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Equilibrium charge state distributions for boron and carbon ions emerging from carbon and aluminum targets
Nuclear Instruments and Methods in Physics Research B 268 (2010) 1551–1557
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
Equilibrium charge state distributions for boron and carbon ions emerging from carbon and aluminum targets Chris Schmitt a, Jay A. LaVerne a,b,*, Daniel Robertson a, Matthew Bowers a, Wenting Lu a, Philippe Collon a a b
Department of Physics, University of Notre Dame, Notre Dame, IN 46556, USA Radiation Laboratory, University of Notre Dame, Notre Dame, IN 46556, USA
a r t i c l e
i n f o
Article history: Received 7 January 2010 Received in revised form 29 January 2010 Available online 11 February 2010 Keywords: Equilibrium charge state distributions Carbon targets Aluminum targets Carbon ions Boron ions
a b s t r a c t Equilibrium charge state distributions of boron and carbon ions through carbon and aluminum targets were measured with an energy range of 3–6 MeV. Comparisons of the data with relevant semi-empirical models for the equilibrium mean charge states and for the charge state distribution widths could provide valuable insight on the underlying mechanisms for a fast ion to lose or capture electrons. In-depth examinations of the experimental results in combination with semi-empirical models suggest that equilibrium charge state distributions are well represented by Gaussian distributions. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction Fast ions lose energy by Coulombic interactions with the electrons of a medium so knowledge of the charge state of the ion is essential to describing a number of fundamental properties including the stopping power of the medium and the range of the ions [1]. Radiation effects and dosimetry are two of many areas that are ultimately dependent on the charge state of the incident ion. Applications of importance range from accelerator design and accelerator mass spectrometry to medical therapy. Unfortunately, significant gaps exist in the data for the charge state distribution of low-energy ions in solid materials, which makes it difficult to determine the trustworthiness of stopping power and range compilations. Significant amounts of data exist on equilibrium charge states for ions of low atomic number, Z, in carbon targets. This information can be found in several reviews [2–6]. Such systematic data does not exist for many other targets, which is the case with aluminum. The few studies on the equilibrium charge states of low Z ions in aluminum have explored energy in the keV range and sporadic energy increments between 10 and 100 MeV [4]. Aluminum is a typical lightweight target material that is used extensively for windows. The much needed information on equilibrium charge states of ions often has to be extrapolated and there is not enough confi-
* Corresponding author. Address: Department of Physics, University of Notre Dame, Notre Dame, IN 46556, USA. E-mail address: [email protected] (J.A. LaVerne). 0168-583X/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2010.02.003
dence in the models that such a procedure can be performed reliably. In this work, equilibrium charge state distribution measurements have been performed for boron1 and carbon ions emerging from carbon and aluminum foils in the energy range of 3–6 MeV. The present data is compared to the predictions of relevant semiempirical formalisms provided by Schiwietz et al. and Ziegler–Biersack-Littmark based on their inclusion of target dependence terms [7–9]. In addition, the Gaussian behavior of the charge state fractions is explored and charge state distribution widths are evaluated and discussed.
2. Experimental procedure The experimental procedures have been discussed in a previous paper, but will be briefly described below [10]. Boron and carbon ions are produced by a Source of Negative Ions by Cesium Sputtering negative-ion source, SNICS, and accelerated by the FN Tandem Van de Graaff in the Nuclear Structure Laboratory at the University of Notre Dame. The incident ions pass through the accelerator mass spectrometry (AMS) beamline and through target foils in a target chamber. Some of the ion beam is Rutherford scattered into a silicon (monitor) detector located in the target chamber, which acts as an ion beam monitor and a normalization tool. The un-scattered beam is sent directly into a Browne–Buechner Spectrograph where the charge state fractions are separated magnetically and mea1 10
B was used and then scaled to
11
B to compare with established data.
1552
C. Schmitt et al. / Nuclear Instruments and Methods in Physics Research B 268 (2010) 1551–1557
sured by an electron suppressed Faraday cup. This experimental procedure is repeated as a function of energy for each ion and target combination. When a charge state distribution for a given energy has been measured, the charge fractions, Fq, can be calculated using F q ¼ N q =RN q , where Nq is given by N q ¼ Iq =qeW with Iq being the current read from the Faraday cup for a specific charge state q, e is 1:6 1019 C and W is the normalization counts from the monitor. Following the determination of the charge fractions, the mean P qF q and the distribution charge can be determined using q ¼ P 1 2 width, d ¼ ½ ðq qÞ F q 2 , can also be calculated. This procedure was applied to both carbon and boron ions in carbon and aluminum targets and the experimental results are summarized in Table 1. Charge equilibrium is reached when sufficient collisions have occurred and the electron loss rate equals the electron capture rate. The attainment of equilibrium is based on previously reported experimental results or determined from charge state fraction measurements for several targets of different thicknesses [4,5,11]. The charge state fractions will remain constant for sufficiently thick targets, provided significant energy loss by the incident ions does not occur. Both the carbon and aluminum target had a thickness of 20 lg cm2 as confirmed by energy loss measurements with an alpha particle source.
q¼Z
ð8:29x þ x4 Þ ; 0:06=x þ 4 þ 7:4x þ x4
ð1Þ
where
x ¼ c1 ðv~ =c2 =1:54Þ1þ1:83=Z ;
ð2Þ
is a reformulated reduced velocity and the power term is used to adjust the steepness of the charge state response as a function of x with the following correction terms:
c1 ¼ 1 0:26 expðZ t =11Þ expððZ t ZÞ2 =9Þ and c2 ¼ 1 þ 0:030v~ lnðZ t Þ:
ð3Þ
The first term in Eq. (3) accounts for resonant electron capture, which reduces the mean charge state or similarly x for symmetrical ion-target combinations, while the second correction term allows for a target dependent deformation at high velocities. The final component in the reformulated reduced velocity is the scaled projectile velocity
v~ ¼ Z0:543 v p =v B :
ð4Þ
The sub- and superscripts have ‘‘p” for projectile, ‘‘B” for Bohr and ‘‘t” for target. The limitation noted for this model is that the ratio of the projectile velocity to the Bohr velocity must be >0.4 for Z P 3. The second relative ionization expression to be examined here is from the Ziegler–Biersack–Littmark model that is used in the well known SRIM and TRIM codes [9]. For ions of Z > 2 the Ziegler, Biersack and Littmark formula can be written as:
3. Results and discussion 3.1. Comparison with empirical models for mean charge states A variety of semi-empirical models have been developed to predict the experimental mean charge states. These models were usually constructed from data for a limited number of ions and targets and were optimized over a finite energy range. Effective charge can be obtained from stopping powers and does not always correspond to the mean charge state. The following formulae for mean charge states are examined in more detail since they are valid for the energy range covered in this work and they allow for target dependence in their formalism. In both models, the relative ionization (q=Z) is given. This quantity is defined as the mean charge state of the ion divided by its Z. A formula created by Schiwietz et al. [8] uses a highly parameterized least-square fit built from an array of over 800 data points that span a wide variety of ions and targets. The expression for the relative ionization is given as
0:6 3 q ¼ Z 1 expð0:803y0:3 r 1:3167yr 0:381557yr 0:008983yr Þ ; ð5Þ where yr is the reduced velocity as given by relative velocity as given by
vr ¼ v for
v r =v B Z2=3 and vr is the
1 þ v 2F =5v 2 ;
ð6aÞ
v > vF and
v r ¼ 3v F =4
1 þ 2v 2 =3v 2F v 4 =15v 4F ;
ð6bÞ
for v 6 vF where v is ion velocity and vF is the Fermi velocity of the medium. An examination of the relative ionization as a function of energy can be used to compare predictions of the Schiwietz model with
Table 1 Experimental charge fractions, mean charges, distribution widths and skewness for carbon and boron ions in carbon and aluminum targets. 11 2+
Incident Energy (MeV)
q
d
s
2+
3+
4+
5+
Z2 = 6
5.5 6.05 6.6
3.73 3.83 3.89
0.64 0.64 0.65
0.19 0.06 0.08
0.46 0.41 0.94
36.6 29.42 23.98
52.85 57.32 59.98
10.09 12.84 15.26
Z2 = 13
3.3 4.4 5.5 6.05
3.24 3.56 3.73 3.84
0.65 0.67 0.68 0.68
0.14 0.004 0.02 0.06
9.99 4 1.99 1.38
57.78 42.81 34.43 28.09
30.32 46.88 52.17 55.68
1.9 6.31 11.41 14.86
12 2+
C Z2 = 6
Incident energy (MeV) 3 4 5 5.5 6
q 3.39 3.78 3.96 4.09 4.18
d 0.65 0.63 0.58 0.63 0.63
s 0.03 0.12 0.06 0.07 0.24
2+ 6.19 1.55 0.66 0.46 0
3+ 51.63 28.64 16.36 13.67 10.47
4+ 39.38 60.3 70.38 63.65 62.12
5+ 2.8 9.36 11.73 21.13 25.87
6+ 0 0.15 .87 1.11 1.54
Z2 = 13
3 4 5 5.5 6
3.46 3.87 4.04 4.18 4.27
0.69 0.68 0.66 0.67 0.68
0.12 0.01 0.33 0.07 0.23
7 1.32 0.59 0.34 0
43.8 26.21 15.31 12.45 10.03
45.1 56.77 65.57 58.26 55.9
4.1 15.37 16.22 27.01 30.79
0 0.35 2.31 1.94 3.28
B
C. Schmitt et al. / Nuclear Instruments and Methods in Physics Research B 268 (2010) 1551–1557
those of the Ziegler–Biersack–Littmark model. The relative ionization is shown in Fig. 1 as a function of the specific ion energy for carbon targets (Fig. 1a) and for aluminum targets (Fig. 1b). The open symbols are the data from Wittkower [4] and the closed symbols represent the present work. The predictions of the Schiwietz model in Fig. 1a almost perfectly match the data, whereas the predictions of the Ziegler model are slightly shifted to higher values of relative ionization. Both models describe the ion-target systems well. The largest error between experiment and the Schiwietz model is 1.1%, while for that for the Ziegler model approaches 4%. Interestingly, the model prediction with the aluminum targets as shown in Fig. 1b have the opposite response as the carbon targets in that the Ziegler model predicts slightly lower values for the relative ionization than the Schiwietz model. The predictions of the Ziegler model have a better agreement with the data for aluminum targets than the Schiwietz model. For the aluminum targets, the Schiwietz model is not as good as it is with the carbon target data. The error between the experimental boron ion data and the Schiwietz model as high as 4.5% for the lowest energy ions and drops to 3.2% at greater ion energies. The Ziegler model has an error approaching 3% at the lowest boron ion energies and drops to <1.6% at higher boron ion energies. For carbon ions, the error between data and model predictions is <5.8% for the Schiwietz model and <1.7% for the Ziegler model. Although both models are reasonably good at predicting the relative ionization for low Z ions in low Z targets, the discrepancies are disturbing. 3.2. Comparison of charge state fractions with a Gaussian distribution In addition to developing semi-empirical models for equilibrium mean charge states, there is also a need to calculate the equilibrium charge fractions. Equilibrium charge state fractions give information on the relative rate of change of charge states that can be used to investigate the underlying physical processes responsible for charge equilibrium. Charge state distributions for ions greater than beryllium are often approximated by a Gaussian (symmetric) distribution. The expression for a Gaussian distribution can be described in terms related to charge states as the following:
pffiffiffiffiffiffiffi 2 F q ¼ ð1=d 2pÞ exp½ðq qÞ2 =2d
ð7Þ
where Fq is the charge state fraction, d is the charge state distribution width, q is the charge state and q is the mean charge state [5].
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The physical description is that when the outermost electrons of the ion are distributed over a shell n, where n is the principal quantum number, then the distribution can be approximated as a Gaussian [6]. There are instances when the charge state distributions are not symmetric. Asymmetric charge state distributions occur when the outermost electrons are spread between two adjacent n shells as has been shown by Shima et al. [12] to be approximated by two Gaussians. However, this phenomenon has only been demonstrated with high Z ions where this shell effect seems to be more enhanced [12–15]. Shell effects have been observed in both equilibrium and non-equilibrium charge states with closed-shell electron structures. In addition, there are special cases where asymmetric behavior has been observed with low and high velocity ions. According to Baudinet-Robinet, charge state distribution observed behind carbon targets can be approximated by v2, Gaussian, and reduced v2 distributions for low-, intermediate-, and highcharge ions, respectively [5,14]. P 3 The skewness, s ¼ q ðq qÞ3 F q =d , of the experimentally derived charge state distribution can be used to determine if a Gaussian representation is a proper choice for the charge state fraction. Table 1 clearly shows that the skewness of the charge state distribution in all instances suggest that a Gaussian distribution is a suitable choice for representing the charge state distribution. For a Gaussian distribution, the skewness is ideally equal to zero and the distribution width can be related to the full width at half maximum as 2d(2ln(2))1/2 2.35d. [3,4]. The comparisons of mean charge states show excellent agreement with only a 0.03% difference in the boron ion data and 0.04% for the carbon ion data. The distribution widths show varied and somewhat inconsistent results and the values for the Gaussian distribution are lower in magnitude than the experimental results, especially for the carbon ion data. Fig. 2 (boron ions) and Fig. 3 (carbon ions) show the evolution of the charge state distributions as functions of incident ion energy. Despite the difference in targets there is a striking similarity in the respective ion/target systems. Another interesting observation is that the combination of previously published data and the current data for the charge state distributions in Fig. 2a and b shows identical behaviors in the transition from low to high energies. This transition displays a clear break where the lower charge states lose their dominance and shell effects occur. In Fig. 2a this transition is observed between 3.5 and 4.5 MeV for boron ions traversing a car-
Fig. 1. Relative ionization for boron and carbon ions through carbon (a) and aluminum (b) targets as a function of specific energy: boron ions (d) this work; (s) boron ions from [4,11]; carbon ions () this work; and carbon ions (}) from [4]. The solid lines are from the Schiwietz model [7] and the dashed lines are from the Ziegler–Biersack– Littmark model [9].
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C. Schmitt et al. / Nuclear Instruments and Methods in Physics Research B 268 (2010) 1551–1557
Fig. 2. Evolution of charge state distributions for boron ions in carbon targets (a) and aluminum targets (b) with incident ion energy: (j) 5.5 MeV, (d) 6.05 MeV and (N) 6.6 MeV in carbon targets this work; (.) 2.5 MeV, (/) 3.0 MeV, (}) 3.5 MeV and (r) 4.5 MeV in carbon targets from [4]; (j) 3.3 MeV, (d) 4.4 MeV, (N) 5.5 and (.) 6.05 MeV in aluminum targets, this work.
Fig. 3. Evolution of charge state distributions for carbon ions in carbon targets (a) and aluminum targets (b) with incident ion energy: (j) 3.0 MeV, (d) 4.0 MeV and (N) 5.0 MeV, (.) 5.5 MeV, and () 6.0 MeV, this work.
bon target. Shell effects become more dominant at the lower energies of 2.5 and 3 MeV that are near the K–L shell boundary. A similar shell effect is seen in Fig. 2b when boron ions traverse an aluminum target. For carbon ions in Fig. 3 the charge state distribution does not show as drastic a change, but for both the carbon and aluminum target at 5 MeV the Gaussian distribution shape narrows in comparison to its neighbors. This energy also corresponds to the K–L shell boundary. This shell effect translates to the charge state distribution width where there is a large shift in d. In Fig. 6a (boron data) this shift is seen at X < 1 corresponding to the aforementioned shell effect, but the effect is subtle in Fig. 6b (carbon data) where clearly at X = 1.1 there is a distinct shift in d. Both values of X correspond to where the mean number of electrons (Z q) is approximately two, which reflects the tightly bound 1s electron shell. When there are cases, such as this one, where a Gaussian distribution can describe the equilibrium mean charge state well then ideally other information can be extracted successfully. One such
valuable piece of information is the equilibrium charge state fraction. Reliable models that can predict the equilibrium mean charge state and can describe the distribution width will then have the predictive power needed to determine equilibrium charge fractions using Eq. (7). The distribution width is the more problematic part to estimate, which will be discussed in more detail in the following section. Another approach for determining charge state distributions, as outlined in ref [16], is to fit polynomial curves to each of the charge fraction data as a function of energy. The charge fraction can then be written as F q ð%Þ ¼ 10½Y , where Y ¼ A þ B1 EðMeV=uÞ þ B2 EðMeV=uÞ2 . Such an approach has limitations in that no insight is gained on the nature of charge exchange, the expressions can only be used within the energy range employed for constructing them and the expressions have to be based on existing experimental data. The resulting fits for boron and carbon ions are shown in Figs. 4 and 5, respectively. In addition there is a dashed line representing charge state fractions calculated from an energy loss program
C. Schmitt et al. / Nuclear Instruments and Methods in Physics Research B 268 (2010) 1551–1557
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Fig. 4. The charge state fractions for boron ions as a function of specific energy for carbon (a) and aluminum targets (b). The solid data points represent current data and open data points come from reference [4]. The solid lines are from the polynomial fit and the dashed lines are from the CasP program [7,8].
Fig. 5. The charge state fractions for carbon ions as a function of specific energy for carbon (a) and aluminum targets (b). The solid data points represent current data and open data points come from reference [4]. The solid lines are from the polynomial fit and the dashed lines are from the CasP program [7,8].
called CasP,2 Convolution approximation for swift Particles, developed by Schiwietz et al. [7,8]. The standard deviation between the sets of data for the boron ions in carbon targets is as high as 3.3%, while for boron ions in aluminum targets it is 1.3%. The data for carbon ions in carbon targets have only one instance where the standard deviation is as high as 2.3% while the rest of the data has a deviation of <1.7%. The data for carbon ions in aluminum have similarly good 2 The CasP software is windows based and can be located at http://www.hmi.de/ people/schiwietz/casp.html with additional references and publications.
results with one instance of a deviation of 2.6% and the rest at <1.6%. Comparing the present work with the results from CasP there is in general terrific agreement on the behavior of the charge state trends. For the boron ions the difference is <6%, while with carbon ions it is <15%. These high percent differences are rare where in many instances the agreement is much more reasonable. The instances where the dashed lines abruptly end are where CasP does not calculate charge state fractions <1%. Overall there is good agreement between the Gaussian fit and the polynomial fit to the experimental data and both techniques can provide useful results.
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C. Schmitt et al. / Nuclear Instruments and Methods in Physics Research B 268 (2010) 1551–1557
Fig. 6. The charge state distribution widths for boron (a) and carbon (b) ions as functions of reduced velocity X. The closed symbols are from the present work, open symbols are from [4]. Symbols with an ‘‘x” are extrapolated from Gaussian fits, where squares are carbon and circles are aluminum target data. The dashed line is from the Nikolaev and Dmitriev model [18] and the solid line is from Baudinet-Robinet [14].
3.3. Comparison with empirical formulae for charge state distribution width
of equilibrium charge fractions will give more information of the distribution widths, and should be further studied.
The distribution width for a Gaussian distribution is given by the standard deviation. Obviously, charge state distribution widths can be readily obtained for those systems that can be represented by Gaussian distributions. Several semi-empirical models have been developed for estimating the charge state distribution width, but all are valid for a short range of energy with the exception of some of the carbon target systems or for high Z projectiles. In addition to the general lack of experimental information, there is a lack of systematic data with various targets. Shima et al. [17] have published some data that displays a dependence of distribution width on various targets. This target dependence will make it difficult to develop a universal formula to describe an ion/target system successfully. A model suggested by Nikolaev and Dmitriev [18] for the width of the charge state distribution based on carbon target data can be used in energy range (X < 1):
4. Conclusions
d ¼ 0:5Z 0:5 ð1 þ X 1=0:6 Þ0:8 X 1=1:2 ;
ð8Þ
and another model for carbon targets by Baudinet-Robinet [14] works in the 1 < X < 2.5 range and given by:
d ¼ Z 0:4 ð0:426 0:0571XÞ;
ð9Þ
where X is the reduced velocity. Fig. 6 shows the distribution widths as functions of the reduced velocity for (a) boron and (b) carbon ions. No strong dependence on the distribution widths is observed with respect to the reduced velocity. Neither theoretical model describes the observed distribution widths well. The general conclusion is that the distribution width is deceptively complex to quantify, unlike the mean charge state. Much more systematic data is needed to gather the most information out of the distribution width. There is already a large amount of data for several ions in carbon targets that suggest the distribution width is dependent on atomic structure, which makes development of simple semiempirical models very problematic. Even with the large amount of carbon target data, it is not possible to describe how the charge distribution width varies with target material. An accurate description
The current work expands on the knowledge of charge state distributions of low Z ions (boron and carbon) with the standard carbon targets and the less well studied aluminum targets. This new data allows an extensive comparison between two relevant semiempirical models and elucidates the limitations of each model. In addition, the CasP program was used to calculate charge state fractions and the results agree well with experimental values. The Gaussian nature of the charge state distribution was verified for the given ion/target systems within the specified energy range. The complicated nature of the charge state distribution width was revealed leading to questions that may provide some insight on the nature of charge exchange. Acknowledgments The authors would like to give special thanks to Larry Lamm for all of his technical assistance before, during and after the experiment and Michael Carilli’s assistance in making the aluminum targets. This work was supported by the National Science Foundation Grant NSF-PHY07-58110. The work of J.A.L. was supported by the Office of Basic Energy Sciences of the US Department of Energy and this document is NDRL-4840 of the Notre Dame Radiation Laboratory. References [1] [2] [3] [4] [5] [6] [7] [8]
N. Bohr, Dan. Vid. Selsk. Mat. Fys. Medd. 18 (8) (1948) 1–144. S.K. Allison, Rev. Mod. Phys. 30 (1958) 1137. H.D. Betz, Rev. Mod Phys. 44 (1972) 465. A.B. Wittkower, H.D. Betz, Atom. Data 5 (1973) 113. K. Shima, T. Mikumo, H. Tawara, Atom. Data Nucl. Data Tabl. 34 (1986) 357. K. Shima, N. Kuno, M. Yamanouchi, Atom. Data Nucl. Data Tabl. 51 (1992) 173. G. Schiwietz, P.L. Grande, Nucl. Instr. Meth. Phys. Res. B 3 (2001) 175–177. G. Schiwietz, K. Czerski, M. Roth, F. Staufenbiel, P.L. Grande, Nucl. Instr. Meth. Phys. Res. B 226 (2004) 683. [9] J.F. Ziegler, J.P. Biersack, U. Littmark, The Stopping and Range of Ions in Solids, Pergamon Press, New York, 1985. [10] C. Schmitt, J. LaVerne, D. Robertson, M. Bowers, W. Lu, P. Collon, Phys. Rev. A 80 (2009) 052711-1.
C. Schmitt et al. / Nuclear Instruments and Methods in Physics Research B 268 (2010) 1551–1557 [11] W.N. Lennard, T.E. Jackman, D. Phillips, Phys. Rev. A 24 (5) (1981) 2809–2813. [12] K. Shima, T. Ishihara, T. Miyoshi, T. Mikumo, Phys. Rev. A 28 (4) (1983) 2162– 2168. [13] C.D. Moak, H.O. Lutz, L.B. Bridwell, L.C. Northcliffe, S. Datz, Phys. Rev. Lett. 18 (1967) 41. [14] Y. Baudinet-Robinet, Nucl. Instr. Meth. 190 (1981) 197–202.
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[15] E.J. Knystautas, M. Jomphe, Phys. Rev. A 23 (2) (1981) 679–682. [16] A.E. Stuchbery, A.N. Wilson, P.M. Davidson, Nucl. Instr. Meth. B 243 (2006) 265–271. [17] K. Shima, N. Kuno, M. Yamanouchi, Phys. Rev. A 40 (7) (1989) 3557–3570. [18] V.S. Nikolaev, I.S. Dmitriev, Phys. Lett. 28A (4) (1968) 277–278.
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
Equilibrium charge state distributions for boron and carbon ions emerging from carbon and aluminum targets Chris Schmitt a, Jay A. LaVerne a,b,*, Daniel Robertson a, Matthew Bowers a, Wenting Lu a, Philippe Collon a a b
Department of Physics, University of Notre Dame, Notre Dame, IN 46556, USA Radiation Laboratory, University of Notre Dame, Notre Dame, IN 46556, USA
a r t i c l e
i n f o
Article history: Received 7 January 2010 Received in revised form 29 January 2010 Available online 11 February 2010 Keywords: Equilibrium charge state distributions Carbon targets Aluminum targets Carbon ions Boron ions
a b s t r a c t Equilibrium charge state distributions of boron and carbon ions through carbon and aluminum targets were measured with an energy range of 3–6 MeV. Comparisons of the data with relevant semi-empirical models for the equilibrium mean charge states and for the charge state distribution widths could provide valuable insight on the underlying mechanisms for a fast ion to lose or capture electrons. In-depth examinations of the experimental results in combination with semi-empirical models suggest that equilibrium charge state distributions are well represented by Gaussian distributions. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction Fast ions lose energy by Coulombic interactions with the electrons of a medium so knowledge of the charge state of the ion is essential to describing a number of fundamental properties including the stopping power of the medium and the range of the ions [1]. Radiation effects and dosimetry are two of many areas that are ultimately dependent on the charge state of the incident ion. Applications of importance range from accelerator design and accelerator mass spectrometry to medical therapy. Unfortunately, significant gaps exist in the data for the charge state distribution of low-energy ions in solid materials, which makes it difficult to determine the trustworthiness of stopping power and range compilations. Significant amounts of data exist on equilibrium charge states for ions of low atomic number, Z, in carbon targets. This information can be found in several reviews [2–6]. Such systematic data does not exist for many other targets, which is the case with aluminum. The few studies on the equilibrium charge states of low Z ions in aluminum have explored energy in the keV range and sporadic energy increments between 10 and 100 MeV [4]. Aluminum is a typical lightweight target material that is used extensively for windows. The much needed information on equilibrium charge states of ions often has to be extrapolated and there is not enough confi-
* Corresponding author. Address: Department of Physics, University of Notre Dame, Notre Dame, IN 46556, USA. E-mail address: [email protected] (J.A. LaVerne). 0168-583X/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2010.02.003
dence in the models that such a procedure can be performed reliably. In this work, equilibrium charge state distribution measurements have been performed for boron1 and carbon ions emerging from carbon and aluminum foils in the energy range of 3–6 MeV. The present data is compared to the predictions of relevant semiempirical formalisms provided by Schiwietz et al. and Ziegler–Biersack-Littmark based on their inclusion of target dependence terms [7–9]. In addition, the Gaussian behavior of the charge state fractions is explored and charge state distribution widths are evaluated and discussed.
2. Experimental procedure The experimental procedures have been discussed in a previous paper, but will be briefly described below [10]. Boron and carbon ions are produced by a Source of Negative Ions by Cesium Sputtering negative-ion source, SNICS, and accelerated by the FN Tandem Van de Graaff in the Nuclear Structure Laboratory at the University of Notre Dame. The incident ions pass through the accelerator mass spectrometry (AMS) beamline and through target foils in a target chamber. Some of the ion beam is Rutherford scattered into a silicon (monitor) detector located in the target chamber, which acts as an ion beam monitor and a normalization tool. The un-scattered beam is sent directly into a Browne–Buechner Spectrograph where the charge state fractions are separated magnetically and mea1 10
B was used and then scaled to
11
B to compare with established data.
1552
C. Schmitt et al. / Nuclear Instruments and Methods in Physics Research B 268 (2010) 1551–1557
sured by an electron suppressed Faraday cup. This experimental procedure is repeated as a function of energy for each ion and target combination. When a charge state distribution for a given energy has been measured, the charge fractions, Fq, can be calculated using F q ¼ N q =RN q , where Nq is given by N q ¼ Iq =qeW with Iq being the current read from the Faraday cup for a specific charge state q, e is 1:6 1019 C and W is the normalization counts from the monitor. Following the determination of the charge fractions, the mean P qF q and the distribution charge can be determined using q ¼ P 1 2 width, d ¼ ½ ðq qÞ F q 2 , can also be calculated. This procedure was applied to both carbon and boron ions in carbon and aluminum targets and the experimental results are summarized in Table 1. Charge equilibrium is reached when sufficient collisions have occurred and the electron loss rate equals the electron capture rate. The attainment of equilibrium is based on previously reported experimental results or determined from charge state fraction measurements for several targets of different thicknesses [4,5,11]. The charge state fractions will remain constant for sufficiently thick targets, provided significant energy loss by the incident ions does not occur. Both the carbon and aluminum target had a thickness of 20 lg cm2 as confirmed by energy loss measurements with an alpha particle source.
q¼Z
ð8:29x þ x4 Þ ; 0:06=x þ 4 þ 7:4x þ x4
ð1Þ
where
x ¼ c1 ðv~ =c2 =1:54Þ1þ1:83=Z ;
ð2Þ
is a reformulated reduced velocity and the power term is used to adjust the steepness of the charge state response as a function of x with the following correction terms:
c1 ¼ 1 0:26 expðZ t =11Þ expððZ t ZÞ2 =9Þ and c2 ¼ 1 þ 0:030v~ lnðZ t Þ:
ð3Þ
The first term in Eq. (3) accounts for resonant electron capture, which reduces the mean charge state or similarly x for symmetrical ion-target combinations, while the second correction term allows for a target dependent deformation at high velocities. The final component in the reformulated reduced velocity is the scaled projectile velocity
v~ ¼ Z0:543 v p =v B :
ð4Þ
The sub- and superscripts have ‘‘p” for projectile, ‘‘B” for Bohr and ‘‘t” for target. The limitation noted for this model is that the ratio of the projectile velocity to the Bohr velocity must be >0.4 for Z P 3. The second relative ionization expression to be examined here is from the Ziegler–Biersack–Littmark model that is used in the well known SRIM and TRIM codes [9]. For ions of Z > 2 the Ziegler, Biersack and Littmark formula can be written as:
3. Results and discussion 3.1. Comparison with empirical models for mean charge states A variety of semi-empirical models have been developed to predict the experimental mean charge states. These models were usually constructed from data for a limited number of ions and targets and were optimized over a finite energy range. Effective charge can be obtained from stopping powers and does not always correspond to the mean charge state. The following formulae for mean charge states are examined in more detail since they are valid for the energy range covered in this work and they allow for target dependence in their formalism. In both models, the relative ionization (q=Z) is given. This quantity is defined as the mean charge state of the ion divided by its Z. A formula created by Schiwietz et al. [8] uses a highly parameterized least-square fit built from an array of over 800 data points that span a wide variety of ions and targets. The expression for the relative ionization is given as
0:6 3 q ¼ Z 1 expð0:803y0:3 r 1:3167yr 0:381557yr 0:008983yr Þ ; ð5Þ where yr is the reduced velocity as given by relative velocity as given by
vr ¼ v for
v r =v B Z2=3 and vr is the
1 þ v 2F =5v 2 ;
ð6aÞ
v > vF and
v r ¼ 3v F =4
1 þ 2v 2 =3v 2F v 4 =15v 4F ;
ð6bÞ
for v 6 vF where v is ion velocity and vF is the Fermi velocity of the medium. An examination of the relative ionization as a function of energy can be used to compare predictions of the Schiwietz model with
Table 1 Experimental charge fractions, mean charges, distribution widths and skewness for carbon and boron ions in carbon and aluminum targets. 11 2+
Incident Energy (MeV)
q
d
s
2+
3+
4+
5+
Z2 = 6
5.5 6.05 6.6
3.73 3.83 3.89
0.64 0.64 0.65
0.19 0.06 0.08
0.46 0.41 0.94
36.6 29.42 23.98
52.85 57.32 59.98
10.09 12.84 15.26
Z2 = 13
3.3 4.4 5.5 6.05
3.24 3.56 3.73 3.84
0.65 0.67 0.68 0.68
0.14 0.004 0.02 0.06
9.99 4 1.99 1.38
57.78 42.81 34.43 28.09
30.32 46.88 52.17 55.68
1.9 6.31 11.41 14.86
12 2+
C Z2 = 6
Incident energy (MeV) 3 4 5 5.5 6
q 3.39 3.78 3.96 4.09 4.18
d 0.65 0.63 0.58 0.63 0.63
s 0.03 0.12 0.06 0.07 0.24
2+ 6.19 1.55 0.66 0.46 0
3+ 51.63 28.64 16.36 13.67 10.47
4+ 39.38 60.3 70.38 63.65 62.12
5+ 2.8 9.36 11.73 21.13 25.87
6+ 0 0.15 .87 1.11 1.54
Z2 = 13
3 4 5 5.5 6
3.46 3.87 4.04 4.18 4.27
0.69 0.68 0.66 0.67 0.68
0.12 0.01 0.33 0.07 0.23
7 1.32 0.59 0.34 0
43.8 26.21 15.31 12.45 10.03
45.1 56.77 65.57 58.26 55.9
4.1 15.37 16.22 27.01 30.79
0 0.35 2.31 1.94 3.28
B
C. Schmitt et al. / Nuclear Instruments and Methods in Physics Research B 268 (2010) 1551–1557
those of the Ziegler–Biersack–Littmark model. The relative ionization is shown in Fig. 1 as a function of the specific ion energy for carbon targets (Fig. 1a) and for aluminum targets (Fig. 1b). The open symbols are the data from Wittkower [4] and the closed symbols represent the present work. The predictions of the Schiwietz model in Fig. 1a almost perfectly match the data, whereas the predictions of the Ziegler model are slightly shifted to higher values of relative ionization. Both models describe the ion-target systems well. The largest error between experiment and the Schiwietz model is 1.1%, while for that for the Ziegler model approaches 4%. Interestingly, the model prediction with the aluminum targets as shown in Fig. 1b have the opposite response as the carbon targets in that the Ziegler model predicts slightly lower values for the relative ionization than the Schiwietz model. The predictions of the Ziegler model have a better agreement with the data for aluminum targets than the Schiwietz model. For the aluminum targets, the Schiwietz model is not as good as it is with the carbon target data. The error between the experimental boron ion data and the Schiwietz model as high as 4.5% for the lowest energy ions and drops to 3.2% at greater ion energies. The Ziegler model has an error approaching 3% at the lowest boron ion energies and drops to <1.6% at higher boron ion energies. For carbon ions, the error between data and model predictions is <5.8% for the Schiwietz model and <1.7% for the Ziegler model. Although both models are reasonably good at predicting the relative ionization for low Z ions in low Z targets, the discrepancies are disturbing. 3.2. Comparison of charge state fractions with a Gaussian distribution In addition to developing semi-empirical models for equilibrium mean charge states, there is also a need to calculate the equilibrium charge fractions. Equilibrium charge state fractions give information on the relative rate of change of charge states that can be used to investigate the underlying physical processes responsible for charge equilibrium. Charge state distributions for ions greater than beryllium are often approximated by a Gaussian (symmetric) distribution. The expression for a Gaussian distribution can be described in terms related to charge states as the following:
pffiffiffiffiffiffiffi 2 F q ¼ ð1=d 2pÞ exp½ðq qÞ2 =2d
ð7Þ
where Fq is the charge state fraction, d is the charge state distribution width, q is the charge state and q is the mean charge state [5].
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The physical description is that when the outermost electrons of the ion are distributed over a shell n, where n is the principal quantum number, then the distribution can be approximated as a Gaussian [6]. There are instances when the charge state distributions are not symmetric. Asymmetric charge state distributions occur when the outermost electrons are spread between two adjacent n shells as has been shown by Shima et al. [12] to be approximated by two Gaussians. However, this phenomenon has only been demonstrated with high Z ions where this shell effect seems to be more enhanced [12–15]. Shell effects have been observed in both equilibrium and non-equilibrium charge states with closed-shell electron structures. In addition, there are special cases where asymmetric behavior has been observed with low and high velocity ions. According to Baudinet-Robinet, charge state distribution observed behind carbon targets can be approximated by v2, Gaussian, and reduced v2 distributions for low-, intermediate-, and highcharge ions, respectively [5,14]. P 3 The skewness, s ¼ q ðq qÞ3 F q =d , of the experimentally derived charge state distribution can be used to determine if a Gaussian representation is a proper choice for the charge state fraction. Table 1 clearly shows that the skewness of the charge state distribution in all instances suggest that a Gaussian distribution is a suitable choice for representing the charge state distribution. For a Gaussian distribution, the skewness is ideally equal to zero and the distribution width can be related to the full width at half maximum as 2d(2ln(2))1/2 2.35d. [3,4]. The comparisons of mean charge states show excellent agreement with only a 0.03% difference in the boron ion data and 0.04% for the carbon ion data. The distribution widths show varied and somewhat inconsistent results and the values for the Gaussian distribution are lower in magnitude than the experimental results, especially for the carbon ion data. Fig. 2 (boron ions) and Fig. 3 (carbon ions) show the evolution of the charge state distributions as functions of incident ion energy. Despite the difference in targets there is a striking similarity in the respective ion/target systems. Another interesting observation is that the combination of previously published data and the current data for the charge state distributions in Fig. 2a and b shows identical behaviors in the transition from low to high energies. This transition displays a clear break where the lower charge states lose their dominance and shell effects occur. In Fig. 2a this transition is observed between 3.5 and 4.5 MeV for boron ions traversing a car-
Fig. 1. Relative ionization for boron and carbon ions through carbon (a) and aluminum (b) targets as a function of specific energy: boron ions (d) this work; (s) boron ions from [4,11]; carbon ions () this work; and carbon ions (}) from [4]. The solid lines are from the Schiwietz model [7] and the dashed lines are from the Ziegler–Biersack– Littmark model [9].
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Fig. 2. Evolution of charge state distributions for boron ions in carbon targets (a) and aluminum targets (b) with incident ion energy: (j) 5.5 MeV, (d) 6.05 MeV and (N) 6.6 MeV in carbon targets this work; (.) 2.5 MeV, (/) 3.0 MeV, (}) 3.5 MeV and (r) 4.5 MeV in carbon targets from [4]; (j) 3.3 MeV, (d) 4.4 MeV, (N) 5.5 and (.) 6.05 MeV in aluminum targets, this work.
Fig. 3. Evolution of charge state distributions for carbon ions in carbon targets (a) and aluminum targets (b) with incident ion energy: (j) 3.0 MeV, (d) 4.0 MeV and (N) 5.0 MeV, (.) 5.5 MeV, and () 6.0 MeV, this work.
bon target. Shell effects become more dominant at the lower energies of 2.5 and 3 MeV that are near the K–L shell boundary. A similar shell effect is seen in Fig. 2b when boron ions traverse an aluminum target. For carbon ions in Fig. 3 the charge state distribution does not show as drastic a change, but for both the carbon and aluminum target at 5 MeV the Gaussian distribution shape narrows in comparison to its neighbors. This energy also corresponds to the K–L shell boundary. This shell effect translates to the charge state distribution width where there is a large shift in d. In Fig. 6a (boron data) this shift is seen at X < 1 corresponding to the aforementioned shell effect, but the effect is subtle in Fig. 6b (carbon data) where clearly at X = 1.1 there is a distinct shift in d. Both values of X correspond to where the mean number of electrons (Z q) is approximately two, which reflects the tightly bound 1s electron shell. When there are cases, such as this one, where a Gaussian distribution can describe the equilibrium mean charge state well then ideally other information can be extracted successfully. One such
valuable piece of information is the equilibrium charge state fraction. Reliable models that can predict the equilibrium mean charge state and can describe the distribution width will then have the predictive power needed to determine equilibrium charge fractions using Eq. (7). The distribution width is the more problematic part to estimate, which will be discussed in more detail in the following section. Another approach for determining charge state distributions, as outlined in ref [16], is to fit polynomial curves to each of the charge fraction data as a function of energy. The charge fraction can then be written as F q ð%Þ ¼ 10½Y , where Y ¼ A þ B1 EðMeV=uÞ þ B2 EðMeV=uÞ2 . Such an approach has limitations in that no insight is gained on the nature of charge exchange, the expressions can only be used within the energy range employed for constructing them and the expressions have to be based on existing experimental data. The resulting fits for boron and carbon ions are shown in Figs. 4 and 5, respectively. In addition there is a dashed line representing charge state fractions calculated from an energy loss program
C. Schmitt et al. / Nuclear Instruments and Methods in Physics Research B 268 (2010) 1551–1557
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Fig. 4. The charge state fractions for boron ions as a function of specific energy for carbon (a) and aluminum targets (b). The solid data points represent current data and open data points come from reference [4]. The solid lines are from the polynomial fit and the dashed lines are from the CasP program [7,8].
Fig. 5. The charge state fractions for carbon ions as a function of specific energy for carbon (a) and aluminum targets (b). The solid data points represent current data and open data points come from reference [4]. The solid lines are from the polynomial fit and the dashed lines are from the CasP program [7,8].
called CasP,2 Convolution approximation for swift Particles, developed by Schiwietz et al. [7,8]. The standard deviation between the sets of data for the boron ions in carbon targets is as high as 3.3%, while for boron ions in aluminum targets it is 1.3%. The data for carbon ions in carbon targets have only one instance where the standard deviation is as high as 2.3% while the rest of the data has a deviation of <1.7%. The data for carbon ions in aluminum have similarly good 2 The CasP software is windows based and can be located at http://www.hmi.de/ people/schiwietz/casp.html with additional references and publications.
results with one instance of a deviation of 2.6% and the rest at <1.6%. Comparing the present work with the results from CasP there is in general terrific agreement on the behavior of the charge state trends. For the boron ions the difference is <6%, while with carbon ions it is <15%. These high percent differences are rare where in many instances the agreement is much more reasonable. The instances where the dashed lines abruptly end are where CasP does not calculate charge state fractions <1%. Overall there is good agreement between the Gaussian fit and the polynomial fit to the experimental data and both techniques can provide useful results.
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Fig. 6. The charge state distribution widths for boron (a) and carbon (b) ions as functions of reduced velocity X. The closed symbols are from the present work, open symbols are from [4]. Symbols with an ‘‘x” are extrapolated from Gaussian fits, where squares are carbon and circles are aluminum target data. The dashed line is from the Nikolaev and Dmitriev model [18] and the solid line is from Baudinet-Robinet [14].
3.3. Comparison with empirical formulae for charge state distribution width
of equilibrium charge fractions will give more information of the distribution widths, and should be further studied.
The distribution width for a Gaussian distribution is given by the standard deviation. Obviously, charge state distribution widths can be readily obtained for those systems that can be represented by Gaussian distributions. Several semi-empirical models have been developed for estimating the charge state distribution width, but all are valid for a short range of energy with the exception of some of the carbon target systems or for high Z projectiles. In addition to the general lack of experimental information, there is a lack of systematic data with various targets. Shima et al. [17] have published some data that displays a dependence of distribution width on various targets. This target dependence will make it difficult to develop a universal formula to describe an ion/target system successfully. A model suggested by Nikolaev and Dmitriev [18] for the width of the charge state distribution based on carbon target data can be used in energy range (X < 1):
4. Conclusions
d ¼ 0:5Z 0:5 ð1 þ X 1=0:6 Þ0:8 X 1=1:2 ;
ð8Þ
and another model for carbon targets by Baudinet-Robinet [14] works in the 1 < X < 2.5 range and given by:
d ¼ Z 0:4 ð0:426 0:0571XÞ;
ð9Þ
where X is the reduced velocity. Fig. 6 shows the distribution widths as functions of the reduced velocity for (a) boron and (b) carbon ions. No strong dependence on the distribution widths is observed with respect to the reduced velocity. Neither theoretical model describes the observed distribution widths well. The general conclusion is that the distribution width is deceptively complex to quantify, unlike the mean charge state. Much more systematic data is needed to gather the most information out of the distribution width. There is already a large amount of data for several ions in carbon targets that suggest the distribution width is dependent on atomic structure, which makes development of simple semiempirical models very problematic. Even with the large amount of carbon target data, it is not possible to describe how the charge distribution width varies with target material. An accurate description
The current work expands on the knowledge of charge state distributions of low Z ions (boron and carbon) with the standard carbon targets and the less well studied aluminum targets. This new data allows an extensive comparison between two relevant semiempirical models and elucidates the limitations of each model. In addition, the CasP program was used to calculate charge state fractions and the results agree well with experimental values. The Gaussian nature of the charge state distribution was verified for the given ion/target systems within the specified energy range. The complicated nature of the charge state distribution width was revealed leading to questions that may provide some insight on the nature of charge exchange. Acknowledgments The authors would like to give special thanks to Larry Lamm for all of his technical assistance before, during and after the experiment and Michael Carilli’s assistance in making the aluminum targets. This work was supported by the National Science Foundation Grant NSF-PHY07-58110. The work of J.A.L. was supported by the Office of Basic Energy Sciences of the US Department of Energy and this document is NDRL-4840 of the Notre Dame Radiation Laboratory. References [1] [2] [3] [4] [5] [6] [7] [8]
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[15] E.J. Knystautas, M. Jomphe, Phys. Rev. A 23 (2) (1981) 679–682. [16] A.E. Stuchbery, A.N. Wilson, P.M. Davidson, Nucl. Instr. Meth. B 243 (2006) 265–271. [17] K. Shima, N. Kuno, M. Yamanouchi, Phys. Rev. A 40 (7) (1989) 3557–3570. [18] V.S. Nikolaev, I.S. Dmitriev, Phys. Lett. 28A (4) (1968) 277–278.