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amu uranium ions passing through argon and carbon targets
Nuclear Instruments and Methods in Physics Research B 227 (2005) 251–260 www.elsevier.com/locate/nimb
Cross sections for charge change in argon and equilibrium charge states of 3.5 MeV/amu uranium ions passing through argon and carbon targets A.N. Perumal, V. Horvat *, R.L. Watson, Y. Peng, K.S. Fruchey Cyclotron Institute and Department of Chemistry, Texas A&M University, College Station, TX 77843-3366, USA Received 13 July 2004; received in revised form 10 September 2004
Abstract Cross sections for single and multiple electron capture and loss were measured for 3.5 MeV/amu uranium ions, traveling in argon gas, as a function of incident charge state. Multiple electron loss in single collisions was found to contribute significantly to the total loss cross section. The measured cross sections were used to determine the average equilibrium charge in argon by three different methods. The resulting charges were in good agreement with each other and with the effective charge calculated from stopping powers. In order to investigate the gas–solid (density) effect on the average equilibrium charge, the charge distributions of 3.5 MeV/amu uranium ions emerging from carbon foils of different thicknesses were measured. It was found that the average equilibrium charge of the uranium ions emerging from the solid is 41% larger than that of the uranium ions emerging from the gas. The energy dependences of the average equilibrium charges for uranium ions exiting carbon and argon targets were examined by combining the present results with previous results of other investigators and compared with the predictions of a semiempirical formula developed recently by Schiwietz and Grande. 2004 Elsevier B.V. All rights reserved. PACS: 34.50.s; 34.50.Fa; 34.70.+e Keywords: Electron capture; Electron loss; Equilibrium charge distribution; Average equilibrium charge
1. Introduction
*
Corresponding author. Tel.: +1 979 845 1411; fax: +1 979 845 1899. E-mail address: [email protected] (V. Horvat).
Beginning with the observation of Lassen [1] that the density of the target medium influences the distribution of emerging projectile ion charge states, this so called density effect has been extensively studied both experimentally and theoretically. In the case of gas targets, the time between
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collisions is often sufficient for excited projectiles to return to their ground states before entering into subsequent collisions. However, this is not always the case with solid targets, where the time between collisions may be shorter than the lifetime of the projectile excited states. Because the cross sections for electron transfer and ionization are not the same for excited and ground states, the charge distribution of projectiles traveling in solids and gases are expected to differ. The density effect is often discussed in terms of two different models, developed by Bohr and Lindhard (BL) [2] and Betz and Grodzins (BG) [3]. In the BL model it is argued that an increase in the average charge of the projectile occurs while it is inside a solid target due to the higher cross sections for ionization of electrons in excited states. On the other hand, the BG model attributes the higher charge states observed for projectiles exciting solid targets to Auger decay of excited states produced at the exit surface of the target. However, it has been recently established that neither mechanism alone gives a fully satisfactory explanation of the density effect. Objections to the BL model are based upon the lack of experimental evidence that electron loss cross sections increase with target density, and the large number of Auger electrons predicted by the BG model have never been detected experimentally [4–7]. A great deal of work has been published on charge state distributions for many combinations of projectile atomic number and energy, and target atomic number [7–13]. Various empirical or semiempirical relations have been formulated [7–14] that can be used to determine the average equilibrium charge distribution parameters without the knowledge of charge changing cross sections. Based on a large number of experimental results, Schiwietz and Grande (SG) [14] recently derived two semiempirical formulas for average equilibrium charge; one for solid targets and the other for gaseous targets. As a result of these semiempirical relations and other earlier measurements [7,15], two important aspects of the data have been noticed. The difference between the average equilibrium charge for solid and gas targets is very small for light ions, but it increases with increasing projectile atomic number. Furthermore, the gas–
solid charge difference depends on the incident energy of the projectile. At very low and high projectile energies, the average projectile equilibrium charge is nearly the same for gas and solid targets, whereas in the intermediate energy region the difference may be as high as a factor of 2. The primary objectives of the present work were to extend the investigation of the density effect on the average equilibrium charge of uranium ions up to 3.5 MeV/amu and to further demonstrate a new method for determining average equilibrium charges of fast ions in gases. Measurements of the cross sections for electron capture and loss in single collisions of uranium ions with argon atoms have been performed for six incident charge states ranging from 28+ to 51+. These cross sections were used to deduce the gaseous state average equilibrium charge. In addition, the charge fractions of 3.5 MeV/amu uranium ions emerging from carbon foils were measured as a function of foil thickness and used to establish the solid-state equilibrium charge. The results of the present study, as well as those obtained previously, are compared with the predictions of the SG formulas.
2. Experimental procedure Technical details of the cross section measurements can be found in [16]. Therefore, only a brief description is provided here. A beam of 3.5 MeV/ amu U28+ was extracted from the Texas A&M K500 superconducting cyclotron and directed to an analyzing magnet located upstream from the target chamber. Before entering the magnet, the beam was allowed to pass through either a carbon stripper foil or a high residual gas pressure region in the vicinity of the magnet to produce a distribution of projectile charge states. Ions having the desired charge state were then selected using the analyzing magnet and passed through a series of collimators before entering a differentially pumped, windowless gas cell of effective length 2.08 cm containing Ar gas. Projectiles emerging from the gas cell passed through a charge-dispersing magnet and were subsequently detected by a one-dimensional position sensitive microchannel plate detec-
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tor (PSD). The pressure of Ar gas inside the cell was monitored and maintained by a Baratron pressure transducer and a motorized flow valve, operating in conjunction with an automatic control system. The emerging ion charge distribution was measured at six different gas pressures, ranging from 0 to 64 mTorr, for each of the six incident projectile charge states. Cross sections for charge change were determined using the well-known growth rate method [10]. Charge fractions (F) were measured for each contributing charge state and plotted against the target thickness (p), which is the product of the atom density and the effective gas length. The data points were then fitted with second order polynomials of the form Fi(p) = a + bp + cp2. In this equation, the first term represents the background fraction of ions created in collisions with residual gas in the beam line, the second term represents the fraction of ions created in single collisions with the target gas, and the last term represents the fraction of ions created in double collisions with the target gas. The cross section for charge change was obtained from the best-fit value of the parameter b. A similar series of charge distribution measurements was performed with carbon foils as targets. Self-supporting foils having thicknesses of 10, 45,
Fig. 1. Charge spectrum of 3.5 MeV/amu uranium ions after passing thorough a 10 lg/cm2 carbon foil.
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100, 182, 293 and 636 lg/cm2 were mounted on a wheel that could be rotated to position each target in front of the beam without breaking the vacuum. Projectiles emerging from the target were charge dispersed and detected in the same manner as for the gas target measurements. The charge spectrum obtained with the 10 lg/cm2 target is shown in Fig. 1. By plotting the average charge as a function of target thickness, the average equilibrium charge was determined to be 52.5 ± 0.3. For 3.5 MeV/ amu uranium ions emerging from carbon, the SG formula [14] predicts an average equilibrium charge of 54.9.
3. Results and discussion 3.1. Capture and loss cross sections Measured cross sections for the capture and loss of single and multiple electrons by 3.5 MeV/u Uq+ (q = 28, 31, 33, 39, 42 and 51) ions undergoing single collisions with argon target atoms are shown in Fig. 2 as a function of charge change (Dq) for up to 13 electrons. Positive Dq is for loss and negative Dq is for capture. The cross sections are also listed in Table 1. Estimated errors given in Table 1 include statistical uncertainties and uncertainties associated with the extraction of charge fractions from the measured position distributions, determination of the gas cell pressures, and the PSD response function. The electron loss cross sections display an overall decrease with increasing Dq as well as with increasing projectile incident charge (q). On the other hand, the capture cross sections generally increase with projectile incident charge, as expected, but decrease sharply with Dq. It is evident that multiple electron loss is a highly probable outcome of heavy ion–atom collisions when the incident charge of the projectile is below its average equilibrium value. The cross sections for double electron loss are only about 50% smaller than those for single electron loss, while the total cross sections for the loss of more than P one electron ð Dq>1 rq;qþDq Þ are approximately two times larger than the single loss cross sections. This high probability of multiple electron loss has a profound influence on the charge state evolution
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Fig. 2. Cross sections for single and multiple electron capture and loss by 3.5 MeV/amu uranium ions in argon gas as a function of charge change (Dq).
Table 1 Single and multiple electron capture and loss cross sections for 3.5 MeV/amu uranium ions in argon Dq
Projectile incident charge 28
31
33
39
42
52
1 2 3 4
12.6 (1.2)
19.7 (1.8) 1.1 (0.1)
25.0 (2.2) 2.0 (0.2)
52.3 (4.5) 8.1 (0.8) 0.30 (0.05)
61.6 (5.3) 16.1 (1.4) 2.0 (0.2)
82.5 35.3 10.6 1.4
13.4 (1.2) 6.8 (0.7) 4.6 (0.5) 3.7 (0.4) 3.0 (0.3) 2.3 (0.3) 1.8 (0.2) 1.5 (0.2) 1.3 (0.2) 0.86 (0.12) 0.62 (0.09) 0.46 (0.07) 0.26 (0.04)
12.5 (1.1) 5.9 (0.6) 3.9 (0.4) 3.4 (0.4) 2.5 (0.3) 2.0 (0.2) 1.6 (0.2) 1.1 (0.1) 0.84 (0.12) 0.57 (0.08) 0.36 (0.06)
capture capture capture capture
1 loss 2 loss 3 loss 4 loss 5 loss 6 loss 7 loss 8 loss 9 loss 10 loss 11 loss 12 loss 13 loss
8.7 (0.8) 4.4 (0.5) 3.5 (0.4) 2.8 (0.3) 2.2 (0.3) 1.5 (0.2) 1.3 (0.2) 0.91 (0.12) 0.61 (0.09) 0.42 (0.07)
8.0 (0.8) 4.1 (0.4) 2.9 (0.3) 2.0 (0.2) 1.5 (0.2) 0.81 (0.11) 0.37 (0.06)
(7.0) (3.1) (1.0) (0.2)
6.7 (0.8) 3.2 (0.4) 2.0 (0.2) 1.2 (0.2) 0.68 (0.10)
Cross sections are given in units of 1018 cm2 (Mb) and numbers in parentheses are estimated errors.
of the projectile since higher stages of ionization can be readily achieved in a single collision rather
than proceeding through a sequence of single electron removal steps.
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Another interesting feature, a change of slope in the electron loss cross section as a function of Dq, has been reported earlier [17,18] and convincingly suggested to be due to the process of sequential ionization passing from the most weakly bound outermost shell of electrons on the projectile to the next (inner) subshell. In the present case of uranium, the outermost orbitals are 4f8, 4f5, 4f3, 5p5, 5p2 and 4d7 for the corresponding incident charge states 28, 31, 33, 39, 42 and 51, respectively. Slope changes are therefore anticipated in the electron loss cross sections at Dq = 8, 5, 3, 5, 2 and 7 for the corresponding incident charge states. The data for q = 28 and q = 39 in Fig. 2 do appear to exhibit slope changes at the predicted values of Dq, however, there are no clear indications of slope changes for the other incident charges. The absence of significant atomic structure effects may be due to the fact that electron binding energies for the outermost orbitals of heavy elements do not display very large shell effects. 3.2. Average equilibrium charge in argon
DqðqÞ ¼
X
DqP Dq ðqÞ;
255
ð1Þ
Dq
where PDq(q) is the probability of charge change, defined as P Dq ðqÞ ¼
rq;qþDq rtot
ðDq 6¼ 0Þ:
Here rtot is the total cross section, X rq;qþDq : rtot ¼
ð2Þ
ð3Þ
Dq6¼0
The calculated average charge change is plotted in Fig. 3 as a function of projectile incident charge. The solid curve shows a least squares fit to the function DqðqÞ ¼ A þ B tanhðq qc Þ=d:
ð4Þ
The best-fit parameter values were found to be A = 2.19, B = 3.96, qc = 29.0 and d = 13.6. The value of q for which the average charge change is equal to zero (37.5) is interpreted to be the average equilibrium charge.
Each projectile charge state fraction eventually reaches an equilibrium value that does not change as the target thickness is further increased and is independent of the projectile incident charge. This is a consequence of the equalization of the electron capture and loss rates. On the basis of experimentally observed regularities, various empirical and semiempirical formulas have been proposed that predict the average equilibrium charge reasonably well for ions emerging from solid targets. However, much less experimental data are available for gaseous targets and the few existing semiempirical methods of predicting average equilibrium charges in gases have restricted ranges of validity. In this section, three methods of determining average equilibrium charges in gases from the present thin target measurements are described. 3.2.1. Method 1 (from average charge change) The average equilibrium charge can be obtained using a method described in [19]. The average charge change in a single collision of a projectile with incident charge q may be defined as
Fig. 3. Average charge change of 3.5 MeV/amu uranium ions in argon as a function of projectile incident charge. The solid curve has been fit to the data using Eq. (4).
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3.2.2. Method 2 (from weighted probabilities) Weighted probabilities of electron loss and capture as a function of projectile incident charge, P wL ðqÞ and P wC ðqÞ, respectively, are defined as follows: X 1 P wL ðqÞ ¼ Dqrq;qþDq ; ð5Þ rwtot Dq>0 P wC ðqÞ ¼ where rwtot ¼
X
1 rwtot
X
jDqjrq;qþDq ;
ð6Þ
Dq<0
jDqjrq;qþDq :
ð7Þ
Dq
The calculated weighted probabilities plotted against the incident projectile charge are shown in Fig. 4. The solid curves show least squares fits to the function q q i 1h a P ðqÞ ¼ 1 tanh ; ð8Þ 2 s where the (+) sign in Eq. (8) applies to electron capture and the (–) sign applies to electron loss.
Fig. 4. Weighted probabilities for electron capture and loss as a function of projectile incident charge. The solid curves have been fit to the data using Eq. (8).
The best-fit parameter values were found to be qa = 37.3 and s = 7.61. The parameter qa is interpreted to be the average equilibrium charge, qeq, since it is the charge for which the two weighted probabilities are each equal to 0.5. An interesting feature here is that the capture and loss fitting functions [Eq. (8)] are mirror images of each other reflected about the horizontal line defined by P(q) = 0.5. 3.2.3. Method 3 (from the rate equations) The evolution of the charge distribution of ions traveling in matter may be described by the following set of rate equations: 1 dF k X ¼ ðF j rjk F k rkj Þ ðj; k ¼ 0 to Z 1 Þ; dp j¼0
Z
ð9Þ
where Fk is the fraction of ions with charge k, rjk is the cross section for charge change from j to k in a single collision and p is the target thickness expressed in atoms/cm2. The indices range from 0 to Z1, the projectile atomic number. In Eq. (9), the first term on the right-hand side represents the formation of ions with charge state k from those of charge state j, while the second term represents the transformation of the ions with charge k into all other possible charge states j. If all of the cross sections rjk are known, then the charge fractions Fk can be determined by numerical solution of Eq. (9). However, in a typical situation, such as the present, only a limited number of cross sections are measured directly, and so the rest must be estimated. For an accurate representation of the charge distribution via Eq. (9), cross sections for charge change should be measured for projectile incident charge ranging from qmin, the incident charge at which electron capture cross sections become negligible, to qmax, the incident charge at which electron loss cross sections become negligible. Typically, however, the cross sections display a smooth dependence on the number of electrons captured or lost and hence, it is not necessary to measure them for every charge state between qmin and qmax, since the missing cross sections can be estimated by interpolation or extrapolation with sufficient accuracy for the present purpose.
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Fig. 5. Equilibrium charge distribution of 3.5 MeV/amu uranium ions in argon calculated using Eq. (9). The solid line shows the results of a Gaussian fit to the calculated charge fractions.
The equilibrium condition P dFk/dp = 0, together with the requirement that F k ¼ 1, were used to solve Eq. (9) for the equilibrium charge fractions. The calculated equilibrium charge distribution is shown in Fig. 5. A Gaussian fit to the data points (solid curve in Fig. 5) reveals a small asymmetry in the tail of the charge distribution on the low charge state side. The average equilibrium charge was determined to be 37.2 (based on the data points), while the Gaussian centroid was 0.2 charge units lower. In both cases, the distribution width (variance) was found to be 3.6 charge units. Eq. (9) was used also to calculate the average charge, qav, of 3.5 MeV/amu U ions traveling in Ar as a function of incident charge and target thickness. The results are shown in Fig. 6 for incident charge states ranging from 22+ to 51+. A striking feature of this figure is that qav approaches its equilibrium value qeq along a nearly pure exponential curve (i.e., a nearly straight line in a semilog plot). Furthermore, its decay constant (slope in a semi-log plot) does not depend significantly on the projectile incident charge. However, as may be seen in Fig. 6, the thickness at which equilibrium is achieved does depend on the projectile incident charge. For example, the value of d0.1, defined as the thickness at which jqav qeqj = 0.1, ranges roughly from 3 lg/cm2 (4.5 · 1016 atoms/cm2) for q = 37 (incident charge closest to qeq in Fig. 6) to
257
Fig. 6. Calculated absolute difference between the average charge and the average equilibrium charge as a function of target thickness for a variety of projectile incident charge states.
Fig. 7. Target thickness at which the average charge is within 0.1 charge units of its equilibrium value (d0.1) as a function of projectile incident charge.
23 lg/cm2 (3.5 · 1017 atoms/cm2) for q = 51 (incident charge farthest from qeq in Fig. 6). Fig. 7 shows how d0.1 depends on q. The value of d0.1 for jq qeqj > 0.1 may be estimated using the formula d 0:1 ¼ 10:6ðlog jq qeq j þ 1Þ lg=cm2 :
ð10Þ
3.3. Comparison of results The average equilibrium charges determined for 3.5 MeV/amu uranium ions traveling in argon
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gas using the three methods described above are in good agreement with each other ( qeq ¼ 37:3 0:2). This value is very close to the effective charge defined in terms of the stopping powers of 3.5 MeV/amu uranium ions and protons [20], sffiffiffiffiffiffi SU qeff ¼ ¼ 37:4: ð11Þ Sp The average equilibrium charge predicted for 3.5 MeV/amu uranium ions in Ar by the SG semiempirical formula [14] is 43.5, which is 6.2 units higher than the present experimental result. Over the restricted range of low pressures used in the present measurements, it was found that the average charge (determined from the measured charge distributions) varied linearly with target thickness. This is shown in Fig. 8, where straight lines have been fit through the data points and extended to large thicknesses. The intriguing feature of this figure is that all of the straight lines for the different projectile incident charges converge (near a thickness of 8 · 1016 atoms/cm2) over a small region corresponding to an average charge in the range of 37.7–38.6, which is close to the average equilibrium charge. It is not known whether this
Fig. 8. Projectile average charge plotted as a function of target thickness. Straight lines fit to the data points have been extended to determine the average charge in the region of convergence.
is a general characteristic or merely a result of serendipity. Comparing the results obtained for argon gas and solid carbon, it is found that the average equilibrium charge is 15.2 units higher for the solid target. This 41% increase in average charge in going from gas to solid is a clear manifestation of the density effect. 3.4. Energy dependence of average equilibrium charge In order to examine the projectile energy dependence of the density effect, the average equilibrium charges for uranium ions in carbon and argon targets obtained by other investigators [5,11,15,21,22] have been plotted in Fig. 9 along with the present results. Also shown are the values predicted by the SG formula [14]. From this figure,
Fig. 9. Projectile energy dependence of the average equilibrium charge of uranium ions in gaseous argon and solid carbon. The filled symbols show experimental results taken from the literature [5,11,21,22], while the empty symbols show the results of the present measurements. The solid and dashed lines delineate the average equilibrium charges predicted for argon and carbon, respectively, using the semiempirical formula of Schiwietz and Grande [14]. The error bars are plotted only if they are larger than the size of the symbols.
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it is apparent that the difference in the experimental average equilibrium charge values for gas and solid targets increases with increasing projectile energy. The experimental average equilibrium charges for solid carbon targets are well represented by the SG semiempirical formula. However, (as was pointed out above) the present result for argon is 14% lower than the semiempirical prediction. It should be noted that the present measurement is in a region of the scaling parameter x [14] where there is considerable scatter among the experimental data points upon which the SG formula is based.
4. Conclusions Cross sections for single, and multiple electron capture and loss were measured for 3.5 MeV/amu uranium ions in argon gas. Multiple electron loss in single collisions was found to be highly probable. No striking atomic structure effects were observed in the loss cross sections, contrary to expectations based on earlier measurements. This was attributed to the fact that the binding energies of electrons in the outermost shells of uranium do not display very large changes from one subshell to the next. The average equilibrium charge was determined using a recently proposed method that relies only on the electron capture and loss cross sections for a few selected projectile charge states [19]. These cross sections were used to determine the average charge change in single collisions as a function of projectile incident charge. The average equilibrium charge was interpreted to be the projectile incident charge for which the average charge change is zero. Two other methods were also applied to the determination of the average equilibrium charge in argon: (1) by plotting the weighted probabilities for electron capture and loss as a function of projectile incident charge and determining the incident charge at which the two are equal, and (2) by solving the charge fraction rate equations [Eq. (9)] to obtain the equilibrium charge distribution and its centroid. The average equilibrium charges obtained by these three methods were in good agreement with each
259
other ðqeq ¼ 37:3 0:2Þ and with the effective charge calculated from stopping powers. In order to establish the magnitude of the density effect for 3.5 MeV/amu uranium ions, the average equilibrium charge after passing through solid carbon was measured and found to be 52.5 ± 0.3. The large difference in average equilibrium charge values for gaseous argon and solid carbon (15.2 charge units) presumably is a consequence of higher net rates of electron loss and larger probabilities of inner-shell vacancy production in the solid state. The projectile energy dependence of the density effect was examined by combining the present results with data from other investigators. The gas–solid difference increases with increasing projectile energy and was found to be well represented over the energy range from approximately 0.01 to 1.4 MeV/amu by the semiempirical formula of Schiwietz and Grande [14]. However, the present result for the 3.5 MeV/amu uranium ion gas–solid difference is 33% larger than that predicted by this formula.
Acknowledgements This work was supported by the Robert A. Welch Foundation.
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Cross sections for charge change in argon and equilibrium charge states of 3.5 MeV/amu uranium ions passing through argon and carbon targets A.N. Perumal, V. Horvat *, R.L. Watson, Y. Peng, K.S. Fruchey Cyclotron Institute and Department of Chemistry, Texas A&M University, College Station, TX 77843-3366, USA Received 13 July 2004; received in revised form 10 September 2004
Abstract Cross sections for single and multiple electron capture and loss were measured for 3.5 MeV/amu uranium ions, traveling in argon gas, as a function of incident charge state. Multiple electron loss in single collisions was found to contribute significantly to the total loss cross section. The measured cross sections were used to determine the average equilibrium charge in argon by three different methods. The resulting charges were in good agreement with each other and with the effective charge calculated from stopping powers. In order to investigate the gas–solid (density) effect on the average equilibrium charge, the charge distributions of 3.5 MeV/amu uranium ions emerging from carbon foils of different thicknesses were measured. It was found that the average equilibrium charge of the uranium ions emerging from the solid is 41% larger than that of the uranium ions emerging from the gas. The energy dependences of the average equilibrium charges for uranium ions exiting carbon and argon targets were examined by combining the present results with previous results of other investigators and compared with the predictions of a semiempirical formula developed recently by Schiwietz and Grande. 2004 Elsevier B.V. All rights reserved. PACS: 34.50.s; 34.50.Fa; 34.70.+e Keywords: Electron capture; Electron loss; Equilibrium charge distribution; Average equilibrium charge
1. Introduction
*
Corresponding author. Tel.: +1 979 845 1411; fax: +1 979 845 1899. E-mail address: [email protected] (V. Horvat).
Beginning with the observation of Lassen [1] that the density of the target medium influences the distribution of emerging projectile ion charge states, this so called density effect has been extensively studied both experimentally and theoretically. In the case of gas targets, the time between
0168-583X/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2004.09.005
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collisions is often sufficient for excited projectiles to return to their ground states before entering into subsequent collisions. However, this is not always the case with solid targets, where the time between collisions may be shorter than the lifetime of the projectile excited states. Because the cross sections for electron transfer and ionization are not the same for excited and ground states, the charge distribution of projectiles traveling in solids and gases are expected to differ. The density effect is often discussed in terms of two different models, developed by Bohr and Lindhard (BL) [2] and Betz and Grodzins (BG) [3]. In the BL model it is argued that an increase in the average charge of the projectile occurs while it is inside a solid target due to the higher cross sections for ionization of electrons in excited states. On the other hand, the BG model attributes the higher charge states observed for projectiles exciting solid targets to Auger decay of excited states produced at the exit surface of the target. However, it has been recently established that neither mechanism alone gives a fully satisfactory explanation of the density effect. Objections to the BL model are based upon the lack of experimental evidence that electron loss cross sections increase with target density, and the large number of Auger electrons predicted by the BG model have never been detected experimentally [4–7]. A great deal of work has been published on charge state distributions for many combinations of projectile atomic number and energy, and target atomic number [7–13]. Various empirical or semiempirical relations have been formulated [7–14] that can be used to determine the average equilibrium charge distribution parameters without the knowledge of charge changing cross sections. Based on a large number of experimental results, Schiwietz and Grande (SG) [14] recently derived two semiempirical formulas for average equilibrium charge; one for solid targets and the other for gaseous targets. As a result of these semiempirical relations and other earlier measurements [7,15], two important aspects of the data have been noticed. The difference between the average equilibrium charge for solid and gas targets is very small for light ions, but it increases with increasing projectile atomic number. Furthermore, the gas–
solid charge difference depends on the incident energy of the projectile. At very low and high projectile energies, the average projectile equilibrium charge is nearly the same for gas and solid targets, whereas in the intermediate energy region the difference may be as high as a factor of 2. The primary objectives of the present work were to extend the investigation of the density effect on the average equilibrium charge of uranium ions up to 3.5 MeV/amu and to further demonstrate a new method for determining average equilibrium charges of fast ions in gases. Measurements of the cross sections for electron capture and loss in single collisions of uranium ions with argon atoms have been performed for six incident charge states ranging from 28+ to 51+. These cross sections were used to deduce the gaseous state average equilibrium charge. In addition, the charge fractions of 3.5 MeV/amu uranium ions emerging from carbon foils were measured as a function of foil thickness and used to establish the solid-state equilibrium charge. The results of the present study, as well as those obtained previously, are compared with the predictions of the SG formulas.
2. Experimental procedure Technical details of the cross section measurements can be found in [16]. Therefore, only a brief description is provided here. A beam of 3.5 MeV/ amu U28+ was extracted from the Texas A&M K500 superconducting cyclotron and directed to an analyzing magnet located upstream from the target chamber. Before entering the magnet, the beam was allowed to pass through either a carbon stripper foil or a high residual gas pressure region in the vicinity of the magnet to produce a distribution of projectile charge states. Ions having the desired charge state were then selected using the analyzing magnet and passed through a series of collimators before entering a differentially pumped, windowless gas cell of effective length 2.08 cm containing Ar gas. Projectiles emerging from the gas cell passed through a charge-dispersing magnet and were subsequently detected by a one-dimensional position sensitive microchannel plate detec-
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tor (PSD). The pressure of Ar gas inside the cell was monitored and maintained by a Baratron pressure transducer and a motorized flow valve, operating in conjunction with an automatic control system. The emerging ion charge distribution was measured at six different gas pressures, ranging from 0 to 64 mTorr, for each of the six incident projectile charge states. Cross sections for charge change were determined using the well-known growth rate method [10]. Charge fractions (F) were measured for each contributing charge state and plotted against the target thickness (p), which is the product of the atom density and the effective gas length. The data points were then fitted with second order polynomials of the form Fi(p) = a + bp + cp2. In this equation, the first term represents the background fraction of ions created in collisions with residual gas in the beam line, the second term represents the fraction of ions created in single collisions with the target gas, and the last term represents the fraction of ions created in double collisions with the target gas. The cross section for charge change was obtained from the best-fit value of the parameter b. A similar series of charge distribution measurements was performed with carbon foils as targets. Self-supporting foils having thicknesses of 10, 45,
Fig. 1. Charge spectrum of 3.5 MeV/amu uranium ions after passing thorough a 10 lg/cm2 carbon foil.
253
100, 182, 293 and 636 lg/cm2 were mounted on a wheel that could be rotated to position each target in front of the beam without breaking the vacuum. Projectiles emerging from the target were charge dispersed and detected in the same manner as for the gas target measurements. The charge spectrum obtained with the 10 lg/cm2 target is shown in Fig. 1. By plotting the average charge as a function of target thickness, the average equilibrium charge was determined to be 52.5 ± 0.3. For 3.5 MeV/ amu uranium ions emerging from carbon, the SG formula [14] predicts an average equilibrium charge of 54.9.
3. Results and discussion 3.1. Capture and loss cross sections Measured cross sections for the capture and loss of single and multiple electrons by 3.5 MeV/u Uq+ (q = 28, 31, 33, 39, 42 and 51) ions undergoing single collisions with argon target atoms are shown in Fig. 2 as a function of charge change (Dq) for up to 13 electrons. Positive Dq is for loss and negative Dq is for capture. The cross sections are also listed in Table 1. Estimated errors given in Table 1 include statistical uncertainties and uncertainties associated with the extraction of charge fractions from the measured position distributions, determination of the gas cell pressures, and the PSD response function. The electron loss cross sections display an overall decrease with increasing Dq as well as with increasing projectile incident charge (q). On the other hand, the capture cross sections generally increase with projectile incident charge, as expected, but decrease sharply with Dq. It is evident that multiple electron loss is a highly probable outcome of heavy ion–atom collisions when the incident charge of the projectile is below its average equilibrium value. The cross sections for double electron loss are only about 50% smaller than those for single electron loss, while the total cross sections for the loss of more than P one electron ð Dq>1 rq;qþDq Þ are approximately two times larger than the single loss cross sections. This high probability of multiple electron loss has a profound influence on the charge state evolution
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Fig. 2. Cross sections for single and multiple electron capture and loss by 3.5 MeV/amu uranium ions in argon gas as a function of charge change (Dq).
Table 1 Single and multiple electron capture and loss cross sections for 3.5 MeV/amu uranium ions in argon Dq
Projectile incident charge 28
31
33
39
42
52
1 2 3 4
12.6 (1.2)
19.7 (1.8) 1.1 (0.1)
25.0 (2.2) 2.0 (0.2)
52.3 (4.5) 8.1 (0.8) 0.30 (0.05)
61.6 (5.3) 16.1 (1.4) 2.0 (0.2)
82.5 35.3 10.6 1.4
13.4 (1.2) 6.8 (0.7) 4.6 (0.5) 3.7 (0.4) 3.0 (0.3) 2.3 (0.3) 1.8 (0.2) 1.5 (0.2) 1.3 (0.2) 0.86 (0.12) 0.62 (0.09) 0.46 (0.07) 0.26 (0.04)
12.5 (1.1) 5.9 (0.6) 3.9 (0.4) 3.4 (0.4) 2.5 (0.3) 2.0 (0.2) 1.6 (0.2) 1.1 (0.1) 0.84 (0.12) 0.57 (0.08) 0.36 (0.06)
capture capture capture capture
1 loss 2 loss 3 loss 4 loss 5 loss 6 loss 7 loss 8 loss 9 loss 10 loss 11 loss 12 loss 13 loss
8.7 (0.8) 4.4 (0.5) 3.5 (0.4) 2.8 (0.3) 2.2 (0.3) 1.5 (0.2) 1.3 (0.2) 0.91 (0.12) 0.61 (0.09) 0.42 (0.07)
8.0 (0.8) 4.1 (0.4) 2.9 (0.3) 2.0 (0.2) 1.5 (0.2) 0.81 (0.11) 0.37 (0.06)
(7.0) (3.1) (1.0) (0.2)
6.7 (0.8) 3.2 (0.4) 2.0 (0.2) 1.2 (0.2) 0.68 (0.10)
Cross sections are given in units of 1018 cm2 (Mb) and numbers in parentheses are estimated errors.
of the projectile since higher stages of ionization can be readily achieved in a single collision rather
than proceeding through a sequence of single electron removal steps.
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Another interesting feature, a change of slope in the electron loss cross section as a function of Dq, has been reported earlier [17,18] and convincingly suggested to be due to the process of sequential ionization passing from the most weakly bound outermost shell of electrons on the projectile to the next (inner) subshell. In the present case of uranium, the outermost orbitals are 4f8, 4f5, 4f3, 5p5, 5p2 and 4d7 for the corresponding incident charge states 28, 31, 33, 39, 42 and 51, respectively. Slope changes are therefore anticipated in the electron loss cross sections at Dq = 8, 5, 3, 5, 2 and 7 for the corresponding incident charge states. The data for q = 28 and q = 39 in Fig. 2 do appear to exhibit slope changes at the predicted values of Dq, however, there are no clear indications of slope changes for the other incident charges. The absence of significant atomic structure effects may be due to the fact that electron binding energies for the outermost orbitals of heavy elements do not display very large shell effects. 3.2. Average equilibrium charge in argon
DqðqÞ ¼
X
DqP Dq ðqÞ;
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ð1Þ
Dq
where PDq(q) is the probability of charge change, defined as P Dq ðqÞ ¼
rq;qþDq rtot
ðDq 6¼ 0Þ:
Here rtot is the total cross section, X rq;qþDq : rtot ¼
ð2Þ
ð3Þ
Dq6¼0
The calculated average charge change is plotted in Fig. 3 as a function of projectile incident charge. The solid curve shows a least squares fit to the function DqðqÞ ¼ A þ B tanhðq qc Þ=d:
ð4Þ
The best-fit parameter values were found to be A = 2.19, B = 3.96, qc = 29.0 and d = 13.6. The value of q for which the average charge change is equal to zero (37.5) is interpreted to be the average equilibrium charge.
Each projectile charge state fraction eventually reaches an equilibrium value that does not change as the target thickness is further increased and is independent of the projectile incident charge. This is a consequence of the equalization of the electron capture and loss rates. On the basis of experimentally observed regularities, various empirical and semiempirical formulas have been proposed that predict the average equilibrium charge reasonably well for ions emerging from solid targets. However, much less experimental data are available for gaseous targets and the few existing semiempirical methods of predicting average equilibrium charges in gases have restricted ranges of validity. In this section, three methods of determining average equilibrium charges in gases from the present thin target measurements are described. 3.2.1. Method 1 (from average charge change) The average equilibrium charge can be obtained using a method described in [19]. The average charge change in a single collision of a projectile with incident charge q may be defined as
Fig. 3. Average charge change of 3.5 MeV/amu uranium ions in argon as a function of projectile incident charge. The solid curve has been fit to the data using Eq. (4).
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3.2.2. Method 2 (from weighted probabilities) Weighted probabilities of electron loss and capture as a function of projectile incident charge, P wL ðqÞ and P wC ðqÞ, respectively, are defined as follows: X 1 P wL ðqÞ ¼ Dqrq;qþDq ; ð5Þ rwtot Dq>0 P wC ðqÞ ¼ where rwtot ¼
X
1 rwtot
X
jDqjrq;qþDq ;
ð6Þ
Dq<0
jDqjrq;qþDq :
ð7Þ
Dq
The calculated weighted probabilities plotted against the incident projectile charge are shown in Fig. 4. The solid curves show least squares fits to the function q q i 1h a P ðqÞ ¼ 1 tanh ; ð8Þ 2 s where the (+) sign in Eq. (8) applies to electron capture and the (–) sign applies to electron loss.
Fig. 4. Weighted probabilities for electron capture and loss as a function of projectile incident charge. The solid curves have been fit to the data using Eq. (8).
The best-fit parameter values were found to be qa = 37.3 and s = 7.61. The parameter qa is interpreted to be the average equilibrium charge, qeq, since it is the charge for which the two weighted probabilities are each equal to 0.5. An interesting feature here is that the capture and loss fitting functions [Eq. (8)] are mirror images of each other reflected about the horizontal line defined by P(q) = 0.5. 3.2.3. Method 3 (from the rate equations) The evolution of the charge distribution of ions traveling in matter may be described by the following set of rate equations: 1 dF k X ¼ ðF j rjk F k rkj Þ ðj; k ¼ 0 to Z 1 Þ; dp j¼0
Z
ð9Þ
where Fk is the fraction of ions with charge k, rjk is the cross section for charge change from j to k in a single collision and p is the target thickness expressed in atoms/cm2. The indices range from 0 to Z1, the projectile atomic number. In Eq. (9), the first term on the right-hand side represents the formation of ions with charge state k from those of charge state j, while the second term represents the transformation of the ions with charge k into all other possible charge states j. If all of the cross sections rjk are known, then the charge fractions Fk can be determined by numerical solution of Eq. (9). However, in a typical situation, such as the present, only a limited number of cross sections are measured directly, and so the rest must be estimated. For an accurate representation of the charge distribution via Eq. (9), cross sections for charge change should be measured for projectile incident charge ranging from qmin, the incident charge at which electron capture cross sections become negligible, to qmax, the incident charge at which electron loss cross sections become negligible. Typically, however, the cross sections display a smooth dependence on the number of electrons captured or lost and hence, it is not necessary to measure them for every charge state between qmin and qmax, since the missing cross sections can be estimated by interpolation or extrapolation with sufficient accuracy for the present purpose.
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Fig. 5. Equilibrium charge distribution of 3.5 MeV/amu uranium ions in argon calculated using Eq. (9). The solid line shows the results of a Gaussian fit to the calculated charge fractions.
The equilibrium condition P dFk/dp = 0, together with the requirement that F k ¼ 1, were used to solve Eq. (9) for the equilibrium charge fractions. The calculated equilibrium charge distribution is shown in Fig. 5. A Gaussian fit to the data points (solid curve in Fig. 5) reveals a small asymmetry in the tail of the charge distribution on the low charge state side. The average equilibrium charge was determined to be 37.2 (based on the data points), while the Gaussian centroid was 0.2 charge units lower. In both cases, the distribution width (variance) was found to be 3.6 charge units. Eq. (9) was used also to calculate the average charge, qav, of 3.5 MeV/amu U ions traveling in Ar as a function of incident charge and target thickness. The results are shown in Fig. 6 for incident charge states ranging from 22+ to 51+. A striking feature of this figure is that qav approaches its equilibrium value qeq along a nearly pure exponential curve (i.e., a nearly straight line in a semilog plot). Furthermore, its decay constant (slope in a semi-log plot) does not depend significantly on the projectile incident charge. However, as may be seen in Fig. 6, the thickness at which equilibrium is achieved does depend on the projectile incident charge. For example, the value of d0.1, defined as the thickness at which jqav qeqj = 0.1, ranges roughly from 3 lg/cm2 (4.5 · 1016 atoms/cm2) for q = 37 (incident charge closest to qeq in Fig. 6) to
257
Fig. 6. Calculated absolute difference between the average charge and the average equilibrium charge as a function of target thickness for a variety of projectile incident charge states.
Fig. 7. Target thickness at which the average charge is within 0.1 charge units of its equilibrium value (d0.1) as a function of projectile incident charge.
23 lg/cm2 (3.5 · 1017 atoms/cm2) for q = 51 (incident charge farthest from qeq in Fig. 6). Fig. 7 shows how d0.1 depends on q. The value of d0.1 for jq qeqj > 0.1 may be estimated using the formula d 0:1 ¼ 10:6ðlog jq qeq j þ 1Þ lg=cm2 :
ð10Þ
3.3. Comparison of results The average equilibrium charges determined for 3.5 MeV/amu uranium ions traveling in argon
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gas using the three methods described above are in good agreement with each other ( qeq ¼ 37:3 0:2). This value is very close to the effective charge defined in terms of the stopping powers of 3.5 MeV/amu uranium ions and protons [20], sffiffiffiffiffiffi SU qeff ¼ ¼ 37:4: ð11Þ Sp The average equilibrium charge predicted for 3.5 MeV/amu uranium ions in Ar by the SG semiempirical formula [14] is 43.5, which is 6.2 units higher than the present experimental result. Over the restricted range of low pressures used in the present measurements, it was found that the average charge (determined from the measured charge distributions) varied linearly with target thickness. This is shown in Fig. 8, where straight lines have been fit through the data points and extended to large thicknesses. The intriguing feature of this figure is that all of the straight lines for the different projectile incident charges converge (near a thickness of 8 · 1016 atoms/cm2) over a small region corresponding to an average charge in the range of 37.7–38.6, which is close to the average equilibrium charge. It is not known whether this
Fig. 8. Projectile average charge plotted as a function of target thickness. Straight lines fit to the data points have been extended to determine the average charge in the region of convergence.
is a general characteristic or merely a result of serendipity. Comparing the results obtained for argon gas and solid carbon, it is found that the average equilibrium charge is 15.2 units higher for the solid target. This 41% increase in average charge in going from gas to solid is a clear manifestation of the density effect. 3.4. Energy dependence of average equilibrium charge In order to examine the projectile energy dependence of the density effect, the average equilibrium charges for uranium ions in carbon and argon targets obtained by other investigators [5,11,15,21,22] have been plotted in Fig. 9 along with the present results. Also shown are the values predicted by the SG formula [14]. From this figure,
Fig. 9. Projectile energy dependence of the average equilibrium charge of uranium ions in gaseous argon and solid carbon. The filled symbols show experimental results taken from the literature [5,11,21,22], while the empty symbols show the results of the present measurements. The solid and dashed lines delineate the average equilibrium charges predicted for argon and carbon, respectively, using the semiempirical formula of Schiwietz and Grande [14]. The error bars are plotted only if they are larger than the size of the symbols.
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it is apparent that the difference in the experimental average equilibrium charge values for gas and solid targets increases with increasing projectile energy. The experimental average equilibrium charges for solid carbon targets are well represented by the SG semiempirical formula. However, (as was pointed out above) the present result for argon is 14% lower than the semiempirical prediction. It should be noted that the present measurement is in a region of the scaling parameter x [14] where there is considerable scatter among the experimental data points upon which the SG formula is based.
4. Conclusions Cross sections for single, and multiple electron capture and loss were measured for 3.5 MeV/amu uranium ions in argon gas. Multiple electron loss in single collisions was found to be highly probable. No striking atomic structure effects were observed in the loss cross sections, contrary to expectations based on earlier measurements. This was attributed to the fact that the binding energies of electrons in the outermost shells of uranium do not display very large changes from one subshell to the next. The average equilibrium charge was determined using a recently proposed method that relies only on the electron capture and loss cross sections for a few selected projectile charge states [19]. These cross sections were used to determine the average charge change in single collisions as a function of projectile incident charge. The average equilibrium charge was interpreted to be the projectile incident charge for which the average charge change is zero. Two other methods were also applied to the determination of the average equilibrium charge in argon: (1) by plotting the weighted probabilities for electron capture and loss as a function of projectile incident charge and determining the incident charge at which the two are equal, and (2) by solving the charge fraction rate equations [Eq. (9)] to obtain the equilibrium charge distribution and its centroid. The average equilibrium charges obtained by these three methods were in good agreement with each
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other ðqeq ¼ 37:3 0:2Þ and with the effective charge calculated from stopping powers. In order to establish the magnitude of the density effect for 3.5 MeV/amu uranium ions, the average equilibrium charge after passing through solid carbon was measured and found to be 52.5 ± 0.3. The large difference in average equilibrium charge values for gaseous argon and solid carbon (15.2 charge units) presumably is a consequence of higher net rates of electron loss and larger probabilities of inner-shell vacancy production in the solid state. The projectile energy dependence of the density effect was examined by combining the present results with data from other investigators. The gas–solid difference increases with increasing projectile energy and was found to be well represented over the energy range from approximately 0.01 to 1.4 MeV/amu by the semiempirical formula of Schiwietz and Grande [14]. However, the present result for the 3.5 MeV/amu uranium ion gas–solid difference is 33% larger than that predicted by this formula.
Acknowledgements This work was supported by the Robert A. Welch Foundation.
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