Schematic layout of the experimental setup for channeling studies at the fragment separator FRS. 

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Context 1

... to 300 MeV/u bare uranium ions is 2, i.e., greater than unity; a classical approach to calculation of energy transfers to target electrons is thus permitted ͓ 6 ͔ . The principal aims of the experiment presented in this paper are ͑ i ͒ to deduce EII cross sections from F C ( q out ) measurements for very high Z ions, for which simple perturbation treatments are questionable even at high velocities, ͑ ii ͒ to obtain experimental energy loss spectra as a function of q out for comparison with semiclassical calculation of energy loss in channeling. Our results on EII will be compared to various theoretical predictions and to other measurements, in particular to those of Claytor et al . ͓ 7 ͔ , who also performed channeling measurements with uranium ions at similar energies, but using nearly stripped incident ions. More generally, an extended review of charge exchange processes for heavy ions in channeling was recently given by Krause and Datz ͓ 8 ͔ . In this review, the EII results obtained by channeling and by other methods such as electron beam ion trap ͑ EBIT ͒ are compared to theory. In Sec. II we describe the experimental setup that allows charge state and energy analysis of the transmitted ions. In Sec. III we present our experimental data that include charge state distributions F ( q out ) and energy loss spectra for given q out , g ( ⌬ E ͉ q out ). In Sec. IV we describe a simulation code that enables us to calculate both F ( q out ) and g ( ⌬ E ͉ q out ). The comparison of simulated and measured profiles is presented in Sec. V and the results are compared to theoretical calculations. We have used relativistic uranium ion beams ͓ Z 1 ϭ 92, M 1 ϭ 238, E o ϭ 300 MeV/u, i.e., ␤ ϭ v / c ϭ 0.654 and ␥ ϭ (1 Ϫ ␤ 2 ) Ϫ 1/2 ϭ 1.32] with q in ϭ 73. The thickness t ϭ 120 ␮ m of the silicon ( Z 2 ϭ 14) single crystal target along the ͗ 110 ͘ axis was large enough to ensure a broad emergent charge state distribution F C ( q out ) ͑ i.e., a good E Ќ selection ͒ and measurable energy loss distributions for each emergent charge state. The experiment was performed at the heavy ion synchro- tron SIS at GSI ͑ Gesellschaft f ̈ r Schwerionenforschung, Darmstadt ͒ , which provided a 300 MeV/u U 73 ϩ beam. The projectiles were injected into a high-resolution magnetic spectrometer ͑ fragment separator FRS ͓ 9 ͔͒ , which consists of four ion optical stages each having one dipole magnet, five quadrupole magnets, and four hexapole magnets. The first two stages were used to prepare a beam of small angular divergence ͑ see Fig. 1 ͒ , which is a most important parameter in the experiment. The spot size on the silicon crystal was of the order of 10 ϫ 15 mm 2 . This large spot size was a necessary consequence of the optimization of the beam angular divergence. It had, however, minor influence on the experimental results as the rather thick silicon single crystal was large (20-mm diameter ͒ and x-ray topography measurements indicated negligible misalignment or mosaic spread. The beam intensity from SIS was typically Ϸ 10 6 ions per second. After charge and emittance selection by slits, it was Ϸ 10 2 ions per second on the crystal. The beam dose was calibrated and monitored using a scintillator outside the vacuum sys- tem, which measured secondary radiation due to the fraction of U ions hitting the slits. The Si crystal target was mounted on a remotely con- trolled, high-resolution, three-axis goniometer designed for ultrahigh vacuum, placed at S 2 , which could be moved with an accuracy of 0.01mrad. In the focal planes after the third and fourth stages of the FRS, the charge state distribution was measured using multiwire chambers ͓ multiwire proportional counter ͑ MWPC ͔͒ in S 3 and S 4 . The wire chambers were used to determine the integral, position, and shape of the peak of a specific exit charge q out state, giving information on the emergent charge state distribution, energy loss, and energy loss straggling. The correspondence between the position w ͑ in mm ͒ on the MWPC and the longitudinal momentum p of the particles, which depends on the measured rigidity Br ( B , magnetic field; r , bending radius ͒ , is given by the dispersion D ͑ in mm ͒ via ⌬ p / p ϭ ⌬ w / D for our ion optical setting. In the FRS, one has D 3 ϭ 1380 mm ͑ for S 3 ) and D 4 ϭ 9530 mm ͑ for S 4 ). The charge state distributions were measured at S 3 and energy loss spectra at S 4 . The variations ⌬ p of momentum and variations ⌬ E of energy are related by ⌬ E / E ϭ (1 ϩ 1/ ␥ ) ⌬ p / p ϭ 1.756 ⌬ p / p , which gives ⌬ E 4 ϭ 0.0553 MeV/u and ⌬ E 3 ϭ 0.38 MeV/u for ⌬ w ϭ 1 mm, respectively, in S 4 and S 3 . The alignment of the crystal for ͗ 110 ͘ channeling was achieved either by maximizing the frozen charge state 73 ϩ yield or by minimizing the 90 ϩ emerging charge state yield ͑ these yields were measured using a scintillator placed after the MWPC in S 4 , in relation to the monitor scintillator near S 1 ). In reality, the observation of the frozen charge state yield gives here a much more precise alignment than given by the observation of the 90 ϩ charge state yield ͑ see Sec. III ͒ . In Fig. 2 we present the measured yields of the emergent charge states q out ϭ 73 ͑ frozen ͒ and q out ϭ 90 as a function of the tilt angle ␸ o between the beam direction and the ͗ 110 ͘ axis. The angular distribution of the 73 ϩ ions is very narrow, with a half width at half maximum of ⌿ 73 1/2 ϩ ϭ 0.11 mrad. The observed angular scan for 90 ϩ emergent ions is dominated by particles with high transverse energy, which were mostly ionized by close encounters with target nuclei ͑ NII ͒ . The half width ⌿ ϭ 0.285 mrad of the 90 ϩ scan is thus, of course, larger than that of the 73 scan. 1/2 is clearly related to the transverse energy E Ќ c required to approach atomic strings at a distance of the order of ␳ th , the 2D rms of the thermal displacements of atoms perpendicular to the strings: E ⌿ 2 1/2 Ӎ E Ќ c Ӎ U s ( ␳ ), U s being the string potential. The value of the Lindhard relativistic critical angle ͓ 10 ͔ ␺ 1 ϭ ͱ 4 Z 1 Z 2 e 2 / p v d is 0.39 mrad. Thus, we find a value ⌿ 1/2 ϭ 0.73 ␺ 1 , somewhat smaller than extrapolated from numerical simulations ͓ 11 ͔ for a trial charge in silicon at room temperature, which gives ⌿ 1/2 ϭ 0.85 ␺ 1 . An upper limit of the critical transverse energy E Ќ 73 c ϩ to emerge in the frozen charge state q out ϭ 73 is related to E Ќ c by E Ќ 73 c ϩ ϭ ( ⌿ 73 1/2 ϩ / ⌿ 1/2 ) 2 E Ќ c Ӎ 0.15 E Ќ c . In fact, ⌿ 73 1/2 ϩ is mainly determined by the beam angular divergence, as will be demonstrated in Secs. IV and V, and thus E Ќ 73 c ϩ is certainly signifi- cantly smaller than this upper limit. The particles emerging with q out ϭ 73 are very well-channeled ions, but E Ќ selection through q is less accurate than that in Ref. ͓ 5 ͔ : the q 73 ions, which represent 3.8% of the emergent beam, are not all hyperchanneled ͑ see Secs. IV and V ͒ . The minimum yield of the 90 ϩ scan is Ӎ 5%. This is somewhat higher than the Ӎ 2% yield obtained in Monte Carlo calculations by Barrett ͓ 11 ͔ or measured with MeV light ions on the same crystal and axis ͓ 12 ͔ for close encounter events. This may be explained, at least partially, by the fact that NII is not the only process for q out ϭ 90 production; the binding energy of the L electrons of uranium is B L Ӎ 20 keV ͑ it depends on the charge state ͒ and the maximum energy transfer in a close encounter EII process is Ӎ Em e / M U ϭ 164 keV. Hence, EII alone may produce 90 ϩ uranium ions. Another important ...

Context 2

... second and third term in the bracket of Eq. 8 represent the usual relativistic correction of the first-order quan- tum perturbation theory. ⌬ L LS ( ␥ ) is a correction term representing the deviation to the perturbation theory, which has been recently calculated by Lindhard and S ” rensen ͓ 29 ͔ ͑ see Fig. 1 of this reference ͒ . The predicted value ⌬ L LS ( ␥ ) has been confirmed experimentally by Datz et al . ͓ 39 ͔ for ultrarelativistic Pb ions. In our case, ⌬ L LS represents ϩ 3% of the overall bracket term in Eq. ͑ 8 ͒ . Then, with q ϭ 73, Eq. ͑ 8 ͒ gives ⌬ E max v al ϭ 155.5 MeV ϭ A ϫ 0.654 MeV/u. This value, calculated with v al 4 electrons/atom, is an upper limit for the energy loss to the target valence electrons of channeled U 73 ϩ ͑ it corresponds to the random case ͒ . (b) Mean energy loss at E Ќ ϭ 0: contribution of valence and core electrons. A precise determination of the mean energy loss ⌬ E v al for a channeled beam may be reached by an impact parameter approach ͑ which is possible for ␬ Ͼ 1) and by integration in the channel using local electronic densities ␳ e ( r ជ Ќ ) ͑ here, averaging on ␳ e is performed along the ͗ 110 ͘ direction only ͒ . For symmetry reasons ͑ cylindrical geometry may be used ͒ , this type of calculation is tractable for ion trajectories just in the middle of the channel. This approach, leading to a calculated value based on the variations of ␳ v al with the distance from the channel center, was already used in ͓ 5 ͔ and is presented in detail in Ref. ͓ 3 ͔ . Using Eq. ͑ 23 ͒ of ͓ 3 ͔ , one ...