Previously it was shown that by crystallizing each of the two counter-circulating beams, a much larger beam-beam tune shift can be tolerated during the beam-beam collisions; thus a higher luminosity can be reached for colliding beams [1]. On the other hand, crystalline beams can only be formed at energies below the transition energy ({gamma}{sub T}) of the accelerators [2]. In this paper, we investigate the formation of crystals in a high-{gamma}{sub T} lattice that also satisfies the maintenance condition for a crystalline beam [3]

MACHIDA, S.

OKAMOTO, H.

SESSLER, A.

WEI, J.

YURI, Y.

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/---? ,/4 ---)' BROOKHAWEN NATI 0wK-L LAB 0 R A T 0  RY I BNL-75482-2006-CP Crystalline Beams At High Energies J. Wei Presented at the I dh Biennial European Particle Accelerator Conference (EPAC) Edinburgh, UK June26 - June30,2006 June 2006 Collider-Accelerator Department Brookhaven National Laboratory P.O. Box 5000 Upton, NY I 1973-5000 www.bnl.gov Notice: This manuscript has been authored by employees of Brookhaven Science Associates, LLC under Contract No. DE-AC02-98CH10886 with the U.S. Department of Energy. The publisher by accepting the manuscript for publication acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes. This preprint is intended for publication in a journal or proceedings. Since changes may be made before publication, it may not be cited or reproduced without the author's permission. DISCLAIMER This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, nor any of their contractors, subcontractors, or their employees, I makes my warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or any third party’s use or the results of such use of any information, apparatus, product, or process disclosed, or represents that its use would not infi.inge privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof or its contractors or subcontractors. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof CRYSTALLINE BEAMS AT HIGH ENERGIES* Jie Weit, Brookhaven National Laboratory, USA Hiromi Okamoto, Shuta Ochi, Hiroshima University, Japan Yosuke Yuri, Japan Atomic Energy Agency, Japan Andrew Sessler, Lawrence Berkeley National Laboratory, USA Shinji Machida, Rutherford Appleton Laboratory, United Kingdom Abstract Previously it was shown that by crystallizing each of the two counter-circulating beams, a much larger beam-beam tune shift can be tolerated during the beam-beam colli- sions; thus a higher luminosity can be reached for colliding beams [l]. On the other hand, crystalline beams can only be formed at energies below the transition energy ( Y T )  of the accelerators [2]. In this paper, we investigate the for- mation of crystals in a high-yT lattice that also satisfies the maintenance condition for a crystalline beam [3]. INTRODUCTION For high-energy colliders, the luminosity is usually lim- ited by the Coulomb interactions between particles of ei- ther the opposite beams (the beam-beam effect) or the same beam (intra-beam scattering) causing growth in the beam emittance. Beam cooling is often considered to reduce the emittance and enhance the luminosity performance. Crystalline-beam state is the ultimate state when the beam emittance approaches zero. Previously it was shown that by crystallizing each of the two counter-circulating beams, a much larger beam-beam tune shift can be tolerated during the beam-beam collisions;. thus a higher luminosity can be reached for colliding beams [l]. On the other hand, present studies on beam crystalliza- tion is limited to very low energies, due partly to lack of effective cooling techniques at high energies, and partly to conditions that prohibits crystal formation above the tran- sition energy of the accelerator ring [2]. In this paper, we explore crystal formation in ring lattices where the transi- tion energy is made either much higher than the transverse tunes of the ring or imaginary. CONDITIONS FOR CRYSTALLIZATION The Hamiltonian of particles in an AG-focusing machine can be expressed in terms of the action-angle variables J,, Jy and momentum p, as [4] . where the transverse tunes u, and uy are positive for a sta- ble machine lattice, VC is the Coulomb potential, and F, is *Work performed under the auspices of the U.S. Department of En- t jwei@bnl.gov ergy. related to the normalized horizontal dispersion D as [4] D + DD" $: (0')' (bending section) DD" + (D')2 (straight section). F,= { (2) The average value of F, can be obtained as 1 Ddt 3 2, (3 )  TT (F,) = &{Fzd t=  - P where y~ is the transition energy of the machine. For a stable crystalline beam, the Coulomb force must on the av- erage provide focusing in the azimuthal direction, . (4) where k2 2 0 is the effective Coulomb focusing strength. It can thus be seen from Eqs. 1 and 3 that the azimuthal mo- tion will not be bounded if y 2 YT. Hence, the crystalline beam can not form when the beam energy y is above the transition energy TT.  This is the so-called crystal forma- tion condition or the first condition [2]. Eq. l also indicates that crystals may be formed if the transition energy is imag- inary (negative momentum-compaction). The second condition is on the crystal maintenance. The condition derived from the linear-resonance criteria states that the frequency of ring lattice variation seen by the circu- lating crystalline beam must not equal the sum of any two phonon frequencies of the crystal [5]. This condition is sat- isfied for beams of any density when the transverse phase advance across the lattice superperiod is less than 7 r /  fi (or 127") [3,5]. Considering more practical aspects of cooling a high- temperature beam and the need to suppress linear reso- nances over the whole density and temperature region, a more stringent version of the second condition is that the transverse phase advance across the lattice superperiod is less than 7r/4 (or 90") [6,71. HIGH-TT LATTICES 3-cell Missing-dipole Lattice A conventional method to achieve small or negative momentum-compaction (high or imaginary T T )  is to use modules of three FODO cells with missing dipole in the middle cell (F.g 1 [8, 91). The horizontal phase advance of about 90" per cell excites the dispersion oscillation so that high dispersion occurs only at locations of missing dipole (Eq. 3). However, this lattice is not suitable for beam crystallization since the phase advance (-270") across the module is much higher than 127". 1.5 - 1.0 - Azimuthal displacement along ring, s .., .. .' . .. . .. .... Figure 1: 3-cells missing-dipole low-momentum- compaction lattice module. B, QF, and QD denote dipole, focusing, and defocusing quadrupole magnets, respec- tively. The horizontal phase advance across the module is approximately 270". Negative-bend Lattice A low-momentum-compaction lattice that satisfies the maintenance condition is shown .in Fig. 2. The short, negative-bend dipoles at the high-dispersion region com- pensate for the long, regular dipoles at the low-dispersion region. Such lattice was proposed in 1955 to avoid transi- tion crossing [lo]. A variation of this structure is proposed for the Fixed Field Alteniating Gradient (FFAG) accelera- tor combining B and QD as one magnet while keeping the cell phase advance below 90" [ 111. MOLECULAR-DYNAMICS RESULTS We evaluate the formation and stability of crystalline beams with various types of lattices using the molecular- dynamics (MD) computer simulation. With the 3-cell missing-dipole lattice, only 1D crystalline string may be formed. In the following, we discuss three cases of the negative-bend lattice: imaginary-yT with combined- function magnets and phase advance per cell below 90"; high-yT with combined-function magnets and phase be- low 90"; and high-yT with separate-function magnets and phase above 90". 24Mg+ ions of various densities are stored in rings each consisting of 12 superperiods. Positive and negative bends are of equal bending radius. Strong, tapered cooling is ap- plied [ 121. If a crystalline state is reached, the cooling force is removed to test the stability of the formed crystal. Imaginary- yT Lattice Lattice functions of each of the 12 ring superperiods is shown in Fig. 3. The ring circumference is 84 m. The phase advances per cell are px = 87" and pv = 87". The tunes (us, p y )  are (2.9,2.9). The transition energy is imaginary, SJpoc  = 0. Table name = TCVISS s ( I n )  * BN Figure 3: Imaginary-yT negative-bend lattice with 87" phase advance. The middle positive bend is defocusing. ~~ Azimuthal displacement along ring, s Figure 2: Negative-bend low-momentum-compaction lat- tice. B, BN, and QD denote dipole, negative-bend dipole, and defocusing quadrupole magnets, respectively. The hor- izontal phase advance across the module is usually below 127". Fig. 4 shows a stable, multi-shell (3D) crystal formed at energy y = 1.45 with a line density of 2.1x107/m. For- mation of stable 3D crystals becomes increasingly difficult for higher beam energies. When ?.is higher than the hor- izontal tune ux of the ring, only 2D crystals are formed. Due to reduction of the effective horizontal focusing (ef- fective focusing strength is u; - y2 in the horizontal and ui in the vertical directions under the smooth approximation [5]), the zig-zag structure extends in the horizontal plane. Fig. 5 shows a stable, 2D crystal formed at y = 5.5 with a line density of 1 . 6 ~  10G/m. l o g  Figure 4: A multi-shell crystalline beam formed with lattice in Fig. 3 with y = 1.45 and density 2.1 x 107/m. the Figure 5: A 2D zig-zag crystalline beam formed with the lattice in Fig. 3 with y = 5.5 and density 1 . 6 ~  106/m. High-yT, Low-tune Lattice With a structure similar to Fig. 3, a high-yT lattice is used for the MD study. The phase advances per cell are px = 88" and py = 85". The tunes (ux, pzI) are (2.9,2.8). The transition energy is real, YT = 102. The crystal prop- erties are similar to those formed with the imaginary-yT lattice. High-yT, High-tune Lattice Fig. 6 shows a lattice with separate- function positive and negative bending and defocusing magnets. The phase ad- vances per cell are pZ = 96.6" and py = 90.3" exceeding the 90" vaIue. The tunes (ux, py) are (3.21,3.01). The transition energy is real, YT = 165. The crystal properties are similar to those formed with lattices of cell phase advance below 90". At a high energy, stable 2D structures in the horizontal plane are formed at y = 9, much higher than the transverse tunes. DISCUSSIONS. AND SUMMARY We develop low-momentum-compaction lattices that would allow the formation of crystalline beams at high en- ergies (y higher than the machine tunes). Lattices con- taining negative bending are adopted to keep the transverse phase advance per lattice superperiod below r /  4. I HIGH-GT NEG.-BEND SEPARATE FODO Win32 version 8.51/15 20/06/06 12.26.19 8. 3. 2 9;'.., ' , .  . 5 7 . -  ; 6. :  I ,,' 2. 3. 4. 3. &. 3. s (eiJ SJpcc  = 0. Table nasre = TWISS Figure 6: High-yT negative-bend lattice with 96.6" phase advance. With these lattices, we formed multidimensional crystals at energies (y) much higher than the machine tunes ( v , , ~ ) .  However, when y > u,,,, stable crystals turn to extend in the horizontal plane due to reduction of the effective hori- zontal focusing. Under the smooth approximation, the transition YT is about equal to the horizontal tune of the ring. This approx- imation no longer holds for low-momentum-compaction lattices whose TT deviates significantly from the tune. More systematic analysis is in progress to evaluate the change of effective focusing with energy in such lattices. We thank X.-P. Li and D. Trbojevic for many helpful dis- cussions. REFERENCES [l] J. Wei, A.M. Sessler, EPAC 1998, Stockholm (1998) 862 [2] J. Wei, X-P. Li, A. M. Sessler, Phys. Rev. Lett., 73, (1994) [3] J. Wei, H. Okamoto, and A.M. Sessler, Phys. Rev. Lett., 80, [4] 3. Wei, X.-P. Li, A.M. Sessler, Proc. 6th Adv. Accel. Conf. [5] X.-P. Li, H. Enokizono, H. Okamoto, Y. Yuri, A.M. Sessler, [6] K. Okabe, H. Okamoto, Jpn. J. Appl. Phys. 42 (2003) 4584 [7] I. Hofmann, L. Laslett, L. Smith, I. Haber, Part. Accel. 13 [8] R.C. Gupta, J.I. Botman, M.K. Craddock, IEEE Trans. Nucl. [9J D. Trbojevic, D. Finley, R. Gerig, S. HoImes, EPAC90 (1990) [IO] V.V. Vladimirski, E.K. Tarasov, Theoretical problems of the [ l  I] C. Johhnstone, E. Keil, A.M. Sessler, and D. Trbojevic, pri- [I21 H. Okamoto, J. Wei, Phys. Rev. E, 58 (1998) 3817 3089 (1998) 2606 (1994) 224 J. Wei, Phys. Rev. ST-AB, 9 (2006) 034201 (1983) 145 Sci. NS-32 (1985) 2308 1536 ring accelerators, USSR Academy of Science (1955) vate communications 

CRYSTALLINE BEAMS AT HIGH ENERGIES.

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CRYSTALLINE BEAMS AT HIGH ENERGIES.

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