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A new form of accelerator for charged particles
Nuclear Instruments and Methods in Physics Research A268 (1988) 273-274 North-Holland, Amsterdam
273
Letter to the Editor
A NEW FORM OF ACCELERATOR FOR CHARGED PARTICLES H. DANIEL Physics Department, Technical University of Munich, D-8046 Garching, FRG
Received 29 July 1987 and in revised form 28 December 1987
A new form of accelerator is proposed which employs an electric field of constant magnitude rotating in a plane perpendicular to a static magnetic guidance field. The proposed new form of an accelerator for charged particles is to some extent related to cyclotron and omegatron. It is most easily understood in the case of a homogeneous magnetic field Bo as permanent guidance field. An electric rf field E of constant magnitude E and frequency wo rotates perpendicular to Bo . In order to analyze the particle motion we project the trajectory of an ion with mass m and charge q in parallel projection along Bo on a plane f perpendicular to Bo and call this projection s (fig. 1) . Due to the Lorentz force the tangent unit vector u on s with angle a to the x axis rotates in f with constant angular velocity wc if we are in the nonrelativistic regime and neglect the B field related to E as well as the E component perpendicular to u. This is true for any
value of the tangential component of E. If E has a normal component E,, the angular frequency of the ion will be qB Z 2m
da wt --= dt
4ZBZ 4M2
qE. mr '
where BZ is the Z component of Bo in a cylindrical coordinate system R, ©, Z with the Z axis antiparallel to Bo and r the momentary radius of curvature of s [1]; the minus sign in front of the square root is valid for mr
'
in the case of wo > 0. It is easily seen (cf. fig. 1) that E acts in such a way as to bring u and e in parallel for q > 0 and antiparallel for q < 0, hence in any case to finally accelerate the ion in the cyclotron resonance case 'WO
= wt =wr .
A v component along Bo remains unchanged. In particular, charged particles with no momentum in the By direction at time t _ to will stay in the plane f perpendicular to Bo where they were at to . Particles with no v component in f at to are fully accelerated for t to and are thus automatically in phase with E. For the case u and e form an angle ß = a - 0 at t = to we derive the following differential equation for ß confining ourselves to the only practically reasonable sign in front of the square root in eq. (1): w Fig. 1. Sketch of the proposed accelerator form ; projection plane f and projected orbit s seen from the top (cf. text). Bo perpendicular to plane. Spiral : part of s. a and e are unit vectors in point P on s and along E, respectively . The particle and the electric field rotate counter-clockwise. 0168-9002/88/$03 .50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
d,E dt
r 2
wr _ qE sin 8
+V4
mr
(3)
For r-> 0 (small ion energy) and ß :# 0 (mod 7r) dß/dt tends toward infinitely, i.e. the ion "instantaneously" becomes directed parallel to E. In the case of a small second term under the square root of eq . (3) compared
274
H. Daniel / New form of accelerator for charged particles
to the first we expand the square root. This yields in first order _dß__ gEsin8 mrwr dt For large r (high ion energy) we may neglect the r and m changes, i .e. neglect the acceleration along u, and obtain as solution [2] tan[ß(t)/21 =expI - qE(t-to) mrwr tan[ ß(to)/2]
I.
If there are two electric fields E l and EZ having different frequencies w0( l ) and woe) acting simulta neously, each field will strongly affect only particles in resonance with it as described above if the other field is not so strong as to "destroy" the motion ; the effect of the other field will average out . If we confine ourselves now to a setup with a small "source volume", i.e. the volume where the particles "start" with v - 0 at to , and allow a nonuniform Bo field, we may provide axial focussing with the help of an azimuth-dependent magnetic field, as in the case of an isochronous cyclotron, and may also allow m to increase with increasing distance from the center of the source volume . If we do not limit the starting time t o to certain time intervals correlated to the phase of E we shall, however, have no radial orbit separation .
The production of approximately the desired electric field is possible although efficient production may turn out to be difficult . Two waves polarized perpendicularly to each other traveling in the Bo direction or the corresponding standing waves can be used which will produce an E field with no rf B field in a plane f. The new accelerator form will allow a number of applications, for example as high-current cyclotron, high-current selective ion source, mass separator and muon reactivator in a (future) power reactor based on muon-catalyzed dt fusion .
Acknowledgements R. Callies contributed the idea of the two polarized E waves . It is a pleasure to thank him, F .J . Hartmann and U . Schmidt-Rohr for discussion .
References [1] H . Daniel, Nucl . Instr. and Meth. A235 (1985) 424 . [2] E . Kamke, Differentialgleichungen - Lösungsmethoden und Lösungen (Akad . Verlagsges., Leipzig, 1942) p. 15 ; I .N . Bronstein and K.A. Semendjajew, Taschenbuch der Mathematik, 20th ed . (Harri Deutsch, Thun and Frankfurt/M, 1981) p . 104 .
273
Letter to the Editor
A NEW FORM OF ACCELERATOR FOR CHARGED PARTICLES H. DANIEL Physics Department, Technical University of Munich, D-8046 Garching, FRG
Received 29 July 1987 and in revised form 28 December 1987
A new form of accelerator is proposed which employs an electric field of constant magnitude rotating in a plane perpendicular to a static magnetic guidance field. The proposed new form of an accelerator for charged particles is to some extent related to cyclotron and omegatron. It is most easily understood in the case of a homogeneous magnetic field Bo as permanent guidance field. An electric rf field E of constant magnitude E and frequency wo rotates perpendicular to Bo . In order to analyze the particle motion we project the trajectory of an ion with mass m and charge q in parallel projection along Bo on a plane f perpendicular to Bo and call this projection s (fig. 1) . Due to the Lorentz force the tangent unit vector u on s with angle a to the x axis rotates in f with constant angular velocity wc if we are in the nonrelativistic regime and neglect the B field related to E as well as the E component perpendicular to u. This is true for any
value of the tangential component of E. If E has a normal component E,, the angular frequency of the ion will be qB Z 2m
da wt --= dt
4ZBZ 4M2
qE. mr '
where BZ is the Z component of Bo in a cylindrical coordinate system R, ©, Z with the Z axis antiparallel to Bo and r the momentary radius of curvature of s [1]; the minus sign in front of the square root is valid for mr
'
in the case of wo > 0. It is easily seen (cf. fig. 1) that E acts in such a way as to bring u and e in parallel for q > 0 and antiparallel for q < 0, hence in any case to finally accelerate the ion in the cyclotron resonance case 'WO
= wt =wr .
A v component along Bo remains unchanged. In particular, charged particles with no momentum in the By direction at time t _ to will stay in the plane f perpendicular to Bo where they were at to . Particles with no v component in f at to are fully accelerated for t to and are thus automatically in phase with E. For the case u and e form an angle ß = a - 0 at t = to we derive the following differential equation for ß confining ourselves to the only practically reasonable sign in front of the square root in eq. (1): w Fig. 1. Sketch of the proposed accelerator form ; projection plane f and projected orbit s seen from the top (cf. text). Bo perpendicular to plane. Spiral : part of s. a and e are unit vectors in point P on s and along E, respectively . The particle and the electric field rotate counter-clockwise. 0168-9002/88/$03 .50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
d,E dt
r 2
wr _ qE sin 8
+V4
mr
(3)
For r-> 0 (small ion energy) and ß :# 0 (mod 7r) dß/dt tends toward infinitely, i.e. the ion "instantaneously" becomes directed parallel to E. In the case of a small second term under the square root of eq . (3) compared
274
H. Daniel / New form of accelerator for charged particles
to the first we expand the square root. This yields in first order _dß__ gEsin8 mrwr dt For large r (high ion energy) we may neglect the r and m changes, i .e. neglect the acceleration along u, and obtain as solution [2] tan[ß(t)/21 =expI - qE(t-to) mrwr tan[ ß(to)/2]
I.
If there are two electric fields E l and EZ having different frequencies w0( l ) and woe) acting simulta neously, each field will strongly affect only particles in resonance with it as described above if the other field is not so strong as to "destroy" the motion ; the effect of the other field will average out . If we confine ourselves now to a setup with a small "source volume", i.e. the volume where the particles "start" with v - 0 at to , and allow a nonuniform Bo field, we may provide axial focussing with the help of an azimuth-dependent magnetic field, as in the case of an isochronous cyclotron, and may also allow m to increase with increasing distance from the center of the source volume . If we do not limit the starting time t o to certain time intervals correlated to the phase of E we shall, however, have no radial orbit separation .
The production of approximately the desired electric field is possible although efficient production may turn out to be difficult . Two waves polarized perpendicularly to each other traveling in the Bo direction or the corresponding standing waves can be used which will produce an E field with no rf B field in a plane f. The new accelerator form will allow a number of applications, for example as high-current cyclotron, high-current selective ion source, mass separator and muon reactivator in a (future) power reactor based on muon-catalyzed dt fusion .
Acknowledgements R. Callies contributed the idea of the two polarized E waves . It is a pleasure to thank him, F .J . Hartmann and U . Schmidt-Rohr for discussion .
References [1] H . Daniel, Nucl . Instr. and Meth. A235 (1985) 424 . [2] E . Kamke, Differentialgleichungen - Lösungsmethoden und Lösungen (Akad . Verlagsges., Leipzig, 1942) p. 15 ; I .N . Bronstein and K.A. Semendjajew, Taschenbuch der Mathematik, 20th ed . (Harri Deutsch, Thun and Frankfurt/M, 1981) p . 104 .