Ag K-shell ionization by electron impact: New cross-section measurements between 50 and 100 keV and review of previous experimental data

Radiation Physics and Chemistry 119 (2016) 14–23

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Ag K-shell ionization by electron impact: New cross-section measurements between 50 and 100 keV and review of previous experimental data V.R. Vanin a,n, M.V. Manso Guevara b, N.L. Maidana a, M.N. Martins a, J.M. Fernández-Varea c,a a

Laboratório do Acelerador Linear, Instituto de Física, Universidade de São Paulo, Travessa R 187, Cidade Universitária, CEP: 05508-900 São Paulo, SP, Brazil Departamento de Ciências Exatas e Tecnológicas, Universidade Estadual de Santa Cruz, Campus Soane Nazaré de Andrade, Rodovia Jorge Amado km 16, CEP: 45662-900 Ilhéus, BA, Brazil c Facultat de Física (ECM and ICC), Universitat de Barcelona, Diagonal 645, E-08028 Barcelona, Spain b

H I G H L I G H T S







Thin targets employed, dispensing with electron path length and angular deflection corrections. Cross-section deduced from characteristic to bremsstrahlung radiation. Evaluation of the bremsstrahlung yield by an accurate method. Critical review of available experimental data on Ag K-shell ionization by electron impact.

art ic l e i nf o

a b s t r a c t

Article history: Received 5 June 2015 Accepted 8 September 2015 Available online 10 September 2015

We report the measurement of Ag K-shell ionization cross-section by electron impact in the range 50– 100 keV and review the experimental data found in the literature. The sample consisted in a thin film of Ag evaporated on a thin C backing. The x-ray spectra generated by electron bombardment in the São Paulo Microtron were observed with a planar HPGe detector. The ratios between characteristic and bremsstrahlung x-ray yields were transformed to ionization cross sections with the help of theoretical atomic-field bremsstrahlung cross sections. The measured cross sections are compared with existing experimental values and calculations based on the semi-relativistic distorted-wave Born approximation. According to our experiment, the ratio of Ag Kβ to Kα x-ray intensities is 0.2018(24). & 2015 Elsevier Ltd. All rights reserved.

Keywords: Atomic K shell Electron-impact ionization Distorted-wave Born approximation Ag K x-rays

1. Introduction We have recently measured the K-shell ionization cross sections of Au and Bi by electron impact from threshold to 100 keV (Fernández-Varea et al., 2014). It was the first measurement on Bi near the ionization threshold. When we compared our results for Au with those from other experiments, we found that the available data form a discrepant set, with variations that reach a factor of four, i.e. much larger than the quoted uncertainties. These incompatibilities in the experimental results are actually not restricted to Au. In a recent review, Llovet et al. (2014a) realized that the data sets of experimental inner-shell ionization cross-sections n

Corresponding author. Fax: þ55 30916640. E-mail address: [email protected] (V.R. Vanin).

http://dx.doi.org/10.1016/j.radphyschem.2015.09.005 0969-806X/& 2015 Elsevier Ltd. All rights reserved.

are discrepant for almost all elements and inner shells, and pointed out that those for Cu K and Ag K show the smallest inconsistencies. Therefore, Cu and Ag seem to be good candidates to check the accuracy of new experimental arrangements for measuring K-shell ionization. This paper describes new measurements of Ag K ionization cross sections by electron impact between 50 and 100 keV. The results were deduced from the ratio of characteristic to bremsstrahlung x-ray intensities in the photon energy spectrum and theoretical bremsstrahlung cross sections. The experimental procedure coincides with that employed in our previous article (Fernández-Varea et al., 2014) and improves upon that applied by Schneider et al. (1993) in a similar work on Ag K ionization from threshold to 65 keV; our data extend the studied energy interval up to four times the ionization threshold and have smaller uncertainties. We compare the results to the previous experimental

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values and the prediction of the distorted-wave Born approximation (DWBA) (Bote and Salvat, 2008; Llovet et al., 2014b). We made these measurements in the São Paulo Microtron electron-beam facility as part of a long-term research plan that aims to develop the knowledge of the fundamental electron interaction processes in the 10 keV to 6 MeV energy interval. We are designing our beam lines and irradiation chambers seeking to minimize interferences and reduce the uncertainties in the measurement of the quantities of interest. In this context, the present article has two purposes. On one hand, it explains the analysis method in great detail, quantifies the uncertainties and discusses possible sources of systematic errors and the interference of target backing in the measurement. On the other hand, we critically review the previous Ag K-shell electron impact ionization experiments described in the literature, which was beyond the scope of the review by Llovet et al. (2014a). However, this analysis faces limitations because some publications do not provide information that is relevant to assess the quality of the measurement, such as sample thickness, beam charge collection efficiency, and photon counting rate.

2. Absolute and relative measurement methods Although the experimentalists used many different types of electron sources, they invariably deduced the cross section from the observed characteristic K x-rays. Hence, we will need the relation between the x-ray production cross section in the beam of energy Ee , σ x (Ee ), and Nx , the number of K x-rays observed in the detector due to a radiative atomic transition in the element of atomic number Zj, with Ne being electrons impinging on the target, which has 5j atoms per unit volume:

Nx =

Cj ε (Ex ) σ x (Ee ) 4π sr

per incident photon of energy E′. The intensity per electron recorded by the detector in the energy interval (E1, E2 ) produced by bremsstrahlung is given by

IB (Zj ; E1, E2 ) =

∫E

E2

dE

1

∫0

∞

dE′ R (E, E′) B (Zj ; Ee, se; E′),

(4)

where B (Zj ; Ee, se; E′) is the average doubly differential cross section (DDCS) for the emission of a bremsstrahlung photon with energy E′ towards the solid angle subtended by the detector. Then, the total number of bremsstrahlung photons produced by the atoms in the target and observed by the detector in the energy interval (E1, E2 ) is ν

NB =

∑ Cj IB (Zj; E1, E2 ), j=1

(5)

where the factors Cj are the same given by formula (2) and we allowed for a target with ν elements. The “relative method” consists in defining some ratio between Eqs. (1) and (5) to eliminate the parameters contained in Eq. (2). This procedure also simplifies the efficiency calibration because only the relative photon detection efficiency is required. The experiments differed in the way they evaluated expression (5); see the Appendix of Fernández-Varea et al. (2014) for the discussion of various possible approximations, and Section 5.1 for our approach. The relative method is immune to many experimental problems found in the application of the absolute method because it does not need measurements of the irradiation charge, target mass, detector solid angle and dead time. However, the experimental result is affected by the uncertainty in the adopted bremsstrahlung differential cross section as well as any approximation in the evaluation of formula (5), particularly difficult when the target is not too thin and the electron beam experiences elastic collisions when passing through the sample.

(1)

with

3. Literature review

Cj = Ne 5j (d/cos α ) Ω,

(2)

where Ω is the solid angle subtended by the detector, ε (Ex ) is the intrinsic detection efficiency at energy Ex , d is the film thickness and α is the angle between the direction normal to the sample surface and the electron beam. In turn, the K-shell ionization cross section, σ K , is related to the cross section for the production of a specific Ki x-ray peak by the relation

σKxi =

15

ΓKi ω K σ K, ΓK,tot

(3)

where the atomic relaxation parameters ωK and ΓKi/ΓK,tot are, respectively, the fluorescence yield and the ratio of Ki emission rate to the total rate. These atomic data can be found in existing compilations. Most authors had recourse to an “absolute method” to determine the ionization cross section, meaning that the target atom surface density 5j d and Ω come from other experiments, and Ne is deduced from the electron beam current measured by a method that did not rely on the x-ray spectrum observed during the target irradiation. On the other hand, a few works normalize Nx by the photon intensity from the electron bremsstrahlung in the atomic field (see Section 3), in which case Cj (which comprises 5j d , Ne and Ω) is determined from the measured intensity in the continuous bremsstrahlung spectrum and its theoretical cross section — the “relative method”. Consider an electron beam with average energy Ee and dispersion se that hits a target of an element with atomic number Zj and a spectrometer with a response function R (E, E′), normalized

Ag K-shell ionization by electron impact at energies below 1 MeV has been measured many times (Schneider et al., 1993; Webster et al., 1933; Clark, 1935; Rester and Dance, 1966; Hansen and Flammersfeld, 1966; Fischer and Hoffmann, 1967; Hübner et al., 1972; Davis et al., 1972; Seif el Nasr et al., 1974; Schlenk et al., 1976; Ricz et al., 1977; Shima et al., 1981; Kiss et al., 1981; Westbrook and Quarles, 1987; Zhou et al., 2001), see Fig. 1. It suggests that the cross section attains a maximum around 100 keV, and has a quite smooth behavior above that energy owing to the increasing contribution of the transverse interaction that nearly compensates the decrease of the longitudinal term. When compared to the data compiled by Llovet et al. (2014a) (see their Fig. 7), this figure omits a few data, for reasons explained below, and values extracted from papers that gave the results exclusively in graphical form may be affected by small errors introduced by the digitalization process. Webster et al. (1933) provide the oldest Ag K ionization cross section measurement found in the literature, published in 1933. They determined the relative cross section from threshold up to 180 keV and, in their article, only plot the shape of the σ K (E ) curve, in the form of graphs without error bars. Their results were not included in Fig. 1, although they follow the observed trend, due to the difficulty in assigning reliable uncertainties. Shortly afterwards, Clark (1935) conducted the first absolute measurement at 70 keV and presented the most detailed experimental description available in the literature. The result is an average of the measurements with 10 self-supporting targets manufactured from commercial Ag foils, with non-uniform thickness, on average equal to 0.17 μm , equivalent to 187 μg/cm2, which

16

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Ag K

σK/ b

60

40

Cl35 Re66 Ha66 Da72 Hu72 Se74 Ri77 Ki81 Sh81 We87 Sc93 Zh01

20

0

10

100

1000

E / keV Fig. 1. Experimental Ag K electron-impact ionization cross sections from threshold to 1 MeV, selected from the available literature. The “key [reference]” pairs are Cl35 (Clark, 1935), Re66 (Rester and Dance, 1966), Ha66 (Hansen and Flammersfeld, 1966), Da72 (Davis et al., 1972), Hu72 (Hübner et al., 1972), Se74 (Seif el Nasr et al., 1974), Ri77 (Ricz et al., 1977), Ki81 (Kiss et al., 1981), Sh81 (Shima et al., 1981), We87 (Westbrook and Quarles, 1987), Sc93 (Schneider et al., 1993), and Zh01 (Zhou et al., 2001).

was measured by x-ray transmission through a 6–8 foils package and direct weighing. The repeated measurement of several foils circumvented the problem arising from thickness non-uniformity. Considering the experimental methods and the characteristics of the target, the author corrected all possible distortions caused by concurrent phenomena. In 1966, Rester and Dance (1966) reported measurements for electron energies ranging from 100 keV to 1 MeV, using the absolute method. Self-supporting thin and thick targets were employed. The x-rays were observed with NaI(Tl) scintillation detectors, hence the subtraction of the continuum part of the photon energy spectrum under the peak, mostly arising from bremsstrahlung, was one of the important issues in their data-reduction procedure. The authors corrected for the attenuation of x-rays, both in the air and in the irradiation-chamber's mylar spectroscopy window, but they did not take into account the increase of the electron path length caused by elastic scattering of the electrons inside the thick target, now impossible to evaluate since target thicknesses were not stated quantitatively. They also did not specify how they measured the electron beam current. The following year, Fischer and Hoffmann (1967) applied the relative method for the first time, normalizing their result to the bremsstrahlung DDCS calculated by Kirkpatrick and Wiedmann (1945). They acquired the x-ray spectrum with a thin NaI(Tl) scintillation detector and obtained a result with 50% relative uncertainty. Since other cross sections with higher precision are available, we did not bother to correct their data for the currently accepted bremsstrahlung DDCS and did not include their result in Fig. 1. Three experiments carried out between 1966 and 1974 employed electrons from radioactive sources with energy selected by beta-ray spectrometers: Hansen and Flammersfeld (1966); Hübner et al. (1972), and Seif el Nasr et al. (1974), the first one using not only electrons but also positrons from β þ decay. They have many characteristics in common: targets two or three orders of magnitude thicker than those used in electron-beam experiments; x-rays detected with NaI(Tl) scintillators; counted the number of coincidences between the x-ray and the impinging electron,

observed by a Si detector positioned under the target. The low electron current yielded by the radioactive source allowed the coincidence measurement, and the net production of x-rays was determined by the difference between coincidence spectra from a run with target and another without target. This procedure eliminates the strong bremsstrahlung part of the x-ray spectra and effectively solves the problem created by the poor resolution of the x-ray detector. Angular deflection of electrons and photon selfabsorption were taken into account by all authors. In contrast, only Hansen and Flammersfeld (1966) corrected for photoionization by bremsstrahlung, with a 4.82-mg/cm2-thick target and 100– 400 keV electrons. Hübner et al. (1972) reasonably neglected photoexcitation because their target had a mass thickness of 1.77 mg/cm2 and electrons of 114 keV. Seif el Nasr et al. (1974) used targets between 20 and 50 mg/cm2, with 490 and 670 keV electrons, and accounted only for electron angular deflection and x-ray self-absorption; these were very significant corrections difficult to evaluate and, consequently, the resulting cross sections have relative uncertainties of about 25%, greater than the expected correction from photoionization by bremsstrahlung. In 1972, Davis et al. (1972) published Ag K-shell ionization cross section for electrons between 30 and 100 keV, employing a Ge(Li) detector to acquire the x-ray spectra, placed at 90° with respect to the beam direction. Gathering the x-ray spectroscopic data with a high-resolution semiconductor detector, as done in all other works described below, solves the problem of subtracting the continuous energy component under the characteristic peaks in x-ray spectra. They corrected for photon attenuation by the mylar window of the irradiation chamber and stated that the target was thin, but did not give its thickness. A possible explanation for their apparently overestimated cross section may be the unaccounted increase in the electron path length produced by elastic scattering inside the target, which was tilted, because they placed the detector at 90° relative to the beam axis. It is worth recalling that HPGe x-ray detectors need relatively long times for pulse processing, hence we suspect that the dead time fraction was large, leading to a first-order correction in the cross section through the reduction in the number of electrons producing the x-rays effectively counted. Since the authors did not write about this correction and did not indicate the dead time fraction and the counting rate, it is impossible to find out whether this characteristic of the experiment affected their results. Anyway, dead time effects, which are more important for HPGe spectrometers than for scintillation detectors, were not mentioned either in all the subsequent works, reviewed below. Schlenk et al. (1976) and Ricz et al. (1977) had recourse to an electrostatic accelerator and a Si(Li) detector to measure σ K using films with mass thicknesses from 150 to 300 μg/cm2 that can be considered thin for the electron energies of their experiment, from 300 to 600 keV. These two papers published in successive years by the same authors report results differing by 5–12%, comparable to the relative uncertainties of 9%. Owing to the absence of a remark on these differences by the authors, we included in Fig. 1 only the results of their second publication (Ricz et al., 1977). Shima et al. (1981) stressed the requisites for precise measurements near threshold by the absolute method, which include a precise knowledge of the average energy of the electron beam. They irradiated samples with Ag mass thicknesses in the 4–6 μg/cm2 range over a 7-μg/cm2-thick C backing, hence the correction for increased electron path arising from elastic scattering inside the target was deemed negligible. After Clark (1935), this is the first work performed with an accelerator that brings back the issue of photoionization by bremsstrahlung inside the target which, although negligible in this case, may distort the results from thicker samples because photoionization cross sections are orders of magnitude larger than electron-impact ionization cross sections.

Meanwhile, Kiss et al. (1981) measured K-shell ionization cross sections for Ag, among other elements, in the 60–100 keV energy interval by the absolute method, with 12% uncertainties. Thin and thick samples were prepared to infer the increase in electron path length with target thickness. Nevertheless, in their paper there is no quantitative information on the ensuing corrections. Corrections associated to the presence of the backing are not discussed either. It can be deduced, however, that the thick targets had thicknesses of tens of mg/cm2, because they estimated the selfabsorption of Ag K x-rays to be about 15%. In 1987, Westbrook and Quarles (1987) measured σ K for many elements, bombarding targets with thicknesses in the range 35– 60 μg/cm2 over 15 μg/cm2 C backings with 100 keV electrons. They normalized the results to the bremsstrahlung yield, assuming an uncertainty of about 15% in the absolute scale of the theoretical bremsstrahlung differential cross section. They measured the x-rays emitted at 90° and positioned the sample at an angle of 45°, both angles measured with respect to the beam direction. Hence, the target was sufficiently thick to produce an appreciable angular dispersion of the electrons in the beam, with two consequences: an increase in characteristic x-ray production originated by the increased electron path length with respect to the foil thickness, and also an increase in the effective bremsstrahlung cross section, which varies rapidly with the photon emission angle relative to the initial direction of the radiating electron. However, this angular dispersion was apparently not taken into account by the authors. The comparison of their results with those from other experiments suggests that both effects led to similar relative increases in the production of characteristic and bremsstrahlung x-rays, which seem to have cancelled out when calculating the cross section by the relative method. Note that, contrary to what happens in measurements by the absolute method, detector dead-time effects do not interfere with this result, which is deduced from the ratio of characteristic to bremsstrahlung x-rays. In the early nineties, Schneider et al. (1993) measured K-shell ionization cross sections for electron energies from 30 keV up to 65 keV, normalizing the cross section with respect to the bremsstrahlung calculated values. Their aim was to compare positronand electron-impact ionization processes, with special attention to the absolute values of the respective cross sections. Relatively thick targets, in the range 100–500 μg/cm2, tilted 45° with respect to the beam direction, were used, which was an experimental necessity dictated by the low current (10 pA) of the positron beam; the paper does not identify clearly the target thickness used in the electron-impact cross section measurements. Anyway, the angular dispersion of the electrons in this relatively thick target leads to increased characteristic x-ray and bremsstrahlung production at 90° with respect to the beam, in a proportion much greater than in the experiment of Westbrook and Quarles (1987). We extracted the data depicted in Fig. 1 from Fig. 2a of Schneider et al. (1993), a log–log plot, along with uncertainties of 25% as stated in the text. More recently, Zhou et al. (2001) measured σ K near the threshold by the absolute method irradiating a target with 31 μg/cm2 of Ag deposited over a 70-mg/cm2-thick C backing, which stopped the electron beam. They corrected the contribution to the Ag K x-ray production by the electrons reflected at the backing, which amounted up to 20% of the observed intensity. Although they quote the precision in the incident electron energy to be 0.1 keV, this uncertainty was not included in the budget, which would increase the standard deviation of the cross section at 26.0 keV. Their results seem to be compatible with the other experimental values. Nevertheless, the comparison between the fitted detection efficiency with the calibration points using 241Am and 137Cs radioactive sources in their publication (Zhou et al., 2001, Fig. 1) shows systematic differences of about 10%; this

counts/channel

V.R. Vanin et al. / Radiation Physics and Chemistry 119 (2016) 14–23

10

5

10

4

17

103 10

2

10

1

10

0

0

20

40

60

80

100

E / keV Fig. 2. X-ray spectrum recorded in the 80 keV electron beam run, contracted to 0.2 keV/channel.

suggests that the authors extracted incorrectly the solid angle Ω, directly related to the cross section as given in formulas (1) and (2). Notice that the authors state that the contribution of detection efficiency to the relative uncertainty in the cross section result is 5%, half of the observed bias.

4. Experimental method 4.1. Sample and electron beam We prepared an Ag film by metal vapor deposition over a 10 − μg/cm2 − thick C backing. The latter was previously evaporated onto a glass slide, and removed afterwards by floating off in distilled water. We fixed the backings on frames also made of C, with dimensions of 15  30 mm2 and 0.7 mm thickness, covering a hole 10 mm in diameter. Although this large unsupported area renders the sample fragile, it is needed to reduce the interference of the electron-beam halo, which was also the reason for employing C as the material for the frames instead of the traditional stainless steel that makes frames much easier to handle. The thickness of the C backing was determined by the response of the vacuum chamber quartz oscillator during the film production process and verified by light transmission curves. We deduced the mass thickness of the Ag film, about 7 μg/cm2, from the intensity of the observed bremsstrahlung spectra, as will be described in Section 5.5. The beam was delivered by the electron gun of the São Paulo Microtron, operated between 50 and 100 keV, and transported through the Microtron injector, with its resonant cavities disabled, to the irradiation chamber (Vanin et al., 2011). The electron gun rms voltage dispersion was about 260 V (see Section 5.2, in particular Fig. 7). The target was located inside the cylindrical irradiation chamber perpendicular to the direction of the incident electron beam (i.e. α ¼0°). The beam spot was ellipsoidal with main axes measuring about 2 and 3 mm. The current varied in the range 100–200 nA and 20–80 min irradiation runs were made for each beam energy. 4.2. Charge measurement The beam was collected by a relatively small graphite Faraday cup and led to a current integrator. Since the semi-aperture angle observed from the target was about 2°, even our thin target scattered an appreciable fraction of the electron beam out of the cup. Hence, we measured a charge collection factor, fq,

18

V.R. Vanin et al. / Radiation Physics and Chemistry 119 (2016) 14–23

corresponding to the average of the ratio of charge collected with beam on target to that directly hitting the Faraday cup, i.e. with the target retracted from the beam. This target in/out collectedcharge ratio measurement was repeated 10 times, in 10 s shifts, for each run to provide fq and its uncertainty at the beam energy. Notice that the purpose of the collected charge was to furnish an estimate of the target density and to evaluate the yield of bremsstrahlung photons from the C backing, not from the Ag film. The bremsstrahlung in C is sufficiently small to contribute a negligible amount to the Ag K ionization cross section standard deviation, therefore we did not delve into the evaluation of the charge measurement precision. 4.3. Detector efficiency and response function The x-ray spectra generated by the target were measured with a planar HPGe detector (Ortec 1000 Series Hyperpure Germanium Low Energy Photon Spectrometer) placed at 120(2)° with respect to the direction of the incident beam, and coupled to an analog amplifier with pile-up rejection (ORTEC 572) and an ADC (analog to digital converter) model ORTEC-927 using a Gedcke–Hale live time clock based on extended dead time correction. Fig. 2 displays the spectrum acquired with the 80 keV electron beam, where the Ag Kα and Kβ x-ray peaks can be clearly seen over the bremsstrahlung continuum. The Fe K peaks from the stainless steel irradiation chamber and the Ge K escape peaks from Ag x-rays are barely visible in the log scale. We kept the counting rate in the range 3000–5000 counts/s, which gave a fraction of dead time below 7%, with standard deviation assumed as 0.5% of the livetime. The detector full-energy (FE) peak efficiency curve was established by means of radioactive sources in the position of the target, and we found an unusual energy dependence. This prompted us to undertake a complete study of the detector efficiency, and the procedure and results were published in Maidana et al. (2013). In Fig. 3 we show the experimental intrinsic efficiency values obtained with the radioactive sources and the fitted ε (E ) curve. The efficiency bump around 80 keV was assigned to an irregular Ge dead layer—thin in the central region, surrounded by an annular region with a thick dead layer. The intrinsic efficiency is then the sum of the intrinsic efficiencies of these two regions, weighted by

0.8

In what follows, we model the detection of photons from an anisotropic source, in this case the bremsstrahlung radiation, by a real detector whose efficiency is not the same for photons hitting the different points of its surface, due to both the escape of scattered photons through the sides of the detector's active volume and its internal structure (Martin and Burns, 1992; Boson et al., 2008). We choose a reference frame with origin at the center of the target, where θ and ϕ are, respectively, the polar and azimuthal angles with respect to the beam direction, and r defines a point in space at a distance r from the origin. The detector center is placed in position rd = (rd, θd, ϕd ) in this coordinate system. The detection efficiency for a photon of energy E is given by a function ϵ(E; r′) defined on each point of the detector surface A ( r′ belongs to A). ^ , assumed to point The inward normal to A is the unit vector n approximately in the photon flux direction, which is the usual experimental arrangement. When the photon flux is F (E, r) r^ , the detection rate is

τ (E ) =

∫A F (E, r)ϵ(E; r′) r^·n^ dA.

(6)

Photons from radioactive point sources have an isotropic distribution, hence F (E, r) = F0, E r^/4πr 2 for a point source with emission rate for photons of energy E equal to F0, E , placed in the target position. Therefore, the FE peak efficiency calibrated with radioactive sources is

ε FE (E ) =

τ (E ) = F0, E =

ε (E ) =

∫A ϵ(E; r′) 4π1r2 r^·n^ dA ∫Ω ϵ(E; r′) d4Ωπ

=

Ω ε (E ), 4π

(7)

1 Ω

∫Ω ϵ(E; r′) dΩ.

Notice the change of coordinates in the integral, from those of the detector to those of the target. In the experiment, the detector axis was aligned along θ = θd and ϕ = ϕd , and the radioactive source was placed far from the ^ ≈ 1 for all r^ inside Ω. In this case, when the detector, hence r^·n

0.6

ε(E )

4.4. Modeling the detection of an anisotropic source

where the detector subtends the solid angle Ω at the source and we define the so-called intrinsic efficiency as the average efficiency, i.e.

1

0.4

detector efficiency exhibits axial symmetry around the detector axis and F (E, r) varies linearly with θ and ϕ inside Ω, expression (6) can be simplified to

0.2

0

the fractions of the crystal surface area that they occupy on the crystal front-face. Refer to the published paper for the details. The detector response function R (E, E′) represents the number of events recorded at energy E for each photon of energy E′. For this function, we chose the simple step-and-tail shape already described in detail in Fernández-Varea et al. (2014) because we did the measurement runs for this experiment with the same detection system.

0

20

40

60

80

100

120

140

E / keV Fig. 3. Intrinsic efficiency of the planar HPGe detector. The symbols with error bars (one standard deviation) were obtained using activity-calibrated radioactive sources and the detector size indicated by the manufacturer. The continuous curve was calculated with Seltzer's model (Seltzer, 1981) and the procedure described by Maidana et al. (2013). The dot-dashed and dotted curves are, respectively, the intrinsic efficiencies of the inner circle with radius ρ¼ 8.0 mm and the annular region with 8.0 mm < ρ ≤ 9.3 mm .

τ (E ) ≈ F (E, rd) Ωε (E ).

(8)

Since the bremsstrahlung DDCS varies with angle and our detector presented important efficiency variations on its surface, we calculated formula (6) to check the validity of approximation (8). We have found that the differences amount to less than 0.1%. The reason why these differences are negligibly small in our case is that the bremsstrahlung intensity at θd = 120°, although varying more than 10% over the detector surface, depends almost linearly

V.R. Vanin et al. / Radiation Physics and Chemistry 119 (2016) 14–23

on the polar angle along the detector semi-aperture angle of 1.8°. Furthermore, although there is an important variation in detection efficiency with the position where the photon hits the surface, due to its peculiar dead layer structure (Maidana et al., 2013), its symmetry around the detector axis allied to the approximate linearity of the bremsstrahlung DDCS attenuates the differences between formulas (6) and (8); consequently, we adopted the approximation (8) in the analysis of this experiment. We point out that formula (6) must be evaluated whenever the source anisotropy is important along the detector surface, and always works in the sense of attenuating the angular distribution (see, for instance, Rose, 1953), an instrumental effect that is overlooked when using formula (8).

The observed spectrum y is the superposition of the characteristic x-rays, the bremsstrahlung, the natural background radiation in the laboratory and secondary detection effects. The components yi of y are numbered according to the channel number i recorded by the acquisition system. We adapt the formula for pile-up in do Nascimento et al. (2011) to the case where the photon energy calibration function is Ep = a + b i with a ≠ 0 and the counting rate is constant. In this situation, the pile-up events pi are given by

5.1. Bremsstrahlung intensity from the target The data-reduction procedure was similar to that outlined in Fernández-Varea et al. (2014) for Au and Bi cross sections. However, in the present experiment the C backing is expected to play a more prominent role because Ag has an intermediate atomic number. Hence, many equations are similar to those in the previous paper but generalized to a multi-elemental target. Neglecting the change in direction of the electrons when they cross the target, and modeling the electron beam energy dispersion by a Gaussian function with average Ee and standard deviation se , G (Ee, se; E″), the average bremsstrahlung cross section is

∫0

∞

dE″

d2 σ (Zj ; E″, E′; θ ) G (Ee, se; E″), dE dΩ

(11)

in the notation already defined with Eq. (4). Note that the small energy loss and straggling in the target are absorbed in the parameters Ee and se , respectively; electrons of 50 keV lose less than 30 eV in our target, according to the mass stopping powers recommended by ICRU (1984) and the measured mass thickness. The theoretical bremsstrahlung DDCSs were taken from tabulations based on partial-wave calculations (Seltzer and Berger, 1986; Kissel et al., 1983) (see Fernández-Varea et al., 2014 for more details). 5.2. Least-squares fit to the bremsstrahlung tip

pi = η ∑ yi − j − k yj

(9)

j

with k = Round (a/b) and η chosen to yield a pile-up spectrum p that fits the region above the bremsstrahlung tip, where this effect dominates. Fig. 4 compares the experimental spectrum recorded in the 53 keV run to that predicted by Eq. (9). The background spectrum, g, can be neglected except for the determination of the factor η. We deduce the spectrum corrected for secondary effects as

y′ = y − p − g

(10)

which we model by the bremsstrahlung from the elements in the target in the region near the tip, where the characteristic x-rays do not interfere.

counts/channel

5. Data analysis

B (Zj ; Ee, se; E′) =

4.5. Model of the energy spectrum

19

In this experiment, the target has only two atomic components, Ag and C, both with mass thicknesses of the order of 10 μg/cm2, which means that most of the observed bremsstrahlung photons come from Ag. Below, we identify the Ag film by the superscript ‘f’ and the C in the backing by ‘b’, since the formulas are valid regardless of the chemical element when Zf ⪢Zb . We determine the backing bremsstrahlung using the theoretical DDCS with Cb calculated from the estimated mass thickness and beam current, and leave Cf as an adjustable parameter in a least-squares procedure. We minimize the merit function

Q (Cf ; Ee, se ) =

∑′ i

( yf,i − Cf IB,i )2 , var (yf, i )

(12)

where

105

⎛ Δ Δ⎞ yf, i = yi′ − Cb IB ⎜ Zb; Ei − , Ei + ⎟ ⎝ 2 2⎠

104

is the experimental net spectrum in channel i assigned to the Ag film according to Eq. (5) and IB, i ≡ IB (Zf ; Ei − Δ/2, Ei + Δ/2) is the intensity of the bremsstrahlung spectrum from the Ag film, integrated over the energy interval (Ei − Δ/2, Ei + Δ/2) corresponding to channel i. The prime in the summation symbol indicates the restriction to channels in the selected region

103

(13)

10

2

10

1

(E1, E2 ) = (Ee − 2.3 keV, Ee + 1.0 keV)

10

0

10

−1

around the bremsstrahlung tip. The denominator in the right-hand side of Eq. (12) is the variance of the number of counts per channel in the net spectrum, which is well approximated by

0

20

40

60

80

E / keV Fig. 4. Photon energy spectrum in the run with 53 keV electrons, contracted to 0.2 keV/channel. The dots are the experimental spectrum while the continuous curve is the model of formula (9). The crosses indicate the laboratory background, scaled to the duration of the acquisition time.

var (yf, i ) ≈ var (yi′) ≈ yi + pi2

var (η) η2

(14)

considering that the uncertainty in the backing bremsstrahlung affects similarly all channels near the tip and the natural background is completely negligible in the energy region of the bremsstrahlung tip.

V.R. Vanin et al. / Radiation Physics and Chemistry 119 (2016) 14–23

64.9

101

64.8

100.9

Ee / keV

Ee / keV

20

64.7

100.7

64.6

64.5

100.8

100.6

0

0.1

0.2

0.3

0.4

0.5

0

0.1

0.2

101.5

102

25

80

counts/channel

counts/channel

0.6

30

100

60 40 20

20 15 10 5 0

0 −20 62.5

63

63.5

64

64.5

65

65.5

−5 98.5

66

Ee (keV)

99.5

100

100.5

101

Fig. 6. Same as Fig. 5 but for the 101 keV run. Here χ 2 = 108 with 123 d.f.

0.5 0.4 se / keV

Notice that the dependence on Ee and se in the right-hand side of Eq. (12) is embodied in IB . Since Q is quadratic in Cf , given a point (Ee, se ) we can determine the value Cf′ that minimizes Q through 2 ⎤−1 yf, i IB, i ⎡ IB, i ⎢ ∑′ ⎥ . var (yf, i ) ⎢⎣ i var (yf, i ) ⎥⎦

99

Ee (keV)

Fig. 5. Sum of squares contour plot (above) and x-ray spectra (below, energy dispersion 25 eV/channel) in the run with 65 keV electrons. The level curves were drawn at χ 2 + j2 with j = 1, 2, …, where χ 2 = 105 with 123 d.f. The points with error bars in the x-ray spectrum are the experimental data, and the continuous curve is the calculated spectrum using the fitted parameters (E^e, s^e ) found in the upper figure. The dashed and dotted curves, which are barely distinguishable, correspond to bremsstrahlung from Ag and C, respectively.

i

0.5

35

120

∑′

0.4

se / keV

se / keV

Cf′ =

0.3

0.3 0.2 0.1

(15)

With this result, we can plot the function Q (Cf′; Ee, se ) as a function of Ee and se and find the minimum of Q, hence the least-squares ^ estimates E^e , s^e , Cf , from the corresponding 2-dimensional plots. ^ ^ Moreover, Q (Cf ; Ee, s^e ) = χ 2 is approximately distributed like chisquare because most of the channels have numbers of counts much greater than 1, meaning that it is possible to approximate their probability density functions by normal distributions. Figs. 5 and 6 present two of such plots along with the comparison between the experimental spectra and those calculated with the fitted parameters. When the conditions for unbiasedness, normality, and efficiency for the non-linear least-squares estimator hold, these level curves should be concentric ellipses around the point of minimum (Eadie et al., 1971). This happens for the run at 65 keV, Fig. 5. In the 101 keV run case, Fig. 6, the level curves for j ≥ 3 become open lines as a consequence of the smaller counting statistics in that run; if this were the only spectrum to analyze it would have been

0 50

60

70

80 Ee / keV

90

100

Fig. 7. Fitted values of the beam energy dispersion parameter, s^e , as a function of ^ the fitted average electron-beam energy, Ee .

difficult to trust the fitted parameters. However, the fitted values of s^e for all runs are consistent, as can be seen in Fig. 7. The observed chi-squares varied in the range 87–161, with 123 or 124 degrees of freedom (d.f.), with a grand total of 2165 with 2103 d.f., which has a 17% probability of being exceeded. 5.3. Ag K x-ray peak areas and atomic parameters The Kα and Kβ x-rays peak areas were determined as the sum of the number of counts of the respective peak regions in the energy spectrum, corrected for counts in the continuum

V.R. Vanin et al. / Radiation Physics and Chemistry 119 (2016) 14–23

1 0.9

fq

component, represented by a second-degree polynomial in the channel number with parameters fitted to large regions of the energy spectrum extending at both sides of the group of peaks. We considered that all counts above the polynomial component of the spectra from 1 keV below the KM1 forbidden transition up to 1 keV above the K edge (24.8–26.5 keV) belonged to the Kβ group even if there were no well-defined peaks in the endpoints of this interval. The K-shell fluorescence yield was taken from Krause (1979), assuming a 2% accuracy. The partial and total K emission probabilities were taken from Scofield (1974), and its relative uncertainty was estimated as 2% by Madison et al. (1975). From the Kα and Kβ x-ray peak areas and the detector efficiency we evaluated the intensity ratio IKβ /IKα .

0.6 50

∑′ yf, i

(16)

i

with yf, i taken from Eq. (13). From Eq. (5) we see that 5B,f is the experimental estimate of the bremsstrahlung intensity from the film:

NB,f ≈ Cf IB (Z f ; E1, E2 ).

(17)

In the two preceding formulas, the prime in the summation and the energy interval (E1, E2 ) have the same meaning defined in the minimization procedure described in Section 5.2. Using the number of x-rays counted, 5x , in place of Nx in Eq. (1), and dividing the resulting expression by Eq. (17), we find

σ x (Ee ) =

5x 4π sr IB (Z f ; E1, E2 ). 5B,f ε (Ex )

(18)

Substituting this result into Eq. (3) yields σ K . 5.5. Mass thickness of the Ag film We deduced the mass thickness of the Ag film from the ratio between the number of photons observed in the (E1, E2 ) energy interval and the bremsstrahlung yield calculated using the fitted parameters and the detector response function. From Eqs. (5), (13) and (2) it can be derived that

⎛ Δ Δ⎞ yf, i = Ne 5f dΩIB ⎜ Z f ; Ei − , Ei + ⎟. ⎝ 2 2⎠

(19)

⎛N t ⎞−1 5f d = 5B,f ⎜⎜ e live 4π ∑′ IB, i ⎟⎟ , ⎝ fq tclock ⎠ i

(20)

where fq is the charge-collection factor defined in Section 4.2 and tlive/tclock is the live time to clock time ratio, evaluated according to Section 4.3. The bremsstrahlung integrals were calculated from the theoretical DDCSs with the fitted values (E^ , s^ ) and Ω was ine

e

cluded in the detector response function R employed to calculate IB, i .

6. Results Fig. 8 displays the ratio of charge collection by the Faraday cup with target to without target, fq, as a function of the beam energy,

70

80

90

100

Fig. 8. Measured charge collection factor, fq, as a function of the electron-beam energy.

measured following the procedure described in Section 4.2. This ratio increases smoothly with energy, as expected for increasing electron energy. Since fq > 0.7 for all energies and the electron angular distribution median is not much different from the mean scattering angle, it can be deduced that the average scattering angle is smaller than the semi-aperture of the Faraday cup, ∼2°. This justifies the underlying approximation that the electron path length is equal to the film thickness and the angle of bremsstrahlung emission is the detector angle with respect to the beam direction. Fig. 9 shows the Ag mass thickness deduced with the procedure presented in Section 5.5, using the values of the charge collected by the Faraday cup corrected by the charge collection factors of Fig. 8. The average value is 7.6 (9) μg/cm2, where the standard deviation is practically determined by the uncertainty in the bremsstrahlung cross section and angular distribution. We collect the K-shell ionization cross sections for Ag measured in this work in Table 1 and Fig. 10. Uncertainties about 13% were found after combining in quadrature the relative uncertainties of the K emission rates (Scofield, 1974), the fluorescence (Krause, 1979), the counting statistics ( < 3%), the efficiency (Maidana et al., 2013), the bremsstrahlung DDCS (E10% Seltzer and Berger, 1986), and ∼ 6% from the 2° measurement angle uncertainty. Although we evaluated the ionization cross sections from the Kα peaks because they are better defined than the Kβ peaks, the IKβ /IKα 10

2

Integrating both sides of the equations along the same channels selected in the least-squares procedure, the Ag atom surface density of the film can be isolated, and we get

60

Ee / keV

Mass thickness / ( μ g/cm )

5B,f =

0.8 0.7

5.4. Cross-section evaluation We determined the ionization cross section by the relative method. To this end, we chose the tip region of the spectrum as mentioned above to count the number of bremsstrahlung photons:

21

9

8

7

6

5 50

60

70

80

90

100

Ee / keV Fig. 9. Mass thickness of the Ag film as evaluated in each energy run. The horizontal lines indicate the average and its standard deviation.

22

V.R. Vanin et al. / Radiation Physics and Chemistry 119 (2016) 14–23

Table 1 Ag K-shell ionization by electron impact cross-section as a function of the electron beam energy. The third column is the Kβ/Kα ratio of peak areas, corrected for detection efficiency. The fourth column is the fraction of the observed bremsstrahlung in (E1, E2 ) produced by the C backing.

Ee (keV) 51.53(4) 52.90(4) 57.35(4) 61.22(4) 64.71(4) 68.68(4) 72.76(7) 76.41(4) 80.23(4) 84.01(5) 88.31(5) 92.05(5) 96.46(5) 100.77(6)

σ K (b)

IKβ /IKα

Bremsstrahlung backing/total (%)

44(5) 43(5) 46(5) 49(6) 51(6) 52(6) 55(7) 54(7) 56(7) 57(7) 54(7) 58(7) 58(7) 61(7)

0.204(5) 0.199(5) 0.201(5) 0.204(5) 0.204(5) 0.204(5) 0.200(3) 0.202(6) 0.204(4) 0.205(5) 0.198(6) 0.202(5) 0.200(6) 0.198(5)

6.2 6.2 6.4 6.4 6.4 5.8 5.8 5.2 5.4 5.5 5.8 5.4 5.3 5.3

7. Discussion

80

Ag K

σK / b

60

40

DWBA Cl35 Re66 Ha66 Da72 Ki81 Sh81 We87 Sc93 Zh01 This work

20

0 20

40

60

the curve calculated in the framework of the DWBA by Bote and Salvat (2008) and Llovet et al. (2014b). Our experimental results are in satisfactory agreement with the average experimental values and also with the theoretical curve.

80

100

E / keV Fig. 10. Electron-impact K-shell ionization cross section from threshold to 100 keV. The continuous curve is the DWBA calculation of Bote and Salvat (2008) taken from Llovet et al. (2014b). Our results are plotted as black closed circles; see Fig. 1 for the legend of the other symbols.

ratio of peak areas was determined in all runs. These ratios are summarized in Table 1, where the uncertainty in the detection efficiency, common to all values, was propagated to the uncertainties of IKβ /IKα . The weighted average is 0.2018(24), taking into account the statistical correlation caused by the detection efficiency. Our experimental result falls between Scofield's theoretical value, ΓKM2,3/ΓKL2,3 = 0.1964 (39) (Scofield, 1974; Madison et al., 1975), and the value 0.210(17) measured by Ertugral et al. (2007). In Fig. 10, all available experimental K-shell ionization cross section data from threshold up to 100 keV are displayed, as well as

Even if the differences between the experimental results are somewhat greater than expected from their error bars, the datasets from all articles are consistent with the DWBA when the experimental uncertainties are considered. This finding suggests that the DWBA predicts correctly the cross section and that the experimentalists underestimated their uncertainties. The literature examination reveals that the experiments were, in general, not well documented. This makes it difficult to understand why the experimental Ag K ionization cross sections are more consistent than the data set available for other elements, as found by Llovet et al. (2014a). Although the characteristics of the target element facilitate the experiment (Ag gives good targets, has a medium atomic number, with high cross section and low K x-ray self-absorption) it seems likely that the good quality of the first measurement (Clark, 1935) helped us to direct the subsequent experiments. All works analyzed in Section 3 that used the absolute method and quoted relative standard deviations ∼10% did not account for a number of experimental uncertainties, namely target non-uniformity, electron beam current measurement, and detector live time measurement, which were not mentioned in the respective papers. Experiments where the relative method was applied did not take into account the electron angular dispersion when calculating the bremsstrahlung cross section, even when thick targets were employed. The electron beam energy precision was almost never accounted for in the results, irrespective of the analysis method, although it plays a role only near threshold. These omissions likely contributed to the observed discrepancies between the experimental results. The electron beam halo and target frame material have never been mentioned in the publications reviewed in Section 3. In a direct measurement using an isolated void frame in our experimental arrangement, with a 10 mm central hole, identical to those used in our targets, we have found a frame current of about 2 × 10−4 of the beam current measured with the Faraday cup. Since the CSDA range of 100 keV electrons is smaller than the frame thickness (ICRU, 1984), the total bremsstrahlung yield from these electrons can be appreciable if a medium-Z material is employed, which led us to replace them by C frames. Since the relative method limits the precision in the crosssection to that of the bremsstrahlung cross section, about 10% (Seltzer and Berger, 1986; Kissel et al., 1983), it is inevitable to resort to the absolute method to get more precise results. This implies a precise determination of the mass thickness of the target, a difficult task with films in the μg/cm2 range, that may demand a specially designed experiment, possibly involving Rutherford Backscattering Spectrometry. The measurement of the irradiation charge requires the collection of the charge scattered by the target to the irradiation chamber walls, and the determination of the photon detection live-time. Note that neglecting the detector dead time fraction leads to an error in the cross section directly proportional to its value, according to Eqs. (1) and (2), and the errors in its determination must be taken into account in the cross-section uncertainty budget, then it is important that the experimentalists quote this quantity in their publications.

V.R. Vanin et al. / Radiation Physics and Chemistry 119 (2016) 14–23

8. Conclusions We have measured the Ag K electron-impact ionization cross section between 50 and 100 keV, relying on a procedure based on normalization to the theoretical atomic-field bremsstrahlung cross section. The overall uncertainty was about 13%, arising mainly from uncertainties in the theoretical bremsstrahlung differential cross section. Corrections for the detector response and backing bremsstrahlung contribution to the observed bremsstrahlung were considered. The present results agree with experimental values published by other authors and the predictions of the DWBA (Bote and Salvat, 2008). The literature review was of limited usefulness in the design of an improved experiment, as most of the experiments were not well documented. The absolute method can achieve a high precision in the cross section, but this demands good accuracy in the irradiation charge, a thin target with well-known thickness, evaluation of photon detector dead-time, accurate calibration of detection efficiency and, near the ionization threshold, a precise electron energy measurement. We are now developing the São Paulo Microtron experimental facilities to meet most of these requisites.

Acknowledgments We thank Dr. R.R. Lima, Dr. A.A. Malafronte, A.L. Bonini and M. Sc. W.G.P. Engel for their assistance, technical support during the Microtron operation and help with the preparation of the samples. The preparation and calibration of radioactive sources by Dr. M. Koskinas at the LMN-IPEN, CNEN-SP is greatly appreciated. The financial support given to the project by FAPESP, CAPES and CNPq is also acknowledged. J.M. Fernández-Varea thanks the financial support from the Brazilian Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (Program CAPES-PVE) as well as the Generalitat de Catalunya (Project no. 2014 SGR 846).

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