Calculated electron impact cross sections for the K-shellionization of Fe, Co, Mn, Ti, Zn, Nb, and Mo atoms using theDM formalismH. Deutscha, K. Beckerb, B. Gstirc, and T.D. Ma¨rkc,*aInstitut fu¨r Physik, Ernst-Moritz-Arndt Universita¨t, D-17487 Greifswald, GermanybDepartment of Physics and Engineering Physics, Stevens Institute of Technology, Hoboken, NJ 07030, USAcInstitut fu¨r Ionenphysik, Leopold-Franzens Universita¨t, A-6020 Innsbruck, AustriaReceived 25 June 2001; accepted 24 July 2001AbstractWe used the Deutsch-Ma¨rk (DM) formalism to calculate atomic K-shell electron impact ionization cross sections for theelements Fe, Co, Mn, Ti, Zn, Nb, and Mo. The calculated K-shell ionization cross sections are compared with recentlymeasured K-shell ionization cross sections. Good to satisfactory agreement was found for all atoms with the exception of Ti.Moreover, when compared to other available K-shell ionization cross sections for these atoms, calculated using othertheoretical methods and semiempirical formulae, the predictions of the DM formalism achieve a level of agreement withexperimental data that is as good or better than the predictions from the other methods. (Int J Mass Spectrom 213 (2002) 5–8)© 2002 Elsevier Science B.V. PACS Numbers: 34.80, 52.20Keywords: Electron ionization; K-shell ionization; Cross sections; Metal atoms1. IntroductionCross sections for the removal of electrons fromthe innermost shell of atoms by electron impact areneeded in many fields such as atomic physics, plasmaphysics, materials and surface science, and radiationchemistry [1]. Recently, new measurements of K-shell ionization cross sections were reported for theelements Sc and V [2] and Cr, Ni, and Cu [3]. Allthese metal atoms are of particular importance infusion plasmas [4]. The authors of references [2] and[3] also compared the experimentally determinedcross sections with a variety of predictions fromvarious theoretical models and semiempirical formu-lae. An et al. [2] also mentioned the Deutsch-Ma¨rk(DM) formalism (see [4]) as a method to calculateK-shell ionization cross sections and said that thepredictions based on the DM formalism are inferior tothe results from many other methods. We would liketo point out that this statement is erroneous. As shownrecently and discussed in detail [4], the Deutsch-Ma¨rk(DM) formalism can be used to calculate K-shellionization cross sections for these five atoms, andgood to satisfactory agreement was found between theDM calculations and the measured K-shell ionizationcross sections for these five atoms.In this letter, we extend the calculations of K-shell* Corresponding author. E-mail: [email protected]/02/$20.00 © 2002 Elsevier Science B.V. All rights reservedPII S1387-3806(01)00515-2International Journal of Mass Spectrometry 213 (2002) 5–8 www.elsevier.com/locate/ijmsionization cross sections using the DM formalism tothe atoms Fe, Co, Mn, Ti, Zn, Nb, and Mo, for whichexperimental data are also available [5–10]. The DMformalism expresses the K-shell ionization cross sec-tion1sas1s⫽ g1s共r1s兲21sf共U兲 F共U兲 (1)where (r1s)2is the radius of maximum radial densityof the atomic 1s-shell taken from the tables ofDesclaux [11];1s, the number of electrons in the1s-shell, is equal to 2; and g1sis a weighting factor(see references [12–14] for further details). The en-ergy dependence of the K-shell ionization cross sec-tion is given by the product of the two functions f(U)and F(U). Here U refers to the reduced impact energy,U ⫽ E/E1s, where E is the energy of the incidentelectron and E1srefers to the binding energy of 1selectrons. The function f(U) is similar (but not iden-tical) to the energy dependence first given by Gryz-inski [15] and has the formf共U兲 ⫽ d 共1/U兲关共U ⫺ 1兲/共U ⫹ 1兲兴a兵b ⫹ c关1⫺ 共2U兲⫺1兴 ln 关2.7 ⫹ 共U ⫺ 1兲1/2兴其, (2)where the parameters a, b, c, and d have the followingvalues: a ⫽ 1.06, b ⫽ 0.23, c ⫽ 1.00, and d ⫽ 1.1.The function F(U) is a relativistic correction factor,which is again similar (but not identical) to the oneintroduced by Gryzinski [15] and has the form, in ourcase, ofF共U兲 ⫽ R共U兲关1 ⫹ 2 共U兲1/4/共 J兲2兴,with J ⫽ (mec2)/E1s, and with mebeing the electronmass. The function R(U) is given byR共U兲 ⫽ 共1 ⫹ 2J兲/共U ⫹ 2J兲 ⫻ 关共U ⫹ J兲/共1 ⫹ J兲兴2兵关共1 ⫹ U兲共U ⫹ 2J兲共1 ⫹ J兲2兴/关J2共1 ⫹ 2J兲 ⫹ U共U ⫹ 2J兲共1 ⫹ J兲2兴其3/2. (3)We used the DM formula of Eq. (1) to calculate theK-shell ionization cross sections for the atoms Fe, Co,Mn, Ti, Zn, Nb, and Mo, and we compare thecalculation to recently measured experimental crosssections and to predictions from the empirical formulaof Casnati et al. [16] and the theoretical calculationsof Luo and Joy [17]. The advantage of the DMformalism lies in the simplicity of its use comparedwith other theoretical models. However, the DMformula is superior to purely empirical formulae as itcontains quantum mechanically calculated informa-tion, and all parameters in Eq. (1) have a real physicalmeaning.Fig. 1 shows the experimentally determined K-shell ionization cross section data for Fe of Luo et al.[6] in comparison with the calculated cross sections ofLuo and Joy [17], Casnati et al. [16], and with thepresent result. It is apparent that all three calculatedcross sections represent the experimental data quitewell over the entire range of impact energies depictedin the figure. In the case of Co (Fig. 2), the DMcalculation lies somewhat below the other two calcu-lated cross section curves and the measured crosssection [5] for energies from threshold to about twicethe threshold energy but agrees with the experimentwithin the quoted margin of error of the measureddata. For higher energies, the DM calculation is closerto the experiment than the other two calculations. TheDM calculation provides the best description of themeasured K-shell ionization cross section for Mn [6]over the entire range of impact energies (Fig. 3). Fig.Fig. 1. K-shell ionization cross sectionfor Fe as a function of thereduced impact energy u ⫽ E/Eks. The experimental data are fromLuo et al. [6]. The solid line represents the present DM calculation,the dashed line denotes the prediction of the empirical formula ofCasnati et al. [16], and the filled circles are the calculated crosssections of Luo and Joy [17].6 H. Deutsch et
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