- Email: [email protected]
Tandem-method for measurement of destruction cross-sections of neutral projectiles at intermediate and high velocities
ARTICLE IN PRESS
Nuclear Instruments and Methods in Physics Research A 577 (2007) 349–352 www.elsevier.com/locate/nima
Tandem-method for measurement of destruction cross-sections of neutral projectiles at intermediate and high velocities M.M. Sant’Anna, B.F. Magnani, R.S. Correˆa, L.F.S. Coelho Instituto de Fı´sica, Universidade Federal do Rio de Janeiro, Cx. Postal 68528, Rio de Janeiro 21941-972, Brazil Available online 21 February 2007
Abstract We have recently presented destruction cross-section data for negative ions obtained with a technique that uses the gas stripper of a tandem accelerator as the collision target. In this work, we develop an extension of that technique to measure destruction cross-sections for neutral projectiles, important parameters to estimate neutral beam attenuation in Heavy Ion Fusion applications. Measurements for the H þ N2 collision system are used to exemplify and discuss the capabilities and limitations of the proposed experimental method. r 2007 Elsevier B.V. All rights reserved. PACS: 34.50.Fa Keywords: Heavy Ion Fusion; Collisions; Destruction cross-sections; Charge-changing cross-sections; Neutral projectiles
1. Introduction The use of high-velocity neutral beams has been considered for Heavy Ion Fusion (HIF) possible scenarios (e.g. Refs. [1–6]). Neutralization at some stage after final focusing can, in principle, overcome problems due to the huge space charge of intense ion beams. Once neutralized, it is desirable that these high-velocity projectiles do not lose electrons. However, interaction of beam particles with the fusion-chamber environment can lead to beam ionization by several mechanisms [7]. Beam electron stripping in collisions with background gas is one of these mechanisms and destruction cross-sections, sd0 (also called stripping or total electron-loss cross-sections) are parameters needed to quantify the fraction of the initially neutral beam that can reach the fusion target [1,2]. There is a considerable amount of experimental data for these cross-sections at low projectile velocities, v, (compared to the Bohr velocity vB , equivalent to E ¼ 25 keV=u). On the other hand, high-velocity data are very scarce in the literature (e.g. Refs. [8,9]). The reason for this asymmetry is basically the difficulty of preparing a good-quality neutral beam at Corresponding author.
E-mail address: [email protected] (M.M. Sant’Anna). 0168-9002/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2007.02.028
high velocities that can be subsequently used as an incoming beam in a collision experiment. Producing neutral beams at vovB is relatively simple, as positive projectiles present huge electron capture cross-sections at these velocities. Therefore, a neutral beam can be efficiently created through neutralization of positive ions going through a gaseous chamber. A rather distinct picture arises at intermediate and high velocities, since capture cross-sections decrease very sharply with projectile velocity. Thus, alternative strategies must be used to measure sd0 . In the traditional growth-rate method [8,10] (hereafter called traditional method), a negative-ion beam is prepared and crosses a gas target with controlled variable pressure. Final-charge-state fractions, F j , are measured as a function of the gas-cell line density (the product of the gas density and effective length of the cell), x. Qualitatively, the idea is to prepare the neutral beam inside the collision target itself (through a first collision) and let it collide, at least once more, with other atoms in the gas cell. The difficulty here is that, for usual experimental setups, a high-velocity anion beam is more difficult to obtain than the corresponding equivelocity positive beam. Ranges of all the available data [8,11,12], to our knowledge, for the N2 target and any neutral projectile are shown in Fig. 1. The highestvelocity available datum for multielectron projectiles is that for 1 MeV/u Li0þ þ N2 [11,12].
ARTICLE IN PRESS M.M. Sant’Anna et al. / Nuclear Instruments and Methods in Physics Research A 577 (2007) 349–352
350 3
0
2
ERange(MeV)
+
N
X +N2 → X or Σj=1 X
N
a
j+
Tandem limit (1.7MeV): UFRJ - Rio de Janeiro Ne He B
1
K
Kr
Ba
Sr
U
Al Fe
O
Xe Ag Te I
C
0
b
Tandem limit: Tandar - Buenos Aires
20
ERange(MeV)
Tandem limit: ANU - Canberra
15 H
10 Li
5
Tandem limit: UFRJ - Rio de Janeiro
0 0
20
40
60
80
100
ZP Fig. 1. Cross-section energy ranges for electron loss of neutral projectiles impinging on N2 , as a function of the atomic number of the projectile. Dotted lines represent examples of energy limits of accelerators: (a) the UFRJ Tandem limit is compared to ranges of available data; (b) highlights the lack of data at high velocities and show limits of other accelerators.
Our group at UFRJ has developed a variation of the growth-rate method that uses the gas stripper of a tandem accelerator as the collision target to determine the anion destruction cross-section, sd1 [13]. The line density is calibrated against the pressure at the exit of the accelerator column. One high-velocity well-known sd1 value is needed for normalization. If a H iþ beam is used for the calibration, compiled values for the six possible chargeexchange cross-sections sij (i; j ¼ 1; 0; 1) [9] can be used (sij is the charge-changing cross-section from i to j) and the measurement of F 0 ðxÞ improves the calibration [13]. The inversion of this procedure, i.e., obtaining individual crosssections from fitting parameters is not straightforward, particularly for many electron projectiles, and is discussed in the next section. We have studied several anionic-projectile collision systems (e.g. Ref. [14]) including halogen anions incident on N2 [15], motivated in the latter case by a proposal to produce neutral beams for HIF from the photoionization of halogen anions [1]. In this paper, we present and discuss an extension of that method (hereafter called Tandem method) to measure destruction cross-sections sd0 for any neutral atomic projectile, provided that it has a stable anion. 2. Tandem method versus traditional method The Tandem method has advantages and disadvantages when compared to the traditional method. The main
advantage is the possibility of use of standard tandem accelerators to measure sd0 at energies up to the maximum terminal voltage times the electron charge. At UFRJ 1.7 MeV is our energy limit. However, there are open laboratories currently operating tandems (with gaseous strippers) up to 20 MV (Fig. 1). A second crucial characteristic of the Tandem method is that relatively small analyzing magnets can be used, even for collision of very heavy projectiles at the highest available energies. This happens because after the collision at the center of the accelerator, the anionic beam is decelerated down to its injection energy (typically tens of keV) and can be easily bent to be detected by a Faraday cup. The outgoing neutral projectile is simply detected (through secondary electron emission) at the central exit of the magnet. The main disadvantage of our method is that negative and neutral detection necessarily have distinct collection efficiencies and beam transmission due to different focusing conditions by the accelerator itself. Thus, we do have beam transmission and detection efficiencies different from 100% [13]. However, these unknown values are not necessary for the determination of the destruction cross-sections sd . We do not need to know the beam transmission, it is sufficient to guarantee that (for each terminal voltage) the transmission is constant during a single measurement (while we vary the pressure at the gas stripper). The determination of absolute values for sd is possible, in the high-velocity limit, because these parameters appear in the arguments of exponentials (see Section 2.1). Positive outgoing projectiles can, in principle, be detected in a separate run but that would require a strong magnet, since cations are accelerated in the second accelerator stage. Under these conditions we cannot determine the cross-sections for electron loss to a particular final charge state. A quantitative assessment of the proposed method is presented in the rest of this section. For a gas target with a finite thickness, the emerging beam fractions, F i , satisfy the Allison’s equations [10] N X dF i þ F i sdi ¼ sji F j dx jai
with sdi ¼
N X
sij .
(1)
jai
The set of coupled differential equations above can be transformed into a set of linear algebraic equations through the substitution F i ðxÞ ¼ F 1i þ
N X
Aik expðsk xÞ.
(2)
k¼1
This system can in principle be solved with the constants F 1i , Aik and sk expressed as functions of the cross-sections sji . However, the expressions are cumbersome and fluctuations in experimental data for F i ðxÞ make data reduction not straightforward in the determination of the sji from fitted F 1i , Aik and sk , even for projectiles with only two electrons [10,16–18]. Important simplifications apply in the high-velocity regime.
ARTICLE IN PRESS M.M. Sant’Anna et al. / Nuclear Instruments and Methods in Physics Research A 577 (2007) 349–352
2.1. The high-velocity limit
351
15
(3)
Σj= i+ 1 j
2 MeV
-16
10
d
F 1 ¼ expðsd1 xÞ
3
i
Li + N2 -> Li
σi (10 cm2)
The projectile with a definite charge state can be destroyed by either losing or capturing electrons, with P respective total cross-sections given by sli ¼ N s and ij j4i P c s . For high velocities, s is negligible and sci ¼ N i joi ij sdi sli . The first two projectile final charge states have particularly simple solutions:
5
and s10 ½expðsd0 xÞ expðsd1 xÞ. sd1 sd0
Neutral projectile
(4) 0
-1
0
1
2
i Fig. 2. Destruction cross-sections as a function of initial projectile charge state, i, for Liiþ þ N2 at 2 MeV. Obtained from growth curves of Ref. [12]: squares, traditional growth-rate analysis; circles, simplified Tandem growth-rate analysis.
σ-11+ σ-10 i
-15
10
H +N2 → H
σ10
j
σ01
2
For an arbitrary F i with vbvB , sk ¼ sdk and F 1i ¼ diN , where N is the maximum number of electrons removable from the anion. In Table 1 we give the parameters for Eq. (2) that give the high-velocity solutions for the first four F i with an incident anionic beam. In the traditional method F i ðxÞ is measured as a function of x (through pressure measurement) for several i and relations between coefficients of Table 1 allow the determination of individual sij and also decrease the uncertainties in the fitted sdi . In the Tandem method F i ðxÞ curves are measured apart from a constant prefactor and only the parameters in the exponentials can be extracted. Fitting uncertainties are larger for i40, but for i ¼ 0 the only effect of different detection efficiencies is that s10 (see Eq. (4)) cannot be determined in the Tandem method. As an example, we have used published growth curves and cross-sections for Liiþ þ N2 at 2 MeV [11,12] to determine sdi using relations of Table 1 (traditional method analysis) and considering A PikN1as free adjusting parameters, with AiN ¼ diN j¼1 Aij to satisfy F i ð0Þ ¼ d1i (Tandem method analysis). Results are shown in Fig. 2 and appreciable differences are not seen until i ¼ 2.
σij ( cm )
F0 ¼
-16
10
Normalization point -17
10
σ0-1 σ-11
1
2.2. Breakdown at low v: the H þ N 2 example
10
v (atomic units)
The collision system H þ N2 is used as a test case for the experimental method proposed in this work with measurements performed at UFRJ and shown in Fig. 3. The line-
Fig. 3. Charge-changing cross-sections for Hiþ þ N2 as a function of the collision velocity: full symbols, present results; lines, data compilation of Ref. [9]; diamonds, Ref. [17].
Table 1 Parameters for the four first charge-state fractions F i ðxÞ in Eq. (2), with F 1 ð0Þ ¼ 1, vbvB and koN k
sk
Aik i ¼ 1
i¼0 s10 D10
1
sd1
1
0
sd0
0
1
sd1
0
s10 D10 0
2
sd2
0
0
If i ¼ No3, ANN in table is substituted by 1
PN1
k¼1 ANk
i¼1
i¼2
s10 s01 s11 D10 D11 D11 s10 s01 D10 D01 s10 s01 s11 þ D11 D01 D11 0
s10 s01 s12 s11 s12 s10 s02 s12 þ þ þ D10 D11 D12 D11 D12 D10 D12 D12 s10 s01 s12 s10 s02 D10 D01 D02 D10 D02 s10 s01 s12 s11 s12 D11 D01 D12 D11 D12 s10 s01 s12 s11 s12 s10 s02 s12 þ þ D12 D02 D12 D12 D12 D02 D12 D12
(Dmn ¼ sdm sdn ).
ARTICLE IN PRESS 352
M.M. Sant’Anna et al. / Nuclear Instruments and Methods in Physics Research A 577 (2007) 349–352
density, x, was calibrated using sd1 of Ref. [17] at 400 keV. The sd0 error bars are estimated as the difference between data compilation results [9] and our measured sd0 . We deliberately pushed the method too far on the low-velocity side to see its breakdown on the intermediate-to-low velocity region. For v44vB measured and compilation values are proportional and agree within the 20% dispersion of the compilation [9]. Below v ¼ 2vB the error bars clearly increase. This is consistent with the velocity dependence of the capture cross-section s10 , which plunges above v ¼ 2vB . For heavier projectiles estimating the lower velocity limit of the Tandem method from capture cross-section values may not be trivial, because of lack of experimental data for capture cross-sections. However, in principle, the limitation should be less stringent than for H for three reasons: (i) for equivelocity neutral projectiles sd0 should be larger for projectiles with more (and more loosely bound) electrons, (ii) for equivelocity singly-charged projectiles s10 should be smaller for projectiles with more electrons, because of Pauli blocking of occupied projectile orbitals [19,20], and (iii) multiple capture cross-sections drop even faster than single capture. It is important to note that the beam emittance at the entrance of the stripping canal increases with decreasing beam energy. However, previous studies for the Li þ N2 collision system [16] showed that even at velocities as low as v ¼ 0:34vB , the measurements are only overestimated by about 20% due to elastic scattering. Electron capture by the projectile increases fast with decreasing beam energy and causes the breakdown of the method before elastic scattering becomes a relevant factor.
the accelerator, is used as the target gas cell for the experiment. We have previously applied the Tandem method to determine anion destruction cross-sections for several collision systems at intermediate projectile velocities. The extension of the method for neutral-projectile destruction is shown to be feasible. However, the suitable velocity region is narrower for neutral projectiles than for anions. This shortcoming of the method is discussed in detail for Hiþ þ N2 . Acknowledgments This work was partially supported by the Brazilian agencies MCT/CNPq (CT-Energ), FINEP, CAPES, FAPERJ, and FUJB. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
3. Conclusion A method for measuring destruction cross-sections for neutral projectiles at intermediate-to-high and high velocities is presented and discussed. It has the potential to extend widely the range of experimental data currently available. Our method is similar to the traditional growthrate method with an anionic initial beam. The main difference is that the gas stripper of a standard Tandem electrostatic accelerator, inside the high-voltage terminal of
[15] [16] [17] [18]
[19] [20]
L.R. Grisham, Nucl. Instr. and Meth. A 464 (2001) 315. L.R. Grisham, Fusion Sci. Technol. 43 (2003) 191. D.R. Welch, et al., Phys. Plasmas 5 (2002) 2344. W.M. Sharp, et al., Nucl. Fusion 44 (2004) S221. P.K. Roy, et al., Phys. Plasmas 11 (2004) 2890. P.K. Roy, et al., Phys. Rev. Lett. 95 (2005) 234801. B. Sharkov, Plasma Phys. Control Fusion 43 (2001) A229. R.C. Dehmel, H.K. Chau, H.H. Fleischmann, At. Data 5 (1973) 231. Y. Nakai, T. Shirai, T. Tabata, R. Ito, At. Data Nucl. Data Tables 37 (1987) 69. S.K. Allison, Rev. Mod. Phys. 30 (1958) 1137. L.R. Grisham, et al., Rev. Sci. Instr. 53 (1982) 281. B.M. Johnson, Nucl. Instr. and Meth. 214 (1983) 73. J.C. Acquadro, et al., Nucl. Instr. and Meth. B 171 (2000) 373. H. Luna, et al., Phys. Rev. A 63 (2001) 022705; H. Luna, et al., Phys. Rev. A 63 (2001) 052716; F. Zappa, et al., Phys. Rev. A 64 (2001) 032701; F. Zappa, et al., Phys. Rev. A 67 (2003) 012702; F. Zappa, et al., Phys. Rev. A 69 (2004) 012703. M.M. Sant’Anna, et al., Plasma Phys. Control. Fusion 46 (2004) 1009. E. Horsdal Pedersen, P. Hvelplund, J. Phys. B 6 (1973) 1277. J. Heinemeier, P. Hvelplund, F.R. Simpson, J. Phys. B 9 (1976) 2669. T.J. Kvale, J.S. Allen, X.D. Fang, A. Sen, R. Matuoliene, Phys. Rev. A 51 (1995) 1351; T.J. Kvale, J.S. Allen, A. Sen, X.D. Fang, R. Matuoliene, Phys. Rev. A 51 (1995) 1360. M.M. Sant’Anna, et al., J. Phys. B 33 (2000) 353. T. Kirchner, M. Horbatsch, Phys. Rev. A 63 (2001) 062718.
Nuclear Instruments and Methods in Physics Research A 577 (2007) 349–352 www.elsevier.com/locate/nima
Tandem-method for measurement of destruction cross-sections of neutral projectiles at intermediate and high velocities M.M. Sant’Anna, B.F. Magnani, R.S. Correˆa, L.F.S. Coelho Instituto de Fı´sica, Universidade Federal do Rio de Janeiro, Cx. Postal 68528, Rio de Janeiro 21941-972, Brazil Available online 21 February 2007
Abstract We have recently presented destruction cross-section data for negative ions obtained with a technique that uses the gas stripper of a tandem accelerator as the collision target. In this work, we develop an extension of that technique to measure destruction cross-sections for neutral projectiles, important parameters to estimate neutral beam attenuation in Heavy Ion Fusion applications. Measurements for the H þ N2 collision system are used to exemplify and discuss the capabilities and limitations of the proposed experimental method. r 2007 Elsevier B.V. All rights reserved. PACS: 34.50.Fa Keywords: Heavy Ion Fusion; Collisions; Destruction cross-sections; Charge-changing cross-sections; Neutral projectiles
1. Introduction The use of high-velocity neutral beams has been considered for Heavy Ion Fusion (HIF) possible scenarios (e.g. Refs. [1–6]). Neutralization at some stage after final focusing can, in principle, overcome problems due to the huge space charge of intense ion beams. Once neutralized, it is desirable that these high-velocity projectiles do not lose electrons. However, interaction of beam particles with the fusion-chamber environment can lead to beam ionization by several mechanisms [7]. Beam electron stripping in collisions with background gas is one of these mechanisms and destruction cross-sections, sd0 (also called stripping or total electron-loss cross-sections) are parameters needed to quantify the fraction of the initially neutral beam that can reach the fusion target [1,2]. There is a considerable amount of experimental data for these cross-sections at low projectile velocities, v, (compared to the Bohr velocity vB , equivalent to E ¼ 25 keV=u). On the other hand, high-velocity data are very scarce in the literature (e.g. Refs. [8,9]). The reason for this asymmetry is basically the difficulty of preparing a good-quality neutral beam at Corresponding author.
E-mail address: [email protected] (M.M. Sant’Anna). 0168-9002/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2007.02.028
high velocities that can be subsequently used as an incoming beam in a collision experiment. Producing neutral beams at vovB is relatively simple, as positive projectiles present huge electron capture cross-sections at these velocities. Therefore, a neutral beam can be efficiently created through neutralization of positive ions going through a gaseous chamber. A rather distinct picture arises at intermediate and high velocities, since capture cross-sections decrease very sharply with projectile velocity. Thus, alternative strategies must be used to measure sd0 . In the traditional growth-rate method [8,10] (hereafter called traditional method), a negative-ion beam is prepared and crosses a gas target with controlled variable pressure. Final-charge-state fractions, F j , are measured as a function of the gas-cell line density (the product of the gas density and effective length of the cell), x. Qualitatively, the idea is to prepare the neutral beam inside the collision target itself (through a first collision) and let it collide, at least once more, with other atoms in the gas cell. The difficulty here is that, for usual experimental setups, a high-velocity anion beam is more difficult to obtain than the corresponding equivelocity positive beam. Ranges of all the available data [8,11,12], to our knowledge, for the N2 target and any neutral projectile are shown in Fig. 1. The highestvelocity available datum for multielectron projectiles is that for 1 MeV/u Li0þ þ N2 [11,12].
ARTICLE IN PRESS M.M. Sant’Anna et al. / Nuclear Instruments and Methods in Physics Research A 577 (2007) 349–352
350 3
0
2
ERange(MeV)
+
N
X +N2 → X or Σj=1 X
N
a
j+
Tandem limit (1.7MeV): UFRJ - Rio de Janeiro Ne He B
1
K
Kr
Ba
Sr
U
Al Fe
O
Xe Ag Te I
C
0
b
Tandem limit: Tandar - Buenos Aires
20
ERange(MeV)
Tandem limit: ANU - Canberra
15 H
10 Li
5
Tandem limit: UFRJ - Rio de Janeiro
0 0
20
40
60
80
100
ZP Fig. 1. Cross-section energy ranges for electron loss of neutral projectiles impinging on N2 , as a function of the atomic number of the projectile. Dotted lines represent examples of energy limits of accelerators: (a) the UFRJ Tandem limit is compared to ranges of available data; (b) highlights the lack of data at high velocities and show limits of other accelerators.
Our group at UFRJ has developed a variation of the growth-rate method that uses the gas stripper of a tandem accelerator as the collision target to determine the anion destruction cross-section, sd1 [13]. The line density is calibrated against the pressure at the exit of the accelerator column. One high-velocity well-known sd1 value is needed for normalization. If a H iþ beam is used for the calibration, compiled values for the six possible chargeexchange cross-sections sij (i; j ¼ 1; 0; 1) [9] can be used (sij is the charge-changing cross-section from i to j) and the measurement of F 0 ðxÞ improves the calibration [13]. The inversion of this procedure, i.e., obtaining individual crosssections from fitting parameters is not straightforward, particularly for many electron projectiles, and is discussed in the next section. We have studied several anionic-projectile collision systems (e.g. Ref. [14]) including halogen anions incident on N2 [15], motivated in the latter case by a proposal to produce neutral beams for HIF from the photoionization of halogen anions [1]. In this paper, we present and discuss an extension of that method (hereafter called Tandem method) to measure destruction cross-sections sd0 for any neutral atomic projectile, provided that it has a stable anion. 2. Tandem method versus traditional method The Tandem method has advantages and disadvantages when compared to the traditional method. The main
advantage is the possibility of use of standard tandem accelerators to measure sd0 at energies up to the maximum terminal voltage times the electron charge. At UFRJ 1.7 MeV is our energy limit. However, there are open laboratories currently operating tandems (with gaseous strippers) up to 20 MV (Fig. 1). A second crucial characteristic of the Tandem method is that relatively small analyzing magnets can be used, even for collision of very heavy projectiles at the highest available energies. This happens because after the collision at the center of the accelerator, the anionic beam is decelerated down to its injection energy (typically tens of keV) and can be easily bent to be detected by a Faraday cup. The outgoing neutral projectile is simply detected (through secondary electron emission) at the central exit of the magnet. The main disadvantage of our method is that negative and neutral detection necessarily have distinct collection efficiencies and beam transmission due to different focusing conditions by the accelerator itself. Thus, we do have beam transmission and detection efficiencies different from 100% [13]. However, these unknown values are not necessary for the determination of the destruction cross-sections sd . We do not need to know the beam transmission, it is sufficient to guarantee that (for each terminal voltage) the transmission is constant during a single measurement (while we vary the pressure at the gas stripper). The determination of absolute values for sd is possible, in the high-velocity limit, because these parameters appear in the arguments of exponentials (see Section 2.1). Positive outgoing projectiles can, in principle, be detected in a separate run but that would require a strong magnet, since cations are accelerated in the second accelerator stage. Under these conditions we cannot determine the cross-sections for electron loss to a particular final charge state. A quantitative assessment of the proposed method is presented in the rest of this section. For a gas target with a finite thickness, the emerging beam fractions, F i , satisfy the Allison’s equations [10] N X dF i þ F i sdi ¼ sji F j dx jai
with sdi ¼
N X
sij .
(1)
jai
The set of coupled differential equations above can be transformed into a set of linear algebraic equations through the substitution F i ðxÞ ¼ F 1i þ
N X
Aik expðsk xÞ.
(2)
k¼1
This system can in principle be solved with the constants F 1i , Aik and sk expressed as functions of the cross-sections sji . However, the expressions are cumbersome and fluctuations in experimental data for F i ðxÞ make data reduction not straightforward in the determination of the sji from fitted F 1i , Aik and sk , even for projectiles with only two electrons [10,16–18]. Important simplifications apply in the high-velocity regime.
ARTICLE IN PRESS M.M. Sant’Anna et al. / Nuclear Instruments and Methods in Physics Research A 577 (2007) 349–352
2.1. The high-velocity limit
351
15
(3)
Σj= i+ 1 j
2 MeV
-16
10
d
F 1 ¼ expðsd1 xÞ
3
i
Li + N2 -> Li
σi (10 cm2)
The projectile with a definite charge state can be destroyed by either losing or capturing electrons, with P respective total cross-sections given by sli ¼ N s and ij j4i P c s . For high velocities, s is negligible and sci ¼ N i joi ij sdi sli . The first two projectile final charge states have particularly simple solutions:
5
and s10 ½expðsd0 xÞ expðsd1 xÞ. sd1 sd0
Neutral projectile
(4) 0
-1
0
1
2
i Fig. 2. Destruction cross-sections as a function of initial projectile charge state, i, for Liiþ þ N2 at 2 MeV. Obtained from growth curves of Ref. [12]: squares, traditional growth-rate analysis; circles, simplified Tandem growth-rate analysis.
σ-11+ σ-10 i
-15
10
H +N2 → H
σ10
j
σ01
2
For an arbitrary F i with vbvB , sk ¼ sdk and F 1i ¼ diN , where N is the maximum number of electrons removable from the anion. In Table 1 we give the parameters for Eq. (2) that give the high-velocity solutions for the first four F i with an incident anionic beam. In the traditional method F i ðxÞ is measured as a function of x (through pressure measurement) for several i and relations between coefficients of Table 1 allow the determination of individual sij and also decrease the uncertainties in the fitted sdi . In the Tandem method F i ðxÞ curves are measured apart from a constant prefactor and only the parameters in the exponentials can be extracted. Fitting uncertainties are larger for i40, but for i ¼ 0 the only effect of different detection efficiencies is that s10 (see Eq. (4)) cannot be determined in the Tandem method. As an example, we have used published growth curves and cross-sections for Liiþ þ N2 at 2 MeV [11,12] to determine sdi using relations of Table 1 (traditional method analysis) and considering A PikN1as free adjusting parameters, with AiN ¼ diN j¼1 Aij to satisfy F i ð0Þ ¼ d1i (Tandem method analysis). Results are shown in Fig. 2 and appreciable differences are not seen until i ¼ 2.
σij ( cm )
F0 ¼
-16
10
Normalization point -17
10
σ0-1 σ-11
1
2.2. Breakdown at low v: the H þ N 2 example
10
v (atomic units)
The collision system H þ N2 is used as a test case for the experimental method proposed in this work with measurements performed at UFRJ and shown in Fig. 3. The line-
Fig. 3. Charge-changing cross-sections for Hiþ þ N2 as a function of the collision velocity: full symbols, present results; lines, data compilation of Ref. [9]; diamonds, Ref. [17].
Table 1 Parameters for the four first charge-state fractions F i ðxÞ in Eq. (2), with F 1 ð0Þ ¼ 1, vbvB and koN k
sk
Aik i ¼ 1
i¼0 s10 D10
1
sd1
1
0
sd0
0
1
sd1
0
s10 D10 0
2
sd2
0
0
If i ¼ No3, ANN in table is substituted by 1
PN1
k¼1 ANk
i¼1
i¼2
s10 s01 s11 D10 D11 D11 s10 s01 D10 D01 s10 s01 s11 þ D11 D01 D11 0
s10 s01 s12 s11 s12 s10 s02 s12 þ þ þ D10 D11 D12 D11 D12 D10 D12 D12 s10 s01 s12 s10 s02 D10 D01 D02 D10 D02 s10 s01 s12 s11 s12 D11 D01 D12 D11 D12 s10 s01 s12 s11 s12 s10 s02 s12 þ þ D12 D02 D12 D12 D12 D02 D12 D12
(Dmn ¼ sdm sdn ).
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density, x, was calibrated using sd1 of Ref. [17] at 400 keV. The sd0 error bars are estimated as the difference between data compilation results [9] and our measured sd0 . We deliberately pushed the method too far on the low-velocity side to see its breakdown on the intermediate-to-low velocity region. For v44vB measured and compilation values are proportional and agree within the 20% dispersion of the compilation [9]. Below v ¼ 2vB the error bars clearly increase. This is consistent with the velocity dependence of the capture cross-section s10 , which plunges above v ¼ 2vB . For heavier projectiles estimating the lower velocity limit of the Tandem method from capture cross-section values may not be trivial, because of lack of experimental data for capture cross-sections. However, in principle, the limitation should be less stringent than for H for three reasons: (i) for equivelocity neutral projectiles sd0 should be larger for projectiles with more (and more loosely bound) electrons, (ii) for equivelocity singly-charged projectiles s10 should be smaller for projectiles with more electrons, because of Pauli blocking of occupied projectile orbitals [19,20], and (iii) multiple capture cross-sections drop even faster than single capture. It is important to note that the beam emittance at the entrance of the stripping canal increases with decreasing beam energy. However, previous studies for the Li þ N2 collision system [16] showed that even at velocities as low as v ¼ 0:34vB , the measurements are only overestimated by about 20% due to elastic scattering. Electron capture by the projectile increases fast with decreasing beam energy and causes the breakdown of the method before elastic scattering becomes a relevant factor.
the accelerator, is used as the target gas cell for the experiment. We have previously applied the Tandem method to determine anion destruction cross-sections for several collision systems at intermediate projectile velocities. The extension of the method for neutral-projectile destruction is shown to be feasible. However, the suitable velocity region is narrower for neutral projectiles than for anions. This shortcoming of the method is discussed in detail for Hiþ þ N2 . Acknowledgments This work was partially supported by the Brazilian agencies MCT/CNPq (CT-Energ), FINEP, CAPES, FAPERJ, and FUJB. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
3. Conclusion A method for measuring destruction cross-sections for neutral projectiles at intermediate-to-high and high velocities is presented and discussed. It has the potential to extend widely the range of experimental data currently available. Our method is similar to the traditional growthrate method with an anionic initial beam. The main difference is that the gas stripper of a standard Tandem electrostatic accelerator, inside the high-voltage terminal of
[15] [16] [17] [18]
[19] [20]
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