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A super-periodic radio frequency quadrupole focused structure for the general part of the ion linac
Nuclear Instruments and Methods in Physics Research A 489 (2002) 45–58
A super-periodic radio frequency quadrupole focused structure for the general part of the ion linac S. Minaev* Institute of Theoretical and Experimental Physics (ITEP), B. Cheremushkinskaya ul. 25, Moscow 117218, Russia Received 2 June 2001; received in revised form 5 December 2001; accepted 17 March 2002
Abstract A spatially periodic radio frequency quadrupole focused structure is investigated with the aim of improving the RFQ accelerating efficiency at the second stage of the ion linac. Compared to the conventional RFQ, higher RF voltage is applied to the structure with the enlarged quadrupole aperture. The period of focusing longer than bl (super-period) is necessary to keep the transverse beam stability because the focusing gradient is reduced due to aperture increase. The RFQ structure with the period length of 2bl is studied in detail. The analytical theory promises the high values of accelerating rate and beam current; the results of numerical calculations are in good agreement with the theory prediction. That is, beam dynamics simulation in the 37 cell 5.4 m long section behind conventional RFQ gives the average accelerating gradient of around 3.3 MV/m for the medium ion beam with the current of 150 mA and normalized emittance of around 4 pmm mrad. The absence of magnetic lenses simplifies the design and reduces the total cost of the linac. r 2002 Elsevier Science B.V. All rights reserved. PACS: 41.75.Lx; 29.17.+w Keywords: Accelerators; Ion beam; Radio frequency quadrupole focusing
1. Introduction The well-known RFQ structure has been developed as an initial part of the ion linac which can deliver an intense bunched beam with the energy of several MeV; then RFQ accelerating efficiency falls off because the fixed RF voltage is applied to the increasing sell length. The drift tube structures using the magnetic quadrupole focusing become more efficient in the range of higher energy, being *Tel.: +7-095-123-0292. E-mail address: [email protected] (S. Minaev).
traditionally the main instruments for the general part of the linac. Magnetic lenses with powering and cooling systems, however, contribute to the total cost of the structure considerably. This is the reason why the idea of beam focusing by means of RF accelerating field itself is still attractive behind RFQ at least for some applications. For example, the drift tubes supplied with the ‘‘horns’’ [1,2] or ‘‘fingers’’ [3,4] as well as the ‘‘chain-like structure’’ [5] which create the quadrupole component of the RF field were under investigation in Russia, Germany, USA and Japan. In spite of high-order spatial harmonics and sparking problems, the
0168-9002/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 0 0 2 ( 0 2 ) 0 0 8 9 4 - X
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S. Minaev / Nuclear Instruments and Methods in Physics Research A 489 (2002) 45–58
‘‘horn’’ supplied structure has been successfully put into operation as a proton injector for IHEP synchrotron at Protvino [6]. On the other hand, the accelerating gradient of around 5 MV/m can be reached in the optimized alternating phase focused (APF) structure [7], where the beam stability is also obtained by means of RF field without magnets. Beam focusing by decelerating RF field in a drift tube linac [8] can be mentioned as well. The present work aims to develop the theory and application of the spatially periodic RF quadrupole focused structure (SP-RFQ) which could provide efficient acceleration at the second stage behind a conventional RFQ. In comparison with the first stage RFQ, the idea is to apply higher RF voltage to the structure with the enlarged quadrupole aperture keeping the surface field strength. The reduction of the focusing gradient caused by the aperture increase is compensated by the longer period of focusing (super-period). Following the traditional way of RFQ design, the super-period should be preferably close to the ideal geometry formed by the equipotential surfaces of the basic accelerating and focusing components. Such an approach implies a good compromise between the content of high-order spatial harmonics and surface field strength. The design principles, the features and parameters of the SP-RFQ are investigated in detail in the next sections. In order to distinguish different RFQ structures at the first and second stages, the original name of spatially uniform quadrupole focusing (or SU-RFQ) is used below for conventional first stage RFQ.
2. A simplified model of RFQ beam dynamics: accelerating rate against transverse stability The motivation for the longer period of RF focusing may be supported by using the simple linear model of beam dynamics which takes into account only one travelling wave accelerating harmonic and the main harmonic of the quadrupole periodic force. The particle bunch is assumed to be a uniformly charged ellipsoid. With this
assumption, the equation of the transverse motion can be written in the following form: d2 x eZ 2p pEz0 sinðfÞ G sin þ t þ 2 dt m0 g Tf bs l 3eZIb Mr x¼0 ð1Þ e0 cr2b Dfb bs where x is one of the transverse coordinates, G is the transverse gradient of the main focusing harmonic, Tf is the time period of the quadrupole focusing, Ez0 is the amplitude of the accelerating travelling wave harmonic, f is the current phase of an arbitrary particle in the field of accelerating harmonic, bs is the synchronous particle velocity related to the speed of light c; l is the RF wavelength, Ib is the time-averaged beam current over RF period; rb ; Dfb ; Mr are the radius, the phase extent and the form factor of the bunch, respectively, eZ; m0 ; g are the charge, the rest mass and the relativistic mass factor of the ion, respectively; e0 is the permittivity of the vacuum. Eq. (1) can be easily reduced to Mathieu’s equation: d2 x þ p2 ½a þ 2q sinð2ptÞx ¼ 0 dt2
ð2Þ
where t ¼ t=Tf is the normalized time variable. The phase advance of particle transverse oscillations per focusing period, which is the main parameter of motion stability, can be approximated as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffi q2 mx Dp a þ 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi eZ eZTf2 G2 pEz0 sinðfÞ 3Ib Mr : ¼ Tf þ m0 g 8p2 m0 g bs l e0 cr2b Dfb bs ð3Þ The right part of Eq. (3) contains under the square root one focusing term proportional to G 2 and two defocusing terms caused by the repulsing action of accelerating field and space charge, respectively. When the beam is bunched and accelerated at the first stage of RFQ until fs EconstE 301; the phase extent of the bunch Dfb does not vary considerably as well as the form factor Mr :
S. Minaev / Nuclear Instruments and Methods in Physics Research A 489 (2002) 45–58
Consider the family of p-mode structures (Wideroe or IH type) which is the most suitable for medium and heavy ion acceleration. Both accelerating harmonic and focusing gradient are created there by the same RF voltage DV applied to the accelerating cell. Neglecting the non-linear terms, the proportionality can be written as Ez0 BDV =bs l; GBDV =r20 where r0 is the distance from the axis to the electrode surface at the exactly quadrupole cross-section (for SU-RFQ, r0 is the average aperture). In practice, focusing gradient is limited by the surface field strength at the minimum gap between the adjacent electrodes of opposite polarity. Let us examine the spatially uniform RFQ where the focusing period is fixed, Tf ¼ l=c: If the RF voltage is also fixed, the accelerating amplitude reduces as 1=bs along the structure. The obvious way to improve the acceleration at the RFQ exit part is merely to raise the RF voltage together with the average aperture, r0 BDV ; keeping the similarity of the cross-section profile and surface field strength. But the focusing gradient will be reduced in this case as GBDV =r20 B1=DV : Consequently, the focusing term in Eq. (3) reduces as 1=DV 2 while the defocusing one grows as DV for the given synchronous velocity. It means that the higher RF voltage applied to the larger quadrupole aperture at the final part of SU-RFQ may cause, in practice, the dramatic reduction of transverse phase advance. As Eq. (3) shows, only lengthened period of focusing, Tf > l=c, can compensate for a weaker gradient when the accelerating voltage is gained considerably. The SP-RFQ structure with the spatial period longer than bl allows to apply more voltage to the enlarged aperture cell at the second stage behind SU-RFQ providing higher acceleration. The disadvantage of this concept is that the amplitude of transverse oscillations certainly grows with a weaker gradient and longer period, but a larger aperture can accept it. Since, the beam is preaccelerated at the first RFQ stage, some aperture increase does not lead to any beam distortions, reducing at the same time space charge density. One should notice that the RF focusing period longer than bl is used in the RFD structure [4]
47
where the RF quadrupole lenses are created within the drift tubes of Alvarez linac. Unfortunately, the cavity based on TM010 mode becomes too large for medium and heavy ions.
3. Paraxial beam dynamics analysis in the SP-RFQ structure Suppose that the RF quadrupole super-period includes p accelerating periods, Lf ¼ pbs l: The general spatial distribution of the RF potential within the accelerating channel can be represented by the infinite sum of the spatial components (harmonics) in the following form: N N X X nk r Uðr; z; yÞ ¼ U0 B0;q ðkrÞq þ Bn;q Iq p q¼0 n¼1 nk sin z þ Fn cosðqyÞ ð4Þ p where U0 is the potential amplitude; k ¼ 2p=bs l is the longitudinal wave number; Fn is the initial phase of nth longitudinal harmonic. The values of n; q are the numbers of harmonic variations over the longitudinal period and over the azimuth angle of 2p; respectively. The common azimuth number for all axisymmetric harmonics is q ¼ 0 while the numbers of q ¼ 2; 6; 10y2ð2i 1Þ; i ¼ 1; 2; 3y correspond to the quadrupole and higher symmetries. The potential terms with the coefficients B0;q do not depend on z-coordinate being responsible for beam focusing in the SU-RFQ structure. If can be easily seen that the longitudinal number of accelerating harmonic is p ¼ Lf =bl: Unlike in the SU-RFQ, accelerating harmonic is not always the lowest one in the structure, that is why the number of p appears as a denominator of the longitudinal wave number k: In the particular case of p ¼ 1; Eq. (4) gives the potential expansion for SU-RFQ channel where the longitudinal quadrupole harmonics shorter than bl are additionally included. So far as the spatially periodic RFQ is the actual subject of investigation, z-independent terms are neglected below. Simplifying the analysis, consider the RF potential as a super-position of the accelerating and focusing parts oscillating in time
S. Minaev / Nuclear Instruments and Methods in Physics Research A 489 (2002) 45–58
48
at the angular frequency o Uðr; y; z; tÞ ¼ U0 ½Aðr; zÞ þ F ðr; y; zÞcosðotÞ
ð5Þ
where only one axisymmetric harmonic and a set of the quadrupole harmonics with the amplitudes Ap ¼ Bp;0 ; Fn ¼ Bn;2 ; respectively, are taken into account Aðr; zÞ ¼ Ap I0 ðkrÞsinðkzÞ F ðr; z; yÞ ¼
nk r p n¼1 nk cos ð2yÞ sin z þ Fn : p N X
ð6Þ
Fn I2
ð7Þ
The longitudinal coordinate is chosen so that the initial phase of the axisymmetric harmonic is zero, Fp ¼ 0: The autophased longitudinal motion is described near the axis by the ordinary equation: d2 c þ m2z max ðcosðfs þ cÞ cosðfs ÞÞ ¼ 0 dt2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi eZU0 Ap mz max ¼ 2p 2m0 c2 b2s g3
ð8Þ
ð9Þ
where t ¼ ct=l is the dimensionless time variable normalized to the RF period, fs ; c are the synchronous phase and the phase deviation of an arbitrary particle from the synchronous position, respectively, mz max is the maximum phase advance of the small longitudinal oscillations which can be reached at the given accelerating amplitude near fs ¼ 901: A linear analysis of the transverse motion is available for the small deviations from the axis where I1 ðxÞDx=2; I2 ðxÞDx2 =8: Since, the quadrupole symmetry occurs in the structure, it is enough to explore only one transverse direction, for example, x at y =0. The transverse motion equation is determined by the set of the electric field harmonics derived from the potential function (5) together with Eqs. (6) and (7) N 2 d2 x eZU0 k2 1X n A sinðkzÞ þ p dt2 2 n¼1 p 2m0 g nk z þ Fn cosðotÞx ¼ 0: Fn sin ð10Þ p
Every standing wave harmonic from Eq. (10) may be represented as a super-position of direct and reverse travelling wave components sinðkzÞcosðotÞ ¼ 12½sinðkz otÞ þ sinðkz þ otÞ n sin kz þ Fn cosðotÞ p 1 n ¼ sin kz ot þ Fn 2 p n þsin kz þ ot þ Fn : p
ð11Þ
ð12Þ
Substituting the particle coordinate z ¼ z0 þ bct; one can see that the direct axisymmetric component with the number of p ¼ Lf =bl is resonant for the particles producing the timeindependent effect whereas the reverse axisymmetric component acts with the duplicated frequency of 2o and can be neglected on average. At the same time, all focusing components are kept in the structure. The period of nth harmonic action on the particles is in this case Tn ¼
pl : jp njc
ð13Þ
Allowing the index n to become both positive and negative, the polyharmonic equation of the transverse motion can be written in the following form: " N 2 d2 x eZU0 k2 1 X n A þ sinðfÞ7 p dt2 2 n¼N p 4m0 g n pn ot8Fn x ¼ 0 Fn sin f þ ð14Þ p p where f ¼ 2pz0 =bl is the phase of the particle in the field of the accelerating harmonic; jnj is the number of the quadrupole standing wave harmonic. The positive and negative signs ‘‘7’’ in front of the sum and ‘‘8’’ in front of the initial phase correspond to the direct and reverse quadrupole travelling components whose numbers are n > 0 and no0; respectively. Note that the direct components within the number interval of 0onop are faster than the particles while the others are slower ðn > pÞ or reversed ðno0Þ:
S. Minaev / Nuclear Instruments and Methods in Physics Research A 489 (2002) 45–58
Eq. (14) comprises the focusing terms acting on the particle with different frequencies. It can be investigated by using the averaging method, but with some considerable limitations (forced oscillations must be fast and small enough) resulting in unwieldy final expressions. In order to simplify the analysis, assume that one of the focusing harmonics is dominant in the structure. It seems reasonable to expect that the focusing is mainly accomplished by the quadrupole harmonic with the low number, preferably by the first one, n ¼ 1: But for every harmonic number from the interval 0onop; one can find another component with the number of 2p n which acts on the particles with the same period. Therefore, the main focusing harmonic with the largest amplitude is always supplemented by the synchronously acting additional component. Since the period is the same, both of them may be included in the simple analysis. One should especially emphasize that the direct quadrupole component with the number of p resonant for the particles is, therefore, the most dangerous for beam stability. Introduce the new time variable normalized to the focusing period t ¼ ðp nÞct=pl: Motion Eq. (14) can be reduced in this case to Mathieu’s equation taking into account both the main and the supplementary quadrupole focusing harmonics d2 x þ p2 ½aðfÞ þ 2qðfÞsinð2ptÞx ¼ 0 dt2
ð15Þ
49
namely F2pn 2 2p n 4 SðfÞ ¼ n Fn F2pn 2p n 2 2 n Fn
cosð2f þ Fn þ F2pn Þ
ð17Þ
Fn ; F2pn are the initial phases of the main and supplementary focusing harmonics, respectively. The phase advance of the transverse oscillations per focusing period can be approximated from Eq. (15) as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qðfÞ2 mr ðfÞDp aðfÞ þ ð18Þ 2 with the accuracy of around 10% within the range of 0omr o901: The exact simulations show that the approximation error grows from 3% up to 18% when the value of mr varies from 401 to 1201. According to Kapchinsky theory [2], the normalized transverse acceptance of the channel Vc and the limit current Im can be evaluated by using the following expressions: Vc ¼
jp njgmr min r2max plð1 þ qmax =2Þ2
ð19Þ
Im ¼
2pe0 m0 c3 jp njbs g2 mr min Vc kb eZ pl
ð20Þ
where rmax is the maximum possible radius of the beam envelope which is usually assumed equal to the minimum aperture, mr min is the minimum transverse phase advance within the bunch, kb is the coefficient of the beam bunching.
with the coefficients
aðfÞ ¼
qðfÞ ¼
4. Ideal geometry and parameters of 2bk super-periodic structure
eZU0 Ap sinðfÞ p2 m0 c2 b2s g ðp nÞ2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi eZU0 Fn 1 þ SðfÞ 4m0 c2 b2s g
2
n ðp nÞ2
ð16Þ
where the current phase of the particle f is supposed as an adiabatic parameter; SðfÞ is the factor of the supplementary harmonic action,
Let us examine the structure with the focusing period length of 2bl (p ¼ 2) which seems expedient to be applied directly behind RFQ; the accelerating and main focusing harmonics are A2 ; F1 ; respectively, in this case. Third quadrupole harmonic, F3 ; can be taken into account as the supplementary focusing term due to the same period of action on the particles. At the same time,
S. Minaev / Nuclear Instruments and Methods in Physics Research A 489 (2002) 45–58
50
the second quadrupole term, F2 ; must be certainly forbidden in the structure because the direct travelling wave component of this term moves synchronously with the particles (similar to the accelerating harmonic), deflecting them permanently from the axis. Note that the amplitude of this harmonic is fortunately zeroed for symmetric focusing period. Consider the symmetry when the total RF potential is always zero at z ¼ 0 meaning Fn ¼ 0: Uðr; y; zÞU0 A2 I0 ðkrÞsinðkzÞ: k k þ F1 I2 r sin z 2 2 3k 3k þF3 I2 r sin z cosð2yÞ : ð21Þ 2 2 Defining the current aperture of the channel at the given cross-section as a minimum distance from the axis to the closest electrode, suppose that the aperture is a at the maximum potential of the axisymmetric harmonic, z ¼ bl=4; and ma at the maximum of the main quadrupole harmonic, z ¼ bl=2: Unlike in the conventional RFQ, the factor of m determines the modulation of the channel on the whole but not of the single electrode. It means that the maximum aperture occurs at the symmetric cross-section. If the ratio of the supplementary harmonic amplitude to the amplitude of the main focusing harmonic hs ¼ F3 =F1 is determined beforehand, the main harmonic amplitudes can be found from two boundary equations, namely
Uða; 0; bl=4Þ¼U0 ; Uðma; 0; bl=2Þ ¼ U0 ; as follows: A2 ¼
1 pffiffiffi k 3k a 2 I 2 a þ hs I 2 C 1 B 2 B1 2 C @ A k 3k I0 ðkaÞ 2 I2 ma hs I2 ma 2 2 F1 ¼
0
1 : k 3k ma I2 ma hs I2 2 2
a=8mm
a=8mm 200
10mm 12mm 14mm 16mm
a=16mm F1
A2
ð23Þ
Fig. 1 shows how these amplitudes depend on the modulation factor m for the different aperture values if only one focusing harmonic exists in the structure, hs ¼ 0: As an example, consider the ion beam with the charge to mass ratio of 13 behind of 0.8 MeV/u SURFQ linac operating at the frequency of 80 MHz. The values of a and m which generally define the channel geometry have to be specified for the given beam parameters from the desired phase advance, mr ; taking into account the RF voltage. But the larger the quadrupole aperture ma is, the highest is the RF voltage U0 that can be applied to the structure. For this reason, the calculations were carried out in terms of the fixed field strength at the symmetric cross-section. The maximum RF voltage pffiffiffi can be estimated as 2U0 DEm D; where D ¼ mað 2ð1 þ er Þ 2er Þ is the minimum space between the adjacent electrodes at the symmetric cross-section, Em is the average electric field within this space; er ¼ re =ma is the coefficient of the
0.8
0.4
0
100
0 1.0
1.5
2.0
m
2.5
ð22Þ
1.0
1.5
2.0
2.5
m
Fig. 1. Amplitudes of basic accelerating and focusing harmonics against modulation factor.
S. Minaev / Nuclear Instruments and Methods in Physics Research A 489 (2002) 45–58
150
a=8mm 10mm 12mm 14mm 16mm
µr s ,deg
100
50
choice of a ¼ 10 mm, mrs ¼ 601 gives m ¼ 1:97; A2 ¼ 0:79; F1 ¼ 49: The right part of Fig. 2 shows the phase advance plotted versus arbitrary particle phase at the different supplementary harmonic amplitudes. It is seen that the dependence mr ðfÞ which is originally caused by the varied defocusing action of the accelerating field (see curve h ¼ 0) is compensated considerably near the synchronous phase of 301 when the supplementary harmonic ratio is around hs ¼ 0:021: Being the exclusive feature of RF focusing, the reduced dependence of
120
80
h=0.042 h=0.021 h=0
µr s ,deg
electrode curvature, re is the radius of the curvature. The curvature coefficient, er ¼ 0:75; typical for SU-RFQ is chosen for the present example (note that the value of ma is the same as the average aperture radius R0 in the existing RFQ theory [2]). The average electric field Em ¼ 25 MV/m is assumed at the minimum gap between the electrodes. The transverse phase advance in the two harmonics structure calculated for the synchronous phase of 301 at different aperture radii is shown by a graph in the left part of Fig. 2. The
51
40
0
0 1.0
1.5
2.0
2.5
-60
-30
0
30
, deg
m
Fig. 2. Phase advance of the transverse oscillations against modulation factor and current phase of an arbitrary particle.
Table 1 Evaluated parameters of 2bl superperiodic structure behind of conventional RFQ Charge to mass ratio Beam energy Operating frequency Electric field estimated at the symmetric cross-section Aperture radius Synchronous phase Bunch extent Transverse phase advance
1/3 MeV/u MHz MV/m Mm Deg. Deg. Deg.
0.8 80 25 10 30 60 60
Type of the structure
Two harmonics
Three harmonics
Supplementary harmonic ratio Modulation factor Radiofrequency voltage Acceleration rate Minimum transverse phase advance Normalized transverse acceptance Limit current
0 1.97 240 3.3 42 5.7p 177
0.021 2.25 274 3.7 60 7.1p 321
7; kV MV/m Deg. mm mrad mA
52
S. Minaev / Nuclear Instruments and Methods in Physics Research A 489 (2002) 45–58
YS /
XS /
the transverse phase advance along the bunch may improve the intense beam stability and moderate the emittance growth. An example of evaluated parameters of 2bl super-periodic structure without and with the supplementary quadrupole focusing harmonic is shown in Table 1. The average electric field of 25 MV/m at the minimum gap in the symmetric cross-section is used in this table. Numerical calculations give the maximum surface field strength of around 28 MV/m in this case. One can see that the three harmonics (3H) structure
takes more RF voltage at the same surface electric field resulting in larger channel acceptance and higher limit current. The longitudinal profiles of the ideal equipotential electrodes in the xoz and yoz planes are shown at the upper and lower parts of Fig. 3, respectively. More complicated 3 H configuration leads to higher capacitive loading. The question whether the advantages given by the supplementary harmonic really cost additional RF power and more complicated geometry, should be answered after an exact simulation of the intense beam dynamics.
−U0
+U0
0.2
3H profile 0
2a
2H profile
2ma −U0
0.2 0
0.25
0.5
0.75
+U0 1.0
1.25
1.5
1.75
2.0
z / Fig. 3. Longitudinal profiles of ideal equipotential electrodes without and with the supplementary focusing harmonics.
+Uo sin (t)
Symmetriccross-sectionS-S
S
S −Uo sin (t)
(a)
2
(b) Fig. 4. Super-period formed by the electrodes with constant cross-section curvature (type 2).
S. Minaev / Nuclear Instruments and Methods in Physics Research A 489 (2002) 45–58
5. Realistic design of the accelerating channel The ideal equipotential geometry has to be approximated by the real electrodes available for the conventional manufacturing technique. Fig. 4 shows how the 2bl super-period may look in reality for the simple two harmonics case. The couples of identical electrodes form the gaps of two kinds, one of them being mostly accelerating while the other one combining the accelerating and focusing functions. One has to emphasize that the acceleration is nearly the same in these different gaps. Generally, the accelerating channel can be interpreted as a sequence of the RF quadrupole lenses integrated to every other gap of the p-mode structure. Unlike the known magnetic focused structures, the RF quadrupole super-period has no drift space for the longitudinal motion and accelerates the particles two times at every RF period. In order to keep the beam quality, the highorder azimuthal harmonics of RF potential given by the terms q > 2 in the series (4) should be minimized because just these components are mainly responsible for beam distortion and cause the emittance growth. At the same time, the terms specified by the numbers of n > 1; q ¼ 2 determine the longitudinal distribution of the quadrupole potential component and may even improve the focusing conditions as it was shown by the example of the supplementary harmonic. Fourth axisymmetric harmonic, n ¼ 4; q ¼ 0 determines the longitudinal asymmetry of the accelerating period bl; i.e. different acceleration in the gaps of different kind when the total acceleration is specified by the main accelerating harmonic, n ¼ 2; q ¼ 0: The higher axisymmetric terms, n > 4; q ¼ 0; do not disturb the particle motion, being just useless both for acceleration and focusing. In practice, the single electrode can be either machined by means of rotation around some axis producing the varied cross-section radius as Fig. 4a shows (type 1) or manufactured by a special cutter with the semicircular cross-section of the fixed curvature similar to the conventional RFQ production (type 2, Fig. 4b). The potential distributions created by the electrodes of different types have been investigated and compared by
53
using the specially developed 3D electrostatic code. The results clearly show that the constant curvature electrodes fit to the ideal geometry better than those of conical shape do. For example, the same longitudinal profile of an ideal equipotential geometry formed by two harmonics, A2 ¼ 0:787 and F1 ¼ 49:25; had been applied to the electrodes of both types. The realistic accelerating and focusing amplitudes calculated for the electrodes of type 1 differ from the ideal values by +10% and 12%, respectively, whereas the electrodes of type 2 give the difference of around +1.5% and 2%, respectively. Fig. 5 shows an example of the equipotential lines at the longitudinal and transverse cross-sections for the fixed curvature electrodes of type 2.
6. Beam dynamics simulation in the SP-RFQ section Beam dynamics in the test SP-RFQ section has been numerically simulated in order to verify the analytic theory. A special computer code was developed where the particles are assumed as the uniformly charged spherical clouds, one transparent for another. Particle 3D trajectories are calculated in time at the electric field given by the set of oscillating spatial harmonics. The whole structure is subdivided into elementary bl=2 cells with the accelerating gap at the center of each of them (remember two kinds of the gap exist in the structure). At the first step of simulation, the electric field in every cell is defined in terms of the main geometric parameters a and ma; the particle dynamics is simulated in the field of accelerating and focusing harmonics only. When a satisfactory two-term beam dynamics is used, a final simulation in the full electric field has to be done. Fig. 6 shows the values of minimum aperture a and quadrupole aperture ma along the structure. The transverse phase advance calculated for zero current is shown in the same plot. The realistic electrodes with the fixed crosssection curvature have been chosen for the test section. Three different radii of the electrode curvature, namely 16, 20 and 24 mm are used in
54
S. Minaev / Nuclear Instruments and Methods in Physics Research A 489 (2002) 45–58
Fig. 5. Equipotential map calculated for the realistic electrodes with the constant cross-section curvature (type 2).
60 µr , deg
ma, mm
40
20 a, mm 0
1
2
3
4
5
L, m
Fig. 6. Transverse phase advance and channel aperture along the simulated structure.
the simulated structure. The RF voltage DU ¼ 2U0 applied to the accelerating cells grows smoothly from 0.5 MV at the beginning to 1.18 MV at the end of the structure as Fig. 7 shows. Since the quadrupole aperture ma also grows, the averaged electric field at the symmetric cross-section does not exceed 25 MV/m. The electric field at the axis of 37-cell section is shown in Fig. 7 as well. Different shapes of the field distribution can be seen at the gaps of different kinds.
In order to test the structure capability, high values of normalized transverse emittance and beam current, namely 4.4 pmrad mm and 150 mA, respectively, have been simulated; these values seem to be close to the SU-RFQ limit at the given charge to mass ratio and operating frequency. Table 2 shows the simulated parameters of the section. Fig. 8 demonstrates two 95% transverse envelopes for beam current values of 40 and 150 mA. The beam size increases a little along the
S. Minaev / Nuclear Instruments and Methods in Physics Research A 489 (2002) 45–58
55
U, MV
1.5
1.0
0.5
0
0
1
2
3
4
5
L, m
0
1
2
3
4
5
L, m
Ez , MV/m
10
0
-10
Fig. 7. RF potential applied to the cells and electric field distribution at the axis of SP-RFQ section.
Table 2 Simulated parameters of SP RFQ section 1 3
Charge to mass ratio Operating frequency Input/output beam energy Number of the cells Total length Synchronous phase RF voltage applied to the cell Radii of electrode curvature Electric field estimated at the symmetric cross-sections Normalized transverse beam emittance Input/output bunch extent Beam current
MHz MeV/u m Deg. MV mm MV/m
81.4 0.8/6.7 37 5.42 30 0.5–1.18 16, 20, 24 23–25
mm mrad
4.4p
Deg. mA
52/30 40, 150
around 3.3 MV/m. The results of numerical simulation shown in Table 2 are in good agreement with the parameters evaluated from the analytic theory.
7. Different RFQ structures—a comparison The accelerating efficiency of different RFQ structures should be estimated and compared at the same conditions that have been used for SPRFQ structure shown in Table 1. A conventional SU-RFQ where Lf ¼ bl; p ¼ 1 may be evaluated at the beginning. The two-term RF potential (5) includes in this case the axisymmetric and zindependent quadrupole components Aðr; zÞ ¼ A1 I0 ðkrÞsinðkzÞ F ðr; zÞ ¼ F0 k2 r2 cosð2yÞ:
structure in both cases, but the growth of transverse RMS emittance was not clearly seen. First beam losses of around 2% appear in the structure when the beam current is close to 150 mA. The longitudinal stability is easily obtained at the synchronous phase of 301. Finally, the total energy of 17.67 MeV per charge unit is gained on the length of 5.42 m producing an average accelerating gradient of
ð24Þ
The normalized amplitudes A1 ; F0 can be specified by using the minimum and maximum distances from the axis to the electrode denoted as a and am ; respectively, a2m a2 a2m I0 ðkaÞ þ a2 I0 ðkam Þ 1 A1 I0 ðkaÞ : F0 ¼ k 2 a2 A1 ¼
ð25Þ
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S. Minaev / Nuclear Instruments and Methods in Physics Research A 489 (2002) 45–58
x, mm
20 150 mA 10
40 mA y, mm
0
10
20 0
1
2
4
3
5
L, m
Fig. 8. 95% transverse envelopes for different beam current of 40 and 150 mA.
The Mathieu’s Eq. (15) is valid for SU-RFQ structure with the following coefficients: eZU0 A1 sinðfÞ m0 c2 b2s g 4eZU0 F0 : q¼ m0 c2 b2s g
aðfÞ ¼
B1;0 ; F0 ¼ B0;2 ; F2 ¼ B2;2 : In contrast to the SURFQ structure, a second quadrupole harmonic is added to the focusing part of the RF potential F ðr; y; zÞ ¼ ðF0 k2 r2 þ F2 I2 ð2krÞ cosð2kzÞÞcosð2yÞ
ð26Þ
Similar to the example given by Table 1, the values of er ¼ 0:75; Em ¼ 25 MV/m are chosen for the SU-RFQ structure. Formula (18) with the coefficients (26) gives the transverse phase advance of mr ¼ 601 at the maximum RF voltage U0 ¼ 151 kV applied to the average aperture of R0 ¼ 11:7 mm (a ¼ 8 mm, am ¼ 15:4 mm). With these parameters, the maximum accelerating gradient of 1.4 MV/m can be achieved in the SU-RFQ structure. At the next step, consider the RF quadrupole focused p-mode ‘‘split coaxial’’ structure described in Ref. [3]. The period of focusing is as long as bl including two similar accelerating gaps each of which is supplied with two couples of ‘‘fingers’’. At the fixed moment of time, all gaps produce either focusing or defocusing action together for the same direction, x or y; whereas the electric field is opposite at the adjacent gaps. Analyzing the spatial potential distribution, one can find two main components of the quadrupole field: zindependent component and longitudinally varied one with the number of n ¼ 2 and period length of bl=2: The number of the accelerating harmonic is p ¼ 1: Related to potential expression (4), the amplitudes of the main components are A1 ¼
ð27Þ
bringing the phase dependence to the ‘‘focusing’’ coefficient of Mathieu’s equation: qðfÞ ¼
eZU0 m0 c2 b2s g
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 16F02 þ F22 þ 8F0 F2 cosð2fÞ:
ð28Þ
One should mention that only a structure with low content of high-order harmonic might be correctly analyzed by the simple theory. When the ‘‘fingers’’ or ‘‘horns’’ are introduced, the highorder terms contribute to the RF potential considerably. Therefore, the amplitudes of the main harmonics can be specified keeping in mind that the analysis is very approximate 1 1 1 A1 ¼ ; F0 ¼ 2 2 ; F2 ¼ : ð29Þ I0 ðkaÞ 2k a 2I2 ð2kaÞ The accelerating potential Aðr; zÞ; which is the same as in the SU-RFQ (see Eq. (24)), equals zero at the center of the gap, Aðr; 0Þ ¼ 0; becoming maximum at the tube center, Aða; bl=4Þ ¼ 1; while the quadrupole potential (27) behaves vice versa, F ða; 0; 0Þ ¼ 1; F ða; y; bl=4Þ ¼ 0: The uniform and periodic quadrupole components cancel each other out at the tube center, acting together at the center of the gap. Keeping the finger radius of 0:75a; average the field between the fingers Em ¼ 25 MV/ m and transverse phase advance, mr ¼ 601; the
S. Minaev / Nuclear Instruments and Methods in Physics Research A 489 (2002) 45–58
maximum accelerating gradient can be estimated as 1.7 MV/m. Note that the last value may be difficult to reach in practice because of high field strength at the fingertips. The RF focused drift tube linac (RFD) is based on the 2p-mode Alvarez structure where the RF quadrupole lenses are created by two pairs of fingers installed within the drift tubes. Unlike in the p-mode structures, the RF voltage applied to the accelerating gap, 2Ua ; differs from the finger voltage, 2Uf : The ratio of Ua =Uf may be varied by changing the geometry of the cavity, namely the space between the blades of the tube stem. Similar to the SP-RFQ structure discussed above, the focusing and accelerating harmonics are first and second, F1 and A2 ; respectively. When both accelerating field and focusing gradient are distributed uniformly within the gap and lens, equivalent harmonic amplitudes may be specified as follows: 4 kg A2 ¼ sin pkg 2 4 kL F1 ¼ sin pI2 ðka=2Þ 4
ð30Þ
where g and L are the lengths of the accelerating gap and focusing lens, respectively. The expressions for Mathieu’s coefficients (16) are valid in this case with n ¼ 1; p ¼ 2; SðfÞ ¼ 0 and Ua ; Uf instead of U0 : In order to roughly estimate the RFD abilities, suppose that the accelerating gap is as long as g ¼ bl=4 with the average field strength of 10 MV/m. The effective length of the RF lens is assumed as L ¼ bl=2: Since, the focusing voltage does not depend on the accelerating one, beam focusing may be optimized at the lower voltage and smaller aperture. In particular, phase advance of 601 is obtained at the aperture of 10 mm and finger–finger voltage of 114 kV only. The total accelerating potential of 390 kV produces the average accelerating gradient of around 1.9 MV/m independent of the focusing parameters. Finally, the RFD structure seems more flexible in focusing than SP-RFQ, but the accelerating efficiency is limited because only one accelerating gap is possible at the length of bl: Besides, the
57
Alvarez structure becomes too big in diameter for medium and heavy ions. The accelerating channel of SP-RFQ structure supplied with the symmetric RF voltage 7U0 has been investigated above. In practice, the structure on the whole can be based on TE11 or TE12 mode cavity in this case. As an example, IH-DTL structure of GSI high current linac [9] may be mentioned where 920 kW RF power is dissipated in the walls of the 10.3 m long tank generating the maximum RF voltage amplitude of 1.3 MV. The voltage increase along the structure may be realized by the special tuning when the inductance or/and capacitance per length unit is ramped up from the entrance to the exit. Alternatively, the growing RF voltage per cell can be naturally obtained at the modified TM010 structure proposed in Ref. [5]. The capacitance of the SP-RFQ channel is obviously larger than that for drift tube channel because of focusing regions with quadrupole configuration, especially compared to IHDTL where the tubes are minimized. Therefore, some additional RF power is always necessary for RF quadrupole focusing. The optimistic factor of the power growth might be very roughly estimated as around 1.5. The correct quantity of the RF power consumption has to be specified from the numerical simulation and experimental study of the chosen cavity.
8. Conclusion A theory developed in this paper shows how the RFQ accelerating efficiency can be improved at the second stage of the ion linac. Higher RF voltage applied to the structure with the enlarged quadrupole aperture raises the acceleration rate behind the conventional RFQ. At the same time, the focusing period longer than bl; or superperiod, is necessary in this case to keep the transverse stability of the beam because the gradient is reduced due to aperture increase. The polyharmonic model of the beam dynamics allows to find the optimized geometry of the accelerating channel and to analyze the main parameters of the structure.
58
S. Minaev / Nuclear Instruments and Methods in Physics Research A 489 (2002) 45–58
The spatially periodic RFQ structure with the period length of 2bl has been investigated in detail. Analytical theory predicts high values of the channel acceptance, beam current limit and accelerating gradient more than 3 MV/m at the realistic surface field strength. Numerical simulation of the beam dynamics in the 37 cell 5.4 m long section confirms the capability of the structure, being in good agreement with the analytical evaluations. In particular, 150 mA ion beam is accelerated with the average effective gradient of around 3.3 MV/ m, which is much higher than a conventional RFQ can provide at the first stage of the linac. An approximate comparison with the other RFQ structures shows that the SP-RFQ may provide the highest accelerating rate among them. On the other hand, the SP-RFQ seems quite competitive with the magnetic focusing behind SU-RFQ. The absence of magnetic lenses simplifies the design reducing the total cost of the linac considerably. The super-period length of 2bl is only one particular case, which can be regarded as FODORFQ lattice. The super-period may contain more accelerating gaps and RF quadrupole lenses
integrated to the structure; the doublets and triplets can be created if necessary. The theory developed in this paper can be applied to longer period as well for parameter evaluation.
References [1] V.A. Teplyakov, Prib. Tekn. Eksp. 6 (1964) 24. [2] I.M. Kapchinsky, Theory of Resonance Linear Accelerators, Harwood, New York, 1985. [3] R.W. Muller, et al., Proceedings of the 1984 Linear Accelerator Conference, GSI-Darmstadt, Germany, 1984, p. 77. [4] D.A. Swenson, et al., Proceedings of the 1998 LINAC Conference, Chicago, IL, 1998, p. 648. [5] M. Odera, M. Hemmi, Proceedings of the 1984 Linear Accelerator Conference, GSI-Darmstadt, Germany, 1984, p. 346. [6] V.A. Zenin, et al., Proceedings of the 1994 International Linac Conference, Tsukuba, Japan, 1994, p. 156. [7] S. Minaev, U. Ratzinger, B. Schlitt, Proceedings of the 1999 Particle Accelerator Conference, New York, 1999, p. 3555. [8] A.I. Balabin, G.N. Kropachev, Nucl. Instr. and Meth. A 459 (1–2) (2001) 87. [9] U. Ratzinger, Nucl. Instr. and Meth. A 464 (2001) 636.
A super-periodic radio frequency quadrupole focused structure for the general part of the ion linac S. Minaev* Institute of Theoretical and Experimental Physics (ITEP), B. Cheremushkinskaya ul. 25, Moscow 117218, Russia Received 2 June 2001; received in revised form 5 December 2001; accepted 17 March 2002
Abstract A spatially periodic radio frequency quadrupole focused structure is investigated with the aim of improving the RFQ accelerating efficiency at the second stage of the ion linac. Compared to the conventional RFQ, higher RF voltage is applied to the structure with the enlarged quadrupole aperture. The period of focusing longer than bl (super-period) is necessary to keep the transverse beam stability because the focusing gradient is reduced due to aperture increase. The RFQ structure with the period length of 2bl is studied in detail. The analytical theory promises the high values of accelerating rate and beam current; the results of numerical calculations are in good agreement with the theory prediction. That is, beam dynamics simulation in the 37 cell 5.4 m long section behind conventional RFQ gives the average accelerating gradient of around 3.3 MV/m for the medium ion beam with the current of 150 mA and normalized emittance of around 4 pmm mrad. The absence of magnetic lenses simplifies the design and reduces the total cost of the linac. r 2002 Elsevier Science B.V. All rights reserved. PACS: 41.75.Lx; 29.17.+w Keywords: Accelerators; Ion beam; Radio frequency quadrupole focusing
1. Introduction The well-known RFQ structure has been developed as an initial part of the ion linac which can deliver an intense bunched beam with the energy of several MeV; then RFQ accelerating efficiency falls off because the fixed RF voltage is applied to the increasing sell length. The drift tube structures using the magnetic quadrupole focusing become more efficient in the range of higher energy, being *Tel.: +7-095-123-0292. E-mail address: [email protected] (S. Minaev).
traditionally the main instruments for the general part of the linac. Magnetic lenses with powering and cooling systems, however, contribute to the total cost of the structure considerably. This is the reason why the idea of beam focusing by means of RF accelerating field itself is still attractive behind RFQ at least for some applications. For example, the drift tubes supplied with the ‘‘horns’’ [1,2] or ‘‘fingers’’ [3,4] as well as the ‘‘chain-like structure’’ [5] which create the quadrupole component of the RF field were under investigation in Russia, Germany, USA and Japan. In spite of high-order spatial harmonics and sparking problems, the
0168-9002/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 0 0 2 ( 0 2 ) 0 0 8 9 4 - X
46
S. Minaev / Nuclear Instruments and Methods in Physics Research A 489 (2002) 45–58
‘‘horn’’ supplied structure has been successfully put into operation as a proton injector for IHEP synchrotron at Protvino [6]. On the other hand, the accelerating gradient of around 5 MV/m can be reached in the optimized alternating phase focused (APF) structure [7], where the beam stability is also obtained by means of RF field without magnets. Beam focusing by decelerating RF field in a drift tube linac [8] can be mentioned as well. The present work aims to develop the theory and application of the spatially periodic RF quadrupole focused structure (SP-RFQ) which could provide efficient acceleration at the second stage behind a conventional RFQ. In comparison with the first stage RFQ, the idea is to apply higher RF voltage to the structure with the enlarged quadrupole aperture keeping the surface field strength. The reduction of the focusing gradient caused by the aperture increase is compensated by the longer period of focusing (super-period). Following the traditional way of RFQ design, the super-period should be preferably close to the ideal geometry formed by the equipotential surfaces of the basic accelerating and focusing components. Such an approach implies a good compromise between the content of high-order spatial harmonics and surface field strength. The design principles, the features and parameters of the SP-RFQ are investigated in detail in the next sections. In order to distinguish different RFQ structures at the first and second stages, the original name of spatially uniform quadrupole focusing (or SU-RFQ) is used below for conventional first stage RFQ.
2. A simplified model of RFQ beam dynamics: accelerating rate against transverse stability The motivation for the longer period of RF focusing may be supported by using the simple linear model of beam dynamics which takes into account only one travelling wave accelerating harmonic and the main harmonic of the quadrupole periodic force. The particle bunch is assumed to be a uniformly charged ellipsoid. With this
assumption, the equation of the transverse motion can be written in the following form: d2 x eZ 2p pEz0 sinðfÞ G sin þ t þ 2 dt m0 g Tf bs l 3eZIb Mr x¼0 ð1Þ e0 cr2b Dfb bs where x is one of the transverse coordinates, G is the transverse gradient of the main focusing harmonic, Tf is the time period of the quadrupole focusing, Ez0 is the amplitude of the accelerating travelling wave harmonic, f is the current phase of an arbitrary particle in the field of accelerating harmonic, bs is the synchronous particle velocity related to the speed of light c; l is the RF wavelength, Ib is the time-averaged beam current over RF period; rb ; Dfb ; Mr are the radius, the phase extent and the form factor of the bunch, respectively, eZ; m0 ; g are the charge, the rest mass and the relativistic mass factor of the ion, respectively; e0 is the permittivity of the vacuum. Eq. (1) can be easily reduced to Mathieu’s equation: d2 x þ p2 ½a þ 2q sinð2ptÞx ¼ 0 dt2
ð2Þ
where t ¼ t=Tf is the normalized time variable. The phase advance of particle transverse oscillations per focusing period, which is the main parameter of motion stability, can be approximated as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffi q2 mx Dp a þ 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi eZ eZTf2 G2 pEz0 sinðfÞ 3Ib Mr : ¼ Tf þ m0 g 8p2 m0 g bs l e0 cr2b Dfb bs ð3Þ The right part of Eq. (3) contains under the square root one focusing term proportional to G 2 and two defocusing terms caused by the repulsing action of accelerating field and space charge, respectively. When the beam is bunched and accelerated at the first stage of RFQ until fs EconstE 301; the phase extent of the bunch Dfb does not vary considerably as well as the form factor Mr :
S. Minaev / Nuclear Instruments and Methods in Physics Research A 489 (2002) 45–58
Consider the family of p-mode structures (Wideroe or IH type) which is the most suitable for medium and heavy ion acceleration. Both accelerating harmonic and focusing gradient are created there by the same RF voltage DV applied to the accelerating cell. Neglecting the non-linear terms, the proportionality can be written as Ez0 BDV =bs l; GBDV =r20 where r0 is the distance from the axis to the electrode surface at the exactly quadrupole cross-section (for SU-RFQ, r0 is the average aperture). In practice, focusing gradient is limited by the surface field strength at the minimum gap between the adjacent electrodes of opposite polarity. Let us examine the spatially uniform RFQ where the focusing period is fixed, Tf ¼ l=c: If the RF voltage is also fixed, the accelerating amplitude reduces as 1=bs along the structure. The obvious way to improve the acceleration at the RFQ exit part is merely to raise the RF voltage together with the average aperture, r0 BDV ; keeping the similarity of the cross-section profile and surface field strength. But the focusing gradient will be reduced in this case as GBDV =r20 B1=DV : Consequently, the focusing term in Eq. (3) reduces as 1=DV 2 while the defocusing one grows as DV for the given synchronous velocity. It means that the higher RF voltage applied to the larger quadrupole aperture at the final part of SU-RFQ may cause, in practice, the dramatic reduction of transverse phase advance. As Eq. (3) shows, only lengthened period of focusing, Tf > l=c, can compensate for a weaker gradient when the accelerating voltage is gained considerably. The SP-RFQ structure with the spatial period longer than bl allows to apply more voltage to the enlarged aperture cell at the second stage behind SU-RFQ providing higher acceleration. The disadvantage of this concept is that the amplitude of transverse oscillations certainly grows with a weaker gradient and longer period, but a larger aperture can accept it. Since, the beam is preaccelerated at the first RFQ stage, some aperture increase does not lead to any beam distortions, reducing at the same time space charge density. One should notice that the RF focusing period longer than bl is used in the RFD structure [4]
47
where the RF quadrupole lenses are created within the drift tubes of Alvarez linac. Unfortunately, the cavity based on TM010 mode becomes too large for medium and heavy ions.
3. Paraxial beam dynamics analysis in the SP-RFQ structure Suppose that the RF quadrupole super-period includes p accelerating periods, Lf ¼ pbs l: The general spatial distribution of the RF potential within the accelerating channel can be represented by the infinite sum of the spatial components (harmonics) in the following form: N N X X nk r Uðr; z; yÞ ¼ U0 B0;q ðkrÞq þ Bn;q Iq p q¼0 n¼1 nk sin z þ Fn cosðqyÞ ð4Þ p where U0 is the potential amplitude; k ¼ 2p=bs l is the longitudinal wave number; Fn is the initial phase of nth longitudinal harmonic. The values of n; q are the numbers of harmonic variations over the longitudinal period and over the azimuth angle of 2p; respectively. The common azimuth number for all axisymmetric harmonics is q ¼ 0 while the numbers of q ¼ 2; 6; 10y2ð2i 1Þ; i ¼ 1; 2; 3y correspond to the quadrupole and higher symmetries. The potential terms with the coefficients B0;q do not depend on z-coordinate being responsible for beam focusing in the SU-RFQ structure. If can be easily seen that the longitudinal number of accelerating harmonic is p ¼ Lf =bl: Unlike in the SU-RFQ, accelerating harmonic is not always the lowest one in the structure, that is why the number of p appears as a denominator of the longitudinal wave number k: In the particular case of p ¼ 1; Eq. (4) gives the potential expansion for SU-RFQ channel where the longitudinal quadrupole harmonics shorter than bl are additionally included. So far as the spatially periodic RFQ is the actual subject of investigation, z-independent terms are neglected below. Simplifying the analysis, consider the RF potential as a super-position of the accelerating and focusing parts oscillating in time
S. Minaev / Nuclear Instruments and Methods in Physics Research A 489 (2002) 45–58
48
at the angular frequency o Uðr; y; z; tÞ ¼ U0 ½Aðr; zÞ þ F ðr; y; zÞcosðotÞ
ð5Þ
where only one axisymmetric harmonic and a set of the quadrupole harmonics with the amplitudes Ap ¼ Bp;0 ; Fn ¼ Bn;2 ; respectively, are taken into account Aðr; zÞ ¼ Ap I0 ðkrÞsinðkzÞ F ðr; z; yÞ ¼
nk r p n¼1 nk cos ð2yÞ sin z þ Fn : p N X
ð6Þ
Fn I2
ð7Þ
The longitudinal coordinate is chosen so that the initial phase of the axisymmetric harmonic is zero, Fp ¼ 0: The autophased longitudinal motion is described near the axis by the ordinary equation: d2 c þ m2z max ðcosðfs þ cÞ cosðfs ÞÞ ¼ 0 dt2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi eZU0 Ap mz max ¼ 2p 2m0 c2 b2s g3
ð8Þ
ð9Þ
where t ¼ ct=l is the dimensionless time variable normalized to the RF period, fs ; c are the synchronous phase and the phase deviation of an arbitrary particle from the synchronous position, respectively, mz max is the maximum phase advance of the small longitudinal oscillations which can be reached at the given accelerating amplitude near fs ¼ 901: A linear analysis of the transverse motion is available for the small deviations from the axis where I1 ðxÞDx=2; I2 ðxÞDx2 =8: Since, the quadrupole symmetry occurs in the structure, it is enough to explore only one transverse direction, for example, x at y =0. The transverse motion equation is determined by the set of the electric field harmonics derived from the potential function (5) together with Eqs. (6) and (7) N 2 d2 x eZU0 k2 1X n A sinðkzÞ þ p dt2 2 n¼1 p 2m0 g nk z þ Fn cosðotÞx ¼ 0: Fn sin ð10Þ p
Every standing wave harmonic from Eq. (10) may be represented as a super-position of direct and reverse travelling wave components sinðkzÞcosðotÞ ¼ 12½sinðkz otÞ þ sinðkz þ otÞ n sin kz þ Fn cosðotÞ p 1 n ¼ sin kz ot þ Fn 2 p n þsin kz þ ot þ Fn : p
ð11Þ
ð12Þ
Substituting the particle coordinate z ¼ z0 þ bct; one can see that the direct axisymmetric component with the number of p ¼ Lf =bl is resonant for the particles producing the timeindependent effect whereas the reverse axisymmetric component acts with the duplicated frequency of 2o and can be neglected on average. At the same time, all focusing components are kept in the structure. The period of nth harmonic action on the particles is in this case Tn ¼
pl : jp njc
ð13Þ
Allowing the index n to become both positive and negative, the polyharmonic equation of the transverse motion can be written in the following form: " N 2 d2 x eZU0 k2 1 X n A þ sinðfÞ7 p dt2 2 n¼N p 4m0 g n pn ot8Fn x ¼ 0 Fn sin f þ ð14Þ p p where f ¼ 2pz0 =bl is the phase of the particle in the field of the accelerating harmonic; jnj is the number of the quadrupole standing wave harmonic. The positive and negative signs ‘‘7’’ in front of the sum and ‘‘8’’ in front of the initial phase correspond to the direct and reverse quadrupole travelling components whose numbers are n > 0 and no0; respectively. Note that the direct components within the number interval of 0onop are faster than the particles while the others are slower ðn > pÞ or reversed ðno0Þ:
S. Minaev / Nuclear Instruments and Methods in Physics Research A 489 (2002) 45–58
Eq. (14) comprises the focusing terms acting on the particle with different frequencies. It can be investigated by using the averaging method, but with some considerable limitations (forced oscillations must be fast and small enough) resulting in unwieldy final expressions. In order to simplify the analysis, assume that one of the focusing harmonics is dominant in the structure. It seems reasonable to expect that the focusing is mainly accomplished by the quadrupole harmonic with the low number, preferably by the first one, n ¼ 1: But for every harmonic number from the interval 0onop; one can find another component with the number of 2p n which acts on the particles with the same period. Therefore, the main focusing harmonic with the largest amplitude is always supplemented by the synchronously acting additional component. Since the period is the same, both of them may be included in the simple analysis. One should especially emphasize that the direct quadrupole component with the number of p resonant for the particles is, therefore, the most dangerous for beam stability. Introduce the new time variable normalized to the focusing period t ¼ ðp nÞct=pl: Motion Eq. (14) can be reduced in this case to Mathieu’s equation taking into account both the main and the supplementary quadrupole focusing harmonics d2 x þ p2 ½aðfÞ þ 2qðfÞsinð2ptÞx ¼ 0 dt2
ð15Þ
49
namely F2pn 2 2p n 4 SðfÞ ¼ n Fn F2pn 2p n 2 2 n Fn
cosð2f þ Fn þ F2pn Þ
ð17Þ
Fn ; F2pn are the initial phases of the main and supplementary focusing harmonics, respectively. The phase advance of the transverse oscillations per focusing period can be approximated from Eq. (15) as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qðfÞ2 mr ðfÞDp aðfÞ þ ð18Þ 2 with the accuracy of around 10% within the range of 0omr o901: The exact simulations show that the approximation error grows from 3% up to 18% when the value of mr varies from 401 to 1201. According to Kapchinsky theory [2], the normalized transverse acceptance of the channel Vc and the limit current Im can be evaluated by using the following expressions: Vc ¼
jp njgmr min r2max plð1 þ qmax =2Þ2
ð19Þ
Im ¼
2pe0 m0 c3 jp njbs g2 mr min Vc kb eZ pl
ð20Þ
where rmax is the maximum possible radius of the beam envelope which is usually assumed equal to the minimum aperture, mr min is the minimum transverse phase advance within the bunch, kb is the coefficient of the beam bunching.
with the coefficients
aðfÞ ¼
qðfÞ ¼
4. Ideal geometry and parameters of 2bk super-periodic structure
eZU0 Ap sinðfÞ p2 m0 c2 b2s g ðp nÞ2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi eZU0 Fn 1 þ SðfÞ 4m0 c2 b2s g
2
n ðp nÞ2
ð16Þ
where the current phase of the particle f is supposed as an adiabatic parameter; SðfÞ is the factor of the supplementary harmonic action,
Let us examine the structure with the focusing period length of 2bl (p ¼ 2) which seems expedient to be applied directly behind RFQ; the accelerating and main focusing harmonics are A2 ; F1 ; respectively, in this case. Third quadrupole harmonic, F3 ; can be taken into account as the supplementary focusing term due to the same period of action on the particles. At the same time,
S. Minaev / Nuclear Instruments and Methods in Physics Research A 489 (2002) 45–58
50
the second quadrupole term, F2 ; must be certainly forbidden in the structure because the direct travelling wave component of this term moves synchronously with the particles (similar to the accelerating harmonic), deflecting them permanently from the axis. Note that the amplitude of this harmonic is fortunately zeroed for symmetric focusing period. Consider the symmetry when the total RF potential is always zero at z ¼ 0 meaning Fn ¼ 0: Uðr; y; zÞU0 A2 I0 ðkrÞsinðkzÞ: k k þ F1 I2 r sin z 2 2 3k 3k þF3 I2 r sin z cosð2yÞ : ð21Þ 2 2 Defining the current aperture of the channel at the given cross-section as a minimum distance from the axis to the closest electrode, suppose that the aperture is a at the maximum potential of the axisymmetric harmonic, z ¼ bl=4; and ma at the maximum of the main quadrupole harmonic, z ¼ bl=2: Unlike in the conventional RFQ, the factor of m determines the modulation of the channel on the whole but not of the single electrode. It means that the maximum aperture occurs at the symmetric cross-section. If the ratio of the supplementary harmonic amplitude to the amplitude of the main focusing harmonic hs ¼ F3 =F1 is determined beforehand, the main harmonic amplitudes can be found from two boundary equations, namely
Uða; 0; bl=4Þ¼U0 ; Uðma; 0; bl=2Þ ¼ U0 ; as follows: A2 ¼
1 pffiffiffi k 3k a 2 I 2 a þ hs I 2 C 1 B 2 B1 2 C @ A k 3k I0 ðkaÞ 2 I2 ma hs I2 ma 2 2 F1 ¼
0
1 : k 3k ma I2 ma hs I2 2 2
a=8mm
a=8mm 200
10mm 12mm 14mm 16mm
a=16mm F1
A2
ð23Þ
Fig. 1 shows how these amplitudes depend on the modulation factor m for the different aperture values if only one focusing harmonic exists in the structure, hs ¼ 0: As an example, consider the ion beam with the charge to mass ratio of 13 behind of 0.8 MeV/u SURFQ linac operating at the frequency of 80 MHz. The values of a and m which generally define the channel geometry have to be specified for the given beam parameters from the desired phase advance, mr ; taking into account the RF voltage. But the larger the quadrupole aperture ma is, the highest is the RF voltage U0 that can be applied to the structure. For this reason, the calculations were carried out in terms of the fixed field strength at the symmetric cross-section. The maximum RF voltage pffiffiffi can be estimated as 2U0 DEm D; where D ¼ mað 2ð1 þ er Þ 2er Þ is the minimum space between the adjacent electrodes at the symmetric cross-section, Em is the average electric field within this space; er ¼ re =ma is the coefficient of the
0.8
0.4
0
100
0 1.0
1.5
2.0
m
2.5
ð22Þ
1.0
1.5
2.0
2.5
m
Fig. 1. Amplitudes of basic accelerating and focusing harmonics against modulation factor.
S. Minaev / Nuclear Instruments and Methods in Physics Research A 489 (2002) 45–58
150
a=8mm 10mm 12mm 14mm 16mm
µr s ,deg
100
50
choice of a ¼ 10 mm, mrs ¼ 601 gives m ¼ 1:97; A2 ¼ 0:79; F1 ¼ 49: The right part of Fig. 2 shows the phase advance plotted versus arbitrary particle phase at the different supplementary harmonic amplitudes. It is seen that the dependence mr ðfÞ which is originally caused by the varied defocusing action of the accelerating field (see curve h ¼ 0) is compensated considerably near the synchronous phase of 301 when the supplementary harmonic ratio is around hs ¼ 0:021: Being the exclusive feature of RF focusing, the reduced dependence of
120
80
h=0.042 h=0.021 h=0
µr s ,deg
electrode curvature, re is the radius of the curvature. The curvature coefficient, er ¼ 0:75; typical for SU-RFQ is chosen for the present example (note that the value of ma is the same as the average aperture radius R0 in the existing RFQ theory [2]). The average electric field Em ¼ 25 MV/m is assumed at the minimum gap between the electrodes. The transverse phase advance in the two harmonics structure calculated for the synchronous phase of 301 at different aperture radii is shown by a graph in the left part of Fig. 2. The
51
40
0
0 1.0
1.5
2.0
2.5
-60
-30
0
30
, deg
m
Fig. 2. Phase advance of the transverse oscillations against modulation factor and current phase of an arbitrary particle.
Table 1 Evaluated parameters of 2bl superperiodic structure behind of conventional RFQ Charge to mass ratio Beam energy Operating frequency Electric field estimated at the symmetric cross-section Aperture radius Synchronous phase Bunch extent Transverse phase advance
1/3 MeV/u MHz MV/m Mm Deg. Deg. Deg.
0.8 80 25 10 30 60 60
Type of the structure
Two harmonics
Three harmonics
Supplementary harmonic ratio Modulation factor Radiofrequency voltage Acceleration rate Minimum transverse phase advance Normalized transverse acceptance Limit current
0 1.97 240 3.3 42 5.7p 177
0.021 2.25 274 3.7 60 7.1p 321
7; kV MV/m Deg. mm mrad mA
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S. Minaev / Nuclear Instruments and Methods in Physics Research A 489 (2002) 45–58
YS /
XS /
the transverse phase advance along the bunch may improve the intense beam stability and moderate the emittance growth. An example of evaluated parameters of 2bl super-periodic structure without and with the supplementary quadrupole focusing harmonic is shown in Table 1. The average electric field of 25 MV/m at the minimum gap in the symmetric cross-section is used in this table. Numerical calculations give the maximum surface field strength of around 28 MV/m in this case. One can see that the three harmonics (3H) structure
takes more RF voltage at the same surface electric field resulting in larger channel acceptance and higher limit current. The longitudinal profiles of the ideal equipotential electrodes in the xoz and yoz planes are shown at the upper and lower parts of Fig. 3, respectively. More complicated 3 H configuration leads to higher capacitive loading. The question whether the advantages given by the supplementary harmonic really cost additional RF power and more complicated geometry, should be answered after an exact simulation of the intense beam dynamics.
−U0
+U0
0.2
3H profile 0
2a
2H profile
2ma −U0
0.2 0
0.25
0.5
0.75
+U0 1.0
1.25
1.5
1.75
2.0
z / Fig. 3. Longitudinal profiles of ideal equipotential electrodes without and with the supplementary focusing harmonics.
+Uo sin (t)
Symmetriccross-sectionS-S
S
S −Uo sin (t)
(a)
2
(b) Fig. 4. Super-period formed by the electrodes with constant cross-section curvature (type 2).
S. Minaev / Nuclear Instruments and Methods in Physics Research A 489 (2002) 45–58
5. Realistic design of the accelerating channel The ideal equipotential geometry has to be approximated by the real electrodes available for the conventional manufacturing technique. Fig. 4 shows how the 2bl super-period may look in reality for the simple two harmonics case. The couples of identical electrodes form the gaps of two kinds, one of them being mostly accelerating while the other one combining the accelerating and focusing functions. One has to emphasize that the acceleration is nearly the same in these different gaps. Generally, the accelerating channel can be interpreted as a sequence of the RF quadrupole lenses integrated to every other gap of the p-mode structure. Unlike the known magnetic focused structures, the RF quadrupole super-period has no drift space for the longitudinal motion and accelerates the particles two times at every RF period. In order to keep the beam quality, the highorder azimuthal harmonics of RF potential given by the terms q > 2 in the series (4) should be minimized because just these components are mainly responsible for beam distortion and cause the emittance growth. At the same time, the terms specified by the numbers of n > 1; q ¼ 2 determine the longitudinal distribution of the quadrupole potential component and may even improve the focusing conditions as it was shown by the example of the supplementary harmonic. Fourth axisymmetric harmonic, n ¼ 4; q ¼ 0 determines the longitudinal asymmetry of the accelerating period bl; i.e. different acceleration in the gaps of different kind when the total acceleration is specified by the main accelerating harmonic, n ¼ 2; q ¼ 0: The higher axisymmetric terms, n > 4; q ¼ 0; do not disturb the particle motion, being just useless both for acceleration and focusing. In practice, the single electrode can be either machined by means of rotation around some axis producing the varied cross-section radius as Fig. 4a shows (type 1) or manufactured by a special cutter with the semicircular cross-section of the fixed curvature similar to the conventional RFQ production (type 2, Fig. 4b). The potential distributions created by the electrodes of different types have been investigated and compared by
53
using the specially developed 3D electrostatic code. The results clearly show that the constant curvature electrodes fit to the ideal geometry better than those of conical shape do. For example, the same longitudinal profile of an ideal equipotential geometry formed by two harmonics, A2 ¼ 0:787 and F1 ¼ 49:25; had been applied to the electrodes of both types. The realistic accelerating and focusing amplitudes calculated for the electrodes of type 1 differ from the ideal values by +10% and 12%, respectively, whereas the electrodes of type 2 give the difference of around +1.5% and 2%, respectively. Fig. 5 shows an example of the equipotential lines at the longitudinal and transverse cross-sections for the fixed curvature electrodes of type 2.
6. Beam dynamics simulation in the SP-RFQ section Beam dynamics in the test SP-RFQ section has been numerically simulated in order to verify the analytic theory. A special computer code was developed where the particles are assumed as the uniformly charged spherical clouds, one transparent for another. Particle 3D trajectories are calculated in time at the electric field given by the set of oscillating spatial harmonics. The whole structure is subdivided into elementary bl=2 cells with the accelerating gap at the center of each of them (remember two kinds of the gap exist in the structure). At the first step of simulation, the electric field in every cell is defined in terms of the main geometric parameters a and ma; the particle dynamics is simulated in the field of accelerating and focusing harmonics only. When a satisfactory two-term beam dynamics is used, a final simulation in the full electric field has to be done. Fig. 6 shows the values of minimum aperture a and quadrupole aperture ma along the structure. The transverse phase advance calculated for zero current is shown in the same plot. The realistic electrodes with the fixed crosssection curvature have been chosen for the test section. Three different radii of the electrode curvature, namely 16, 20 and 24 mm are used in
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S. Minaev / Nuclear Instruments and Methods in Physics Research A 489 (2002) 45–58
Fig. 5. Equipotential map calculated for the realistic electrodes with the constant cross-section curvature (type 2).
60 µr , deg
ma, mm
40
20 a, mm 0
1
2
3
4
5
L, m
Fig. 6. Transverse phase advance and channel aperture along the simulated structure.
the simulated structure. The RF voltage DU ¼ 2U0 applied to the accelerating cells grows smoothly from 0.5 MV at the beginning to 1.18 MV at the end of the structure as Fig. 7 shows. Since the quadrupole aperture ma also grows, the averaged electric field at the symmetric cross-section does not exceed 25 MV/m. The electric field at the axis of 37-cell section is shown in Fig. 7 as well. Different shapes of the field distribution can be seen at the gaps of different kinds.
In order to test the structure capability, high values of normalized transverse emittance and beam current, namely 4.4 pmrad mm and 150 mA, respectively, have been simulated; these values seem to be close to the SU-RFQ limit at the given charge to mass ratio and operating frequency. Table 2 shows the simulated parameters of the section. Fig. 8 demonstrates two 95% transverse envelopes for beam current values of 40 and 150 mA. The beam size increases a little along the
S. Minaev / Nuclear Instruments and Methods in Physics Research A 489 (2002) 45–58
55
U, MV
1.5
1.0
0.5
0
0
1
2
3
4
5
L, m
0
1
2
3
4
5
L, m
Ez , MV/m
10
0
-10
Fig. 7. RF potential applied to the cells and electric field distribution at the axis of SP-RFQ section.
Table 2 Simulated parameters of SP RFQ section 1 3
Charge to mass ratio Operating frequency Input/output beam energy Number of the cells Total length Synchronous phase RF voltage applied to the cell Radii of electrode curvature Electric field estimated at the symmetric cross-sections Normalized transverse beam emittance Input/output bunch extent Beam current
MHz MeV/u m Deg. MV mm MV/m
81.4 0.8/6.7 37 5.42 30 0.5–1.18 16, 20, 24 23–25
mm mrad
4.4p
Deg. mA
52/30 40, 150
around 3.3 MV/m. The results of numerical simulation shown in Table 2 are in good agreement with the parameters evaluated from the analytic theory.
7. Different RFQ structures—a comparison The accelerating efficiency of different RFQ structures should be estimated and compared at the same conditions that have been used for SPRFQ structure shown in Table 1. A conventional SU-RFQ where Lf ¼ bl; p ¼ 1 may be evaluated at the beginning. The two-term RF potential (5) includes in this case the axisymmetric and zindependent quadrupole components Aðr; zÞ ¼ A1 I0 ðkrÞsinðkzÞ F ðr; zÞ ¼ F0 k2 r2 cosð2yÞ:
structure in both cases, but the growth of transverse RMS emittance was not clearly seen. First beam losses of around 2% appear in the structure when the beam current is close to 150 mA. The longitudinal stability is easily obtained at the synchronous phase of 301. Finally, the total energy of 17.67 MeV per charge unit is gained on the length of 5.42 m producing an average accelerating gradient of
ð24Þ
The normalized amplitudes A1 ; F0 can be specified by using the minimum and maximum distances from the axis to the electrode denoted as a and am ; respectively, a2m a2 a2m I0 ðkaÞ þ a2 I0 ðkam Þ 1 A1 I0 ðkaÞ : F0 ¼ k 2 a2 A1 ¼
ð25Þ
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S. Minaev / Nuclear Instruments and Methods in Physics Research A 489 (2002) 45–58
x, mm
20 150 mA 10
40 mA y, mm
0
10
20 0
1
2
4
3
5
L, m
Fig. 8. 95% transverse envelopes for different beam current of 40 and 150 mA.
The Mathieu’s Eq. (15) is valid for SU-RFQ structure with the following coefficients: eZU0 A1 sinðfÞ m0 c2 b2s g 4eZU0 F0 : q¼ m0 c2 b2s g
aðfÞ ¼
B1;0 ; F0 ¼ B0;2 ; F2 ¼ B2;2 : In contrast to the SURFQ structure, a second quadrupole harmonic is added to the focusing part of the RF potential F ðr; y; zÞ ¼ ðF0 k2 r2 þ F2 I2 ð2krÞ cosð2kzÞÞcosð2yÞ
ð26Þ
Similar to the example given by Table 1, the values of er ¼ 0:75; Em ¼ 25 MV/m are chosen for the SU-RFQ structure. Formula (18) with the coefficients (26) gives the transverse phase advance of mr ¼ 601 at the maximum RF voltage U0 ¼ 151 kV applied to the average aperture of R0 ¼ 11:7 mm (a ¼ 8 mm, am ¼ 15:4 mm). With these parameters, the maximum accelerating gradient of 1.4 MV/m can be achieved in the SU-RFQ structure. At the next step, consider the RF quadrupole focused p-mode ‘‘split coaxial’’ structure described in Ref. [3]. The period of focusing is as long as bl including two similar accelerating gaps each of which is supplied with two couples of ‘‘fingers’’. At the fixed moment of time, all gaps produce either focusing or defocusing action together for the same direction, x or y; whereas the electric field is opposite at the adjacent gaps. Analyzing the spatial potential distribution, one can find two main components of the quadrupole field: zindependent component and longitudinally varied one with the number of n ¼ 2 and period length of bl=2: The number of the accelerating harmonic is p ¼ 1: Related to potential expression (4), the amplitudes of the main components are A1 ¼
ð27Þ
bringing the phase dependence to the ‘‘focusing’’ coefficient of Mathieu’s equation: qðfÞ ¼
eZU0 m0 c2 b2s g
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 16F02 þ F22 þ 8F0 F2 cosð2fÞ:
ð28Þ
One should mention that only a structure with low content of high-order harmonic might be correctly analyzed by the simple theory. When the ‘‘fingers’’ or ‘‘horns’’ are introduced, the highorder terms contribute to the RF potential considerably. Therefore, the amplitudes of the main harmonics can be specified keeping in mind that the analysis is very approximate 1 1 1 A1 ¼ ; F0 ¼ 2 2 ; F2 ¼ : ð29Þ I0 ðkaÞ 2k a 2I2 ð2kaÞ The accelerating potential Aðr; zÞ; which is the same as in the SU-RFQ (see Eq. (24)), equals zero at the center of the gap, Aðr; 0Þ ¼ 0; becoming maximum at the tube center, Aða; bl=4Þ ¼ 1; while the quadrupole potential (27) behaves vice versa, F ða; 0; 0Þ ¼ 1; F ða; y; bl=4Þ ¼ 0: The uniform and periodic quadrupole components cancel each other out at the tube center, acting together at the center of the gap. Keeping the finger radius of 0:75a; average the field between the fingers Em ¼ 25 MV/ m and transverse phase advance, mr ¼ 601; the
S. Minaev / Nuclear Instruments and Methods in Physics Research A 489 (2002) 45–58
maximum accelerating gradient can be estimated as 1.7 MV/m. Note that the last value may be difficult to reach in practice because of high field strength at the fingertips. The RF focused drift tube linac (RFD) is based on the 2p-mode Alvarez structure where the RF quadrupole lenses are created by two pairs of fingers installed within the drift tubes. Unlike in the p-mode structures, the RF voltage applied to the accelerating gap, 2Ua ; differs from the finger voltage, 2Uf : The ratio of Ua =Uf may be varied by changing the geometry of the cavity, namely the space between the blades of the tube stem. Similar to the SP-RFQ structure discussed above, the focusing and accelerating harmonics are first and second, F1 and A2 ; respectively. When both accelerating field and focusing gradient are distributed uniformly within the gap and lens, equivalent harmonic amplitudes may be specified as follows: 4 kg A2 ¼ sin pkg 2 4 kL F1 ¼ sin pI2 ðka=2Þ 4
ð30Þ
where g and L are the lengths of the accelerating gap and focusing lens, respectively. The expressions for Mathieu’s coefficients (16) are valid in this case with n ¼ 1; p ¼ 2; SðfÞ ¼ 0 and Ua ; Uf instead of U0 : In order to roughly estimate the RFD abilities, suppose that the accelerating gap is as long as g ¼ bl=4 with the average field strength of 10 MV/m. The effective length of the RF lens is assumed as L ¼ bl=2: Since, the focusing voltage does not depend on the accelerating one, beam focusing may be optimized at the lower voltage and smaller aperture. In particular, phase advance of 601 is obtained at the aperture of 10 mm and finger–finger voltage of 114 kV only. The total accelerating potential of 390 kV produces the average accelerating gradient of around 1.9 MV/m independent of the focusing parameters. Finally, the RFD structure seems more flexible in focusing than SP-RFQ, but the accelerating efficiency is limited because only one accelerating gap is possible at the length of bl: Besides, the
57
Alvarez structure becomes too big in diameter for medium and heavy ions. The accelerating channel of SP-RFQ structure supplied with the symmetric RF voltage 7U0 has been investigated above. In practice, the structure on the whole can be based on TE11 or TE12 mode cavity in this case. As an example, IH-DTL structure of GSI high current linac [9] may be mentioned where 920 kW RF power is dissipated in the walls of the 10.3 m long tank generating the maximum RF voltage amplitude of 1.3 MV. The voltage increase along the structure may be realized by the special tuning when the inductance or/and capacitance per length unit is ramped up from the entrance to the exit. Alternatively, the growing RF voltage per cell can be naturally obtained at the modified TM010 structure proposed in Ref. [5]. The capacitance of the SP-RFQ channel is obviously larger than that for drift tube channel because of focusing regions with quadrupole configuration, especially compared to IHDTL where the tubes are minimized. Therefore, some additional RF power is always necessary for RF quadrupole focusing. The optimistic factor of the power growth might be very roughly estimated as around 1.5. The correct quantity of the RF power consumption has to be specified from the numerical simulation and experimental study of the chosen cavity.
8. Conclusion A theory developed in this paper shows how the RFQ accelerating efficiency can be improved at the second stage of the ion linac. Higher RF voltage applied to the structure with the enlarged quadrupole aperture raises the acceleration rate behind the conventional RFQ. At the same time, the focusing period longer than bl; or superperiod, is necessary in this case to keep the transverse stability of the beam because the gradient is reduced due to aperture increase. The polyharmonic model of the beam dynamics allows to find the optimized geometry of the accelerating channel and to analyze the main parameters of the structure.
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S. Minaev / Nuclear Instruments and Methods in Physics Research A 489 (2002) 45–58
The spatially periodic RFQ structure with the period length of 2bl has been investigated in detail. Analytical theory predicts high values of the channel acceptance, beam current limit and accelerating gradient more than 3 MV/m at the realistic surface field strength. Numerical simulation of the beam dynamics in the 37 cell 5.4 m long section confirms the capability of the structure, being in good agreement with the analytical evaluations. In particular, 150 mA ion beam is accelerated with the average effective gradient of around 3.3 MV/ m, which is much higher than a conventional RFQ can provide at the first stage of the linac. An approximate comparison with the other RFQ structures shows that the SP-RFQ may provide the highest accelerating rate among them. On the other hand, the SP-RFQ seems quite competitive with the magnetic focusing behind SU-RFQ. The absence of magnetic lenses simplifies the design reducing the total cost of the linac considerably. The super-period length of 2bl is only one particular case, which can be regarded as FODORFQ lattice. The super-period may contain more accelerating gaps and RF quadrupole lenses
integrated to the structure; the doublets and triplets can be created if necessary. The theory developed in this paper can be applied to longer period as well for parameter evaluation.
References [1] V.A. Teplyakov, Prib. Tekn. Eksp. 6 (1964) 24. [2] I.M. Kapchinsky, Theory of Resonance Linear Accelerators, Harwood, New York, 1985. [3] R.W. Muller, et al., Proceedings of the 1984 Linear Accelerator Conference, GSI-Darmstadt, Germany, 1984, p. 77. [4] D.A. Swenson, et al., Proceedings of the 1998 LINAC Conference, Chicago, IL, 1998, p. 648. [5] M. Odera, M. Hemmi, Proceedings of the 1984 Linear Accelerator Conference, GSI-Darmstadt, Germany, 1984, p. 346. [6] V.A. Zenin, et al., Proceedings of the 1994 International Linac Conference, Tsukuba, Japan, 1994, p. 156. [7] S. Minaev, U. Ratzinger, B. Schlitt, Proceedings of the 1999 Particle Accelerator Conference, New York, 1999, p. 3555. [8] A.I. Balabin, G.N. Kropachev, Nucl. Instr. and Meth. A 459 (1–2) (2001) 87. [9] U. Ratzinger, Nucl. Instr. and Meth. A 464 (2001) 636.