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Critical fluctuations of the field emission current
,:
.\ .,
:
,
4
.:‘ ~:i, ;
,
,! j
‘*
;
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1*
surface science
\~ ~ :. :;:_ i,.’ 1 ,:,, ::,, j c~... .I, I,;,;:‘. :>,
ELSEVIER
Surface Science 317 (1994) 253-258
Critical fluctuations of the field emission current E.S. Shikhovtseva a,*, K. Skwarek b, J. Beben ay* a Institute of Experimental Physics, Wroclaw University,pl. Maxa Borna 9, PL 50-204 Wroclaw, Poland b Institute of Theoretical Physics, Wroclaw University,pl. Maxa Borna 9, PL SO-204 Wroclaw, Poland Received 7
February 1994;accepted for publication 7 June 1994
Abstract
The cross-correlation and spectral density functions of the adsorbate density fluctuations have been calculated at the transition temperature of a second order phase transition. Comparison with hydrodynamic fluctuations was made. Our calculations showed that for t = 0 the cross-correlation functions of the critical fluctuations are positive for any distance between the probed regions (they decay with distance as X-I/~). The hydrodynamic cross-correlation functions for t = 0 are positive when the probed regions overlap and are zero when they are separated. The spectral density for the critical fluctuations reveals faster decay for low frequencies in comparison with the hydrodynamic one.
1. Introduction
The field emission microscope technique enables us to measure fluctuations of a number of adparticles in a small surface region and can be used to examine surface diffusion. The fluctuations cause local work-function changes and give rise to field emission current fluctuations. The theory of density fluctuations by Smoluchowski [ 1,2] can be applied to describe the current fluctuations [ 371. The measurement of the autocorrelation [ 41, cross-correlation [ 81 or spectral density [ 3,9, lo] functions is a direct method for measuring the surface diffusion coefficient. In adsorption systems some phases and transitions between them occur, the existence of which have to be considered, when we examine surface diffusion. The hydrodynamic description of the density fluctuations, which assumes the validity of the diffusion equation, can be applied for both the non-interacting and interacting adatoms unless the system is inside a critical region. The occurrence of a second order phase transition in an adsorbed layer was examined by Mazenko et al. [ 5,111 and the autocorrelation function of the density fluctuations at the transition temperature * Corresponding author. 1 On leave from Bashkir State University, Center, Russian Federation.
Ufa, Russian Federation;
Current address: Department
0039-6028/94/$07.00@ 1994 Elsevier Science B.V. All rights reserved SSDI0039-6028(94)00358-G
of Physics, Ufa Scientific
ES. Shikhovtseva et al. /Surface Science 317 (1994) 253-258
254
was derived for the long time limit. In this paper we extend the idea to both the cross-correlation and spectral density functions.
2. Cross-correlation The hydrodynamic description of the adsorbate density fluctuations structure factor S(r, t) fulfills the diffusion equation [ $12,131.
assumes that the dynamical
-&S(r, t) = DV%s(r, t),
(1)
with sb.2
-0,t)
=
(~p(~2,0)~p(rl,t)),
and
dptr, t) = p(r, t) - P, where t is the delay time, p(r, t) is the adsorbate density and D is the diffusion space Fourier transform of S(r, t) resulting from Eq. (1) is given by S(k, t) = So exp(-k2Dt),
coefficient.
The (2)
where k is the length of the wave vector. When the system is near a second order phase transition Eq. (2) is no longer valid. The corresponding formula for critical fluctuations at the transition temperature was given by Mazenko et al. [ 51 S(k,t)
= &
exp(-Ak’t),
(3)
where the c and A are constant, l/4 and 15/4, respectively). The cross-correlation function CC(A,,A,,t)
G $
drl
s Al
s
and the q and z are critical indices
(for the Ising model equal to
is defined as drzS(r2
-rl,t)
A2
=&Jdrl Jd*2 J dk Al
exp [ik(rz - r1 )]S(k,
t),
(4)
A2
where Ai and A2 are the areas of the probed regions (patches). Substituting Eq. (2) into Eq. (4) and integrating over circular areas of radius r at a distance x away, we obtain the cross-correlation functions for hydrodynamic fluctuations: D=
CCHW
= &(2n)2
J J
k-‘J,(xk)Jf(rk)
exp(-k2Dt)
dk
0
00
= S,(27c)2
0
C'J~ht)Jf(t)
exp
(5)
E.S. Shikhovtsevaet al. /Surface Science 317 (I 994) 253-258
(4
255
(b)
hydrodynamic
critical
fluctuations
fluctuations
3
NORMALIZED
TIME
Fig. 1. Cross-correlation functions for hydrodynamic (a) and critical (b) fluctuations. The probed region geometry is shown on the diagrams. The parameter m is the ratio of the distance between patches and patch radius.
where m = x/r; Jo and Jt are the Bessel functions. The corresponding equation for critical fluctuations is obtained from Eqs. ( 3) and (4) CCC(t) = c(27r)2
J
k-11/4Je(xk)J:(rk)
exp(-kt5j4At)
dk
0
= c(2a)2r7/4
Jm~p11/4J~(m~)Jf(~) exp (-C15’4--&)
dc.
0
Fig. 1 shows the cross-correlations for both hydrodynamic and critical fluctuations plotted after numerical calculation of Eqs. (5) and (6). The curves for m = 0 correspond to the autocorrelation functions (the integral was calculated for the same probe regions). The curves are normalized (in respect to those for m = 0) and they are shown in identical axis scales in order to glean a better comparison. The distance dependence of the cross-correlation at t = 0 can be expressed analytically. For hydrodynamic fluctuations we have 2n2 - ma&iZG-
CCH(0)=
OLmc2, 25 m.
47~arcsin(m/2),
o
,
i
(7)
For critical fluctuations,
’ r(g>r(~),F,({-~,-~,~},{-~, 8d333U3 CCC(O)=
l},m2/4
m11/4r(-+)pFq({-~, 4, $},{jf, y},m2/4) 4
+
223hr
3
($ )
m-l/4U&Fq ({Q,i, t}, {2,3), d/m21 \
2w-(;)
3
OImc2,
2 I m,
(8)
E.S. Shiwlovtseva et al. /Surface Science 317 (1994) 253-258
256
IO
2 H g
1 10-l 10 -I
2 G
. 0.5
10 -'
Fg lo-' (0 10 -& 2 i
0
lo-'lo-'lo-'
1
10
10'
NORMALIZED FREQUENCY On Fig. 2. (Left) Distance dependence of the cross-correlation functions at t = 0. The solid and dashed lines correspond hydrodynamic and critical fluctuations, respectively. m is ratio of the distance between patches and patch radius. Fig. 3. (Right) Spectral density functions for the hydrodynamic (solid line) and critical (dashed line) fluctuations. normalized frequency is wn = (r2/D)w for the hydrodynamic and wn = (r15/4/A)w for critical fluctuations.
where r is the gamma function , pFq is the generalized hypergeometric function which correspond to Eqs. (7) and (8) are plotted in Fig. 2. For hydrodynamic overlapping patches only give non-zero cross-correlation (m = 2 corresponds patches). For critical fluctuations, the cross-correlation CC’(O) is always positive distance between patches as X-I/~.
to the
The
[ 141. The curves fluctuations the to the adjacent and decays with
3. Spectral density A spectral density function is defined as a time Fourier transform of an autocorrelation function. Taking m = 0 in Eqs. (5) and (6) we obtain the autocorrelation for both the hydrodynamic AH(t ) and the critical AC(t ) fluctuations
J
AH(t) = S0(27r)~ W<_‘Jf(<) exp 0
J r-1’/4.1f(<) 00
F(t)
= C(27+7/4
exp ( -<‘5/4 $&)
d&
0
The time Fourier transform of AH( t ) and AC(t ) are cm
2r2
J
sHbd = S”(2n) 5 o and
kJ:W k4 + ($/~)+2
dk
>
(11)
ES. Shikhovtseva et al. /Surface Science 317 (1994) 253-253
dk +kJ;Vd (rls/4/A)2~2 J0kW2
257
r29/4 O” SC(o)
=
c(27+x
(12)
respectively. The calculated spectral densities S* (11)) and SC (co) for hydrodynamic (solid line) and critical (dashed line) fluctuations are presented in Fig. 3. The normalized frequency is o,, = (r2/D)o for hydrodynamic and wn = (r15/4/A)o for critical fluctuations. The full form of the integrals (9) and (11) can be found in Ref. [7].
4. Discussion In the density fluctuation method, the comparison of the experimental and theoretical autocorrelation, cross~~elation or spectral density functions yields the surface diffusion coefficient D. The magnitude of D depends on the shape of these functions. There are various factors which can influence their shape and hence D, namely: ( 1) The finite resolution of the field emission microscope requires an appropriate correction [ 151; (2) The boundary conditions influence the long time limit of auto- or cross-correlations and the low frequency range of spectral densities [7]; (3) The anisotropy of the examined adsorption system can be considered introducing the surface diffusion tensor [6,7]; (4) The effect of low and high frequency cut-off should be included according to the band width of the amplifier used [ 16- 181. The occurrence of critical fluctuations in the adlayer is the next effect which can considerably change the shape of the autocorrelation, cross-correlation or spectral density functions. In the ~lculations presented in the previous sections the values of the critical indices, 9 = l/4 and z = 15/4, were arbitrarily chosen. A real system might have different exponents. Especially controversial is the value of the dynamical critical index z. These values may influence the long time limit of the correlation functions which decay as t- VIZ.Nevertheless, the calculated results can be useful when interpreting experimental data. For hydrodynamic fluctuations, the correlation functions decay as t- I. A slower decay can indicate that we are inside a critical region. Similar qualitative conclusions can be drawn when we analyze experimental spectral density functions, Another characteristic feature of critical fluctuations is a positive value of the cross-correlation at r = 0 (for separated probe regions) that should be 0 for hydrodynamic fluctuations. We can only roughly estimate the transition temperature if we do not know the exact values of the critical indices for a real system. An attempt to estimate the transition temperature was done for ~~ssiurn adlayers on a ( 112 ) tungsten plane [ 19 1. The decay of the experimental cross-correlation functions was determined by fitting the Gaussian-like curve C(x, t) = t-p exp(-x2/4Dt) to the experimental cross-correlations (C(x, t) decays like t-p for a long time). The fitting procedure gave /3, which became smaller with a decreasing temperature as if the system approached a critical point from above. j? M l/ 15 was achieved. The estimated transition temperatures were about 350 and 380 K at potassium coverages of 0.30 and 0.38, respectively. For a similar adsorption system, potassium on Mo( 112), a transition tem~rature of about 350 K was obtained from the LEED spots intensity experiments related to the c(2 x 2) surface structure 1201. The Monte Carlo experiments which serve extremely well in describing critical fluctuations in the adlayer should support the experimental results. In the paper of Tringides and Gomer [ 2 11, the density fluctuations method was simulated and diffusion coefficients were determined by fitting the theoretical autocorrelations functions to those obtained in the Monte Carlo experiments. The authors
258
E.S. Shikhovtseva et al. /Surface Science 317 (1994) 253-258
reported that for attractive interactions J, J/kT > 1.5, there were deviations from hydrodynamic autoco~elation functions for long times. The hydrodynamic cross-correlation for separated probe regions is zero for t = 0. It can be understood when interpreting it in terms of probability. The cross-correlation defined by Eq. (4) is given by the double integral of the dynamical structure factor S(r, t) which can be interpreted as the “after effect” of Smoluchowski. S(r, t) is a probability density to find a particle in position r at time t when its initial position was r = 0 at t = 0. The integral over the area A2 gives the probability to find a particle inside an area A2 when it started from position r = 0 at time t = 0. The second integral divided by Ai gives an average over the area At. Hence, we can interpret the cross-correlation as a probability to find a particle inside an area Al after time t when its initial position was any place inside A 1. In a critical regime the calculations show non-zero cross-correlation for any distance (Fig. lb). This is caused by the existence of adsorbed phases, such as islands for example, which are in thermal equilib~um. The experimental cross-correlations at t = 0 can differ from the theoretical ones because of the time shift resulting from the finite resolution of the field emission microscope [ 151. Further experiments and Monte Carlo calculations should contribute to better understanding the critical fluctuations observed in the field emission microscope.
Acknowledgements We wish to thank Professor A. Pekalski and Professor R. Meclewski for their helpful comments. The help of MS Malinda Schmiechen in preparing the final version of the manuscript is also gratefully acknowledged. This study was supported by the MEN, under Project No 2016/W/IFD/93, which is gratefully acknowledged. References [I ] .M. Smoluchowski, Bull. Acad. Cracoviae (1906) 203; Wien. Ber. 123 (19 14) 238 1. [2 ] S. Chandrasekhar, Rev. Mod. Phys. I5 ( 1943) 1. [3] G.W. Timm and A. van der Ziel, Physica 32 (1966) 1333. (41 R. Gomet, Surf. Sci. 38 (1973) 373. [ 51 G. Mazenko, J.R. Banavar and R. Gomer, Surf. Sci. 107 ( 198 1) 459. [6] D.R. Bowman, R. Gomer, K. Muttalib and M. Tringides, Surf. Sci. 138 ( 1984) 58 1. [7 ] M.A. Gesley and L.W. Swanson, Phys. Rev. I3 32 (1985) 7703. [8] A.M. Dabrowski and Ch. Kleint, Surf. Sci. 172 (1986) 372. [9] J. Beben, Ch. Kleint and R. Meclewski, Appl. Phys. A 40 (1986) 79. f lo] J. Beben, Ch. Kleint and R. Meclewski, Z. Phys. B (Condens. Matter) 69 (1987) 3 19. [ 11 J G. Mazenko, in: Surface Mobilities on Solid Materials, Ed. Vu Thien Binh (Plenum, New York, 198 1) pp. 27-62. [ 121 D. Forster, Hydrodynamic Fluctuations, Broken Symmetry and Correlation Functions (Benjamin, Reading, MA, 1975). [ 131 R. Gomer, Rep. Prog. Phys. 53 (1990) 9 17. [ 141 A.P. Prudnikov, Yu.A. Brychkhov and 0.1. Marichev, Integrals and Series, Supplement (Nauka, Moscow, 1986) [in Russian J. [ 151 R. Gomer and A. Auerbach, Surf Sci. 167 (1986) 493. [ 161 Y. Song and R. Gomer, Surf. Sci. 290 ( 1993) 1. f 171 Y. Song and R. Gomer, Surf. Sci. 295 ( 1993) 174. [ 181 J. Beben and W. Gubernator, Surf. Sci. 304 (1994) 59. [ 191 J. Beben, Solid State Phenom. 12 ( 1990) 17. [ 201 M. S. Gupalo, Fiz. Tverd. Tela 22 ( 1980) 23 11. [21] M.C. Tringides and R. Gomer, Surf. Sci. 265 (1992) 283.
.\ .,
:
,
4
.:‘ ~:i, ;
,
,! j
‘*
;
,>,;
1*
surface science
\~ ~ :. :;:_ i,.’ 1 ,:,, ::,, j c~... .I, I,;,;:‘. :>,
ELSEVIER
Surface Science 317 (1994) 253-258
Critical fluctuations of the field emission current E.S. Shikhovtseva a,*, K. Skwarek b, J. Beben ay* a Institute of Experimental Physics, Wroclaw University,pl. Maxa Borna 9, PL 50-204 Wroclaw, Poland b Institute of Theoretical Physics, Wroclaw University,pl. Maxa Borna 9, PL SO-204 Wroclaw, Poland Received 7
February 1994;accepted for publication 7 June 1994
Abstract
The cross-correlation and spectral density functions of the adsorbate density fluctuations have been calculated at the transition temperature of a second order phase transition. Comparison with hydrodynamic fluctuations was made. Our calculations showed that for t = 0 the cross-correlation functions of the critical fluctuations are positive for any distance between the probed regions (they decay with distance as X-I/~). The hydrodynamic cross-correlation functions for t = 0 are positive when the probed regions overlap and are zero when they are separated. The spectral density for the critical fluctuations reveals faster decay for low frequencies in comparison with the hydrodynamic one.
1. Introduction
The field emission microscope technique enables us to measure fluctuations of a number of adparticles in a small surface region and can be used to examine surface diffusion. The fluctuations cause local work-function changes and give rise to field emission current fluctuations. The theory of density fluctuations by Smoluchowski [ 1,2] can be applied to describe the current fluctuations [ 371. The measurement of the autocorrelation [ 41, cross-correlation [ 81 or spectral density [ 3,9, lo] functions is a direct method for measuring the surface diffusion coefficient. In adsorption systems some phases and transitions between them occur, the existence of which have to be considered, when we examine surface diffusion. The hydrodynamic description of the density fluctuations, which assumes the validity of the diffusion equation, can be applied for both the non-interacting and interacting adatoms unless the system is inside a critical region. The occurrence of a second order phase transition in an adsorbed layer was examined by Mazenko et al. [ 5,111 and the autocorrelation function of the density fluctuations at the transition temperature * Corresponding author. 1 On leave from Bashkir State University, Center, Russian Federation.
Ufa, Russian Federation;
Current address: Department
0039-6028/94/$07.00@ 1994 Elsevier Science B.V. All rights reserved SSDI0039-6028(94)00358-G
of Physics, Ufa Scientific
ES. Shikhovtseva et al. /Surface Science 317 (1994) 253-258
254
was derived for the long time limit. In this paper we extend the idea to both the cross-correlation and spectral density functions.
2. Cross-correlation The hydrodynamic description of the adsorbate density fluctuations structure factor S(r, t) fulfills the diffusion equation [ $12,131.
assumes that the dynamical
-&S(r, t) = DV%s(r, t),
(1)
with sb.2
-0,t)
=
(~p(~2,0)~p(rl,t)),
and
dptr, t) = p(r, t) - P, where t is the delay time, p(r, t) is the adsorbate density and D is the diffusion space Fourier transform of S(r, t) resulting from Eq. (1) is given by S(k, t) = So exp(-k2Dt),
coefficient.
The (2)
where k is the length of the wave vector. When the system is near a second order phase transition Eq. (2) is no longer valid. The corresponding formula for critical fluctuations at the transition temperature was given by Mazenko et al. [ 51 S(k,t)
= &
exp(-Ak’t),
(3)
where the c and A are constant, l/4 and 15/4, respectively). The cross-correlation function CC(A,,A,,t)
G $
drl
s Al
s
and the q and z are critical indices
(for the Ising model equal to
is defined as drzS(r2
-rl,t)
A2
=&Jdrl Jd*2 J dk Al
exp [ik(rz - r1 )]S(k,
t),
(4)
A2
where Ai and A2 are the areas of the probed regions (patches). Substituting Eq. (2) into Eq. (4) and integrating over circular areas of radius r at a distance x away, we obtain the cross-correlation functions for hydrodynamic fluctuations: D=
CCHW
= &(2n)2
J J
k-‘J,(xk)Jf(rk)
exp(-k2Dt)
dk
0
00
= S,(27c)2
0
C'J~ht)Jf(t)
exp
(5)
E.S. Shikhovtsevaet al. /Surface Science 317 (I 994) 253-258
(4
255
(b)
hydrodynamic
critical
fluctuations
fluctuations
3
NORMALIZED
TIME
Fig. 1. Cross-correlation functions for hydrodynamic (a) and critical (b) fluctuations. The probed region geometry is shown on the diagrams. The parameter m is the ratio of the distance between patches and patch radius.
where m = x/r; Jo and Jt are the Bessel functions. The corresponding equation for critical fluctuations is obtained from Eqs. ( 3) and (4) CCC(t) = c(27r)2
J
k-11/4Je(xk)J:(rk)
exp(-kt5j4At)
dk
0
= c(2a)2r7/4
Jm~p11/4J~(m~)Jf(~) exp (-C15’4--&)
dc.
0
Fig. 1 shows the cross-correlations for both hydrodynamic and critical fluctuations plotted after numerical calculation of Eqs. (5) and (6). The curves for m = 0 correspond to the autocorrelation functions (the integral was calculated for the same probe regions). The curves are normalized (in respect to those for m = 0) and they are shown in identical axis scales in order to glean a better comparison. The distance dependence of the cross-correlation at t = 0 can be expressed analytically. For hydrodynamic fluctuations we have 2n2 - ma&iZG-
CCH(0)=
OLmc2, 25 m.
47~arcsin(m/2),
o
,
i
(7)
For critical fluctuations,
’ r(g>r(~),F,({-~,-~,~},{-~, 8d333U3 CCC(O)=
l},m2/4
m11/4r(-+)pFq({-~, 4, $},{jf, y},m2/4) 4
+
223hr
3
($ )
m-l/4U&Fq ({Q,i, t}, {2,3), d/m21 \
2w-(;)
3
OImc2,
2 I m,
(8)
E.S. Shiwlovtseva et al. /Surface Science 317 (1994) 253-258
256
IO
2 H g
1 10-l 10 -I
2 G
. 0.5
10 -'
Fg lo-' (0 10 -& 2 i
0
lo-'lo-'lo-'
1
10
10'
NORMALIZED FREQUENCY On Fig. 2. (Left) Distance dependence of the cross-correlation functions at t = 0. The solid and dashed lines correspond hydrodynamic and critical fluctuations, respectively. m is ratio of the distance between patches and patch radius. Fig. 3. (Right) Spectral density functions for the hydrodynamic (solid line) and critical (dashed line) fluctuations. normalized frequency is wn = (r2/D)w for the hydrodynamic and wn = (r15/4/A)w for critical fluctuations.
where r is the gamma function , pFq is the generalized hypergeometric function which correspond to Eqs. (7) and (8) are plotted in Fig. 2. For hydrodynamic overlapping patches only give non-zero cross-correlation (m = 2 corresponds patches). For critical fluctuations, the cross-correlation CC’(O) is always positive distance between patches as X-I/~.
to the
The
[ 141. The curves fluctuations the to the adjacent and decays with
3. Spectral density A spectral density function is defined as a time Fourier transform of an autocorrelation function. Taking m = 0 in Eqs. (5) and (6) we obtain the autocorrelation for both the hydrodynamic AH(t ) and the critical AC(t ) fluctuations
J
AH(t) = S0(27r)~ W<_‘Jf(<) exp 0
J r-1’/4.1f(<) 00
F(t)
= C(27+7/4
exp ( -<‘5/4 $&)
d&
0
The time Fourier transform of AH( t ) and AC(t ) are cm
2r2
J
sHbd = S”(2n) 5 o and
kJ:W k4 + ($/~)+2
dk
>
(11)
ES. Shikhovtseva et al. /Surface Science 317 (1994) 253-253
dk +kJ;Vd (rls/4/A)2~2 J0kW2
257
r29/4 O” SC(o)
=
c(27+x
(12)
respectively. The calculated spectral densities S* (11)) and SC (co) for hydrodynamic (solid line) and critical (dashed line) fluctuations are presented in Fig. 3. The normalized frequency is o,, = (r2/D)o for hydrodynamic and wn = (r15/4/A)o for critical fluctuations. The full form of the integrals (9) and (11) can be found in Ref. [7].
4. Discussion In the density fluctuation method, the comparison of the experimental and theoretical autocorrelation, cross~~elation or spectral density functions yields the surface diffusion coefficient D. The magnitude of D depends on the shape of these functions. There are various factors which can influence their shape and hence D, namely: ( 1) The finite resolution of the field emission microscope requires an appropriate correction [ 151; (2) The boundary conditions influence the long time limit of auto- or cross-correlations and the low frequency range of spectral densities [7]; (3) The anisotropy of the examined adsorption system can be considered introducing the surface diffusion tensor [6,7]; (4) The effect of low and high frequency cut-off should be included according to the band width of the amplifier used [ 16- 181. The occurrence of critical fluctuations in the adlayer is the next effect which can considerably change the shape of the autocorrelation, cross-correlation or spectral density functions. In the ~lculations presented in the previous sections the values of the critical indices, 9 = l/4 and z = 15/4, were arbitrarily chosen. A real system might have different exponents. Especially controversial is the value of the dynamical critical index z. These values may influence the long time limit of the correlation functions which decay as t- VIZ.Nevertheless, the calculated results can be useful when interpreting experimental data. For hydrodynamic fluctuations, the correlation functions decay as t- I. A slower decay can indicate that we are inside a critical region. Similar qualitative conclusions can be drawn when we analyze experimental spectral density functions, Another characteristic feature of critical fluctuations is a positive value of the cross-correlation at r = 0 (for separated probe regions) that should be 0 for hydrodynamic fluctuations. We can only roughly estimate the transition temperature if we do not know the exact values of the critical indices for a real system. An attempt to estimate the transition temperature was done for ~~ssiurn adlayers on a ( 112 ) tungsten plane [ 19 1. The decay of the experimental cross-correlation functions was determined by fitting the Gaussian-like curve C(x, t) = t-p exp(-x2/4Dt) to the experimental cross-correlations (C(x, t) decays like t-p for a long time). The fitting procedure gave /3, which became smaller with a decreasing temperature as if the system approached a critical point from above. j? M l/ 15 was achieved. The estimated transition temperatures were about 350 and 380 K at potassium coverages of 0.30 and 0.38, respectively. For a similar adsorption system, potassium on Mo( 112), a transition tem~rature of about 350 K was obtained from the LEED spots intensity experiments related to the c(2 x 2) surface structure 1201. The Monte Carlo experiments which serve extremely well in describing critical fluctuations in the adlayer should support the experimental results. In the paper of Tringides and Gomer [ 2 11, the density fluctuations method was simulated and diffusion coefficients were determined by fitting the theoretical autocorrelations functions to those obtained in the Monte Carlo experiments. The authors
258
E.S. Shikhovtseva et al. /Surface Science 317 (1994) 253-258
reported that for attractive interactions J, J/kT > 1.5, there were deviations from hydrodynamic autoco~elation functions for long times. The hydrodynamic cross-correlation for separated probe regions is zero for t = 0. It can be understood when interpreting it in terms of probability. The cross-correlation defined by Eq. (4) is given by the double integral of the dynamical structure factor S(r, t) which can be interpreted as the “after effect” of Smoluchowski. S(r, t) is a probability density to find a particle in position r at time t when its initial position was r = 0 at t = 0. The integral over the area A2 gives the probability to find a particle inside an area A2 when it started from position r = 0 at time t = 0. The second integral divided by Ai gives an average over the area At. Hence, we can interpret the cross-correlation as a probability to find a particle inside an area Al after time t when its initial position was any place inside A 1. In a critical regime the calculations show non-zero cross-correlation for any distance (Fig. lb). This is caused by the existence of adsorbed phases, such as islands for example, which are in thermal equilib~um. The experimental cross-correlations at t = 0 can differ from the theoretical ones because of the time shift resulting from the finite resolution of the field emission microscope [ 151. Further experiments and Monte Carlo calculations should contribute to better understanding the critical fluctuations observed in the field emission microscope.
Acknowledgements We wish to thank Professor A. Pekalski and Professor R. Meclewski for their helpful comments. The help of MS Malinda Schmiechen in preparing the final version of the manuscript is also gratefully acknowledged. This study was supported by the MEN, under Project No 2016/W/IFD/93, which is gratefully acknowledged. References [I ] .M. Smoluchowski, Bull. Acad. Cracoviae (1906) 203; Wien. Ber. 123 (19 14) 238 1. [2 ] S. Chandrasekhar, Rev. Mod. Phys. I5 ( 1943) 1. [3] G.W. Timm and A. van der Ziel, Physica 32 (1966) 1333. (41 R. Gomet, Surf. Sci. 38 (1973) 373. [ 51 G. Mazenko, J.R. Banavar and R. Gomer, Surf. Sci. 107 ( 198 1) 459. [6] D.R. Bowman, R. Gomer, K. Muttalib and M. Tringides, Surf. Sci. 138 ( 1984) 58 1. [7 ] M.A. Gesley and L.W. Swanson, Phys. Rev. I3 32 (1985) 7703. [8] A.M. Dabrowski and Ch. Kleint, Surf. Sci. 172 (1986) 372. [9] J. Beben, Ch. Kleint and R. Meclewski, Appl. Phys. A 40 (1986) 79. f lo] J. Beben, Ch. Kleint and R. Meclewski, Z. Phys. B (Condens. Matter) 69 (1987) 3 19. [ 11 J G. Mazenko, in: Surface Mobilities on Solid Materials, Ed. Vu Thien Binh (Plenum, New York, 198 1) pp. 27-62. [ 121 D. Forster, Hydrodynamic Fluctuations, Broken Symmetry and Correlation Functions (Benjamin, Reading, MA, 1975). [ 131 R. Gomer, Rep. Prog. Phys. 53 (1990) 9 17. [ 141 A.P. Prudnikov, Yu.A. Brychkhov and 0.1. Marichev, Integrals and Series, Supplement (Nauka, Moscow, 1986) [in Russian J. [ 151 R. Gomer and A. Auerbach, Surf Sci. 167 (1986) 493. [ 161 Y. Song and R. Gomer, Surf. Sci. 290 ( 1993) 1. f 171 Y. Song and R. Gomer, Surf. Sci. 295 ( 1993) 174. [ 181 J. Beben and W. Gubernator, Surf. Sci. 304 (1994) 59. [ 191 J. Beben, Solid State Phenom. 12 ( 1990) 17. [ 201 M. S. Gupalo, Fiz. Tverd. Tela 22 ( 1980) 23 11. [21] M.C. Tringides and R. Gomer, Surf. Sci. 265 (1992) 283.