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Magic-angle thermal desorption mass spectroscopy
250
Surface
MAGIC-ANGLE Steven W. PAULS
THERMAL
DESORPTION
Science 226 (1990) 250-256 North-Holland
MASS SPECTROSCOPY
and Charles T. CAMPBELL
*
Chemistry Department, University of Washington, Seattle, WA 98195, USA Received
12 June 1989; accepted
for publication
9 November
1989
Accurate quantitative measurements of desorption rates or adsorbate coverages in thermal desorption mass spectroscopy (TDS) using line-of-sight mass spectrometers are hindered by the fact that the angular distributions of desorption flux can vary widely from desorbate to desorbate, ranging from cost + to cos’ Q, for most species studied to date (# = polar angle from surface normal). These differences can easily lead to errors exceeding 400% in measuring the relative desorption rates of different species. We show here that. by placing the mass spectrometer’s ion source or entrance aperture at a “magic-angle” &,,, these errors can be reduced to less than 26% maximum deviation (or f 7% standard deviation). Depending upon the sample-to-detector distance, &,, varies from - 42” to for improvement in the quantitative accuracy of 34O. It is recommended that TDS experiments be performed at this “magic-angle” coverage
or rate measurements.
1. Introduction Thermal desorption mass spectroscopy (TDS), also commonly referred to as temperature-programmed reaction spectroscopy (TPRS) and temperature-programmed desorption (TPD), is a widely-used method in surface chemical studies of planar surfaces such as polished metal foils or oriented single-crystal surfaces [l-6]. In this method the surface is heated, usually at a constant rate, and the desorbing gaseous species are monitored with a mass spectrometer. Frequently the ionization source of the mass spectrometer or its entrance aperture is located in line-of-sight, near and normal to the surface to enhance the signal of the desorbing species being monitored. In wellpumped systems this has the advantages that (1) the signal from the front surface of the sample is greatly enhanced compared to its back and edges, and (2) the signal for a given species is directly proportional to the flux of desorbing molecules (provided the common assumptions hold that all the molecules desorb with the same velocity distribution and that their internal energy distribution does not change, or at least that the changes in * Camille
and Henry
~39-6028/~/~03.50 (North-Holland)
Dreyfus
Teacher-Scholar.
0 Elsevier
Science
Publishers
B.V.
this distribution do not effect the ionization probability). It is necessary for kinetic analyses to be able to relate the measured signal (flux) at any time, 1, to the total instantaneous desorption rate, R(r) [l-6]. Commonly. it is simply assumed that the desorption rate is directly proportional to the instantaneous mass spectrometer signal for a given species, S(t). R(t)
= kS(t).
(1)
Referring to the experimental geometry shown schematically in fig. 1, the ionization source of the mass spectrometer is commonly located normal to the surface, at polar angle $I = 0, and at a fixed distance d from the surface. The sample surface is taken as circular, of radius r. Commonly, desorption occurs in a cosine angular distribution such that the desorption flux at any polar angle ($) from the surface normal varies according to: F(&
t) = CR(t)
cos rp,
(2)
where C is a constant. (This is strictly true only for the case where the sample is considered as a point source, or where d B- r.) The advantage of normal detection (+ = 0) is then obvious since it provides the maximum signal and, to the extent the mass spectrometer acts as a flux (rather than
S. W. Pauls, C. T. Campbell / Magic-angle desorption
z
/ Detector
and substituting the expression of eq. (4) for F(+, t) in eq. (5), the normalization constant, C, can be shown to be: C = (n + 1)/(2md2).
Fig. 1. Schematic representation of the ultrahigh vacuum TDS experiment in which species desorb from a planar sample surface into a mass spectrometer ion source or entrance aperture located at polar angle (9) from the surface normal and distance (d) from the surface.
density) detector, it provides a direct obtaining the desorption rate: R(t)
= c-V(t)
= U(t).
means
for
(3)
In this case the simple and desirable condition of eq. (1) is achieved. However, this in only the case when the species actually desorb in a cosine or otherwise fixed angular distribution. More and more frequently it is being found that angular distributions of desorbing species are far from cosine, often being much more sharply peaked normal to the surface [7-381. However, experimentally observed angular distributions of desorption flux [7-381 can all be fairly well fitted with co?‘+ distributions where n is a constant which we will refer to as the “sharpness parameter” (see qualifier in section 3). For most systems, the value of n is between 1 and 9 [7-381. (In a few exceptional cases, n exceeds 9 [15,31-371.) Thus the desorption flux is more generally given by: F(&
t) = CR(t)
cos”cp,
(4)
where usually 1 I n I 9. (For an excellent discussion of the physical meaning of such distributions, see ref. [9].) By equating the total angular-integrated desorption flux to the total desorption rate according to the requirement of mass balance: R(t)
= j2* l”/2F(*, 8=0 +=o = 2rd2
T’2F(+, J +=o
t)d*
sin (p de d$
t) sin + de,
251
(5)
(6)
Let us suppose we wish to compare the desorption rates, R(t), for two different experiments where the angular distribution is not necessarily the same. This commonly occurs, for example, while monitoring two distinct molecules in a single TDS spectrum, or while monitoring desorption of the same molecule in two different TDS experiments where the surface structures or coverages are different [8,9]. This also occurs indirectly when comparing coverages if the TDS peak area (timeintegrated rate) is used as a measure of coverage. Let us assume for the sake of argument that in case A the angular distribution is cos’+ and in case B it is cos’+, but that in both cases the desorption rate, R(t) is the same. If the mass spectrometer is set at $ = O” as usual, eqs. (4) and (5) show that the measured flux will be F(t) = (l/md2)R(t) in case A, but F(t) = (5/md2)R(t) in case B. Thus, the mass spectrometer would give a signal in case B that was five times as large as in case A. Without prior knowledge of the angular distribution, this would lead to a 400% error if using these signals to compare desorption rates or coverages. Recognizing this complication, we decided to see if there is an optimum polar angle 9 for location of the mass spectrometer in TDS which could minimize these differences in signal intensities due to differences in angular distributions of desorption flux. This paper presents the results of simple calculations which show that indeed there in TDS, and that this is such a “magic-angle” angle should be used whenever possible.
2. Calculations and results First, we calculated the variation in the desorption flux, F(+, t), with the sharpness parameter n for various detection angles ($) assuming a cos”+ desorption angular distribution, where the total rate of desorption R(t), and the distance from the surface to the detector (d) were both held con-
S. W. Pnuls, C. T. Campbell
252
0.0
1
I
I
I
I
I
2
3
4
5
6
n = Exponent
in
cosRO
7
8
9’
Distribution
Fig. 2. Variation in the desorption flux with n for cos”+ angular distributions at various detection angles (#) for a fixed total desorption rate (R(t)) and detector distance (d). Here we assume the sample is a point source (d/r z 1000).
stant. Here, we assumed that the sample is a point source, so that d/r = co. Here we have used eq. (4) to calculate the desorption flux. The results are shown in fig. 2 for several detection angles tp. Here, we have normalized the entire curve F(+, t) versus n at each angle cp with a multiplicative constant k(+) to achieve the least-squares difference between the flux curve and the true desorption rate, R(t), which is independent of n. The detection angle Cpwhose flux curve shows the least deviation from a flat line at 1.0 in these units then provides the best approximation to a true measure of the total desorption rate, R(t), for randomly chosen sharpness parameters n between 1 and 9, reflecting the changes in angular distribution for different desorbates. As can be seen, the detection angle of 34” provides for a flux measurement which is a reasonably good approximation to a horizontal line (i.e., the true angularintegrated desorption rate), displaying a maximum deviation of less than 30% between any two values of the sharpness parameter n from 1 to 9. This compares to a 400% maximum deviation for normal detection (+ = 0 o ) and even worse errors for +260°. Thus far, we have considered the sample as only a point source. However, most mass spec-
/ Magic-angle
de-sorption
trometers do not see the sample just as a point source (d/r = co), but are placed much closer to the sample. Typical values of d/r are - I to 5 for surface science experiments (see fig. I). in this case one must integrate the desorption flux over the surface of the sample, recognizing that the polar angle (p and distance to the detector vary significantly with position on the sample surface. We repeated the calculations such as demonstrated in fig. 2 for a range of d/r ratios by numerically integrating the desorption flux over the surface of the sample. At d/r = 1000, the results were indistinguishable from those in fig. 2, which are valid for d/r = co. This supports the accuracy of our computational methods. The results for d/r = 2 are shown in fig. 3. Again as in fig. 2 the total desorption rate R(t) at a fixed detection angle (+) is held constant while the desorption flux F(+, t) at the detector was cakulated as a function of the sharpness parameter n in ~0s”~. This was then repeated for a range of detection angles. As shown in fig. 3, the detection angle of 42O provides a reasonable approximation to a horizontal line (i.e., the true angular-integrated desorption rate), deviating by less than 26% be-
1
1.6.
I
,
1
I
,
I
1.6
d/r=2 g 1.4p:
0242
0.01 1
- 1.4
I
I
I
1
I
t
#
2
3
4
5
6
7
8
n = Exponent
in ms”pl
10.0 9
Distribution
Fig. 3. Variation in the desorption flux with n for cos”cp angular distributions at various detection angles (+) for a fixed total desorption rate ( R(t)) and detector distance (d). Here we assume that the sample radius is one-half the detector distance (d/r = 2).
S. W. Pa&, C. T. Campbeit / Magic-angle desorption
= 20P a
Ratioof Distance of Mass Spec: Radius of Sample
(d/r1
Fig. 4. Optimum or magic polar angle (cp,) with respect to the
surface normal recommended for TDS experiments as a function of the ratio of the mass spectrometer distance to the sampie radius (d/r).
tween any two values of the sharpness parameter in the range n = 1-9. Again, this compares to a 250% maximum deviation for normal detection (#B= 0” ) and even worse errors when cp2 55 O. Since the detector distance and sample radius vary for different experimental systems, the optimum or magic angle (em) has been calculated as a function of the ratio d/r, and is plotted in fig. 4. This optimum or magic angle +,,, is defined as the polar angle which gives the least-squares difference between the normalized flux versus n curve (for n = l-9) and the true desorption rate {which is a horizontal line, independent of n). Again, the curve at each angle cp was first multiplied by a normalization constant to achieve a least-squares difference at that angle. As can be seen for typical ratios d/r between 2 and 5, the optimum angle changes from 42” to 34” respectively, while giving an optimum angle of 34” at a point source approximating a ratio of d/r = 1000. We recommend, therefore, that line-of-site thermal desorption mass spectroscopy be performed at the “magic-angle” (p, taken from fig. 4 when possible, so that more reliable estimates of the relative desorption rates of different species will be provided by the measured mass spectrometer
253
intensities. Caution must be exercised even when using fig. 4, of course, since the mass spectrometer signal intensity, which is really proportional to the molecule’s number density, in only an accurate measure of the relative desorption flux for systems where the translational energy distribution of the desorbing particles are relatively similar for all species being detected. The most common current detectors for TDS use electron impact ionizers, usually with mass filtering. The ion source is frequently capped, with a small circular entrance aperture [6,42]. In this case, the desorbing molecules which enter this aperture make many wall collisions and many passes through the ionizer before exiting it. They are therefore probably nearly randomized and thermalized at the detector temperature, so that the detector signal is truly proportional to the desorption flux at the entrance aperture, independent of the velocity distribution with which the molecules enter this aperture. (See ref. [9] for an excellent discussion of velocity distributions and detector types.) Using such a detector at the “magic-angle” of fig. 4 is a very accurate and recommended method for obtaining quantitative TDS rates. However, even when the molecules only make a single pass through the detector so that errors arise due to differences in velocity distribution, using the magic angle of fig. 4 is a better first approximation than neglecting corrections due to angular dist~butions. Let us define r, as the radius of the entrance aperture of a capped detector or the effective radius of any uncapped detector’s cross section. The calculations and discussion presented so far are strictly true only for a “point detector” (that is, for the case when rd K d, or d/rd = 00). More typical aperture sizes are - 25-100% of the sample surface, so that d/r, ranges from - 1 to 20. When d/rd -=z3, the detector is accepting signals from a rather large range of polar angles even with a “point source” (i.e. when d/r = co). When d/r,, -c 3, one could also consider performing another integration over the area of the detector, similar to that which we have performed over the area of the sample. However, when d/rd r 2, the correction due to the finite size of the detector should be rather small if the axis of the detector aperture is aimed directly at the center of the sample. In this
254
S. W. Pa&. C. T. Campbell / Magic-angle desorptron
geometry, at least for the point source, no side of the detector aperture is preferentially sampled as occurs for a large sample whose axis is not aimed directly at even a tiny detector (e.g. fig. 1). This preferential sampling is what leads to the corrections in c$,,, of fig. 4 for small d/r. Such corrections for small d/rd will be much less important. We have not performed similar numerical integrations over the area of the detector because of the smaller corrections involved and also due to the large number of possible combinations of d/r, d/rd and detector directionality. Of course, when d/rd drops below about one, a large fraction of the total solid angle is being accepted by the detector. In the limiting case, some experimentalists [42] make rd = r and d -=cr so that almost all the desorbing molecules enter the detector. In this case, it is obviously best to place the detector apparatus directly over the sample (+ = 0 o ). This seems in contradiction to the trend in fig. 4. However, one must remember that on fig. 4 (which is only strictly appropriate for a tiny detector area) in the limit of closest detector (d/r = 0), any detector angle + performs equally well (i.e. +, = O-90 o ) provided the detector falls within the sample surface area. The signal of course drops to zero if it does not. Geometries such as described above where the detector aperture captures almost all the molecules desorbing from the surface are of course ideal since they will undoubtedly provide the largest signal, and magic-angle corrections such as described here can be neglected. A good description of such a detector is found in ref. [42]. However, it is frequently impossible to achieve such a close detector (d/r 2 0.2) due to geometric constraints imposed by the vacuum chamber and its associated hardware. In those cases, consideration must be given to the optimum angle for measuring quantitative desorption rates. Fig. 4 can be used for predicting this angle when d/r exceeds about 0.75 and d/rd exceeds about 2.
3. Discussion method has a potential This “ magic-angle” limitation for very sharp angular distributions in
that the signal will be much smaller than that normal to the surface. However, the signal will still be of the same size as that for a cosine distribution of the same desorption rate (figs. 2 and 3). Since signals from cosine distributions are easy to measure at this angle (and still more than half as large as the signal perpendicular to the surface), we can envision no real signal-size limitation at the magic-angle even for distributions as sharp as cosi4@ In any case, the magic-angle of fig. 4 will give a signal which is a much closer approximation to the true desorption rate for varying angular distributions then will a detector perpendicular to the surface. We should qualify one of our statements in the introduction concerning the measured angular distributions of desorbing species. We stated that these angular distributions could all be fairly well fitted by cos”$ angular distributions. In fact, in some cases the measured flux is only well fitted by a linear combination of two COS”C$distributions using two separate sharpness parameters n [9,28,29,39]. (In those cases it is generally thought [9] that there are two separate reaction pathways which lead to the two components in desorption.) While not explicitly treated above, it is straightforward to show that the main conclusion of this paper still holds in these case (provided n is between 1 and 9 for both components). That is, the magic-angle from fig. 4 should still be used to minimize differences in TDS signal intensities which are due to differences between species (or surface conditions) in angular distributions of desorption flux. This is true because, in order to minimize differences in such linear combinations, it is necessary to first minimize differences between the individual components COS”C+and cosm+, if n and m are independent and randomly selected. Fig. 4 shows the optimum angle for minimizing the differences between the individual components. The advantages of “magic-angle” detection for quantitative analysis is not only limited to TDS, but can be extended to any particle-desorption technique where the particles are emitted in cos”+ angular distributions. For example, the angular distribution of neutrals sputtered from metals and alloys using Ar+ ions follow nearly a cos’+ distri-
S. W. PaA,
C. T. Campbell / Magic-angle desorption
bution when the primary ion beam energy exceeds 3 keV [40]. The electron-impact or laser ionization and mass-filtered detection of such neutrals form the basis for several emerging, ultra-sensitive surface analytical methods [40], which we will refer to collectively here as secondary neutral mass spectroscopy (SNMS). The need to eliminate problems associated with angular variations for accurate quantitative analysis by SNMS has already been pointed out [40], and a detection angle of approximately 45 o was recommended [41]. Below a primary energy of 3 keV, the angular distributions of sputtered neutrals show angular distributions which are not close to cos”+ [40]. Fig. 5 of ref. [40] shows the variation with angle in the measured relative SNMS signals for the two elements constituting a series of binary alloys. There, the signal ratios are normalized to the ratio of their total SNMS signals integrated over the total solid angle (hemisphere) above the sample. This normalization constant really represents their true (angular-integrated) sensitivity factor ratio. Note that the ratio R plotted there should therefore remain at unity if there were no differences between the angular distributions of two elements. As can be seen in ref. [40], there are strong differences between R and unity, when the primary energy is below 3 keV. Consider a cut through those curves at the magic angle we found above (34-42O, depending upon the sample to detector distance). Interestingly, R is very close to unity near this angle for all samples studied there. In any case, R is much closer to unity at the magicangle than at the usual detector position, $I = 0 O. Thus, by detecting the neutrals at or near the one can effectively minimize the “ magic-angle”, intensity variations associated with angular distributions which change between elements and between samples. This fact suggests that “magicangle” detection may have even more general applicability than for systems with cos”+ behavior. Indeed, one can expect that a “magic-angle” near 42” will be close to the “average” desorption angle for many angular distributions of desorption flux F(+) which are circularly symmetric (independent of the azimuthal angle) and for which F( +) approaches zero as + approached 90”. This is because the multiple of the flux F(9) times the
255
area element sin cp d+ de will drop smoothly to zero at both 0 and 90 O, but maximize somewhere in between. Finally, we should point out that Winkler et al. [2] have recently demonstrated a simple method for determining the sharpness parameter n in cos”+ angular distributions of desorption by varying only the crystal-to-detector distance d.
Acknowledgements The authors would like to thank the US Department of Energy, Office of Basic Energy Sciences, Chemical Sciences Division, for partial support of this work.
Note added in proof In one case for CO, production on a fee (110) surface [43], the sharpness parameter, n, was found to vary with azimuthal angle, 8. The “magic angle” of fig. 4 is still recommended for quantitative TDS even in such cases.
References 111J.N. Russell, Jr., I. Chorkendorff, A.M. Lanzillotto, M.D. Alvey and J.T. Yates, Jr., J. Chem. Phys. 85 (1986) 6186.
121A. Winkler and J.T. Yates, Jr., J. Vat. Sci. Technol. A 6 (1988) 2929.
[31 R.J. Madix, in: Chemistry and Physics of Solid Surfaces, Vol. II, Ed. R. Vanselow (CRC Press, Boca Raton, 1979) p. 63. [41 J. Fair and R.J. Madix, J. Chem. Phys. 73 (1986) 3480. 151 D.A. King, Surf. Sci. 47 (1975) 384. 161 D. Menzel, in: Chemistry and Physics of Solid Surfaces, Vol. IV (CRC Press, Boca Raton, 1982) p. 389. [71 E.I. Ko, J.B. Benziger and R.J. Madix, J. Catal. 62 (1980) 264. PI M. BaIooch, M.J. Cardillo, D.R. Miller and R.E. Stickney, Surf. Sci. 46 (1974) 358. [91 G. Comsa and R. David, Surf. Sci. Rep. 5 (1985) 145. WI H.P. Steinruck, A. Winkler and K.D. Redulic, J. Phys. Chem. 17 (1984) L311; A.E. Dabiri, T.J. Lee and R.E. Stickney, Surf. Sci. 26 (1971) 522; G. Comsa, in: Proc. 7th Int. Vacuum Congress and 3rd Int. Congress on Solid Surfaces, Vienna, 1977, Ed. R. Dobrozemsky, p. 80.
S. W. Pauls, C. T. Campbell / Magic-angle desorpiion
256 [ll]
T. Matsushima, Z. Phys. Chem. 158 (1988) 175; Surf. Sci. 197 (1988) L287. [12] D.R. Horton and R.I. Masel, Surf. Sci. 125 (1983) 699. 1131 G. Comsa, R. David and B. Schumacher, Surf. Sci. 85 (1979) 45. 1141 R.L. Palmer 1453.
and J.N. Smith, Jr., J. Chem.
[15] H.H. 996.
and
Sawin
R.P. Merrill,
J. Chem.
Phys. 60 (1974) Phys. 73 (1980)
[16] T.H. Lin and G.A. Somorjai, J. Chem. Phys. 81 (1984) 704. [17] W. van Willigen, Phys. Lett. A 28 (1968) 80. [18] K.C. Janda, Surf. Sci. 93 (1980) 270. [19] L.L. Verheij, M.B. Hugenschmidt, A.B. Anton, B. Poelsema and G. Comsa, Surf. Sci. 210 (1989) 1. [20] R.C. Cosser, S.R. Bare, S.M. Francis and D.A. King, Vacuum 31 (1981) 503. [21] [22] [23] [24] [25]
T. Engel, J. Chem. Phys. 69 (1978) 373. T. Engel and G. Ertl, J. Chem. Phys. 69 (1978) 1267. T. Matsushima, J. Catal. 83 (1983) 446. T. Matsushima, J. Phys. Chem. 88 (1984) 202. CT. Campbell, G. Ertl and J. Segner, Surf. Sci. 115 (1982) 309. [26] C.T. Campbell, G. Ertl, H. Kuipers and J. Segner, Surf. Sci. 107 (1981) 220. [27] C.T. Campbell, G. Ertl, H. Kuipers and J. Segner, Surf. Sci. 107 (1981) 207. [28] C.T. Campbell, G. Ertl, H. Kuipers and J. Segner, J. Chem. Phys. 73 (1980) 5862.
[29] J. Segner, C.T. Campbell, G. Doyen and G. Ertl, Surf. Sci. 138 (1984) 505. [30] J. Segner, W. Vielhaber and G. Ertl, Isr. J. Chem. 22 (1982) 375. [31] T. Matsushima, Surf. Sci. 127 (1983) 403. [32] T. Matsushima and H. Asada, J. Chem. Phys. 85 (1986) 1658. [33] T. Matsushima, Surf. Sci. 123 (1982) L663. [34] T. Matsushima, T. Matsui and M. Hashimoto, J. Chem. Phys. 81 (1984) 5151. [35] T. Matsushima (1985) 412.
and
H. Asada,
Chem.
Phys.
Lett.
120
[36] T. Matsushima, J. Phys. Chem. 91 (1987) 6192. [37] T. Matsushima, Z. Phys. Chem. 158 (1988) 175. [38] M. Asscher, W.L. Guthrie, T.H. Lin and G.A. Somorjai, J. Phys. Chem. 88 (1984) 3234. [39] G. Comsa and R. David, Chem. Phys. Lett. 49 (1977) 512. [40] A. Wucher, F. Novak and W. Reuter, J. Vat. Sci. Technol. A 6 (1988) 2265. [41] A. Wucher and W. Reuter, J. Vat. Sci. Technol. A 6 (1988) 2316. [42] P. Feulner and D. Menzel, J. Vat. Sci. Technol. 17 (1980) 662; V. Smentkowski and J.T. Yates, Jr., J. Vat. Sci. Technol. 7 (1989) 3325. [43] T. Matsushima, private communication.
Surface
MAGIC-ANGLE Steven W. PAULS
THERMAL
DESORPTION
Science 226 (1990) 250-256 North-Holland
MASS SPECTROSCOPY
and Charles T. CAMPBELL
*
Chemistry Department, University of Washington, Seattle, WA 98195, USA Received
12 June 1989; accepted
for publication
9 November
1989
Accurate quantitative measurements of desorption rates or adsorbate coverages in thermal desorption mass spectroscopy (TDS) using line-of-sight mass spectrometers are hindered by the fact that the angular distributions of desorption flux can vary widely from desorbate to desorbate, ranging from cost + to cos’ Q, for most species studied to date (# = polar angle from surface normal). These differences can easily lead to errors exceeding 400% in measuring the relative desorption rates of different species. We show here that. by placing the mass spectrometer’s ion source or entrance aperture at a “magic-angle” &,,, these errors can be reduced to less than 26% maximum deviation (or f 7% standard deviation). Depending upon the sample-to-detector distance, &,, varies from - 42” to for improvement in the quantitative accuracy of 34O. It is recommended that TDS experiments be performed at this “magic-angle” coverage
or rate measurements.
1. Introduction Thermal desorption mass spectroscopy (TDS), also commonly referred to as temperature-programmed reaction spectroscopy (TPRS) and temperature-programmed desorption (TPD), is a widely-used method in surface chemical studies of planar surfaces such as polished metal foils or oriented single-crystal surfaces [l-6]. In this method the surface is heated, usually at a constant rate, and the desorbing gaseous species are monitored with a mass spectrometer. Frequently the ionization source of the mass spectrometer or its entrance aperture is located in line-of-sight, near and normal to the surface to enhance the signal of the desorbing species being monitored. In wellpumped systems this has the advantages that (1) the signal from the front surface of the sample is greatly enhanced compared to its back and edges, and (2) the signal for a given species is directly proportional to the flux of desorbing molecules (provided the common assumptions hold that all the molecules desorb with the same velocity distribution and that their internal energy distribution does not change, or at least that the changes in * Camille
and Henry
~39-6028/~/~03.50 (North-Holland)
Dreyfus
Teacher-Scholar.
0 Elsevier
Science
Publishers
B.V.
this distribution do not effect the ionization probability). It is necessary for kinetic analyses to be able to relate the measured signal (flux) at any time, 1, to the total instantaneous desorption rate, R(r) [l-6]. Commonly. it is simply assumed that the desorption rate is directly proportional to the instantaneous mass spectrometer signal for a given species, S(t). R(t)
= kS(t).
(1)
Referring to the experimental geometry shown schematically in fig. 1, the ionization source of the mass spectrometer is commonly located normal to the surface, at polar angle $I = 0, and at a fixed distance d from the surface. The sample surface is taken as circular, of radius r. Commonly, desorption occurs in a cosine angular distribution such that the desorption flux at any polar angle ($) from the surface normal varies according to: F(&
t) = CR(t)
cos rp,
(2)
where C is a constant. (This is strictly true only for the case where the sample is considered as a point source, or where d B- r.) The advantage of normal detection (+ = 0) is then obvious since it provides the maximum signal and, to the extent the mass spectrometer acts as a flux (rather than
S. W. Pauls, C. T. Campbell / Magic-angle desorption
z
/ Detector
and substituting the expression of eq. (4) for F(+, t) in eq. (5), the normalization constant, C, can be shown to be: C = (n + 1)/(2md2).
Fig. 1. Schematic representation of the ultrahigh vacuum TDS experiment in which species desorb from a planar sample surface into a mass spectrometer ion source or entrance aperture located at polar angle (9) from the surface normal and distance (d) from the surface.
density) detector, it provides a direct obtaining the desorption rate: R(t)
= c-V(t)
= U(t).
means
for
(3)
In this case the simple and desirable condition of eq. (1) is achieved. However, this in only the case when the species actually desorb in a cosine or otherwise fixed angular distribution. More and more frequently it is being found that angular distributions of desorbing species are far from cosine, often being much more sharply peaked normal to the surface [7-381. However, experimentally observed angular distributions of desorption flux [7-381 can all be fairly well fitted with co?‘+ distributions where n is a constant which we will refer to as the “sharpness parameter” (see qualifier in section 3). For most systems, the value of n is between 1 and 9 [7-381. (In a few exceptional cases, n exceeds 9 [15,31-371.) Thus the desorption flux is more generally given by: F(&
t) = CR(t)
cos”cp,
(4)
where usually 1 I n I 9. (For an excellent discussion of the physical meaning of such distributions, see ref. [9].) By equating the total angular-integrated desorption flux to the total desorption rate according to the requirement of mass balance: R(t)
= j2* l”/2F(*, 8=0 +=o = 2rd2
T’2F(+, J +=o
t)d*
sin (p de d$
t) sin + de,
251
(5)
(6)
Let us suppose we wish to compare the desorption rates, R(t), for two different experiments where the angular distribution is not necessarily the same. This commonly occurs, for example, while monitoring two distinct molecules in a single TDS spectrum, or while monitoring desorption of the same molecule in two different TDS experiments where the surface structures or coverages are different [8,9]. This also occurs indirectly when comparing coverages if the TDS peak area (timeintegrated rate) is used as a measure of coverage. Let us assume for the sake of argument that in case A the angular distribution is cos’+ and in case B it is cos’+, but that in both cases the desorption rate, R(t) is the same. If the mass spectrometer is set at $ = O” as usual, eqs. (4) and (5) show that the measured flux will be F(t) = (l/md2)R(t) in case A, but F(t) = (5/md2)R(t) in case B. Thus, the mass spectrometer would give a signal in case B that was five times as large as in case A. Without prior knowledge of the angular distribution, this would lead to a 400% error if using these signals to compare desorption rates or coverages. Recognizing this complication, we decided to see if there is an optimum polar angle 9 for location of the mass spectrometer in TDS which could minimize these differences in signal intensities due to differences in angular distributions of desorption flux. This paper presents the results of simple calculations which show that indeed there in TDS, and that this is such a “magic-angle” angle should be used whenever possible.
2. Calculations and results First, we calculated the variation in the desorption flux, F(+, t), with the sharpness parameter n for various detection angles ($) assuming a cos”+ desorption angular distribution, where the total rate of desorption R(t), and the distance from the surface to the detector (d) were both held con-
S. W. Pnuls, C. T. Campbell
252
0.0
1
I
I
I
I
I
2
3
4
5
6
n = Exponent
in
cosRO
7
8
9’
Distribution
Fig. 2. Variation in the desorption flux with n for cos”+ angular distributions at various detection angles (#) for a fixed total desorption rate (R(t)) and detector distance (d). Here we assume the sample is a point source (d/r z 1000).
stant. Here, we assumed that the sample is a point source, so that d/r = co. Here we have used eq. (4) to calculate the desorption flux. The results are shown in fig. 2 for several detection angles tp. Here, we have normalized the entire curve F(+, t) versus n at each angle cp with a multiplicative constant k(+) to achieve the least-squares difference between the flux curve and the true desorption rate, R(t), which is independent of n. The detection angle Cpwhose flux curve shows the least deviation from a flat line at 1.0 in these units then provides the best approximation to a true measure of the total desorption rate, R(t), for randomly chosen sharpness parameters n between 1 and 9, reflecting the changes in angular distribution for different desorbates. As can be seen, the detection angle of 34” provides for a flux measurement which is a reasonably good approximation to a horizontal line (i.e., the true angularintegrated desorption rate), displaying a maximum deviation of less than 30% between any two values of the sharpness parameter n from 1 to 9. This compares to a 400% maximum deviation for normal detection (+ = 0 o ) and even worse errors for +260°. Thus far, we have considered the sample as only a point source. However, most mass spec-
/ Magic-angle
de-sorption
trometers do not see the sample just as a point source (d/r = co), but are placed much closer to the sample. Typical values of d/r are - I to 5 for surface science experiments (see fig. I). in this case one must integrate the desorption flux over the surface of the sample, recognizing that the polar angle (p and distance to the detector vary significantly with position on the sample surface. We repeated the calculations such as demonstrated in fig. 2 for a range of d/r ratios by numerically integrating the desorption flux over the surface of the sample. At d/r = 1000, the results were indistinguishable from those in fig. 2, which are valid for d/r = co. This supports the accuracy of our computational methods. The results for d/r = 2 are shown in fig. 3. Again as in fig. 2 the total desorption rate R(t) at a fixed detection angle (+) is held constant while the desorption flux F(+, t) at the detector was cakulated as a function of the sharpness parameter n in ~0s”~. This was then repeated for a range of detection angles. As shown in fig. 3, the detection angle of 42O provides a reasonable approximation to a horizontal line (i.e., the true angular-integrated desorption rate), deviating by less than 26% be-
1
1.6.
I
,
1
I
,
I
1.6
d/r=2 g 1.4p:
0242
0.01 1
- 1.4
I
I
I
1
I
t
#
2
3
4
5
6
7
8
n = Exponent
in ms”pl
10.0 9
Distribution
Fig. 3. Variation in the desorption flux with n for cos”cp angular distributions at various detection angles (+) for a fixed total desorption rate ( R(t)) and detector distance (d). Here we assume that the sample radius is one-half the detector distance (d/r = 2).
S. W. Pa&, C. T. Campbeit / Magic-angle desorption
= 20P a
Ratioof Distance of Mass Spec: Radius of Sample
(d/r1
Fig. 4. Optimum or magic polar angle (cp,) with respect to the
surface normal recommended for TDS experiments as a function of the ratio of the mass spectrometer distance to the sampie radius (d/r).
tween any two values of the sharpness parameter in the range n = 1-9. Again, this compares to a 250% maximum deviation for normal detection (#B= 0” ) and even worse errors when cp2 55 O. Since the detector distance and sample radius vary for different experimental systems, the optimum or magic angle (em) has been calculated as a function of the ratio d/r, and is plotted in fig. 4. This optimum or magic angle +,,, is defined as the polar angle which gives the least-squares difference between the normalized flux versus n curve (for n = l-9) and the true desorption rate {which is a horizontal line, independent of n). Again, the curve at each angle cp was first multiplied by a normalization constant to achieve a least-squares difference at that angle. As can be seen for typical ratios d/r between 2 and 5, the optimum angle changes from 42” to 34” respectively, while giving an optimum angle of 34” at a point source approximating a ratio of d/r = 1000. We recommend, therefore, that line-of-site thermal desorption mass spectroscopy be performed at the “magic-angle” (p, taken from fig. 4 when possible, so that more reliable estimates of the relative desorption rates of different species will be provided by the measured mass spectrometer
253
intensities. Caution must be exercised even when using fig. 4, of course, since the mass spectrometer signal intensity, which is really proportional to the molecule’s number density, in only an accurate measure of the relative desorption flux for systems where the translational energy distribution of the desorbing particles are relatively similar for all species being detected. The most common current detectors for TDS use electron impact ionizers, usually with mass filtering. The ion source is frequently capped, with a small circular entrance aperture [6,42]. In this case, the desorbing molecules which enter this aperture make many wall collisions and many passes through the ionizer before exiting it. They are therefore probably nearly randomized and thermalized at the detector temperature, so that the detector signal is truly proportional to the desorption flux at the entrance aperture, independent of the velocity distribution with which the molecules enter this aperture. (See ref. [9] for an excellent discussion of velocity distributions and detector types.) Using such a detector at the “magic-angle” of fig. 4 is a very accurate and recommended method for obtaining quantitative TDS rates. However, even when the molecules only make a single pass through the detector so that errors arise due to differences in velocity distribution, using the magic angle of fig. 4 is a better first approximation than neglecting corrections due to angular dist~butions. Let us define r, as the radius of the entrance aperture of a capped detector or the effective radius of any uncapped detector’s cross section. The calculations and discussion presented so far are strictly true only for a “point detector” (that is, for the case when rd K d, or d/rd = 00). More typical aperture sizes are - 25-100% of the sample surface, so that d/r, ranges from - 1 to 20. When d/rd -=z3, the detector is accepting signals from a rather large range of polar angles even with a “point source” (i.e. when d/r = co). When d/r,, -c 3, one could also consider performing another integration over the area of the detector, similar to that which we have performed over the area of the sample. However, when d/rd r 2, the correction due to the finite size of the detector should be rather small if the axis of the detector aperture is aimed directly at the center of the sample. In this
254
S. W. Pa&. C. T. Campbell / Magic-angle desorptron
geometry, at least for the point source, no side of the detector aperture is preferentially sampled as occurs for a large sample whose axis is not aimed directly at even a tiny detector (e.g. fig. 1). This preferential sampling is what leads to the corrections in c$,,, of fig. 4 for small d/r. Such corrections for small d/rd will be much less important. We have not performed similar numerical integrations over the area of the detector because of the smaller corrections involved and also due to the large number of possible combinations of d/r, d/rd and detector directionality. Of course, when d/rd drops below about one, a large fraction of the total solid angle is being accepted by the detector. In the limiting case, some experimentalists [42] make rd = r and d -=cr so that almost all the desorbing molecules enter the detector. In this case, it is obviously best to place the detector apparatus directly over the sample (+ = 0 o ). This seems in contradiction to the trend in fig. 4. However, one must remember that on fig. 4 (which is only strictly appropriate for a tiny detector area) in the limit of closest detector (d/r = 0), any detector angle + performs equally well (i.e. +, = O-90 o ) provided the detector falls within the sample surface area. The signal of course drops to zero if it does not. Geometries such as described above where the detector aperture captures almost all the molecules desorbing from the surface are of course ideal since they will undoubtedly provide the largest signal, and magic-angle corrections such as described here can be neglected. A good description of such a detector is found in ref. [42]. However, it is frequently impossible to achieve such a close detector (d/r 2 0.2) due to geometric constraints imposed by the vacuum chamber and its associated hardware. In those cases, consideration must be given to the optimum angle for measuring quantitative desorption rates. Fig. 4 can be used for predicting this angle when d/r exceeds about 0.75 and d/rd exceeds about 2.
3. Discussion method has a potential This “ magic-angle” limitation for very sharp angular distributions in
that the signal will be much smaller than that normal to the surface. However, the signal will still be of the same size as that for a cosine distribution of the same desorption rate (figs. 2 and 3). Since signals from cosine distributions are easy to measure at this angle (and still more than half as large as the signal perpendicular to the surface), we can envision no real signal-size limitation at the magic-angle even for distributions as sharp as cosi4@ In any case, the magic-angle of fig. 4 will give a signal which is a much closer approximation to the true desorption rate for varying angular distributions then will a detector perpendicular to the surface. We should qualify one of our statements in the introduction concerning the measured angular distributions of desorbing species. We stated that these angular distributions could all be fairly well fitted by cos”$ angular distributions. In fact, in some cases the measured flux is only well fitted by a linear combination of two COS”C$distributions using two separate sharpness parameters n [9,28,29,39]. (In those cases it is generally thought [9] that there are two separate reaction pathways which lead to the two components in desorption.) While not explicitly treated above, it is straightforward to show that the main conclusion of this paper still holds in these case (provided n is between 1 and 9 for both components). That is, the magic-angle from fig. 4 should still be used to minimize differences in TDS signal intensities which are due to differences between species (or surface conditions) in angular distributions of desorption flux. This is true because, in order to minimize differences in such linear combinations, it is necessary to first minimize differences between the individual components COS”C+and cosm+, if n and m are independent and randomly selected. Fig. 4 shows the optimum angle for minimizing the differences between the individual components. The advantages of “magic-angle” detection for quantitative analysis is not only limited to TDS, but can be extended to any particle-desorption technique where the particles are emitted in cos”+ angular distributions. For example, the angular distribution of neutrals sputtered from metals and alloys using Ar+ ions follow nearly a cos’+ distri-
S. W. PaA,
C. T. Campbell / Magic-angle desorption
bution when the primary ion beam energy exceeds 3 keV [40]. The electron-impact or laser ionization and mass-filtered detection of such neutrals form the basis for several emerging, ultra-sensitive surface analytical methods [40], which we will refer to collectively here as secondary neutral mass spectroscopy (SNMS). The need to eliminate problems associated with angular variations for accurate quantitative analysis by SNMS has already been pointed out [40], and a detection angle of approximately 45 o was recommended [41]. Below a primary energy of 3 keV, the angular distributions of sputtered neutrals show angular distributions which are not close to cos”+ [40]. Fig. 5 of ref. [40] shows the variation with angle in the measured relative SNMS signals for the two elements constituting a series of binary alloys. There, the signal ratios are normalized to the ratio of their total SNMS signals integrated over the total solid angle (hemisphere) above the sample. This normalization constant really represents their true (angular-integrated) sensitivity factor ratio. Note that the ratio R plotted there should therefore remain at unity if there were no differences between the angular distributions of two elements. As can be seen in ref. [40], there are strong differences between R and unity, when the primary energy is below 3 keV. Consider a cut through those curves at the magic angle we found above (34-42O, depending upon the sample to detector distance). Interestingly, R is very close to unity near this angle for all samples studied there. In any case, R is much closer to unity at the magicangle than at the usual detector position, $I = 0 O. Thus, by detecting the neutrals at or near the one can effectively minimize the “ magic-angle”, intensity variations associated with angular distributions which change between elements and between samples. This fact suggests that “magicangle” detection may have even more general applicability than for systems with cos”+ behavior. Indeed, one can expect that a “magic-angle” near 42” will be close to the “average” desorption angle for many angular distributions of desorption flux F(+) which are circularly symmetric (independent of the azimuthal angle) and for which F( +) approaches zero as + approached 90”. This is because the multiple of the flux F(9) times the
255
area element sin cp d+ de will drop smoothly to zero at both 0 and 90 O, but maximize somewhere in between. Finally, we should point out that Winkler et al. [2] have recently demonstrated a simple method for determining the sharpness parameter n in cos”+ angular distributions of desorption by varying only the crystal-to-detector distance d.
Acknowledgements The authors would like to thank the US Department of Energy, Office of Basic Energy Sciences, Chemical Sciences Division, for partial support of this work.
Note added in proof In one case for CO, production on a fee (110) surface [43], the sharpness parameter, n, was found to vary with azimuthal angle, 8. The “magic angle” of fig. 4 is still recommended for quantitative TDS even in such cases.
References 111J.N. Russell, Jr., I. Chorkendorff, A.M. Lanzillotto, M.D. Alvey and J.T. Yates, Jr., J. Chem. Phys. 85 (1986) 6186.
121A. Winkler and J.T. Yates, Jr., J. Vat. Sci. Technol. A 6 (1988) 2929.
[31 R.J. Madix, in: Chemistry and Physics of Solid Surfaces, Vol. II, Ed. R. Vanselow (CRC Press, Boca Raton, 1979) p. 63. [41 J. Fair and R.J. Madix, J. Chem. Phys. 73 (1986) 3480. 151 D.A. King, Surf. Sci. 47 (1975) 384. 161 D. Menzel, in: Chemistry and Physics of Solid Surfaces, Vol. IV (CRC Press, Boca Raton, 1982) p. 389. [71 E.I. Ko, J.B. Benziger and R.J. Madix, J. Catal. 62 (1980) 264. PI M. BaIooch, M.J. Cardillo, D.R. Miller and R.E. Stickney, Surf. Sci. 46 (1974) 358. [91 G. Comsa and R. David, Surf. Sci. Rep. 5 (1985) 145. WI H.P. Steinruck, A. Winkler and K.D. Redulic, J. Phys. Chem. 17 (1984) L311; A.E. Dabiri, T.J. Lee and R.E. Stickney, Surf. Sci. 26 (1971) 522; G. Comsa, in: Proc. 7th Int. Vacuum Congress and 3rd Int. Congress on Solid Surfaces, Vienna, 1977, Ed. R. Dobrozemsky, p. 80.
S. W. Pauls, C. T. Campbell / Magic-angle desorpiion
256 [ll]
T. Matsushima, Z. Phys. Chem. 158 (1988) 175; Surf. Sci. 197 (1988) L287. [12] D.R. Horton and R.I. Masel, Surf. Sci. 125 (1983) 699. 1131 G. Comsa, R. David and B. Schumacher, Surf. Sci. 85 (1979) 45. 1141 R.L. Palmer 1453.
and J.N. Smith, Jr., J. Chem.
[15] H.H. 996.
and
Sawin
R.P. Merrill,
J. Chem.
Phys. 60 (1974) Phys. 73 (1980)
[16] T.H. Lin and G.A. Somorjai, J. Chem. Phys. 81 (1984) 704. [17] W. van Willigen, Phys. Lett. A 28 (1968) 80. [18] K.C. Janda, Surf. Sci. 93 (1980) 270. [19] L.L. Verheij, M.B. Hugenschmidt, A.B. Anton, B. Poelsema and G. Comsa, Surf. Sci. 210 (1989) 1. [20] R.C. Cosser, S.R. Bare, S.M. Francis and D.A. King, Vacuum 31 (1981) 503. [21] [22] [23] [24] [25]
T. Engel, J. Chem. Phys. 69 (1978) 373. T. Engel and G. Ertl, J. Chem. Phys. 69 (1978) 1267. T. Matsushima, J. Catal. 83 (1983) 446. T. Matsushima, J. Phys. Chem. 88 (1984) 202. CT. Campbell, G. Ertl and J. Segner, Surf. Sci. 115 (1982) 309. [26] C.T. Campbell, G. Ertl, H. Kuipers and J. Segner, Surf. Sci. 107 (1981) 220. [27] C.T. Campbell, G. Ertl, H. Kuipers and J. Segner, Surf. Sci. 107 (1981) 207. [28] C.T. Campbell, G. Ertl, H. Kuipers and J. Segner, J. Chem. Phys. 73 (1980) 5862.
[29] J. Segner, C.T. Campbell, G. Doyen and G. Ertl, Surf. Sci. 138 (1984) 505. [30] J. Segner, W. Vielhaber and G. Ertl, Isr. J. Chem. 22 (1982) 375. [31] T. Matsushima, Surf. Sci. 127 (1983) 403. [32] T. Matsushima and H. Asada, J. Chem. Phys. 85 (1986) 1658. [33] T. Matsushima, Surf. Sci. 123 (1982) L663. [34] T. Matsushima, T. Matsui and M. Hashimoto, J. Chem. Phys. 81 (1984) 5151. [35] T. Matsushima (1985) 412.
and
H. Asada,
Chem.
Phys.
Lett.
120
[36] T. Matsushima, J. Phys. Chem. 91 (1987) 6192. [37] T. Matsushima, Z. Phys. Chem. 158 (1988) 175. [38] M. Asscher, W.L. Guthrie, T.H. Lin and G.A. Somorjai, J. Phys. Chem. 88 (1984) 3234. [39] G. Comsa and R. David, Chem. Phys. Lett. 49 (1977) 512. [40] A. Wucher, F. Novak and W. Reuter, J. Vat. Sci. Technol. A 6 (1988) 2265. [41] A. Wucher and W. Reuter, J. Vat. Sci. Technol. A 6 (1988) 2316. [42] P. Feulner and D. Menzel, J. Vat. Sci. Technol. 17 (1980) 662; V. Smentkowski and J.T. Yates, Jr., J. Vat. Sci. Technol. 7 (1989) 3325. [43] T. Matsushima, private communication.