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Energy loss corrections for MeV ions in tandem accelerator stripping
886
Nuclear
Instruments
and Methods
in Physics
Research
B56/57
(1991) 886-888 North-Holland
Energy loss corrections for MeV ions in tandem accelerator stripping * A.M. Arrale, S. Matteson, Departmeni
F.D. McDaniel
and J.L. Duggan
of Physics and Center for Materials Characterization, University of North Texas, Denton, TX 76203, USA
Ion energy losses occur during passage of an ion through the stripping gas or foil in the terminal of a tandem accelerator. The energy loss is frequently ignored. For accelerator mass spectrometty (AMS) [J.M. Anthony et al., Nucl. Instr. and Meth. B50 (1990) 2621, the energy loss, if not properly accounted for, may result in significant reductions in ion transmission. We have obtained an approximate expression for the energy loss of l-4 MeV ions in nitrogen gas and carbon foils for all ions. By using the scaling theory of Lindhard, Scharff, and Schiott (LSS) [K. Dan Vidensk. Selsk. Mat. Fys. Medd. 33 (1963) no. 141 a semi-empirical formula is found for the energy loss as a function of the reduced energy c. Calculated energy losses of all ions as a function of atomic mass are plotted for each energy, and the result is compared to experimental data measured in an AMS system. Other effects such as kinematic energy loss during electron stripping and relativistic corrections to the energy are also evaluated and discussed.
1. Introduction One of the most significant parameters which characterizes an ion beam is its energy. In frequent cases the ion’s energy can be determined and controlled by means of an energy dispersive spectrometer. In the case of accelerator mass spectrometry (AMS), however, it is imperative that the ion energy be known a priori, in order to assure an optimum and consistent transmission of the ion. The level of precision required is determined by the resolution of the system. For the AMS system at the University of North Texas, the resolution requires that the energy of the ion be determined to better than approximately 1 keV. Several phenomena affect the energy of an ion undergoing acceleration in a tandem accelerator. The stability and accuracy of the potential measuring instrumentation is, of course, essential and is assumed in this work. Fundamental ion physics phenomena such as dissociation of molecular ions, elastic scattering, electron stripping and inelastic electron energy loss may also be important. The kinematics of the dissociation and stripping of an ion will have a significant effect on the final energy of the ion. The energy after reaching the terminal is e(U + U,), with U the terminal potential and Ua the injection potential. If the ion (which may be a molecule)
* Work supported
in part by the State of Texas Coordinating Board, Texas Instruments Inc., National Science Foundation Grant No. DMR-8812331. Office of Naval Research Grants No. N00014-89-J-1309, N00014-90-J-1344 and N00014-90-J-1691, the Robert A, Welch Foundation. Texas Utilities Electric Co., International Digital Modeling Corp.. Texas Utilities Electric Co., and the University of North Texas Organized Research Fund.
0168-583X/91/$03.50
0 1991 - Elsevier Science Publishers
dissociates and is stripped to a state of q + , then the residual energy of the fragment of neutral mass m, at the terminal after breakup and stripping, is e[u + Ua](m - qm,)/(mo + m,), where the quantities m,, m,, m refer to the mass of the neutralized injected ion, the mass of the attached (stripped electron) and the mass of the neutral atomic ion, respectively. For hydrogen ions the kinematic energy loss due to stripping alone amounts to 1%0 (part-per-mil), a 3 keV error in energy for a 3 MV terminal potential. For heavier ions, the kinematic energy loss effect due to electron stripping is not as dramatic, but nevertheless, presents a systematic error in the energy. Thus, the final ion energy neglecting other effects is E = eqU + e[ U + U,]( m - qm,)/( m, + m,). Molecular dissociation, on the other hand, does present a significant energy loss for all ions. Fortunately, this phenomenon is also accurately accounted for in the preceding analysis. At the level of precision of the present work, otherwise negligible effects must be considered. One such effect is the relativistic correction to the apparent energy which can be important, especially for protons. The correct energy is given by E,, = [l + ( E/2mc2)]E, where m is understood to be the rest mass of the ion, and E is the energy determined strictly from the potentials and kinematics [l]. Whenever a fast ion passes through matter, inelastic excitation of the material results, with a consequent energy loss. Although no complete theory exists, nor is there comprehensive data for all cases, some scaling laws seem to work adequately. In the following section we examine a method to estimate a universal semi-empirical stopping power for all ions in nitrogen gas and carbon foils. Later the major predictions of the theory will be compared to selected measurements of the energy loss of ions in the stripping gas of a tandem accelerator.
B.V. (North-Holland)
A.M. Arrale et al. / Energy loss corrections 4E+4,
2. Theory of stopping powers There are numerous stopping power data in the literature. Among these are the range and stoppingpower tables of Northcliffe and Schilling [2] and the stopping cross sections of Ziegler [3]. In the present work we investigate the energy loss of an ion of given energy (l-4 MeV) as a function of the incident ion’s atomic muss. Experimental data were extracted from ref. [3], assuming standard temperature and pressure and extrapolating to lower energies, where necessary, using a power law. A typical plot of the energy loss of all ions incident on a nitrogen gas versus the atomic mass for 2 MeV is shown in fig. 1. There extracted data are represented by points. A universal stopping power curve is drawn as the solid line. Fig. 2 illustrates the similar case for carbon films. Examination of the plot reveals that for small values of the ion’s mass (ml < 35 u) the stopping power increases with the ion mass following the trend of the ion’s nuclear charge (Z: to first order). However, as mi gets larger (> 35 u), the stopping power declines. This decline results from two effects: a lower effective ionic charge and less participation of the outer shell electrons of the stripper gas or foil. The net effect is to decrease the stopping power for ions of the same energy but larger mass. We used the scaling theory of Lindhard, Scharff and Schiott [4-61 (LSS) to compute a universal stopping power for all energies (1-4 MeV). In this theory the range and energy can be parameterized by p = 4~a2XNm,m2/(ml c = am2E/[
887
(1) (2)
+ m2)‘,
ZlZ2e2( ml + m2)],
8,
3E+4s zz -2E+4-
lE+4 __ ***.
l
Scaling Theory Experimental Data
OE+O ,,,,,,,~,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,~,,,,~, 0 50 100 M
150
200
250
(u)
Fig. 2. Tabulated energy loss of 2 MeV ions incident on a carbon foil vs atomic mass of the ion (points) with empirical fit (solid line).
where a = 0.8853ao(Zf/3 + Zz’3)-1/2, with a, the Bohr radius and Z2 the atomic number of the target atom, while X is the range [cm] for the material of number density N, and E is the ion energy in the laboratory frame. The Thomas-Fermi potential leads to a universal dimensionless stopping power, dr/dp, which can be expressed in terms of energy loss, dE/d X in [MeV/cm] as follows: dc/dp
= a dE/dX,
(3)
where a: = (ml + m2)/(4~NaZ1Z2e2m,) with N and m2 the number density and the mass of the target material, respectively. The atomic number of the ion, Z,, can be expressed as a function of the ion’s atomic mass (m,) using the most stable isobar of Z, given as [7,8]: Z, = 40m,/(80
+ 0.6mf/3).
(4)
The electronic energy loss (for a free electron gas) is expressed in reduced quantities as (dr/dp), = kc’/‘, for slow ions and where the constant k varies only slowly with Z, and Z,: k = 0.0793Z:/3Z:/2( my2my2
~ .**==
Scaling Theory Experimental Data
Fig. 1. Tabulated energy loss of 2 MeV ions incident on nitrogen vs atomic mass of the ion (points) with empirical fit (solid line). Standard temperature and pressure are assumed.
ml + m2)3’2
( zv3
+ z;/3
)3’4
.
(5)
The reduced energy loss de/dp was tabulated using eqs. (3) and (4), together with experimental and extrapolated data from the literature. A universal (empirical) equation was found that has the appropriate asymptotic behavior (a power law dr/dp a tP, p > 0 for small c and tending toward zero for large e). This equation is dc/dp
= kC,&/g(
c)
(6) X. MASS SPECTROMETRY
888
A.M. Arrale et al. / Energy loss corrections
Table 1 Parameters
p, and b, given in eq. (8).
Target
P2
Nitrogen Carbon
-0.02266 -0.01482
-0.0286 -0.0260
P3
-0.00212 -0.00130
P4
b,
- 1.749~ 1O-6 -7.616~10-~
7.436~ lo- ” 1.936x10-”
and g(e)
= (1 + c,r + c,t* + C,r3).
For the curves shown in this paper, five values of the coefficients C, were fitted for each energy. It was found that the coefficients were also a function of energy given by: C,=p,E+b,,
(8)
where C, is the respective coefficient (C,, C,, C,, C,, or C,), and E is the energy of the ion in MeV. The corresponding values of CL,and b, are given in table 1.
3. Experimental In order to test the validity and significance of the formalism described above, the relative energy loss for a variety of ions was measured for various gas stripper pressures. The energy loss was measured using a high resolution electrostatic energy analyzer described elsewhere [9]. The relative gas pressure in the stripper canal was inferred from the pressure indication at the injector end of the accelerator column pinj. The energy loss was found to be a linear function of the pressure pinj, for the range of values used (5 x 10m8 to 8 X 10e7 Torr). The slope (energy loss per l.tTorr) of this curve is proportional to the stopping power. After the energy loss was measured for several ions spanning the mass
0.94791 0.79834
bz
0.84838 0.85244
b,
0.01236 0.01043
b,
7.968 x 1O-6 2.920x 1O-6
- 3.023 x 10-t’ - 3.896 x10-r’
range from 1 to 80 u, a single parameter fit was made. The parameter was the average gas pressure in the stripper corresponding to a given Pinj, which was significantly lower. A pressure of 3.3 mTorr in the stripper was found to correspond to 1 x 10m7 Torr at the injector end of the column, a typical operating value. This value is quite reasonable. Fig. 3 reveals the qualitatively good agreement between the predictions of the empirical formula and the measured energy correction. Note that the maximum energy loss does occur for m, near 35 u. Energy losses of one to several keV are predicted for gas strippers, while for carbon foils of 2-5 pg/cm* one can expect energy losses of as much as lo-20 keV.
4. Conclusion We have obtained an empirical stopping power formula (parameterized with 10 parameters) that to an accuracy of better than - 10% gives stopping powers for all ions in nitrogen and in carbon for the energy range l-4 MeV. Comparisons with actual energy losses in a tandem accelerator are in good relative agreement. The measurements further indicate that for high and reproducible energy resolution the energy loss cannot be neglected.
References
__ l l l
0
20
Scaling Theory = l Experimental Data
40
60
80,
100
M (u) Fig. 3. Theoretical energy loss of 2 MeV ion, dE/dpi,,j, incident on nitrogen gas vs atomic mass of the ion. Also shown are the experimental data points for the energy loss in the gas stripper of a tandem accelerator. The quantity pinj is the accelerator injector pressure, which is proportional to the pressure of the gas in the stripper canal.
[l] M. Szilagyi, Electron and Ion Optics (Plenum, New York, 1988) p. 28. [2] L.C. Northcliffe and R.F. Schilhng, Nucl. Data Tables A7 (1970) 233. [3] J.F. Ziegler, Stopping Cross-section For Energetic Ions In AU Metals, vol. 5 (Pergamon, New York, 1980) p. 91. [4] J. Lindhard, M. Scharff and H. Schiott, K. Dan. Vidensk. Selsk. 33 (1963) no. 14. [5] J. Lindhard, V. Nielsen and M. Scharff, K. Dan. Vidensk. Selsk. 36 (1968) no. 10. [6] J. Lindhard and M. Scharff, Phys. Rev. 124 (1961) 128. [7] B.G. Harvey, Nuclear Physics and Chemistry, 2nd ed. (Prentice-Hall, Englewood Cliffs, NJ, 1969) p. 78. [8] P.A. Seeger, Nucl. Phys. 25 (1961) 1. [9] S. Matteson, F.D. McDaniel, J.L. Duggan, J.M. Anthony, T.J. Bennett and R.L. Beavers, Nucl. Instr. and Meth. B40/41 (1989) 759.
Nuclear
Instruments
and Methods
in Physics
Research
B56/57
(1991) 886-888 North-Holland
Energy loss corrections for MeV ions in tandem accelerator stripping * A.M. Arrale, S. Matteson, Departmeni
F.D. McDaniel
and J.L. Duggan
of Physics and Center for Materials Characterization, University of North Texas, Denton, TX 76203, USA
Ion energy losses occur during passage of an ion through the stripping gas or foil in the terminal of a tandem accelerator. The energy loss is frequently ignored. For accelerator mass spectrometty (AMS) [J.M. Anthony et al., Nucl. Instr. and Meth. B50 (1990) 2621, the energy loss, if not properly accounted for, may result in significant reductions in ion transmission. We have obtained an approximate expression for the energy loss of l-4 MeV ions in nitrogen gas and carbon foils for all ions. By using the scaling theory of Lindhard, Scharff, and Schiott (LSS) [K. Dan Vidensk. Selsk. Mat. Fys. Medd. 33 (1963) no. 141 a semi-empirical formula is found for the energy loss as a function of the reduced energy c. Calculated energy losses of all ions as a function of atomic mass are plotted for each energy, and the result is compared to experimental data measured in an AMS system. Other effects such as kinematic energy loss during electron stripping and relativistic corrections to the energy are also evaluated and discussed.
1. Introduction One of the most significant parameters which characterizes an ion beam is its energy. In frequent cases the ion’s energy can be determined and controlled by means of an energy dispersive spectrometer. In the case of accelerator mass spectrometry (AMS), however, it is imperative that the ion energy be known a priori, in order to assure an optimum and consistent transmission of the ion. The level of precision required is determined by the resolution of the system. For the AMS system at the University of North Texas, the resolution requires that the energy of the ion be determined to better than approximately 1 keV. Several phenomena affect the energy of an ion undergoing acceleration in a tandem accelerator. The stability and accuracy of the potential measuring instrumentation is, of course, essential and is assumed in this work. Fundamental ion physics phenomena such as dissociation of molecular ions, elastic scattering, electron stripping and inelastic electron energy loss may also be important. The kinematics of the dissociation and stripping of an ion will have a significant effect on the final energy of the ion. The energy after reaching the terminal is e(U + U,), with U the terminal potential and Ua the injection potential. If the ion (which may be a molecule)
* Work supported
in part by the State of Texas Coordinating Board, Texas Instruments Inc., National Science Foundation Grant No. DMR-8812331. Office of Naval Research Grants No. N00014-89-J-1309, N00014-90-J-1344 and N00014-90-J-1691, the Robert A, Welch Foundation. Texas Utilities Electric Co., International Digital Modeling Corp.. Texas Utilities Electric Co., and the University of North Texas Organized Research Fund.
0168-583X/91/$03.50
0 1991 - Elsevier Science Publishers
dissociates and is stripped to a state of q + , then the residual energy of the fragment of neutral mass m, at the terminal after breakup and stripping, is e[u + Ua](m - qm,)/(mo + m,), where the quantities m,, m,, m refer to the mass of the neutralized injected ion, the mass of the attached (stripped electron) and the mass of the neutral atomic ion, respectively. For hydrogen ions the kinematic energy loss due to stripping alone amounts to 1%0 (part-per-mil), a 3 keV error in energy for a 3 MV terminal potential. For heavier ions, the kinematic energy loss effect due to electron stripping is not as dramatic, but nevertheless, presents a systematic error in the energy. Thus, the final ion energy neglecting other effects is E = eqU + e[ U + U,]( m - qm,)/( m, + m,). Molecular dissociation, on the other hand, does present a significant energy loss for all ions. Fortunately, this phenomenon is also accurately accounted for in the preceding analysis. At the level of precision of the present work, otherwise negligible effects must be considered. One such effect is the relativistic correction to the apparent energy which can be important, especially for protons. The correct energy is given by E,, = [l + ( E/2mc2)]E, where m is understood to be the rest mass of the ion, and E is the energy determined strictly from the potentials and kinematics [l]. Whenever a fast ion passes through matter, inelastic excitation of the material results, with a consequent energy loss. Although no complete theory exists, nor is there comprehensive data for all cases, some scaling laws seem to work adequately. In the following section we examine a method to estimate a universal semi-empirical stopping power for all ions in nitrogen gas and carbon foils. Later the major predictions of the theory will be compared to selected measurements of the energy loss of ions in the stripping gas of a tandem accelerator.
B.V. (North-Holland)
A.M. Arrale et al. / Energy loss corrections 4E+4,
2. Theory of stopping powers There are numerous stopping power data in the literature. Among these are the range and stoppingpower tables of Northcliffe and Schilling [2] and the stopping cross sections of Ziegler [3]. In the present work we investigate the energy loss of an ion of given energy (l-4 MeV) as a function of the incident ion’s atomic muss. Experimental data were extracted from ref. [3], assuming standard temperature and pressure and extrapolating to lower energies, where necessary, using a power law. A typical plot of the energy loss of all ions incident on a nitrogen gas versus the atomic mass for 2 MeV is shown in fig. 1. There extracted data are represented by points. A universal stopping power curve is drawn as the solid line. Fig. 2 illustrates the similar case for carbon films. Examination of the plot reveals that for small values of the ion’s mass (ml < 35 u) the stopping power increases with the ion mass following the trend of the ion’s nuclear charge (Z: to first order). However, as mi gets larger (> 35 u), the stopping power declines. This decline results from two effects: a lower effective ionic charge and less participation of the outer shell electrons of the stripper gas or foil. The net effect is to decrease the stopping power for ions of the same energy but larger mass. We used the scaling theory of Lindhard, Scharff and Schiott [4-61 (LSS) to compute a universal stopping power for all energies (1-4 MeV). In this theory the range and energy can be parameterized by p = 4~a2XNm,m2/(ml c = am2E/[
887
(1) (2)
+ m2)‘,
ZlZ2e2( ml + m2)],
8,
3E+4s zz -2E+4-
lE+4 __ ***.
l
Scaling Theory Experimental Data
OE+O ,,,,,,,~,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,~,,,,~, 0 50 100 M
150
200
250
(u)
Fig. 2. Tabulated energy loss of 2 MeV ions incident on a carbon foil vs atomic mass of the ion (points) with empirical fit (solid line).
where a = 0.8853ao(Zf/3 + Zz’3)-1/2, with a, the Bohr radius and Z2 the atomic number of the target atom, while X is the range [cm] for the material of number density N, and E is the ion energy in the laboratory frame. The Thomas-Fermi potential leads to a universal dimensionless stopping power, dr/dp, which can be expressed in terms of energy loss, dE/d X in [MeV/cm] as follows: dc/dp
= a dE/dX,
(3)
where a: = (ml + m2)/(4~NaZ1Z2e2m,) with N and m2 the number density and the mass of the target material, respectively. The atomic number of the ion, Z,, can be expressed as a function of the ion’s atomic mass (m,) using the most stable isobar of Z, given as [7,8]: Z, = 40m,/(80
+ 0.6mf/3).
(4)
The electronic energy loss (for a free electron gas) is expressed in reduced quantities as (dr/dp), = kc’/‘, for slow ions and where the constant k varies only slowly with Z, and Z,: k = 0.0793Z:/3Z:/2( my2my2
~ .**==
Scaling Theory Experimental Data
Fig. 1. Tabulated energy loss of 2 MeV ions incident on nitrogen vs atomic mass of the ion (points) with empirical fit (solid line). Standard temperature and pressure are assumed.
ml + m2)3’2
( zv3
+ z;/3
)3’4
.
(5)
The reduced energy loss de/dp was tabulated using eqs. (3) and (4), together with experimental and extrapolated data from the literature. A universal (empirical) equation was found that has the appropriate asymptotic behavior (a power law dr/dp a tP, p > 0 for small c and tending toward zero for large e). This equation is dc/dp
= kC,&/g(
c)
(6) X. MASS SPECTROMETRY
888
A.M. Arrale et al. / Energy loss corrections
Table 1 Parameters
p, and b, given in eq. (8).
Target
P2
Nitrogen Carbon
-0.02266 -0.01482
-0.0286 -0.0260
P3
-0.00212 -0.00130
P4
b,
- 1.749~ 1O-6 -7.616~10-~
7.436~ lo- ” 1.936x10-”
and g(e)
= (1 + c,r + c,t* + C,r3).
For the curves shown in this paper, five values of the coefficients C, were fitted for each energy. It was found that the coefficients were also a function of energy given by: C,=p,E+b,,
(8)
where C, is the respective coefficient (C,, C,, C,, C,, or C,), and E is the energy of the ion in MeV. The corresponding values of CL,and b, are given in table 1.
3. Experimental In order to test the validity and significance of the formalism described above, the relative energy loss for a variety of ions was measured for various gas stripper pressures. The energy loss was measured using a high resolution electrostatic energy analyzer described elsewhere [9]. The relative gas pressure in the stripper canal was inferred from the pressure indication at the injector end of the accelerator column pinj. The energy loss was found to be a linear function of the pressure pinj, for the range of values used (5 x 10m8 to 8 X 10e7 Torr). The slope (energy loss per l.tTorr) of this curve is proportional to the stopping power. After the energy loss was measured for several ions spanning the mass
0.94791 0.79834
bz
0.84838 0.85244
b,
0.01236 0.01043
b,
7.968 x 1O-6 2.920x 1O-6
- 3.023 x 10-t’ - 3.896 x10-r’
range from 1 to 80 u, a single parameter fit was made. The parameter was the average gas pressure in the stripper corresponding to a given Pinj, which was significantly lower. A pressure of 3.3 mTorr in the stripper was found to correspond to 1 x 10m7 Torr at the injector end of the column, a typical operating value. This value is quite reasonable. Fig. 3 reveals the qualitatively good agreement between the predictions of the empirical formula and the measured energy correction. Note that the maximum energy loss does occur for m, near 35 u. Energy losses of one to several keV are predicted for gas strippers, while for carbon foils of 2-5 pg/cm* one can expect energy losses of as much as lo-20 keV.
4. Conclusion We have obtained an empirical stopping power formula (parameterized with 10 parameters) that to an accuracy of better than - 10% gives stopping powers for all ions in nitrogen and in carbon for the energy range l-4 MeV. Comparisons with actual energy losses in a tandem accelerator are in good relative agreement. The measurements further indicate that for high and reproducible energy resolution the energy loss cannot be neglected.
References
__ l l l
0
20
Scaling Theory = l Experimental Data
40
60
80,
100
M (u) Fig. 3. Theoretical energy loss of 2 MeV ion, dE/dpi,,j, incident on nitrogen gas vs atomic mass of the ion. Also shown are the experimental data points for the energy loss in the gas stripper of a tandem accelerator. The quantity pinj is the accelerator injector pressure, which is proportional to the pressure of the gas in the stripper canal.
[l] M. Szilagyi, Electron and Ion Optics (Plenum, New York, 1988) p. 28. [2] L.C. Northcliffe and R.F. Schilhng, Nucl. Data Tables A7 (1970) 233. [3] J.F. Ziegler, Stopping Cross-section For Energetic Ions In AU Metals, vol. 5 (Pergamon, New York, 1980) p. 91. [4] J. Lindhard, M. Scharff and H. Schiott, K. Dan. Vidensk. Selsk. 33 (1963) no. 14. [5] J. Lindhard, V. Nielsen and M. Scharff, K. Dan. Vidensk. Selsk. 36 (1968) no. 10. [6] J. Lindhard and M. Scharff, Phys. Rev. 124 (1961) 128. [7] B.G. Harvey, Nuclear Physics and Chemistry, 2nd ed. (Prentice-Hall, Englewood Cliffs, NJ, 1969) p. 78. [8] P.A. Seeger, Nucl. Phys. 25 (1961) 1. [9] S. Matteson, F.D. McDaniel, J.L. Duggan, J.M. Anthony, T.J. Bennett and R.L. Beavers, Nucl. Instr. and Meth. B40/41 (1989) 759.