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Relative intensities in photoelectron spectroscopy of atoms and molecules
Journal of Electron Spectroscopy and Related Phenomena, 8 (1976) 389-394 @ Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands
RELATIVE INTENSITIES ATOMS AND MOLECULES*
IN
ROBERT
MSEZANE
F. REILMAN,
Department
ALFRED
PHOTOELECTRON
and STEVEN
SPECTROSCOPY
OF
T. MANSON
of Physics, Georgiu Sk&e University, Atlantu, Gewgiu 30303 (U.S.A.)
(Received l6 September 1975)
ABSTRACT
The relative intensities of photoelectron lines is discussed. The relationship of observed intensities to angIe of observation is considered as are the errors introduced by ignoring the fact that different lines may have different angular distributions. Tables of theoretical results for the angular distribution asymmetry parameter, /3, are presented for incident Al K,, Mg K,, and Zr M, radiation for all atomic ground state subshells of non-zero angular momentum. The application of these results to molecules is discussed. I. INTRODUCTION
Photoelectron spectroscopy has been a very rapidly expanding field since the pioneering ESCA work of Siegbahn et al. ‘. This technique provides a great deal of basic information to aid in our understanding of atoms and molecules, in addition to its numerous applications. In a photoelectron spectroscopic (or ESCA) measurement, a beam of photons is incident on a target and the photoelectrons are detected at an angle 8 with respect to the photon beam. In such a measurement, each photoelectron line represents a particular photoionization channel. Information is obtained primarily from the energy of the photoelectrons and the relative intensities of the various photoelectron lines. The energy is not dependent on the angle of detection of the photoelectron; the relative intensities are, however. To interpret the relative intensities in terms of photoionization cross-sections, a knowledge of the angular distribution of the photoelectrons is necessary. Thus, in this paper, calculated angular distributions of atomic photoelectrons are presented for the commonly used photon energies of 1486.6 eV (Al K,), 1253.6 eV (Mg K,), and 151.4 eV (Zr M,} over a broad range of Z spanning the periodic table, The use of the results of calculations on atoms * Supported by NSF and by the U.S. Army Research Office.
390 to interpret ESCA experiments on molecules is justified for core electrons since they are very atomic-like, e.g., photoabsorption measurements on solids show excellent agreement with atomic calculations in the energy region where photoionization of core electrons dominates’. In addition, at high energies, the major contribution to the photoionization matrix element will come from the region very close to the nucleus3 where the wave functions of even valence electrons should be very atomic-like. Thus, far above the threshold region it is expected that an atomic calculation will be useful for molecules. II. THEORY
The angular distribution of photoelectrons impinging on an nLsubshel1 is given by4
ME> -
__
2
I
P, (COSO)
ionized
by unpolarized
light
(1)
where E is the photoelectron energy, ~J”~(E)is the cross-section for photoionizing an rzl electron, 8 is the angle between photon and photoelectron direction, and P2(x) = (3.x2 - 1)/2. Equation (1) depends only upon the photoabsorption going via an electric dipole process, not upon the details of the wave functions employed. The asymmetry parameter Pnr(&) does, however. In this paper we present results of P,,[(E) calculated using Hartree-Slate? wave functions in a formulation given in detail elsewhere6. These calculations have been shown to be quite reliableZm4 except in the vicinity of a Cooper minimum’, i.e., where G,,~(E)has a minimum due to the vanishing of the matrix element in the nl + E, I + 1 photoionization channel. This generally occurs quite close to threshold and only for valence and near-valence subshells2p 3. For X-rays in the keV range, the dipole approximations for photoionization begin to break down. For photoelectrons of speed v, corrections to (T,~ go as v”/c” while corrections to the angular distribution go as v/c, c being the speed of light. Note that this does not imply that BIIl is incorrect, but rather, that extra term(s) are added to eqn. (1) such as P,(cos 0) when electric quadrupole becomes nonnegligible. For electrons of 1 keV energy, then, the correction to O,~ is only -0.4% while the correction to the angular distribution is - 6 %; for 2.5 keV the corrections are -1% and - lo%, respectively. Thus, these corrections are not serious for the photon energies which we consider, but they can be very important for much higher energies. The asymmetry parameter Pnr is 2 for s-subshells (except near a Cooper minimum7) and varies from - 1 to 2 for others. The relative intensities of two photoelectron lines, measured at an angle 8 with respect to the photon beam, is given by the ratio of the differential cross-section
391
1_+ [
c 1 -
B
+
P, (cm e) 1 --.
P, (cm 9)
I
(4
If 8 is chosen such that P,(cos 0) = 0 (0 = 54” 44’) the terms in eqn. (2) dependent on /?drop out and one simply gets the ratio of total cross-section, i.e., the same result one would get if photoelectrons were collected with 47cgeometry. Thus, it would be best if measurements were made at this “magic angle”’ where no corrections for angular distribution are necessary. At 8 = 90”,
13)
so that, if jnr = 2 and pnPzJ= - 1, the observed relative intensities become 2~r,,~/~,,~~~, off by a factor of 2 from the “true” relative intensities. Thus serious errors can occur by assuming observed relative intensities at 90” are the “true” relative intensities. III. RESULTS
AND
DISCUSSlON
Calculations of j? have been performed for all of the p-, d-, and f-subshells appearing in the ground states of atoms. No s-subshells are included since, within the framework of our theoretical model, /I,, = 2. The calculations are done at every fifth Z, but since p varies smoothly as a function of Z6, one can interpoIate quite well to determine the value of p for any Z’s not included. In that sense then, the results are inclusive. At this point, it is worthwhile to discuss the accuracy of our results. For core shells, far above threshold, it is expected that these results will be excellent. For core shells, near threshold, spin-orbit splittings; (which are not included in this nonrelativistic calculation) and level shifts owing to molecular binding will combine to introduce some errors into the calculated j3’s. For valence shells, significantly above threshold, our results will probably be reasonable, but it is not known how far above is necessary for the photoionization to become atomic-like. Finally, near the threshold of valence shells, there is no good reason to expect that our results have any relevance at all for molecules so such cases are not included in our tables. The asymmetry parameter, Bnl, is presented for 2p, 3p, 4p, 5p, 6p, 3d, 4d, 56, and 4fsubshells in the tables. In this paper, no attempt is made to discuss the variation of p with E and Z; this has been done extensively and adequately elsewhere6, lo. We have included a sufficient number of points so that to find Pnr for any Z not given a plot of finz vs. Z (at the desired X-ray energy) will be quite accurate.
392 IV. CONCLUDING
REMARKS
It is hoped that the table presented herein will prove useful in determining the “true” relative intensities when photoelectrons are detected at 90” or some other angle. Far X-ray energies not among the three for which the calculations have been TABLE
1
ASYMMETRY
2p
3p
3d
PARAMETER,
B
Z
B
Z
P
5 10 15
1.10 1.49 0.53
35 40 45 50
1.05 0.80 0.44 -0.13
15 20 25 30
1.61 1.48 1.11 0.24
60 65 70 7s 80
0.48 0.39 0.33 0.36 1.05
2s 30 3s
1.08 0.84 0.28
40 4s 50 55 60
0.49 -0.02 -0.71 0.40 0.96
50 55 60 65 70 75 80 85
0.82 0.75 0.60 0.42 0.23 0.05 -0.10 -0.26
4p
46
PARAMETER,
5p
b, FOR Mg Ka (1253.6 eV) X-RAYS
Z
B
Z
B
5 10
25 30
0.19 0.77 1.17 1.39 1.46 1.26
15 20 25 30 35 40 45 50 55
1.13 1.35 1.49 1.56 1.60 1.61 1.59 1.52 1.30
25 30 35 40 45 50 55 60
0.90 1.04 1.14 1.19 1.21 1.17 1.04 0.72
35 40 45 50 55 60 65 70 75 80 85 90 95
1.62 1.65 1.68 1.69 1.70 1.69 1.67 1.65 1.61 1.55 1.46 1.28 0.93
;;
4f
ON ATOMS
Z
B
75 80 85 5d 90 95 100
1.09 1.62 1.88 1.85 1.28 0.36
85 90 9s 100
0.35 0.33 0.25 0.18
@
_
2
ASYMMETRY
3p
INCIDENT
Z
TABLE
2p
B, FOR Zr Mc (151.4 eV) X-RAYS
36
4p
_--l-.
.
i
’
B
40 45 50 5s 60 65 4d 70 7s 80 85 90 95 100
1.22 1.27 1.31 1.33 1.33 1.31 1.27 1.24 1.15 I.04 0.87 0.63 0.24
60 65 70 75 4f 80 85 90 95 100
I .05 1.06 1.05 1.03 1.oo 0.95 0.88 0.79 0.66
INDICENT
ON ATOMS
Z
B
50 55 60 65 70 5p 7s 80 85 90 9s 100
1.70 1.72 1.73 1.73 1.72 1.72 1.71 1.70 1.69 1.67 1.65
75 80 85 5d 90 95 100
1.34 1.33 1.31 1.29 1.26 1.23
85 90 6p 95 100
1.72 1.72 1.72 1.71
393 TABLE
3
ASYMMETRY
2P
@, FOR
AI K, (1484.6 eV) X-RAYS
Z
B
Z
B
5 10
0.38 0.69 1.09 1.35 1.46 1.40
25 30 3s 40 4s 50 55 60 65
0.83 0.98 1.10 1.16 1.20 1.20 1.14 0.97 0.62
:; 25 i30
I15
3fJ
PARAMETER,
20 25 30 35 40 45 50 55 60
1.05 1.28 1.43 1.52 1.58 1.61 1.61 1.58 1.48 1.19
3d
35 40 4s 50 4p 55 60 65 70 7s SO 85 90 95 100
1.59 1.63 1.67 1.69 1.70 1.70 1.69 1.68 1.65 1.62 1.56 1.47 1.33 0.99
Z
40 45 50 55 60 4d 65 70 75 80 85 90 9s 100
4f
60 65 70 75 80 85 90 95 100
INCIDENT
Z
B
SO
55 60 5~ 65 70 75 80 8.5 90 95 100
1.71 1.71 1.73 1.73 1.73 1.73 1.72 1.72 1.71 1.70 1.69
7s 80 Sd 85 90 95 100
1.35 1.35 1.35 1.34 1.32 1.31
6p
1.74 1.74 1.74 1.73
B 1.16 1.23 1.28 1.32 1.33 1.33 1.31 I .26 1.23 1.16 1.06 0.91 0.69 1.02 1.04 1.05 1.04 1.03 1.00 CL95 0.89 0.81
ON ATOMS
85 90 95 100
performed, results can be obtained by contacting the authors. The accuracy of these results, for molecular applications, is not completely known. It would, therefore, be of great utility to have experimental measurements of Pnl for various cases for comparison purposes. This type of experiment could be done on any ESCA apparatus whose photoelectron detector is moveable. Such experiments would serve to fully assess the theoretical results presented and to give insight into any limitations of their application. REFERENCES 1
2 3 4 5 6
K. Siegbahn, C. Nordling, A. Fahlman, R. Nordberg, K. Hamrin, J. Hedman, G. Johansson, T. Bergmark, S. Karlsson, I. Lindgren and B. Lindberg, Nova Acta Reg. Sot. Sci. Upsaliensis, Ser. IV, 20 (1967) 1. S. T. Manson and J. W. Cooper, Phys. Rev., 165 (1968) 128. U. Fano and J. W. Cooper, Rev. Mod. Phys., 40 (1968) 441. D. J. Kennedy and S. T. Manson, P/zys. Rev. A, 5 (1972) 237. F. Herman and S. Skillman, Afomic Structure Calcuhtions, Prentice-Hall, Englewood Cliffs, N.J., 1963. S. T. Manson, J. Electron Specfrosc., 1 (1973) 413.
394 7 8
9 10
J. W. Cooper, Phys. Rev. Lett., 13 (1964) 762. l-l. A. Bethe and E. E. Salpeter, Quantum Mechanics of One- and Two-Electron Verlag, Berlin, 1957, pp. 308-315. J. A. R. Samson, J. Opt. Sot. Am., 59 (1969) 356. S. T. Manson, J. Electron Spectrosc., to be published.
Atoms,
Springer-
RELATIVE INTENSITIES ATOMS AND MOLECULES*
IN
ROBERT
MSEZANE
F. REILMAN,
Department
ALFRED
PHOTOELECTRON
and STEVEN
SPECTROSCOPY
OF
T. MANSON
of Physics, Georgiu Sk&e University, Atlantu, Gewgiu 30303 (U.S.A.)
(Received l6 September 1975)
ABSTRACT
The relative intensities of photoelectron lines is discussed. The relationship of observed intensities to angIe of observation is considered as are the errors introduced by ignoring the fact that different lines may have different angular distributions. Tables of theoretical results for the angular distribution asymmetry parameter, /3, are presented for incident Al K,, Mg K,, and Zr M, radiation for all atomic ground state subshells of non-zero angular momentum. The application of these results to molecules is discussed. I. INTRODUCTION
Photoelectron spectroscopy has been a very rapidly expanding field since the pioneering ESCA work of Siegbahn et al. ‘. This technique provides a great deal of basic information to aid in our understanding of atoms and molecules, in addition to its numerous applications. In a photoelectron spectroscopic (or ESCA) measurement, a beam of photons is incident on a target and the photoelectrons are detected at an angle 8 with respect to the photon beam. In such a measurement, each photoelectron line represents a particular photoionization channel. Information is obtained primarily from the energy of the photoelectrons and the relative intensities of the various photoelectron lines. The energy is not dependent on the angle of detection of the photoelectron; the relative intensities are, however. To interpret the relative intensities in terms of photoionization cross-sections, a knowledge of the angular distribution of the photoelectrons is necessary. Thus, in this paper, calculated angular distributions of atomic photoelectrons are presented for the commonly used photon energies of 1486.6 eV (Al K,), 1253.6 eV (Mg K,), and 151.4 eV (Zr M,} over a broad range of Z spanning the periodic table, The use of the results of calculations on atoms * Supported by NSF and by the U.S. Army Research Office.
390 to interpret ESCA experiments on molecules is justified for core electrons since they are very atomic-like, e.g., photoabsorption measurements on solids show excellent agreement with atomic calculations in the energy region where photoionization of core electrons dominates’. In addition, at high energies, the major contribution to the photoionization matrix element will come from the region very close to the nucleus3 where the wave functions of even valence electrons should be very atomic-like. Thus, far above the threshold region it is expected that an atomic calculation will be useful for molecules. II. THEORY
The angular distribution of photoelectrons impinging on an nLsubshel1 is given by4
ME> -
__
2
I
P, (COSO)
ionized
by unpolarized
light
(1)
where E is the photoelectron energy, ~J”~(E)is the cross-section for photoionizing an rzl electron, 8 is the angle between photon and photoelectron direction, and P2(x) = (3.x2 - 1)/2. Equation (1) depends only upon the photoabsorption going via an electric dipole process, not upon the details of the wave functions employed. The asymmetry parameter Pnr(&) does, however. In this paper we present results of P,,[(E) calculated using Hartree-Slate? wave functions in a formulation given in detail elsewhere6. These calculations have been shown to be quite reliableZm4 except in the vicinity of a Cooper minimum’, i.e., where G,,~(E)has a minimum due to the vanishing of the matrix element in the nl + E, I + 1 photoionization channel. This generally occurs quite close to threshold and only for valence and near-valence subshells2p 3. For X-rays in the keV range, the dipole approximations for photoionization begin to break down. For photoelectrons of speed v, corrections to (T,~ go as v”/c” while corrections to the angular distribution go as v/c, c being the speed of light. Note that this does not imply that BIIl is incorrect, but rather, that extra term(s) are added to eqn. (1) such as P,(cos 0) when electric quadrupole becomes nonnegligible. For electrons of 1 keV energy, then, the correction to O,~ is only -0.4% while the correction to the angular distribution is - 6 %; for 2.5 keV the corrections are -1% and - lo%, respectively. Thus, these corrections are not serious for the photon energies which we consider, but they can be very important for much higher energies. The asymmetry parameter Pnr is 2 for s-subshells (except near a Cooper minimum7) and varies from - 1 to 2 for others. The relative intensities of two photoelectron lines, measured at an angle 8 with respect to the photon beam, is given by the ratio of the differential cross-section
391
1_+ [
c 1 -
B
+
P, (cm e) 1 --.
P, (cm 9)
I
(4
If 8 is chosen such that P,(cos 0) = 0 (0 = 54” 44’) the terms in eqn. (2) dependent on /?drop out and one simply gets the ratio of total cross-section, i.e., the same result one would get if photoelectrons were collected with 47cgeometry. Thus, it would be best if measurements were made at this “magic angle”’ where no corrections for angular distribution are necessary. At 8 = 90”,
13)
so that, if jnr = 2 and pnPzJ= - 1, the observed relative intensities become 2~r,,~/~,,~~~, off by a factor of 2 from the “true” relative intensities. Thus serious errors can occur by assuming observed relative intensities at 90” are the “true” relative intensities. III. RESULTS
AND
DISCUSSlON
Calculations of j? have been performed for all of the p-, d-, and f-subshells appearing in the ground states of atoms. No s-subshells are included since, within the framework of our theoretical model, /I,, = 2. The calculations are done at every fifth Z, but since p varies smoothly as a function of Z6, one can interpoIate quite well to determine the value of p for any Z’s not included. In that sense then, the results are inclusive. At this point, it is worthwhile to discuss the accuracy of our results. For core shells, far above threshold, it is expected that these results will be excellent. For core shells, near threshold, spin-orbit splittings; (which are not included in this nonrelativistic calculation) and level shifts owing to molecular binding will combine to introduce some errors into the calculated j3’s. For valence shells, significantly above threshold, our results will probably be reasonable, but it is not known how far above is necessary for the photoionization to become atomic-like. Finally, near the threshold of valence shells, there is no good reason to expect that our results have any relevance at all for molecules so such cases are not included in our tables. The asymmetry parameter, Bnl, is presented for 2p, 3p, 4p, 5p, 6p, 3d, 4d, 56, and 4fsubshells in the tables. In this paper, no attempt is made to discuss the variation of p with E and Z; this has been done extensively and adequately elsewhere6, lo. We have included a sufficient number of points so that to find Pnr for any Z not given a plot of finz vs. Z (at the desired X-ray energy) will be quite accurate.
392 IV. CONCLUDING
REMARKS
It is hoped that the table presented herein will prove useful in determining the “true” relative intensities when photoelectrons are detected at 90” or some other angle. Far X-ray energies not among the three for which the calculations have been TABLE
1
ASYMMETRY
2p
3p
3d
PARAMETER,
B
Z
B
Z
P
5 10 15
1.10 1.49 0.53
35 40 45 50
1.05 0.80 0.44 -0.13
15 20 25 30
1.61 1.48 1.11 0.24
60 65 70 7s 80
0.48 0.39 0.33 0.36 1.05
2s 30 3s
1.08 0.84 0.28
40 4s 50 55 60
0.49 -0.02 -0.71 0.40 0.96
50 55 60 65 70 75 80 85
0.82 0.75 0.60 0.42 0.23 0.05 -0.10 -0.26
4p
46
PARAMETER,
5p
b, FOR Mg Ka (1253.6 eV) X-RAYS
Z
B
Z
B
5 10
25 30
0.19 0.77 1.17 1.39 1.46 1.26
15 20 25 30 35 40 45 50 55
1.13 1.35 1.49 1.56 1.60 1.61 1.59 1.52 1.30
25 30 35 40 45 50 55 60
0.90 1.04 1.14 1.19 1.21 1.17 1.04 0.72
35 40 45 50 55 60 65 70 75 80 85 90 95
1.62 1.65 1.68 1.69 1.70 1.69 1.67 1.65 1.61 1.55 1.46 1.28 0.93
;;
4f
ON ATOMS
Z
B
75 80 85 5d 90 95 100
1.09 1.62 1.88 1.85 1.28 0.36
85 90 9s 100
0.35 0.33 0.25 0.18
@
_
2
ASYMMETRY
3p
INCIDENT
Z
TABLE
2p
B, FOR Zr Mc (151.4 eV) X-RAYS
36
4p
_--l-.
.
i
’
B
40 45 50 5s 60 65 4d 70 7s 80 85 90 95 100
1.22 1.27 1.31 1.33 1.33 1.31 1.27 1.24 1.15 I.04 0.87 0.63 0.24
60 65 70 75 4f 80 85 90 95 100
I .05 1.06 1.05 1.03 1.oo 0.95 0.88 0.79 0.66
INDICENT
ON ATOMS
Z
B
50 55 60 65 70 5p 7s 80 85 90 9s 100
1.70 1.72 1.73 1.73 1.72 1.72 1.71 1.70 1.69 1.67 1.65
75 80 85 5d 90 95 100
1.34 1.33 1.31 1.29 1.26 1.23
85 90 6p 95 100
1.72 1.72 1.72 1.71
393 TABLE
3
ASYMMETRY
2P
@, FOR
AI K, (1484.6 eV) X-RAYS
Z
B
Z
B
5 10
0.38 0.69 1.09 1.35 1.46 1.40
25 30 3s 40 4s 50 55 60 65
0.83 0.98 1.10 1.16 1.20 1.20 1.14 0.97 0.62
:; 25 i30
I15
3fJ
PARAMETER,
20 25 30 35 40 45 50 55 60
1.05 1.28 1.43 1.52 1.58 1.61 1.61 1.58 1.48 1.19
3d
35 40 4s 50 4p 55 60 65 70 7s SO 85 90 95 100
1.59 1.63 1.67 1.69 1.70 1.70 1.69 1.68 1.65 1.62 1.56 1.47 1.33 0.99
Z
40 45 50 55 60 4d 65 70 75 80 85 90 9s 100
4f
60 65 70 75 80 85 90 95 100
INCIDENT
Z
B
SO
55 60 5~ 65 70 75 80 8.5 90 95 100
1.71 1.71 1.73 1.73 1.73 1.73 1.72 1.72 1.71 1.70 1.69
7s 80 Sd 85 90 95 100
1.35 1.35 1.35 1.34 1.32 1.31
6p
1.74 1.74 1.74 1.73
B 1.16 1.23 1.28 1.32 1.33 1.33 1.31 I .26 1.23 1.16 1.06 0.91 0.69 1.02 1.04 1.05 1.04 1.03 1.00 CL95 0.89 0.81
ON ATOMS
85 90 95 100
performed, results can be obtained by contacting the authors. The accuracy of these results, for molecular applications, is not completely known. It would, therefore, be of great utility to have experimental measurements of Pnl for various cases for comparison purposes. This type of experiment could be done on any ESCA apparatus whose photoelectron detector is moveable. Such experiments would serve to fully assess the theoretical results presented and to give insight into any limitations of their application. REFERENCES 1
2 3 4 5 6
K. Siegbahn, C. Nordling, A. Fahlman, R. Nordberg, K. Hamrin, J. Hedman, G. Johansson, T. Bergmark, S. Karlsson, I. Lindgren and B. Lindberg, Nova Acta Reg. Sot. Sci. Upsaliensis, Ser. IV, 20 (1967) 1. S. T. Manson and J. W. Cooper, Phys. Rev., 165 (1968) 128. U. Fano and J. W. Cooper, Rev. Mod. Phys., 40 (1968) 441. D. J. Kennedy and S. T. Manson, P/zys. Rev. A, 5 (1972) 237. F. Herman and S. Skillman, Afomic Structure Calcuhtions, Prentice-Hall, Englewood Cliffs, N.J., 1963. S. T. Manson, J. Electron Specfrosc., 1 (1973) 413.
394 7 8
9 10
J. W. Cooper, Phys. Rev. Lett., 13 (1964) 762. l-l. A. Bethe and E. E. Salpeter, Quantum Mechanics of One- and Two-Electron Verlag, Berlin, 1957, pp. 308-315. J. A. R. Samson, J. Opt. Sot. Am., 59 (1969) 356. S. T. Manson, J. Electron Spectrosc., to be published.
Atoms,
Springer-