11 Application of Accelerators and Storage Rings
763
physically reasonable criterion has been proven to be a good first-order approximation for estimating the charge states which can be reached. Semi-empirical
approximation formulas were developed later to describe the energy and target
dependence of the equilibrium charge and the width of charge-state distributions.
For the low energy range, that means for energies from about 1 to some 10 MeV/u,
the equilibrium charge state for heavy ions can be approximately described by a
simple parameterized formula of the type
1 − (q/Z P ) = C exp (−δβ/α) ,
(11.90)
with q = average charge state, Z P = nuclear charge of the ion, β = v/c, α = 1/137,
δ = Z
−γ
P , see among others [194, 195]. For the stripping data gathered for example
in the operation of the Unilac (GSI) over the years, this formula is used with
C = C ∗ + 140/Z 2
P , C ∗ = 1.0285. Averaging the experimental data, γ-values of
0.56 for foil and 0.65 for nitrogen-gas-jet data have been calculated [196]. Foil
strippers deliver higher charge-states, because the higher collision frequency in
solid materials leads to higher electron loss probability [195], but also suffer from
degradation due to beam intensity. For the range of 0.2 ≤ q/Z P ≥ 0.8, which means
if the ion has a sufficient number of electrons with comparable binding energy, the
charge state distribution is roughly Gaussian in shape. The width of the Gaussian
curve is proportional to Z
1/2
P (see e.g. [195]).
Simple exponential relations for the average charge begin to deviate from
measured data, and charge state distributions are no longer Gaussian-like in shape,
when the electron configuration of the ion approaches inner shells with large steps
in ionization energies. This was discovered in the case of bromine ions by C.D.
Moak et al. in 1967 when approaching the L-shell by the stripping process in the
energy range between 1 and 2 MeV/u [197]. Therefore, after estimations with simple
semi-empirical formulas, experimental data should be taken into account whenever
available.
For beams of energies in the range from 10 MeV/u to 80 MeV/u, the computer
program ETACHA was developed by Rozet et al. to calculate charge-exchange
cross-sections, charge-state evolutions in targets, and equilibrium charge-state
distributions for highly charged ions, taking into account the electronic structure of
inner shells [198]. Experimental results in this energy range can be found in [199],
for example.
An overview of theoretical and experimental results for Xe, Au, and U projectiles
impinging with kinetic energies from 80 to 1000 MeV/u on solid and gaseous targets
ranging from Be to U is given in [200]. The calculations are compared to data from
experiments carried out at the BEVALAC (LBL) and at the heavy ion synchrotron
SIS (GSI) and associated facilities. Measured equilibrium charge state distributions
for uranium in the energy range from 1 to 1000 MeV/u are shown in Fig. 11.42
[200].
For high intensity beams, gas or liquid strippers have advantages compared to foil
strippers concerning their durability. Gas strippers lead to much lower equilibrium
763
physically reasonable criterion has been proven to be a good first-order approximation for estimating the charge states which can be reached. Semi-empirical
approximation formulas were developed later to describe the energy and target
dependence of the equilibrium charge and the width of charge-state distributions.
For the low energy range, that means for energies from about 1 to some 10 MeV/u,
the equilibrium charge state for heavy ions can be approximately described by a
simple parameterized formula of the type
1 − (q/Z P ) = C exp (−δβ/α) ,
(11.90)
with q = average charge state, Z P = nuclear charge of the ion, β = v/c, α = 1/137,
δ = Z
−γ
P , see among others [194, 195]. For the stripping data gathered for example
in the operation of the Unilac (GSI) over the years, this formula is used with
C = C ∗ + 140/Z 2
P , C ∗ = 1.0285. Averaging the experimental data, γ-values of
0.56 for foil and 0.65 for nitrogen-gas-jet data have been calculated [196]. Foil
strippers deliver higher charge-states, because the higher collision frequency in
solid materials leads to higher electron loss probability [195], but also suffer from
degradation due to beam intensity. For the range of 0.2 ≤ q/Z P ≥ 0.8, which means
if the ion has a sufficient number of electrons with comparable binding energy, the
charge state distribution is roughly Gaussian in shape. The width of the Gaussian
curve is proportional to Z
1/2
P (see e.g. [195]).
Simple exponential relations for the average charge begin to deviate from
measured data, and charge state distributions are no longer Gaussian-like in shape,
when the electron configuration of the ion approaches inner shells with large steps
in ionization energies. This was discovered in the case of bromine ions by C.D.
Moak et al. in 1967 when approaching the L-shell by the stripping process in the
energy range between 1 and 2 MeV/u [197]. Therefore, after estimations with simple
semi-empirical formulas, experimental data should be taken into account whenever
available.
For beams of energies in the range from 10 MeV/u to 80 MeV/u, the computer
program ETACHA was developed by Rozet et al. to calculate charge-exchange
cross-sections, charge-state evolutions in targets, and equilibrium charge-state
distributions for highly charged ions, taking into account the electronic structure of
inner shells [198]. Experimental results in this energy range can be found in [199],
for example.
An overview of theoretical and experimental results for Xe, Au, and U projectiles
impinging with kinetic energies from 80 to 1000 MeV/u on solid and gaseous targets
ranging from Be to U is given in [200]. The calculations are compared to data from
experiments carried out at the BEVALAC (LBL) and at the heavy ion synchrotron
SIS (GSI) and associated facilities. Measured equilibrium charge state distributions
for uranium in the energy range from 1 to 1000 MeV/u are shown in Fig. 11.42
[200].
For high intensity beams, gas or liquid strippers have advantages compared to foil
strippers concerning their durability. Gas strippers lead to much lower equilibrium