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Particle optics for a plasma-based beam focusing and transport system
Nuclear Instruments and Methods in Physics Research A 415 (1998) 496—502
Particle optics for a plasma-based beam focusing and transport system M. de Magistris!,*, A. Tauschwitz" ! Universita% di Napoli Federico II, Dipartimento di Ingegneria Elettrica Via Claudio 21, 80125 Napoli, Italy " Universita( t Erlangen-Nu( rnberg, Physikalisches Institut Erwin-Rommel-Str. 1, 91058 Erlangen, Germany
Abstract The ion optical properties of plasma-based final focusing and transport for a heavy ion ICF reactor are discussed. The advantages of plasma-based final focusing are mainly the structure of the field, the beam charge- and current neutralization and the reduction of beam rigidity due to stripping of electrons in the plasma. Two possible focusing modes are considered: the first one, more traditional, is the so-called coherent mode; it is based on the symmetric strong focusing which can be obtained with a cylindrical, or eventually shape optimized, “short” lens. The optical properties are in this case favorable compared to quadrupole lenses. The lens is, in particular, highly insensitive to chromatic aberrations. The disadvantage of this focusing scheme, in an ICF scenario, is the relatively short focal length, which requires a design with the lens integrated in the reactor chamber. A very interesting alternative is the use of a transport plasma channel, which can provide the required standoff distance from the pellet. In this case one can exploit the incoherent focusing of an adiabatic lens. The adiabatic lens has an extremely large acceptance both of beam emittance and momentum spread. Two examples of final focusing and transport systems adapted to different accelerator and target designs are presented in detail. ( 1998 Elsevier Science B.V. All rights reserved.
1. Introduction Plasma-based focusing and transport systems have often been considered as an alternative to traditional focusing devices in special applications that demand high focusing strength. In spite of the fact that they have been proven to be quite effective in a number of different situations, they are often still considered a risky option in beam line design.
* Corresponding author. Tel.: #39 81 768 3253; fax: #39 81 139 6897
This is mainly based on the insufficient understanding of processes related to beam—plasma interaction processes, as collisions, instabilities, charge and current neutralization, and on the assumption of a relatively poor stability and reliability of pulsed plasma discharges compared to traditional DC magnets. Nevertheless the advantages presented by the plasma lens solution can play a decisive role in the design of special accelerator applications [1]. We will devote our attention to the possibilities offered by plasma lenses and plasma channels in the design of a driver for a heavy ion ICF reactor. In particular, we will refer our examples to the
0168-9002/98/$19.00 ( 1998 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 0 0 2 ( 9 8 ) 0 0 4 2 4 - 0
M.D. Magistris, A. Tauschwitz/Nucl. Instr. and Meth. in Phys. Res. A 415 (1998) 496—502
typical parameters of the European Heavy Ion Driven Inertial Fusion (HIDIF) study group [2] and to a final focusing and transport system proposed from Berkeley [3]. In principle, plasma lens focusing can be considered for both, light and heavy ions. In the latter case plasma lenses have an additional advantage since, due to the stripping process in the plasma, the beam rigidity for heavy ions can be reduced by almost two orders of magnitude which leads to lower discharge currents. An important feature of plasma lens focusing is the charge neutralization of the beam ions in the discharge plasma. Going into the details of this process exceeds the scope of this paper by far and is a topic of open research. Let us recall that simplified models [4] as well as numerical simulations justify the assumption of practically instantaneous and complete beam neutralization in the plasma within the range of parameters considered for HIF. This has a tremendous impact on the design of the final transport and focusing since it allows to consider a much lower number of beam lines to the target. Because of the mentioned advantages, the choice of plasma lenses strongly relaxes the requirements for the accelerator, which can lead to much cheaper and simpler technical solutions for the construction of the driver. Moreover, the consideration of this kind of final focusing scheme is advantageous for the target chamber design since it offers the possibility to transport the whole driver beam in only one ore two narrow channels, depending on the target requirements.
2. Requirements of final focusing and transport for ICF It is quite clear that final focusing is a crucial issue for any ion-driven ICF design. The final lens has to handle extremely dense, space-charge-dominated beams and focus them to very small spots, of the order of some millimeters. It needs a large acceptance in both, the transverse and longitudinal phase space, it has to be sufficiently insensitive to any kind of aberration, and it has to provide the required standoff distance from the last focusing element to the target.
497
One of the most challenging problems are the high space-charge forces that appear for the required beam intensity close to the small target. For light ion beams this problem is far beyond the technological limits of present conventional multipole systems. For singly charged heavy ions with their higher mass-to-charge ratio, space charge is less but still represents a difficult problem which can only be solved by splitting the total required beam intensity into tenth of beams, to keep the space charge sufficiently low. Exploiting the space charge and current neutralization in a plasma lens it is possible to design completely different schemes which use, for example, two-sided illumination of the target, reducing the overall complexity and cost of the driver significantly. A second relevant topic are aberrations which have to be kept small enough to achieve a specified focal spot size. The focal spot is normally related to emittance and the effects of chromatic and other aberrations. Large focusing angles are required for a small emittance limit whereas small focusing angles are beneficial to reduce chromatic aberrations. The overall acceptance of the final focus system, once the required spot size at the target is fixed, is then often determined by these contradictory requirements. The plasma lens, as we will see in the following sections, has a large acceptance; in particular, the scheme with an adiabatic lens followed by a plasma channel is highly insensitive to beam emittance and momentum spread. One further important feature of final focusing based on a plasma discharge is the possibility to merge several individual beams in a neutralizing environment, reducing the number of beam paths to the target. This possibility is mainly related to the high acceptance of the system, which allows the transmission of beams with an initial angle to the focusing axis.
3. Focusing modes 3.1. Focusing with a short lens The plasma lens is based on an externally produced plasma discharge, coaxial with the beam. By
V. BEAMS
498
M.D. Magistris, A. Tauschwitz/Nucl. Instr. and Meth. in Phys. Res. A 415 (1998) 496—502
Fig. 1. Principal operation of a short plasma lens.
means of the azimuthal magnetic field produced by the discharge current, it is capable of focusing the beam symmetrically. In Fig. 1 this principle is sketched. From the ion optics point of view this configuration has some intrinsic advantages [5]. Assuming uniform distribution of the plasma current it is possible to calculate the particle trajectories analytically according to the simple equation [6] xA#k2x"0,
(1)
where k2"(k /2n)(I/R) with x the transverse coor0 dinate, I the plasma current, R the beam rigidity and k the vacuum magnetic permeability. If the 0 betatron wavelength k is long compared to the lens length, we have a focusing mode that is similar to conventional focusing. We want to call this mode “coherent” focusing to distinguish it from a different concept that we will introduce in the next section. From Eq. (1) it is straightforward to derive the major scalings for the focusing. In this regime the focusing is to the first order limited by the emittance and the plasma current according to the relationship x "g eI~1@2 (2) .*/ # where g "g (d/l) is a factor depending on the # # drift-over-length ratio, e.g. the focusing phase u"kl. In particular, the best focusing is achieved for a zero drift (u"p/2). The performance of such a lens is favorable for short drift distances. Moreover, tapering the lens opportunely toward the exit, e.g. k"k(z) improves the performance with all other parameters fixed; in fact, the spot scaling is again of the form expressed in Eq. (2) but with a reduced factor g, depending now additionally on the lens shape [7]. The last interesting point in the description of the coherent mode is the analysis of aberrations. Due
Fig. 2. Comparison of the chromatic aberrations limit to the emittance limit for the HIDIF parameters.
to the symmetric structure the lens has small geometric aberrations, which are noticeable only at very high radius over length ratios [8]. The field related aberrations are dependent on the degree of homogeneity in the discharge. For beams of high emittance these field-related aberrations can in special cases even have positive effects on the focal spot [9]. The reduced number of focusing elements, compared to multipole systems, as well as the reduced aperture for the same spot size, significantly minimizes the chromatic aberrations in this kind of system. This point could be of particular importance in the design of the HIDIF final focusing system, since the limitation on the spot size due to chromatic aberrations is far in excess of the limitations due to the emittance. A comparison of the chromatic aberrations limit to the emittance limit for the HIDIF parameters is shown in Fig. 2. 3.2. Adiabatic focusing It is possible to reduce the beam size in a way different from the traditional, coherent focusing. This second type of focusing can be achieved by slowly increasing the focusing strength along the lens. Correspondingly, the size of the beam envelope is reduced, regardless of any phase relation in the motion of the particles. The advantage of this incoherent focusing which is roughly sketched in Fig. 3, is that it works even for beams with a large momentum spread and high emittance. The obtainable reduction ratio x /x of the final & * beam radius x to the initial beam radius x is & * mainly determined by increasing the focusing
M.D. Magistris, A. Tauschwitz/Nucl. Instr. and Meth. in Phys. Res. A 415 (1998) 496—502
499
Fig. 3. Schematic of an adiabatic plasma lens with a transport channel.
strength from the entrance to the exit of the lens. Two possible mechanisms determine the focusing strength of a lens with a fixed current for a certain beam. The first is the current density of the discharge, which can be changed along the lens by tapering the discharge tube. The second is the charge state of the ions, which will, in general, change along the lens since the beam is stripped in the discharge gas. The connection between the focusing ratio and the tapering factor R /R , which is * & the ratio between the initial tube radius R and the * final tube radius R , is given in Eq. (3) &
AB AB
x R 1@2 Z 1@4 &" & * . x R Z * * &
(3)
To fulfill the adiabaticity condition, the lens has to be long compared to the betatron wavelength of the particles in the lens. A lower boundary for the length of the lens can be set by the condition that the length is at least equal to one betatron wavelength of the ions at the lens entrance. This corresponds to R /R oscillations in the higher field * & gradients at the end of the lens. This adiabaticity condition is given in Eq. (4)
A
¸"
B
8p3 R2R 1@2 * . k I 0
(4)
A minimum value for the discharge current, that is required to reduce the beam to the final discharge radius R is given by Liouville’s theorem of phase & space conservation, which sets the emittance limit for particle beam focusing Eq. (5)
AB
2p e 2 I5 R. k R 0 &
(5)
If a beam enters the lens under an angle to the lens axis, this angle can be converted to a corresponding emittance e in Eq. (5) which has then to be added to the emittance of the beam. This conversion is simply done be multiplying the angle by the radius of the beam at the lens entrance. The maximum angle under which a beam can enter the lens is important if several beams shall be combined in the lens. It is limited to values smaller than the focusing angle at the end of the lens. In general, the beam ions have a different phase of their betatron oscillation when they leave the lens. In a short lens this leads to chromatic aberrations. In the case of an adiabatic lens or a transport channel we will rather refer to this effect as “phase mixing”. A strong phase mixing usually leads to a concentration of the beam intensity in the central region of the discharge. There are several reasons for the phase mixing to occur. Even in the case of an ideal beam with zero emittance, without momentum spread, and a lens with a perfectly linear focusing field the effect does exist, although many oscillations and large angles of the ion trajectories with the lens axis are necessary, to lead to a noticeable effect. The reason for the phase mixing is in this case the coupling of longitudinal to transversal motion, which occurs predominantly for large angles [8]. Obviously, the phase mixing depends on the momentum spread of the beam. The dependence of the phase shift per oscillation on the momentum spread is very simple. With a betatron wavelength j of B j "2p(Rr/BU)1@2, (6) B the total phase shift at the end of a transport channel can be derived by calculating Lj /Lp. For B a channel with N oscillations the phase shift *j B V. BEAMS
500
M.D. Magistris, A. Tauschwitz/Nucl. Instr. and Meth. in Phys. Res. A 415 (1998) 496—502
becomes *j N*p B" . j 2 p B
(7)
To get a complete phase mixing it is necessary to have a total phase shift in the order of at least one betatron oscillation. In this case only a weak dependence of the beam intensity distribution remains on the length of the channel. For a channel with 20 oscillations this criterion is met for a momentum spread of $5%. Another source for phase mixing are non-linearities in the focusing field. In this case an analytical treatment is difficult because the effect depends on the kind and degree of the non-linearity. Monte-Carlo simulations with different field profiles show that realistic non-linearities, as they may occur in a transport channel, tend to form a halo around the beam. With increasing number of oscillations and increasing degree of non-linearity more and more ions are found in this halo, but the ions can still be confined in the focusing field around the transport channel.
4. Technical limitations In recent years five major plasma lens developments have taken place, covering a large variety of
discharge types and parameters as well as ion species and beam energies. The different conditions under which these plasma lenses have so far been successfully tested are summarized in Table 1. The experiments are sorted by the energy of the focused ion beam. The limits concerning the discharge parameters are not completely explored. Although the discharge currents in different plasma lens experiments were varying from 5 to 350 kA, the corresponding current densities range only from 10 to 75 kA/cm2. Higher current densities are of special interest for the adiabatic lens, but have not been investigated so far. There are no theoretical limits to the focal spot size of an adiabatic lens. In practice, the highest possible focusing ratio will be mainly determined by the fact that an excessive tapering of the discharge tube will result in a complicated hydrodynamic behavior which may lead to instabilities in the discharge plasma. In the only experimental test of adiabatic focusing moderate tapering ratios from 20 to 7 mm and from 20 to 5 mm have been tested successfully. Higher ratios are probably possible but have not been investigated. There are several other open questions. It is for example not known how far the diameter of a wall-stabilized discharge in a thin lens can be increased, without loosing the stabilizing influence of the wall and working in a classical z-pinch
Table 1 Characteristics of recent plasma lens developments
Berkeley 1995—1996 [X] GSI, 1990—1992 [X] GSI, 1991—1994 [X] GSI, 1994—1997 [X] CERN, 1991—1992 [X]
Accelerator
Ion
Energy
Discharge
ESQ-injector
K#
45 keV/u
Unilac
Au24#
11.4 MeV/u
Unilac
Au24#, Ar11#, Ni14#
5.6—15.4 MeV/u
SIS
Ne10#
100—300 MeV/u
PS/AC
protons
1.8 GeV
Adiabatic lens, Wall stabilized, 3.12 kA Short lens, z-pinch, 80—400 kA Short lens, Wall stabilized 10—25 kA Short lens, Wall stabilized, 150—350 kA Short lens, z-pinch 400—500 kA
501
M.D. Magistris, A. Tauschwitz/Nucl. Instr. and Meth. in Phys. Res. A 415 (1998) 496—502
regime. The largest beam aperture of a wall-stabilized discharge was 18 mm in diameter at a discharge diameter of 24 mm. Off-line magnetic field measurements indicate that higher apertures are possible. Additional problems have to be considered thinking of a reactor scenario with laser preionized transport channels. The most serious are electrical breakdown problems of the transport channels in the reactor chamber. The consequence of these problems are that long pulse rise times of the order of several microseconds have to be taken into account to keep the inductive voltage drop along the channel sufficiently small and that the number of channels in the reactor chamber is limited to a few. Finally, the lifetime of the discharge tube and the electrodes of the adiabatic lens have to be long enough to allow a repetitive use for at least several weeks. From the experience with the CERN plasma lens which underwent 20 000 discharges of 55 kJ energy during a one-week beam test [10] a lifetime of several million discharges can be expected.
5. Results of simulations A Monte-Carlo code calculating single-particle trajectories as well as an envelope code were used to study the final intensity distribution at the end of a transport channel, the effects of phase mixing, nonlinearities in the focusing field profile, and large momentum spread on the focusing process. Two
different options of interest for the work of the European HIDIF group [2] and to the plasma lens-based final focus and transport scheme from the Heavy Ion Fusion groups in Berkeley and Livermore [3] are summarized in Table 2. Both have used a Bi ion beam of 50 p mm mrad, stripped to the equilibrium charge state [11] for the used ion energy. In both cases the system consists of a 1 m long adiabatic lens and a 4 m long transport channel. For the HIDIF scenario a 10 GeV beam with a momentum spread of $1% is assumed, that is focused to a circular spot of 6 mm radius at the lens entrance with an angle of 10 mrad. The adiabatic lens is tapered from an initial radius of 20 mm to a final radius of 1.7 mm which reduces the beam diameter by a factor of 3.5. A total discharge current of 60 kA with a homogeneous current distribution is sufficient to focus the beam and transport it through the channel without particle losses. The Berkeley scenario uses a target design by Tabak [12] that requires beams of 3 and 4 GeV ion energy. Therefore, the simulation was done for 3.5 GeV beam energy with a momentum spread of $15%. The simulation assumes that four beams, each focused with an angle of 10 mrad coming from directions of $30 mrad to the lens axis are overlaid in a 10 mm radius spot at the entrance of the lens. The adiabatic section of the focusing system is tapered from 40 to 5 mm radius and reduces the beam diameter by a factor of two. A total current of 100 kA with a distribution that is pinched to half of
Table 2 Possible final focus scenarios adapted to the HIDF and Berkeley final focus concepts
Ion Energy Charge state [11] Rigidity Emittance *p/p Plasma current Lens length Channel length Spot radius Entrance beam radius
HIDIF
Berkeley
Determined by
Bi 10 GeV 79.2 2.6 Tm 50 p mm mrad $1% 60 kA 1m 4m 1.7 mm 6 mm
Bi 3.5 GeV 68.0 1.8 Tm 50 p mm mrad $15% 100 kA 1m 4m 5 mm 10 mm
Heaviest stable element Target design Stripping in plasma Equilibrium charge state Accelerator Accelerator and target design Channel driving voltage Adiabaticity condition Reactor diameter Target design Tapering limitations
V. BEAMS
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M.D. Magistris, A. Tauschwitz/Nucl. Instr. and Meth. in Phys. Res. A 415 (1998) 496—502
the insulator diameter is producing a sufficient focusing field to avoid particle losses. These very different examples show the large flexibility of the adiabatic lens that can be adapted to a wide variety of beam conditions and target requirements. References [1] E. Boggasch, B. Heimrich, D.H.H. Hoffmann, Nucl. Instr. Meth. 336 (1993) 438—41. [2] G. Plass, Nucl. Instr. and Meth. A 415 (1998) 204. [3] S.S. Yu et al., Nucl. Instr. and Meth. A 415 (1998) 174. [4] R.B. Miller, Intense Charged Particle Beams, Plenum Press, New York, 1982.
[5] M. Reiser, Theory and Design of Charged Particle Beams, Wiley, New York, 1994. [6] J.D. Lawson, The Physics of Charged Particle Beams, Clarendon Press, Oxford, 1988. [7] M. de Magistris, D.H.H Hoffmann, E. Boggasch, A. Tauschwitz, Il Nuovo Cimento 106 11 (1993) 1643—8. [8] M. de Magistris, L. De Menna, G. Miano, Proc. Europ. Part. Accel. Conf., June 1996. [9] M. de Magistris, L. De Menna, G. Miano, C. Serpico, Fus. Eng. Des. 1996, 32. [10] R. Kowalewicz et al., Proc. Int. Conf. on High Energy Acceleration, Hamburg, 1992 in Int. J. Mod. Phys. (Proc. Suppl.) (2A) (1993) 182. [11] H.D. Betz, in: H. Massey, E. Mcdaniel, B. Bederson (Eds.), Applied Atomic Collision Physics, Academic Press, Orlando, 1983, p. 1. [12] M. Tabak, D. Callahan-Miller, Nucl. Instr. and Meth. A 415 (1998) 75.
Particle optics for a plasma-based beam focusing and transport system M. de Magistris!,*, A. Tauschwitz" ! Universita% di Napoli Federico II, Dipartimento di Ingegneria Elettrica Via Claudio 21, 80125 Napoli, Italy " Universita( t Erlangen-Nu( rnberg, Physikalisches Institut Erwin-Rommel-Str. 1, 91058 Erlangen, Germany
Abstract The ion optical properties of plasma-based final focusing and transport for a heavy ion ICF reactor are discussed. The advantages of plasma-based final focusing are mainly the structure of the field, the beam charge- and current neutralization and the reduction of beam rigidity due to stripping of electrons in the plasma. Two possible focusing modes are considered: the first one, more traditional, is the so-called coherent mode; it is based on the symmetric strong focusing which can be obtained with a cylindrical, or eventually shape optimized, “short” lens. The optical properties are in this case favorable compared to quadrupole lenses. The lens is, in particular, highly insensitive to chromatic aberrations. The disadvantage of this focusing scheme, in an ICF scenario, is the relatively short focal length, which requires a design with the lens integrated in the reactor chamber. A very interesting alternative is the use of a transport plasma channel, which can provide the required standoff distance from the pellet. In this case one can exploit the incoherent focusing of an adiabatic lens. The adiabatic lens has an extremely large acceptance both of beam emittance and momentum spread. Two examples of final focusing and transport systems adapted to different accelerator and target designs are presented in detail. ( 1998 Elsevier Science B.V. All rights reserved.
1. Introduction Plasma-based focusing and transport systems have often been considered as an alternative to traditional focusing devices in special applications that demand high focusing strength. In spite of the fact that they have been proven to be quite effective in a number of different situations, they are often still considered a risky option in beam line design.
* Corresponding author. Tel.: #39 81 768 3253; fax: #39 81 139 6897
This is mainly based on the insufficient understanding of processes related to beam—plasma interaction processes, as collisions, instabilities, charge and current neutralization, and on the assumption of a relatively poor stability and reliability of pulsed plasma discharges compared to traditional DC magnets. Nevertheless the advantages presented by the plasma lens solution can play a decisive role in the design of special accelerator applications [1]. We will devote our attention to the possibilities offered by plasma lenses and plasma channels in the design of a driver for a heavy ion ICF reactor. In particular, we will refer our examples to the
0168-9002/98/$19.00 ( 1998 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 0 0 2 ( 9 8 ) 0 0 4 2 4 - 0
M.D. Magistris, A. Tauschwitz/Nucl. Instr. and Meth. in Phys. Res. A 415 (1998) 496—502
typical parameters of the European Heavy Ion Driven Inertial Fusion (HIDIF) study group [2] and to a final focusing and transport system proposed from Berkeley [3]. In principle, plasma lens focusing can be considered for both, light and heavy ions. In the latter case plasma lenses have an additional advantage since, due to the stripping process in the plasma, the beam rigidity for heavy ions can be reduced by almost two orders of magnitude which leads to lower discharge currents. An important feature of plasma lens focusing is the charge neutralization of the beam ions in the discharge plasma. Going into the details of this process exceeds the scope of this paper by far and is a topic of open research. Let us recall that simplified models [4] as well as numerical simulations justify the assumption of practically instantaneous and complete beam neutralization in the plasma within the range of parameters considered for HIF. This has a tremendous impact on the design of the final transport and focusing since it allows to consider a much lower number of beam lines to the target. Because of the mentioned advantages, the choice of plasma lenses strongly relaxes the requirements for the accelerator, which can lead to much cheaper and simpler technical solutions for the construction of the driver. Moreover, the consideration of this kind of final focusing scheme is advantageous for the target chamber design since it offers the possibility to transport the whole driver beam in only one ore two narrow channels, depending on the target requirements.
2. Requirements of final focusing and transport for ICF It is quite clear that final focusing is a crucial issue for any ion-driven ICF design. The final lens has to handle extremely dense, space-charge-dominated beams and focus them to very small spots, of the order of some millimeters. It needs a large acceptance in both, the transverse and longitudinal phase space, it has to be sufficiently insensitive to any kind of aberration, and it has to provide the required standoff distance from the last focusing element to the target.
497
One of the most challenging problems are the high space-charge forces that appear for the required beam intensity close to the small target. For light ion beams this problem is far beyond the technological limits of present conventional multipole systems. For singly charged heavy ions with their higher mass-to-charge ratio, space charge is less but still represents a difficult problem which can only be solved by splitting the total required beam intensity into tenth of beams, to keep the space charge sufficiently low. Exploiting the space charge and current neutralization in a plasma lens it is possible to design completely different schemes which use, for example, two-sided illumination of the target, reducing the overall complexity and cost of the driver significantly. A second relevant topic are aberrations which have to be kept small enough to achieve a specified focal spot size. The focal spot is normally related to emittance and the effects of chromatic and other aberrations. Large focusing angles are required for a small emittance limit whereas small focusing angles are beneficial to reduce chromatic aberrations. The overall acceptance of the final focus system, once the required spot size at the target is fixed, is then often determined by these contradictory requirements. The plasma lens, as we will see in the following sections, has a large acceptance; in particular, the scheme with an adiabatic lens followed by a plasma channel is highly insensitive to beam emittance and momentum spread. One further important feature of final focusing based on a plasma discharge is the possibility to merge several individual beams in a neutralizing environment, reducing the number of beam paths to the target. This possibility is mainly related to the high acceptance of the system, which allows the transmission of beams with an initial angle to the focusing axis.
3. Focusing modes 3.1. Focusing with a short lens The plasma lens is based on an externally produced plasma discharge, coaxial with the beam. By
V. BEAMS
498
M.D. Magistris, A. Tauschwitz/Nucl. Instr. and Meth. in Phys. Res. A 415 (1998) 496—502
Fig. 1. Principal operation of a short plasma lens.
means of the azimuthal magnetic field produced by the discharge current, it is capable of focusing the beam symmetrically. In Fig. 1 this principle is sketched. From the ion optics point of view this configuration has some intrinsic advantages [5]. Assuming uniform distribution of the plasma current it is possible to calculate the particle trajectories analytically according to the simple equation [6] xA#k2x"0,
(1)
where k2"(k /2n)(I/R) with x the transverse coor0 dinate, I the plasma current, R the beam rigidity and k the vacuum magnetic permeability. If the 0 betatron wavelength k is long compared to the lens length, we have a focusing mode that is similar to conventional focusing. We want to call this mode “coherent” focusing to distinguish it from a different concept that we will introduce in the next section. From Eq. (1) it is straightforward to derive the major scalings for the focusing. In this regime the focusing is to the first order limited by the emittance and the plasma current according to the relationship x "g eI~1@2 (2) .*/ # where g "g (d/l) is a factor depending on the # # drift-over-length ratio, e.g. the focusing phase u"kl. In particular, the best focusing is achieved for a zero drift (u"p/2). The performance of such a lens is favorable for short drift distances. Moreover, tapering the lens opportunely toward the exit, e.g. k"k(z) improves the performance with all other parameters fixed; in fact, the spot scaling is again of the form expressed in Eq. (2) but with a reduced factor g, depending now additionally on the lens shape [7]. The last interesting point in the description of the coherent mode is the analysis of aberrations. Due
Fig. 2. Comparison of the chromatic aberrations limit to the emittance limit for the HIDIF parameters.
to the symmetric structure the lens has small geometric aberrations, which are noticeable only at very high radius over length ratios [8]. The field related aberrations are dependent on the degree of homogeneity in the discharge. For beams of high emittance these field-related aberrations can in special cases even have positive effects on the focal spot [9]. The reduced number of focusing elements, compared to multipole systems, as well as the reduced aperture for the same spot size, significantly minimizes the chromatic aberrations in this kind of system. This point could be of particular importance in the design of the HIDIF final focusing system, since the limitation on the spot size due to chromatic aberrations is far in excess of the limitations due to the emittance. A comparison of the chromatic aberrations limit to the emittance limit for the HIDIF parameters is shown in Fig. 2. 3.2. Adiabatic focusing It is possible to reduce the beam size in a way different from the traditional, coherent focusing. This second type of focusing can be achieved by slowly increasing the focusing strength along the lens. Correspondingly, the size of the beam envelope is reduced, regardless of any phase relation in the motion of the particles. The advantage of this incoherent focusing which is roughly sketched in Fig. 3, is that it works even for beams with a large momentum spread and high emittance. The obtainable reduction ratio x /x of the final & * beam radius x to the initial beam radius x is & * mainly determined by increasing the focusing
M.D. Magistris, A. Tauschwitz/Nucl. Instr. and Meth. in Phys. Res. A 415 (1998) 496—502
499
Fig. 3. Schematic of an adiabatic plasma lens with a transport channel.
strength from the entrance to the exit of the lens. Two possible mechanisms determine the focusing strength of a lens with a fixed current for a certain beam. The first is the current density of the discharge, which can be changed along the lens by tapering the discharge tube. The second is the charge state of the ions, which will, in general, change along the lens since the beam is stripped in the discharge gas. The connection between the focusing ratio and the tapering factor R /R , which is * & the ratio between the initial tube radius R and the * final tube radius R , is given in Eq. (3) &
AB AB
x R 1@2 Z 1@4 &" & * . x R Z * * &
(3)
To fulfill the adiabaticity condition, the lens has to be long compared to the betatron wavelength of the particles in the lens. A lower boundary for the length of the lens can be set by the condition that the length is at least equal to one betatron wavelength of the ions at the lens entrance. This corresponds to R /R oscillations in the higher field * & gradients at the end of the lens. This adiabaticity condition is given in Eq. (4)
A
¸"
B
8p3 R2R 1@2 * . k I 0
(4)
A minimum value for the discharge current, that is required to reduce the beam to the final discharge radius R is given by Liouville’s theorem of phase & space conservation, which sets the emittance limit for particle beam focusing Eq. (5)
AB
2p e 2 I5 R. k R 0 &
(5)
If a beam enters the lens under an angle to the lens axis, this angle can be converted to a corresponding emittance e in Eq. (5) which has then to be added to the emittance of the beam. This conversion is simply done be multiplying the angle by the radius of the beam at the lens entrance. The maximum angle under which a beam can enter the lens is important if several beams shall be combined in the lens. It is limited to values smaller than the focusing angle at the end of the lens. In general, the beam ions have a different phase of their betatron oscillation when they leave the lens. In a short lens this leads to chromatic aberrations. In the case of an adiabatic lens or a transport channel we will rather refer to this effect as “phase mixing”. A strong phase mixing usually leads to a concentration of the beam intensity in the central region of the discharge. There are several reasons for the phase mixing to occur. Even in the case of an ideal beam with zero emittance, without momentum spread, and a lens with a perfectly linear focusing field the effect does exist, although many oscillations and large angles of the ion trajectories with the lens axis are necessary, to lead to a noticeable effect. The reason for the phase mixing is in this case the coupling of longitudinal to transversal motion, which occurs predominantly for large angles [8]. Obviously, the phase mixing depends on the momentum spread of the beam. The dependence of the phase shift per oscillation on the momentum spread is very simple. With a betatron wavelength j of B j "2p(Rr/BU)1@2, (6) B the total phase shift at the end of a transport channel can be derived by calculating Lj /Lp. For B a channel with N oscillations the phase shift *j B V. BEAMS
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becomes *j N*p B" . j 2 p B
(7)
To get a complete phase mixing it is necessary to have a total phase shift in the order of at least one betatron oscillation. In this case only a weak dependence of the beam intensity distribution remains on the length of the channel. For a channel with 20 oscillations this criterion is met for a momentum spread of $5%. Another source for phase mixing are non-linearities in the focusing field. In this case an analytical treatment is difficult because the effect depends on the kind and degree of the non-linearity. Monte-Carlo simulations with different field profiles show that realistic non-linearities, as they may occur in a transport channel, tend to form a halo around the beam. With increasing number of oscillations and increasing degree of non-linearity more and more ions are found in this halo, but the ions can still be confined in the focusing field around the transport channel.
4. Technical limitations In recent years five major plasma lens developments have taken place, covering a large variety of
discharge types and parameters as well as ion species and beam energies. The different conditions under which these plasma lenses have so far been successfully tested are summarized in Table 1. The experiments are sorted by the energy of the focused ion beam. The limits concerning the discharge parameters are not completely explored. Although the discharge currents in different plasma lens experiments were varying from 5 to 350 kA, the corresponding current densities range only from 10 to 75 kA/cm2. Higher current densities are of special interest for the adiabatic lens, but have not been investigated so far. There are no theoretical limits to the focal spot size of an adiabatic lens. In practice, the highest possible focusing ratio will be mainly determined by the fact that an excessive tapering of the discharge tube will result in a complicated hydrodynamic behavior which may lead to instabilities in the discharge plasma. In the only experimental test of adiabatic focusing moderate tapering ratios from 20 to 7 mm and from 20 to 5 mm have been tested successfully. Higher ratios are probably possible but have not been investigated. There are several other open questions. It is for example not known how far the diameter of a wall-stabilized discharge in a thin lens can be increased, without loosing the stabilizing influence of the wall and working in a classical z-pinch
Table 1 Characteristics of recent plasma lens developments
Berkeley 1995—1996 [X] GSI, 1990—1992 [X] GSI, 1991—1994 [X] GSI, 1994—1997 [X] CERN, 1991—1992 [X]
Accelerator
Ion
Energy
Discharge
ESQ-injector
K#
45 keV/u
Unilac
Au24#
11.4 MeV/u
Unilac
Au24#, Ar11#, Ni14#
5.6—15.4 MeV/u
SIS
Ne10#
100—300 MeV/u
PS/AC
protons
1.8 GeV
Adiabatic lens, Wall stabilized, 3.12 kA Short lens, z-pinch, 80—400 kA Short lens, Wall stabilized 10—25 kA Short lens, Wall stabilized, 150—350 kA Short lens, z-pinch 400—500 kA
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regime. The largest beam aperture of a wall-stabilized discharge was 18 mm in diameter at a discharge diameter of 24 mm. Off-line magnetic field measurements indicate that higher apertures are possible. Additional problems have to be considered thinking of a reactor scenario with laser preionized transport channels. The most serious are electrical breakdown problems of the transport channels in the reactor chamber. The consequence of these problems are that long pulse rise times of the order of several microseconds have to be taken into account to keep the inductive voltage drop along the channel sufficiently small and that the number of channels in the reactor chamber is limited to a few. Finally, the lifetime of the discharge tube and the electrodes of the adiabatic lens have to be long enough to allow a repetitive use for at least several weeks. From the experience with the CERN plasma lens which underwent 20 000 discharges of 55 kJ energy during a one-week beam test [10] a lifetime of several million discharges can be expected.
5. Results of simulations A Monte-Carlo code calculating single-particle trajectories as well as an envelope code were used to study the final intensity distribution at the end of a transport channel, the effects of phase mixing, nonlinearities in the focusing field profile, and large momentum spread on the focusing process. Two
different options of interest for the work of the European HIDIF group [2] and to the plasma lens-based final focus and transport scheme from the Heavy Ion Fusion groups in Berkeley and Livermore [3] are summarized in Table 2. Both have used a Bi ion beam of 50 p mm mrad, stripped to the equilibrium charge state [11] for the used ion energy. In both cases the system consists of a 1 m long adiabatic lens and a 4 m long transport channel. For the HIDIF scenario a 10 GeV beam with a momentum spread of $1% is assumed, that is focused to a circular spot of 6 mm radius at the lens entrance with an angle of 10 mrad. The adiabatic lens is tapered from an initial radius of 20 mm to a final radius of 1.7 mm which reduces the beam diameter by a factor of 3.5. A total discharge current of 60 kA with a homogeneous current distribution is sufficient to focus the beam and transport it through the channel without particle losses. The Berkeley scenario uses a target design by Tabak [12] that requires beams of 3 and 4 GeV ion energy. Therefore, the simulation was done for 3.5 GeV beam energy with a momentum spread of $15%. The simulation assumes that four beams, each focused with an angle of 10 mrad coming from directions of $30 mrad to the lens axis are overlaid in a 10 mm radius spot at the entrance of the lens. The adiabatic section of the focusing system is tapered from 40 to 5 mm radius and reduces the beam diameter by a factor of two. A total current of 100 kA with a distribution that is pinched to half of
Table 2 Possible final focus scenarios adapted to the HIDF and Berkeley final focus concepts
Ion Energy Charge state [11] Rigidity Emittance *p/p Plasma current Lens length Channel length Spot radius Entrance beam radius
HIDIF
Berkeley
Determined by
Bi 10 GeV 79.2 2.6 Tm 50 p mm mrad $1% 60 kA 1m 4m 1.7 mm 6 mm
Bi 3.5 GeV 68.0 1.8 Tm 50 p mm mrad $15% 100 kA 1m 4m 5 mm 10 mm
Heaviest stable element Target design Stripping in plasma Equilibrium charge state Accelerator Accelerator and target design Channel driving voltage Adiabaticity condition Reactor diameter Target design Tapering limitations
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the insulator diameter is producing a sufficient focusing field to avoid particle losses. These very different examples show the large flexibility of the adiabatic lens that can be adapted to a wide variety of beam conditions and target requirements. References [1] E. Boggasch, B. Heimrich, D.H.H. Hoffmann, Nucl. Instr. Meth. 336 (1993) 438—41. [2] G. Plass, Nucl. Instr. and Meth. A 415 (1998) 204. [3] S.S. Yu et al., Nucl. Instr. and Meth. A 415 (1998) 174. [4] R.B. Miller, Intense Charged Particle Beams, Plenum Press, New York, 1982.
[5] M. Reiser, Theory and Design of Charged Particle Beams, Wiley, New York, 1994. [6] J.D. Lawson, The Physics of Charged Particle Beams, Clarendon Press, Oxford, 1988. [7] M. de Magistris, D.H.H Hoffmann, E. Boggasch, A. Tauschwitz, Il Nuovo Cimento 106 11 (1993) 1643—8. [8] M. de Magistris, L. De Menna, G. Miano, Proc. Europ. Part. Accel. Conf., June 1996. [9] M. de Magistris, L. De Menna, G. Miano, C. Serpico, Fus. Eng. Des. 1996, 32. [10] R. Kowalewicz et al., Proc. Int. Conf. on High Energy Acceleration, Hamburg, 1992 in Int. J. Mod. Phys. (Proc. Suppl.) (2A) (1993) 182. [11] H.D. Betz, in: H. Massey, E. Mcdaniel, B. Bederson (Eds.), Applied Atomic Collision Physics, Academic Press, Orlando, 1983, p. 1. [12] M. Tabak, D. Callahan-Miller, Nucl. Instr. and Meth. A 415 (1998) 75.