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Universal empirical fit to L-shell X-ray production cross sections in ionization by protons
Nuclear Instruments and Methods in Physics Research B xxx (2017) xxx–xxx
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
Universal empirical fit to L-shell X-ray production cross sections in ionization by protons G. Lapicki a,⇑, J. Miranda b,c a
Department of Physics, East Carolina University, Grenville, NC 27858, USA Instituto de Física, Universidad Nacional Autónoma de México, A.P. 20-364, Álvaro Obregón, Ciudad de México 01000, Mexico c Centro de Ciencias de la Atmósfera, Universidad Nacional Autónoma de México, Coyoacán Ciudad de México 04510, Mexico b
a r t i c l e
i n f o
Article history: Received 3 July 2017 Received in revised form 26 July 2017 Accepted 26 July 2017 Available online xxxx Keywords: L-shell X-ray production Protons Universal empirical fit
a b s t r a c t A compilation published in 2014, with a recent 2017 update, contains 5730 experimental total L-shell X-ray production cross sections (XRPCS). The database covers an energy range from 10 keV to 1 GeV, and targets from 18Ar to 95Am. With only two adjustable parameters, universal fit to these data normalized to XRPCS calculated at proton velocity v1 equal to the electron velocity in the L-shell v2L, is obtained in terms of a single ratio of v1/v2L. This fit reproduces 97% of the compiled XRPCS to within a factor of 2. Ó 2017 Elsevier B.V. All rights reserved.
1. Introduction The relevance of X-ray production cross sections (XRPCS) and the related ionization cross sections (ICS) in many research areas has been described at length and analyzed in detail by Miranda and Lapicki [1]. Particle Induced X-ray Emission (PIXE) strongly requires trustworthy databases for XRPCS. X-ray emission by ion impact is a relevant phenomenon in many other areas. In particular, reliable L-shell cross sections in ionization by protons are necessary for better input in track structure calculations [2,3], simulation of delta rays in lithography [4], radiation dosimetry with 1 MeV to 2.5 MeV protons where L-shell ionization dominates X-ray production in 28Ni [5], charged particle dose and track studies in DNA [6], as well as inter-atomic Auger processes during charged-particle penetration through solid targets that involve L-shell vacancies [7]. Table 1 in a compilation of Miranda and Lapicki [1], with their recent 2017 errata and update [8], contains more than 5730 total L-shell XRPCS for target elements ranging from 18Ar to 95Am; this compilation has 105% more total L-shell XRPCS than the sum of these cross sections from previous databases of Hardt and Watson [9], Sokhi and Crumpton [10], and Orlic
⇑ Corresponding author. E-mail address: [email protected] (G. Lapicki).
et al. [11]. The data search in Miranda and Lapicki [1] terminated in December 2012. With an exception of XRPCS from Miranda et al. [12] that were already in Table 1, the subsequent update [8] listed total L-shell XRPCS from Ferree [13], Simpson [14], Toten [15], Harrigan [16], Joseph et al. [17], Zhou et al. [18], Bertol et al. [19], Batyrbekov et al. [20], Mohan et al. [21], Bertol et al. [22], and Zhou et al. [23]. Fig. 1 displays a distribution of total L-shell XRPCS by protons versus target element published until April 2017. With all the new experimental points, the cumulative number of these cross sections is expected to saturate at 5850 as predicted by a similar logistic curve shown in Fig. 2 of [1] for all data compiled therein. Eliminating 38 XRPCS for 18Ar because they are affected by multiple ionization, the goal of the present work is to develop a formula that fits the 5699 XRPCS for 10 keV to 1 GeV protons and across the periodic table of target elements with a single scaling variable and a minimum number of adjustable parameters. 2. Universal empirical fit to total L-shell XRPCS by protons Theories are proposed and improved to predict required cross sections. In order to check if theories are accurate across the periodic table of target elements and a large range of projectile energies – equally comprehensive experimental databases are essential, and a universal fit for them is desired. This work focuses
http://dx.doi.org/10.1016/j.nimb.2017.07.029 0168-583X/Ó 2017 Elsevier B.V. All rights reserved.
Please cite this article in press as: G. Lapicki, J. Miranda, Universal empirical fit to L-shell X-ray production cross sections in ionization by protons, Nucl. Instr. Meth. B (2017), http://dx.doi.org/10.1016/j.nimb.2017.07.029
2
G. Lapicki, J. Miranda / Nuclear Instruments and Methods in Physics Research B xxx (2017) xxx–xxx
Fig. 1. Total L-shell X-ray production cross sections per target element [1,8].
on total L-shell XRPCS rLX. A universal fit to all experimental cross sections should be in terms of a variable by which they scale with a minimum of adjustable parameters. For each target element, the compiled XRPCS tend to follow a single curve when plotted versus the ratio of the proton velocity v1 to the electron orbital velocity v2L = Z2L/n, where Z2L = Z2 4.15 is the screened nuclear charge in the L shell with the principal quantum number n = 2:
v 1 6:33E1=2 1 ¼ ; v 2L ðZ 2L =2Þ
ð1Þ
where E1 is the proton laboratory energy in MeV. Furthermore, the compiled cross sections reach a maximum when v1 = v2L. This indicates that, with
v logðv 1 =v 2L Þ;
ð2Þ
for each Z2L the compiled data can be fitted as
rLX ¼ Fðv Þ; where F(v) r
rLX ¼ r
ð3Þ max LX
and F(0) = r
max LX ðZ 2L Þ exp½ð1
max LX .
þ a1 Z 2L Þv 2 þ a2 v 7 ;
ð4Þ
satisfies Eq. (3) subject to the constraints F(v) rmax and F(0) LX = rmax LX . For small v, the chosen exponential function with just two terms in its argument proportional to v2 and v7 has the observed shape of an inverse parabola 1 (1 + a1Z2L)v2 near v = 0. Striving for the universality of our fit formula, Eq. (4) was constructed so it applies to all target elements with the least dependence on Z2. This is why this dependence is only linear with Z2L in front of v2 and none in front of v7. Overall, the best fit to 5699 XRPCS obtains with a1=0.00484 and a2 = 0.005 in Eq. (4). rLXmax(Z2L) in this equation can be calculated by any inner-shell ionization theory because practically for all of them rLX peaks
Fig. 2. ECUSAR [27] cross sections for each Z @ v = 0 shown as points are fitted over three intervals of Z.
around v = 0. These theories include: Plane Wave Born Approximation (PWBA) [24], Binary Encounter Approximation (BEA) [25], Energy-loss Coulomb-deflection Perturbed-Stationary-State Relativistic (ECPSSR) [26], Energy-loss Coulomb-deflection United Separated Atom Relativistic (ECUSAR) [27], Local Plasma Approximation (LPA) [28], Classical trajectory Monte Carlo or Atomic Orbital Coupled-State (AOCC) [29], and Continuum Distorted Wave (CDW) [30]. rmax LX (Z2L) for our fit is calculated using the ECUSAR theory [27]. Instead of a table, displayed in Fig. 2 as points, the ECUSAR cross sections for each Z at v = 0 are cross sections fitted over three intervals of Z as shown therein. XRPCS data compiled in [1,8], fitted with Eq. (4), and their ratios to that universal empirical fit are displayed for selected elements in Figs. 3–7.
Please cite this article in press as: G. Lapicki, J. Miranda, Universal empirical fit to L-shell X-ray production cross sections in ionization by protons, Nucl. Instr. Meth. B (2017), http://dx.doi.org/10.1016/j.nimb.2017.07.029
G. Lapicki, J. Miranda / Nuclear Instruments and Methods in Physics Research B xxx (2017) xxx–xxx
3
Fig. 3. XRPCS for Cu (Z = 29) and Br (Z = 35) as data compiled in [1,8], fitted with Eq. (4) and the ratios of these experimental data to this empirical fit.
Fig. 4. XRPCS for Ag (Z = 47) and Sn (Z = 50) as data compiled in [1,8], fitted with Eq. (4) and the ratios of these experimental data to this empirical fit.
Please cite this article in press as: G. Lapicki, J. Miranda, Universal empirical fit to L-shell X-ray production cross sections in ionization by protons, Nucl. Instr. Meth. B (2017), http://dx.doi.org/10.1016/j.nimb.2017.07.029
4
G. Lapicki, J. Miranda / Nuclear Instruments and Methods in Physics Research B xxx (2017) xxx–xxx
Fig. 5. XRPCS for Tb (Z = 65) and Au (Z = 79) as data compiled in [1,8], fitted with Eq. (4) and the ratios of these experimental data to this empirical fit.
Fig. 6. XRPCS for Bi (Z = 83) and U (Z = 92) as data compiled in [1,8], fitted with Eq. (4) and the ratios of these experimental data to this empirical fit.
Please cite this article in press as: G. Lapicki, J. Miranda, Universal empirical fit to L-shell X-ray production cross sections in ionization by protons, Nucl. Instr. Meth. B (2017), http://dx.doi.org/10.1016/j.nimb.2017.07.029
G. Lapicki, J. Miranda / Nuclear Instruments and Methods in Physics Research B xxx (2017) xxx–xxx
5
Fig. 7. XRPCS for Np (Z = 94) and Am (Z = 95) as data compiled in [1,8], fitted with Eq. (4) and the ratios of these experimental data to this empirical fit.
Fig. 8. With a standard deviation of 32%, the mean value of the ratio of all 5699 data compiled in [1,8], to the universal empirical fit according to Eq. (4) is 1.01. 97% of these ratios are within a factor of 2 of the ideal ratio of 1.0.
Please cite this article in press as: G. Lapicki, J. Miranda, Universal empirical fit to L-shell X-ray production cross sections in ionization by protons, Nucl. Instr. Meth. B (2017), http://dx.doi.org/10.1016/j.nimb.2017.07.029
6
G. Lapicki, J. Miranda / Nuclear Instruments and Methods in Physics Research B xxx (2017) xxx–xxx
In Fig. 6a, the outlying point at v1/v2L = 0.33 is the XRPCS for Bi bombarded by 4 MeV protons from Joseph et al. [17]. This datum is 67 times smaller than fitted with Eq. (4); as such it is an exceptional and the largest outlier among all 5699 compiled data. 3. Conclusions As shown in Fig. 8, the ratios of XRPCS compiled for all elements are in excellent agreement with the universal fit to these data as given in Eq. (4). Only 3.4% of Data/Fit ratios differ from 1 by more than a factor of 2. Only 0.7% Data/Fit ratios differ from 1 by more than a factor of 4. With the standard deviation (SD) equal to 32%, a Gaussian curve gives 5.6% for a factor of 2, while our sharper fit of Eq. (4) has only 3.4% of 5699 ratios outside of that factor. This fit is indeed universal because it: covers a wide segment of target elements from the periodic table 24 Z2 95 and spans five orders of magnitude range of proton energies with 0.026 MeV E11 GeV; with SD = 32%, reproduces 97% of all 5699 XRPCS within a factor of 2; while a Gaussian distribution would have to assume an unrealistic SD = 14% for typical XRPCS to be equally successful.
References [1] J. Miranda, G. Lapicki, Experimental cross sections for L-shell x-ray production and ionization by protons, At. Data Nucl. Data Tables 100 (2014) 651–780. [2] H.G. Paretzke, Physical events of heavy ion interactions with matter, Adv. Space. Res. 6 (1986) 67–73. [3] G. Bäckström, M.E. Galassi, N. Tilly, A. Ahnesjö, J.M. Fernández-Varea, Track structure of protons and other light ions in liquid water: applications of the L IonTrack code at the nanometer scale, Med. Phys. 40 (2013) 064101–106411. [4] C.N.B. Udalagama, A.A. Bettiol, F. Watt, A Monte Carlo study of the extent of proximity effects in e-beam and p-beam writing of PMMA, Nucl. Instrum. Meth. B 260 (2007) 384–389. [5] H.W. Choi, H.J. Woo, J.H. Park, G.H. Lee, J.D. Kim, J.K. Kim, W. Hong, C.M. Eum, Low energy ion beam monitoring system by dosimetry film and particle induced X-ray, Nucl. Instrum. Meth. B 261 (2007) 102–105. [6] H. Lekadir, I. Abbas, C. Champion, J. Hanssen, Total cross sections for ionizing processes induced by proton impact on molecules of biological interest: a Classical Trajectory Monte Carlo approach, Nucl. Instrum. Meth. B 267 (2009) 1011–1014. [7] N.A. Medvedev, A.E. Volkov, B. Rethfeld, N.S. Shchlebanov, Effect of interatomic Auger processes on relaxation of electronic vacancies at deep levels of highly ionized atoms in swift heavy ion tracks, Nucl. Instrum. Meth. B 268 (2009) 2870–2873. [8] J. Miranda, G. Lapicki, Errata and update to Experimental cross sections for Lshell x-ray production and ionization by protons, At. Data Nucl. Data Tables (2017), http://dx.doi.org/10.1016/j.adt.2017.05.004.
[9] T.L. Hardt, R.L. Watson, Cross sections for L-Shell X-Ray and auger-electron production by heavy ions, At. Data Nucl. Data Tables 17 (1976) 107–125. [10] R.S. Sokhi, D. Crumpton, Experimental L-shell x-ray production and ionization cross sections for proton impact, At. Data Nucl. Data Tables 30 (1984) 49–124. [11] I. Orlic, C.H. Sow, S.M. Tang, Experimental L-Shell X-Ray production and ionization cross sections for proton impact, At. Data Nucl. Data Tables 56 (1994) 159–210. [12] J. Miranda, G. Murillo, B. Méndez, J. López-Monroy, R.V. Díaz, J. Aspiazu, P. Villaseñor, J.C. Pineda, J. Reyes-Herrera, Measurement of L X-ray production cross sections by impact of proton beams on Hf, Ir, and Tl, Nucl. Instrum. Meth. B 316 (2013) 113–122. [13] D.V. Ferree, K-shell Ionization Cross Sections for Ti, Fe, Co, Ni, Ge, Rb, Zr, Pd, Ag, Cd, and Sb, and L-shell Ionization Cross Sections for Ba and Tl for Protons from 1.0 to 6.0 MeV (M. Sc. thesis), University of Tennesee, Knoxville, 1972. [14] A.E. Simpson, Elemental Analysis by X-ray Methods and Some Medical Applications (Ph.D. thesis), University of Birmingham, Birmingham, 1977. [15] A.D. Toten, Electron-Ion Time of Flight Coincidence Measurements of K-K Electron Capture, Cross Sections, for Nitrogen, Methane, Ethylene, Ethane, Carbon Dioxide and Argon (L-K) Targets (Ph.D. thesis), North Texas State University, Denton, 1986. [16] M.F. Harrigan, Proton and Helium Induced L Subshell Ionization Cross Section Measurements of Medium to High Z Atoms (Ph.D. thesis), University of Melbourne, 1986. [17] D. Joseph, S.V.S. Nageshwara Rao, S. Kailas, Measurement of L X-ray production cross-sections of Au, Ho, Bi and K-X-ray cross sections of Nb, Sn, Sb by using protons of energy 4 MeV, Mapana J. Sci. 12 (2013) 1–8. [18] X. Zhou, Y. Zhao, R. Cheng, Y. Wang, Y. Lei, X. Wang, Y. Sun, K and L-shell X-ray production cross sections for 50–250 keV proton impact on elements with Z = 26–30, Nucl. Instrum. Meth. B 299 (2013) 61–67. [19] A.P.L. Bertol, J. Trincavelli, R. Hinrichs, M.A.Z. Vasconcellos, L-shell X-ray production cross sections induced by protons and alpha-particles in the 0.7– 2.0 MeV/amu range for Ru and Ag, Nucl. Instrum. Meth. B 318 (2014) 19–22. [20] E. Batyrbekov, I. Gorlachev, I. Ivanov, A. Platov, K-, L- and M-shell x-ray production cross sections by 1–1.3 MeV protons, Nucl. Instrum. Meth. B 325 (2014) 84–88. [21] H. Mohan, A.K. Jain, M. Kaur, P.S. Singh, S. Sharma, Cross section for induced L X-ray emission by protons of energy < 400 keV, Nucl. Instrum. Meth. B 332 (2014) 103–105. [22] A.P.L. Bertol, R. Hinrichs, M.A.Z. Vasconcellos, Proton induced L1, L2, L3-subshell X-ray production cross sections of Hf and Au, Nucl. Instrum. Meth. B 363 (2015) 28–32. [23] X. Zhou, R. Cheng, Y. Wang, Y. Lei, Y. Chen, X. Chen, Z. Yongtao, G. Xiao, L x-ray production in ionization of 60 Nd by 100–250 keV protons, Nucl. Instrum. Meth. B (2017), http://dx.doi.org/10.1016/j.nimb.2017.03.102. [24] E. Merzbacher, H.W. Lewis, Handbuch der Physik, Springer, Berlin, 1958, p. 166. [25] J.H. McGuire, P. Richard, Procedure for computing cross sections for single and multiple ionization of atoms in the binary-encounter approximation by the impact of heavy charged particles, Phys. Rev. A 8 (1973) 1374–1384. [26] W. Brandt, G. Lapicki, Energy-loss effect in inner-shell Coulomb ionization by heavy charged particles, Phys. Rev. A 23 (1981) 1717–1729. [27] G. Lapicki, The status of theoretical L-subshell ionization cross sections for protons, Nucl. Instrum. Meth. B 189 (2002) 8–20. [28] C.C. Montanari, J.N. Miraglia, N.R. Arista, Dynamics of solid inner-shell electrons in collisions with bare and dressed swift ions, Phys. Rev. A 66 (2002) 042902–042907. [29] G. Schiwietz, K. Czerski, M. Roth, F. Staufenbiel, P.L. Grande, Femtosecond dynamics – snapshots of the early ion-track evolution, Nucl. Instrum. Meth. B 225 (2004) 4–26. [30] D. Belkic, Quantum Theory of High-Energy Ion-Atom Collisions, CRC Press, Boca Raton, FL USA; Taylor and Francis, London, UK, 2009.
Please cite this article in press as: G. Lapicki, J. Miranda, Universal empirical fit to L-shell X-ray production cross sections in ionization by protons, Nucl. Instr. Meth. B (2017), http://dx.doi.org/10.1016/j.nimb.2017.07.029
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
Universal empirical fit to L-shell X-ray production cross sections in ionization by protons G. Lapicki a,⇑, J. Miranda b,c a
Department of Physics, East Carolina University, Grenville, NC 27858, USA Instituto de Física, Universidad Nacional Autónoma de México, A.P. 20-364, Álvaro Obregón, Ciudad de México 01000, Mexico c Centro de Ciencias de la Atmósfera, Universidad Nacional Autónoma de México, Coyoacán Ciudad de México 04510, Mexico b
a r t i c l e
i n f o
Article history: Received 3 July 2017 Received in revised form 26 July 2017 Accepted 26 July 2017 Available online xxxx Keywords: L-shell X-ray production Protons Universal empirical fit
a b s t r a c t A compilation published in 2014, with a recent 2017 update, contains 5730 experimental total L-shell X-ray production cross sections (XRPCS). The database covers an energy range from 10 keV to 1 GeV, and targets from 18Ar to 95Am. With only two adjustable parameters, universal fit to these data normalized to XRPCS calculated at proton velocity v1 equal to the electron velocity in the L-shell v2L, is obtained in terms of a single ratio of v1/v2L. This fit reproduces 97% of the compiled XRPCS to within a factor of 2. Ó 2017 Elsevier B.V. All rights reserved.
1. Introduction The relevance of X-ray production cross sections (XRPCS) and the related ionization cross sections (ICS) in many research areas has been described at length and analyzed in detail by Miranda and Lapicki [1]. Particle Induced X-ray Emission (PIXE) strongly requires trustworthy databases for XRPCS. X-ray emission by ion impact is a relevant phenomenon in many other areas. In particular, reliable L-shell cross sections in ionization by protons are necessary for better input in track structure calculations [2,3], simulation of delta rays in lithography [4], radiation dosimetry with 1 MeV to 2.5 MeV protons where L-shell ionization dominates X-ray production in 28Ni [5], charged particle dose and track studies in DNA [6], as well as inter-atomic Auger processes during charged-particle penetration through solid targets that involve L-shell vacancies [7]. Table 1 in a compilation of Miranda and Lapicki [1], with their recent 2017 errata and update [8], contains more than 5730 total L-shell XRPCS for target elements ranging from 18Ar to 95Am; this compilation has 105% more total L-shell XRPCS than the sum of these cross sections from previous databases of Hardt and Watson [9], Sokhi and Crumpton [10], and Orlic
⇑ Corresponding author. E-mail address: [email protected] (G. Lapicki).
et al. [11]. The data search in Miranda and Lapicki [1] terminated in December 2012. With an exception of XRPCS from Miranda et al. [12] that were already in Table 1, the subsequent update [8] listed total L-shell XRPCS from Ferree [13], Simpson [14], Toten [15], Harrigan [16], Joseph et al. [17], Zhou et al. [18], Bertol et al. [19], Batyrbekov et al. [20], Mohan et al. [21], Bertol et al. [22], and Zhou et al. [23]. Fig. 1 displays a distribution of total L-shell XRPCS by protons versus target element published until April 2017. With all the new experimental points, the cumulative number of these cross sections is expected to saturate at 5850 as predicted by a similar logistic curve shown in Fig. 2 of [1] for all data compiled therein. Eliminating 38 XRPCS for 18Ar because they are affected by multiple ionization, the goal of the present work is to develop a formula that fits the 5699 XRPCS for 10 keV to 1 GeV protons and across the periodic table of target elements with a single scaling variable and a minimum number of adjustable parameters. 2. Universal empirical fit to total L-shell XRPCS by protons Theories are proposed and improved to predict required cross sections. In order to check if theories are accurate across the periodic table of target elements and a large range of projectile energies – equally comprehensive experimental databases are essential, and a universal fit for them is desired. This work focuses
http://dx.doi.org/10.1016/j.nimb.2017.07.029 0168-583X/Ó 2017 Elsevier B.V. All rights reserved.
Please cite this article in press as: G. Lapicki, J. Miranda, Universal empirical fit to L-shell X-ray production cross sections in ionization by protons, Nucl. Instr. Meth. B (2017), http://dx.doi.org/10.1016/j.nimb.2017.07.029
2
G. Lapicki, J. Miranda / Nuclear Instruments and Methods in Physics Research B xxx (2017) xxx–xxx
Fig. 1. Total L-shell X-ray production cross sections per target element [1,8].
on total L-shell XRPCS rLX. A universal fit to all experimental cross sections should be in terms of a variable by which they scale with a minimum of adjustable parameters. For each target element, the compiled XRPCS tend to follow a single curve when plotted versus the ratio of the proton velocity v1 to the electron orbital velocity v2L = Z2L/n, where Z2L = Z2 4.15 is the screened nuclear charge in the L shell with the principal quantum number n = 2:
v 1 6:33E1=2 1 ¼ ; v 2L ðZ 2L =2Þ
ð1Þ
where E1 is the proton laboratory energy in MeV. Furthermore, the compiled cross sections reach a maximum when v1 = v2L. This indicates that, with
v logðv 1 =v 2L Þ;
ð2Þ
for each Z2L the compiled data can be fitted as
rLX ¼ Fðv Þ; where F(v) r
rLX ¼ r
ð3Þ max LX
and F(0) = r
max LX ðZ 2L Þ exp½ð1
max LX .
þ a1 Z 2L Þv 2 þ a2 v 7 ;
ð4Þ
satisfies Eq. (3) subject to the constraints F(v) rmax and F(0) LX = rmax LX . For small v, the chosen exponential function with just two terms in its argument proportional to v2 and v7 has the observed shape of an inverse parabola 1 (1 + a1Z2L)v2 near v = 0. Striving for the universality of our fit formula, Eq. (4) was constructed so it applies to all target elements with the least dependence on Z2. This is why this dependence is only linear with Z2L in front of v2 and none in front of v7. Overall, the best fit to 5699 XRPCS obtains with a1=0.00484 and a2 = 0.005 in Eq. (4). rLXmax(Z2L) in this equation can be calculated by any inner-shell ionization theory because practically for all of them rLX peaks
Fig. 2. ECUSAR [27] cross sections for each Z @ v = 0 shown as points are fitted over three intervals of Z.
around v = 0. These theories include: Plane Wave Born Approximation (PWBA) [24], Binary Encounter Approximation (BEA) [25], Energy-loss Coulomb-deflection Perturbed-Stationary-State Relativistic (ECPSSR) [26], Energy-loss Coulomb-deflection United Separated Atom Relativistic (ECUSAR) [27], Local Plasma Approximation (LPA) [28], Classical trajectory Monte Carlo or Atomic Orbital Coupled-State (AOCC) [29], and Continuum Distorted Wave (CDW) [30]. rmax LX (Z2L) for our fit is calculated using the ECUSAR theory [27]. Instead of a table, displayed in Fig. 2 as points, the ECUSAR cross sections for each Z at v = 0 are cross sections fitted over three intervals of Z as shown therein. XRPCS data compiled in [1,8], fitted with Eq. (4), and their ratios to that universal empirical fit are displayed for selected elements in Figs. 3–7.
Please cite this article in press as: G. Lapicki, J. Miranda, Universal empirical fit to L-shell X-ray production cross sections in ionization by protons, Nucl. Instr. Meth. B (2017), http://dx.doi.org/10.1016/j.nimb.2017.07.029
G. Lapicki, J. Miranda / Nuclear Instruments and Methods in Physics Research B xxx (2017) xxx–xxx
3
Fig. 3. XRPCS for Cu (Z = 29) and Br (Z = 35) as data compiled in [1,8], fitted with Eq. (4) and the ratios of these experimental data to this empirical fit.
Fig. 4. XRPCS for Ag (Z = 47) and Sn (Z = 50) as data compiled in [1,8], fitted with Eq. (4) and the ratios of these experimental data to this empirical fit.
Please cite this article in press as: G. Lapicki, J. Miranda, Universal empirical fit to L-shell X-ray production cross sections in ionization by protons, Nucl. Instr. Meth. B (2017), http://dx.doi.org/10.1016/j.nimb.2017.07.029
4
G. Lapicki, J. Miranda / Nuclear Instruments and Methods in Physics Research B xxx (2017) xxx–xxx
Fig. 5. XRPCS for Tb (Z = 65) and Au (Z = 79) as data compiled in [1,8], fitted with Eq. (4) and the ratios of these experimental data to this empirical fit.
Fig. 6. XRPCS for Bi (Z = 83) and U (Z = 92) as data compiled in [1,8], fitted with Eq. (4) and the ratios of these experimental data to this empirical fit.
Please cite this article in press as: G. Lapicki, J. Miranda, Universal empirical fit to L-shell X-ray production cross sections in ionization by protons, Nucl. Instr. Meth. B (2017), http://dx.doi.org/10.1016/j.nimb.2017.07.029
G. Lapicki, J. Miranda / Nuclear Instruments and Methods in Physics Research B xxx (2017) xxx–xxx
5
Fig. 7. XRPCS for Np (Z = 94) and Am (Z = 95) as data compiled in [1,8], fitted with Eq. (4) and the ratios of these experimental data to this empirical fit.
Fig. 8. With a standard deviation of 32%, the mean value of the ratio of all 5699 data compiled in [1,8], to the universal empirical fit according to Eq. (4) is 1.01. 97% of these ratios are within a factor of 2 of the ideal ratio of 1.0.
Please cite this article in press as: G. Lapicki, J. Miranda, Universal empirical fit to L-shell X-ray production cross sections in ionization by protons, Nucl. Instr. Meth. B (2017), http://dx.doi.org/10.1016/j.nimb.2017.07.029
6
G. Lapicki, J. Miranda / Nuclear Instruments and Methods in Physics Research B xxx (2017) xxx–xxx
In Fig. 6a, the outlying point at v1/v2L = 0.33 is the XRPCS for Bi bombarded by 4 MeV protons from Joseph et al. [17]. This datum is 67 times smaller than fitted with Eq. (4); as such it is an exceptional and the largest outlier among all 5699 compiled data. 3. Conclusions As shown in Fig. 8, the ratios of XRPCS compiled for all elements are in excellent agreement with the universal fit to these data as given in Eq. (4). Only 3.4% of Data/Fit ratios differ from 1 by more than a factor of 2. Only 0.7% Data/Fit ratios differ from 1 by more than a factor of 4. With the standard deviation (SD) equal to 32%, a Gaussian curve gives 5.6% for a factor of 2, while our sharper fit of Eq. (4) has only 3.4% of 5699 ratios outside of that factor. This fit is indeed universal because it: covers a wide segment of target elements from the periodic table 24 Z2 95 and spans five orders of magnitude range of proton energies with 0.026 MeV E11 GeV; with SD = 32%, reproduces 97% of all 5699 XRPCS within a factor of 2; while a Gaussian distribution would have to assume an unrealistic SD = 14% for typical XRPCS to be equally successful.
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Please cite this article in press as: G. Lapicki, J. Miranda, Universal empirical fit to L-shell X-ray production cross sections in ionization by protons, Nucl. Instr. Meth. B (2017), http://dx.doi.org/10.1016/j.nimb.2017.07.029