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UDK 621.384.6+51.73+51.74+517.928+517.938
Yu. V. Senichev, A. N. Chechenin, S. F. Kostromin
VARIABLE TRANSITION ENERGY LATTICES
BASED ON DIFFERENT PERIODIC CELLS
WITH VARIOUS TYPES OF DISPERSION SUPPRESSOR
1. Introduction. The “resonant” lattice [1, 2] developed to vary the transition energy is based on the resonantly correlated curvature and gradient modulations in arcs with integer tunes in horizontal or both planes. This method was first adopted for the TRIUMF and Moscow Kaon Factories project [3,4]. It was then applied in the SSC low energy booster [5], the CERN Neutrino Factory [6], and in the Main Ring of the Japanese proton accelerator research complex facility [7, 8]. For the superconducting option of the High-Energy Storage Ring lattice in the FAIR project, the same principle was also applied [9]. Besides, it is one of the candidates for PS2 in CERN [10]. Due to its special features, the “resonant” lattice can also be used as a lattice with stochastic cooling where different arcs have different mixing factors, while the dynamic aperture is conserved for the whole machine [11]. At present the new project of multi-particle synchrotron NICA in JINR is considered to search for possible manifestation of signs of the mixed phase and critical endpoint in heavy ion collisions and polarized proton-proton and deuteron-deuteron beams. In this case the “resonant” structure is very much suitable, as due to variable transition energy it can give the dual properties which are meeting the requirements of acceleration of particles with different rigidity [12]. The “resonant” lattice also appears to be useful for electron machines. In synchrotron-light sources, in particular, the minimum momentum-compaction factor and the minimum modulation of the dispersion function are both simultaneously required to have a small horizontal emittance [13,14].
But it is necessary to notice, that demanded properties of lattice depend on type of cells forming the super-period and how cells are incorporated in the super-period itself. In our previous paper we always considered the super-period based on the singlet FODO cells. In this paper we consider the “resonant” lattice based on the doublet, the triplet and compare their advantages and disadvantages with the singlet cell lattice. Another subject solved in this paper is the dispersion suppressor. Earlier we always set in a suppression of dispersion by means of the integer 2n condition of radial oscillation on arc, so-called the incorporated
Сеничев Юрий Валерьевич — доктор физико-математических наук, ведущий научный сотрудник, позиция А1, профессор Научного центра г. Юлиха, Германия. Количество опубликованных работ: более 100. Научное направление: ускорительная физика. E-mail:y.senichev@fz-juelich.de.
Чеченин Алексей Николаевич — кандидат физико-математических наук, ассистент кафедры компьютерного моделирования и многопроцессорных систем факультета прикладной математики-процессов управления Санкт-Петербургского государственного университета. Количество опубликованных работ: 10. Научные направления: ускорительная физика, математическое моделирование. E-mail: achg03@gmail.com.
Костромин Сергей Александрович — кандидат физико-математических наук, научный сотрудник Объединенного института ядерных исследований (г. Дубна). Количество опубликованных работ: 46. Научное направление: ускорительная техника. E-mail:kostromin@jinr.ru.
© Yu.Senichev, A.Chechenin, S.Kostromin, 2011
in arc suppressor. Here we shall consider various schemes of the allocated suppressors of dispersion on the ends of arcs and compare them with incorporated ones. Furthermore we analyze different schemes of sextupoles location in the lattice based on the singlet, doublet and triplet cells and with different types of suppressor.
2. Transition energy variation. With a specially correlated modulation of the quadrupole gradient K(s) and orbit curvature p(s), and a particular choice of betatron oscillation frequencies v, the theory of “resonant” lattices makes it possible to get interrelated dispersion variations D(s) and 1/p(s) along the equilibrium orbit and a negative momentum compaction factor (MCF)
2n
1 r D(-d)
2n J p($)
0
The general principles of construction of “resonant” lattices detailed in [1,2] are based on the solution of the equation for the dispersion D(d-) in the bi-periodic structure with the cell periodicity and super-periodicity
D" + K (d)D = 1/p(&).
In the optics with the horizontal tune v and the super-periodical curvature modulation 1/p(d) ~ Belu,d + 1/i?, the dispersion solution is
D _
D№ ~ Aeiv* + ------jre^ + D.
v2 — U2
Since MCF is the average value of the function D(^)/p(^), it can be written as the sum
D D(-d) ■ f($)
“ “ 1+ I ’
where D(-d) = Belu>§/(v2 — u2) and r = BRe1^^ are the functions oscillating about the
averages. In an ordinary regular FODO lattice without the super-periodic modulation, the
oscillating component gives D(-d) ■ f($) = 0, and the minimum value of MCF a = D/R = 1/v2 is limited by the horizontal tune v.
In the “resonant” lattice, the oscillating components contribute the term D(-d) ■ r($) = B2/2(v2 —u2), which in the case v ^ u can result in a negative MCF. The same can be shown for the gradient modulation K(^)+e^k(^). Here, we omit the long intermediate computations
and quote the final formula for the case when the functions of gradients sk('d) = ^ gk cos kfl
k=0
and/or orbit curvature 1/p($) = 1/R (1^2 rn cos nfi) are jointly modulated:
V n=1 J
4(1 — u/v)
___ 2 '
v) [1 - (1 - uj/is)2} Tk
2
(1)
where gk and rk are fundamental harmonics producing the maximum effect on MCF. Indeed, if both the gradient function and the orbit curvature function are modulated with identical frequency, the second term in (1) may make an appreciable negative contribution to MCF provided that the value |1 — u/v\ is small and v < u.
In addition, from (1), there follows an obvious condition of antiphase modulation of the gradient and curvature function gkrk < 0, which allows correlated variation
a
of the momentum compaction factor with the aid of these functions. Based on the above reasoning, the “resonant” lattice method with simultaneous orbit curvature and quadrupole gradient modulation with an identical frequency of the fundamental harmonics and an approximately identical contribution of both modulations to the final value of the momentum compaction factor is most effective. From (1) it is easy to derive the following equality for an arbitrary fundamental harmonics gk and rk giving a « —\/v2:
i 7] 77^ 1 ~ rk — ±2 {kSarc/varc — 1) / ,
\V ) 1 — (1 — kSarc/Varc)2
2
gk
1 — (1 — kSarc/Varc)2
Apparently, that if we modulate one of functions the modulation value should be double.
3. Microstructure of the “resonant” lattice. In common case there are two types of lattices used in accelerators with inserted straight sections, the so-called circular lattices with S identical super-periods and the lattice consisting of arcs with Sarc super-periods per arc separated by straight sections.
In the former lattices the momentum compaction factor completely coincides with its value for one super-period. In the lattices consisting of arcs with Sarc super-periods of length Ls and separated by straight sections of length Lstr, the momentum compaction factor for the entire accelerator atotai and for a super-period as are related by the equation
Sarc ' Ls /0\
Q-total = as 7 i • \^)
Sarc ' Ls + Lstr
Thus, knowing the momentum compaction factor for one super-period, one can easily find its value for the entire accelerator. Since the straight sections do not affect essentially on the
momentum compaction value, just as a coefficient in expression (2), we concentrate on the
arc structure for the most part.
The arcs are based either on the regular cell-periodical structure or on super-periods. A super-period is usually formed by varying parameters of a regular lattice consisting of singlet FODO cells, doublet FDO cells, or triplet FDFO cells (F is the focusing quadrupole, D is the defocusing quadrupole, and O is the drift space for bend magnet location), each having its advantages and drawbacks. Fig. 1 shows the microstructure of one super-period based on singlet, the doublet and triplet. You can see that in first option in each space between quadrupoles one bend magnet is placed and in two other ones we have two magnets in each gap. However from the point of view of realization of focusing force modulation K($) + e • k($) all options are identical, since it does not depend on the cell structure.
Nevertheless we should mention, that for each of options there is own scope of usage. So for example, considering the chromaticity compensation requirement, the FODO lattice is most preferable because the other two lack good separation of the horizontal and vertical ^-functions, which results in a decreased efficiency of the sextupoles and accordingly in a decreased dynamic aperture. In addition, the FODO super-period with mirror symmetry about its center provides most favorable conditions for independent control of the betatron frequencies, chromaticity in both planes, and momentum compaction factor, which makes a lattice like this superior to any other.
In the same time on belief of the experts working with the heavy ion machines (FAIR in GSI) the structure based on doublet has greater efficiency at separation of particles with
Fig. 1. Microstructure: super-period based on singlet, doublet and triplet B - bending magnet; QF 1,2- focusing quadrupoles; QD - defocusing quadrupole; SF and SD
and defocusing sextupoles.
focusing
different mass-charge ratio. Concerning of the triplet structure it gives smoother envelope behavior and provides the lower intra-beam scattering, but this feature has to be traded off together with all other [12].
We have studied all structures and we can tell, that from the point of view of realization of high value of transition energy all structures are almost equivalent. In fig. 2 examples of calculations of these structures of one cell and one super-period for the same values of transition energy are shown. You can see that in spite of different envelope on cells the averaged dispersion has similar character along the super-period.
The number of cells in a super-period Nceii is dictated by the required phase advance of radial oscillations. Following the theory of “resonant” lattices, we should try to construct a lattice with the horizontal tune v as close to the number of super-periods as possible. In this case, the phase advance of horizontal oscillations per cell will be about 2nvarc/(Sarc • Ncell). At the same time it is known that from the point of view of minimization of the 3-functions for a cell the phase advance of radial oscillations should fall within the range 60-100°. Thus, in a lattice with the fundamental harmonic of the modulation of the super-period parameters k = 1 and with varc < Sarc the number of cells turns out to be 3-5 per super-period. Since an increase in the number of cells requires greater splitting of the super-period and entails an increase in the number of magnetic optical elements, we exclude the five-cell option from consideration and confine ourselves to analysis of a super-period comprising 3-4 cells.
P, m |
jnrn
nlTI
X.
S, m h
---------------------------Py
S, m
Dx
Fig. 2. Twiss parameters for FODO (a), FDO (b) and FDFO (c)
Obviously, strict periodicity of cells does not make it possible to get the required value of the momentum compaction factor, which is fixed by the value of the horizontal betatron frequency in this case. According to the theory of “resonant” structures, either modulation of the lens gradients or modulation of the orbit curvature, or simultaneous modulation of both functions is introduced to get the required momentum compaction factor [1, 2]. As you can see fig. 2 shows a super-period made up of three periodical cells with gradient modulation alone and mirror symmetry about the center, where two quadrupole families form the required fundamental harmonic k = 1. The arc is supposed to consist of 4 super-periods with almost
the same total length. However, to get the required high gamma-transition this modulation method requires a great change of the field in the quadrupoles. Note that strong modulation of the gradients leads to a considerable increase in ^-functions, and chromaticity of the entire accelerator, which results in a reduced dynamic aperture.
In case of the curvature-varying lattices ^-functions became smaller and chromaticity is kept lower. Fig. 3, a shows example of the super-period with modulated curvature by “missing magnet” in center of super-period PS2 machine. But unfortunately, the orbit curvature modulation method does not always provide the required value of gamma-transition.
a b
, m -10 - 8 - 6
- 4
- 2 - 0 -2 -4
-6
Fig. 3. Superperiod with modulation a — the orbit curvature modulation; b — the orbit curvature and quadrupole gradients modulation.
Therefore the only possibility is a modulation of both the orbit curvature and the quadrupole gradients to get the required momentum compaction factor (see fig. 3, b). The former method allows controlling the momentum compaction factor with the minimum increase in the /3x function and Dx and, compared with gradient modulation lattices, does not require strong sextupoles for chromaticity correction. However, the gradient modulation method is the only method allowed the momentum compaction factor of the periodic lattice to be varied, which one plays crucial role in some machine, for instance in the dual structure for NICA [12].
4. Macrostructure of the “resonant” lattice. The macrostructure basically is defined by how we build a dispersion suppressor and as we compensate nonlinear influence of sextupoles. Following the later there are three possible versions (see fig. 4, a-c). In the first version (fig. 4, a) the suppressor is incorporated in arcs, and a dispersion zeroing due to integer 2n of horizontal phase advance. This means that the phase advance in one superperiod should be 2nvarc/Sarc. On the other hand, for MCF to be controlled, it is reasonable to take the minimum possible difference varc — Sarc = —1. Thus, many ratios exist between Sarc and varc: (4:3), (6:5), (8:7), (10:9), ...It is obvious that in all ratios, the number of super-periods Sarc is taken to be even, while the betatron oscillation frequency takes on odd values. In this case, the phase advance of the radial oscillations between the cells located in different super-periods and separated by Sarc/2 super-periods is
p. varc Sarc ~ varc , ~
27T—---- • —- = 27T^— = 7T + 27Til,
Sarc 2 2
80
Dx, m
‘..nnOTnn..n nUin |—I.nrfUnn..
%------------------------------------Pv
---------Dx
S, n
superperiod
n rnnlm n r -i L _
| y: y Lt \ short coupling of sextupole AO = n
D,D_’ = 0_
AO = 37i
r 1 r 1
3 cells 3 cells 3 cells 3 cells
long coupling of sextupole AO = 37C
D,D’ = 0
D,D’£0
\
b D, O' = 01 DS
D,D’= 0
sextupoles in 4 superperiods
3 cells
X
3 cells
3 cells
X
3 cells
> regular cells with magnets,
V
D,D'± 0
_D,D’=0
D,D’=0
^1 1 1 1 1 1^
- 3 cells 3 cells 3 cells 3 cells
A_
2 cells with sextupoles w/o magnets
□
B
QF1
QF2
[] QD D SF | SD | DS
dispersion suppressor (no sextupoles)
Fig. 4. Macrostructure based on different types of dispersion suppressor a - variant w/o dispersion suppressor; b - variant with dispersion suppressor; c - variant with half-magnet
dispersion suppressor.
which corresponds to the condition of first-approximation compensation for the nonlinear effects of sextupoles located in these cells. This remarkable property also applies to higher multipoles in any element. This coupling through Sarc/2 number of super periods will be referred to as the long coupling. In the first version the sextupoles are placed on arcs in all super-periods, and the condition of compensation is fulfilled for all of them. Advantage of such structure is absence of a special suppressor of a dispersion. However it leads to additional growth of a dispersion in the center of an arc due to arch modulation of dispersion.
To exclude the last in second version the special suppressors of a dispersion at edges of an arc are entered. In this version on a periodic part of an arc the dispersion has no modulation because of absence of the requirement to have a zero dispersion at its edges. To keep properties of a dispersion suppressor in wide region of energy the suppression is done by variation of strength in two quadrupoles of suppressors. Sextupoles are installed only in a periodic part of arc in every super-period and they are absent in the dispersion suppressors in order to avoid a violation of the sextupole compensation condition. In this version at the 90° advanced phase per cell we have two types of sextupole compensation: the long coupling and the short coupling acting through two cells with total phase advance 180°. The general advantage of these two versions is the possibility of a transition energy variation: in the first
due to variation of gradients of quadrupoles on whole arc, in second version on its periodic part only.
If to refuse from two important features, a variation of critical energy and dispersion suppressors working in wide energy region, the number of super-periods per an arc may be arbitrary, and dispersion suppressors are constructed by a principle half- or missing-magnets. In the third version (fig. 4, c) the dispersion suppressors based on a principle “half-magnet” are located on the ends of arcs. The periodic part of arc can be shorter, which one exclude the sextupole long coupling. Nevertheless the condition of the mutual paired compensation sextupoles forces us to have them outside arcs.
5. Nonlinear tune shift control. In case of great chromaticity contribution from the straight section a method of the second order non-linearity compensation has to be provided. Taking into account nonlinear terms no higher than y, the average Hamiltonian in the
vicinity of the resonance can be written as:
(k2 + k2)1/2 (k2 + k2)1/2
H(Ix,0x,Iv,0v)= K x Ly> -A-Ix+ ,y -A-Jy +
ky
y
+ 2 hkx,ky p Ikx /2lyV/2 cos (kx flx + ky fly ) + Cx12 + Cxy1 x Iy + Cy12
1/2
where A is the detuning from resonance p+kxvx+kyvy _ A-(k^ + kjj) . The coefficients Zx,
Zy, (xy determine the nonlinear tune shift. In the first order of the perturbation theory, the nonlinear tune shifts arise due to the octupoles only, but in the second order, the sextupoles also contribute to the nonlinear tune shift:
Z ____ zsex I ZOCt
Zx _ Zx + Zx ,
Z ____ zsex I ZOCt
Zxy — Zxy ' Zxy ,
Z ____ Zsex i zoct
Zy _ Zy + Zy ■
Let us consider the Hamiltonian of the system near the third-order resonance in one of the planes. In the system without the friction force, there are two types of fixed points. The fixed point Ix, flx is the saddle, if the roots of the characteristic equation are real and it is the center when the roots are imaginary. The point itself is derived from the equations:
A + ^30P^/2 cos Mx + 2CJx + Cxyly = 0, sin Mx = 0.
The latter gives plenty of combinations with the fixed points depending on the ratio between the parameters A, h30p, Zx and Zxy. The influence of nonlinearity is specified by the discriminant in the expression:
p/2 _ 3h30p COS 3i9x 1/9 OA A
1x —-------------^ %\l 4 30p - ^ (- c>Xy1y)- \A>
Substituting 1, Ix in the characteristic equation, we can define what kind of fixed point this is. From the lattice design point of view, there are three interesting cases: Zx ^ h30p\ Zx ~ h3oP and Zx ^ h3oP. In the “resonant” lattice, the value of h30p is effectively suppressed. Therefore when the nonlinearity remains under control and strong, we get Zx ^ h30p. It is obvious from (3) that if the sign of the detuning A coincides with the sign of the tune shift Zx, the discriminant is negative and the system has only one center at Ix _ 0. Therefore, this case corresponds to the maximum stable region and the lattice with these features is the most promising.
In principle, the sextupole nonlinear tune cannot be controlled after the sextupole location has been fixed. Therefore, the sign of total chromaticity is controlled by the octupoles, which are located in the multipole correctors. Thus, after chromaticity correction, the nonlinear tune shift is measured and then using the correcting octupole, we adjust the required sign and value of the nonlinear tune shift, as described in our previous paper.
6. “Resonant” lattice application. The most important application of the “resonant” lattice is the magneto-optic structure without jtr crossing. Based on the above reasoning, the “resonant” lattice method with simultaneous orbit curvature and quadrupole gradient modulation with an approximately identical contribution of both modulations to the final value of the momentum compaction factor is most effective. From (1) it is easy to derive the following equation for arbitrary fundamental harmonics gk and rk giving a « — 1/v2 and
Ytr ~ iv
/ —x 2
gk
1 — (1 — w/v)2
Another application of the “resonant” lattice is as an advanced lattice for stochastic cooling. It is known that to intensify the stochastic cooling process, the mixing factor between the pick-up and kicker should ideally be as large as possible. On the other hand, in the case of mixing between the kicker and pick-up, the mixing factor should be smaller. It can be seen that the “resonant” lattice has a remarkable feature: the gradient and the curvature modulation amplify each other when they have opposite signs, while they can compensate for each other when they have the same signs (see formula (1)):
1 ^ 9k Tk|^) « 2 or 0.
4(w/v — 1) y [1 — (1 — w/v)2]
Thus, in such a lattice, we can make two arcs (see fig. 5) with different slip factors: npk _ 1/v2 — 1/y2; nkp _ — 1/v2 — 1/y2. In case y « v, one of the arcs is isochronous when the slip factor is npk ~ 0 and the other slip factor is nkp ~ —2/v2.
Third application of the “resonant” lattice is the dual structure of NICA [12]. To meet the NICA requirements to work with different magnetic rigidity beams Au-ions in range 1-4,5 GeV/u and with proton 6-13 GeV we consider the possible lattice with variation of both the beam energy and the transition energy. Such lattice has to have the feature of duality, when the same optic structure provides the minimum IBS heating for Au-ions and the increased transition energy for proton modes. In the proton mode the number of cells in one super-period Ncen and number of super-periods Sarc per arc is dictated by the required phase advance of radial oscillations. Following the theory of resonant lattices, we construct a lattice with the horizontal frequency on arc varc as close to the number of super-periods Sarc as possible keeping them both integers. In the Au option any modulation of gradient leads to IBS heating. Therefore taking all arguments we choose the arc with 12 FODO cells grouping in 4 super-periods for the proton by the gradient modulation and remaining the regular structure (no modulation) for Au-ions (see fig. 6, a, b). Transition from one option to another is done just by gradient change in two focusing quadrupole families on arcs and then matching with the straight sections.
The fourth application of the “resonant” lattice is the synchrotron light source. Since the horizontal emittance depends upon the horizontal dispersion function nx, as exm (H)dipole, where H _ Yxnx + 2ax^xnfx + f3xn^, the lattice with small electron emittances therefore
S, m
------------ P*------------------Py----------------------Dx
Fig. 5. The advanced HESR lattice with two different arcs
requires the smaller dispersion function and, as a consequence, the stronger sextupoles are needed in order to correct the chromaticity. At the same time, it is well known that the sextupoles dramatically decrease the dynamic aperture due to their nonlinear action. In this case, we can use the “resonant” lattice where the sextupolar terms are aimed to be smaller, and each pair of sextupoles have the same effect as one octupole. The nonlinear tune then plays the role of a stabilizing factor. Such a lattice has to be classified as a special lattice [13], since the sextupolar term is effectively suppressed, but the nonlinearity remains under control and strong. This method together with the smallest emittance results in a large dynamic aperture. For comparison, the dynamic aperture of a modified circular Chasman-Green lattice with the same number of sextupole families is smaller by a factor of four.
Furthermore, there is currently much interest in magnetic lattices, which can be operated over a range of momentum compaction factors. This provides several advantages and the possibility of working without sextupoles.
Fig. 7 shows the arc with a tuneable momentum compaction factor. In this lattice, one cell is formed by two adjoining cells, and each second focusing quadrupole is replaced by a short combined function bend magnet with positive gradient BG (G > 0). As a result, the arc consists of four cells and each half cell has a structure: trim QF+short BG (G > 0)+long BG (G < 0) + QF. The trim quadrupole is used for fine tuning. The dispersion function is equal to zero at the ends of the arc. This is automatically caused by an integer tune number for the arc. The matching sections are not needed. Due to the low beta function and zero dispersion function in the middle of each long bending magnet, the electron beam emittance has a rather low value [14].
7. Conclusion. We have developed a multi-application “resonant” lattice based on singlet, doublet and triplet with the following distinguishing features: ability to achieve the negative momentum compaction factor using resonantly correlated curvature and gradient modulations; gamma transition variation in a wide region from jt ~ vx to jt ~ ivx with quadrupole strength variation only; dispersion-free straight section; independent optical parameters of arcs and straight sections; two families of focusing and one of defocusing
Fig. 6. Dual lattice of one super-period and arc tuned for protons (a) and Au79+ (b) beams
S, m
Fig. 7. The arc lattice with variable momentum compaction factor
quadrupoles; separated adjustment of gamma transition, horizontal and vertical tunes; convenient chromaticity correction method using sextupoles; first-order self-compensating scheme of multipoles and a large dynamic aperture.
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Статья рекомендована к печати проф. Д. А. Овсянниковым.
Статья принята к печати 14 октября 2010 г.
