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Limitations on the validity of the non-relativistic dipole approximation for photoelectron angular distributions
Volume
47, number 2
CHEMICAL PHYSICS LETPERS
LIMITATIONS ON THE VALIDITY OF THE NON-RELATI~S~C
15 April 1977
DffOLE ~PRO~~~ON-
FOR PHOTOELECTRON ANGULAR DISTRIBUTIONS* Akiva RON Racah Institute of Physics, Hebrew UniveMy Jerusalem, Israel R.H.
of Jerusalem.
PRATT
Department of PJzysks and Astronomy, Pittsburgh, Pennsylvanirr ZS260, USA
University of~ttsbur~,
and H-K. TSENG Department of Physics, National Central Univers&y, Ckng-Li, Taiwan, Republic of China Received 29 October 1976 Revised manuscript received 8 January 1977
We present results which exhibit hazards in the use of the non-relativistic dipole approximation in the interpretation of experimental photoelectron angular distributions. Significant deviations from the non-relativistic dipole approximation occur for both low and high Z atoms. In Lightelements these effects are found in the keV range and even beLow. In heavy eLements they are found even for energies very close to threshold. For total cross sections, in contrast to this anguLardistribution situation, the surviving integrated relativistic and m~tipole corrections tend to cancel, so that the non-reLativisticdipoie approximation hoids to SurprisingIy high energies_
For some time now photoeffect has proved a useful tool in many investigations. A detailed account and references are given by Krause El]. Applications include accurate measurements of linewidths f2], determination of chemical shifts [3,4], and locating surface impurities in crystals [S] . More recently there has been interest in the use of photoelectric angular distributions, advocated by Manson and others [6,7], as in the determination of X-ray intensities [ 1 ,g] and the study of molecular orbitals [9]. The photon energies of concern for such investigations are generally in the range of a few keV and lower. Usually ~l~ulations performed for such energies disregard both relativistic and multi* Supported in part by the NationaL Science Foundation under Grant MPS74-03531 and Grant OlP7503599 and in part by the National Science Council of the Republic of China.
pole effects. This non-relativistic (NR) dipole approximation yields the simple result (for the case ofunpolarized radiation) [10,6] do/da = (ffo~4~) [I - 4 flP(cos
S)] =Xf
Ysin28.
(1)
Here 00 is the total photoeiectric cross section,Pz is the second Legendre polynomial and 8 is the angIe between the direction of the incident photon and the direction of the outgoing photoelectron, and fl is the asymmetry parameter, a function of photon energy and shell from which the electron is ejected, For emission from s states @ Z 2. One may expect corrections to the NR dipole approximation, characterized by the parameters u/c and Zol, originating both from higher multipoles and from relativistic effects. For total cross sections, however, we have made both numerical and ant&tical calcuIa377
Volume 47, number 2
CHEMICAL PHYSICS LETTERS
tions [l l] which demonstrate that after integration over angles these effects tend to cancel, both for low Z and high Z atoms. The consequence is that the nonrelativistic dipole approximation remains valid to quite high energy, above 50 keV in the K shell case. Multipole and relativistic effects do nut cancel in the angular distribution, and deviations from the nonrelativistic dipole approximation can be significant even at Iow energies. When ZCY4 v/c, we can show that, as in the point Coulomb case, the corrections to the angular distributions are U(u/c), and our numerical computations, using the methods of our previous work on total cross sections [I 2 3 confirm this estimate, showing corrections which grow with v/c_ However when Zff % u/c our numerical results do not show a correction ci(u/c); it appears that, in the screened case, near thresholds corrections persist as effects in Z~L In particular we do not confirm the correction [I + (4 f I)(u/c)cos 61 to the sin% term in eq. (1) suggested by Cooper and Manson [6j. Even the sign of the correction term varies with energy. In summary, what we find is that the deviations of angular distributions from the NR dipoIe approximation can persist for low Z atoms in the few keV region and for high 2 atoms near thresholds. Therefore the simple form (1) should be used with caution. We illustrate these remarks with some examples. In fig 1 we compare with experiments of Wuilleumier and Krause [13] for the 2p shell of Ne. One sees clearly the deviation from the NR dipole results [eq. (I)], especially for small angles, and generally better agreement of the experimental data with our compIete relativistic results than the suggested point Coulomb corrections. Fig. 2 shows Krause’s [14] results for krypton experiments. A compfete cancellation of relativistic and multipole effects takes place for the angular distribution of the 3s and 3p but not 3d subshell electrons at this energy, in agreement with our numerical calculations. Krause had already noted that the CooperManson prediction was unsatisfactory in these cases. Samson and Gardner [ 1S] have suggested using the fact that P,(cos 54O44’) = 0 to transform eq. (1) to (du/dS2)54044,
=
uo/4a.
(2)
Thus one wouId obtain the total cross section by measuring the photoelectron intensity at this particular angle. However, if the angular distribution deviates from the NR dipole approximation eq. (2) is incorrect, as
15 April 1977
00 0
30
60
120
90
150
6, Angle (deg ) Fig. 1. Comparisons of normalized angular cross sections J(0) = 100 [do(@)/dn J/[da@ = 90°1/dnJ of Ne 2p photo&cuons between the experimental data (solid circIes) of WuiIIeumier and Krause (I 3 1 and our relativistic results
IZOr . . , k = I .2540
=‘ o!Jl’l”~‘ff”l’**xJ 0 30
60
90
e,
120
150
keV
180
Angle (deg)
Fig. 2. Comparisons of the normalized angular cross section J(8) of Kr 3s, 3p and 3d photoelectrons ejected by Mg Ku X-rays (i.e. k = 1.254 keV) between the experimental data (circles) of Krause 1141 and our relativistic results (crosses). The experimental results were fitted bjr Krause according to J(e) =X+ Y sin*13(solid lines) with X f Y = 100.
Volume 47, number 2
CHEMICAL PHYSICS LETTERS
Table 1 Comparison between our numerical total cross section and results of eq. (2) for neon and krypton. Here a(n) means II X lOn, as in standard computational notation (nor an error limit) c__-._^ Z
____ _ .__-_ Subshell
k
(keV)
Photocffect cross section (barns/atom) this
work 10
36
2p
3s 3P 3d
0.1089 0.1323 1.2536 1.4866 1.2540 1.2540 1.2540
2.86(6) 1.86(6) 2.52(3) 1.40(3) 2.99(4) 1.08(S) 6.21(4)
approx. eq. (2) -2.91(6) 1.91(a) 2.87(3) 1.62(3) 2.99(4) 1.09(S) 9.20(4)
approx. this work 1.02 1.03 1.14 1.16 1.00 1.01 1.12
shown in table 1 for Krause’s experiments. Niehaus and Ruf [ 161 measured the asymmetry parameter fl for the angular distribution of 10.78 eV photoelectrons from the 6s subshell of mercury, obtaining fl= 1.68 f: 0.1. In a more general formulation one may write do/da = (oo/47Q nqo B,$‘,(cosQ , with BO z 1. We may identify the asymmetry parameter /I = -2B2, and this characterization by one number will be meaningful if the other Bn’s are small. For this case the other Bn’s are indeed small (B, = 1.70 X 10b3, B3=-1.69X10-3,B4=-1.41 X10-6,B5=--1.25X 10eg) and our numerical calculation gives /3= 1.80, in good agreement with the experiment and in contrast with the non-relativistic dipole approximation fl G 2 for s-states. We plan to subsequently present more detailed discussion of the corrections to the non-relativistic dipole approximation, leading to a systematic analysis of the situations for which they must be considered. Here we conclude by again urging caution in using eq. (1) for photoeiectron angular distribuiionl. Ii is dear from our examples that, even at low energies, relativistic and multipole corrections should in some cases be considered together with many electron effects. Significant
15 April 1977
errors are possible if the non-relativistic dipole approxi-
mation is used to interpret the photoelectron angular distribution resulting from Al KU and Mg KU radiaticn incident on any target. The impact of these corrections might be further enhanced near a Cooper minimum, where the non-relativistic dipole result is small. We would like to acknowledge helpful discussions ’ wirh Dr. MO. Krause, Professor S.T. Manson, Professor D. Shirley, Dr. CM. Lee and Dr. Simon Yu.
References [ 1) MO. Krause, in: Atomic inner-shell processes, Vol. 2. Experimental approaches and applications, ed. B. Crasemann (Academic Press, New York, 1975) p. 33. [2] U. Gelius, E. Basilier. S. Svensson, T. Bergmark and K. Siegbahn, J. Electron Spectry. 2 (1974) 405. [3] S.B.M. HagstrBm, C. -Nordling and K. Siegbahn, Z. Physik 178 (1964) 439. ]4] K. Siegbahn, C. Nordling. A. Fahlman, R. Nordberg, K. Hamrin, J. Hedman, G. Johansson, T. Bergmark, S.-E. Karlsson, I. Lindgren and B. Lindberg, Nova Acta Regiae Sot. Sci. Upsal. 20 (1967). [5] C.S. Fadley and S.A.L. BergstrGm, Phys. Letters 35A (1971) 375. [6] J.W. Cooper and S.T. Manson, Phys. Rev. 177 (1969) 157. 171D.J. Kennedy and S.T. Manson, Phys. Rev. A5 (1972) 221; A.F. Starace, S.T. Manson and D.J. Kennedy, Phys. Rev. A9 (1974) 2A53; D. Dill, A.F. Starace and S.T. Manson, Phys. Rev. AI 1 (1975) 1596. 181M.O. Krause, unpublished data (1974). I91 T.A. Carlson and C.P. Anderson, Chem. Phys. Letters 10 (1971) 561. 1101H.A. Bethe and E.E. Salpeter, Quantum mechanics of one- and two-electron atoms (Academic Press, New York, 1957) pp. 308-310. 1111A. Ronn, R.H. Pratt and H.K. Tsrng, Phys. Rev., to bc submitted for publication. [I21 R.H. Pratt, A. Ron and H.K. Tseng, Rev. Mod. Phys. 45
(1973) 273. ]I31 F. Wuilleumier and M.O. Krause, Phys. Rev. A10 (1973)
[I41 1151
242. M.O. Krause, Phys. Rev. 1’77(1969) 151. J.A.R. §a !so7 and J.L. timher, i. Opt. 5%. Am. 62
[lb1
(1972) &Zb. A. Niehnus and M.W. Ruf, Z. Physik 252 (1972) 84.
47, number 2
CHEMICAL PHYSICS LETPERS
LIMITATIONS ON THE VALIDITY OF THE NON-RELATI~S~C
15 April 1977
DffOLE ~PRO~~~ON-
FOR PHOTOELECTRON ANGULAR DISTRIBUTIONS* Akiva RON Racah Institute of Physics, Hebrew UniveMy Jerusalem, Israel R.H.
of Jerusalem.
PRATT
Department of PJzysks and Astronomy, Pittsburgh, Pennsylvanirr ZS260, USA
University of~ttsbur~,
and H-K. TSENG Department of Physics, National Central Univers&y, Ckng-Li, Taiwan, Republic of China Received 29 October 1976 Revised manuscript received 8 January 1977
We present results which exhibit hazards in the use of the non-relativistic dipole approximation in the interpretation of experimental photoelectron angular distributions. Significant deviations from the non-relativistic dipole approximation occur for both low and high Z atoms. In Lightelements these effects are found in the keV range and even beLow. In heavy eLements they are found even for energies very close to threshold. For total cross sections, in contrast to this anguLardistribution situation, the surviving integrated relativistic and m~tipole corrections tend to cancel, so that the non-reLativisticdipoie approximation hoids to SurprisingIy high energies_
For some time now photoeffect has proved a useful tool in many investigations. A detailed account and references are given by Krause El]. Applications include accurate measurements of linewidths f2], determination of chemical shifts [3,4], and locating surface impurities in crystals [S] . More recently there has been interest in the use of photoelectric angular distributions, advocated by Manson and others [6,7], as in the determination of X-ray intensities [ 1 ,g] and the study of molecular orbitals [9]. The photon energies of concern for such investigations are generally in the range of a few keV and lower. Usually ~l~ulations performed for such energies disregard both relativistic and multi* Supported in part by the NationaL Science Foundation under Grant MPS74-03531 and Grant OlP7503599 and in part by the National Science Council of the Republic of China.
pole effects. This non-relativistic (NR) dipole approximation yields the simple result (for the case ofunpolarized radiation) [10,6] do/da = (ffo~4~) [I - 4 flP(cos
S)] =Xf
Ysin28.
(1)
Here 00 is the total photoeiectric cross section,Pz is the second Legendre polynomial and 8 is the angIe between the direction of the incident photon and the direction of the outgoing photoelectron, and fl is the asymmetry parameter, a function of photon energy and shell from which the electron is ejected, For emission from s states @ Z 2. One may expect corrections to the NR dipole approximation, characterized by the parameters u/c and Zol, originating both from higher multipoles and from relativistic effects. For total cross sections, however, we have made both numerical and ant&tical calcuIa377
Volume 47, number 2
CHEMICAL PHYSICS LETTERS
tions [l l] which demonstrate that after integration over angles these effects tend to cancel, both for low Z and high Z atoms. The consequence is that the nonrelativistic dipole approximation remains valid to quite high energy, above 50 keV in the K shell case. Multipole and relativistic effects do nut cancel in the angular distribution, and deviations from the nonrelativistic dipole approximation can be significant even at Iow energies. When ZCY4 v/c, we can show that, as in the point Coulomb case, the corrections to the angular distributions are U(u/c), and our numerical computations, using the methods of our previous work on total cross sections [I 2 3 confirm this estimate, showing corrections which grow with v/c_ However when Zff % u/c our numerical results do not show a correction ci(u/c); it appears that, in the screened case, near thresholds corrections persist as effects in Z~L In particular we do not confirm the correction [I + (4 f I)(u/c)cos 61 to the sin% term in eq. (1) suggested by Cooper and Manson [6j. Even the sign of the correction term varies with energy. In summary, what we find is that the deviations of angular distributions from the NR dipoIe approximation can persist for low Z atoms in the few keV region and for high 2 atoms near thresholds. Therefore the simple form (1) should be used with caution. We illustrate these remarks with some examples. In fig 1 we compare with experiments of Wuilleumier and Krause [13] for the 2p shell of Ne. One sees clearly the deviation from the NR dipole results [eq. (I)], especially for small angles, and generally better agreement of the experimental data with our compIete relativistic results than the suggested point Coulomb corrections. Fig. 2 shows Krause’s [14] results for krypton experiments. A compfete cancellation of relativistic and multipole effects takes place for the angular distribution of the 3s and 3p but not 3d subshell electrons at this energy, in agreement with our numerical calculations. Krause had already noted that the CooperManson prediction was unsatisfactory in these cases. Samson and Gardner [ 1S] have suggested using the fact that P,(cos 54O44’) = 0 to transform eq. (1) to (du/dS2)54044,
=
uo/4a.
(2)
Thus one wouId obtain the total cross section by measuring the photoelectron intensity at this particular angle. However, if the angular distribution deviates from the NR dipole approximation eq. (2) is incorrect, as
15 April 1977
00 0
30
60
120
90
150
6, Angle (deg ) Fig. 1. Comparisons of normalized angular cross sections J(0) = 100 [do(@)/dn J/[da@ = 90°1/dnJ of Ne 2p photo&cuons between the experimental data (solid circIes) of WuiIIeumier and Krause (I 3 1 and our relativistic results
IZOr . . , k = I .2540
=‘ o!Jl’l”~‘ff”l’**xJ 0 30
60
90
e,
120
150
keV
180
Angle (deg)
Fig. 2. Comparisons of the normalized angular cross section J(8) of Kr 3s, 3p and 3d photoelectrons ejected by Mg Ku X-rays (i.e. k = 1.254 keV) between the experimental data (circles) of Krause 1141 and our relativistic results (crosses). The experimental results were fitted bjr Krause according to J(e) =X+ Y sin*13(solid lines) with X f Y = 100.
Volume 47, number 2
CHEMICAL PHYSICS LETTERS
Table 1 Comparison between our numerical total cross section and results of eq. (2) for neon and krypton. Here a(n) means II X lOn, as in standard computational notation (nor an error limit) c__-._^ Z
____ _ .__-_ Subshell
k
(keV)
Photocffect cross section (barns/atom) this
work 10
36
2p
3s 3P 3d
0.1089 0.1323 1.2536 1.4866 1.2540 1.2540 1.2540
2.86(6) 1.86(6) 2.52(3) 1.40(3) 2.99(4) 1.08(S) 6.21(4)
approx. eq. (2) -2.91(6) 1.91(a) 2.87(3) 1.62(3) 2.99(4) 1.09(S) 9.20(4)
approx. this work 1.02 1.03 1.14 1.16 1.00 1.01 1.12
shown in table 1 for Krause’s experiments. Niehaus and Ruf [ 161 measured the asymmetry parameter fl for the angular distribution of 10.78 eV photoelectrons from the 6s subshell of mercury, obtaining fl= 1.68 f: 0.1. In a more general formulation one may write do/da = (oo/47Q nqo B,$‘,(cosQ , with BO z 1. We may identify the asymmetry parameter /I = -2B2, and this characterization by one number will be meaningful if the other Bn’s are small. For this case the other Bn’s are indeed small (B, = 1.70 X 10b3, B3=-1.69X10-3,B4=-1.41 X10-6,B5=--1.25X 10eg) and our numerical calculation gives /3= 1.80, in good agreement with the experiment and in contrast with the non-relativistic dipole approximation fl G 2 for s-states. We plan to subsequently present more detailed discussion of the corrections to the non-relativistic dipole approximation, leading to a systematic analysis of the situations for which they must be considered. Here we conclude by again urging caution in using eq. (1) for photoeiectron angular distribuiionl. Ii is dear from our examples that, even at low energies, relativistic and multipole corrections should in some cases be considered together with many electron effects. Significant
15 April 1977
errors are possible if the non-relativistic dipole approxi-
mation is used to interpret the photoelectron angular distribution resulting from Al KU and Mg KU radiaticn incident on any target. The impact of these corrections might be further enhanced near a Cooper minimum, where the non-relativistic dipole result is small. We would like to acknowledge helpful discussions ’ wirh Dr. MO. Krause, Professor S.T. Manson, Professor D. Shirley, Dr. CM. Lee and Dr. Simon Yu.
References [ 1) MO. Krause, in: Atomic inner-shell processes, Vol. 2. Experimental approaches and applications, ed. B. Crasemann (Academic Press, New York, 1975) p. 33. [2] U. Gelius, E. Basilier. S. Svensson, T. Bergmark and K. Siegbahn, J. Electron Spectry. 2 (1974) 405. [3] S.B.M. HagstrBm, C. -Nordling and K. Siegbahn, Z. Physik 178 (1964) 439. ]4] K. Siegbahn, C. Nordling. A. Fahlman, R. Nordberg, K. Hamrin, J. Hedman, G. Johansson, T. Bergmark, S.-E. Karlsson, I. Lindgren and B. Lindberg, Nova Acta Regiae Sot. Sci. Upsal. 20 (1967). [5] C.S. Fadley and S.A.L. BergstrGm, Phys. Letters 35A (1971) 375. [6] J.W. Cooper and S.T. Manson, Phys. Rev. 177 (1969) 157. 171D.J. Kennedy and S.T. Manson, Phys. Rev. A5 (1972) 221; A.F. Starace, S.T. Manson and D.J. Kennedy, Phys. Rev. A9 (1974) 2A53; D. Dill, A.F. Starace and S.T. Manson, Phys. Rev. AI 1 (1975) 1596. 181M.O. Krause, unpublished data (1974). I91 T.A. Carlson and C.P. Anderson, Chem. Phys. Letters 10 (1971) 561. 1101H.A. Bethe and E.E. Salpeter, Quantum mechanics of one- and two-electron atoms (Academic Press, New York, 1957) pp. 308-310. 1111A. Ronn, R.H. Pratt and H.K. Tsrng, Phys. Rev., to bc submitted for publication. [I21 R.H. Pratt, A. Ron and H.K. Tseng, Rev. Mod. Phys. 45
(1973) 273. ]I31 F. Wuilleumier and M.O. Krause, Phys. Rev. A10 (1973)
[I41 1151
242. M.O. Krause, Phys. Rev. 1’77(1969) 151. J.A.R. §a !so7 and J.L. timher, i. Opt. 5%. Am. 62
[lb1
(1972) &Zb. A. Niehnus and M.W. Ruf, Z. Physik 252 (1972) 84.