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or transfer width?
Surface Science 81 (1979) 57-68 0 North-Holland Publishing Company
COHERENCE LENGTH AND/OR TRANSFER
WIDTH?
George COMSA Institut fiir Grenzjl~chenforschung D-51 70 Jtilich, Germany
und Vakuumphysik,
Kernforschungsanlage
Jiilich,
Received 22 April 19’78; manuscript received in final form 8 September 1978
A dissatisfactory correlation results between beam widths obtained in electron and molecule diffraction experiments and coherence length values calculated by the consequent use of some of the pertinent formulae. Arguments are presented for the necessity to emphasize that the diffraction patterns are ultimately produced by the interference of each particle with itself. These lead to the definition of a length on the surface, which, in analogy with the transfer width introduced by Park et al., characterizes the obscuring effect of the instrument. Formulae for the transfer width in molecular beam experiments are presented and compared with experimental results.
1. Introduction The concept of coherence length (area) was introduced in the diffraction of low energy electrons (LEED) a relatively short time after this method became a practical instrument for the study of the surface structure. The initial question was: how large is the area from which the structural LEED information is coming? Or even a more practical one: how perfect has a periodic surface to be in order to produce “good” LEED patterns? The first estimation of the dimensions of the coherence area was obtained by Lander and Morrison [l], who assumed that the area is limited by the first Fresnel zone. The figure they got for the linear dimension was -3 X lo4 A. One year later, the same authors [2] considered this value as far too large to be the limiting factor for the coherence area. They were confirmed in their views by Chutjian [3]. Already in 1964, Heidenrich [4], using a very general definition, deduced a couple of formulae for the components of the coherence length as a function of the angular dimension and the energy spread of the source. Since that time, these formulae (in most cases only one of them) were currently used in discussing electron and molecular beam diffraction experiments. The values obtained ranged in general between 100 and 500 A. In his recent book, Pendry [5] defined the coherence length somewhat differently and deduced a corresponding formula, which includes, besides the source 57
58
G. Comsa f Coherence length and/or transfer width?
parameters mentioned above, also the angle of incidence. Further Pendry [6] analyzed the part played by the coherence in the appearance or non-appearance of extra spots due to adsorbate overlayers. This analysis led him to the estimation of the relative dimensions of adsorbate domains and coherence length. Park et al. [7] chose a different approach by calculating an instrument response function, From this function they deduced a length in the real space and, in analogy to the coherence length, they considered it as defining the range over which the instrument behaves as an interference detector. This length, which they called “transfer width”, is a function of a great number of instrument parameters and has in usual LEED systems values around 100 8. The existence of three different approaches refering to the same subject is not particularly unusual. Unusual was only the fact that, when a new approach was introduced, the preceding one was not commented (in fact not even cited). As a result, experimentalists needing a figure for the coherence length relevant in their experiment, took the formulae they incidentally encountered first. Because, in the case of the usual LEED experiments the three approaches lead to figures of the same order of magnitude nobody was particularly concerned. However, discussions around recent molecular beam diffraction (h4BD) experiments [8], in which the influence of the energy spread of the source is predominant, revealed that different approaches lead to figures differing by two orders of magnitude. We propose in this paper to analyze the coherence length concept and the philosophy of its use in particle diffraction experiments. The main argument concerns the necessity to emphasize the fact that the diffraction intensity pattern is ultimately resulting from the incoherent superposition of individual probability (intensity) patterns, each produced by the interference of an individual particle with itself. Thus the diffraction pattern eventually observed is determined by both the surface structure within the large coherence area of the individual particles and the broadening effect of the instrument. The transfer width is only a measure of this broadening effect. Besides the discussions with many colleagues, acknowledged in part at the end, the comments of the reviewers (Rev. 1 and Rev. 2) have contributed to the present form of the paper and will be cited correspondingly.
2. The coherence length formulae and the experiment Heidenreich [4], following the classical treatise on optics by Born and Wolf [9], “The coherence length is the spatial extent of the gave the following definition: wave packet and, for identical packets, the maximum separation at which the packets can overlap.” Doubts concerning the efficiency of this definition arise when considering that, in contrast to the photon wave packets, the wave packets describing electrons or atoms are spreading out when moving in a field free space.
G. Comsa / Coherence length and/or transfer width?
59
Using pertinent formulae (see, e.g., ref. [IO] or [ll J) one obtains that, after covering a distance of say 10 cm (source-surface), a wave packet, with an average wave length of 1 A, will have dimensions exceeding lo4 A regardless its initial dimensions. Moreover, Rev. 1 considers the identification of the size of the wave packet with coherent length to be incorrect. Indeed, he remarks that the correlation function
which characterizes the degree of interference between wave packets, is conserved in free flight in spite of the spreading of the wave packets. It results anyhow that the wave packet size is not a good measure for the coherence length. Pendry [5] gave a somewhat different definition of the coherence length: “There is a characteristic length called the coherence length such that atoms in the surface within a coherence length of one another can be thought of as illuminated by the simple wave (A exp(i k . r)). Atoms further apart must be regarded as illuminated by waves whose phase relationship is arbitrary.” Pendry considers two points in the surface, separated by a distance 1 on which the beam is incident. He calculates the mean square deviation of the phase difference between these two points and obtains (A&)* = (Ae)* i IkI* IlI* COS*ei t (AQ*(ZE,)-* IkI* II* sin*er , (1) where (A0)* and (A!?)* are the mean square angular and energy spread of the beam, respectively; E the average energy of the beam and k the wave vector; finally Bi the angle between incident beam and surface normal. He considers further that when the mean square spread in phase becomes
@jp-n*,
(2),
the phase relationship between the two points has vanished and thus, according to his definition the coherence length will be: 1,~ h/[2 cos20,(A8)* t sin*Br(AE/f?)*]“*.
(3)
An easy way to estimate whether the numerical values of the coherence length obtained from eq. (3) are reasonable or not is to correlate them with the angular spread of the diffraction beams. It is well-known that the angular spread of a beam emitted at a polar angle t9r by a great number of coherent sources placed along an interval L is As,-x(L
cos ef)-l
.
(4)
Introducing in eq. (4) for the length L values of ZCobtained from eq. (3) one gets the minimal angular spreads of the diffraction beams which can be obtained with a
G. Coma / Coherence
60
length and/or transfer width?
given instrument. Combin~g in this way the two equations, the shortcom~~ of eq. (3) become apparent. Indeed, from eqs. (3) and (4) we deduce that both, coherence length and angular spread of the diffracted beams should be, at normal incidence (0i = 0”), independent of the relative energy spread (A/Z/E) of the source. The origin of this obviously incorrect result has to be found in the coherence length definition. Although the coherence length is defined on the surface, the condition for non-arbitrary phase relationship has to be imposed at the detector where the interference is really measured and not at the surface. One has only to calculate the additional mean square deviation of the phase difference which appears when the waves reemitted by the two points in the surface are covering the distance from surface to the detector. This mean square de~tion (A~~)2 is obtained from eq. (1) by substituting for Bi, the angle Bf between the scattered beam and surface normal. Condition (2) becomes now (A@
+ (A@f)2 = n2 .
coherence length will be: __EAN X/[2 cOs20i(A0)’ + (sin’@t + sinz0f)(~/K)2]1’Z
(2’)
The corresponding
.
(3’)
(We neglected the additional angular contributions which are of no importance at this point .) As expected eq, (3’) leads to a dependence of the coherence length on the relative energy spread even at normal incidence. However, despite of this improvement even eq. (3’) leads to incorrect results in the case of the specular beam (0, = 0,). This becomes particularly evident for molecular beams where the relative energy spread is more important. For instance, considering a well collimated Maxwellian beam ((Ae)2 <10m5 radians2 and (m/E)’ = 0.5), incident at 45’, we get from eq. (3’)
With these values introduced in eq, (4), the angular spread of the specular beam is: ABf = 57’. Even for a nozzle beam with (&/f12 - 6 X IOe3, the figure is Af?f = 6.2’. These values are at least 300 respectively 30 times larger than the assumed width of the incident beam. It is however well known that the specular beam scattered from a surface of good quality has almost the same FWHM as the incident beam, regardless its energy spread.
3. The build-up of a diffraction pattern The shortcomings of the coherence length formulae have actually a common origin. By considering in a coherence length formula A/Z and A0 as characterizing the whole ensemble of the particles contained in the beam, one implicitly assumes
G. Comsa / Coherence
length and/or transfer width?
61
that one measures a coherent interference between perfectly incoherent particles. The electrons (or the molecules) leave the source at time intervals distributed completely at random and so they arrive at the detector. The contribution of any incidental interference between any two particles will average out during the measuring time, which corresponds to the arrival of many particles. Therefore the diffraction pattern is exclusively the result of the interference of each particle with itself. Simply by emphasizing this obvious and well known fact, the actual build-up of the diffraction pattern in a given experiment clearly emerges (this build-up process can be taken as a model for the calculating procedure): Each particle produces its diffraction probability pattern (squared amplitudes), the shape of which depends on surface properties. The probability patterns of the different particles are not identical, the peaks being shifted against each other due to the energy and angular spread of the incident beam. The diffraction pattern produced by all the particles results from the incoherent superposition (incoherent because probabilities and not amplitudes add together) of the individual probability patterns weighted according to the energy and angular distributions of the particles. This superposition and the finite acceptance angle of the detector lead to the angular width of the peaks, A0f, which is in fact measured. Let us first see what can be said about the angular width 68f of the peaks in the diffraction probability pattern of an individual particle, scattered on an infinite perfect crystal. If the particle is represented by a plane wave, the angular broadness of the peaks will be zero, i.e. the momentum of the particle is perfectly determined. However, the particles are emitted from a source, which is placed at a finite distance D from the surface. Thus the particle must in fact be represented by a spherical wave (its position is somewhat more precised). As a consequence the momentum is not perfectly determinated and the peaks broaden somewhat. The Fresnel zone construction gives an intuitive way to evaluate this broadening. It is well known (see, e.g., ref. [9]) that if between a wave source and detector a screen is placed covering all but half of the first Fresnel zone, the signal at the detector will be the same as without screen. In the absence of the screen, the waves propagating outside the half of the first Fresnel zone anihilate each other at the detector. The signal at the detector being the same, we can consider here that the particle is scattered only by an area of the dimensions of the half of the first Fresnel zone. These dimensions are: I = (Dh/2)“’
,
(5)
i.e.forD=lOcmandh=lA,about 104A. From eqs. (4) and (5) we may infer that the angular spread of the peaks in the diffraction probability pattern of the individual particles is about 6Br N lo-“ radians , i.e. negligible when compared with the angular spreads obtained in LEED and MBD experiments. Accordingly, we can consider for most practical purposes the
62
G. Comsa [Coherence length an&x
transfer width?
momenta of the individu~ particles as perfectly dete~ined, i.e. the corresponding probability peaks as S functions; precisely for the cases when the crystal is periodic over dimensions of at least lo4 A. If these dimensions become smaller the momenta of the particles become less and less well determined and, according to eq. (4), the diffraction probability peaks of the individual particles broaden. The diffraction pattern produced by a real instrument on an infinite perfect crystal gives us the angular broadening of the peaks due to the instrument. However, we were looking here for a length on the surface which characterizes the effect of the instrument, and primarily not for an angular broadening. This length w can be obtained fo~Q1~ from eq. (4) (with w instead of L): w = hf(A@, cos 0,) .
(4’)
The length w can be thought of as the fmite dimension of an otherwise perfect crystal from which a plane wave (the most perfect instrument) produces a diffraction peak with an angular spread A0f equal to the spread of the corresponding peak produced by the real instrument on an infinite perfect crystal (see, e.g., ref. [12]). This interpretation leads to a clear statement: A real instrument will be able to produce resolved diffraction beams in the direction f3r only if the period, d, of the grating of the perfect crystal is smaller than w; otherwise the peaks will be smeared out. Indeed, it is obvious that even the most perfect instrument cannot produce a diffraction pattern unless there are at least two scatterers present within the length W.
This statement can be put in a somewhat more general form: the instrument obscures info~at~on concem~g dimensions on the surface which are larger than w. Let us illustrate this by an example. We assume that the surface is covered with antiphase adsorbate domains. Let the period d of the adsorbate lattice be smaller and the dimensions d: of the domains larger than w. The extra peaks in the diffraction probability pattern of the individual particles, due to the scattering on the adsorbate layer, will split as a result of the presence of antiphase domains. The splitting corresponds to the distance L’ between the centers of the domains. As always X’ > &in and thus L’ > w, the instrument will smear out the splitting completely and no direct evidence of it will be seen in the diffraction pattern. Pendry [6] had predicted that in the case of randomly distributed antiphase domains, an extinction of extra beams is possible. In the light of the present analysis this would imply that even for domains with dimensions larger than w an extinction of extra beams could be observed, because non-existent peaks cannot be smeared out. However, as Rev. 2 pointed out, Pendry’s prediction is not correct because no privision is made for the transfer of the extinguished intensity in other channels, (The error originates probably in the assumption that “diffraction processes within a single domain are essentially those taking place in an infinite space” [6] ; this led to 6 function shaped beams which would be in fact extinguished. However, the finite dimensions d: and L’ and their relative magnitude L’ > d: are in this case essential and preclude the extinction of extra beams.) To the author’s know-
G. Comsa / Coherence
length and/or transfer width?
63
ledge there is no example where the presence of domains would lead to the extinction of diffracted beams. There is thus no straightforward way to demonstrate experimentally that the individual particles are scattered coherently far beyond the transfer width, Nevertheless, the measured diffracted beams do contain information supplied by the coherent scattering of the individual particles which takes place on very large areas limited practically only by surface heterogeneity. The information is only obscured by the instrument, but still there. Wang and Lagally have very recently shown that by knowing accurately the instrument response function - i.e. the obscuring effect of the instrument - one may extract from the measured diffraction beams information concerning regions on the surface whose linear dimension is at least 2 times larger than the length w (see ref. [ 131 quoted below). How should we call the length w? If the coherence length is considered only as a criterion for the “visibility” of interferences, one might call it so. It is, however, misleading to use the idea of coherence in this connection. Indeed, we have seen that for a perfect crystal and for any real instrument the individual particles are coherently scattered on areas far beyond w. Moreover, the length w is nothing else than a measure of the broadening effect of the instrument resulting from the incoherent superposition of intensities. We will see below that w is in fact identical to the transfer width introduced by Park et al. [7]. Accordingly we will call it so. Let us summarize the approach of Park et al. [7] and compare it briefly with the arguments presented above. They handled the problem in the same way as wave form distortions introduced by electrical measuring equipment are analysed. The measured intensity is expressed as the convolution product of the intensity function i(k), which is the pattern one would measure with a perfect instrument, and the instruments response function T(k), which represent the response of the instrument to a diffraction beam from a perfect crystal: i(k) meaS= i(k) * 7’(k) .
i(k) corresponds
to our diffraction probability pattern of an individual particle, which is the most perfect instrument if it originates at infinity (plane wave). Taking ‘the Fourier transform of the measured intensity they obtain F{i(k),,,,)
= F(G))
FIT(k))
= Q(r) t(r> p
where 6(t) is the autocorrelation function and t(r), the Fourier transform of the response function, is defined as the transfer function. r is a vector connecting grating points. The equation shows that no correlation between grating points separated by distances r for which t(r) is zero can be observed [13]. Accordingly, Park et al. introduce the transfer width wt, the width of the function t(r) in the real space, as “the range over which the instrument is effective as an interference detector”. Thus wt is indeed equivalent to the length w we deduced above and its interpretation corresponds to the meaning of the information we consider that a given instrument is able to give, without making use of a detailed knowledge of the transfer function.
64
G. Comsa /Coherence
length and/or transfer width?
For the special case of the LEED instruments (01= 0”, focussing effects, etc.), Park et al. [7] have calculated the width WT of the response function T(k) in k-space. By assuming that T(k) is Gaussian they obtained the transfer width from
It is in fact surprising that in spite of the existence of this rigorous approach and of these “ready to use” formulae for LEED experiments, the Heidenreich and Pendry formulae are still used by LEED people. One may speculate about the reasons: the not very intuitive character of the approach or the lack of any mutual recognition between the authors of the different approaches.
4. The transfer width in MBD experiments We will deduce now “ready to use” formulae for MBD experiments and discuss some aspects of the influence of the energy spread which is predominant in this kind of experiments. The contribution of the finite source detector and aperture dimensions to the transfer width will be calculated by following the procedure presented in section 3: we will calculate first the broadening effect of these parameters and then we correlate it by means of eq. (4’) with a length on the surface. An approximate expression for the broadening can be obtained from elementary geometrical considerations by using the scheme in fig. 1:
(6) The four terms represent the contribution of the source dimension, of the aperture, of the spot size and of the detector opening, respectively. The result is a rough approximation because the broadenings were added as if they were gaussian and
Fig. 1. Scheme
of an MBD experiment.
G. Coma / Coherence
length and/or transfer width?
65
because in the first two terms some contribution are considered in excess. However, (6) is a useful expression for estimating the contribution of the various instrumental dimensions. From eqs. (6) and (4’) we obtain the corresponding contribution to the transfer width: we = h/( lA,&l cos 0,) .
(6’)
More interesting, and as already mentioned much more important in MBD, is the contribution of the energy spread of the source. We may use the same procedure as above, i.e. to calculate first the angular spread resulting from the energy spread. However, we will choose a procedure showing similarities to that used by Pendry [5] in order to emphasize the differences: Let us consider the phase difference @ resulting at the detector by the scattering of one molecule (plane wave, with wave vector k) on two points in the surface separated by a distance w: $ = kw(sin Br - sin 0,) .
(7)
The diffraction probability pattern of the molecule will have in the direction Br a maximum or a minimum depending on whether @ is an even or an odd multiple of n. Consistent with the definition of the transfer width w, we impose the condition that, due to the spread in energy, among the different probability patterns of the molecules constituting the beam, one may find patterns with maxima and others with minima in the same direction of; correspondingly, the peaks around Br in the measured diffraction pattern will be smeared out. The condition is: zA@ =n’.
(8)
This is in fact identical to Pendry’s condition (2). However condition (8) is imposed after the interference of each particle with itself had taken place, and not before. From eqs. (7) and (8) we obtain
21 h/{ lsin 01 - sin
efl[m/E2]
l12}.
Using eq. (4) the contribution obtained Afir
AOf =
of the energy spread to the angular broadening
2: { lsin Bi - sin ~~l[(AE)‘/E’]“‘}/(cos
The total angular broadening KA,ef)2
+ @df)
is obtained
2 l/2
1
(9)
0,) .
is
(10)
from (6) and (10)
(11)
3
and finally the transfer width is w = h/[(A,0f)2 It is worthwhile term disappears.
c0s2ef t (sin Bi - sin ef)2(aE)z/h-2]l’2
.
(12)
to examine eqs. (9) to (12) more closely: For Bi= fQ the energy Thus, in accordance with experiment, the energy spread has no
G. Comsa / Coherence length and/or transfer width?
66
influence on the broadness of the specular beam. In section 2, the anomalous broadening of the specular beam predicted by Pendry’s formula led us to its rejection. Using now the Bragg condition, eq. (9) may be written: WE~d/{InI[(~2/E2]“2}
)
(9’)
where d is the period of the grating and n the order of diffraction. Remembering that the instrument can resolve diffraction peaks only if d G w, we obtain the maximum order of diffraction which a given instrument is able to resolve Inl < l/[(A_!?)2/E2]1’2 .
(13)
This upper limit is due to the energy spread alone, the angular imprecisions leading to a further reduction. (a) For Maxwellian beams ((A&‘)“/E2 = OS), condition (13) limits the well resolved diffraction beams to the zero and first order beams. To our knowledge, all Maxwellian beam experiments except one confirm this conclusion so far. Indeed, the diffraction patterns, presented in all classical MBD papers published in the thirties (see, e.g., Estermann and Stem [14], Estermann et al. [15], Johnson [ 161 and Zabel [17]), as well as in the modern ones in which Maxwellian beams were still used (see, e.g., Crews [18], O’Keefe et al. [19], Hoinkes et al. [20], Hayward and Walters [21] and Finzel et al. [22]), contain besides the specular beam only first order beams. Only Weinberg and Merrill [23] obtained, when using a Maxwellian beam, well defined [20] and [02] beams. However, this exception confirms in fact the rule because the beam shapes had also another troublesome peculiarity which could not be explained by the authors: The half-widths of the diffracted beams were less than 50% of the incident beam half-width. (b) After new molecular beam sources came in use (multi-arrays, nozzles), which met less and less the Knudsen cell conditions leading to always narrower energy spreads, the order of resolved diffraction beams continuously increased. The largest order yet resolved is shown in a recent paper by Lapujoulade and Lejay [241 with diffraction beams up to n 2: 11-12. They give for the energy spread of their beam: (A,!?)“/&? = 3 X 10-3. F rom (13) we deduce an upper limit Inl<
17 )
which is in fair agreement with the experiment.
5. Conclusion The analysis shows that neither Heidenreichs’ [4] nor Pendry’s [5] formulae properly describe the role of the instrument in obscuring the diffraction pattern features, in partictdar the instrumental broadening of the diffraction peaks. The reason for this failure is probably the fact that they imply the existence of a coherence of an ensemble of perfectly incoherent particles.
G. Comsa /Coherence length and/or transfer width?
61
It is assumed here that the measured diffraction pattern is ultimately the result of the coherent interference of each particle with itself; more precisely, that the diffraction pattern results from the incoherent summation of the diffraction probability patterns of the individual particles. Due to the instrumental energy and angular spreads, the individual probability patterns are shifted against each other leading to a broadening of the measured diffraction beams. In the case of a perfect infinite crystal, this angular broadening may be correlated to a length on the surface, named “transfer width” which is thus only a measure of broadening effect of the instrument. The transfer width corresponds to the largest period of the grating which can be straightforwardly resolved with a given instrument. These results are essentially equivalent to those obtained from the approach by Park et al. [7]. The formulae for the transfer width and angular spread in the particular case of molecular beam diffraction which have been derived, compare well with experiments. We hope that this discussion will reduce the arbitrariness in choosing the “right coherence formula”.
Acknowledgements The conclusions I reached in this paper were preceded by discussions with many colleagues from this and other laboratories. I am particularly indebted to Harald Ibach for his suggestion concerning the influence of the detector and for the critical reading of the manuscript. It is a pleasure to acknowledge the valuable discussions with Hans Bonzel, R. Feder, Jan Fremerey and Heribert Wagner.
References [ 1) [2] [3] [4] [S] [6] [7] [8]
[9] [ 101 [ll]
J.J. Lander and .I. Morrison, J. Chem. Phys. 37 (1962) 729. J.J. Lander and J. Morrison, J. Appl. Phys. 34 (1963) 3517. A. Chutjian, Phys. Letters 24A (1967) 615. R.D. Heidenreich, Fundamentals of Transmission Electron Microscopy (Interscience, New York, 1964) pp. 101-105. J.B. Pendry, Low Energy Electron Diffraction (Academic Press, London, 1974) pp. 5-6. See ref. [S], pp. 228-236. R.L. Park, J.E. Houston and D.G. Schreiner, Rev. Sci. Instr. 42 (1971) 60. J. Lapujoulade and Y. Lejay, in: Compt. Rend. 3Bme ColIoq. Intern. du Physique et Chimie des Surfaces, Grenoble, 1977 (Suppl. Le Vide No. 185) pp. 328-334; Y. Lejay and J. Lapujoulade, in: Proc. 7th Intern. Vacuum Congr. and 3rd Intern. Conf. on Solid Surfaces, Vienna, 1977, Eds. R. Dobrozemsky, F. Riidenauer, F.P. ViehbGck and A. Breth, Vol. II, pp. 1373-1375. M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon, Oxford, 1965). L.I. Shiff, Quantum Mechanics, 2nd ed. (McGraw-Hill,New York, 1955) pp. 57-58. A. Messiah, MBcanique Quantique (Dunod, Paris, 1959) p. 187.
68
G. Comsa / Coherence length and/or transfer width?
[12] M. Henzler, in: Electron
Spectroscopy for Surface Analysis, Ed. H. Ibach (Springer, Berlin, 1977) pp. 120-121. [ 131 G.-C. Wang and M.G. Lagally, Surface Sci. 81 (1979) 69. The authors show that: “Only if t(r) is non-zero in a region of finite width.(e.g. a rectangular function) wiB the determination of e(r) be limited to finite r. Since T(k) and hence t(r), is typically between a Gaussian and a Lorentzian function, only the noise level on these functions, . . ., wili limit the range r over which a(r) can be determined.” And further that: “It is the uncertainty in the determination of both i(k),,,s and T(k) that results in a minimum angular width for i(k) that can be determined with any given level of confidence. i(k) functions with a FWHM of less than 50% of the FWHM of T(k) are possible”. I am indebted to Max Lagaiiy for sending me the manuscript prior publication. Unfortunately, I got it only when the revised form of this paper was accepted for publication so that a deeper integration in this paper of their very clear ideas was hardly possible. [14] I. Estermann and 0. Stern, Z. Physik 61 (1930) 95. [ 151 I. Estermann, R. Frisch and 0. Stern, Z. Physik 73 (1931) 348. [ 161 T.H. Johnson, Phys. Rev. 37 (1931) 847. [ 171 R.M. Zabel, Phys. Rev. 42 (1932) 218. [ 181 J.C. Crews, J. Chem. Phys. 37 (1962) 2004. [19] D.R. O’Keefe, R.L. Palmer, H. Saltsburg and J.N. Smith, Jr., J. Chem. Phys. 49 (1968) 5194; D.R. O’Keefe, J.N. Smith, Jr., R.L. Palmer and H. Saltsburg, Surface Sci. 20 (1970) 27. [20] H. Hoinkes, H. Nahr and H. Wilsch, Surface Sci. 30 (1972) 363. [21] D.O. Hayward and M.R. Walters, in: Proc. 2nd Intern. Conf. on Solid Surfaces, Japan. J. Appl. Phys. Suppl. 2 (1974) 587. [22] H.-U. Finzel, H. Frank, H. Hoinkes, M. Luschka, H. Nahr, H. Wilsch and U. Wonka, Surface Sci. 49 (1975) 577. [23] W.H. Weinberg and R.P. Merrill, J. Chem. Phys. 56 (1972) 2893. [24] J. Lapujoulade and Y. Lejay, Surface Sci. 69 (1977) 354.
COHERENCE LENGTH AND/OR TRANSFER
WIDTH?
George COMSA Institut fiir Grenzjl~chenforschung D-51 70 Jtilich, Germany
und Vakuumphysik,
Kernforschungsanlage
Jiilich,
Received 22 April 19’78; manuscript received in final form 8 September 1978
A dissatisfactory correlation results between beam widths obtained in electron and molecule diffraction experiments and coherence length values calculated by the consequent use of some of the pertinent formulae. Arguments are presented for the necessity to emphasize that the diffraction patterns are ultimately produced by the interference of each particle with itself. These lead to the definition of a length on the surface, which, in analogy with the transfer width introduced by Park et al., characterizes the obscuring effect of the instrument. Formulae for the transfer width in molecular beam experiments are presented and compared with experimental results.
1. Introduction The concept of coherence length (area) was introduced in the diffraction of low energy electrons (LEED) a relatively short time after this method became a practical instrument for the study of the surface structure. The initial question was: how large is the area from which the structural LEED information is coming? Or even a more practical one: how perfect has a periodic surface to be in order to produce “good” LEED patterns? The first estimation of the dimensions of the coherence area was obtained by Lander and Morrison [l], who assumed that the area is limited by the first Fresnel zone. The figure they got for the linear dimension was -3 X lo4 A. One year later, the same authors [2] considered this value as far too large to be the limiting factor for the coherence area. They were confirmed in their views by Chutjian [3]. Already in 1964, Heidenrich [4], using a very general definition, deduced a couple of formulae for the components of the coherence length as a function of the angular dimension and the energy spread of the source. Since that time, these formulae (in most cases only one of them) were currently used in discussing electron and molecular beam diffraction experiments. The values obtained ranged in general between 100 and 500 A. In his recent book, Pendry [5] defined the coherence length somewhat differently and deduced a corresponding formula, which includes, besides the source 57
58
G. Comsa f Coherence length and/or transfer width?
parameters mentioned above, also the angle of incidence. Further Pendry [6] analyzed the part played by the coherence in the appearance or non-appearance of extra spots due to adsorbate overlayers. This analysis led him to the estimation of the relative dimensions of adsorbate domains and coherence length. Park et al. [7] chose a different approach by calculating an instrument response function, From this function they deduced a length in the real space and, in analogy to the coherence length, they considered it as defining the range over which the instrument behaves as an interference detector. This length, which they called “transfer width”, is a function of a great number of instrument parameters and has in usual LEED systems values around 100 8. The existence of three different approaches refering to the same subject is not particularly unusual. Unusual was only the fact that, when a new approach was introduced, the preceding one was not commented (in fact not even cited). As a result, experimentalists needing a figure for the coherence length relevant in their experiment, took the formulae they incidentally encountered first. Because, in the case of the usual LEED experiments the three approaches lead to figures of the same order of magnitude nobody was particularly concerned. However, discussions around recent molecular beam diffraction (h4BD) experiments [8], in which the influence of the energy spread of the source is predominant, revealed that different approaches lead to figures differing by two orders of magnitude. We propose in this paper to analyze the coherence length concept and the philosophy of its use in particle diffraction experiments. The main argument concerns the necessity to emphasize the fact that the diffraction intensity pattern is ultimately resulting from the incoherent superposition of individual probability (intensity) patterns, each produced by the interference of an individual particle with itself. Thus the diffraction pattern eventually observed is determined by both the surface structure within the large coherence area of the individual particles and the broadening effect of the instrument. The transfer width is only a measure of this broadening effect. Besides the discussions with many colleagues, acknowledged in part at the end, the comments of the reviewers (Rev. 1 and Rev. 2) have contributed to the present form of the paper and will be cited correspondingly.
2. The coherence length formulae and the experiment Heidenreich [4], following the classical treatise on optics by Born and Wolf [9], “The coherence length is the spatial extent of the gave the following definition: wave packet and, for identical packets, the maximum separation at which the packets can overlap.” Doubts concerning the efficiency of this definition arise when considering that, in contrast to the photon wave packets, the wave packets describing electrons or atoms are spreading out when moving in a field free space.
G. Comsa / Coherence length and/or transfer width?
59
Using pertinent formulae (see, e.g., ref. [IO] or [ll J) one obtains that, after covering a distance of say 10 cm (source-surface), a wave packet, with an average wave length of 1 A, will have dimensions exceeding lo4 A regardless its initial dimensions. Moreover, Rev. 1 considers the identification of the size of the wave packet with coherent length to be incorrect. Indeed, he remarks that the correlation function
which characterizes the degree of interference between wave packets, is conserved in free flight in spite of the spreading of the wave packets. It results anyhow that the wave packet size is not a good measure for the coherence length. Pendry [5] gave a somewhat different definition of the coherence length: “There is a characteristic length called the coherence length such that atoms in the surface within a coherence length of one another can be thought of as illuminated by the simple wave (A exp(i k . r)). Atoms further apart must be regarded as illuminated by waves whose phase relationship is arbitrary.” Pendry considers two points in the surface, separated by a distance 1 on which the beam is incident. He calculates the mean square deviation of the phase difference between these two points and obtains (A&)* = (Ae)* i IkI* IlI* COS*ei t (AQ*(ZE,)-* IkI* II* sin*er , (1) where (A0)* and (A!?)* are the mean square angular and energy spread of the beam, respectively; E the average energy of the beam and k the wave vector; finally Bi the angle between incident beam and surface normal. He considers further that when the mean square spread in phase becomes
@jp-n*,
(2),
the phase relationship between the two points has vanished and thus, according to his definition the coherence length will be: 1,~ h/[2 cos20,(A8)* t sin*Br(AE/f?)*]“*.
(3)
An easy way to estimate whether the numerical values of the coherence length obtained from eq. (3) are reasonable or not is to correlate them with the angular spread of the diffraction beams. It is well-known that the angular spread of a beam emitted at a polar angle t9r by a great number of coherent sources placed along an interval L is As,-x(L
cos ef)-l
.
(4)
Introducing in eq. (4) for the length L values of ZCobtained from eq. (3) one gets the minimal angular spreads of the diffraction beams which can be obtained with a
G. Coma / Coherence
60
length and/or transfer width?
given instrument. Combin~g in this way the two equations, the shortcom~~ of eq. (3) become apparent. Indeed, from eqs. (3) and (4) we deduce that both, coherence length and angular spread of the diffracted beams should be, at normal incidence (0i = 0”), independent of the relative energy spread (A/Z/E) of the source. The origin of this obviously incorrect result has to be found in the coherence length definition. Although the coherence length is defined on the surface, the condition for non-arbitrary phase relationship has to be imposed at the detector where the interference is really measured and not at the surface. One has only to calculate the additional mean square deviation of the phase difference which appears when the waves reemitted by the two points in the surface are covering the distance from surface to the detector. This mean square de~tion (A~~)2 is obtained from eq. (1) by substituting for Bi, the angle Bf between the scattered beam and surface normal. Condition (2) becomes now (A@
+ (A@f)2 = n2 .
coherence length will be: __EAN X/[2 cOs20i(A0)’ + (sin’@t + sinz0f)(~/K)2]1’Z
(2’)
The corresponding
.
(3’)
(We neglected the additional angular contributions which are of no importance at this point .) As expected eq, (3’) leads to a dependence of the coherence length on the relative energy spread even at normal incidence. However, despite of this improvement even eq. (3’) leads to incorrect results in the case of the specular beam (0, = 0,). This becomes particularly evident for molecular beams where the relative energy spread is more important. For instance, considering a well collimated Maxwellian beam ((Ae)2 <10m5 radians2 and (m/E)’ = 0.5), incident at 45’, we get from eq. (3’)
With these values introduced in eq, (4), the angular spread of the specular beam is: ABf = 57’. Even for a nozzle beam with (&/f12 - 6 X IOe3, the figure is Af?f = 6.2’. These values are at least 300 respectively 30 times larger than the assumed width of the incident beam. It is however well known that the specular beam scattered from a surface of good quality has almost the same FWHM as the incident beam, regardless its energy spread.
3. The build-up of a diffraction pattern The shortcomings of the coherence length formulae have actually a common origin. By considering in a coherence length formula A/Z and A0 as characterizing the whole ensemble of the particles contained in the beam, one implicitly assumes
G. Comsa / Coherence
length and/or transfer width?
61
that one measures a coherent interference between perfectly incoherent particles. The electrons (or the molecules) leave the source at time intervals distributed completely at random and so they arrive at the detector. The contribution of any incidental interference between any two particles will average out during the measuring time, which corresponds to the arrival of many particles. Therefore the diffraction pattern is exclusively the result of the interference of each particle with itself. Simply by emphasizing this obvious and well known fact, the actual build-up of the diffraction pattern in a given experiment clearly emerges (this build-up process can be taken as a model for the calculating procedure): Each particle produces its diffraction probability pattern (squared amplitudes), the shape of which depends on surface properties. The probability patterns of the different particles are not identical, the peaks being shifted against each other due to the energy and angular spread of the incident beam. The diffraction pattern produced by all the particles results from the incoherent superposition (incoherent because probabilities and not amplitudes add together) of the individual probability patterns weighted according to the energy and angular distributions of the particles. This superposition and the finite acceptance angle of the detector lead to the angular width of the peaks, A0f, which is in fact measured. Let us first see what can be said about the angular width 68f of the peaks in the diffraction probability pattern of an individual particle, scattered on an infinite perfect crystal. If the particle is represented by a plane wave, the angular broadness of the peaks will be zero, i.e. the momentum of the particle is perfectly determined. However, the particles are emitted from a source, which is placed at a finite distance D from the surface. Thus the particle must in fact be represented by a spherical wave (its position is somewhat more precised). As a consequence the momentum is not perfectly determinated and the peaks broaden somewhat. The Fresnel zone construction gives an intuitive way to evaluate this broadening. It is well known (see, e.g., ref. [9]) that if between a wave source and detector a screen is placed covering all but half of the first Fresnel zone, the signal at the detector will be the same as without screen. In the absence of the screen, the waves propagating outside the half of the first Fresnel zone anihilate each other at the detector. The signal at the detector being the same, we can consider here that the particle is scattered only by an area of the dimensions of the half of the first Fresnel zone. These dimensions are: I = (Dh/2)“’
,
(5)
i.e.forD=lOcmandh=lA,about 104A. From eqs. (4) and (5) we may infer that the angular spread of the peaks in the diffraction probability pattern of the individual particles is about 6Br N lo-“ radians , i.e. negligible when compared with the angular spreads obtained in LEED and MBD experiments. Accordingly, we can consider for most practical purposes the
62
G. Comsa [Coherence length an&x
transfer width?
momenta of the individu~ particles as perfectly dete~ined, i.e. the corresponding probability peaks as S functions; precisely for the cases when the crystal is periodic over dimensions of at least lo4 A. If these dimensions become smaller the momenta of the particles become less and less well determined and, according to eq. (4), the diffraction probability peaks of the individual particles broaden. The diffraction pattern produced by a real instrument on an infinite perfect crystal gives us the angular broadening of the peaks due to the instrument. However, we were looking here for a length on the surface which characterizes the effect of the instrument, and primarily not for an angular broadening. This length w can be obtained fo~Q1~ from eq. (4) (with w instead of L): w = hf(A@, cos 0,) .
(4’)
The length w can be thought of as the fmite dimension of an otherwise perfect crystal from which a plane wave (the most perfect instrument) produces a diffraction peak with an angular spread A0f equal to the spread of the corresponding peak produced by the real instrument on an infinite perfect crystal (see, e.g., ref. [12]). This interpretation leads to a clear statement: A real instrument will be able to produce resolved diffraction beams in the direction f3r only if the period, d, of the grating of the perfect crystal is smaller than w; otherwise the peaks will be smeared out. Indeed, it is obvious that even the most perfect instrument cannot produce a diffraction pattern unless there are at least two scatterers present within the length W.
This statement can be put in a somewhat more general form: the instrument obscures info~at~on concem~g dimensions on the surface which are larger than w. Let us illustrate this by an example. We assume that the surface is covered with antiphase adsorbate domains. Let the period d of the adsorbate lattice be smaller and the dimensions d: of the domains larger than w. The extra peaks in the diffraction probability pattern of the individual particles, due to the scattering on the adsorbate layer, will split as a result of the presence of antiphase domains. The splitting corresponds to the distance L’ between the centers of the domains. As always X’ > &in and thus L’ > w, the instrument will smear out the splitting completely and no direct evidence of it will be seen in the diffraction pattern. Pendry [6] had predicted that in the case of randomly distributed antiphase domains, an extinction of extra beams is possible. In the light of the present analysis this would imply that even for domains with dimensions larger than w an extinction of extra beams could be observed, because non-existent peaks cannot be smeared out. However, as Rev. 2 pointed out, Pendry’s prediction is not correct because no privision is made for the transfer of the extinguished intensity in other channels, (The error originates probably in the assumption that “diffraction processes within a single domain are essentially those taking place in an infinite space” [6] ; this led to 6 function shaped beams which would be in fact extinguished. However, the finite dimensions d: and L’ and their relative magnitude L’ > d: are in this case essential and preclude the extinction of extra beams.) To the author’s know-
G. Comsa / Coherence
length and/or transfer width?
63
ledge there is no example where the presence of domains would lead to the extinction of diffracted beams. There is thus no straightforward way to demonstrate experimentally that the individual particles are scattered coherently far beyond the transfer width, Nevertheless, the measured diffracted beams do contain information supplied by the coherent scattering of the individual particles which takes place on very large areas limited practically only by surface heterogeneity. The information is only obscured by the instrument, but still there. Wang and Lagally have very recently shown that by knowing accurately the instrument response function - i.e. the obscuring effect of the instrument - one may extract from the measured diffraction beams information concerning regions on the surface whose linear dimension is at least 2 times larger than the length w (see ref. [ 131 quoted below). How should we call the length w? If the coherence length is considered only as a criterion for the “visibility” of interferences, one might call it so. It is, however, misleading to use the idea of coherence in this connection. Indeed, we have seen that for a perfect crystal and for any real instrument the individual particles are coherently scattered on areas far beyond w. Moreover, the length w is nothing else than a measure of the broadening effect of the instrument resulting from the incoherent superposition of intensities. We will see below that w is in fact identical to the transfer width introduced by Park et al. [7]. Accordingly we will call it so. Let us summarize the approach of Park et al. [7] and compare it briefly with the arguments presented above. They handled the problem in the same way as wave form distortions introduced by electrical measuring equipment are analysed. The measured intensity is expressed as the convolution product of the intensity function i(k), which is the pattern one would measure with a perfect instrument, and the instruments response function T(k), which represent the response of the instrument to a diffraction beam from a perfect crystal: i(k) meaS= i(k) * 7’(k) .
i(k) corresponds
to our diffraction probability pattern of an individual particle, which is the most perfect instrument if it originates at infinity (plane wave). Taking ‘the Fourier transform of the measured intensity they obtain F{i(k),,,,)
= F(G))
FIT(k))
= Q(r) t(r> p
where 6(t) is the autocorrelation function and t(r), the Fourier transform of the response function, is defined as the transfer function. r is a vector connecting grating points. The equation shows that no correlation between grating points separated by distances r for which t(r) is zero can be observed [13]. Accordingly, Park et al. introduce the transfer width wt, the width of the function t(r) in the real space, as “the range over which the instrument is effective as an interference detector”. Thus wt is indeed equivalent to the length w we deduced above and its interpretation corresponds to the meaning of the information we consider that a given instrument is able to give, without making use of a detailed knowledge of the transfer function.
64
G. Comsa /Coherence
length and/or transfer width?
For the special case of the LEED instruments (01= 0”, focussing effects, etc.), Park et al. [7] have calculated the width WT of the response function T(k) in k-space. By assuming that T(k) is Gaussian they obtained the transfer width from
It is in fact surprising that in spite of the existence of this rigorous approach and of these “ready to use” formulae for LEED experiments, the Heidenreich and Pendry formulae are still used by LEED people. One may speculate about the reasons: the not very intuitive character of the approach or the lack of any mutual recognition between the authors of the different approaches.
4. The transfer width in MBD experiments We will deduce now “ready to use” formulae for MBD experiments and discuss some aspects of the influence of the energy spread which is predominant in this kind of experiments. The contribution of the finite source detector and aperture dimensions to the transfer width will be calculated by following the procedure presented in section 3: we will calculate first the broadening effect of these parameters and then we correlate it by means of eq. (4’) with a length on the surface. An approximate expression for the broadening can be obtained from elementary geometrical considerations by using the scheme in fig. 1:
(6) The four terms represent the contribution of the source dimension, of the aperture, of the spot size and of the detector opening, respectively. The result is a rough approximation because the broadenings were added as if they were gaussian and
Fig. 1. Scheme
of an MBD experiment.
G. Coma / Coherence
length and/or transfer width?
65
because in the first two terms some contribution are considered in excess. However, (6) is a useful expression for estimating the contribution of the various instrumental dimensions. From eqs. (6) and (4’) we obtain the corresponding contribution to the transfer width: we = h/( lA,&l cos 0,) .
(6’)
More interesting, and as already mentioned much more important in MBD, is the contribution of the energy spread of the source. We may use the same procedure as above, i.e. to calculate first the angular spread resulting from the energy spread. However, we will choose a procedure showing similarities to that used by Pendry [5] in order to emphasize the differences: Let us consider the phase difference @ resulting at the detector by the scattering of one molecule (plane wave, with wave vector k) on two points in the surface separated by a distance w: $ = kw(sin Br - sin 0,) .
(7)
The diffraction probability pattern of the molecule will have in the direction Br a maximum or a minimum depending on whether @ is an even or an odd multiple of n. Consistent with the definition of the transfer width w, we impose the condition that, due to the spread in energy, among the different probability patterns of the molecules constituting the beam, one may find patterns with maxima and others with minima in the same direction of; correspondingly, the peaks around Br in the measured diffraction pattern will be smeared out. The condition is: zA@ =n’.
(8)
This is in fact identical to Pendry’s condition (2). However condition (8) is imposed after the interference of each particle with itself had taken place, and not before. From eqs. (7) and (8) we obtain
21 h/{ lsin 01 - sin
efl[m/E2]
l12}.
Using eq. (4) the contribution obtained Afir
AOf =
of the energy spread to the angular broadening
2: { lsin Bi - sin ~~l[(AE)‘/E’]“‘}/(cos
The total angular broadening KA,ef)2
+ @df)
is obtained
2 l/2
1
(9)
0,) .
is
(10)
from (6) and (10)
(11)
3
and finally the transfer width is w = h/[(A,0f)2 It is worthwhile term disappears.
c0s2ef t (sin Bi - sin ef)2(aE)z/h-2]l’2
.
(12)
to examine eqs. (9) to (12) more closely: For Bi= fQ the energy Thus, in accordance with experiment, the energy spread has no
G. Comsa / Coherence length and/or transfer width?
66
influence on the broadness of the specular beam. In section 2, the anomalous broadening of the specular beam predicted by Pendry’s formula led us to its rejection. Using now the Bragg condition, eq. (9) may be written: WE~d/{InI[(~2/E2]“2}
)
(9’)
where d is the period of the grating and n the order of diffraction. Remembering that the instrument can resolve diffraction peaks only if d G w, we obtain the maximum order of diffraction which a given instrument is able to resolve Inl < l/[(A_!?)2/E2]1’2 .
(13)
This upper limit is due to the energy spread alone, the angular imprecisions leading to a further reduction. (a) For Maxwellian beams ((A&‘)“/E2 = OS), condition (13) limits the well resolved diffraction beams to the zero and first order beams. To our knowledge, all Maxwellian beam experiments except one confirm this conclusion so far. Indeed, the diffraction patterns, presented in all classical MBD papers published in the thirties (see, e.g., Estermann and Stem [14], Estermann et al. [15], Johnson [ 161 and Zabel [17]), as well as in the modern ones in which Maxwellian beams were still used (see, e.g., Crews [18], O’Keefe et al. [19], Hoinkes et al. [20], Hayward and Walters [21] and Finzel et al. [22]), contain besides the specular beam only first order beams. Only Weinberg and Merrill [23] obtained, when using a Maxwellian beam, well defined [20] and [02] beams. However, this exception confirms in fact the rule because the beam shapes had also another troublesome peculiarity which could not be explained by the authors: The half-widths of the diffracted beams were less than 50% of the incident beam half-width. (b) After new molecular beam sources came in use (multi-arrays, nozzles), which met less and less the Knudsen cell conditions leading to always narrower energy spreads, the order of resolved diffraction beams continuously increased. The largest order yet resolved is shown in a recent paper by Lapujoulade and Lejay [241 with diffraction beams up to n 2: 11-12. They give for the energy spread of their beam: (A,!?)“/&? = 3 X 10-3. F rom (13) we deduce an upper limit Inl<
17 )
which is in fair agreement with the experiment.
5. Conclusion The analysis shows that neither Heidenreichs’ [4] nor Pendry’s [5] formulae properly describe the role of the instrument in obscuring the diffraction pattern features, in partictdar the instrumental broadening of the diffraction peaks. The reason for this failure is probably the fact that they imply the existence of a coherence of an ensemble of perfectly incoherent particles.
G. Comsa /Coherence length and/or transfer width?
61
It is assumed here that the measured diffraction pattern is ultimately the result of the coherent interference of each particle with itself; more precisely, that the diffraction pattern results from the incoherent summation of the diffraction probability patterns of the individual particles. Due to the instrumental energy and angular spreads, the individual probability patterns are shifted against each other leading to a broadening of the measured diffraction beams. In the case of a perfect infinite crystal, this angular broadening may be correlated to a length on the surface, named “transfer width” which is thus only a measure of broadening effect of the instrument. The transfer width corresponds to the largest period of the grating which can be straightforwardly resolved with a given instrument. These results are essentially equivalent to those obtained from the approach by Park et al. [7]. The formulae for the transfer width and angular spread in the particular case of molecular beam diffraction which have been derived, compare well with experiments. We hope that this discussion will reduce the arbitrariness in choosing the “right coherence formula”.
Acknowledgements The conclusions I reached in this paper were preceded by discussions with many colleagues from this and other laboratories. I am particularly indebted to Harald Ibach for his suggestion concerning the influence of the detector and for the critical reading of the manuscript. It is a pleasure to acknowledge the valuable discussions with Hans Bonzel, R. Feder, Jan Fremerey and Heribert Wagner.
References [ 1) [2] [3] [4] [S] [6] [7] [8]
[9] [ 101 [ll]
J.J. Lander and .I. Morrison, J. Chem. Phys. 37 (1962) 729. J.J. Lander and J. Morrison, J. Appl. Phys. 34 (1963) 3517. A. Chutjian, Phys. Letters 24A (1967) 615. R.D. Heidenreich, Fundamentals of Transmission Electron Microscopy (Interscience, New York, 1964) pp. 101-105. J.B. Pendry, Low Energy Electron Diffraction (Academic Press, London, 1974) pp. 5-6. See ref. [S], pp. 228-236. R.L. Park, J.E. Houston and D.G. Schreiner, Rev. Sci. Instr. 42 (1971) 60. J. Lapujoulade and Y. Lejay, in: Compt. Rend. 3Bme ColIoq. Intern. du Physique et Chimie des Surfaces, Grenoble, 1977 (Suppl. Le Vide No. 185) pp. 328-334; Y. Lejay and J. Lapujoulade, in: Proc. 7th Intern. Vacuum Congr. and 3rd Intern. Conf. on Solid Surfaces, Vienna, 1977, Eds. R. Dobrozemsky, F. Riidenauer, F.P. ViehbGck and A. Breth, Vol. II, pp. 1373-1375. M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon, Oxford, 1965). L.I. Shiff, Quantum Mechanics, 2nd ed. (McGraw-Hill,New York, 1955) pp. 57-58. A. Messiah, MBcanique Quantique (Dunod, Paris, 1959) p. 187.
68
G. Comsa / Coherence length and/or transfer width?
[12] M. Henzler, in: Electron
Spectroscopy for Surface Analysis, Ed. H. Ibach (Springer, Berlin, 1977) pp. 120-121. [ 131 G.-C. Wang and M.G. Lagally, Surface Sci. 81 (1979) 69. The authors show that: “Only if t(r) is non-zero in a region of finite width.(e.g. a rectangular function) wiB the determination of e(r) be limited to finite r. Since T(k) and hence t(r), is typically between a Gaussian and a Lorentzian function, only the noise level on these functions, . . ., wili limit the range r over which a(r) can be determined.” And further that: “It is the uncertainty in the determination of both i(k),,,s and T(k) that results in a minimum angular width for i(k) that can be determined with any given level of confidence. i(k) functions with a FWHM of less than 50% of the FWHM of T(k) are possible”. I am indebted to Max Lagaiiy for sending me the manuscript prior publication. Unfortunately, I got it only when the revised form of this paper was accepted for publication so that a deeper integration in this paper of their very clear ideas was hardly possible. [14] I. Estermann and 0. Stern, Z. Physik 61 (1930) 95. [ 151 I. Estermann, R. Frisch and 0. Stern, Z. Physik 73 (1931) 348. [ 161 T.H. Johnson, Phys. Rev. 37 (1931) 847. [ 171 R.M. Zabel, Phys. Rev. 42 (1932) 218. [ 181 J.C. Crews, J. Chem. Phys. 37 (1962) 2004. [19] D.R. O’Keefe, R.L. Palmer, H. Saltsburg and J.N. Smith, Jr., J. Chem. Phys. 49 (1968) 5194; D.R. O’Keefe, J.N. Smith, Jr., R.L. Palmer and H. Saltsburg, Surface Sci. 20 (1970) 27. [20] H. Hoinkes, H. Nahr and H. Wilsch, Surface Sci. 30 (1972) 363. [21] D.O. Hayward and M.R. Walters, in: Proc. 2nd Intern. Conf. on Solid Surfaces, Japan. J. Appl. Phys. Suppl. 2 (1974) 587. [22] H.-U. Finzel, H. Frank, H. Hoinkes, M. Luschka, H. Nahr, H. Wilsch and U. Wonka, Surface Sci. 49 (1975) 577. [23] W.H. Weinberg and R.P. Merrill, J. Chem. Phys. 56 (1972) 2893. [24] J. Lapujoulade and Y. Lejay, Surface Sci. 69 (1977) 354.