Deconvolution of Atomic Photoabsorption Spectra: a Match between Theory and Experiment

Transcription

1 Journal of the Chinese Chemical Society, 2001, 48, Deconvolution of Atomic Photoabsorption Spectra: a Match between Theory Experiment Tu-nan Chang a * ( ), Yuxiang Luo a ( ), Hok-Sum Fung b ( ) Tai-Sone Yih b ( ) a Department of Physics Astronomy, University of Southern California, Los Angeles, CA , U.S.A. b Department of Physics, National Central University, Chung-Li 320, Taiwan, R.O.C. We present in this paper an extension of a recently proposed deconvolution procedure to compare directly the theoretical experimental spectrum of a doubly excited ultra-narrow nearly symmetric resonance in atomic photoabsorption. Our discussion is based on a set of analytical relations in terms of the variations of i) the ratio between the resonance width the experimental energy resolution in the limit when = 1 ii) the column densitynl of the media in a photoabsorption experiment. I. INTRODUCTION In an attempt to estimate the width of a narrow isolated doubly excited resonance from measured spectra in the absence of an ultrahigh-energy resolution, Fang Chang 1 have recently proposed a deconvolution procedure which enables a direct extrapolation to infinite energy resolution using a set of explicit analytical relations in terms of the ratior of the resonant width the experimental energy resolution in the limit of R = = 1. This procedure applies well for a photoionization experiment when the photoion /or the photoelectron are measured directly. Unlike the photoionization, the resonant spectra in a photoabsorption experiment is determined by detecting the light attenuation through a medium. It is known that the measured cross section is significantly affected by the column densitynl of the medium the experimental monochromator (or slit) functionf. 2 The photoabsorption cross-section¾ Pa (E) at a photon energye is determined experimentally using the Beer-Lambert law, I(E) =I o (E)e nl¾pa (E) : (1) wherei o is the intensity of the incident light,i is the attenuated intensity of the transmitted light, nl is the column density. At a photon energye,i o I can be expressed in terms of the slit functionf centered ate characterized by an energy resolution, i.e., I o (E) = i o F(E 0 E; )de 0 (2) I(E) = i o F(E 0 E; )e nl ¾(E) de 0 ; (3) where ¾ is the cross section at an infinite energy resolution (i.e., at = 0). From Eqs. (1-3), the measured resonance structure in a photoabsorption experiment is represented by a convoluted spectrum in the form of ¾ Pa (E) = 1 nl ln( F(E 0 E; )e nl ¾(E0) de 0 ): (4) As expected, whennl! 0, the cross section takes the same form of the photoionization given by Eq. (3) of Ref. 1, i.e., ¾ Pa!¾ Pi, where ¾ Pi (E) = ¾(E 0 )F(E 0 E; )de 0 : (5) The slit function F may be approximated at the center by a Gaussian distributiong modified at its tail by a Lorentzian distributionl. It can be expressed by a weighted combination ofg L, 1;3 i.e., F(E; ;w g ;w`) =w g G(E; ) +w`l(e; ); (6) where the sum ofw g w` equals one. (G L are given explicitly by Eq. (4) of Ref. 1.) There is no well established general procedure to determine the weighting factorsw g w` experimentally in the absence of ultrahigh energy resolution. Based on the analytical relations discussed in the next section, we shall propose a procedure leading to the determination ofw g w`. The density effect in photoabsorption can be easily illus- Special issue for the Fourth Asian International Seminars on Atomic Molecular Physics

2 348 J. Chin. Chem. Soc., Vol. 48, No. 3, 2001 Chang et al. trated by the variation of the convoluted spectra of an isolated resonance with changingnl. For simplicity, we will limit our discussion using a Fano-type of resonance described by an asymmetry parameter q the smoothly varying background cross section¾ b, i.e., 4 ¾(E) =¾ b (q +²) 2 1 +² 2 ; (7) where the reduced energy² = (E E r )=( 1 ) is defined in 2 terms of the energye r the width of the resonance. The cross section ¾ is expected to reach its peak value¾ max = ¾ b (1 +q 2 ) a zero at energies E max =E r ( =q) E min =E r 1 ( q); (8) 2 respectively. Fig. 1 presents a number of selected convoluted photoionization spectra using Eq. (5) with R ranging from 1=10 to 1=25 a slit function F represented either by a Gaussian distributiong or a Lorentzian distribution L. These spectra correspond to a fictitious resonance derived from Eq. (7) with E r = 2:110 Ry, ¾ b = 1:0 Mb, q 2 = 2500, = 10 6 Ry. As expected, for a given ratio R, the peak cross section¾ max corresponding to the spectrum convoluted Fig. 2. Convoluted photoabsorption spectra using Eq. (4) with R ranging from 1/10 to 1/25 a column densitynl = 0:001 Mb 1. using Gaussian distribution is substantially higher than the one using Lorentzian distribution. The density effect in photoabsorption measurement is unambiguous demonstrated by the substantial reduction in peak cross sections shown in Fig. 2 when the same spectrum is convoluted using Eq. (4) with a column densitynl = 0:001Mb 1. II. PEAK CROSS SECTIONS Fig. 1. Convoluted photoionization spectra using Eq. (5) withrranging from 1/10 to 1/25. For an ultra-narrow nearly symmetric resonance, the peak cross section ¾ max is very well approximated by the cross section at E = E r, i.e., ¾ max ¼ ¾(E r ), since, from Eq. (8), the energy corresponding to the peak cross section, i.e., E max, equals approximately the resonant energye r as the difference E max E r is substantially smaller than the resonance width, i.e., asq 2 À 1 q. In a photoionization experiment, ¾max Pi» = ¾(E 0 )F(E 0 E r ; )de 0 : (9) Although Eq. (5) is not in general integrable for an arbi-

3 Deconvolution of Atomic Photoabsorption Spectra J. Chin. Chem. Soc., Vol. 48, No. 3, trary energy E, Eq. (9) can be integrated analytically. For a Lorentzian distribution, ¾ L max =¾ b (1+q 2 R)=(1 +R) (10) for a Gaussian distribution, ¾ G max =¾ b (1+¼ 1=2 (q 2 1))Re R2 F c (R); (11) where F c (x) = 1 (2= p ¼) R x 0 e y2 dy is the complementary error function. For a nearly symmetric ultra-narrow resonance, such as the fictitious resonance shown in Figs. 1 2, our calculation shows that the approximate¾ max ate =E r, derived from Eqs. (10) (11), are within 0.05% of the exact peak cross sections determined from the numerically calculated spectra using Eq. (5). In a photoabsorption experiment, the peak cross section¾ max can also be approximated simi- larly from Eq. (4) as ¾ pa» max = 1 nl ln( F(E 0 E r ; )e nl¾(e0) de 0 ): (12) Fig. 3 shows that the approximate peak cross sections ¾ max, represented by the nearly straight lines obtained from Eq. (12) are in close agreement with the exact peak cross sections at R = 1 25 ; 1 20 ; for a number of column 10 densitiesnl derived directly from the numerically calculated convoluted spectra using Eqs. (4) (5) for photoabsorption photoionization (i.e., whennl = 0), respectively. In general, Eq. (12) can not be integrated analytically due to the exponential terme nl¾. However, whene nl¾ is exped into an infinite series, each individual term becomes integrable ¾ max can be expressed in terms of a polynomial inr, i.e., where ¾ max!¾ Pa (E =E r ) =q 2 ¾ b X(R;nl); (13) X(R;nl) = X i=1( 1) i+1» i (½)R i (14) ½is a parameter given by ½ =nlq 2 ¾ b : (15) For a Lorentzian distribution, the first few expansion coefficients are» 1 = ½ ½ ½ ½4 (16)» 2 = ½ ½ ½ ½4 (17)» 3 = ½ ½ ½ ½4 (18)» 4 = 1 ½+½ ½ ½4 (19) Fig. 3. Comparison between ¾ max (nearly straight lines) obtained from Eq. (12) the exact peak cross sections at R = 25 1; 20 1; for a number of column densitiesnl derived directly from the numerically calculated convoluted spectra using Eqs. (4) (5) for photoabsorption photoionization (i.e., whennl = 0), respectively.» 5 = ½ ½ ½ ½4 ; (20) for a Gaussian distribution» 1 = p ¼(1 1 4 ½ ½ ½ ½4 ) (21)» 2 = 2 ¼( 1 2 ½ 1 4 ½ ½ ½4 + ) (22)

4 350 J. Chin. Chem. Soc., Vol. 48, No. 3, 2001 Chang et al.» 3 = p ¼(1 7 4 ½ + ( ¼ 3 )½2 ( ¼ 4 )½3 +( ¼ 8 )½4 ) (23)» 4 = 4 3 (1 ½) ¼(½ 2½2 + ( ¼ 4 )½3 ( ¼ 4 )½4 + ) (24)» 5 = p ¼( ½ + ( ¼)½2 ( ¼ 4 )½3 +( ¼ 24 +¼2 5 )½4 ): (25) Under a typical experimental condition, even at a fairly low column densitynl, the parameter½may be close to or greater than unity asq 2 À 1. Consequently, the peak cross section ¾ max can be estimated approximately from Eq. (13) only if R is very small the number of contributing» i terms is limited. More discussion will be given in section III. III. PROPOSED PROCEDURES A. Determination ofw` w g in Photoionization When (or, R 1), for a nearly symmetric resonance withq 2 À 1, the observed¾ max =w`¾ L max+w g ¾ G max can be expressed approximately according to Eqs. (10) (11) as ¾ max! (w` +w g ¼ 1=2 )(q 2 ¾ b )R; (26) or,¾ max varies linearly as functions of 1=. As a result, (w` +w g ¼ 1=2 )(q 2 ¾ b ) =S 1 ; (27) wheres 1 is the slope determined experimentally by a plot of ¾ max vs. 1= according to Eq. (12). In addition, according to Eq. (17) of Ref. 1, (1:3282w` + 1:9646w g )(q 2 ¾ b ) =S 2 ; (28) wheres 2 is also a slope determined experimentally by a procedure detailed in Ref. 1. Eqs. (27) (28), together with w` +w g = 1, offer an unambiguous procedure to determine the weighting factorsw` w g in a photoionization experiment. Fig. 4. Comparison of the approximate peak cross sections,¾ E=Er, obtained from the analytical expression Eq. (13), the numerically calculated¾ E=Er from Eq. (12) the exact ¾ max from Eq. (4). B. Determination ofq 2 ¾ b Fig. 4 shows that the approximate peak cross sections, ¾ E=Er, obtained from the analytical expression Eq. (13) remain in close agreement with the numerically calculated ¾ E=Er from Eq. (12) the exact¾ max from Eq. (4) for a value of½ =nlq 2 ¾ b as large as Clearly, as½ increases to a value of 2.5, Eq. (13) is no longer applicable. Since (not R) is an experimentally measured variable, we shall now work with an alternative polynomial, in stead of the polynomialx(r;nl), for¾ max, i.e., ¾ max = (q 2 ¾ b )Y ( ;nl); (29) wherey( ;nl) takes the form Y( ;nl) = X i=1( ) i 1» i (½) i (30) = 1= : (31) Our proposed procedure starts with a best fit of the measured ¾ max at a number of energy resolutions to an expression ¾ max ( ;nl) = X ¹=1 ¹ (nl) ¹: (32) By comparing Eq. (32) to Eq. (29), the fitted coefficients ¹ for a givennl is independent of. In addition, the ratio

5 Deconvolution of Atomic Photoabsorption Spectra J. Chin. Chem. Soc., Vol. 48, No. 3, of two fitted values of ¹ at two different column densities equals to the ratio of two» ¹, i.e., ¹ (nl) ¹ (n 0 l 0 ) =» ¹(½)» ¹ (½ 0 ) : (33) Since ¾ max is a slowly varying function of, a number of experimentally determined ratios between several pairs of 1 obtained at different column densities should be sufficient to fit adequately a value ofq 2 ¾ b. With a best fitted q 2 ¾ b, the resonant width can be determined readily from Eq. (13) or (29). IV. RESULTS AND DISCUSSION Fig. 5 presents the variation of the simulated photoabsorption peak cross sections ¾ max as a function of 1= at several column densities. It is derived from the convoluted spectra numerically calculated from Eq. (4) for a fictitious resonance with a width = Ry, q 2 = 400, ¾ b = 0:015Mb. Following the procedure outlined in section III.B, for eachnl, a parameter 1 is first least-square fitted from Eq. (29). Second, from Eq. (30), we obtain a best fitted value of 5:42Mb 5:54Mb forq 2 ¾ b using the Lorentzian Gaussian distribution, respectively. Finally, Eq. (13) leads to a width of 5: Ry (Lorentzian) 5: Ry (Gaussian). The 10% error introduced in this application is not unexpected due to values of½ which exceed unity for some of the column densities. The deconvolution procedure proposed in this paper works best when q 2 À 1. It clearly posts a difficult experimental challenge as it also requires simultaneously a small parameter ½ =nlq 2 ¾ b when a small column densitynl may adversely reduce the signal to noise ratio in a photoabsorption experiment. In spite of this difficulty, the procedure proposed above offers a realistic possibility to take advantage of the density effect, in an attempt to determine experimentally the width of an ultra-narrow nearly symmetric atomic resonance which can not be measured directly otherwise. ACKNOWLEDGMENTS This work is partially supported by the National Science Council of Taiwan under contract NSC M (TSY) by NSF Grant No. PHY (TNC). Received April 30, Key Words Atomic photoabsorption spectra; Deconvolution. REFERENCES Fig. 5. Variation of simulated photoabsorption peak cross sections¾ max (from Eq. (4)) as a function of 1= at column densities ranging from 0:07Mb 1 to 0:23Mb 1. (Only Lorentzian data are shown.) 1. Fang, T. K.; Chang, T. N. Phys. Rev. 1998, A57, Hudson, R. D.; Carter, V. L. J. Op. Soc. Am. 1968, ; Chan, W. F.; Cooper, G.; Brion, C. E. Phys. Rev. 1991, A44, Schulz, K.; Kaindl, G.; Domke, M.; Bozek, J. D.; Heimann, P. A.; Schlachter, A. S.; Rost, J. M. Phys. Rev. Lett. 1996, 77, Fano, U. Phys. Rev. 1961, 124, 1866.