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A simple model for temperature effects in leed
SURFACE
SCIENCE 28 (1971) 258-266 0 North-Holland
A SIMPLE MODEL FOR TEMPERATURE
Publishing Co.
EFFECTS IN LEED
B. W. HOLLAND School of Physics, University of Warwick, Coventry, Warwickshire CV4 7AL, England
Received 21 June 1971; revised manuscript received 2 August 1971 A theory of the purely elastic multiple scattering of electrons by the surface of a vibrating crystal is given. The result of Duke and Laramore, that the rigid lattice theory is modified simply by the appearance of a temperature dependent factor with each ion-core t-matrix element, is obtained directly when equal time correlations between displacements on different sites are neglected. A simplified model is proposed, in which these temperature dependent factors are made isotropic and an isotropic model adopted for the ion-core scattering amplitude. Using an Einstein model for the lattice, calculations (in which no adjustable parameters appear) describe qualitatively the dramatic temperature effects found by Reid on Cu(OO1).
1. Introduction Recent work of Reidl) shows that variations in temperature can have a dramatic effect on certain features of LEED data. He found that for appropriate orientations of the incident beam falling on a Cu(OO1) surface, the intensity-voltage profile for the specularly reflected beam shows a doublet structure, with a weak peak at about 34 eV and a much stronger peak at about 42 eV. On raising the temperature by a few hundred degrees the higher energy peak dropped well below the lower energy peak in intensity. While such effects do not appear to be common, this result shows that any calculation based on a rigid lattice model may fail to give even a qualitatively correct description of certain aspects of the data. Further, it is not possible to account for the effect by the use of a simple Debye-Waller correction as in work on X-ray diffraction. The reason is that in LEED the temperature effects are complicated by the presence of strong multiple scattering. A theory of LEED that incorporates both multiple scattering and temperature effects has been developed by Duke and Laramore2), but as yet few calculations going beyond the rigid lattice model have been performedz-5). The theory shows that in the multiple scattering series, temperature appears in a rather simple way provided the lattice displacements can be treated in the harmonic approximation, and that the contributions of certain phonon propagators (which are shown to be small in the Debye model in the high 258
MODEL
FOR TEMPERATURE
EFFECTS
IN LEED
259
temperature limit) are neglected. In fact the rigid ion-core t-matrix elements for an incoming state lk) and an outgoing state lk’) are simply multiplied by a factor exp[+ 1 K”(U”UB),KB], where tzK=A(k-k’) is the momentum transfer and (lJ”@), is the thermal average of a product of ion-core displacement components. Notice however, in contrast to the simple Debye-Waller correction of single-scattering theory, that k and k’ are not necessarily wave-vectors of the external incident and diffracted beams, so that the intensity calculation in general involves an integration over one or both of thema). A crude model that has proved rather useful in understanding the qualitative behaviour of LEED data with variation of the experimental parameters such as energy, angle of incidence and azimuthal angle, is the isotropic scatterer version of the Inelastic Collision Models-lo). This is so in spite of the fact that it is well known that at the energies of interest in LEED higher partial waves give an important contribution to the ion-core scattering amplitude so that the scattering is far from isotropicrr,ra). Presumably, the gross features of the systematic behaviour of the data with experimental parameters are determined largely by the geometry, the strength of the inelastic damping, and by the overall strength of the ion-core scattering rather than by the details of how the scattering is distributed among the partial wavesis). This suggests that an extension of the isotropic scatterer version of the Inelastic Collision Model to include the influence of temperature should be capable of giving a qualitative description of Reid’s effect, and our aim in this paper is to demonstrate that this is indeed the case. Of course, if detailed agreement between theory and experiment is sought, one must give a realistic description of the ion-core scattering, but Reid’s result and the theory of Duke and Laramore indicate that not merely accurate rigid ion-core potentials are required, but also the significant modifications arising from the dynamics of the lattice should be included. There is also a question concerning the relevance of bulk potentials to the problem, particularly at low energies. We first derive in a different and rather direct way the result of Duke and Laramore2) that the temperature enters simply by modifying the ion-core t-matrices. In order to do this we have to neglect effects arising from equal time correlations between displacements of different ions, which is the meaning of the neglect of diagrams containing phonon propagators in Duke and Laramore’s approach. The simplification to isotropic scatterers follows, together with details of the calculation, and finally the comparison between theory and experiment is presented and discussed.
260
B. W. HOLLAND
2. Multiple scattering in a vibrating lattice Since the electrons travel so much more rapidly than the ions, we solve for the scattering from the ions in a definite configuration and then average the intensity over configurations. The cross-section can be formally expressed as ck+k’
2l(k’l T lk>12,
=
where the matrix elements of the transition operator T are defined with respect to states of an electron interacting with a uniform electron gas in the quasi-particle model, and are in fact damped plane waves’). Provided that the interaction Hamiltonian can be expressed as a sum of interactions of the quasi-particle with the individual ion-cores, T can be expanded in a multiple scattering series13); T =
c t, + R
where t, is the transition
1
tR,G+tR +...,
(2&
operator
for the ion-core
at R, and
1 G+ = lim ~ ,,+e E - Ho + iq ’ where E is the incident energy, Ho the free quasi-particle Hamiltonian, and yl a real positive parameter. On taking matrix elements of the above equation with respect to the incident and final states we find an infinite series with general terms of the form14Tr5)
c
RI,....R.
(Rz#Rl,
exp[i(k*R,
- k’.R,)]
etc.)
x exp [i(q
- k’)eU,]
x exp[i(q’-q).U,_,]
s
t, lq) x
d3q d3q’...d3p(k’l
w4k(Rn ~~~~~ - 4-J~~~
x ~~
E(4)
w[$~(L~
m~Py
E(q)
cql
t
- 2
-
_
” 1
lql)
x
L2)1~~x
- CP
x ... (pl t, Ik) exp [i (k - p). UJ, where Ri specifies a lattice site, lJi a displacement from that site and ti is the ion-core transition operator taking the origin of coordinates at ith lattice site; l?(q) =E-x(q, E), where c(q, E) is an electron self-energy. The energy denominators ensure that only on energy shell ion-core t-matrix elements contribute. Notice that apart from the exponential factors involving lattice
MODEL
FOR TEMPERATURE
EFFECTS
261
IN LEED
displacement operators associated with each ion-core t-matrix element, the result is the same as for the rigid lattice14). To find the intensity for the chosen configuration we must now multiply this series by its complex conjugate, which will yield another infinite series, each term of which will contain a product of exponential factors involving the ion-core displacements. Each product can be collapsed into a simple exponential factor eiU, where U is of the form
(where Ki is some wave-vector) since all the displacement We must now average the intensity over configurations using a theorem due to Glauber IS) ; (exp(iU)),
= exp(-
operators commute. which is easily done
+,),
since U is a linear combination of phonon annihilation and creation Let us assume for the moment that all displacements contained to different sites. Then ( U2),
operators. in U refer
= C Wi (Ki) + C KTDsKjS 3 I %B i,j
where Wi (Ki) = ((Ki. Ui)2)T, and Dij=(U,“Uf), is an equal times correlation function for the displacement components at different sites. If we now neglect correlations between displacements at different sites the entire temperature dependence can be described by associating a Debye-Waller type of factor, exp [ - fWi (Ki)], with each ion-core r-matrix element. We can now, as shown in detail by Duke and Laramore 2), derive a set of coupled algebraic equations for the planar scattering amplitudes which are essentially similar to those obtained by Beeby14) for the rigid lattice and which can be solved by the same methods79*). Duke and Laramorea) have evaluated essentially the equal times correlation functions, assuming a Debye model, in the high temperature limit, for a momentum transfer AK such that A2K2/2m is about 25 eV, and find that they can be safely neglected though at much higher momentum transfers this conclusion may no longer be valid. However, experiments are usually done at temperatures comparable to the Debye temperature. Further, for the correlations of most importance in LEED, in view of the strong inelastic damping, namely near neighbour correlations, the Debye model is hardly very reliable. We therefore regard the neglect of displacement correlations as a weak point in the theory, but since a more careful treatment would prevent a description of temperature effects by any simple modification of the rigid lattice theory, we shall not attempt it unless comparison with experi-
262
B. W. HOLLAND
ment indicates that it is essential. This is indeed likely to be the case for crystals with more than one atom per primitive unit cell, unless the temperature is so high that kT is greater than the optical phonon energies. Our arguments have been based on the consideration of terms for which no two displacements refer to the same ion. However, many terms in the series will contain repetitions of single ion-core displacements, and then some of the displacement correlations that we have ignored will become identical with the W quantities that we have retained. Thus, with regard to such terms our analysis is not correct. However, for every term in which a certain repetition occurs, there will be of order N similar terms with no such repetition, where N is the number of ions within a sphere of radius of the order of the mean free path of the electron. Hence typically N$ 1 and the error arising from our incorrect treatment of the above terms will be relatively small, though one can indeed imagine scatterers with special properties such that this need not be so.
3. Further simplifications
and details of the calculation
Since we are interested here only in a qualitative understanding of the temperature effects we now revert to the isotropic model [whose deficiencies have been discussed previously 7,g,10)] in which the ion-core on energy shell t-matrix elements are replaced by (27-&/m) [a(E)]*, where a(E) is the ion core cross section at energy E. However, the temperature dependent factors exp [ - +I+‘, (Ki)] still make the effective ion-core scattering amplitude anisotropic. Assume now that we are dealing with a crystal of cubic symmetry and that the ion displacement averages are identical to those of the bulk material. Then Wi (ki) = fK; (Ui’),
,
where ( Uf)T is the mean square displacement of the ion at Ri, and Kf = (ki - I%:)‘, where ki is an incoming and k: an outgoing wave vector labelling the corresponding ion-core t-matrix element. Hence Kf =kf + kj2--22ki.k: gives rise to anisotropic scattering through the term ki.ki. Since ki and k: lie on the energy shell, Kf will have the average value (2r?l/h2) (E+ I/,) =K2, where V, is the inner potential. Replacing Kf by its average value we obtain an isotropic temperature dependent factor exp ( -AK2 ( Uf)T). Since we have already neglected displacement correlations it seems appropriate to adopt an Einstein model in order to calculate the mean square displacements. Hence for a monatomic solid
MODEL
FOR TEMPERATURE
EFFECTS
where A4 is the ionic mass, k is Boltzmann’s temperature. Thus the temperature t-matrix element is
IN LEED
constant
dependent
factor
263
and f3 is the Einstein associated
with each
We require the Einstein frequency to be equal to the average frequency in the Debye model, and hence the Debye temperature serves to determine the Einstein temperature, which in the case of Cu is about 200°K. In order to avoid the introduction of arbitrary parameters in our calculations on copper, we have used Capart’slz) calculations of the energy
r
20
Experiment
Theory
(340”)
n
I
I
30
40 Energy
(e”,
Fig. 1. The intensity-voltage profiles for the (00) beam from Cu(OO1) for a series of angles of incidence. The zeros for different curves are displaced arbitrarily. The experimental results were taken at room temperature.
264
B. W. HOLLAND
dependent phase shifts to determine [a(E)]*. We have also employed his value of the inner potential, V, = 15 eV and chosen a free path of 2a,, where a, =2.55 A is the length of the side of a unit mesh, which agrees with Capart’s value for the imaginary part of the self-energy in the appropriate energy range. Hence there are no adjustable parameters in the calculations, which otherwise proceed exactly as in our previous worklo) except for the (now isotropic) temperature dependent factor associated with each ion-core tmatrix element. 4. Comparison with experiment Fig. 1 shows that the distribution of intensity between the two peaks of the doublet in the (00) beam as the angle of incidence changes is qualitatively
Fig. 2. The effect of temperature changes on an intensity-voltage profile for the (00) beam from Cu(OO1). Corresponding experimental and theoretical curves refer to the same temperature and have the same base-line, but curves for different temperatures have slightly different base lines [see Reidl)].
MODEL
well reproduced
FOR TEMPERATURE
by the calculations.
EFFECTS
IN LEED
The discrepancy
in the absolute
265
angle of
incidence at which the effects occur appears to be characteristic of the isotropic scatterer model as noted previouslyg). Fig. 2 shows further that Reid’s most significant result, namely the switching of the relative intensities of the lower and higher energy peaks as the temperature is raised, is also quite closely reproduced in the calculations, given the appropriate angular correction determined from fig. 1. Fig. 3 shows that at somewhat larger angles of incidence the theory predicts no such reversal in relative intensities with temperature, again in agreement with experiment. The base lines of the different curves in figs. 2 and 3 are displaced somewhatr) for ease of display, but since both empirical and calculated curves have the same base lines it is clear by inspection that the effective Debye temperaturese) calculated from the theory agree reasonably closely with those obtained from experiment, though the calculated values are somewhat too low.
. ,
Fig. 3.
The effect of temperature changes at slightly larger angles of incidence than for fig. 2. Other details as in fig. 2.
266
B. W. HOLLAND
5. Discussion The fact that we have been able to give such a good qualitative description of Reid’s data with a very crude model, in which no parameters characteristic of the surface ions enter, makes it clear that the effect is not connected with surface modes or relaxation of surface planes, but arises from the sensitivity of the multiple scattering amplitude to the effective ion-core cross-section which is modulated in a temperature dependent way. Thus the lower energy peak of the doublet, which is at the position expected for the (004) Bragg peak arises predominantly from single scattering and has an effective Debye temperature of 311 f 15 ‘Kl), not much different from the nominal Debye temperature for Cu of about 330°K. The higher energy peak however, arises from multiple scattering and, having a Debye-Waller, type of factor occurring for each order of scattering, will tend to decay more rapidly with increasing temperature; hence the low effective Debye temperature of about 200”Kl). It is immediately apparent that a low effective Debye temperature does not necessarily imply that surface ions have larger mean square displacements than bulk ions. Such an argument implicitly assumes an interpretation in terms of single scattering events.
Acknowledgements 1 am indebted to Dr. G. Capart for allowing me to use his manuscript before publication, and to Dr. C. J. Hearn for some useful comments.
References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16)
R. J. Reid, Phys. Status Solidi (a) 4 (1971) K211. C. B. Duke and G. E. Laramore, Phys. Rev. 2 (1970) B4765, B4783. C. B. Duke, A. J. Howsmon and G. E. Laramore, J. Vacuum Sci. Technol. 8 (1971) 10. S Y Tong and T. N. Rhodin, Phys. Rev. Letters 26 (1971) 711. C. B. Duke, G. E. Laramore, B. W. Holland and A. M. Gibbons, Surface Sci. 27 (1971) 523. C. B. Duke and C. W. Tucker, Jr., Surface Sci. 15 (1969) 231. C. B. Duke, J. R. Anderson and C. W. Tucker, Jr., Surface Sci. 19 (1970) 127. C. W. Tucker, Jr. and C. B. Duke, Surface Sci. 24 (1971) 31. B. W. Holland, R. W. Hannum and A. M. Gibbons, Surface Sci. 25 (1971) 561. B. W. Holland, R. W. Hannum, A. M. Gibbons and D. P. Woodruff, Surface Sci. 25 (1971) 576. J. B. Pendry, J. Phys. C. 2 (1969) 2273, 2283. G. Capart, Surface Sci. 26 (1971) 429. R. G. Newton, Scattering 771eov.r of Waves and Particles (McGraw-Hill, New York, 1966). J. L. Beeby, J. Phys. C 1 (1968) 82. J. L. Beeby, Proc. Roy. Sot. (London) 279 (1964) 82. R. J. Glauber, Phys. Rev. 98 (1955) 1692.
SCIENCE 28 (1971) 258-266 0 North-Holland
A SIMPLE MODEL FOR TEMPERATURE
Publishing Co.
EFFECTS IN LEED
B. W. HOLLAND School of Physics, University of Warwick, Coventry, Warwickshire CV4 7AL, England
Received 21 June 1971; revised manuscript received 2 August 1971 A theory of the purely elastic multiple scattering of electrons by the surface of a vibrating crystal is given. The result of Duke and Laramore, that the rigid lattice theory is modified simply by the appearance of a temperature dependent factor with each ion-core t-matrix element, is obtained directly when equal time correlations between displacements on different sites are neglected. A simplified model is proposed, in which these temperature dependent factors are made isotropic and an isotropic model adopted for the ion-core scattering amplitude. Using an Einstein model for the lattice, calculations (in which no adjustable parameters appear) describe qualitatively the dramatic temperature effects found by Reid on Cu(OO1).
1. Introduction Recent work of Reidl) shows that variations in temperature can have a dramatic effect on certain features of LEED data. He found that for appropriate orientations of the incident beam falling on a Cu(OO1) surface, the intensity-voltage profile for the specularly reflected beam shows a doublet structure, with a weak peak at about 34 eV and a much stronger peak at about 42 eV. On raising the temperature by a few hundred degrees the higher energy peak dropped well below the lower energy peak in intensity. While such effects do not appear to be common, this result shows that any calculation based on a rigid lattice model may fail to give even a qualitatively correct description of certain aspects of the data. Further, it is not possible to account for the effect by the use of a simple Debye-Waller correction as in work on X-ray diffraction. The reason is that in LEED the temperature effects are complicated by the presence of strong multiple scattering. A theory of LEED that incorporates both multiple scattering and temperature effects has been developed by Duke and Laramore2), but as yet few calculations going beyond the rigid lattice model have been performedz-5). The theory shows that in the multiple scattering series, temperature appears in a rather simple way provided the lattice displacements can be treated in the harmonic approximation, and that the contributions of certain phonon propagators (which are shown to be small in the Debye model in the high 258
MODEL
FOR TEMPERATURE
EFFECTS
IN LEED
259
temperature limit) are neglected. In fact the rigid ion-core t-matrix elements for an incoming state lk) and an outgoing state lk’) are simply multiplied by a factor exp[+ 1 K”(U”UB),KB], where tzK=A(k-k’) is the momentum transfer and (lJ”@), is the thermal average of a product of ion-core displacement components. Notice however, in contrast to the simple Debye-Waller correction of single-scattering theory, that k and k’ are not necessarily wave-vectors of the external incident and diffracted beams, so that the intensity calculation in general involves an integration over one or both of thema). A crude model that has proved rather useful in understanding the qualitative behaviour of LEED data with variation of the experimental parameters such as energy, angle of incidence and azimuthal angle, is the isotropic scatterer version of the Inelastic Collision Models-lo). This is so in spite of the fact that it is well known that at the energies of interest in LEED higher partial waves give an important contribution to the ion-core scattering amplitude so that the scattering is far from isotropicrr,ra). Presumably, the gross features of the systematic behaviour of the data with experimental parameters are determined largely by the geometry, the strength of the inelastic damping, and by the overall strength of the ion-core scattering rather than by the details of how the scattering is distributed among the partial wavesis). This suggests that an extension of the isotropic scatterer version of the Inelastic Collision Model to include the influence of temperature should be capable of giving a qualitative description of Reid’s effect, and our aim in this paper is to demonstrate that this is indeed the case. Of course, if detailed agreement between theory and experiment is sought, one must give a realistic description of the ion-core scattering, but Reid’s result and the theory of Duke and Laramore indicate that not merely accurate rigid ion-core potentials are required, but also the significant modifications arising from the dynamics of the lattice should be included. There is also a question concerning the relevance of bulk potentials to the problem, particularly at low energies. We first derive in a different and rather direct way the result of Duke and Laramore2) that the temperature enters simply by modifying the ion-core t-matrices. In order to do this we have to neglect effects arising from equal time correlations between displacements of different ions, which is the meaning of the neglect of diagrams containing phonon propagators in Duke and Laramore’s approach. The simplification to isotropic scatterers follows, together with details of the calculation, and finally the comparison between theory and experiment is presented and discussed.
260
B. W. HOLLAND
2. Multiple scattering in a vibrating lattice Since the electrons travel so much more rapidly than the ions, we solve for the scattering from the ions in a definite configuration and then average the intensity over configurations. The cross-section can be formally expressed as ck+k’
2l(k’l T lk>12,
=
where the matrix elements of the transition operator T are defined with respect to states of an electron interacting with a uniform electron gas in the quasi-particle model, and are in fact damped plane waves’). Provided that the interaction Hamiltonian can be expressed as a sum of interactions of the quasi-particle with the individual ion-cores, T can be expanded in a multiple scattering series13); T =
c t, + R
where t, is the transition
1
tR,G+tR +...,
(2&
operator
for the ion-core
at R, and
1 G+ = lim ~ ,,+e E - Ho + iq ’ where E is the incident energy, Ho the free quasi-particle Hamiltonian, and yl a real positive parameter. On taking matrix elements of the above equation with respect to the incident and final states we find an infinite series with general terms of the form14Tr5)
c
RI,....R.
(Rz#Rl,
exp[i(k*R,
- k’.R,)]
etc.)
x exp [i(q
- k’)eU,]
x exp[i(q’-q).U,_,]
s
t, lq) x
d3q d3q’...d3p(k’l
w4k(Rn ~~~~~ - 4-J~~~
x ~~
E(4)
w[$~(L~
m~Py
E(q)
cql
t
- 2
-
_
” 1
lql)
x
L2)1~~x
- CP
x ... (pl t, Ik) exp [i (k - p). UJ, where Ri specifies a lattice site, lJi a displacement from that site and ti is the ion-core transition operator taking the origin of coordinates at ith lattice site; l?(q) =E-x(q, E), where c(q, E) is an electron self-energy. The energy denominators ensure that only on energy shell ion-core t-matrix elements contribute. Notice that apart from the exponential factors involving lattice
MODEL
FOR TEMPERATURE
EFFECTS
261
IN LEED
displacement operators associated with each ion-core t-matrix element, the result is the same as for the rigid lattice14). To find the intensity for the chosen configuration we must now multiply this series by its complex conjugate, which will yield another infinite series, each term of which will contain a product of exponential factors involving the ion-core displacements. Each product can be collapsed into a simple exponential factor eiU, where U is of the form
(where Ki is some wave-vector) since all the displacement We must now average the intensity over configurations using a theorem due to Glauber IS) ; (exp(iU)),
= exp(-
operators commute. which is easily done
+
since U is a linear combination of phonon annihilation and creation Let us assume for the moment that all displacements contained to different sites. Then ( U2),
operators. in U refer
= C Wi (Ki) + C KTDsKjS 3 I %B i,j
where Wi (Ki) = ((Ki. Ui)2)T, and Dij=(U,“Uf), is an equal times correlation function for the displacement components at different sites. If we now neglect correlations between displacements at different sites the entire temperature dependence can be described by associating a Debye-Waller type of factor, exp [ - fWi (Ki)], with each ion-core r-matrix element. We can now, as shown in detail by Duke and Laramore 2), derive a set of coupled algebraic equations for the planar scattering amplitudes which are essentially similar to those obtained by Beeby14) for the rigid lattice and which can be solved by the same methods79*). Duke and Laramorea) have evaluated essentially the equal times correlation functions, assuming a Debye model, in the high temperature limit, for a momentum transfer AK such that A2K2/2m is about 25 eV, and find that they can be safely neglected though at much higher momentum transfers this conclusion may no longer be valid. However, experiments are usually done at temperatures comparable to the Debye temperature. Further, for the correlations of most importance in LEED, in view of the strong inelastic damping, namely near neighbour correlations, the Debye model is hardly very reliable. We therefore regard the neglect of displacement correlations as a weak point in the theory, but since a more careful treatment would prevent a description of temperature effects by any simple modification of the rigid lattice theory, we shall not attempt it unless comparison with experi-
262
B. W. HOLLAND
ment indicates that it is essential. This is indeed likely to be the case for crystals with more than one atom per primitive unit cell, unless the temperature is so high that kT is greater than the optical phonon energies. Our arguments have been based on the consideration of terms for which no two displacements refer to the same ion. However, many terms in the series will contain repetitions of single ion-core displacements, and then some of the displacement correlations that we have ignored will become identical with the W quantities that we have retained. Thus, with regard to such terms our analysis is not correct. However, for every term in which a certain repetition occurs, there will be of order N similar terms with no such repetition, where N is the number of ions within a sphere of radius of the order of the mean free path of the electron. Hence typically N$ 1 and the error arising from our incorrect treatment of the above terms will be relatively small, though one can indeed imagine scatterers with special properties such that this need not be so.
3. Further simplifications
and details of the calculation
Since we are interested here only in a qualitative understanding of the temperature effects we now revert to the isotropic model [whose deficiencies have been discussed previously 7,g,10)] in which the ion-core on energy shell t-matrix elements are replaced by (27-&/m) [a(E)]*, where a(E) is the ion core cross section at energy E. However, the temperature dependent factors exp [ - +I+‘, (Ki)] still make the effective ion-core scattering amplitude anisotropic. Assume now that we are dealing with a crystal of cubic symmetry and that the ion displacement averages are identical to those of the bulk material. Then Wi (ki) = fK; (Ui’),
,
where ( Uf)T is the mean square displacement of the ion at Ri, and Kf = (ki - I%:)‘, where ki is an incoming and k: an outgoing wave vector labelling the corresponding ion-core t-matrix element. Hence Kf =kf + kj2--22ki.k: gives rise to anisotropic scattering through the term ki.ki. Since ki and k: lie on the energy shell, Kf will have the average value (2r?l/h2) (E+ I/,) =K2, where V, is the inner potential. Replacing Kf by its average value we obtain an isotropic temperature dependent factor exp ( -AK2 ( Uf)T). Since we have already neglected displacement correlations it seems appropriate to adopt an Einstein model in order to calculate the mean square displacements. Hence for a monatomic solid
MODEL
FOR TEMPERATURE
EFFECTS
where A4 is the ionic mass, k is Boltzmann’s temperature. Thus the temperature t-matrix element is
IN LEED
constant
dependent
factor
263
and f3 is the Einstein associated
with each
We require the Einstein frequency to be equal to the average frequency in the Debye model, and hence the Debye temperature serves to determine the Einstein temperature, which in the case of Cu is about 200°K. In order to avoid the introduction of arbitrary parameters in our calculations on copper, we have used Capart’slz) calculations of the energy
r
20
Experiment
Theory
(340”)
n
I
I
30
40 Energy
(e”,
Fig. 1. The intensity-voltage profiles for the (00) beam from Cu(OO1) for a series of angles of incidence. The zeros for different curves are displaced arbitrarily. The experimental results were taken at room temperature.
264
B. W. HOLLAND
dependent phase shifts to determine [a(E)]*. We have also employed his value of the inner potential, V, = 15 eV and chosen a free path of 2a,, where a, =2.55 A is the length of the side of a unit mesh, which agrees with Capart’s value for the imaginary part of the self-energy in the appropriate energy range. Hence there are no adjustable parameters in the calculations, which otherwise proceed exactly as in our previous worklo) except for the (now isotropic) temperature dependent factor associated with each ion-core tmatrix element. 4. Comparison with experiment Fig. 1 shows that the distribution of intensity between the two peaks of the doublet in the (00) beam as the angle of incidence changes is qualitatively
Fig. 2. The effect of temperature changes on an intensity-voltage profile for the (00) beam from Cu(OO1). Corresponding experimental and theoretical curves refer to the same temperature and have the same base-line, but curves for different temperatures have slightly different base lines [see Reidl)].
MODEL
well reproduced
FOR TEMPERATURE
by the calculations.
EFFECTS
IN LEED
The discrepancy
in the absolute
265
angle of
incidence at which the effects occur appears to be characteristic of the isotropic scatterer model as noted previouslyg). Fig. 2 shows further that Reid’s most significant result, namely the switching of the relative intensities of the lower and higher energy peaks as the temperature is raised, is also quite closely reproduced in the calculations, given the appropriate angular correction determined from fig. 1. Fig. 3 shows that at somewhat larger angles of incidence the theory predicts no such reversal in relative intensities with temperature, again in agreement with experiment. The base lines of the different curves in figs. 2 and 3 are displaced somewhatr) for ease of display, but since both empirical and calculated curves have the same base lines it is clear by inspection that the effective Debye temperaturese) calculated from the theory agree reasonably closely with those obtained from experiment, though the calculated values are somewhat too low.
. ,
Fig. 3.
The effect of temperature changes at slightly larger angles of incidence than for fig. 2. Other details as in fig. 2.
266
B. W. HOLLAND
5. Discussion The fact that we have been able to give such a good qualitative description of Reid’s data with a very crude model, in which no parameters characteristic of the surface ions enter, makes it clear that the effect is not connected with surface modes or relaxation of surface planes, but arises from the sensitivity of the multiple scattering amplitude to the effective ion-core cross-section which is modulated in a temperature dependent way. Thus the lower energy peak of the doublet, which is at the position expected for the (004) Bragg peak arises predominantly from single scattering and has an effective Debye temperature of 311 f 15 ‘Kl), not much different from the nominal Debye temperature for Cu of about 330°K. The higher energy peak however, arises from multiple scattering and, having a Debye-Waller, type of factor occurring for each order of scattering, will tend to decay more rapidly with increasing temperature; hence the low effective Debye temperature of about 200”Kl). It is immediately apparent that a low effective Debye temperature does not necessarily imply that surface ions have larger mean square displacements than bulk ions. Such an argument implicitly assumes an interpretation in terms of single scattering events.
Acknowledgements 1 am indebted to Dr. G. Capart for allowing me to use his manuscript before publication, and to Dr. C. J. Hearn for some useful comments.
References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16)
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