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Impact on electron velocity of hollow electron beam in HIRFL-CSR e-cooler system
ARTICLE IN PRESS
Nuclear Instruments and Methods in Physics Research A 508 (2003) 239–244
Impact on electron velocity of hollow electron beam in HIRFL-CSR e-cooler system G.X. Xia*, J.W. Xia, J.C. Yang, W. Liu, J.X. Wu, X.J. Yin, H.W. Zhao, B.W. Wei Institute of Modern Physics, Chinese Academy of Sciences, Nanchang Road 363#, Lanzhou 730000, People’s Republic of China Received 16 January 2003; received in revised form 7 April 2003; accepted 29 April 2003
Abstract Cooling efficiency in electron cooling systems is closely related to the velocity of electron. The velocity of electron has offset due to the space charge of the intense electron beam in the drift tube of the cooling section and thus increases the temperature of electrons. In order to minimize this effect, a new type of electron gun is adopted to produce a hollow electron beam in HIRFL-CSR e-cooler project. The hot ion beam is cooled by Coulomb interaction with intense and cold hollow electron beams. Using typical parameters of the CSRm e-cooler, theoretical calculations comparing the impact of the space charge field on electron velocity for solid and hollow electron beam are carried out. r 2003 Elsevier B.V. All rights reserved. Keywords: Electron cooling; Space charge; Hollow electron beam
1. Introduction Electron cooling is an effective method to upgrade the beam quality in storage rings or colliders. The cooled electron beam and the ion beam, sharing the same velocity, interact through multiple coulomb collisions in the cooling section. After sufficient cooling time, the transverse emittance and longitudinal momentum spread of ion beam will be reduced. In general, the intensity of electron beam may be up to a few amperes, hence the negative effect of the space charge field produced by the intense electron beam should be taken into account. This field can cause additional velocity offset and increase the temperature of electron beam. *Corresponding author. E-mail address: [email protected] (G.X. Xia).
Cooling theory shows that the cooling time is closely related to the electron velocity. The cooling efficiency is characterized by the reciprocal of the cooling time and cooling time is written as follows [1]: 3 1 dvi 1 A i b 4 g5 tc ¼ ¼ C 2 i i ðy2i þ y2e Þ2 vi dt Qi Zec je
ð1Þ
where tc is the cooling time, vi is the velocity of an ion in the electron moving frame, Ai and Qi are the ion mass number and charge, respectively, bi and gi are relativistic factors, Zec ð¼ Lcooler =Lring Þ is the fraction of the cooling section length to the ring perimeter, je is the density of the electron beam and yi and ye are the dispersive angles of ion and electron in the cooling section, respectively, C is a constant. ye is proportional to the electron velocity or the square root of the electron temperature, so
0168-9002/03/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0168-9002(03)01662-0
ARTICLE IN PRESS 240
G.X. Xia et al. / Nuclear Instruments and Methods in Physics Research A 508 (2003) 239–244
the additional velocity offset of electron will increase the electron temperature and reduce the cooling efficiency according to Eq. (1). The hollow electron beam is adopted in the electron cooler system of the new HIRFLCSR project for the first time [2]. It not only reduces the recombination probability between electrons and ions, but also minimizes the velocity offset caused by the space charge effect of the intense electron beam. In this paper, by using the typical parameters of the e-cooler of CSRm, theoretical calculations comparing the impact on electron velocity of space charge field for solid and hollow electron beams are carried out.
Table 1 Major parameters of e-coolers of HIRFL-CSR Parameters
CSRm
CSRe
Ion energy (MeV/u) Electron energy (keV) Electron beam current (A) Cathode radius (cm) Magnetic expansion factor Max. field in gun region (kG) Magnetic field in collector region (kG) Magnetic field at cooling section (kG) Length of installation (m) Effective cooling section length (m) Deflection angle of toroid (deg) Deflection radius of toroid (m)
8–50 4–35 3 1.25 1–4 2.4 1.2 0.6–1.5 7.2 3:4 m 90 1.0
25–400 10–300 3 1.25 1–10 5 1.2 0.5–1.5 7.2 3:4 m 90 1.0
2. Brief introduction of e-cooler system of HIRFL-CSR The HIRFL-CSR e-cooler system includes two electron coolers for CSRm and CSRe, respectively [2]. The e-cooler is used in CSRm for beam accumulation during injection to increase the beam intensity. In CSRe, the e-cooler is used to compensate the growth of beam emittance in the case of internal-target experiment or to provide high-quality beams for the highresolution mass measurements of nuclei. The major parameters of the two coolers are listed in Table 1. Figs. 1 and 2 show their layouts, respectively. The CSRm cooler could be a new generation cooler with some unique features. Firstly, the 68 independent solenoid coils, which can correct the transverse components of the magnetic field at the cooling section with high precision, are used to produce magnetic field with high parallelism; secondly, electrostatic bending plates compensate the transverse drift of the electron beam in the toroids and clear up the secondary electrons with low energy, which result in low losses of the electron beam; thirdly, it is equipped with a new electron gun that is able to produce a hollow electron beam which can partially solve the problems due to the space charge effect and reduce effect of recombination between the ions and the electron beam in the cooling section.
Fig. 1. Layout of e-cooler of CSRm: 1—electron gun, 2—the solenoid of the gun, 3—toroid, 4—main solenoid, 5—electrostatic deflector, 6—the solenoid of the collector, 7—collector, 8—dipole corrector.
ARTICLE IN PRESS G.X. Xia et al. / Nuclear Instruments and Methods in Physics Research A 508 (2003) 239–244
241
Fig. 2. Layout of e-cooler of CSRe: 1—electron gun, 2—accelerating tube, 3—the solenoid of the gun, 4—toroid, 5—main solenoid, 6—electrostatic deflector, 7—the solenoid of the collector, 8—decelerating tube, 9—collector, 10—300 kV-HV system, 11—dipole corrector.
3. The space charge field of intense electron beam
therefore, the relative kinetic energy difference is given by
3.1. The space charge field of solid electron beam and its impact on electron velocity
DWe eDUsp Ie e r2 ¼ ¼ ; ð5Þ We me c2 ðg 1Þ 4pe0 bc me c2 ðg q1Þffiffiffiffiffiffiffiffiffiffiffiffiffi a2
For the traditional electron cooling system, the solid electron beam is continuously produced from the electron gun. Following the electrostatic Gaussian theorem, in the drift tube of cooling section, the space charge potential at radius r is described as Ie b r 2 Usp ðrÞ ¼ 1 þ 2 ln ; roa; a a 4pe0 bc ð2Þ in which Ie is the electron current and Ie ¼ ne epa2 bc; where ne is the density of electron in drift tube. e0 is the permittivity of free space, c is the speed of light, b is the relativistic factor of electron, a and b are the radii of electron beam and vacuum chamber, respectively. The radial electric field is , Ie r , Esp ðrÞ ¼ u r ; roa; ð3Þ 2pe0 bc a2 ,
where u r is the radial vector unit. The velocity of electron will be increased by this electric field. Compared with the center of beam axis, the potential difference at radius r is Z r , Ie r2 , , , DUsp ð r Þ ¼ Esp ð r Þ d r ¼ ð4Þ 4pe0 bc a2 0
where g is the Lorentz factor, g ¼ 1= 1 b2 ; me is the static mass of electron. The relative velocity offset of an electron at radius r with respect to the axis is DVe 1 DWe Ie e r2 ¼ ¼ : 3 gðg þ 1Þ We Ve 4pe0 b g3 c me c2 a2
ð6Þ
It shows that the relative electron velocity has a parabolic distribution in the radial direction because of space charge field. In addition, the intense electron beam induces an unwanted magnetic field Be according to Ampere theorem, , m Ie , ð7Þ Be ¼ 0 2 r u y : 2 pa , Here m0 is the permeability of free space and u y the vector unit of the azimuthal angle. The total force acting on electrons becomes , , , eIe r , , , ð1 b2 Þ u r f ¼ e½Esp ðrÞ u r þ v 0 Be ¼ 2pe0 a2 bc ð8Þ v0 ð¼ bcÞ is the electron velocity in the cooling section. The ratio of forces caused by the electric field and magnetic field is 1=b2 : In the nonrelativistic case, i.e. b51; the magnetic field effect
ARTICLE IN PRESS G.X. Xia et al. / Nuclear Instruments and Methods in Physics Research A 508 (2003) 239–244
242
can be ignored. With the action of this force, the electron gets the additional velocity variation, namely, the dispersive angle of electron is increased in the cooling section. In most cases of electron coolers, one also has to consider the radial drift velocity of electron in the cross field composed of space charge field and longitudinal magnetic field of solenoid. The drift velocity can be written as [3,4] ,
,
vd ¼ cðEsp b Be Þ=B ¼
Ie 1 r : 2pe0 g2 bc a2 B
Fig. 3. Schematics of hollow electron beam.
ð9Þ
Here, B is the longitudinal magnetic field of the solenoid. Hence the temperature increase of electron beam is kT ¼ 12 me v2d ;
ð10Þ
where k ¼ 1:38066 1023 J; k is the Boltzmann constant. The cooling time increases because of this temperature increment. 3.2. The space charge field of hollow electron beam and its impact on electron velocity In order to reduce the impact of the space charge field, a new type of electron gun was designed to produce hollow electron beam for HIRFL-CSR electron cooler [5]. A control electrode located around the cathode is used to control electron emission from the cathode by changing the potential at the cathode edge, i.e. to suppress the electron emission at the cathode edge by negative voltage on the control electrode, and enhance the electron emission at the cathode edge by positive voltage on the control electrode. The adoption of a hollow electron beam, on the one hand, reduces the recombination probability of intense electrons and ions, and on the other hand partially reduces the space charge field of intense electron beam in the cooling section [6]. Fig. 3 is a schematic view of hollow electron beam in the drift tube of the cooling section. The electric field from a hollow electron beam of uniform density ne can be written as
follows:
8 0; ror0 > > > > 2 2 > < r r0 Ih h ð11Þ ðrÞ ¼ e rða2 r2 Þ; r0 oroa Esp 0 2pe0 bc > > > > 1 > : ; aorob r where r0 is the radius of the hole, a of the electron beam, b of the vacuum chamber, Ieh ¼ ne epða2 r20 Þ: bc is the current of the hollow electron beam, b the electron velocity, c the speed of light. The space charge potential distribution is Ieh b r2 a2 r20 r h Usp ðrÞ ¼ 2 ln 2 þ2 2 ln a a r20 4pe0 bc a r20 a r0 prpa:
ð12Þ
The relative space charge potential difference at radius r differs from that on the beam axis Ieh 1 r h 2 2 DUsp ðrÞ ¼ r r0 1 þ 2 ln : r0 4pe0 bc a2 r20 ð13Þ In the non-relativistic case, the relative velocity offset is DVeh Ieh e 1 ¼ Ve 4pe0 b3 g3 c me c2 a2 r20 r r2 r20 1 þ 2 ln r0 prpa r0
ð14Þ
which is zero at the boundary of the hole. It can be seen from Eq. (14) that the relative velocity variation in the electron beam will become the same as that of the solid electron beam if r0 5a: As before, drift velocity in the cross field composed of space charge field of hollow electron
ARTICLE IN PRESS G.X. Xia et al. / Nuclear Instruments and Methods in Physics Research A 508 (2003) 239–244
beam and solenoidal magnetic field can be obtained as vhd ¼
Ieh 1 1 r2 r20 ; 2pe0 g2 bc B rða2 r20 Þ
the electron beam with energy of 5:486 keV and current of 3 A will be used. In Table 2, the parameters of the electron gun in our calculation are listed. The relative velocity offset in the space charge field of intense electron beam and drift velocity in the cross field are calculated. The impact on electron velocity in different hole radii are compared too. Fig. 4 shows that relative velocity offset due to space charge field with different hole radii of 0, 5 and 10 mm; respectively. It can be seen that the larger the hole radius, the less the relative velocity offset becomes. Fig. 5 shows the drift velocity variation with different hole radii. It indicates that the impact of space charge field on drift velocity is less if the hole radius is larger. However, one cannot make the hole radius too large, as it will result in a decrease of the cooling efficiency.
ð15Þ
the temperature increase due to radial drift velocity is kT h ¼ 12 me vhd :
243
ð16Þ
4. Comparison of the impact on the electron velocity of space charge field for solid and hollow electron beams in HIRFL-CSRm cooler In order to quantitatively compare the impact on the electron velocity by space charge field, the typical parameters of HIRFL-CSRm electron gun is considered. In the first operation of CSRm, 10 MeV=u16 O6þ is injected into the main ring, so
Table 2 Parameters of HIRFL-CSR electron gun design Electron energy ðEe Þ
5:486 keV 8
Electron density ðne Þ Electron beam radius ðaÞ Vacuum chamber radius ðbÞ
1:05 10 cm 2:5 cm 7:5 cm
3
Relativistic factor of electron ðbÞ
0.145
Cathode radius ðrÞ Hole radius ðr0 Þ Solenoid magnetic field ðBÞ
1:25 cm 5–10 mm 1:0 kG
-2
3.0x10
-2
Relative velocity offset
2.4x10
solid electron beam hollow beam with hole radius 5mm hollow beam with hole radius 10mm
-2
1.8x10
-2
1.2x10
-3
6.0x10
0.0 0.0
-3
5.0x10
-2
1.0x10
-2
1.5x10
-2
2.0x10
-2
2.5x10
Radial position of electron beam [m]
Fig. 4. Relative velocity offset as a function of radial position in electron beam.
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G.X. Xia et al. / Nuclear Instruments and Methods in Physics Research A 508 (2003) 239–244 5
Drift velocity in cross field [m/s]
2.5x10
5
2.0x10
solid electron beam hollow beam with hole radius 5mm hollow beam with hole radius 10mm
5
1.5x10
5
1.0x10
4
5.0x10
0.0 0.0
-3
5.0x10
-2
1.0x10
-2
1.5x10
-2
2.0x10
-2
2.5x10
Radial position of electron beam [m] Fig. 5. Drift velocity in cross field as a function of radial position in electron beam.
Acknowledgements We greatly thank Professor V. Parkhomchuk and other BINP colleagues for useful discussion. References [1] I.N. Meshkov, et al., Heavy ion storage ring complex K4– K10, A Technical Proposal, Dubna, 1992, p. 85.
[2] J.W. Xia, et al., Nucl. Instr. and Meth. Phys. Res. A 488 (2002) 11. [3] J. Bosser, et al., Neutralization of the LEAR ECOOL electron beam space charge, CERN PS/AR Note 93-08, 1993. [4] I.N. Meshkov, Physics and Technique of Electron Cooling, RIKEN-AF-AC-2, 1997. [5] Zhao Hongwei, et al., CSR e-Cooler Design Report, 2002. [6] A. Bubley, et al., The electron gun with variable beam profile for optimization of electron cooling, Proceedings of EPAC 2002, Paris, France, p. 1356.
Nuclear Instruments and Methods in Physics Research A 508 (2003) 239–244
Impact on electron velocity of hollow electron beam in HIRFL-CSR e-cooler system G.X. Xia*, J.W. Xia, J.C. Yang, W. Liu, J.X. Wu, X.J. Yin, H.W. Zhao, B.W. Wei Institute of Modern Physics, Chinese Academy of Sciences, Nanchang Road 363#, Lanzhou 730000, People’s Republic of China Received 16 January 2003; received in revised form 7 April 2003; accepted 29 April 2003
Abstract Cooling efficiency in electron cooling systems is closely related to the velocity of electron. The velocity of electron has offset due to the space charge of the intense electron beam in the drift tube of the cooling section and thus increases the temperature of electrons. In order to minimize this effect, a new type of electron gun is adopted to produce a hollow electron beam in HIRFL-CSR e-cooler project. The hot ion beam is cooled by Coulomb interaction with intense and cold hollow electron beams. Using typical parameters of the CSRm e-cooler, theoretical calculations comparing the impact of the space charge field on electron velocity for solid and hollow electron beam are carried out. r 2003 Elsevier B.V. All rights reserved. Keywords: Electron cooling; Space charge; Hollow electron beam
1. Introduction Electron cooling is an effective method to upgrade the beam quality in storage rings or colliders. The cooled electron beam and the ion beam, sharing the same velocity, interact through multiple coulomb collisions in the cooling section. After sufficient cooling time, the transverse emittance and longitudinal momentum spread of ion beam will be reduced. In general, the intensity of electron beam may be up to a few amperes, hence the negative effect of the space charge field produced by the intense electron beam should be taken into account. This field can cause additional velocity offset and increase the temperature of electron beam. *Corresponding author. E-mail address: [email protected] (G.X. Xia).
Cooling theory shows that the cooling time is closely related to the electron velocity. The cooling efficiency is characterized by the reciprocal of the cooling time and cooling time is written as follows [1]: 3 1 dvi 1 A i b 4 g5 tc ¼ ¼ C 2 i i ðy2i þ y2e Þ2 vi dt Qi Zec je
ð1Þ
where tc is the cooling time, vi is the velocity of an ion in the electron moving frame, Ai and Qi are the ion mass number and charge, respectively, bi and gi are relativistic factors, Zec ð¼ Lcooler =Lring Þ is the fraction of the cooling section length to the ring perimeter, je is the density of the electron beam and yi and ye are the dispersive angles of ion and electron in the cooling section, respectively, C is a constant. ye is proportional to the electron velocity or the square root of the electron temperature, so
0168-9002/03/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0168-9002(03)01662-0
ARTICLE IN PRESS 240
G.X. Xia et al. / Nuclear Instruments and Methods in Physics Research A 508 (2003) 239–244
the additional velocity offset of electron will increase the electron temperature and reduce the cooling efficiency according to Eq. (1). The hollow electron beam is adopted in the electron cooler system of the new HIRFLCSR project for the first time [2]. It not only reduces the recombination probability between electrons and ions, but also minimizes the velocity offset caused by the space charge effect of the intense electron beam. In this paper, by using the typical parameters of the e-cooler of CSRm, theoretical calculations comparing the impact on electron velocity of space charge field for solid and hollow electron beams are carried out.
Table 1 Major parameters of e-coolers of HIRFL-CSR Parameters
CSRm
CSRe
Ion energy (MeV/u) Electron energy (keV) Electron beam current (A) Cathode radius (cm) Magnetic expansion factor Max. field in gun region (kG) Magnetic field in collector region (kG) Magnetic field at cooling section (kG) Length of installation (m) Effective cooling section length (m) Deflection angle of toroid (deg) Deflection radius of toroid (m)
8–50 4–35 3 1.25 1–4 2.4 1.2 0.6–1.5 7.2 3:4 m 90 1.0
25–400 10–300 3 1.25 1–10 5 1.2 0.5–1.5 7.2 3:4 m 90 1.0
2. Brief introduction of e-cooler system of HIRFL-CSR The HIRFL-CSR e-cooler system includes two electron coolers for CSRm and CSRe, respectively [2]. The e-cooler is used in CSRm for beam accumulation during injection to increase the beam intensity. In CSRe, the e-cooler is used to compensate the growth of beam emittance in the case of internal-target experiment or to provide high-quality beams for the highresolution mass measurements of nuclei. The major parameters of the two coolers are listed in Table 1. Figs. 1 and 2 show their layouts, respectively. The CSRm cooler could be a new generation cooler with some unique features. Firstly, the 68 independent solenoid coils, which can correct the transverse components of the magnetic field at the cooling section with high precision, are used to produce magnetic field with high parallelism; secondly, electrostatic bending plates compensate the transverse drift of the electron beam in the toroids and clear up the secondary electrons with low energy, which result in low losses of the electron beam; thirdly, it is equipped with a new electron gun that is able to produce a hollow electron beam which can partially solve the problems due to the space charge effect and reduce effect of recombination between the ions and the electron beam in the cooling section.
Fig. 1. Layout of e-cooler of CSRm: 1—electron gun, 2—the solenoid of the gun, 3—toroid, 4—main solenoid, 5—electrostatic deflector, 6—the solenoid of the collector, 7—collector, 8—dipole corrector.
ARTICLE IN PRESS G.X. Xia et al. / Nuclear Instruments and Methods in Physics Research A 508 (2003) 239–244
241
Fig. 2. Layout of e-cooler of CSRe: 1—electron gun, 2—accelerating tube, 3—the solenoid of the gun, 4—toroid, 5—main solenoid, 6—electrostatic deflector, 7—the solenoid of the collector, 8—decelerating tube, 9—collector, 10—300 kV-HV system, 11—dipole corrector.
3. The space charge field of intense electron beam
therefore, the relative kinetic energy difference is given by
3.1. The space charge field of solid electron beam and its impact on electron velocity
DWe eDUsp Ie e r2 ¼ ¼ ; ð5Þ We me c2 ðg 1Þ 4pe0 bc me c2 ðg q1Þffiffiffiffiffiffiffiffiffiffiffiffiffi a2
For the traditional electron cooling system, the solid electron beam is continuously produced from the electron gun. Following the electrostatic Gaussian theorem, in the drift tube of cooling section, the space charge potential at radius r is described as Ie b r 2 Usp ðrÞ ¼ 1 þ 2 ln ; roa; a a 4pe0 bc ð2Þ in which Ie is the electron current and Ie ¼ ne epa2 bc; where ne is the density of electron in drift tube. e0 is the permittivity of free space, c is the speed of light, b is the relativistic factor of electron, a and b are the radii of electron beam and vacuum chamber, respectively. The radial electric field is , Ie r , Esp ðrÞ ¼ u r ; roa; ð3Þ 2pe0 bc a2 ,
where u r is the radial vector unit. The velocity of electron will be increased by this electric field. Compared with the center of beam axis, the potential difference at radius r is Z r , Ie r2 , , , DUsp ð r Þ ¼ Esp ð r Þ d r ¼ ð4Þ 4pe0 bc a2 0
where g is the Lorentz factor, g ¼ 1= 1 b2 ; me is the static mass of electron. The relative velocity offset of an electron at radius r with respect to the axis is DVe 1 DWe Ie e r2 ¼ ¼ : 3 gðg þ 1Þ We Ve 4pe0 b g3 c me c2 a2
ð6Þ
It shows that the relative electron velocity has a parabolic distribution in the radial direction because of space charge field. In addition, the intense electron beam induces an unwanted magnetic field Be according to Ampere theorem, , m Ie , ð7Þ Be ¼ 0 2 r u y : 2 pa , Here m0 is the permeability of free space and u y the vector unit of the azimuthal angle. The total force acting on electrons becomes , , , eIe r , , , ð1 b2 Þ u r f ¼ e½Esp ðrÞ u r þ v 0 Be ¼ 2pe0 a2 bc ð8Þ v0 ð¼ bcÞ is the electron velocity in the cooling section. The ratio of forces caused by the electric field and magnetic field is 1=b2 : In the nonrelativistic case, i.e. b51; the magnetic field effect
ARTICLE IN PRESS G.X. Xia et al. / Nuclear Instruments and Methods in Physics Research A 508 (2003) 239–244
242
can be ignored. With the action of this force, the electron gets the additional velocity variation, namely, the dispersive angle of electron is increased in the cooling section. In most cases of electron coolers, one also has to consider the radial drift velocity of electron in the cross field composed of space charge field and longitudinal magnetic field of solenoid. The drift velocity can be written as [3,4] ,
,
vd ¼ cðEsp b Be Þ=B ¼
Ie 1 r : 2pe0 g2 bc a2 B
Fig. 3. Schematics of hollow electron beam.
ð9Þ
Here, B is the longitudinal magnetic field of the solenoid. Hence the temperature increase of electron beam is kT ¼ 12 me v2d ;
ð10Þ
where k ¼ 1:38066 1023 J; k is the Boltzmann constant. The cooling time increases because of this temperature increment. 3.2. The space charge field of hollow electron beam and its impact on electron velocity In order to reduce the impact of the space charge field, a new type of electron gun was designed to produce hollow electron beam for HIRFL-CSR electron cooler [5]. A control electrode located around the cathode is used to control electron emission from the cathode by changing the potential at the cathode edge, i.e. to suppress the electron emission at the cathode edge by negative voltage on the control electrode, and enhance the electron emission at the cathode edge by positive voltage on the control electrode. The adoption of a hollow electron beam, on the one hand, reduces the recombination probability of intense electrons and ions, and on the other hand partially reduces the space charge field of intense electron beam in the cooling section [6]. Fig. 3 is a schematic view of hollow electron beam in the drift tube of the cooling section. The electric field from a hollow electron beam of uniform density ne can be written as
follows:
8 0; ror0 > > > > 2 2 > < r r0 Ih h ð11Þ ðrÞ ¼ e rða2 r2 Þ; r0 oroa Esp 0 2pe0 bc > > > > 1 > : ; aorob r where r0 is the radius of the hole, a of the electron beam, b of the vacuum chamber, Ieh ¼ ne epða2 r20 Þ: bc is the current of the hollow electron beam, b the electron velocity, c the speed of light. The space charge potential distribution is Ieh b r2 a2 r20 r h Usp ðrÞ ¼ 2 ln 2 þ2 2 ln a a r20 4pe0 bc a r20 a r0 prpa:
ð12Þ
The relative space charge potential difference at radius r differs from that on the beam axis Ieh 1 r h 2 2 DUsp ðrÞ ¼ r r0 1 þ 2 ln : r0 4pe0 bc a2 r20 ð13Þ In the non-relativistic case, the relative velocity offset is DVeh Ieh e 1 ¼ Ve 4pe0 b3 g3 c me c2 a2 r20 r r2 r20 1 þ 2 ln r0 prpa r0
ð14Þ
which is zero at the boundary of the hole. It can be seen from Eq. (14) that the relative velocity variation in the electron beam will become the same as that of the solid electron beam if r0 5a: As before, drift velocity in the cross field composed of space charge field of hollow electron
ARTICLE IN PRESS G.X. Xia et al. / Nuclear Instruments and Methods in Physics Research A 508 (2003) 239–244
beam and solenoidal magnetic field can be obtained as vhd ¼
Ieh 1 1 r2 r20 ; 2pe0 g2 bc B rða2 r20 Þ
the electron beam with energy of 5:486 keV and current of 3 A will be used. In Table 2, the parameters of the electron gun in our calculation are listed. The relative velocity offset in the space charge field of intense electron beam and drift velocity in the cross field are calculated. The impact on electron velocity in different hole radii are compared too. Fig. 4 shows that relative velocity offset due to space charge field with different hole radii of 0, 5 and 10 mm; respectively. It can be seen that the larger the hole radius, the less the relative velocity offset becomes. Fig. 5 shows the drift velocity variation with different hole radii. It indicates that the impact of space charge field on drift velocity is less if the hole radius is larger. However, one cannot make the hole radius too large, as it will result in a decrease of the cooling efficiency.
ð15Þ
the temperature increase due to radial drift velocity is kT h ¼ 12 me vhd :
243
ð16Þ
4. Comparison of the impact on the electron velocity of space charge field for solid and hollow electron beams in HIRFL-CSRm cooler In order to quantitatively compare the impact on the electron velocity by space charge field, the typical parameters of HIRFL-CSRm electron gun is considered. In the first operation of CSRm, 10 MeV=u16 O6þ is injected into the main ring, so
Table 2 Parameters of HIRFL-CSR electron gun design Electron energy ðEe Þ
5:486 keV 8
Electron density ðne Þ Electron beam radius ðaÞ Vacuum chamber radius ðbÞ
1:05 10 cm 2:5 cm 7:5 cm
3
Relativistic factor of electron ðbÞ
0.145
Cathode radius ðrÞ Hole radius ðr0 Þ Solenoid magnetic field ðBÞ
1:25 cm 5–10 mm 1:0 kG
-2
3.0x10
-2
Relative velocity offset
2.4x10
solid electron beam hollow beam with hole radius 5mm hollow beam with hole radius 10mm
-2
1.8x10
-2
1.2x10
-3
6.0x10
0.0 0.0
-3
5.0x10
-2
1.0x10
-2
1.5x10
-2
2.0x10
-2
2.5x10
Radial position of electron beam [m]
Fig. 4. Relative velocity offset as a function of radial position in electron beam.
ARTICLE IN PRESS 244
G.X. Xia et al. / Nuclear Instruments and Methods in Physics Research A 508 (2003) 239–244 5
Drift velocity in cross field [m/s]
2.5x10
5
2.0x10
solid electron beam hollow beam with hole radius 5mm hollow beam with hole radius 10mm
5
1.5x10
5
1.0x10
4
5.0x10
0.0 0.0
-3
5.0x10
-2
1.0x10
-2
1.5x10
-2
2.0x10
-2
2.5x10
Radial position of electron beam [m] Fig. 5. Drift velocity in cross field as a function of radial position in electron beam.
Acknowledgements We greatly thank Professor V. Parkhomchuk and other BINP colleagues for useful discussion. References [1] I.N. Meshkov, et al., Heavy ion storage ring complex K4– K10, A Technical Proposal, Dubna, 1992, p. 85.
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