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Proposal for a variable energy RFQ
Nuclear Instruments and Methods m Physics Research North-Holland, Amsterdam
A273 (1988) 1-4
1
PROPOSAL FOR A VARIABLE ENERGY RFQ A. FABRIS Istttuto Nazionale di Fisica Nucleare, Sezione di Trieste, Trieste, Italy
A. MASSAROTTI Istituto Nazionale di Fisica Nucleare, Sezione di Trieste and Dipartimento di Fisica dell' Unteersità di Trieste, Trieste, Itali
Received 25 January
1988
and m revised form
18
April
1988
A continuous variation of the energy of the particles accelerated by an RFQ can be obtained only if both the voltage and the frequency of the accelerator can be varied . A four-rod RFQ which enables to obtain a continuous variation of the frequency is presented and experimental results of the electrical tests are given. The construction of a prototype to accelerate protons using frequencies in the 50 MHz range is proposed . 1. Introduction The theory and applications of RFQs are well known and have been described elsewhere [1-3]. In these accelerators, once the frequency and the voltage are assigned, the electrode profiles, which establish the energy gain in each cell and the focusing strength, are calculated. Then a resonator working at the given frequency is built. It is well known that, within the classical mechanics' limits, using a structure calculated to accelerate a certain kind of ions at a given energy, with the same frequency and voltage, one can accelerate ions with the same q/A value (e .g . D +, He 2+ , . . .), obtaining the same energy/ nucleon ratio [4]. If one changes the q/A value, the voltage between the electrodes needs to be changed so that the unit cell (whose length L is varying, as is well known) is covered in T/2 (half period of the radiofrequency) . In other words, the unit cell length must always be equal to ßX/2, where /3 is the average normalized speed and X is the free space radiofrequency wavelength [4] . Hence it follows that, if, instead of changing the kind of accelerated particles, one wants to change the output energy, it is necessary to vary the voltage to give more or less energy . Consequently the time needed to pass one cell varies and therefore the frequency needs to be changed in order that the unit cell is covered in T/2 . 2. Discussion Since in RFQs the relative variation of the speed in a cell is usually small, the instantaneous speed in a cell is 0168-9002/88/$03 .50 © Elsevier Science Publishers B.V .
(North-Holland Physics Publishing Division)
rather close to the average speed [11. So one can assume : where A,ßn is the variation of the normalized speed in cell n and /3 is the average normalized speed to the same cell . In this case, keeping in mind that in RFQs particles are not relativistic, classical mechanics applies and the energy gain in each cell is : ~E= 4gAVo cos (p,
where q is the charge, Yo is the voltage between the electrodes (or vanes), q~ is the phase, which may vary along the structure, and : A =
mZ-1 rn 2 7o (ka) + Io (mka) '
where a and m are the parameters characterizing each cell (a is the minimum aperture radius, m is the modulation parameter), li, is the modified Bessel function of order 0 and k = 2zr/,ßA = 77/L is the wave number . A is a quantity that may vary from cell to cell and that depends on structure's geometry . A is usually consid ered as a measure of the acceleration efficiency of the machine. Since L n = ßnX/2 and L n cannot vary, an energy variation should correspond to a frequency variation, i.e . Un f0 _ = constant for any n, ~n f0
(4)
where fe is the working frequency of the accelerator .
2
A Fabris, A Ma.ssarotti / A variable energy RFQ
It follows that, once the geometric structure is fixed and if the phase profile should not vary along the structure, one can obtain energy variation only by varying the voltage and the frequency to satisfy the synchronism condition. Therefore, since the energy goes with the square of the speed, in first approximation, in order to increase by a factor Ft the output energy, the voltage between the electrodes and the input energy must be increased by the same factor, while the frequency must be increased by a factor ti /Z . Let us now consider the equation for the transverse motion for particles near the accelerator axis [1]: d = - [B sin(2aa +¢) +z1]~y~ cos(47ra+ (, P) dat (5) where :
a = t/T,
x/X ,
,
B=moe2\al
2 AVo 9 7 ,1= Z --sm (P . 2
/3 Moe
a
(6)
(7)
A has already been defined and X = 1 -AI,(ka) is again a quantity which can vary from cell to cell and which depends on the geometry . B represents the focusing effect and a the defocusing effect . The solution of the equation for the transverse motion will be convergent or divergent depending on the numerical values of
the parameters B and a . Supposing that the geometric structure is kept unchanged and that the particles' energy is varied by varying the voltage and the frequency in the way suggested above, substituting into eqs. (6) and (7) the new values of Vo and fo, the values of B and 4 are unchanged. Therefore. m first approximation, compatibly with the transverse aperture of the machine, transverse stability still holds. Furthermore, from the discussion, it follows that the ,tt factor can be considered as a scale factor of the accelerator . 3. Proposed structure and experimental results In other works [5,6] an RFQ structure has already been presented which is shown in fig. 1 . The four rods that form the RFQ's electrodes are the capacitive load of the inductance due to the supports. If this basic unit is repeated, the structure in fig. 2 is obtained, which can be considered as a unique resonating structure. Since the rods' capacitance is mainly determined by the accelerator optics, once the frequency is assigned, the inductance is determined too. Then the inductance needs to be made so as to fulfil a reasonable compromise between the support dimensions . One must, for example, make sure that propagation phenomena along the rods do not arise, maintain the dimensions reasonable and so on . All these conditions become very restrictive if a minimization of the losses is desired by varying the height a, the width b and the distance 1 between the supports (see fig. 1) . It should be noted that only two parameters are independent since the resonance frequency is fixed .
Fig. 1 . Basic unit .
A. Fabris, A Massarotti / A uariable energy RFQ
Fig. 2. Fixed frequency model.
Fig. 3. Variable frequency structure.
Several aluminium models have been built and tested . Both electrodes with constant diameter and electrodes shaped approximating a likely modulation were used . The values given here for the resonance frequency f,, the quality factor Q and the shunt impedance Rsh are relative to structures with shaped electrodes . It should be noted that, as in ref. [3], the shunt impedance is defined as : Rsh
1 VO peak LT, 2 W
9
where W is the rf power and LT is the machine length . The dimensions of the model in fig. 2 are : a = 240 mm, b = 80 mm, 1= 240 mm, and the values obtained are: fa = 76 .30 MHz, Q = 2240, Rsh = 75 500 S2 m; with b = 40 mm, one has: fo = 70 .60 MHz, Q = 2900, R,h = 90 750 S2 m.
Fig. 4. Detail of structure m fig. 3. To overcome all the conditions above mentioned, the structure to figs . 3 and 4 has been developed. Two variable capacitances were added to the exterior of the basic structure along both sides. The two parts of each capacitance are movable and connected to alternate supports. The two added capacitances are summed up to the pre-existing one, without affecting the electric
Fig. 5 Support proposed for RFQ Tneste .
4
A Fabros, A Massarotti / A variable energy RFQ
Table 1 RFQ Trieste: parameter list Particles Frequency Voltage between the electrodes Number of cells Total length Electrodes average radius Maximum surface electric field Es _ Input current Input energy Output energy Wh" ,
W.
1
R, o,
protons 50 MHz 18 kV 70 2.23 m 6 mm 5 MV/m 10 mA 30 keV 140 keV -90 °/-45 ° .1 kW 5.5 kW 82 kS2 m
field in the interaction region . In this way the frequency can be changed by simply varying the value of the external capacitance . With a = 240 mm, b = 40 mm, 1= 240 mm, one has : fo = 59 .41 MHz, Q = 2790, (least capacitance) R sh = 71540 9 m; and: f, = 46 .23 MHz, Q = 2430, (greatest capacitance) R,h = 46 400 S2 m. The bandwidth obtained is about 13 MHz. The effect of the least residual capacitance should be noted. 4. RFQ Trieste The construction of an RFQ prototype to accelerate protons using this kind of structure and a frequency in the 50 MHz range has been proposed . It will be built in copper to obtain a 1 .5 (measured) increase in R, h over the aluminium models built for the preliminary studies. In fig. 5 the support shape and dimensions are shown, the distance between the supports will be 255 mm, the capacitances will be built like those shown in fig. 4. The main project parameters at 50 MHz are listed in table 1.
It can be thought that a 60% variation on the output energy will be obtained changing the frequency from 47 to 59 MHz. The RFQ is under construction at the "Sezione INFN" of the Physics Department of the University of Trieste and will probably be operating in the first half of 1989 .
5. Conclusions The frequency change obviously depends on the value of the added capacitance . In order to obtain a structure working between two frequencies fi and f, it will be sufficient to estimate properly the two variable capacitors . The increase in loss caused by the augmented capacitance can be contained within acceptable limits and the continuous frequency change can be obtained in a rather simple way. In conclusion, with electrodes computed in such a way as to obtain transverse focusing in all the needed frequency and voltage ranges, a variable energy RFQ can be built. However, in comparison with "fixed frequency" cavity of four-rod RFQs, one has the great advantage that the frequency and the voltage are not critical anymore.
References [1] M. Puglisi, Hadronic Physics at Intermediate Energy, eds T Bressam and R.A Ricci (Elsevier, 1986) p 387. [2] M. Leo, R.A . Leo. G. Soloani, M. Pugliso, C Rossi and G. Torelli, Phys. Rev A35 (1987) 393. [3] H. Klem, IEEE Trans Nucl . Sci NS-30, (1983) 3313 . [4] T.P Wanglor and R.H . Stokes, IEEE Trans. Nucl . Sci NS-28 (1981) 1494 . [5] A. Fabris, A. Massarotti and M. Vretenar, m: New Techniques for Future Accelerators (Plenum, New York . London, 1987) p. 265 [6] A. Fabros and A Massarotti, INFN/TC-86/16 (October 1986) (in Italian) .
A273 (1988) 1-4
1
PROPOSAL FOR A VARIABLE ENERGY RFQ A. FABRIS Istttuto Nazionale di Fisica Nucleare, Sezione di Trieste, Trieste, Italy
A. MASSAROTTI Istituto Nazionale di Fisica Nucleare, Sezione di Trieste and Dipartimento di Fisica dell' Unteersità di Trieste, Trieste, Itali
Received 25 January
1988
and m revised form
18
April
1988
A continuous variation of the energy of the particles accelerated by an RFQ can be obtained only if both the voltage and the frequency of the accelerator can be varied . A four-rod RFQ which enables to obtain a continuous variation of the frequency is presented and experimental results of the electrical tests are given. The construction of a prototype to accelerate protons using frequencies in the 50 MHz range is proposed . 1. Introduction The theory and applications of RFQs are well known and have been described elsewhere [1-3]. In these accelerators, once the frequency and the voltage are assigned, the electrode profiles, which establish the energy gain in each cell and the focusing strength, are calculated. Then a resonator working at the given frequency is built. It is well known that, within the classical mechanics' limits, using a structure calculated to accelerate a certain kind of ions at a given energy, with the same frequency and voltage, one can accelerate ions with the same q/A value (e .g . D +, He 2+ , . . .), obtaining the same energy/ nucleon ratio [4]. If one changes the q/A value, the voltage between the electrodes needs to be changed so that the unit cell (whose length L is varying, as is well known) is covered in T/2 (half period of the radiofrequency) . In other words, the unit cell length must always be equal to ßX/2, where /3 is the average normalized speed and X is the free space radiofrequency wavelength [4] . Hence it follows that, if, instead of changing the kind of accelerated particles, one wants to change the output energy, it is necessary to vary the voltage to give more or less energy . Consequently the time needed to pass one cell varies and therefore the frequency needs to be changed in order that the unit cell is covered in T/2 . 2. Discussion Since in RFQs the relative variation of the speed in a cell is usually small, the instantaneous speed in a cell is 0168-9002/88/$03 .50 © Elsevier Science Publishers B.V .
(North-Holland Physics Publishing Division)
rather close to the average speed [11. So one can assume : where A,ßn is the variation of the normalized speed in cell n and /3 is the average normalized speed to the same cell . In this case, keeping in mind that in RFQs particles are not relativistic, classical mechanics applies and the energy gain in each cell is : ~E= 4gAVo cos (p,
where q is the charge, Yo is the voltage between the electrodes (or vanes), q~ is the phase, which may vary along the structure, and : A =
mZ-1 rn 2 7o (ka) + Io (mka) '
where a and m are the parameters characterizing each cell (a is the minimum aperture radius, m is the modulation parameter), li, is the modified Bessel function of order 0 and k = 2zr/,ßA = 77/L is the wave number . A is a quantity that may vary from cell to cell and that depends on structure's geometry . A is usually consid ered as a measure of the acceleration efficiency of the machine. Since L n = ßnX/2 and L n cannot vary, an energy variation should correspond to a frequency variation, i.e . Un f0 _ = constant for any n, ~n f0
(4)
where fe is the working frequency of the accelerator .
2
A Fabris, A Ma.ssarotti / A variable energy RFQ
It follows that, once the geometric structure is fixed and if the phase profile should not vary along the structure, one can obtain energy variation only by varying the voltage and the frequency to satisfy the synchronism condition. Therefore, since the energy goes with the square of the speed, in first approximation, in order to increase by a factor Ft the output energy, the voltage between the electrodes and the input energy must be increased by the same factor, while the frequency must be increased by a factor ti /Z . Let us now consider the equation for the transverse motion for particles near the accelerator axis [1]: d = - [B sin(2aa +¢) +z1]~y~ cos(47ra+ (, P) dat (5) where :
a = t/T,
x/X ,
,
B=moe2\al
2 AVo 9 7 ,1= Z --sm (P . 2
/3 Moe
a
(6)
(7)
A has already been defined and X = 1 -AI,(ka) is again a quantity which can vary from cell to cell and which depends on the geometry . B represents the focusing effect and a the defocusing effect . The solution of the equation for the transverse motion will be convergent or divergent depending on the numerical values of
the parameters B and a . Supposing that the geometric structure is kept unchanged and that the particles' energy is varied by varying the voltage and the frequency in the way suggested above, substituting into eqs. (6) and (7) the new values of Vo and fo, the values of B and 4 are unchanged. Therefore. m first approximation, compatibly with the transverse aperture of the machine, transverse stability still holds. Furthermore, from the discussion, it follows that the ,tt factor can be considered as a scale factor of the accelerator . 3. Proposed structure and experimental results In other works [5,6] an RFQ structure has already been presented which is shown in fig. 1 . The four rods that form the RFQ's electrodes are the capacitive load of the inductance due to the supports. If this basic unit is repeated, the structure in fig. 2 is obtained, which can be considered as a unique resonating structure. Since the rods' capacitance is mainly determined by the accelerator optics, once the frequency is assigned, the inductance is determined too. Then the inductance needs to be made so as to fulfil a reasonable compromise between the support dimensions . One must, for example, make sure that propagation phenomena along the rods do not arise, maintain the dimensions reasonable and so on . All these conditions become very restrictive if a minimization of the losses is desired by varying the height a, the width b and the distance 1 between the supports (see fig. 1) . It should be noted that only two parameters are independent since the resonance frequency is fixed .
Fig. 1 . Basic unit .
A. Fabris, A Massarotti / A uariable energy RFQ
Fig. 2. Fixed frequency model.
Fig. 3. Variable frequency structure.
Several aluminium models have been built and tested . Both electrodes with constant diameter and electrodes shaped approximating a likely modulation were used . The values given here for the resonance frequency f,, the quality factor Q and the shunt impedance Rsh are relative to structures with shaped electrodes . It should be noted that, as in ref. [3], the shunt impedance is defined as : Rsh
1 VO peak LT, 2 W
9
where W is the rf power and LT is the machine length . The dimensions of the model in fig. 2 are : a = 240 mm, b = 80 mm, 1= 240 mm, and the values obtained are: fa = 76 .30 MHz, Q = 2240, Rsh = 75 500 S2 m; with b = 40 mm, one has: fo = 70 .60 MHz, Q = 2900, R,h = 90 750 S2 m.
Fig. 4. Detail of structure m fig. 3. To overcome all the conditions above mentioned, the structure to figs . 3 and 4 has been developed. Two variable capacitances were added to the exterior of the basic structure along both sides. The two parts of each capacitance are movable and connected to alternate supports. The two added capacitances are summed up to the pre-existing one, without affecting the electric
Fig. 5 Support proposed for RFQ Tneste .
4
A Fabros, A Massarotti / A variable energy RFQ
Table 1 RFQ Trieste: parameter list Particles Frequency Voltage between the electrodes Number of cells Total length Electrodes average radius Maximum surface electric field Es _ Input current Input energy Output energy Wh" ,
W.
1
R, o,
protons 50 MHz 18 kV 70 2.23 m 6 mm 5 MV/m 10 mA 30 keV 140 keV -90 °/-45 ° .1 kW 5.5 kW 82 kS2 m
field in the interaction region . In this way the frequency can be changed by simply varying the value of the external capacitance . With a = 240 mm, b = 40 mm, 1= 240 mm, one has : fo = 59 .41 MHz, Q = 2790, (least capacitance) R sh = 71540 9 m; and: f, = 46 .23 MHz, Q = 2430, (greatest capacitance) R,h = 46 400 S2 m. The bandwidth obtained is about 13 MHz. The effect of the least residual capacitance should be noted. 4. RFQ Trieste The construction of an RFQ prototype to accelerate protons using this kind of structure and a frequency in the 50 MHz range has been proposed . It will be built in copper to obtain a 1 .5 (measured) increase in R, h over the aluminium models built for the preliminary studies. In fig. 5 the support shape and dimensions are shown, the distance between the supports will be 255 mm, the capacitances will be built like those shown in fig. 4. The main project parameters at 50 MHz are listed in table 1.
It can be thought that a 60% variation on the output energy will be obtained changing the frequency from 47 to 59 MHz. The RFQ is under construction at the "Sezione INFN" of the Physics Department of the University of Trieste and will probably be operating in the first half of 1989 .
5. Conclusions The frequency change obviously depends on the value of the added capacitance . In order to obtain a structure working between two frequencies fi and f, it will be sufficient to estimate properly the two variable capacitors . The increase in loss caused by the augmented capacitance can be contained within acceptable limits and the continuous frequency change can be obtained in a rather simple way. In conclusion, with electrodes computed in such a way as to obtain transverse focusing in all the needed frequency and voltage ranges, a variable energy RFQ can be built. However, in comparison with "fixed frequency" cavity of four-rod RFQs, one has the great advantage that the frequency and the voltage are not critical anymore.
References [1] M. Puglisi, Hadronic Physics at Intermediate Energy, eds T Bressam and R.A Ricci (Elsevier, 1986) p 387. [2] M. Leo, R.A . Leo. G. Soloani, M. Pugliso, C Rossi and G. Torelli, Phys. Rev A35 (1987) 393. [3] H. Klem, IEEE Trans Nucl . Sci NS-30, (1983) 3313 . [4] T.P Wanglor and R.H . Stokes, IEEE Trans. Nucl . Sci NS-28 (1981) 1494 . [5] A. Fabris, A. Massarotti and M. Vretenar, m: New Techniques for Future Accelerators (Plenum, New York . London, 1987) p. 265 [6] A. Fabros and A Massarotti, INFN/TC-86/16 (October 1986) (in Italian) .