Transfer matrix for a high energy cooling system

A transfer matrix for a high-energy electron cooling system, which describes the linear incoherent effects, is derived. Knowing this matrix we can treat an electron cooling section in the ring as an additional focusing element deforming the beam optic functions due to space charge forces This matrix...

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Datum:	2004
Hauptverfasser:	Dolinskii, A., Zenkevich, P.
Format:	Artikel
Sprache:	English
Veröffentlicht:	Національний науковий центр «Харківський фізико-технічний інститут» НАН України 2004
Schriftenreihe:	Вопросы атомной науки и техники
Schlagworte:	Теория и техника ускорения частиц
Online Zugang:	http://dspace.nbuv.gov.ua/handle/123456789/80616
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Zitieren:	Transfer matrix for a high energy cooling system / A. Dolinskii, P. Zenkevich // Вопросы атомной науки и техники. — 2004. — № 5. — С. 155-157. — Бібліогр.: 7 назв. — англ.

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spelling	irk-123456789-806162015-04-20T03:01:59Z Transfer matrix for a high energy cooling system Dolinskii, A. Zenkevich, P. Теория и техника ускорения частиц A transfer matrix for a high-energy electron cooling system, which describes the linear incoherent effects, is derived. Knowing this matrix we can treat an electron cooling section in the ring as an additional focusing element deforming the beam optic functions due to space charge forces This matrix can be implemented in different codes (MAD, MIRKO, SixTrack et al.) for precise calculation of the beam optic and dynamic aperture in storage rings, where a high energy electron cooling system is planed to install. As an example, an optic of the high-energy storage ring [6] calculated by MAD code with derived matrix is given. Виводиться матриця переходу для високоенергетичної системи електронного охолодження, яка описує лiнiйнi некогерентнi ефекти. Така матриця дозволяє розглядати секцію електронного охолодження у кiльцi, як додатковий фокусуючий елемент, який спотворює оптичнi функцii внаслiдок сил просторового заряду. Отримана матриця може використовуватися у рiзних програмах (MAD, MIRKO, SixTrac та iнших) для проведення прецизiйних розрахункiв оптики пучка та динамiчної апертури нагромаджувальних кiлець, де планується планується установка высокоенергетичних систем электронного охлаждения. Як приклад наведено розрахунок оптики високоенергетичного нагромаджувального кiльця [6], виконаний за допомогою програми MAD та виведеної матрицi. Выводится матрица перехода для высокоэнергетической системы электронного охлаждения, которая описывает линейные некогерентные эффекты. Такая матрица позволяет рассматривать секцию электронного охлаждения в кольце, как дополнительный фокусирующий элемент, искажающий оптические функции вследствие сил пространственного заряда. Полученная матрица может использоваться в различных программах (MAD, MIRKO, SixTrac и др.) для проведения прецизионных расчетов оптики пучка и динамической апертуры в накопительных кольцах, где планируется установка высокоэнергетических систем электронного охлаждения. В качестве примера приведен расчет оптики высокоэнергетического накопительного кольца [6], выполненный с помощью программы MAD и выведенной матрицы. 2004 Article Transfer matrix for a high energy cooling system / A. Dolinskii, P. Zenkevich // Вопросы атомной науки и техники. — 2004. — № 5. — С. 155-157. — Бібліогр.: 7 назв. — англ. 1562-6016 PACS: 29.17.+w, 29.27.-a, 29.27.Eg, 02.90.+p http://dspace.nbuv.gov.ua/handle/123456789/80616 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Теория и техника ускорения частиц Теория и техника ускорения частиц
spellingShingle	Теория и техника ускорения частиц Теория и техника ускорения частиц Dolinskii, A. Zenkevich, P. Transfer matrix for a high energy cooling system Вопросы атомной науки и техники
description	A transfer matrix for a high-energy electron cooling system, which describes the linear incoherent effects, is derived. Knowing this matrix we can treat an electron cooling section in the ring as an additional focusing element deforming the beam optic functions due to space charge forces This matrix can be implemented in different codes (MAD, MIRKO, SixTrack et al.) for precise calculation of the beam optic and dynamic aperture in storage rings, where a high energy electron cooling system is planed to install. As an example, an optic of the high-energy storage ring [6] calculated by MAD code with derived matrix is given.
format	Article
author	Dolinskii, A. Zenkevich, P.
author_facet	Dolinskii, A. Zenkevich, P.
author_sort	Dolinskii, A.
title	Transfer matrix for a high energy cooling system
title_short	Transfer matrix for a high energy cooling system
title_full	Transfer matrix for a high energy cooling system
title_fullStr	Transfer matrix for a high energy cooling system
title_full_unstemmed	Transfer matrix for a high energy cooling system
title_sort	transfer matrix for a high energy cooling system
publisher	Національний науковий центр «Харківський фізико-технічний інститут» НАН України
publishDate	2004
topic_facet	Теория и техника ускорения частиц
url	http://dspace.nbuv.gov.ua/handle/123456789/80616
citation_txt	Transfer matrix for a high energy cooling system / A. Dolinskii, P. Zenkevich // Вопросы атомной науки и техники. — 2004. — № 5. — С. 155-157. — Бібліогр.: 7 назв. — англ.
series	Вопросы атомной науки и техники
work_keys_str_mv	AT dolinskiia transfermatrixforahighenergycoolingsystem AT zenkevichp transfermatrixforahighenergycoolingsystem
first_indexed	2025-07-06T04:37:02Z
last_indexed	2025-07-06T04:37:02Z
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fulltext	TRANSFER MATRIX FOR A HIGH ENERGY COOLING SYSTEM A. Dolinskiia,b, P. Zenkevichc a Institute for Nuclear Research, Kiev, Ukraine bGesellschaft für Schwerionenforschung, Darmstadt, Germany c Institute of Theoretical and Experimental Physics, Moscow, Russia e-mail: A. Dolinskii@gsi.de A transfer matrix for a high-energy electron cooling system, which describes the linear incoherent effects, is de- rived. Knowing this matrix we can treat an electron cooling section in the ring as an additional focusing element de- forming the beam optic functions due to space charge forces This matrix can be implemented in different codes (MAD, MIRKO, SixTrack et al.) for precise calculation of the beam optic and dynamic aperture in storage rings, where a high energy electron cooling system is planed to install. As an example, an optic of the high-energy storage ring [6] calculated by MAD code with derived matrix is given. PACS: 29.17.+w, 29.27.-a, 29.27.Eg, 02.90.+p 1. INTRODUCTION The electron cooling method is based on the heat ex- change between the beam of charged heavy particles (ions) circulating in the storage ring and the beam of electrons having the same average velocity [1,2]. By combining an ion beam and a high-intensity electron beam having a small momentum spread in a rectilinear storage section, it is possible to achieve efficient energy exchange between them. The electron beam passing through the cooling section is lost in the collector and takes some of the ion beam thermal energy with it, which leads to an effective reduction in the transverse dimension and the momentum spread in the initial ion beam. At low ion beam intensities electron cooling is successfully used in many devices and a space charge focusing of the electron beam may not significantly per- turb the lattice functions. However, on transition to higher energy ion beam (more then 1 GeV/u) there is tendency to enhancement parameters of an electron cooling section (ECS) to have reasonable small cooling time. In this case the ECS in the ring may treated as an additional focusing element deforming the beam optic functions due to space charge forces. To write the equation of motion of ions in electron beam, from which the transfer matrix for an electron cooling section can be derived, we consider the hydro- dynamic approximation assuming that ion and electron temperatures are zero and the beams undergo simultan- eous coherent transverse motion. The particle density in the beams is radially uniform and the beam radii are the same. The conducting wall is removed to infinity and image charges have no influence on the beam dynamic. The beams are matched for a time shorter than all the other characteristic times of the problem. Outside the cooling section the beam propagates in the storage ring with azimuthally focusing symmetry. We use a simple model to analyse the influence of the finite time of joint beam motion on the interaction of the electron and ion beams. 2. EQUATION The equation of linear motion of the circulation anti- proton beam in the electron cooling section with ac- count of the space charge field and longi-tudinal mag- netic field has the following form [3] 02 2 2 =Ω+Ω ± ±± cc cL c c u ds dui ds ud  . (1) Here subscript c refers to the circulating beam, u±=x±iy describes transverse displacement of the particles in complex notation motion (x, y are the horizontal and vertical displacements respectively), β γ2cAm ZeB p sL =Ω is the Larmour frequency of the circulating beam, Z, A are the charge and mass of the ion, Bs is the longitudinal magnet field in the cooler, mp is the mass of the proton, e is the electron charge, c is the light velocity, β=v/c, v is the ion velocity, γ is the relativistic parameter. 2 , 22 )1( eccc Ω+−=Ω ηηγ , 32 2 , 2 γβ π A Zrn pe ec =Ω are circulating beam frequencies, where ηc=Ic/Ie charac- terises the focusing of ions due to space charge forces in the cooler section (Ic , Ie are ion and electron currents re- spectively), η is the neutralization factor [4], γ2η de- scribes the focusing due to neutralization of ions, ne is the electron density in the cooler, rp is the classical pro- ton radius. Eugenfrequencies of the Eq. (1) are 2 2 2 2,1 22 c LL Ω+        Ω ±        Ω ±=Ω (2) Solution of Eq.(1) can be written in matrix forms:     ′ =    ′ + + + + )()( 0 0 u u M u u ,     ′ =    ′ − − − − )( ~ )( 0 0 u u M u u Here M is the second order complex matrix, M~ is the complex conjugated matrix. The elements of this matrix are given by the following formulae ( ( ) 22 2 c LK Ω+Ω= ): PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2004, № 5. Series: Nuclear Physics Investigations (44), p. 155-157. 155       +Ω±    Ω=+ )cos()sin( 22 exp)(11 KsKs K isisM LL  , (3a) K KssisM L )sin( 2 exp)(12     Ω=+  , (3b) )sin( 2 exp)( 2 21 Ks K sisM c L Ω     Ω=+  , (3c)       +Ω±    Ω=+ )cos()sin( 22 exp)(22 KsKs K isisM LL  , (3d) If K2<0 one should substitute in Eq. (3) instead of cos(Ks) and sin(Ks)/K the expressions )cosh( sK and ( ) KsK /sinh . One can see that the elements of the complex matrix depend on the longitudinal variable s and K, ΩL, Ωc parameters. For practical simulations it is necessary to write a real transfer matrix for the cooling section. The expressions (3) allows us to find the real matrix Tcool of fourth order related with real coordinates by:             ′ ′ =             ′ ′ 0 0 0 0 y y x x T y y x x cool . (4) The matrix Tcool is expressed through the matrix M us- ing the following formula:     − = AB BA Tcool , (5) where A=Re(M), B=-Im(M). The elements of this mat- rix are given by formulae     ΩΩ+    Ω= sKs K sKssA LLL 2 sin)sin( 22 cos)cos()(11 , )sin(1 2 cos)(12 Ks K ssA L     Ω= ,     ΩΩ= sKs K sA L c 2 sin)sin( 2 )( 2 21 ,     ΩΩ−    Ω= sKs K sKssA LLL 2 sin)sin( 22 cos)cos()(22 , (6)     ΩΩ−    Ω= sKs K sKssB LLL 2 cos)sin( 22 sin)cos()(11 , K KsssB L )sin( 2 sin)(12     Ω= ,     ΩΩ= sKs K sB L c 2 sin)sin( 2 )( 2 21 ,     ΩΩ+    Ω= sKs K sKssB LLL 2 cos)sin( 22 sin)cos()(22 . The matrix elements satisfy the relations: det(Tcool)=1,  det(M)=1. The transfer matrix of a circular accelerator is Tper=T1Tcs, where T1 is the transfer matrix from the Electron Cooling Section (ECS) exit to the ECS en- trance, Tcs is the transfer matrix of the cooler section in- cluding fringing fields of the solenoind. The transfer matrix for the cooler section taking account fringing fields of solenoid is written by Tcs=T- sfTcoolT+ sf, where T± sf are matrices for the fringing fields. 3. APPLICATION TO THE HESR A planned high energy storage ring (HESR) for in- ternal experiments with antiprotons at a maximum kin- etic energy of 14.5 GeV is part of the GSI (Gesellschaft für Schwerionenforschung) future project [5]. The beam quality of 2x10-5 and the high average luminosity of 2x1032 c-1cm-2 are main objectives of the HESR [6]. Ob- viously, such parameters can by achieved by strong electron beam cooling, which is mandatory for high lu- minosity as well as for high energy and angular resolu- tion. A cold electron beam at energies up to 8 MeV (corresponding to an antiproton energy of 14 GeV) with an electron current of up to 1 A are required with radius of 3 mm. A high cooling rate can be achieved only by the co-called magnetized cooling. It requires a strong longitudinal magnetic field (B=0.5 T) that guides the electron beam along the entire interaction region of up to 30 m length. The electron cooling system with such parameters is a straight-forward from a physical point of view although the technical realisation seems to be rather challenging. In a storage ring such electron cool- ing system will be as additional focusing element influ- encing on the beam dynamic of antiprotons beams due to space charge forces. In order to determine the complete dynamics of the beam in the storage ring one needs to supplement the matrix of the cooling section with the matrix describing the ion beam motion in the storage ring. That can be done by using ion-optical codes (for instance MAD, TRANSPORT) where the external matrix is possible to introduce. We calculate the lattice functions of the HESR with and without ECS by MAD code, introducing the trans- fer matrix, which elements are calculated by formulae (6). The parameters used to derive proper transfer mat- rix of the ECS are given in the Table. Antiproton energy [GeV] 1.3 Magnet rigidity, BR, [Tm] 6.17 Neutralization factor, η 0.01 Solenoid field strength, Bs [T] 0.0 Electron beam radius,a [mm] 3 Electron beam current, Ie, [A] 1 Length of cooling section, Lcool [m 30 ΩL=Bs/BR 0.044 The lattice functions over the HESR, where the ECS is treated as a drift space, are shown in Fig. 1. This is a desired beta function distribution over the ring. A small value of beta functions (1 m) in both planes is required at the internal target and large ones (100 m) at the ECS. At high energy of antiprotons (more than 3 GeV) the ECS influences on the lattice functions is weak. Hence there is no need to make additional matching of the cooler section with optic of the rest ring. If the energy of antiprotons is below 3 GeV the lattice function of the ring is changed drastically. To demonstrate the influence of the ECS on the lattice functions only due to space charge focusing of the electron beam in the ECS we exclude the longitudinal magnetic field from our simulation. Fig. 1. Lattice functions of the HESR. The ECS is treated as drift space Fig. 2. Lattice functions of the HESR. The ECS with electron beam but without longitudinal magnetic field Fig. 2 shows the beta function perturbation over the ring at the energy of antiproton of 1.3 GeV. It should be noted that according to the matrix elements given by formulae (6) the betatron motion in the ring is coupled. Therefore the Twiss in MAD are computed in coupled functions in the sense of Edwards and Teng paramet- erisation [7]. In this case the additional matching between arcs and a straight section, where the ECS is placed, is needed. 4. CONCLUSIONS The transfer matrix for cooling section with a high electron density and strong longitudinal magnetic field is presented. It was shown on the example of the HESR that space charge focusing may significantly perturb the lattice functions causing a big enough tune shift and ac- ceptance reduction of the circular accelerator. In this case additional matching of the ring optic functions with the ECS depending on the energy should be considered on the designing stage of the machine. REFERENCES 1. Ya.S. Derbenev, A.N. Skrinsky. The effect of an ac- companying magnetic field on electron cooling // Particle Accelerator. 1978, v. 8, p. 235-243. 2. I.N. Meshkov. Electron cooling: Status and perspec- tives // Phys. Part. Nucl. 1994, v. 25(6), p. 631. 3. P.R. Zenkevich,A.E. Bolshakov. Influence of elec- tron cooler on dipole ion oscillations in high-current storage ring // NIM. 2000, v. A441, p.36-39. 4. J. Bosser et al. Experimantal studies of electron beam neutralization // NIM. 1997, v. A391, p. 103- 106. 5. An International Accelerator Facility for Beam of Ion and Antiprotons, GSI, Darmstadt, November, 2001 (see www.gsi.de). 6. B. Franzke et al. Conceptual design of a facility for internal target experiments with antiprotons up to 15 GeV/c. Proc. of EPAC 2002, Paris, France, p. 575. 7. D.A. Edwards and L.C. Teng. Parameterisation of linear coupled motion in periodic systems. IEEE Trans. on Nucl. Sc.20: 885, 1973. МАТРИЦА ПЕРЕХОДА ДЛЯ ВЫСОКОЭНЕРГЕТИЧЕСКОЙ СИСТЕМЫ ЭЛЕКТРОННОГО ОХЛАЖДЕНИЯ А. Долинский, П. Зенкевич Выводится матрица перехода для высокоэнергетической системы электронного охлаждения, которая описывает линейные некогерентные эффекты. Такая матрица позволяет рассматривать секцию электронного охлаждения в кольце, как дополнительный фокусирующий элемент, искажающий оптические функции вследствие сил пространственного заряда. Полученная матрица может использоваться в различных програм- мах (MAD, MIRKO, SixTrac и др.) для проведения прецизионных расчетов оптики пучка и динамической апертуры в накопительных кольцах, где планируется установка высокоэнергетических систем электронного охлаждения. В качестве примера приведен расчет оптики высокоэнергетического накопительного кольца [6], выполненный с помощью программы MAD и выведенной матрицы. МАТРИЦЯ ПЕРЕХОДУ ДЛЯ ВИСОКОЕНЕРГЕТИЧНОЇ СИСТЕМИ ЕЛЕКТРОННОГО ОХОЛОДЖЕННЯ А. Долинський, П. Зенкевич Виводиться матриця переходу для високоенергетичної системи електронного охолодження, яка описує лiнiйнi некогерентнi ефекти. Така матриця дозволяє розглядати секцію електронного охолодження у кiльцi, як додатковий фокусуючий елемент, який спотворює оптичнi функцii внаслiдок сил просторового заряду. Отримана матриця може використовуватися у рiзних програмах (MAD, MIRKO, SixTrac та iнших) для проведення прецизiйних розрахункiв оптики пучка та динамiчної апертури нагромаджувальних кiлець, де планується планується установка высокоенергетичних систем электронного охлаждения. Як приклад наве- дено розрахунок оптики високоенергетичного нагромаджувального кiльця [6], виконаний за допомогою програми MAD та виведеної матрицi. А. Долинский, П. Зенкевич А. Долинський, П. Зенкевич

Transfer matrix for a high energy cooling system

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