibrium (i.e. density ~uctuations). These methods often give quite different values for the diffusion parameters which is not fully understood by now. Monte Carlo experiments were applied to compare the results of both the profile evolution and equilibrium techniques [1,21. The review by Gomer [3] presents results about diffusion on metals and points out the unsolved problems. The experimental methods are still being improved and are
* Corresponding
author.
expected to give the coinciding results in the future. The adsorption system was previously investigated with respect to its field electron emission [4,5] and nucleation properties [5]. The results show that Hf submonolayers in thermal equilibrium with the substrate produce characteristic bright spots in the field emission patterns situated on the (100) planes of tungsten. In this investigation the fluctuations of the field emission current from these spots were examined using the density fluctuation method [6-101. The time autocorrelation functions were measured experimentally and analyzed to obtain the parameters of the surface diffusion of Hf on W. The improved procedure of calculating diffusion coefficients from the experimental autocorrelation function introduced recently by Song and Gomer [ll] was applied.
0039~6028/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved
SSDZ 0039-6028(93)E0847-N
J. BCben, W Gubernator /Surface
60
2. Experimental
The measurements were carried out in a sealed-off, glass field-emission microscope (FEM). The tungsten field emitter was spot-welded to a support loop of a W-wire. Hafnium was evaporated in situ onto the W-tip from a source made of a Hf-wire heated by electric current. Prior to sealing-off, the microscope was thoroughly baked and degassed. The residual gas pressure of the FEM tube was monitored using a Bayard-Alpert gauge and was in the lo-’ Pa range. In order to perform measurements of the field emission current fluctuations, the FE microscope was modified by the addition of a probe hole plate and an electron collector. The FEM image of a W-tip could be observed on a phosphor screen, and the field-electron beam was deflected by a magnetic field using an external magnet. The combination of probe hole and deflection enabled us to probe any surface region of the emission from the W-microcrystal, which had an area of approximately 1 X lo-r2 cm2. A schematic diagram of the experimental arrangement is shown in Fig. 1. Prior to the deposition of Hf the W-substrate surface was cleaned by flashing the emitter over 2300 K. Hafnium was evaporated onto the tungsten substrate held at room temperature for 15 min in each series of measurements. Hf aggregates were thermally equilibrated by heating the emitter at 450-470 K, i.e. about 500 K below the temperature where considerable self-diffusion of
Science 304 (I 994) 59-64
the underlying tungsten occurs [12]. The exact concentration of the deposited atoms is not known. A rough estimate could be made from visual observation of the FE patterns indicating less than a monolayer of Hf. The probed area gave the characteristic bright spots of the (loo} plane of the W-tip. The temperature of the W-tip was controlled by means of resistive heating of the support loop from a stabilized power supply, and was estimated using the resistance-current dependence. The accuracy is believed to be within 25 K. The temperature during the autocorrelation measurements was altered in the range 370-470 K, (i.e. slightly above 0.1 of the melting point of the tungsten substrate or below 0.2T,, of hafnium).
3. Determination
of the diffusion
coefficients
Determination of the diffusion coefficient by the fluctuation method consists of comparison of the experimental and theoretical autocorrelation functions. The general formula for the theoretical autocorrelation function is [7,10]
Xexp( - ‘r1LAz’2), where R(t) is the autocorrelation function, JXZis the area of the probe region, and D is the diffusion coefficient. For a circular probe region the above integral takes the form [lo] R(t)
-
= 1 - e-2az[ Z0(2cy2) - Z,(2a2)],
R(O)
FREAMFIJFIER
H
Fig. 1. Experimental
SELECllM
AMPLIFIER
setup.
where a2 = to/t = ri/4Dt, r,, is the radius of the probe region, and I,, I, are the modified Bessel functions. The finite resolution of the FEM makes that the fluctuations of the field emission current do not exactly reflect the surface density fluctuations of the adsorbate in the probe region. A correction for this effect was given by Gomer and
J. Bqben, WVGubernator /Surface
Auerbach [13]. It consists of replacing R(t/t,) by R(t/t, + 2(A/r,>2> where A = 1.3 X 10-4p(r,/ka~1/2)1/2 cm; p = 1.5 is a compression factor, rt is the emitter radius, (Y= 0.8-0.9 is an image correction, 4 is the emitter work function and k = 3.5 is a field-voltage proportionality factor. The resolution correction calculated for this experiment is 2(A/r,)* = 0.33. In a recent paper of Song and Gomer [ill the way of examining the faster decay regime of the autocorrelation function is presented. The authors have introduced corrections in the autocorrelation shape to compensate the cutting-off low frequencies. The low-frequency cutoff results from the bandwidth of the amplifying filter. In order to incorporate the correction in the autocorrelation function R(t) the spectral density S(w) multiplied by a window function U(w) (a rectangular one was applied) has to be back transformed to obtain the corrected function R’(t). Because the Fourier transform of a product is the convolution of the Fourier transformed functions we can write I’
=
&/~a~(~)U(m) eiwt dw
=
+/1”
~(t-t’)U(t’)
7-r
dt'
m
-
=R(t)
quency cutoff is about 1.2 for to = 5 ms and about 1.05 for to = 1 ms and has been introduced to the calculation of the diffusion coefficient. To determine the diffusion coefficient D the resolution corrected theoretical curves are fitted to the experimental autocorrelation functions A(t) by using the least-mean-squares procedure and to is determined:
2 iz; (F$$ - 1 A(ti)
= minimum,
where A(ti) is an experimental autocorrelation function. to is related to the diffusion coefficient D (to = ri/4Dt>.
2 sin wOt’
dt’
t’ -
Fig. 2. Comparison of R’(t) and R(t) for t,, = 5 ms.
A(O)
1 m =- 2= /_ R( l - t’) 27r6( t’) cc i
61
Science 304 (1994) 59-64
a/:qt-t’) slnjot’ df’. cc
A frequency cutoff below w. gives a faster decay for a small time interval and oscillations for the long time limit. Fig. 2 shows the differences between R’(t) numerically calculated and the resolution corrected Z?(t) for w. = 27~ (which corresponds to a cutoff frequency of 1 Hz according to the amplifying filter used). The solid curve is the resolution corrected autocorrelation function R(t) and the dashed curve represents R’(t). The time scale correction resulting from the fre-
The fitting procedure was applied to the first 50 points only, which is one half of the number of measured values, to reduce the influences of the low-frequency cutoff.
4. Results and discussion The experimental results consist of two series of autocorrelation functions, shown in Fig. 3, measured for similar adsorbed Hf submonolayers and under the conditions described in detail in Section 2. Fig. 4 shows the experimental autocorrelation curves for t < 10 ms and the fitted, resolution
J. &berr, W Gubernaror /Surface Science 304 f1994).59-64
Fig. 3a,b. Two series of normalized autocorrelation deposition.
functions A(t). Both series were measured under similar conditions of Hf
corrected theoretical curves. Only for 370 K a good fit was obtained. The experimenta decay for the other temperatures is for longer times slower than the theoretical one.
A similar observation was reported by Song and Gomer [ill. We believe that this effect cannot be due to boundary conditions because these would modify the autocorrelation function in such
2.0
fbf -
I
1 Delay
Time
[ms]
Fig. 4a,b. The autocorrelation functions Aft) for t < 10 ms (the first 50 measured points). Crosses are experimental points. Solid lines are fitted, resolution-corrected autocorrelation functions. Curves are shifted on the Y-axis by 0.2 each for better viewing.
J. BTben, W. Gubernator /Surface
a way that they would decay more rapidly. The influence of the boundary was investigated by Gesley and Swanson [IO] and they found that a reflecting boundary will reduce the values of the spectral density function only for low frequencies. In its long time limit the autocorrelation function should decay as t-’ [7]. The slower decay may result from no-longer valid assumptions in the hydrodynamic description. This occurs when the mutual interaction of adatoms is sufficiently strong. It was shown by Mazenko et al. [7] that for a system undergoing a second-order phase transition the autocorrelation function decays as t-‘/l5 for the transition temperature. Going from the hydrodynamic region to the transition temperature the autocorrelation should “cross-over” from t-” to t-1/15. Very slow decays of the cross-correlation functions were observed for potassium on tungsten. The order-disorder transition temperature obtained [14] is in accordance with that calculated from the LEED spots intensity measurements related to the ~(2 X 2) surface structure [15]. Decays of the autocorrelation functions slower than t-’ may result from reasonabIy strong mutual interaction of the adatoms, causing that the adsorption system is outside the purely hydrodynamic region. The adsorbed layer may consist of
10 -*
Science 304 (1994159-64
63
a mixture of a structured and a dilute gas. The transition to the dilute gas phase may be of the second-order type. Temperature dependences of the diffusion COefficients D are presented in Fig. 5. Squares and diamonds correspond to different experimental series. Assuming that D = D, exp(E/k,T), the activation energy E and diffusion constant D, were calculated. The values obtained are 0.53 and 0.54 eV, and 2.7 X 10e4 and 3.3 X lob4 cm2/s, respectively.
5. Conclusion The experimental results presented here were analyzed in terms of the hydrodynamic model of adsorbate density fluctuations. A correction to the autocorrelation functions resulting from the finite resolution of the FEM was included. The diffusion coefficients calculated versus l/T in the Arrhenius-like plot enabled us to find the activation energy for surface diffusion of Hf on W(100). Some deviations in the decay of the experimental autocorrelation functions were observed with respect to that theoretically predicted. This may indicate that the Hf layer does not fully satisfy the assumptions of the hydrodynamic model of adsorbate density fluctuations.
_I
6. A~~owiedgements
nnnnn E = OQQQ.0 E = 10
0.53 0.54
The authors express gratitude to Professor Ch. Kleint for enabling them to carry out the fluctuation measurements in Leipzig University as well as for many valuable discussions. We also would like to thank Professor R. Medewski for stimulating discussions. The help of Dr. S. Surma in elaboration of the results is acknowledged. This study was supported by the KE3N Committee, Warsaw under Project No. 201259101, which is gratefully ac~owledged.
eV eV
-‘I.~{ 2.0
2.2
2.4
2.6
2.8
1000/T [K-l] Fig. 5. Temperature dependence of the diffusion coefficients. Squares and diamonds correspond to different series.
7. References [l] M.C. Tringides, J. Chem. Phys. 92 (1990) 2077. [2] C. Uebing and R. Gomer, J. Chem. Phys. 95 (1991) 7626.
64
J. Bqben, FK Gubernator / Surface Science 304 (I994j 59-64
[3] R. Gomer, Rep. Prog. Phys. 53 (1990) 917. [4] R. Mqclewski, Acta Univ. Wratislav. 455 (1983) 119. [S] Z. Szczudko, K. Sendecka, W. Gubernator and A. Ciszewski, to be published. [6] R. Gomer, Surf. Sci. 38 (1973) 373. [?] G. Mazenko, J.R. Banavar and R. Gomer, Surf. Sci. 107 (1981) 459. [8] D.R. Bowman, R. Gomer, K. Muttalib and M. Tringides, Surf. Sci. 138 (1984) 581.
[9] G.W. Timm and A. van der Ziei, Physica 32 (3966) 1333. [lo] M.A. Gesley and L.W. Swanson, Phys. Rev. B 32 (1985) 7703. [ll] Y. Song and R. Gomer, Surf. Sci. 290 (1993) 1. [lZ] D.W. Basset, Proc. R. Sot. A 286 (1965) 191. [13] R. Gomer and A. Auerbach, Surf. Sci. 167 (1986) 493. [14] J. Beben, Solid State Phenom. 12 (1990) 1’7. [IS] M.S. Gupalo, Fiz. Tverd. Tela. 22 (1980) 2311.