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Frequency sweeping process on cyclotron resonance cooling
NUCLEAN INSTRUMENTS
&MErNcmE IN PNYWCS RESEARCH
Nuclear Instruments
ELSJYIER
and Methods
in Physics
Research
A 41 I (1998) SW 505
Sectton A
Frequency sweeping process on cyclotron resonance cooling Toshihiro Taguchi”.*, Kunioki Mimab aDepartment qf Electrical Engineering, Faculw qf‘Engineering. Setsunun Utzircwi~~*. Nevagawa. Osaka 372. Japan h Institute of laser Engineering. Osaka ilniuersi~v, Suita Osuku. 565, Jupan Received
19 December
1997
Abstract
Effect of spatial variation of the magnetic field strength has been investigated in the relativistic cyclotron resonance, which is used for a cyclotron resonance cooling scheme. The result shows that the variation of the field strength is effective in narrowing the resonance width. li;l 1998 Published by Elsevier Science B.V. All rights reserved.
1. Introduction
In the previous paper El], we proposed a concept of a cyclotron resonance cooling. When the electrons propagate along a strong magnetic field with gyrating motion and a circularly polarized electromagnetic wave is irradiated counter to the gyrating beam. cyclotron resonance takes place and the transverse momentum of the resonant electrons selectively increases. Since electrons that acquire large transverse momentum loose their energy faster than non resonant electrons due to radiative damping, it is possible to reduce the phase volume in a specific part of phase space. We numerically calculated the resonance width and showed that the counter irradiated electromagnetic wave is effective for a selective increase of the transverse momentum. Since, however, the cyclotron resonance occurs simultaneously with radiative damping, the resonance
*Correspondingauthor. 016~-9~~~98/$19.~0 ii-, 1998 Published Pit: SO 1~~-900~(98)00~84-~
condition gradually changes. In the relativistic cyclotron motion, the resonance condition is estimated as follows: a,:; - kl,L- (0 = 0.
(1)
Here, 52, is eB,/mc’, where B0 is an applied magnetic field, and k and COare the wave number and the frequency of the incident circularly polarized wave, respectively, and 7 is the Lorentz factor of an electron. Since the electron axial velocity I!, is almost the same as c, the speed of light, the energy reduction mainly gives rise to an increase in the cyclotron frequency, Q,fi_ and the resonance condition will not be satisfied. One solution to compensate the increase of the cyclotron frequency and to maintain the resonance condition further, is a spatial reduction of the magnetic field strength as a frequency sweeping process. In the paper, we add an effect of a spatial variation of the magnetic field to the basic equation described in the previous paper, and show a result of its effect for maintaining the resonance condition.
by Elsevier Science B.V. All rights reserved
504
T. Taguchi, K. Mima/Nucl.
Ins&. and Meth. irr P&s. Res. A 41 I (1998) 503-505
2. Numerical results of the cyclotron resonance in the variable magnetic field The basic equations to analyze the relativistic cyclotron resonance are almost same as the previous paper except that they contain the effect of spatial variation of the axial magnetic field. When the magnetic field is uniform and the radiative damping force can be neglected, the averaged transverse and axial momentum is evaluated as follows:
(PI) = PLO f
2
A~:;;p~ocYo(krYo +2kdo)
490
492
494
496
498
500
(P,)
x (1 - cos A@
Fig. 1. Averaged phase space plots of electrons after 50 m resonance in an uniform magnetic field.
6000
x (1 - cos A+t)
-‘pfo(kC + ky, - kp,,) Aq$sin A#fl.
(3)
5000
Here, k, = 52,/c, t= ct, and 6a is a normalized amplitude of the electromagnetic fields. To obtain this formula, the initial values are set to be the same for all electrons except for an initial gyrating phase, and the average is perfo~ed over the initial phase. Variables with subscripts 0 express their initial values at tl= 0 and A# = k,ly, - kpzo/y - ~1)is an angular velocity of the phase deviation from the resonance point. These formulae show that both the averaged momentums are proportional to 2 near resonance A$ z 0, and the resonance width is inversely proportional to t. Fig. 1 shows the averaged phase space plots, ((pZ), (p+,)), after the electromagnetic wave is irradiated. Here ptp= &2k, is a magnetic moment of an electron. In the calculation, the wavelength of the electromagnetic wave is 10.66 cm, its intensity is 2 x lo5 W/cm2 and the interaction length is 50 m. The strength of an applied axial magnetic field is 1 MG. The initial momentum of electrons is distributed so that the Lorentz factor, y. is around 500 with a width of 6 and the transverse velocity is around O.l2c, with a width of 0.01%. As shown in the Figure, the resonant particles increase its transverse momentum selectively. The resonance width
4000 3000 2000 1000 0t
490
492
494
496
498
500
(P*) Fig. 7. Averaged phase space plots of electrons after 50 m resonance in a linearly decreasing magnetic field.
at 50 m Ay,$ is about 0.2%, and the magnetic moment increases up to about 3 times higher than that of non-resonant electrons. When the radiative damping is switched off numerically, the resonant width is calculated as about 0.1%. Therefore, the resonant width is found to become double due to the radiative damping. Fig. 2 shows the averaged phase space plots, ((p,),
0.3% in total length. The other parameters are the same as Fig. 1. The result shows that the resonance point is slightly lower than the case of a uniform magnetic field, and the resonance width in the vicinity of the peak point is slightly narrower than the uniform case. On the other hand, the width around the bottom of the peak is broader than the uniform case, and the peak magnetic moment is smaller than that of the result of the uniform magnetic field.
3. Conclusion and discussion We have calculated the effect of the spatial variation of the magnetic field strength on the relativistic
cyclotron resonance. As a result, only a 0.3% reduction of the magnetic field affects the resonance width and the location of the resonance point. This means that the resonance point can be controllable by the variation of the magnetic field strength, but, on the other hand, field fluctuation seems to play a crucial role on the cyclotron resonance.
References [I] T. Taguchi, K. Mima. Nucl. Instr. and Meth. A 375 (1996) 610.
&MErNcmE IN PNYWCS RESEARCH
Nuclear Instruments
ELSJYIER
and Methods
in Physics
Research
A 41 I (1998) SW 505
Sectton A
Frequency sweeping process on cyclotron resonance cooling Toshihiro Taguchi”.*, Kunioki Mimab aDepartment qf Electrical Engineering, Faculw qf‘Engineering. Setsunun Utzircwi~~*. Nevagawa. Osaka 372. Japan h Institute of laser Engineering. Osaka ilniuersi~v, Suita Osuku. 565, Jupan Received
19 December
1997
Abstract
Effect of spatial variation of the magnetic field strength has been investigated in the relativistic cyclotron resonance, which is used for a cyclotron resonance cooling scheme. The result shows that the variation of the field strength is effective in narrowing the resonance width. li;l 1998 Published by Elsevier Science B.V. All rights reserved.
1. Introduction
In the previous paper El], we proposed a concept of a cyclotron resonance cooling. When the electrons propagate along a strong magnetic field with gyrating motion and a circularly polarized electromagnetic wave is irradiated counter to the gyrating beam. cyclotron resonance takes place and the transverse momentum of the resonant electrons selectively increases. Since electrons that acquire large transverse momentum loose their energy faster than non resonant electrons due to radiative damping, it is possible to reduce the phase volume in a specific part of phase space. We numerically calculated the resonance width and showed that the counter irradiated electromagnetic wave is effective for a selective increase of the transverse momentum. Since, however, the cyclotron resonance occurs simultaneously with radiative damping, the resonance
*Correspondingauthor. 016~-9~~~98/$19.~0 ii-, 1998 Published Pit: SO 1~~-900~(98)00~84-~
condition gradually changes. In the relativistic cyclotron motion, the resonance condition is estimated as follows: a,:; - kl,L- (0 = 0.
(1)
Here, 52, is eB,/mc’, where B0 is an applied magnetic field, and k and COare the wave number and the frequency of the incident circularly polarized wave, respectively, and 7 is the Lorentz factor of an electron. Since the electron axial velocity I!, is almost the same as c, the speed of light, the energy reduction mainly gives rise to an increase in the cyclotron frequency, Q,fi_ and the resonance condition will not be satisfied. One solution to compensate the increase of the cyclotron frequency and to maintain the resonance condition further, is a spatial reduction of the magnetic field strength as a frequency sweeping process. In the paper, we add an effect of a spatial variation of the magnetic field to the basic equation described in the previous paper, and show a result of its effect for maintaining the resonance condition.
by Elsevier Science B.V. All rights reserved
504
T. Taguchi, K. Mima/Nucl.
Ins&. and Meth. irr P&s. Res. A 41 I (1998) 503-505
2. Numerical results of the cyclotron resonance in the variable magnetic field The basic equations to analyze the relativistic cyclotron resonance are almost same as the previous paper except that they contain the effect of spatial variation of the axial magnetic field. When the magnetic field is uniform and the radiative damping force can be neglected, the averaged transverse and axial momentum is evaluated as follows:
(PI) = PLO f
2
A~:;;p~ocYo(krYo +2kdo)
490
492
494
496
498
500
(P,)
x (1 - cos A@
Fig. 1. Averaged phase space plots of electrons after 50 m resonance in an uniform magnetic field.
6000
x (1 - cos A+t)
-‘pfo(kC + ky, - kp,,) Aq$sin A#fl.
(3)
5000
Here, k, = 52,/c, t= ct, and 6a is a normalized amplitude of the electromagnetic fields. To obtain this formula, the initial values are set to be the same for all electrons except for an initial gyrating phase, and the average is perfo~ed over the initial phase. Variables with subscripts 0 express their initial values at tl= 0 and A# = k,ly, - kpzo/y - ~1)is an angular velocity of the phase deviation from the resonance point. These formulae show that both the averaged momentums are proportional to 2 near resonance A$ z 0, and the resonance width is inversely proportional to t. Fig. 1 shows the averaged phase space plots, ((pZ), (p+,)), after the electromagnetic wave is irradiated. Here ptp= &2k, is a magnetic moment of an electron. In the calculation, the wavelength of the electromagnetic wave is 10.66 cm, its intensity is 2 x lo5 W/cm2 and the interaction length is 50 m. The strength of an applied axial magnetic field is 1 MG. The initial momentum of electrons is distributed so that the Lorentz factor, y. is around 500 with a width of 6 and the transverse velocity is around O.l2c, with a width of 0.01%. As shown in the Figure, the resonant particles increase its transverse momentum selectively. The resonance width
4000 3000 2000 1000 0t
490
492
494
496
498
500
(P*) Fig. 7. Averaged phase space plots of electrons after 50 m resonance in a linearly decreasing magnetic field.
at 50 m Ay,$ is about 0.2%, and the magnetic moment increases up to about 3 times higher than that of non-resonant electrons. When the radiative damping is switched off numerically, the resonant width is calculated as about 0.1%. Therefore, the resonant width is found to become double due to the radiative damping. Fig. 2 shows the averaged phase space plots, ((p,),
0.3% in total length. The other parameters are the same as Fig. 1. The result shows that the resonance point is slightly lower than the case of a uniform magnetic field, and the resonance width in the vicinity of the peak point is slightly narrower than the uniform case. On the other hand, the width around the bottom of the peak is broader than the uniform case, and the peak magnetic moment is smaller than that of the result of the uniform magnetic field.
3. Conclusion and discussion We have calculated the effect of the spatial variation of the magnetic field strength on the relativistic
cyclotron resonance. As a result, only a 0.3% reduction of the magnetic field affects the resonance width and the location of the resonance point. This means that the resonance point can be controllable by the variation of the magnetic field strength, but, on the other hand, field fluctuation seems to play a crucial role on the cyclotron resonance.
References [I] T. Taguchi, K. Mima. Nucl. Instr. and Meth. A 375 (1996) 610.