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The particle dynamics in the electron cooling system based on the modified betatron
Nuclear Instruments and Methods in Physics Research A 441 (2000) 267}270
The particle dynamics in the electron cooling system based on the modi"ed betatron I. Meshkov, A. Sidorin, A. Smirnov*, E. Syresin, G. Trubnikov Joint Institute for Nuclear Research, 141980 Dubna, Russia
Abstract The electron cooling system with circulating electron beam is considered for the cooling of ions of energy of few GeV. The prototype of the electron cooling system based on modi"ed betatron for cooling of antiprotons in the Recycler ring (Fermilab) is under development in the JINR. The peculiarities of the modi"ed betatron are its section structure, the longitudinal guiding magnetic "eld and the quadrupole spiral "eld used to form a closed orbit. The longitudinal magnetic "eld provides the electron magnetisation and long lifetime of the circulating electrons as the strong coupling between the horizontal and vertical degrees of freedom and leads to additional resonances. ( 2000 Elsevier Science B.V. All rights reserved. PACS: 29.27.Bd Keywords: Electron cooling; Modi"ed betatron; Particle dynamics
1. Introduction To avoid the problems of a traditional electron cooling system and to cool down the antiprotons at an energy of 8 GeV, an electron cooling system with circulating electron beams was proposed in Refs. [1,2]. In this system the Recycler ring is equipped with an additional electron one, which is periodically "lled up with a new portion of cold electrons. The electron beam circulates in longitudinal (quasitoroidal) magnetic "eld, and the longterm stability of the beam is provided with additional spiral coils, which form a quadrupole magnetic "eld. Such a focusing system is similar * Corresponding author. Tel.: #7-09261-64495; fax: #709621-66666. E-mail address: [email protected] (A. Smirnov)
to the `stelaratora one. The acceleration of the electron beam without the distortions caused by RF system of linear accelerators is achieved by using induction acceleration. The particle dynamics in a similar circular electron accelerator, the socalled `modi"ed betatrona, was investigated carefully } theoretically and experimentally (see, for instance Refs. [3,4]). In order to test the medium-energy electron cooling system based on modi"ed betatron, the design and manufactory of such a system prototype was started at the Joint Institute for Nuclear Research. The Modi"ed Betatron Prototype (MOBY) [5] is an electron induction accelerator with electron energy of 4.36 MeV and longitudinal magnetic "eld of 1 kG and additional quadrupole spiral magnetic "eld. The general di!erence between the MOBY and the classical modi"ed betatron is its section
0168-9002/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 0 0 2 ( 9 9 ) 0 1 1 4 3 - 2
SECTION V.
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I. Meshkov et al. / Nuclear Instruments and Methods in Physics Research A 441 (2000) 267}270
structure. The MOBY consists of two toroidal and two straight sections connected together as a racetrack. Therefore, it is impossible to use results previously obtained for the classical modi"ed betatron. The usage of the usual software (like MAD) is di$cult due to the presence of the longitudinal magnetic "eld in all optic elements, which produces strong coupling between the vertical and horizontal degrees of freedom. To investigate the particle dynamics in the MOBY the program software named BETATRON was especially elaborated by the JINR electron cooling group. The BETATRON is based on the Beam Optics Library with Integrated Development Environment (BOLIDE) system designed to develop physical programs related to the charged particle beam dynamics in the optic channels, storage rings, etc. The results of the simulations using the BETATRON program are presented in this article.
2. Focusing structure of modi5ed betatron prototype Below we consider the structure in which the radii of both toroidal sections are equal to 1.46 m and both straight sections are of the length of 4.56 m. One of the straight sections is free of the spiral quadrupole winding, the number of the spiral steps being equal to 2 in the other section. Two variants of the betatron focusing structures with the same e!ective value of the quadrupole "eld gradient are considered. First variant (unsmoothed) has
the jump of the focusing "eld gradient between sections, second variant (smoothed) has the adiabatic (linear) changing of a quadrupole "eld gradient at the exist and the entrance of the cooling section. Linear particle dynamics was investigated by the matrix method. The dependencies of dispersion and lattice functions on focusing gradient (Fig. 1a and b) show, that to obtain small dispersion in the cooling section the quadrupole gradient value has to lie between 10 and 15 G/cm. In this case, for smoothed structure the ellipticity of the beam cross section is less than 20%.
3. Angular spread of the circulating electron beam When the stability conditions are satis"ed the angular spread of the circulating electron beam in the section with quadrupole spiral winding could be estimated as the angle between the "eld line and longitudinal axis h+(Gr )/B [6]. This value lies "%!. between 10~3 and 10~2 for typical value of the quadrupole "eld gradient G&5}10 G/cm and the longitudinal magnetic "eld B"1 kG. (Fig. 2, unsmoothed). To e!ectively cool down the antiprotons in the Recycler the angular spread of the electron beam has to be of the order of magnitude 10~4 or even less. It means, that the electron cooling section has to be free on the quadrupole spiral winding. When the particle motion is stable the possibility to have an electron beam of small angular spread is the use
Fig. 1. The dependencies of average magnitudes of dispersion (a) and ratio of the ellipse axes of the beam cross section (b) on quadrupole gradient E"4.3 MeV, I"0 A.
I. Meshkov et al. / Nuclear Instruments and Methods in Physics Research A 441 (2000) 267}270
Fig. 2. The dependence of the maximal angle of the electron in the cooling section on the quadrupole "eld gradient. E"4.3 MeV, I"0 A, B"1 kG, r "1 cm, y "1 cm, "%!. 0 x "x@ "y@ "0, *p/p"0. 0 0 0
of the adiabatic variation of quadrupole "eld gradient at the exit and the entrance of the cooling section, which is free of quadrupole spiral "eld (Fig. 2, smoothed) [7]. Beam injection also has to be performed in the section free of quadrupole "eld.
269
Fig. 3. Computer simulation of the stability diagram, B"1 kG.
below the "rst resonance of the quadrupole rotation can provide the long-term stability of the circulating electron beam.
5. Tune shift 4. The transverse stability of electron motion When the number of the spiral winding steps on each focusing section is an integer, the particle motion in the "rst approximation consists of two independent motions: Larmor rotation of each electron, whose tune is equal to Q +C/2po, and L slow betatron rotation of the beam with a tune Q +(G/B)2¸/2k, where C is the MOBY circum"%5 ference, L is the sum of the lengths of the sections with spiral "eld, k"2p/h, h } step of the quadrupole winding. Stability conditions can be written as follows: n m Q O , Q O , Q $Q Ol, L 2 "%5 L "%5 2
(1)
where n, m, l are integers. The diagram of the particle motion stability can be plotted on the plane `Gradienta of the focusing "eld versus `Energya of the electrons (Fig. 3). The dark lines on this plot correspond to the resonances, where the module of a ring matrix eigenvalue is not equal to unit. The choice of the working point between resonances of the Larmor rotation and
Coulomb tune shift in the uniform modi"ed betatron, which consists of only toroid with quadrupole winding, can be calculated analytically by the expansion of the eigenfrequencies in the Tailor series. In the general case, when the eigenfrequencies are di!erent in di!erent sections of the structure, tune shift is evaluated by the calculation of the eigenvalues of the ring transformation map depending on beam current. The Coulomb tune shift calculated for injection electron energy and round beam of radius of 1 cm is presented in Fig. 4. It should be noted, that the tune shift can be compensated by slow changing of the longitudinal "eld and the quadrupole gradient values during acceleration. At maximal beam energy the Coulomb tune shift is negligible for beam current of several amperes.
6. Beam dynamics during acceleration In the "rst approximation, the betatron tune Q does not depend on particle energy only on the "%5 gradient of the spiral quadrupole "eld. Thus, when the gradient lies between 5 and 20 G/cm the particles do not cross any resonance of the betatron
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Fig. 4. The dependence of coherent tune shift on beam current. E"100 keV, G"10 G/cm.
mode of oscillations during acceleration (Fig. 3). Only at small energy and large beam current these resonances can appear due to space-charge e!ects. These resonances can be avoided by the choice of injection energy of about 100 keV for beam current of about 0.1 A. However, at constant value of the guiding magnetic "eld particles cross a large number of integer and half-integer resonances of fast Larmor mode of oscillations and coupling resonances during acceleration. High-order resonances can be avoided by the choice of energy increase during one revolution in the ring being substantially higher than the resonance width. However, for low-order resonance this way is not realistic due to technical limitations of the acceleration rate. The damage of the beam quality can be prevented by varying the guiding magnetic "eld value in accordance with the condition Q "const. L For such a regime the injection energy of several hundreds of keV is preferable. The magnetic "eld value has to be increased during acceleration from several hundreds to several thousands of Gauss. Therefore, the electron beam radius decreases and transverse momentum spread increases proportionally to the square root of the magnetic "eld value. To obtain the beam with low temperature the cathode of the electron gun has to be placed in the magnetic "eld, which has the value equal to the maximal one in the ring after acceleration.
7. Conclusion In the focusing structure of MOBY with adiabatic variation of quadrupole "eld at the exit
and at the entrance of the cooling section, which is free of quadrupole spiral "eld, the angular spread of the circulating electron beam lies between 10~3 and 10~4 rad. At the focusing gradient value of about 10}15 G/cm the horizontal dispersion in the cooling section is about 40 cm, and beam cross section has almost a round shape. Acknowledgements This work was supported by the Russian Foundation for Basic Research (Grant N 96-02-17211, N 99-02-17716) and INTAS (Grant N 96-0966). References [1] G. Jackson, Modi"ed betatron approach to electron cooling, Proceedings of the International Workshop on Medium Energy Electron Cooling, Novosibirsk, 1997, p. 171. [2] I. Meshkov, A. Sidorin, Electron cooling system with circulating electron beam, Proceedings of the International Workshop on Medium Energy Electron Cooling, Novosibirsk, 1997, p. 183. [3] C.W. Robertson, A. Mondelli, D. Chernin, High-current betatron with stellarator "elds, Phys. Rev. Lett. 50 (7) (1983) 507. [4] C. Kapetanakos et al., Phys. Fluids B 5 (7) (1993) 2295. [5] Yu. Korotaev, I. Meshkov, S. Mironov, A. Sidorin, E. Syresin, The modi"ed betatron prototype dedicated to electron cooling, Proceedings of the Sixth EPAC, Stockholm, 22}26 June, 1998, p. 1061. [6] I. Meshkov, Electron cooling with circulating electron beam in GeV energy range, Proceedings of HEACC'98, Dubna, 1999, p. 409. [7] I. Meshkov, A. Sidorin, A. Smirnov, E. Syresin, E. Musta"n, P. Zenkevich, Stability of the electron beam in the electron cooling system based on the modi"ed betatron, Proceedings of HEACC'98, Dubna, 1999, p. 416.
The particle dynamics in the electron cooling system based on the modi"ed betatron I. Meshkov, A. Sidorin, A. Smirnov*, E. Syresin, G. Trubnikov Joint Institute for Nuclear Research, 141980 Dubna, Russia
Abstract The electron cooling system with circulating electron beam is considered for the cooling of ions of energy of few GeV. The prototype of the electron cooling system based on modi"ed betatron for cooling of antiprotons in the Recycler ring (Fermilab) is under development in the JINR. The peculiarities of the modi"ed betatron are its section structure, the longitudinal guiding magnetic "eld and the quadrupole spiral "eld used to form a closed orbit. The longitudinal magnetic "eld provides the electron magnetisation and long lifetime of the circulating electrons as the strong coupling between the horizontal and vertical degrees of freedom and leads to additional resonances. ( 2000 Elsevier Science B.V. All rights reserved. PACS: 29.27.Bd Keywords: Electron cooling; Modi"ed betatron; Particle dynamics
1. Introduction To avoid the problems of a traditional electron cooling system and to cool down the antiprotons at an energy of 8 GeV, an electron cooling system with circulating electron beams was proposed in Refs. [1,2]. In this system the Recycler ring is equipped with an additional electron one, which is periodically "lled up with a new portion of cold electrons. The electron beam circulates in longitudinal (quasitoroidal) magnetic "eld, and the longterm stability of the beam is provided with additional spiral coils, which form a quadrupole magnetic "eld. Such a focusing system is similar * Corresponding author. Tel.: #7-09261-64495; fax: #709621-66666. E-mail address: [email protected] (A. Smirnov)
to the `stelaratora one. The acceleration of the electron beam without the distortions caused by RF system of linear accelerators is achieved by using induction acceleration. The particle dynamics in a similar circular electron accelerator, the socalled `modi"ed betatrona, was investigated carefully } theoretically and experimentally (see, for instance Refs. [3,4]). In order to test the medium-energy electron cooling system based on modi"ed betatron, the design and manufactory of such a system prototype was started at the Joint Institute for Nuclear Research. The Modi"ed Betatron Prototype (MOBY) [5] is an electron induction accelerator with electron energy of 4.36 MeV and longitudinal magnetic "eld of 1 kG and additional quadrupole spiral magnetic "eld. The general di!erence between the MOBY and the classical modi"ed betatron is its section
0168-9002/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 0 0 2 ( 9 9 ) 0 1 1 4 3 - 2
SECTION V.
268
I. Meshkov et al. / Nuclear Instruments and Methods in Physics Research A 441 (2000) 267}270
structure. The MOBY consists of two toroidal and two straight sections connected together as a racetrack. Therefore, it is impossible to use results previously obtained for the classical modi"ed betatron. The usage of the usual software (like MAD) is di$cult due to the presence of the longitudinal magnetic "eld in all optic elements, which produces strong coupling between the vertical and horizontal degrees of freedom. To investigate the particle dynamics in the MOBY the program software named BETATRON was especially elaborated by the JINR electron cooling group. The BETATRON is based on the Beam Optics Library with Integrated Development Environment (BOLIDE) system designed to develop physical programs related to the charged particle beam dynamics in the optic channels, storage rings, etc. The results of the simulations using the BETATRON program are presented in this article.
2. Focusing structure of modi5ed betatron prototype Below we consider the structure in which the radii of both toroidal sections are equal to 1.46 m and both straight sections are of the length of 4.56 m. One of the straight sections is free of the spiral quadrupole winding, the number of the spiral steps being equal to 2 in the other section. Two variants of the betatron focusing structures with the same e!ective value of the quadrupole "eld gradient are considered. First variant (unsmoothed) has
the jump of the focusing "eld gradient between sections, second variant (smoothed) has the adiabatic (linear) changing of a quadrupole "eld gradient at the exist and the entrance of the cooling section. Linear particle dynamics was investigated by the matrix method. The dependencies of dispersion and lattice functions on focusing gradient (Fig. 1a and b) show, that to obtain small dispersion in the cooling section the quadrupole gradient value has to lie between 10 and 15 G/cm. In this case, for smoothed structure the ellipticity of the beam cross section is less than 20%.
3. Angular spread of the circulating electron beam When the stability conditions are satis"ed the angular spread of the circulating electron beam in the section with quadrupole spiral winding could be estimated as the angle between the "eld line and longitudinal axis h+(Gr )/B [6]. This value lies "%!. between 10~3 and 10~2 for typical value of the quadrupole "eld gradient G&5}10 G/cm and the longitudinal magnetic "eld B"1 kG. (Fig. 2, unsmoothed). To e!ectively cool down the antiprotons in the Recycler the angular spread of the electron beam has to be of the order of magnitude 10~4 or even less. It means, that the electron cooling section has to be free on the quadrupole spiral winding. When the particle motion is stable the possibility to have an electron beam of small angular spread is the use
Fig. 1. The dependencies of average magnitudes of dispersion (a) and ratio of the ellipse axes of the beam cross section (b) on quadrupole gradient E"4.3 MeV, I"0 A.
I. Meshkov et al. / Nuclear Instruments and Methods in Physics Research A 441 (2000) 267}270
Fig. 2. The dependence of the maximal angle of the electron in the cooling section on the quadrupole "eld gradient. E"4.3 MeV, I"0 A, B"1 kG, r "1 cm, y "1 cm, "%!. 0 x "x@ "y@ "0, *p/p"0. 0 0 0
of the adiabatic variation of quadrupole "eld gradient at the exit and the entrance of the cooling section, which is free of quadrupole spiral "eld (Fig. 2, smoothed) [7]. Beam injection also has to be performed in the section free of quadrupole "eld.
269
Fig. 3. Computer simulation of the stability diagram, B"1 kG.
below the "rst resonance of the quadrupole rotation can provide the long-term stability of the circulating electron beam.
5. Tune shift 4. The transverse stability of electron motion When the number of the spiral winding steps on each focusing section is an integer, the particle motion in the "rst approximation consists of two independent motions: Larmor rotation of each electron, whose tune is equal to Q +C/2po, and L slow betatron rotation of the beam with a tune Q +(G/B)2¸/2k, where C is the MOBY circum"%5 ference, L is the sum of the lengths of the sections with spiral "eld, k"2p/h, h } step of the quadrupole winding. Stability conditions can be written as follows: n m Q O , Q O , Q $Q Ol, L 2 "%5 L "%5 2
(1)
where n, m, l are integers. The diagram of the particle motion stability can be plotted on the plane `Gradienta of the focusing "eld versus `Energya of the electrons (Fig. 3). The dark lines on this plot correspond to the resonances, where the module of a ring matrix eigenvalue is not equal to unit. The choice of the working point between resonances of the Larmor rotation and
Coulomb tune shift in the uniform modi"ed betatron, which consists of only toroid with quadrupole winding, can be calculated analytically by the expansion of the eigenfrequencies in the Tailor series. In the general case, when the eigenfrequencies are di!erent in di!erent sections of the structure, tune shift is evaluated by the calculation of the eigenvalues of the ring transformation map depending on beam current. The Coulomb tune shift calculated for injection electron energy and round beam of radius of 1 cm is presented in Fig. 4. It should be noted, that the tune shift can be compensated by slow changing of the longitudinal "eld and the quadrupole gradient values during acceleration. At maximal beam energy the Coulomb tune shift is negligible for beam current of several amperes.
6. Beam dynamics during acceleration In the "rst approximation, the betatron tune Q does not depend on particle energy only on the "%5 gradient of the spiral quadrupole "eld. Thus, when the gradient lies between 5 and 20 G/cm the particles do not cross any resonance of the betatron
SECTION V.
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I. Meshkov et al. / Nuclear Instruments and Methods in Physics Research A 441 (2000) 267}270
Fig. 4. The dependence of coherent tune shift on beam current. E"100 keV, G"10 G/cm.
mode of oscillations during acceleration (Fig. 3). Only at small energy and large beam current these resonances can appear due to space-charge e!ects. These resonances can be avoided by the choice of injection energy of about 100 keV for beam current of about 0.1 A. However, at constant value of the guiding magnetic "eld particles cross a large number of integer and half-integer resonances of fast Larmor mode of oscillations and coupling resonances during acceleration. High-order resonances can be avoided by the choice of energy increase during one revolution in the ring being substantially higher than the resonance width. However, for low-order resonance this way is not realistic due to technical limitations of the acceleration rate. The damage of the beam quality can be prevented by varying the guiding magnetic "eld value in accordance with the condition Q "const. L For such a regime the injection energy of several hundreds of keV is preferable. The magnetic "eld value has to be increased during acceleration from several hundreds to several thousands of Gauss. Therefore, the electron beam radius decreases and transverse momentum spread increases proportionally to the square root of the magnetic "eld value. To obtain the beam with low temperature the cathode of the electron gun has to be placed in the magnetic "eld, which has the value equal to the maximal one in the ring after acceleration.
7. Conclusion In the focusing structure of MOBY with adiabatic variation of quadrupole "eld at the exit
and at the entrance of the cooling section, which is free of quadrupole spiral "eld, the angular spread of the circulating electron beam lies between 10~3 and 10~4 rad. At the focusing gradient value of about 10}15 G/cm the horizontal dispersion in the cooling section is about 40 cm, and beam cross section has almost a round shape. Acknowledgements This work was supported by the Russian Foundation for Basic Research (Grant N 96-02-17211, N 99-02-17716) and INTAS (Grant N 96-0966). References [1] G. Jackson, Modi"ed betatron approach to electron cooling, Proceedings of the International Workshop on Medium Energy Electron Cooling, Novosibirsk, 1997, p. 171. [2] I. Meshkov, A. Sidorin, Electron cooling system with circulating electron beam, Proceedings of the International Workshop on Medium Energy Electron Cooling, Novosibirsk, 1997, p. 183. [3] C.W. Robertson, A. Mondelli, D. Chernin, High-current betatron with stellarator "elds, Phys. Rev. Lett. 50 (7) (1983) 507. [4] C. Kapetanakos et al., Phys. Fluids B 5 (7) (1993) 2295. [5] Yu. Korotaev, I. Meshkov, S. Mironov, A. Sidorin, E. Syresin, The modi"ed betatron prototype dedicated to electron cooling, Proceedings of the Sixth EPAC, Stockholm, 22}26 June, 1998, p. 1061. [6] I. Meshkov, Electron cooling with circulating electron beam in GeV energy range, Proceedings of HEACC'98, Dubna, 1999, p. 409. [7] I. Meshkov, A. Sidorin, A. Smirnov, E. Syresin, E. Musta"n, P. Zenkevich, Stability of the electron beam in the electron cooling system based on the modi"ed betatron, Proceedings of HEACC'98, Dubna, 1999, p. 416.