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Annex I Electron cooling application for luminosity preservation in an

1. tb O ol Ol Characteristic times sec O Beam energy GeV Fig 2 2 The deuteron beam life time due to single scattering on the target atoms 1 and the times of emittance growth up to the acceptance limit in horizontal 2 and in vertical 3 planes For both kinds of particles protons and deuterons the experiment duration is limited by about 5 minutes at the target density of the order of a few 10 Atoms cm and by about 1 hour at the target density of about 10 Atoms cm Correspondingly in the following calculations we will investigate two cases short term beam parameter evolution at high target density and long term evolution at low target density For both beams in the total energy range the particle losses due to aperture limitation play a significant role and suppression of the emittance growth by the beam cooling can increase the integral luminosity of the experiment by several times Ultimate possible efficiency of the cooling decreases with the increase of the beam energy and for deuteron beam at maximum energy the maximum gain in the luminosity is less than two times More accurate estimations of the possible cooling efficiency at high beam intensity have to include consideration of the intrabeam scattering process 2 2 Calculations in presence of intrabeam scattering The intrabeam scattering calculations are performed for the lattice parameters given in Fig 2 3 Initial b
2. The Target form Fig 5 9 includes three tab sheets Target parameters Lattice functions Injection and menu item Target material The tab sheet Injection is visible only when the parameter Stripper foil is checked The menu item Target material calls the form Target material described in the next chapter The parameters of internal target are entered in the tab sheet Target parameters Table 5 4 or calculated from the ion beam and parameters of the target from the Target material form Table 5 4 Target parameters Energy losses Esr eV are used for calculations of formula 2 1 1 ref 1 Stripping angle Osr rad Momentum spread APlPrarget Life time Ti sec is used for calculations of formula 2 1 4 ref 1 Target cross section S cm is used for calculation of the probability of the target crossing by an individual particle in accordance with 2 1 3 122 F Target Target material Energy loss ey Stripping angle rad Momentum spread Life time sec Target cross section em 2 e Stripper foil Lattice functions Injection 15 x 100472 7 Je 110288 5 fs 695802E 6 fo coost 32044 1000 Fig 5 9 The Target form The ion ring lattice functions in the functions Table 5 5 target position are entered in the tab sheet Lattice Table 5 5 Lattice functions Horizontal Beta Bhor m are used for calc
3. eae C q0 d Ds d xB z a 00 where 5 a D B D 6 62 D o2 o and g 28y x x xp Pp y Yo q Y r 13 The integration over z variable can be approximately performed analytically due to small value of the function D u v expl Duv zlin l z dz 2 expl D u v z ln zdz 0 1 DULY 4 1 4 eee er dt D u v t where C is the Euler constant The integration 2 2 9 over other two variables is performed numerically 2 3 Ion multi scattering on residual gas atoms The characteristic time of the emittance grows rate is calculated using the following formula 7 2 2 1 1 oe _ 8n A ea m c 2 3 1 Tt amp of a Bep Here eah Z In 280 a 4z ee ik In 280 a is the multiple scattering density with a given residual gas composition in the program three components of the residual gas atoms can be introduced is the fine structure constant B is the mean beta function in corresponding plane 2 4 Tune resonances of the ion betatron motion The tune resonances are often associated with beam losses but if the resonance is weak enough or it is crossed quickly enough it may lead to an amplitude growth only which is equivalent to heating The maximum amplitude growth time e folding due to a nonlinear one dimensional resonance is 7 pee C 1 e e TAQ Be N 2 N gt 2 2 4 1 Here N is the order of resonance C ring circumfere
4. mio NIN lle 67 134 Cancel fi minus W Fig 2 2 The lattice parameters imported from the MAD output file Paoa Ah Alpha In order to read the lattice functions from the file in text format a special visual form was elaborated Fig 2 2 Using it user can specify positions of the corresponding parameters in input file read the file and check a validity of the data presented in the numerical or graphic format 12 In accordance with the Martini model the growth rates can be inserted into the code in the form of the corresponding characteristic times t l 2 2 6 where angular brackets mean averaging over the ring circumference n 1 for a bunched beam and n 2 for coasting beam Ji a Jl a2er2a 2 2 7 p l n Vno 6 5 5 5 B ys is the linear ion density N L or coasting beam we f 4 l 2 2 8 N i 2 n6 for a bunched beam The functions f are integrals of the following form co TM 27 f k sin Ug U Vv yexp D u v z in t z ldvdudz 2 2 9 00 0 with the coefficients k 1 2 kz a c kz b c and sin u cos v sin ula sinv d cosv i b cos u D u v j 2 2 10 c g u v 1 3sin wcos v 2 2 11 g uy 1 3sin usin v 6d sin y sinv cosv a 2 2 12 2 U V 1 3cos pu 2 2 13 The normalized parameters are to be calculated from the following expressions 0 O aay oO
5. newltems i BItems i newltems BCount B this if BCount delete BlItems Biltems newltems BIndex BCount BCount t All Effect classes are the descendants from BTemplate Simulation tasks do not use standard variable names of different Effect classes They use the pointer array BItems of BTemplate This structure permits to include very easy new effects without any changes in the program 90 Such approach is useful and widely used in our program codes as the base of Object Oriented Programming TimeRate TK Active Effects E M ECOOL M IBS 1 225624802E5 1 TAU_N ao 0 118358036293 1 TAU_Eh RestGas F Stochastic 0 075098676631 1 TAL_Ev Injection fotraar7ea0s 1 TAULP T Target Resonance I Heating i 7 4 Time sect Fig 6 5 The form for visualization of the effect array and output of the sum of the rates of the active effects Another group of objects called Task corresponds to investigation of the beam parameter variation due to common action of the active effects Fig 6 5 These objects get the array of the rates from the list of effects calculate the beam parameters and use the methods of the Beam object for their visualization Only one object of this type is realized now in the program Dynamics which includes method and parameters necessary for numerical solution of the system of differential equations 1 1 and for visualization of the results
6. Timer Interva 2 lepta jinr ru bolide hte ETON IM NonStop 250 me cme Fig 8 12 The Process control window More detail description of the graphs and process control will be done in the manual for BOLIDE system Reference 1 I N Meshkov Yu V Korotaev A L Petrov A O Sidorin A V Smirnov H J Stein E M Syresin G V Trubnikov S V Yakovenko Electron cooling application for luminosity preservation in an experiment with internal targets at COSY Interim report Dubna 2001 2 http nuweb jinr ru lepta 140
7. 0 00314 2 91 10 50 2 75 10 1 96 10 0 00393 5 6 10 60 4 84 10 2 83 10 0 00471 9 85 10 9 70 7 54 10 3 85 10 0 0055 1 54 10 4 80 1 1 10 5 03 10 0 00628 2 32 10 4 At pellet diameter of 40 um electron cooling compensate the particle losses by corresponding decrease of the beam emittance and the experiment luminosity variation during first five minutes is negligible At the same conditions without electron cooling the luminosity decreases by about 3 times At bigger pellet diameter the power of electron cooling is not high enough to suppress the beam heating due to interaction with target and the cooling application leads only to some increase of the luminosity life time The effectiveness of the cooling application can be estimated by comparison of an integral luminosity after certain period of experiment with and without of the cooling The integral luminosity for some period of experiment can be calculated 4 5 N T N Lau i 2 dt TTBB lees 21 Or in the case of numerical integration of the beam dynamics N A amp N t L Se B B 7h ee where At is integration step over time n is the number of the integration steps 3 50E 33 3 00E 33 2 50E 33 2 00E 33 1 50E 33 1 00E 33 Luminosity 1 cm42 sec 5 00E 32 0 00E 00 le 0 om le 0 5A 50 100 150 200 250 300 Time sec 6 00E 33 5 00E 33 4 00E
8. 3 times at minimum and intermediate beam energy At maximum beam energy the effect of cooling application decreases and at the target density below 10 Atoms cm the experiment can be performed 58 without luminosity loss during long period of time without cooling Electron cooling stabilises the beam parameters in the total energy rang and transverse dimensions of the ion beam during experiment can be 2 3 times less than in the case without cooling This permits to decrease the aperture of the gas storage cell in order to increase the target density In the experiment with the pellet target at maximum proton beam energy an electron cooling gives substantial gain in the luminosity up to the luminosity level of about 5 10 cm sec and at experiment duration up to five minutes Using of an electron cooling system one can realise the regime with permanent luminosity during the time of experiment via compensation of the particle losses by corresponding decrease of the beam emittance An electron cooling application gives a possibility to optimise the ion beam parameters and experiment duration in order to obtain maximum integral luminosity taking into account the time of the beam storage and acceleration The proposed scheme of the electron cooling system with circulating electron beam permits to provide the effective ion beam cooling in the total energy range and in the single pass mode of operation the beam cooling during storage pro
9. Doe e A A l 0 1 n 7 Bm c F l A n BY 7 l mrad 5 16 The difference between particle velocity and electron beam one is calculated taking into account the momentum shift of the electron beam 0 0 0 80 5 17 To exclude the electron beam space charge from the calculation the value of Neutralization factor in the tab sheet Cooler parameters is to be equal 1 114 Electron Cooling Fig 5 2 The Electron Cooling form tab sheet Friction force 5 1 2 5 Ion beam space charge The ion beam space charge is taken into account in accordance with the following model When the ion and electron beams merge in the cooling section the kinetic energy of an electron changes according to its position inside the potential well of the ion beam space charge field Let s suppose that we tune the particle initial energy so that the electron velocity is equal to the ion velocity when the electron comes to the center of the ion bunch Then the momentum of the electrons varies across the beam in accordance with the following Ap a __ 2A s r Z p P jes B y 2a 5 18 E ar i where a osPeteePe is the mean bunch radius A is ion linear density and F t is the function which describes the ion beam potential distribution across the beam When the beam density distribution across the beam is Gaussian the function F t can be written as follows _ tl exp u 7 Pp E 5 19 115 The ef
10. lid Mao f2 c F Minor e H po Color P Redraw grids Cancel Fig 8 3 The Grids tab sheet Number of divisions and sub divisions style and color of the grid lines in the plot can be entered using the Grids tab sheet Fig 8 3 134 50 Graph Curve Anis Grid Other Curves Values Background color Filling point color Graphic Font Copy To Clipboard New window Legend width Keep Image on Redraw 0 aje a Cancel Fig 8 4 The Other tab sheet The plot background colour font of the axis and legend legend width can be modified using tab sheet Others Fig 8 3 Note The button Copy To Clipboard does not work in the presented version of the program and to copy the plot to the clipboard can be performed only using Alt PrintScreen buttons combination The button New Window opens new separate window for the same plot The separate window for each plot can be opened by double click of the mouse left button when the cursor is inside the plot region Example of such a separate window is presented in the Fig 8 5 All the points of the curves are displayed in both windows and the separate window for plot can be edited like usual 2D plot 135 5 m pam E z a S m E Lu Time sec Fig 8 5 Separate window for 2D plot The number in the edit window Legend Width determines the width of the legend field which is placed in the
11. the number of last position minus 1 The edit window minus at the panel Column Number of Lattice Functions is used only in the specific case of the lattice calculation with MAD program and usually this value is to be equal to 1 The dispersion function is multiplied with this value In the case when there is incorrect symbol in the range of positions between From to Up to BOLIDE system generates the Warning window Fig 4 5 1 BOLIDE Warnings oO x Lattice file has bad value Row Lattice 1 Fig 4 5 Warning window The parameter Row indicates the string number with incorrect symbol parameter Lattice indicates the parameter number in accordance with numbering at the Column Number of Lattice Functions panel In the case of warning one need to correct the numbers in the edit windows From and or Up to in the corresponding string of the Column Number of Lattice Functions panel When the lattice parameters are loaded successfully the plots of corresponding lattice functions versus longitudinal co ordinate in meters are outputted in the 2D plots in the right part of the form Using the button Save at the Column Number of Lattice Functions panel one can save current numbers in the edit windows of the form in a IIf file 109 5 Main menu Effects 5 1 Effects gt ECool This sub item calls the form for input and analysis of electron cooling system parameters Fig 5 1 Electron Cooling E loj x Circu
12. 1 0E 34 D e le 0 3A E 5 0E 33 0 0E 00 0 50 100 150 200 250 300 Time sec Fig 4 14 Luminosity time dependencies at different electron beam current beam energy is 1 GeV pellet diameter is 80 um At the electron beam current of 0 3 A the beam heating due to interaction with the target prevails on the cooling and beam emittance increases during experiment but not so fast as without cooling This deceleration of the emittance growth gives some gain in the luminosity life time in accordance with formula 4 4 At the electron beam current of 0 4 and 0 5 A the electron cooling completely suppresses the emittance growth and at the first stage of experiment about 20 sec the beam emittance decreases to the equilibrium value The equilibrium is determined by the electron beam current and at 0 5 A it corresponds to about 0 62 m mm mrad in the horizontal plane and 0 88 2 mm mrad in the vertical one At the electron beam current of 0 4 A equilibrium emittances have some higher values 0 7 mmm mrad horizontal and 0 98 m mm mrad vertical After reaching the equilibrium luminosity decreases with the life time determined only by single scattering losses in the target These losses is proportional to the probability of the particle cross the target and therefore inversely proportional to the beam emittance Thus the initial gain in the luminosity when the beam emittance decreases is compensated by the shorter luminosity li
13. M M with m is the electron mass and M the projectile mass amp is a quantity which is proportional to the areal density px of the target p target density in g cm MeV cm Z Z 2 1 13 amp p A E 0 1535 2 2 Intrabeam scattering of the ion beam Two modes for the calculations of the IBS growth rate can be used in the present version of the program At smoothed lattice structure without variation of the beta and dispersion functions the rates are calculated by numerical evaluation of corresponding integrals in accordance with Piwinski model 5 For real lattice of the ring the algorithm described in 6 is used 2 2 1 Piwinski model For the smoothed focusing approximation only the mean values of the lattice functions are used and they are determined as follows R i gt D _ gt Q Q Bis O 0 D 0 2 2 1 In accordance with Piwinski model the growth rates are calculated in accordance with the following expressions 70 1 1 do o A f a b c T 20 dt o do D o 1 xB A f RA 5 P f a b c 2 2 2 T Og dt aaa Oig 1 1 do sA LE T 20 dt b b b a where n 1 for a bunched beam and n 2 for an coasting beam rc 64m 6 0 O 0 0 0 BY bunched beam 2 2 3 3a 4 T N and for coasting beam one needs to use a substitution oO C The standard deviations are determined here as follows t Oa
14. is given by the expression 1 qleU o Y 3nG Z Eyr R N j 5 6 G 1 n U is RF voltage q is RF harmonic number Ep and r are the proton rest a energy and classical radius Z is an ion charge number The tune Q which appears due to RF field is AEN die 6 7 BY 2nE Ay here A is the ion atomic number R m s momentum spread is connected to by the formula o 0 5 8 Now for calculation of the bunch length the tune Q is used the value 5 7 is displayed as well as the values 5 1 5 2 and 5 3 in graphic form for estimation of the influence of the beam space charge on the particles motion 86 6 Description of the program The BETACOOL program developed on the base of BOLIDE system using C Builder 4 and works now under Microsoft Windows operation system The BOLIDE version for Unix is under development presently In the program only the standard C and BOLIDE commands are used for connection of algorithmic and interface parts In future the program can be recompiled to Unix version without any modifications The numerical algorithm is realized on the base of Object Oriented Programming method and the program structure consists of several basic objects For input and visual presentation of parameters of each object special forms were developed The tools of the BOLIDE system to load and save data into hard disc for output the data into two and tree dimensional plots during the
15. 2 7 GeV At the same magnetic rigidity the maximum deuteron beam energy is 2 11 GeV Thus in this report the calculations of the beam parameters were fulfilled at target density from 10 to 10 Atoms cm at lower value the experiment can be performed during long time duration without cooling at the beam intensity up to 10 particles and in the beam energy range from 1 GeV to 2 7 GeV for protons and to 2 11 GeV for deuterons The ring acceptance was taken of 10 m m rad in the both planes which is corresponds to expected aperture of the gas storage sell The initial beam parameters were taken approximately corresponded to these ones after acceleration emittances of 10 m m rad in the both planes and momentum spread of 10 2 ANKE experiment without cooling Efficiency of the cooling application at experiment with internal target is determined by relation between different sources of the particle losses Generals of them are the following single scattering on the target atoms on a large angle the same for residual gas atoms emittance growth and aperture limitation momentum spread growth and limitation of longitudinal acceptance The particle losses due to single scattering on the target atoms restricts the experiment duration and can not be effected by the beam cooling Emittance and momentum growth can be compensated by stochastic or electron cooling In this case luminosity life time is determined only by single scatter
16. 33 3 00E 33 2 00E 33 Luminosity 1 cm 2 sec 1 00E 33 0 00E 00 le 0 mle 0 5A Oo 50 100 150 200 250 300 Time sec 4 6 Fig 4 6 Luminosity time dependence at pellet diameter of 40 um upper plot and 50 um bottom plot proton beam energy is 2 7 GeV The gain in the integral luminosity after five minutes of the experiment provided by the cooling application Fig 4 7 4 8 is several times at pellet diameter less than 50 um and decreases to about 10 at bigger pellet diameter At the pellet diameter less than 40 um the electron beam current of 500 mA is not optimal to obtain maximum luminosity that explains peculiarity of the curve in the Fig 4 7 The dependence of the luminosity on the electron beam current will be discussed more detail below 22 At big target density when the cooling only slightly influences on the beam parameters during experiment the cooling system can be used also for preliminary preparation the beam parameters before the target switching on All the previous estimations were performed at the following initial beam parameters emittances in both planes are 1 m mm mrad and momentum spread is 10 The beam parameters corresponding to the equilibrium between electron cooling and intrabeam scattering are the foolowing horizontal emittance is about 0 3 mm mrad vertical emittance 0 1 1 mm mrad and momentum spread is about 8 10 The equilibrium is r
17. 6 Beta Vertical 7 Alpha Vertical Load Lattice lo i i Long coordinate Fig 4 4 The Load Lattice Form All the data from edit windows are saved in the eta file and then are filled after load of eta file The buttons Open and Save at the panel Column Number of Lattice Functions are related to the Ilf files If the same format of Its file is used in the calculations the load of llf file is not necessary 108 The file 1 f contains the Skip Number of Row Column Number of Lattice Functions and minus parameters in the text format The file with lattice parameters Its has to contain in the text format the table of the longitudinal position of the point in the ring and lattice functions in this point Position in the string of each element of the table has to be fixed it is usually takes place for output file of FORTRAN programs The file can also include the strings with comments below and after the table of lattice parameters In this case this strings are skipped when file is loaded and number of string to be skipped are inputted in the edit windows First number of strings below the table and Last number of strings after the table in the Skip Number of Row position The table in the Load Lattice Functions form Column of Lattice Functions contains specification of the Its file From contains the number of initial position minus 1 of the corresponding lattice parameter The column Up to
18. IT I and IIT are shown in Fig 5 2 Coulomb logarithms are defined by the formulae 2 Pee is p E se 5 3 2 p Pr P min Note that if argument of a logarithm is less than 1 then the logarithm value is set to zero The other parameters are R meee V V a 4tn e min 0 0653540 0 003553 gt the maximal impact parameter 5 4 p jo Rie Anis L eB Pr P T the intermediate impact parameter 5 6 L the Larmor radius of electron 5 5 8 y Ze 1 P m B y7 O 0 P min E 112 2 818 107 333 a 6 the minimal impact parameter 5 7 y WED 7 N 4 1 P EN the number of multiple collisions 5 8 T P 0 y JA k 5123 ks 2 kx 5 9 y9 Figure 5 2 Domains in the velocity space in PRF for 0 Vi vy are electron velocity components in PRF 5 1 2 2 Flattened Non magnetized friction force 0 2Le 0 I r 07 HE 0 I 113 a 0 r 2sgn Le a IT 5 11 L 1 n III 076 a 5 1 2 3 Parkhomchuk formulae 66 2 2 5 12 L a 5 13 5 1 2 4 Electron beam space charge For all the formulae angular and momentum spread of electrons are calculated taking into account self field of the electron beam 1 T _ 1 4003 0 By mc By 107 Tinea mrad 5 14 1 3 RONE T IT pa LE aE sate 10 Tyimey mrad 5 15 B m c B BYa ieni el r I r 1
19. Synchrotron tune are visible only in the case when in the Beam Parameters form Fig 4 2 the option Bunched is chosen The 2D graphics are used for output the time dependencies of the general beam parameters The Bunch length and Synchrotron tune are calculated taking into account the synchrotron tune depression due to the beam space charge Formulae 5 5 and 5 8 in ref 1 The Stability presents the values calculated in accordance with the formulae 5 2 5 3 in ref 1 To edit the plot parameters one need push the right button of the mouse when the cursor position is over the corresponding plot The description of the graphic control interface is given in chapter 8 Left mouse double click when the cursor position is over the plot region opens corresponded plot in the separate window 4 4 Parameters Load Lattice This sub item calls the form for input and visualisation of the ring lattice parameters required for intrabeam scattering calculation Fig 4 4 This form can be called also by the menu item Load Lattice in the Ring parameters form Load Lattice Functions First Last Skip Number of Row Column Number of Lattice Functions 4 From Upto Open sawe H 1 Longitudinal Coordinate 2 Beta Horizontal 3 4lpha Horizontal in be o mo T Wo am om Lry Ey 2 AS wy o 1 H 4 Dispersion Horizontal minus AAI 5 Disperion Derivative
20. be calculated using the following procedure The circulation period is divided into n short intervals of time and we assume that during each of them electron temperature is constant The change of emittance during this period of time can be calculated using cooling time calculated for single pass electron beam Se 4 2 17 Teool sin glepass The change of the ion beam temperature is calculated in accordance with 4 2 15 3AT Mer Me F 7 4 2 18 Assuming that the energy obtained by electron beam is divided between degrees of freedom in the equal parts we can calculate new electron temperature 84 N AT AT 3N 2 N AT AT AT era 4 2 19 e At the next period of time the change of emittance is calculated at new electron temperature and so on Finally the cooling time is calculated using the following expression 1 _ 1 E fin T Ein m 4 2 20 T T cool circ c Here in and Efn are the ion beam emittance value at the begining of the circulation period tT and at it s end 5 Calculations of the beam stability characteristics During the numerical solving of the system 1 1 the following parameters characterizing the beam stability are calculated Incoherent tune shift is calculated using well known formula r N l AQ 2m By e je e B where N is the particle number in the case of coasting beam and product of the particle number per bunch and harmoni
21. be introduced into the program as input parameters We assume that the circulating beam crosses the target during multiturn injection Marget times and the injection pulses follow with the repetition period is Targen Then one can calculate the emittance growth times Ny aro ot NE Ld Mie py 2 1 2 Vi arg et i targ et where P e is the probability of the target crossing by an individual particle see below Formula 2 1 3 The parameters Marget and Target are introduced to have a possibility to calculate beam parameters variation at charge exchange injection when the stored beam crosses the target only during injection period and is cooled between the cycles of injection In the case of an experiment with internal target the particle crosses the target every turn therefore Marget 1 Trarger Trey in this case T ey is the particle revolution period The probability P is introduced to take into account a situation when the target cross section is less than that one of the circulating beam The value P can be estimated as a ratio 68 between beam cross section and the target one without any assumption about real geometry of the target point 1 if S lt S 215 P e z if osos 2 1 3 b where S is the target cross section Sp MOno Overr is the r m s beam cross section Onor and O o Jo DA D are the horizontal and vertical r m s beam dimensions Particle lifetime due to interaction with a ta
22. beam magnetization is an efficient way of electron cooling rate increase using of longitudinal magnetic field 6 for transportation of the electron beam from the gun cathode to the collector through drift chambers including cooling section The criterion of electron beam 33 magnetization has a clear physics meaning radius of electron Larmor spiral in magnetic field has to be smaller of the mean distance between electrons 6 13 p n when gt Bum frim le 3 5 1 Pynra ec Here ne is electron density in the electron rest frame Je electron beam current e m 1 2 i i electron charge and mass Bc electron velocity Y 7 B 2 Computer simulations with single pass electron beam show significant up to several times growth of cooling rate when B Bmin and its saturation at B gt Bmin When electron beam is magnetized electron drift caused by its own electric and guiding magnetic fields is sufficiently suppressed and does not exceed electron thermo velocity if 2I m B gt B cn 2 gt min2 Bya T 5 Comparing the criteria 3 and 4 one can see that Bning gt Bmin2 when Byce la T 3 2 ite r ye 5 3 PT A a a E a where re is electron classic radius For COSY electron cooling system this condition is satisfied in the total range of the electron beam current required for effective cooling at minimum beam energy and minimum beam temperature At reasonable electron beam transverse tem
23. calculations and to control of the calculation process are used Below we discuss briefly general objects of the program Ring parameters iofs Load lattice lon kind Lattice parameters RF system Vacuum Input parameters Atomic number 738 Charge number faz Energy fioo Mevr amu Life time 100 j sec Relativistic factors 2 066038087 B a afo0657142 Fig 6 1 The form for input and output of the parameters of the storage ring in numerical format The object Ring includes the general parameters of the storage ring methods of import of the parameters from external file and control of its validity methods for calculation of required parameter The parameters are divided by several groups in accordance with general systems of the storage ring parameters of the stored ions lattice parameters mean and 87 imported from external file radio frequency and vacuum system parameters View of the form for input of the ring parameters is presented in the Fig 6 1 The object Beam includes general beam parameters particle number values of the emittances and momentum spread methods for calculation of beam parameters characterizing beam stability and luminosity for general parameters visualization in numerical and graphic form The methods use only ring parameters as input Fig 6 2 6 3 present the forms for input and output of the beam parameters Beam Parameters OF x fle Show Luminosity
24. construction now 51 1 Vacuum chamber 2 Kicker plates 3 Feedthroughs Conductors Vacuum chamber Fig 7 4 Schematics of the helical quadrupole winding 52 7 2 2 Study of the circulating beam dynamics Stability diagram in the vicinity of the working point at injection energy of 10 keV is presented schematically in Fig 7 5 The most dangerous resonances are the integer 342 and 343 and half integer 342 5 ones of the fast mode of oscillations the half integer and 1 3 nonlinear of slow mode of oscillations and the coupling resonance of the modes Q fase Qsow 342 Nonlinear resonances of the fast mode of oscillations have very high order and estimations show that they practically do not influence on beam parameters An accurate calculations of the position of the resonances and increments in their centre were performed using especially elaborated computer code BETATRON Power of nonlinear resonance of slow mode of oscillations was calculated using averaging method The nonlinear resonance appears due to sextupole component of the longitudinal magnetic field in the toroidal sections and leads to variation in time of the motion invariant corresponding to the slow mode of oscillation Due to coupling between horizontal and vertical degrees of freedom the variation of slow invariant produces variation in time of the beam cross section In experiment the power of the resonance can be estimated by measurements of the frequ
25. described in accordance with 1 The tools of the BOLIDE system and Windows for loading and saving data to hard disk for output the data into two and tree dimensional plots during the calculations and for controlling the calculation process are used BOLIDE Beam Optics Library amp Interface Development Environment is the kit of program modules developed for quick creation of new applications in Borland C Builder3 under Windows95 98 NT in physical and mathematical applications The main task of this pack is relief of functioning working the physicists and mathematicians during creation of new programs Herewith as ready modules are such possibilities like creation of 2d and 3d graphs management by parallel processes loading and saving of data into files that with the power of visual applications development in the ambience of Borland C Builder3 allows quickly to create and easy develop program software More detail information you can get in the LEPTA web site http nuweb jinr ru lepta bolide htm To start the calculations with BETACOOL please follow next steps start the BetaCool exe file load input parameters from eta file and modify them if necessary using the menu items Parametrs and Effects load the ion ring lattice parameters from Its file which specification can be loaded from the Ilf file specify the effects taken into account in the calculations in the visual form Rates input the step of the integr
26. determine the variable along which the cross section is made Using the edit window Line number one can introduce the variable value in which the cross section is made When the parameter Integral is checked the plot in the tab sheet shows the sum of all the points with the same co ordinate of the surface The 2D plot in the tab sheet is usual 2D graph and its parameters can be controlled as described in the previous chapter For instance the cross section can be displayed in the separate window Fig 8 11 saved or loaded like 2D curve 138 1001 Surface Surfacell Axis Grids Other Surfaces Values Cross ftom shift angle mrad Force mradfurn 1E 10 ae Force mradfurn 1E 9 angle mrad Fig 8 11 New window for cross section plot 8 3 Process control The upper button in the process panel of the Dynamics form chapter calls the Process control window The process control service was elaborated in the frame of BOLIDE system for programs using several separate processes during calculations In the presented version of BETACOOL only one process is used and optimisation of the process parameters is not 139 necessary The button Kill can be used to stop the calculations in the case of cycling the program when it is not stopped by the button Stop in the Dynamics form ed fed Process type p Thread Priorit f Thread Lowest Timer Vee Kill
27. number of divisions is used for integration over the synchrotron phase In the case of coasting beam deviation of longitudinal invariant is calculated through averaging the deviation at plus and minus r m s values The Gaussian beam corresponds to 4 2 8 and 4 2 9 In this case one need to specify the Number of integration steps over invariant parameter which is equal to number of divisions over invariant in the integral 4 2 8 When the parameter Include diffusion is checked the formulae 4 2 3 4 2 5 are calculated completely in the opposite case the terms proportional to square of ion angle deviation are excluded When the parameter Circulating beam is checked the cooling rates are calculated in accordance with chapter 4 2 2 of ref 1 and electron ring parameters are to be entered using menu item Circulating electron beam see next chapter 118 The parameter Coupled motion is not used in the calculations in this version of the program 5 1 4 Menu item Circulating electron beam In presented version of the program only the model corresponded to the model of magnetised electron beam with flattened velocity distribution is realised chapter 4 2 2 ref 1 Therefore the first two parameters in the form Fig 5 5 Maxwelian plasma Flattened distribution are not used Other parameters are listed in the Table 5 3 Electron ring parameters E Masvelian Plasma Flattened distribution Electron fing circumfere
28. of 1 GeV and decreases to 50 at 2 11 GeV In the experiment with cluster beam target additional gain in the luminosity can be obtained due to stabilisation of the horizontal beam dimension and its value is about 1 5 times In the experiment with the pellet target maximum gain in the luminosity is expected at minimum beam energy In this case the experiment duration can be about 5 minutes and gain in the integral luminosity is 2 4 times depending on the pellet diameter The expected average luminosity value exceeds 10 cm sec At these parameters one can realise the regime with permanent luminosity during the time of experiment via compensation of the particle losses by corresponding decrease of the beam emittance At maximum proton beam energy of 2 7 GeV the electron cooling compensates the beam heating due to interaction with target only at pellet diameter of 40 50 um which corresponds to the luminosity value of 3 5 10 cm sec At maximum pellet diameter the electron cooling does not suppress the beam heating but it can be used for initial preparation of the beam parameters before the target switching on In combination with the optimum experiment strategy it can give a gain in the integral luminosity of about 20 It should be noted that all the simulations in this report were performed under assumption that each moment of time one pellet is in the beam In the case of low frequency of the pellet production the effective target
29. of 500 mA and gain in the integral luminosity after five minutes of the experiment is about 50 Fig 4 18 1 0E 35 1 0E 34 Luminosity cm4 2 sec4 1 1 0E 33 0 150 Time sec 30 Fig 4 18 Luminosity time dependence at different electron beam current Deuteron beam energy is 2 11 GeV the pellet diameter is 80 um 1 9E 36 1 8E 36 1 7E 36 1 6E 36 1 5E 36 1 4E 36 1 3E 36 1 2E 36 y lt z gt m S E 3 T D _ S 1 1E 36 1 0E 36 0 2 0 3 Electron current A Fig 4 19 Integral luminosity after five minutes of experiment as function of electron beam current Deuteron beam energy is 2 11 GeV the pellet diameter is 80 um At less deuteron beam energy or at less pellet diameter the maximum gain in the integral luminosity can be achieved at less electron beam current and as in the case with the proton beam maximum expected gain in the luminosity can be 3 4 times 4 3 Possible development of the existing electron cooling system There are two possible design of the new cooling system Traditional configuration of cooling system HV accelerator Cooling system with circulating electron beam Traditional configuration of the cooling system can provide electron beam at low transverse and longitudinal temperature and the presented numerical results demonstrate its ability for ion beam parameters stabilization Elect
30. of the electron beam during the period required for beam circulation At maximum electron beam current the relaxation time is less than circulation period by about three orders of magnitude It means that this process has to be taken into account in estimations of the cooling system efficiency The formula 5 16 describing the electron beam relaxation is half empirical one and due to importance of this process for cooling system with circulating electron beam has to be carefully tested with the real installation The process of the transverse longitudinal relaxation in the cooling system with circulating electron beam will be experimentally investigated at LEPTA ring see chapter 7 42 Relaxation time sec 1 0 1 0 01 0 001 110 4 0 01 b 3 0 001 2 1 10 4 1 10 gt 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 1 Transverse temperature eV Fig 5 7 Characteristic time of the transverse longitudinal relaxation of the electron beam magnetic field is 1 kG 1 1 5 kG 2 2 kG 3 longitudinal temperature is 50 meV electron beam current is 0 1 a and 0 5 A b 5 4 Electron cooling with circulating electron beam in experiments with internal target In experiments with internal target additional limit appears for the electron beam circulation period It is connected with the coherent energy losses of the ion beam due to interaction with the target Due to energy exchange between the ion beam and circulating electron one th
31. preparation of the proton beam parameters for fast extraction in an experiment with external target The ultimate possible efficiency of the beam cooling application at ANKE experiments is investigated in the beginning of this report The existing COSY stochastic cooling system is successfully used now for luminosity preservation in the COSY 11 experiment with cluster beam apparatus at relatively low proton beam intensity Expected target density in ANKE experiment is about one order of magnitude higher and one plans to increase the beam intensity by upgrade the COSY injection system The investigation of the efficiency of the stochastic cooling system at these parameters is one of the goals of this report Efficiency of an electron cooling system in difference with stochastic one does not depend on beam intensity but depends very strongly on beam energy Comparison between two cooling methods is the next goal of this work The optimum choice of the electron cooling system parameters and preliminary design of the cooling system with circulating electron beam are also the topics of the report Further development of this work is experimental investigations of the electron cooling system with circulating electron beam We plan to investigate its problems at LEPTA Low Energy Particle Toroidal Accumulator ring which is under construction in JINR now LEPTA parameters are closed to estimated parameters of COSY electron cooling system This report cont
32. ratio of the cooling section length to the circumference of the particle ring Lc Coulomb logarithm ne the electron density t current time in Laboratory reference frame This approach leads to Budker s formula for friction force 13 To solve the system 4 2 10 4 2 11 we have to introduce two additional parameters of the cooler electron ring circumference C and period of electron circulation te During the electron circulation period the electron beam temperature increases due to relaxation with ion beam after that the electron ring is filled up with new portion of cold electrons The number of electrons in the electron ring is calculated using the expression CI N Bc 4 2 12 Initial ion beam temperature is to be calculated from given values of the transverse and longitudinal emittances of the ion beam The relation between the mean particle kinetic energy and emittance depends on emittance definition used in the calculations Here we use the standard deviation of the particle distribution function in the invariant space as an 82 emittance definition The corresponding phase space contains 63 of particles and relates with r m s beam dimensions in accordance with 4 2 8 The mean kinetic energy for single transverse degree of freedom in the particle rest frame is connected with the beam emittance by the following relations 2 4E e 20 B Px plea oh 4 2 13 pyme wh
33. sheet Friction force and push the button Show force shape The parameter Distance from the bunch center is required in the case of bunched ion beam when the parameter Ion beam space charge is checked 116 Force mradturn Om CO angle mrad Fig 5 3 The 3D plots of the friction force shape Note The Derbenev Skrinsky formula was used in the previous versions of the program Other formulae were not well tested yet 5 1 3 Tab sheet Calculation parameters The parameters required for invariant deviation are inputted in the tab sheet Calculation parameters Fig 5 4 117 f Electron Cooling Mis x Circulating electron beam Cooler parameters Lattice functions Friction force Calculation parameters Single particle f Goussian beam M Include diffusion IY Electron capture Number of integration steps over phase Number of integration steps over invariant P Circulating beam I Coupled motion Fig 5 4 The Electron Cooling form tab sheet Calculation parameters The choice between Single particle and Gaussian beam determines the regime of the friction force averaging The case Single particle corresponds to formula 4 2 6 for invariant deviation and formula 4 2 7 for cooling rate ref 1 In this case one need to input Number of integration steps over phase parameter which is equal to number of divisions over betatron phase in the integral 4 2 6 In the case of bunched beam the same
34. system and helical quadrupole winding 7 2 2 Study of the circulating beam dynamics Summary 1 Role of the cooling in experiments with internal targets 2 The possibility of the existing stochastic cooling system application 3 The possibility of an electron cooling application 4 Electron cooling system with circulating beam Conclusion and recommendations References 13 16 16 20 30 32 32 33 35 35 38 40 41 44 45 47 47 48 48 51 53 53 54 55 55 56 58 Abstract This report is dedicated to investigation of the beam parameter evolution in the experiments with internal target In calculations of the proton and deuteron beams we concentrated on cluster atomic beam storage cell and pellet targets at ANKE experiment mainly In these calculations electron and stochastic cooling intrabeam scattering scattering on the target and residual gas atoms are taken into account Beam parameter evolution is investigated in the long term time scale up to one hour at different beam energies in the range from 1 0 to 2 7 GeV for proton beam and from 1 to 2 11 GeV for deuteron beam The results of numerical simulations of the proton and deuteron beam parameters at different energies obtained using new version of BETACOOL program elaborated at the first stage of this work 1 are presented Optimum parameters of the electron cooling system are estimated The COSY experiment requirements can be satisfied even w
35. that emittance and momentum spread slowly decrease with decrease of the particle number due to losses in the target 16 10 Emittance pi mm mrad 1 0 150 300 Time sec os a fo Oo w E E fo w G 5 0 150 300 Time sec Fig 4 1 Proton beam parameter time dependencies Proton number is 10 energy is 1 GeV target density is 10 Atoms cm Electron beam current is 1 5 A 10 Emittance pi mm mrad 0 150 300 Time sec Fig 4 2 a Proton beam emittance time dependencies Proton number is 10 energy is 2 7 GeV target density is 10 Atoms cm Electron beam current is 3 A 17 Momentum spread 0 1 0 01 o 150 300 Time sec Fig 4 2 b Proton beam momentum spread time dependencies Proton number is 10 energy is 2 7 GeV target density is 10 Atoms cm Electron beam current is 3 A 3 5 3 2 5 2 1 5 1 0 5 0 5 1 1 5 2 2 5 3 Beam energy GeV Fig 4 3 Equilibrium proton beam parameters as a function of the beam energy 1 horizontal emittance in 10 mT m rad 2 vertical emittance in 10 T m rad 3 momentum spread in 10 Proton number is 10 target d
36. the beam heating in all degrees of freedom is of the order of a few Amperes Fig 4 1 presents the emittance and momentum spread evolution during 5 minutes of experiment at electron beam current of 1 5 A After the relaxation of the beam parameters at initial stage the emittances and momentum spread keep to be the constant values during long time and the particle losses are determined only by the single scattering on the target atoms In this range of parameters electron cooling application have a maximum efficiency and after five minutes of experiment it allows to have proton beam of 10 times higher intensity than without cooling At maximum proton energy 2 7 GeV even maximum electron beam current can not provide the stabilisation of the emittance and momentum growth However at electron beam current of 3 A Fig 4 2 the heating processes are substantially suppressed and gain in the particle number after five minutes of experiment is about 20 which is close to maximum achievable value see Fig 2 5 The experiment at high target density will be considered more detail in the next chapter At lower target density the electron cooling stabilises the proton beam parameters in the total energy range In the Fig 4 3 the equilibrium beam parameters as a function of beam energy are presented The electron beam current was chosen to obtain equilibrium proton beam emittance of about 10 m m rad The equilibrium is reached during about 10 minutes and after
37. the effective target density depends on the beam cross section and as a result its value is changed when the beam emittance is varied Correspondingly the efficiency of electron cooling system can be substantially higher in the case when the cooling suppresses the heating effects In the previous chapter the electron cooling efficiency in the experiment with high target density was considered only from the side of the suppression the particle losses due to acceptance limitation and effective target density was assumed to be constant In this chapter we investigate the experiment with the pellet target taking into account variation of the effective target density In this chapter the parameters of the electron cooling system presented in the Table 5 chapter 5 are used in the simulations The proton beam intensity is 10 particles which corresponds to the expected value after installation of new injection system The luminosity of the experiment with pellet target is determined by the expression pap 4 1 ST t rev where N is the particle number in the beam T ey revolution period N is particle number in the target S target cross section P probability of the particle cross the target Assuming that the pellets cross the beam one by one with the period equal to the time duration of the beam crossing by a pellet one can estimate the probability as the following 2 3 4 2 S i T BEBE where Bxy horizontal and vertic
38. up to one order of magnitude at 1 GeV and about 50 at energy of 2 7 GeV For circulating deuteron beam the gain in intensity of about 50 can be obtained at energy of 2 11 GeV which corresponds to the maximum magnetic rigidity At intermediate energies the gain in the beam intensity is determined by suppression of the intrabeam scattering and for 55 deuteron beam is about two times less than the proton one however at energy of 1 GeV the possible gain in the luminosity for deuteron beam is about 5 times Similar dependence on the particle energy takes place at the target density of 10 Atoms cm but maximum expected gain in the beam intensity after five minutes of experiment is about 1 5 times higher The energy dependence of the maximum gain in the experiment luminosity is explained by the input of the intrabeam scattering into emittance growth The role of this process fast decreases with the energy increase and at maximum beam energy emittance growth is determined by the single scattering with the target atoms mainly Thus the potential efficiency of a beam cooling increases with increase of the beam intensity and decreases with increase of the beam energy In the experiment with cluster beam target additional gain in the luminosity can be obtained due to the target dimension in the horizontal plane is 5 mm which is less than beam dimension corresponded to acceptance limitation Stabilisation of the horizontal beam dimension du
39. with the target and energy exchange between beams heating of the longitudinal degree of freedom of the electron beam due to intrabeam scattering In the case of induction acceleration of the electron beam more serious problem is to avoid beam heating during crossing of resonances of transverse particle motion The period of the electron beam circulation is limited by the heating due to interaction with the ion beam When the circulation period is about 0 1 second the efficiency of proposed system is the same as one of the traditional electron cooling system More serious limitation can appear due to transverse longitudinal relaxation in the electron beam This problem as well as the stability of the electron motion and electron beam distortion during injection requires experimental investigations They can be performed at LEPTA installation which construction is in the final stage in JINR We plan to begin the experimental program next year and expected term of its completion is two years The cost of the cooling system with circulating electron beam estimated using the experience of construction of analogous system in JINR can lies between 1 and 1 5 M Conclusion and recommendations In the experiments with gas storage cell and cluster beam targets an electron cooling system can be effectively used for the luminosity preservation when the experiment duration is of the order of 1 hour In this case expected gain in the luminosity is about 2
40. 0 MeV 1 and 2000 MeV 2 45 5 5 Induction acceleration of the electron beam The electron acceleration can be provided using RF linac with further injection into electron ring or inside the electron ring using low frequency betatron acceleration In the second case the energy spread of the accelerated electron beam caused by variations of the acceleration voltage during revolution period is equal to 12 ov dag e ar Trev fin gt 5 21 Ag pete ot where V is the accelerating voltage Trey is the electron revolution period When V Vosin ot and initial and final phases of acceleration are symmetrically disposed around the voltage maximum this formula gives us AE 2 l pe 5 22 Bini B In this case the acceleration voltage amplitude can be obtained by integration of the motion equation Co eV gt c 2B cos t 5 23 where C is the electron ring circumference Binj B initial and average electron velocity in the units of the speed of light Using the existing COSY high voltage power supply for injection one can accelerate electrons 0 5 msec from 100 keV injection up to 1 5 MeV The estimated momentum spread value caused by the voltage variation is negligible Table 5 2 Table 5 2 Parameters of the induction acceleration system Injection energy keV 100 Maximum energy keV 1500 Electron ring circumference m 20 Frequency of the accelerat
41. 1 File Open 3 2 File Save 3 3 File Save as 3 4 File Save On Exit 3 5 File Save DeskTop 3 6 File Graf as Data 3 7 File Open Dialog 3 8 File Exit 4 Main menu Parameters 4 1 Parameters Ring 4 2 Parameters Beam 4 3 Parameters Beam evolution 4 4 Parameters Load Lattice 5 Main menu gt Effects 5 1 Effects ECool 5 1 1 Tab sheets Cooler parameters and Lattice functions 5 1 2 Tab sheet Friction force 5 1 3 Tab sheet Calculation parameters 5 1 4 Menu item Circulating electron beam 5 1 5 Effects ECool Circulating electron beam Show temperature button 5 2 Effects IBS 5 3 Effects RestGas 5 4 Effects Stochastic 5 5 Effects Target 5 5 1 Target form 5 5 2 Target material form 5 6 Effects Injection 5 7 Effects External heating 6 Main menu Task 6 1 Task gt Rates 6 2 Task Dynamics 7 Main Menu gt Tools 8 Graphics and process parameters 8 1 2D graph 8 2 3D graph 8 3 Process control 93 PYRILIA AEDE OT BG eh eK ey eS RBRWW WWW WWNNNNNDND WN NON N NNR rR TOANANAAAMN Introduction BETACOOL program developed on the base of BOLIDE system using C Builder 4 is the Windows 9x NT application dedicated to calculate an ion beam parameter evolution in a storage ring under the common action of different heating and cooling effects The description of the physical model using in the calculation is in the ref 1 All the parameters and references to the formulae in this manual are
42. 1 the ion coordinates are kept constant during ion passing the cooling section 2 the motion of each particle is described by two Courant Snyder invariants 11 1 0 7 7 R is 20 ig ip Bites i x Z 4 2 1 i I where x z are horizontal and vertical co ordinates and B are alpha and beta functions in the cooling section The invariant corresponding to synchrotron motion can be introduced in accordance with 1 2 as follows 2 Ap 1 coasting beam I m m i p 2 bunched beam max I Hion if Ap P max Op Where is the r m s momentum spread The ion betatron coordinates and momentum inside the cooling section can be calculated in accordance with xp V B sing x 5 cos a sino x The same expressions are used for z co ordinate with substitution of corresponding alpha and beta function value The total value of x co ordinate is equal to x x D Ap p x x D Ap p Ae cosh S S O sing P P max where so is longitudinal coordinate of the bunch center x p p A relative change of the ion momentum components after passing the cooling section can be expressed with the following formula re r Op x z s sx z Ap p 4 2 2 where the functions can be calculated in the BETACOOL program using one of three different analytical formulae for friction force of non magnetized 9 or magnetized electron beam 9 10 User of the program can
43. 800 3600 Time sec a Li o EL x eo a E D o D 0 1800 3600 Time sec Fig 3 4 Emittance 1 horizontal 2 vertical and momentum spread evolution Target density is 10 Atoms cm beam energy 2 7 GeV initial beam intensity is 2 10 protons 15 4 Electron cooling application 4 1 Beam parameter evolution with an electron cooling The calculations in this chapter were performed at the electron beam and solenoid parameters corresponding to the existing COSY electron cooling system Table 4 1 Longitudinal and transverse temperature were chosen using the results of the friction force measurements performed at COSY in May 2001 Only the maximum electron beam energy is varied in accordance with the proton energy Table 4 1 Electron cooling system parameters Effective length of the cooling section m 1 4 Maximum magnetic field value kG 1 2 Maximum value of electron beam current A 3 Electron beam radius cm 1 26 Transverse electron temperature me V 300 Longitudinal electron temperature meV 10 Neutralisation factor 30 Beta functions in the cooling section m 13 15 The beam parameter evolution is calculated taking into account all general heating effects interaction with residual gas internal target and intrabeam scattering At minimum proton energy of 1 GeV and target density of 10 Atoms cm the value of the electron beam current required for compensation of
44. Electron cooling application for luminosity preservation in an experiment with internal targets at COSY Final report JINR I N Meshkov A O Sidorin A V Smirnov G V Trubnikov COSY K Fan R Maier D Prasuhn H J Stein Dubna 2001 Contents Abstract Introduction 1 Target and beam parameters 2 ANKE experiment without cooling 2 1 Interaction with the target 2 2 Calculations in presence of intrabeam scattering 3 Beam parameter evolution with existing stochastic cooling system 4 4 Electron cooling application 4 1 Beam parameter evolution with an electron cooling 4 2 Electron cooling in experiments with the pellet target 4 3 Possible development of the existing electron cooling system 5 Electron cooling system with circulating electron beam 5 1 Magnetic field limitations 5 2 Principles of electron cooling with circulating electron beam 5 3 Storage ring with longitudinal magnetic field 5 3 1 Particle dynamics in the ring with longitudinal magnetic field 5 3 2 Current limitation due to microwave instability 5 3 3 Transverse longitudinal relaxation of the electron beam 5 4 Electron cooling with circulating electron beam in experiments with internal target 5 4 Induction acceleration of the electron beam 6 Principles of the electron cooling system design 7 Program of experiments at LEPTA 7 1 General parameters of the LEPTA ring 7 2 Preliminary program of experiments 7 2 1 Tuning of injection
45. General parameters Stability characteristics Bunch parameters Farticle number fi ooo0o0000 Bunched f Coasting Horizontal emittance 7 45E 6 pi m tad fi Es v Collider Vertical emittance Momentum spread Jooo a 4 Mean radius f 3967210986 Longitudinal form factor E 432562474 Space charge impedances Longitudinal 324 6032252 Ohm Transverse ai 31067105 Ohm Peak current B 486904675 A Fig 6 2 The form for input and output of the beam parameters in the numerical format 88 Beam Parameters Oy Momentum Bunch lenath Betatran Tune Synchrotron Tune Stability Particle Number Luminosity 1 Emittance p mm mrad Fig 6 3 The form for output of the beam parameters in the graphic format Each process involved into calculation of the beam parameters evolution is performed as an independent object also This object includes the parameters of the corresponding process or device and methods of these parameters optimization and visualization For instance object Electron Cooling includes parameters of the cooling section parameters of the electron beam and lattice functions in the cooling section To optimize cooling time calculations user can choose physical model of the process and mode of the cooler operation The dependence of friction force on ion angular and momentum deviation is output as a three dimensional plot and for mode with c
46. O E ejat D e lt _ e t S 10 100 1000 10000 Circulation period msec Fig 5 3 Ratio of the characteristic cooling time of transverse degree of freedom providing by the cooling system with circulating electron beam to single pass one Proton beam energy is 2 7 GeV 1 and 1 GeV 2 proton number is 10 initial transverse electron temperature is 300 meV longitudinal 50 meV electron beam radius is 0 5 cm current 0 5 A electron ring circumference is 20 m magnetic field is 1 2 kG cooling section length 1 4 m The limitation of the circulation period is stronger at low beam energy In the Fig 5 3 the dependence of the cooling time on circulating period is presented at proton beam energy of 1 GeV and 2 7 GeV 36 At circulation period shorter than 100 msec efficiency of the cooling system with circulating beam is practically the same as the single pass one at minimum energy increase of the cooling time is less than two times The cooling efficiency decreases during the circulation and after the period of time longer than approximately a few seconds further circulation of electrons does not influence practically on the proton beam parameters The circumference of the electron ring determines the total number of electrons and as a result the thermo capacity of the electron beam In principle the electron beam circumference is to be as long as possible Th
47. Tab sheet General parameters Particle number N Number of ions in the case of bunched beam number of particles in the bunch Horizontal emittance Ex 7 m rad Two sigma emittances Vertical emittance z n m rad eB a Oo 5 O is standard deviation Momentum spread Op R m s momentum spread Calculated parameters Mean radius a cm Mean beam radius is used for calculation of longitudinal form factor GL chapter 5 in ref 1 Longitudinal form factor GL chapter 5 in ref 1 Space charge impedance Longitudinal ZL Ohm Is used for Keil Schnell criterion calculation formula 5 2 in ref 1 Transverse Zi Ohm m Is used for Schnell Zotter criterion calculation formula 5 2 in ref 1 Peak current Imax A Is calculated taking into account bunching factor for coasting beam it is equal to mean value Table 4 6 Tab sheet Stability characteristics Image force correction Is used for tune shift calculation 2 factor if this effect is not taken into account the image force correction factor is to be equal 1 Tune shift AQ Formula 5 1 in ref 1 Factor of distribution FL Is used for Keil Schnell criterion function for Keil Schnell calculation formula 5 2 in ref 1 Fl 1 Longitudinal coupling ZL Ohm This value is added to space charge impedance longitudinal impedance Microvawe instability Formula 5 2 in ref 1 Tune spread for first mode AQn Is
48. ains preliminary program of experiments at LEPTA which are aimed to study an electron cooling system with circulating electron beam Electron beam parameters after injection and adjustment of the septum and kicker coils will be measured at special test bench using optic method of the electron beam diagnostics developed in JINR 3 Dynamics of circulating electron beam will be investigated when LEPTA assembling is finished Resonance behaviour and beam stability at injection energy will be studied using RF acceleration of the electron beam And final test of the cooling system will be performed after installation of the betatron yoke It will include the injection of electron beam its acceleration to maximum energy and extraction for diagnostic of the electron temperature Presently the construction of the general elements of LEPTA magnetic system is in the final stage and we plan to start assembling of the ring and first experiments with electron beam in the next year Expected term of the experimental program completion is two years 1 Target and beam parameters The internal targets used in COSY are listed in the Table 1 1 Table 1 1 Experiments with internal targets Target EDDA COSY 11 ANKE 1 Strip or filament filament strip 2 Cluster x x 3 Pellet X 4 Atomic beam x x 5 Storage cell x x The beam life time in the experiments with solid targets is so short that luminosity preservation using the be
49. al shape of the stability diagram In standard Keil Schnel criterium Fong 1 However when beam energy is less than critical one and momentum distribution function has a long tails Fiong Value can be significantly larger than unit In this case with a good accuracy we can write Fiong 299 207 5 14 40 where Oo is a dynamical aperture of the machine on momentum deviation If the crossing of the half integer resonance is impossible Oo is limited by the condition 5 10 The impedance for cylindrical beam and vacuum chamber is described with the formula 377 b 1 2In mt nf 5 15 Z n n where b and a are the radii of vacuum chamber and electron beam This formula gives us about 250 Ohms at minimum beam energy For described focusing structure the transition energy of the ring can be estimated as yi 20 fast Q slow 8 The sign of Ver is determined by the directions of the longitudinal magnetic field and the qudrupole helical winding rotation what permits to us choose the regime with gt 0 no negative mass instability in the total energy range In this case the instability takes place due to finite conductivity of the vacuum chamber walls The threshold current 5 13 as a function of beam energy and longitudinal magnetic field is presented in the Fig 5 6 the tune value of the slow mode of oscillations was chosen to be equal 0 2 in the total energy range The threshold current decreases with the inc
50. al beta functions in the target position dispersion in the target position is closed to zero x corresponding two sigma emittances As a result we have L NON 4 3 as BeBe 20 The pellet diameter lies in the range from 20 to 80 um the target density is about 1 4 10 Atoms cm Target parameters and luminosity of the experiment are presented in the Table 4 2 The luminosity is calculated under assumption that one pellet is in the beam beam emittance is 1 m mm mrad in the both transverse planes the proton beam intensity is 10 particles and beam energy is 2 7 GeV The luminosity life time is equal to ldL_ 1 dN 1de 1 4 Ld N dt dt dt x y 4 4 Thus in difference with the case of gas storage cell target the luminosity life time depends not only on the particle life time but also on the emittance variation At optimum experiment conditions the electron cooling can provide practically constant luminosity during long period of time by corresponding choice of the electron beam current In the Fig 4 6 the luminosity of the experiment at maximum beam energy is presented for pellet diameter of 40 um and 50 um Table 4 2 The pellet target parameters Pellet diameter Number of Target cross Target length Luminosyty um particles section cm2 cm cm 2 sec 1 20 1 8 10 3 14 10 0 00157 3 66 10 30 5 94 10 4 7 07 10 0 00236 1 21 10 40 1 43 10 1 26 10
51. ale Beam scrapers Fig 6 2 The Dynamics form In order to start the calculations one need to push the button Run in the Process control panel When the calculations are started in the button Set of the panel the time of calculations is indicated The button Pause stops the calculations with keeping all the current data If you push the Run button after Pause the calculations will be prolonged from the stop point Button Stop stops the calculations and sets all the parameters to their initial values Note If calculations were stopped the values of characteristic rates will not be recalculated To have a correct initial values of the rates at the first step of integration it is better to push the button OK in the Rates form before push the button Run For stable work of the numerical algorithm the step of numerical integration has to be small enough in comparison with the characteristic times of the beam parameters evolution In the case when initial step has very big value the program automatically divides it by 2 and makes this procedure fixed number of times If after that the step is big yet the window about numerical mistake Fig 6 3 and BOLIDE Warning window Fig 6 4 will appear BETACOOL EXE 3 grt DOMAIN error Fig 6 3 Mistake window 1 BOLIDE Warnings iof x feet TPoc 22 Fig 6 4 BOLIDE Warning window In this case you need to 128 push the button OK in the Mistake window push the
52. am cooling seems to be unrealistic All the other types of the target will be used in the ANKE experiment and the results of the calculations for them can be extended to other experiments with corresponding corrections the lattice functions in the target position Therefore in this report general attention was directed to the ANKE experiment The cluster beam apparatus of ANKE experiment is very similar to COSY 11 one and maximum designed value of target areal density was equal to 5 x 10 atoms cm However the value really achieved now is substantially higher and can be estimated from of the particle energy loss due to interaction with target using the experiment parameters Table 1 2 The formula for relation between relative energy loss of the proton beam and proton revolution frequency one can calculate the AE value AE 1 y1Af 1 1 E ynf here n is off momentum factor y is Lorenz factor Table 1 2 ANKE experiment with hydrogen cluster beam target at January February 2001 Proton beam kinetic energy GeV 2 65 Revolution frequency MHz 1 5 Off momentum factor n 0 13 Frequency shift during 300 seconds Hz 192 Theoretical value of the mean energy loss during single cross the target is 4 E ae 26 nf Eae p 1 2 here Emax is the maximum energy loss in a head on collision of the projectile with a target electron 2R 2 2 z 2m c By l 1 3 max 2 1 2y Tie pie M M me is the electro
53. ameters Fig 8 8 8 10 has similar structure as for 2D graph The tab sheet Others Fig 8 8 includes additional panel in the right upper corner for rotation of the axis and parameter 3D View if it is not checked the projection of the curve is displayed in the graph The 3D surface also can be saved or loaded from file the file extension is sur The file structure is similar to the numerical presentation of the surface in the tab sheet Values Fig 137 8 9 It is the table of the numbers the first column contains x variable values first string contains y variable values all other the function values in the corresponding points 1001 Surface SurfaceD Anis Grids Other Surfaces Values Cross m ns Rotation 7 Lateral a Frontal Filling point color Graphic Font afe Horizontal Copy To Clipboard Land Legen Newwindow width Iv 3D View f Black Color Fig 8 8 3D graph tab sheet Others 1001 Surface SurfaceD Anis Grids Other Surfaces Values Cross axra OOOO e i 0 0 002 3 8496455E 10 1 2079 78E 9 0 004 V 6298944E 10 2 4459664E 9 0 006 1 1348409E 9 3 7711655E 9 0 008 1 5008467E 9 5 2116606E 9 1 8614099E 9 6 7949596E 9 0 012 22168912E 9 9 5414604E 9 4 cant Fig 8 9 3D graph tab sheet Values Additional tab sheet Cross Fig 8 10 is used for displaying the cross sections of the 3D surface The panel Cross on permits to
54. ance Fig 2 1 2 2 show the characteristic times of the single scattering process and emittance growth up to acceptance limit in both planes for proton and deuteron beam The ring acceptance was taken of 10 m m rad in the both planes The beta functions of the ring at the target position were taken of 2 meters in the horizontal plane and of 3 meters in the vertical one The presented values correspond to the hydrogen target at the areal density of 10 Atoms cm and beam emittances of 10 m m rad in the both planes All the characteristic times are linearly scaled with the target density and can be simply recalculated to each required value The difference in the energy dependence of the single and multiple scattering life times appears because of in the single scattering life time calculation the particle losses due to nuclear scattering are taken into account The difference between horizontal and vertical degrees of freedom is explained by the difference in the beta functions in the target position O oOo A Oo O O O O k O O S no om Q O 200 S 4 dm 4 O O 1 2 Beam energy GeV Fig 2 1 The proton beam life time due to single scattering on the target atoms 1 and the times of emittance growth up to the acceptance limit in horizontal 2 and in vertical 3 planes oO Ol 1 e ee 1 1 5 ie Nh O1 O NO
55. arameters Circumference C m Ion ring circumference Gamma transition Ver Lorenz factor corresponded to transition energy Horizontal Q Qx Horizontal betatron tune Vertical Q Q Vertical betatron tune Horizontal Acc Ax m rad Horizontal and vertical acceptances Vertical Acc m rad that are used for particle losses calculation in accordance with ref 1 chapter 1 2 when the option Beam scrapers is chosen in the Dynamics form chapter 6 2 Horizontal Hromaticity Ex Horizontal and vertical Vertical Hromaticity E hromaticities are used for tune spread calculation formula 5 1 in ref 1 Mean ring parameters output parameters Radius R m The mean ring parameters are used Horizontal beta function Bx m for intrabeam rates calculation in Vertical beta function B m accordance with chapter 2 2 1 Dispersion D ii ref 1 when the option Piwinski is chosen in the Intra Beam Scattering form chapter 5 2 Revolution period Trev Sec Off momentum factor n Wy y 102 103 Table 4 3 Tab sheet RF system Input parameters Harmonic number q Are used for synchrotron tune RF voltage U kV calculation formulae 5 6 5 7 in ref 1 Calculated parameters Separatrix length ep m lsep C q Synchrotron tune Q Formula 5 7 in ref 1 Table 4 4 Tab sheet Vacuum Input param
56. ation at the target areal density of about 10 Atoms em up to ion beam intensity of about 2 3 10 particles Maximum efficiency of the stochastic cooling corresponds to the maximum beam energy due to dependence of the mixing factor on the off momentum factor The gain in the luminosity is close to the theoretical value and in the experiment with the gas jet target is about 20 50 in the experiment with cluster beam target this value can be higher by about 1 5 times due to stabilisation of the horizontal beam dimension At the beam intensity of 10 particles which corresponds to expected value with new COSY injection system or at high target density the existing stochastic cooling system is not powerful enough to expect a substantial effect from its application in the experiment with internal target 56 3 The possibility of an electron cooling application Electron cooling system at realistic parameters of the electron beam stabilises the parameters of the intensive up to 10 particles ion beam at the target areal density of 10 Atoms cm in the total energy range required for experiments Maximum gain in the luminosity corresponds to the minimum beam energy In the experiment with gas jet target for the proton beam expected gain in the luminosity after one hour of experiment is 2 3 times at 1 GeV and decrease to the value of 20 50 at 2 7 GeV For deuteron beam expected gain in the luminosity is about 2 times at energy
57. ation over the time in the visual form Dynamics start the calculations using the button Run in the visual form Dynamics the results of the calculations are presented in the visual form Beam parameters time dependencies of general ion beam parameters and in the visual form Rates time dependencies of characteristic times To stop the calculations click the button Stop in the visual form Dynamics The calculations results can be saved in the graphic format using the buttons combination Alt PrintScreen or in the text format using the graphic interface which is described in the chapter 8 of this manual The edited file of input parameters can be saved with new name or automatically saved with the same name after exit of BetaCool exe file The manual includes description of all visual forms of the BETACOOL program all input parameters using in calculations the tools for graphic and calculation process control elaborated in the frame of BOLIDE system 94 1 The BETACOOL files Executed file of the program BetaCool exe 1 14 Mb requires for its work only standard Windows libraries Input parameters for calculations are presented in the text format file The program does not check extension of the input file and in principle the extension can be arbitrary During the load and save processes the file of parameters in the Open and Save as windows the files in the current directory with extension eta are automa
58. ature due to transverse longitudinal relaxation beam parameter distortion after crossing the resonances of the fast mode of oscillation The aim of the investigations is the final choice of the magnetic field value and the optimal position of the working point of circulating beam in the total range of the beam energy accurate measurements of the maximum beam current which can circulate in the ring without loss the quality during required period of time These experiments will be performed at the special test bench during the LEPTA assembling and with circulating electron beam during the ring commissioning 7 2 1 Tuning of injection system and helical quadrupole winding Injection system of the ring consists of magnetic septum Fig 7 2 Table 7 2 and electrostatic kicker Fig 7 3 Variation of the angle of the electron beam trajectory at the entrance and at the exit of the septum and kicker is about 15 mrad and only a resonance optics of these elements permits to keep electron beam quality after injection In this case the integer number of the electron larmor rotations has to take plase at the length of the kicker and septum Calculate of the effective length of the septum and kicker coils accurately is a complicated numerical problem and these values will be measured experimentally using optic method of the electron transverse temperature measurements 50 Fig 7 2 Septum design 1 Beam from the gun 2 Circulating beam 3 Extracted
59. beam 4 Current lines 5 Magnetic field lines Table 7 2 General parameters of the septum coils Magnetic field value G 120 Current A 1600 Number of turns 6 Conductor cross section mm 4x12 Beam displacement mm 300 Initially the effective lengths of both coil fields will be measured independently by measurement of the magnetic field value corresponding to minimum distortion of the beam parameters after crossing the coil After that the length of the kicker coil will be corrected to have an integer ratio between septum and kicker coil lengths The distortion of the angular spread of the circulating electron beam after crossing the helical quadrupole winding is minimised by adiabatic variation of the quadrupole field gradient at the entrance and at the exit of the coil The designed and constructed helical quadrupole coil Fig 7 4 has in each crossection a geometry of Panofsky lens which provides a maximum linearity of the field The gradient variation at the enetrance and exit of the coil is provided by corresponding variation of the number of winding turns Accurate calculation of the particle dynamics in the coil is practically impossible due to difficulties in measurements of the fringe fields and test of the coil and if it would be necessary correction of its design will be performed also using optic method of beam diagnostics All these investigations will be performed at special test bench which is under
60. button Stop in the Dynamics form correct the Step value push the button OK in the Rates form push the button Run to start the calculations again For some values of the Step such situation can take place when mistake is absent but representation into graphics is also absent It means that step value after the automatic decrease has very small value In this case you need to stop the program then to decrease initial Step value and to start the program once again 129 7 Main Menu Tools The submenu item Calculator calls the Form1 form Fig 7 1 The steps of the program are indicated in the panel in the left bottom corner of the form all other functions are similar to the standard Windows calculator Fig 7 1 Programming calculator The sub menu item Periodic table calls the form Mendeleev s Periodic Table of Elements Fig 7 2 Mendeleev s Periodic Table of Elements Ioj x E Be B C N 0 F Ne 3 Na Mg Al Si P S d Ar K Ca Sc Ti V Gr Mn Fe PENi Cu Zn Ga Ge As Se Br Kr ls Rb Sr y Ze Nb Mo Te Ru Rh Pd Ag Cd In Sn Sb Te I Xe cs Ba La Hf Ta W Re Os Ir Pt Au Hg TI Pb Bi Po At Rn 7 Fr Ra Ac Rf Db s gt Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu 9 gt Th Pa U Np Pu AmGm Bk Cf Es Fm Md No Lw Cobalt atomic number 27 atomic mass 58 9332 atomic radius nm 0 125 boiling K 3230 Fig 7 2 Periodic table The sub menu item Constants calls the Physical Constants form Fig 7 3 Un
61. c number in the case of bunching beam B the bunching 5 1 factor relation of average current to peak current As a characteristics of a longitudinal stability of the beam the following parameter is calculated I A i yB i lt 1 5 2 KS 4F _ 4 Z n Z Here F is the factor depending on the form of distribution function and the impedance phase in Keil Schnell criterion it is assumed that F 1 U 938 MV 1 7 1 y Z is the longitudinal coupling impedance at nQ Criterion for transverse beam stability is calculated as follows U sz sr aes 1 lt 1 ZRZI poso 5 3 85 Here the effective spread of betatron tune for n th mode n o AQ is defined by AQ o onsal AQ gt 5 4 Z is transverse coupling impedance at n Q q and amp is the rind chromaticity F is the factor depending on form of distribution function and the impedance phase At the regime of the storage ring with the bunched ion beam the influence of the space charge fields on synchrotron motion is estimated with numerical calculations also For calculation of linear synchrotron tune Q the model of the beam with linear longitudinal space charge field and parabolic dependence of linear charge density on longitudinal coordinate is applied The synchrotron tune is calculated as follows Q 0 J1 N N 5 5 where WN is ion number per bunch its maximal value NV
62. calculations using a special process called DECAY 76 4 Cooling processes involved into calculations 4 1 Stochastic cooling 4 1 1 Transverse cooling time At the optimum gain of the of the feed back system the optimum cooling rate is 2 1 _lde_ W 4 1 1 Ton Edt N M where W is the bandwidth of the cooling system N is the particle number 3 _fo 4 1 2 4W n5 fo is the mean frequency N L a denotes the absolute relative momentum spread of Yr Y the protons 4 1 2 Longitudinal cooling time At the optimum gain of the of the feed back system the optimum cooling rate is the following dH ee aW 4 1 3 dt N 4 1 3 Power limited cooling In the case when optimum cooling time can not be obtained due to power limitation of the amplifiers the cooling time in first approximation is not dependent of particle number and beam parameters To take into account this case in the program the cooling times can be input directly and kept constant during calculations 4 2 Electron cooling 4 2 1 Electron cooling with single pass electron beam Electron cooling rates are calculated in the BETACOOL program through averaging of the action of the friction force inside the electron cooler over phases of betatron and synchrotron oscillations and over Gaussian distribution function of the particles in the space of the motion invariants The cooling rates calculation was performed under the following general assumption
63. cess at injection energy However the technical design of such a system can not be performed without an experimental test of its general elements 59 References 1 Electron cooling application for luminosity preservation in an experiment with internal targets at COSY Interim report Dubna 2001 2 I Meshkov A Sidorin Electron cooling system with circulating electron beam Workshop on Medium Energy Electron Cooling Novosibirsk 1997 p 183 G Jackson Modified Betatron Approach to Electron Cooling Workshop on Medium Energy Electron Cooling Novosibirsk 1997 p 171 3 V Golubev I Meshkov V Polyakov I Seleznev A Smirnov E Syresin The proceeding of the Workshop on Beam Cooling and Related Topics CERN 94 03 26 April 1994 4 F Hinterberger D Prasuhn Analysis of internal target effects in light ion storage ring NIM A 279 1989 413 422 5 D Anderson at al The development of a prototype multi MeV electron cooling system T Ellison at al Multi MeV electron cooling a tool for increasing the performance of high energy colliders Fifth Annual International Industrial Symposium on the Super Collider 6 8 may 1993 San Francisco California 6 I N Meshkov Fiz Elem Chastits At Yadra 25 1994 1487 Engl transl Phys Part Nucl 25 6 1994 631 I Meshkov preprint RIKEN AF AC 2 January 1997 7 Yu V Korotaev I N Meshkov S V Mironov A O Sidorin E Syresin The Modified Betatron Prototype Dedicated to Electron Co
64. cle protons number in the ring is the particle beam emittance in the corresponding transverse plane i x z For longitudinal degree of freedom invariant of motion Hon is given by the following expression AP coasting beam P H 1 2 on 1 A AP g P gt bunched beam p Q dt p In Eq 1 2 upper line corresponds to coasting beam lower line to bunched beam with constant parameters Thus we limit our consideration with the case of small synchrotron oscillations which corresponds to a well cooled beam A change of the synchrotron frequency Q requires to use adiabatic invariant instead of energy Thus depression of the synchrotron tune due to action of the beam space charge is not taken into account during dynamics simulation in the present consideration Characteristic times are functions of all three emittances and particle number and have positive sign for a heating process and negative for cooling one The negative sign of the lifetime corresponds to the particle loss and the sign of the lifetime can be positive in the presence of particle injection when particle number increases Index j in the system 1 1 is the number of process involved into calculations The program structure is designed in such a way that permits to include into calculation any process which can be described with cooling or heating rates Numerical solution of the system 1 1 is performed using Euler method with automatic step variation T
65. cm after five minutes of experiment without cooling as function of initial beam emittance beam energy is 2 7 GeV pellet diameter is 80 um In order to obtain maximum integral luminosity one can to optimise the experiment duration also The integral luminosity increases approximately as a square root from the experiment duration Fig 4 10 Maximum gain in the luminosity obtained due to beam cooling before experiment corresponds to short term of the experiment and it decreases with increase of the experiment duration Fig 4 11 1 80E 36 1 60E 36 N lt E 140E 36 T 1 20E 36 ar O 1 00E 36 0 3 7 8 00E 35 mr 15 mg 600E 35 ie D 4 00E 35 A 2 00E 35 HA 0 00E 00 oO a oO 100 150 200 250 3 Time sec Q 0 Fig 4 10 Integral Luminosity as function of the experiment duration depending on initial beam emittance in 7 mm mrad beam energy is 2 7 GeV pellet diameter is 80 um cooling is off 24 Thus even at big target density when the cooling power is not enough to suppress the heating due to interaction with the target the experiment luminosity can be improved by preparation the beam parameters before the target switching on and by optimization of the experiment scenario A w a oe N a o Gain in integral luminosity N a 1 10 100 1000 Time sec Fig 4 11 The ratio between t
66. cooling process after refresh the electron beam the coherent momentum shift of the ion beam after circulation period has to be substantially less than its momentum spread Only in this case the model using for cooling process simulation gives correct results for the system with circulating electron beam The condition when the used numerical model is valid for calculation of the cooling time for the system with circulating electron beam can be expressed in the following form 2T o EIA Teire l pip lt T _ S 5 20 The momentum spread of the ion beam Ap p depends on many parameters and can be varied by about one order of magnitude depending on experiment conditions In the Fig 5 9 the coherent energy shift of the proton beam as a function of the circulation period and beam energy is presented One can see that at circulation period of about several hundreds of msec this effect can be ignored at all reasonable ion beam parameters 44 Circulating period sec 16 Circulating period sec Ti 16 Target density Atoms cm 2 Fig 5 8 Maximum circulating period of the electron beam as function of the target density a proton beam at energy 2700 MeV 1 1000 MeV 2 500 MeV 3 b deuteron beam at energy 2100 MeV 1 1000 MeV 2 500 MeV 3 Relative energy losses 0 4 0 6 Circulation period sec Fig 5 9 Relative energy losses of the proton beam at target density of 10 Atoms cm at energy 100
67. cusing system B 1000 G G 20 G cm h 60 cm L 120 cm C 20 m and at electron energy of 500 keV the working point corresponds to Qst 109 4 at maximum beam energy of 1 5 MeV Qst 49 1 Qsiow 0 24 In this case dispersion in the cooling section is relatively small Dy 10 cm ellipticity of the electron beam cross section in the cooling section is less than 1 i e circulating beam in the cooling section has round shape Angular spread of the electron beam due to action of the helical quadrupole field can be reduced to the value less than 1 mrad by adiabatic variation of the gradient value at the entrance and at the exit of the quadrupole field region Calculation of the ring lattice functions and beam parameters in the cooling section are provided numerically and the algorithm and computer code elaborated in JINR for this aim are described in 9 Detailed design of the electron ring optics and accurate choice of the working point in the total range of electron energy can be a topic of further steps of the cooling system design and are not included in this report The motion stability conditions can be written as the following n k Q sast E 2 d Q stow 2 gt Q rsi Ors l 5 8 n k l are integer The resonant condition for the fast mode of oscillation due to large value of its chromaticity limits a dynamic aperture of the ring on momentum deviation The particle momentum spread can not exceed the distance between nearest i
68. d its power supply system The example of the electrostatic kicker design is presented in the next chapter In the case of induction acceleration of the electron beam the existing COSY electron cooling system can be upgraded to medium energy electron cooling system by installation of several new solenoids and betatron yoke for electron acceleration Two of the new solenoids septum solenoids are to be placed vertically above the existing toroidal solenoids In each of these solenoids the coils of magnetic septum and electrostatic kicker are placed Existing electron gun and collector with their solenoids are placed above these two new septum solenoids To form a closed orbit for electron beam two toroidal and straight solenoids are connected with the septum solenoids In the straight solenoid which is placed opposite to the cooling section the betatron yoke and helical quadrupole windings are placed In the regime with circulating electron beam the beam from the gun is directed to the closed orbit by injection septum and kicker plates When the beam fills total circumference of the electron ring kicker voltage is switched off After that beam is accelerated to required energy with the betatron yoke Next current pulse applied to the betatron yoke coil decelerates the circulating electron beam and after deceleration the electron beam is extracted to the collector by the pulse applied to the extraction kicker The cooling system with induction accelera
69. density has to be calculated taking into account the duty factor Final conclusion can be expressed as follows in the experiment with the pellet target at maximum proton beam energy an electron cooling gives substantial gain in the luminosity up to the luminosity level of about 5 10 cm sec For deuteron beam an electron cooling can be effectively used in the total energy range and at beam energy of 2 11 GeV and maximum target density the gain in the integral luminosity after five minutes of experiment is about 50 and it increases to several times at less energy or target density 4 Electron cooling system with circulating electron beam There are two possible design of new COSY cooling system Traditional configuration of cooling system HV accelerator Cooling system with circulating electron beam Traditional configuration of the cooling system can provide electron beam at low transverse and longitudinal temperature and the results of numerical simulations demonstrate its ability for ion beam parameters stabilisation 57 Electron cooling system with circulating electron beam has substantially low cost but in such a system the electron beam quality is expected to be rather less than in the HV system The electron beam quality is determined by a method of the beam acceleration stability of electron motion in the electron storage ring number of electrons and the ring circumference General attention of this report was at
70. e beam velocities in the cooling section are the same Correspondingly the coherent energy variation of the electron and ion beams during the circulation period are connected in accordance with the relation 43 AE AE l T 5 17 During the circulation period the energies of the ion and electron beams have to be inside the dynamic apertures on momentum deviation of corresponding ring The dynamic aperture on momentum deviation of the electron ring with longitudinal magnetic field Go formula 5 10 is less than that one of the ion ring Coherent momentum shift of the electron beam can be calculated as the following T VAP e 5 18 P 2 amp E 1 where AE is the ion energy loss after crossing the target formula 1 2 E is the ion energy and Tcire is the electron beam circulation period From the condition dp p lt o one can write the condition for upper limit of the circulation period TE T max oe 5 19 where C is the electron ring circumference pz is the electron Larmor radius in the longitudinal magnetic field T ey is the ion revolution period The dependencies of the maximum circulation period of the electron beam on the particle energy and the target density are presented in the Fig 5 8 In the total range of the COSY experiment parameters the circulation period has to be less than about 1 second to avoid a resonance of the electron beam Other limitation connected with this effect is related to distortion of the
71. e SA On 2 2 4 B gt 2 Xp gt Z and is the r m s momentum spread are the beam two sigma emittances The function f is the following integral dx f a b c 8n ym gota erh ee 2 2 5 The following relations determine normalized parameters used in the Formula 2 2 5 o o ee Cee E gt a h b c fo peana z EN T gt YO v YO y r O O O g poo x7 1 a q b x 1 0 and the maximum impact parameter d is about 0 5 beam height Integral 2 2 5 is calculated numerically 2 2 2 Martini model To calculate the IBS with the Martini model one needs to use the lattice functions of the COSY for different modes of its operation The lattice parameters are imported to the program from the files presented by COSY see for instance Fig 2 1 71 id 5 x OBR HI JNN Fig 2 1 The optical functions for an ideal COSY lattice with dispersion free telescopic sections obtained by non symmetric operation of the 6 unit cells in the arcs superperiodicity S 2 C sidorin CPP BetaCoohACR 21 0dp0 Its Oo x First Last fe fia Skip Number of Row Column Number of Lattice Functions From Up to Load Save fen fe J 1 Longitudinal Coordinate E Jez J 2 Beta Horizontal jes fre 3 4lpha Horizontal js E 4 Dispersion Horizontal fsa fina 5 Disperion Derivative PER UM a pi Dh l fei AK ine a fios E 6 Beta Vertical
72. e electron cooling system of traditional configuration was performed at COSY earlier 5 The following chapters of this report are dedicated to investigation of cooling efficiency at relatively pure electron beam quality The brief description of the cooling method with circulating electron beam and preliminary design of the installation are given 32 5 Electron cooling system with circulating electron beam The choice of the electron cooling system parameters is restricted by number of limitations Part of them is common for both systems with single pass and with circulating electron beam First of all it is requirement of the beam magnetisation which limits the minimum value of the longitudinal magnetic field and maximum value of electron beam current Other limitations are related only to the system with circulating electron beam and particle dynamics in the storage ring with longitudinal magnetic field Generals of them are the following upper limit of the electron beam circulation period determined by increase of electron temperature during cooling process upper limit of the electron beam momentum spread determined by dynamic aperture of the ring upper limit of the electron beam current determined by threshold of microwave instability limitations of the magnetic field and electron beam current due to longitudinal transverse relaxation of the electron beam 5 1 Magnetic field limitations Electron
73. e hydrogen cross section A 1 3 2 Single scattering on large angles 75 The cross section for single scattering with an angle larger than the acceptance angle 0 is Zr 2 Op mt T 3 2 1 where Z is the charge number of the nucleus of the rest gas atoms r is the proton classic radius The average acceptance angle in eliptical vacuum chamber is given by 2 l l l where lz Pa 3 2 2 o 26 20 OF e accept x z The beam life time due to single Coulomb scattering Y B Ee DE 3 2 3 Tir Anrell B B h oe where n Zin 3 2 4 3 3 Recombination in the electron cooling section The recombination rate in the laboratory frame for the radiative recombination is described by the following formula 1 1 dN _ ann A N dt y 3 3 1 where a is the recombination coefficient see Formula 3 3 2 below n L ena c is electron beam density in the cooler a J electron beam radius and current correspondingly nz is the ring fraction occupied by the electron beam Assuming a flattened electron velocity distribution one can express the formula for in a simple form 8 3 1 3 a 3 02 107 tz EY yf 1322 og BY 3 3 2 s Ti JT Z here T is measured in eV 3 4 Other particle losses All the processes leading to exponential decrease of the particle number can be summarized in the calculations by introducing an equivalent particle lifetime in the ring This value can be included into
74. e ring circumference of 20 meters is the minimum value which permits to install in the ring injection and extraction systems helical quadrupole lens for electron focusing and betatron yoke if it is necessary The electron cooling system parameters determined by the requirements of beam magnetization and cooling efficiency are listed in the Table 5 1 and further calculations are performed at these parameters Table 5 1 General parameters of the electron cooling system with circulating electron beam Electron ring circumference m 20 Magnetic field kG 1 1 5 Electron beam radius cm 0 5 Maximum electron current A 0 5 Effective length of the cooling section m 1 4 Circulation period msec 100 1000 Transverse electron temperature meV 300 1000 Longitudinal electron temperature meV 50 100 5 3 Storage ring with longitudinal magnetic field At electron energy of 10 MeV and higher usual strong focusing ring can be used for storage of electrons In the case of low electron energy the storage ring with longitudinal magnetic field has some advantage like electron magnetization and provides a stable motion of circulating electrons Such electron ring can be used simultaneously for preliminary acceleration of electrons Then the injector delivers electrons of rather low energy So called modified betatron is one of possible schemes of the storage ring with cooling electron beam Combination of th
75. e ring with an injector provides an added bonus 5 3 1 Particle dynamics in the ring with longitudinal magnetic field Focusing system of the ring consists of straight and toroidal solenoids connected together as a racetrack To form a closed orbit for a circulating particle the helical quadrupole winding is used In the cooling section the quadrupole field is absent to avoid distortion of the cooling process The helical winding can be placed in the straight section opposite to the cooling one Due to presence of the longitudinal and helical quadrupole magnetic fields the particle motion in the horizontal and vertical plane are strongly coupled When the number of the spiral winding steps in the focusing section is integer the particle motion in the first approximation consists of two independent modes 8 37 a fast Larmor rotation of every particle around own magnetic field line which tune is equal to C 5 6 Q hast 2p P V 0 eB ymc is the electron cyclotron frequency p is the electron Larmor radius B is the longitudinal magnetic field averaged over the closed orbit C is the ring circumference b slow rotation of the beam as a whole around the axis of the helical quadrupole winding with the tune Qsiow gt a a gt 5 7 L is the length of the sections with quadrupole field G is the gradient of the quadrupole field k 27 h h is the step of the spiral winding For instance at typical parameters of the fo
76. e tab sheet Lattice functions Table 5 2 Table 5 2 Lattice functions Horizontal Beta Bx m Is used for calculation of the Alpha Ox variation of the motion invariants Dispersion D m after single pass the cooling section Dispersion derivative D in accordance with formula 4 2 3 in ref 1 Vertical Beta Bz m Is used for calculation of the Alpha O variation of the motion invariants after single pass the cooling section in accordance with formula 4 2 4 in ref 1 5 1 2 Tab sheet Friction force The deviation of the particle angular spread when passed the cooling section is calculated only in the case when particle distance from electron beam centre is less than electron beam radius and is equal A9 A 0 2r 2 2 I I where A K R 1 8 10 K tA _ m mrad By A a m c A By rae where _ is calculated by using one of the following formulae 111 Flattened non magnetized Derbenev Skrinsky Parkhomchuk The formulae are not presented in ref 1 and here we introduce short description of them 5 1 2 1 Derbenev Skrinsky formulae Q Cline ce 1 ERA n SN 0 o 0 pp epi e A 7 0 4 AL N IT 5 1 col 2 L N gt F col 0 M 0 I 0 Lp k Ly u Z ty i 2 P 8 42sgn0 Lr N oLa ee z k Lu a II 5 2 JA o o Lr N aala H M oi 19 iy 6 Y Domains J I
77. eached during 30 60 sec depending on initial beam parameters Without the target the beam life is higher than 10 sec and permits to perform all required manipulations of the beam parameters 1 80E 36 1 60E 36 1 40E 36 1 20E 36 1 00E 36 e le 0 8 00E 35 m le 0 5A 6 00E 35 4 00E 35 2 00E 35 0 00E 00 Integral luminosity 1 cm 2 oO DS oO D O O foe oO 100 Pellet diameter um Fig 4 7 Integral luminosity after five minutes of experiment as a function of the pellet diameter beam energy is 2 7 GeV Gain in integral luminosity No oO nN oO O O foe oO 100 Pellet diameter um Fig 4 8 Gain in the integral luminosity after five minutes of the experiment at electron beam current of 0 5 A proton beam energy is 2 7 GeV 23 In the Fig 4 9 the integral luminosity was calculated under assumption that emittances in the both planes are the same and momentum spread is 10 The integral luminosity monotonically increases with the decrease of the initial beam emittance and expected gain after five minutes of experiment due to initial preparation the beam before experiment is about 10 1 70E 36 1 65E 36 1 60E 36 1 55E 36 Integral luminosity 1 50E 36 1 45E 36 fo 0 2 0 4 0 6 0 8 1 1 2 1 4 1 6 Initial emittance pi mm mrad Fig 4 9 Integral luminosity in 1
78. eam parameters in the numerical investigations were chosen the following beam emittance in the both planes is equal to 10 m m rad momentum spread is equal to 10 Calculations were performed taking into account the beam interaction with residual gas and target atoms and intrabeam scattering in the ion beam The initial beam intensity was chosen to be equal maximum expected value of 10 particles An example of the calculations of short term variation of the beam parameters at the target areal density of 10 Atoms cm is presented in the Fig 2 4 for proton beam at energy of 2 7 GeV co Dyed Long coordinate Fig 2 3 Lattice parameters at ANKE experiment Q 3 571 Yu 2 243 In the cooling section Beta functions 13 15 m Dispersion 0 in ANKE Beta functions 2 3 m Dispersion 0 10 Enor Emittance pi nim mrad Evert 0 150 300 Time sec Momentum spread 0 1 0 150 300 Time sec tu D a E a z v S 2 wo fai o tu 0 BQ 080 Time sec Fig 2 4 The dependencies of the proton beam emittance a momentum spread b particle number c on time target density is 10 Atoms cm One can see that after five minutes of experiment the beam emittance is limited by the ring acceptance and the particle n
79. ency of beam cross section variation Real width of the linear and coupling resonance is determined by the errors of the focusing fields In the case of focusing by longitudinal magnetic field the errors can not to be calculated through the errors in the position of the focusing elements The errors of the focusing field appear due to those in the solenoid winding and can be evaluated only by measurements of the real field distribution in each section of the solenoid However the field measurements has to be performed in the position of the closed orbit which is known with accuracy of about a few millimetres Qslow Coupling resonance 0 5 1 3 Qhast 342 342 5 343 Fig 7 5 Stability diagram in the vicinity of the working point at beam energy of 10 keV and magnetic field value of 400 G 53 Thus the dynamic behaviour of the ring can be investigated only experimentally And such an investigation is of great importance for proposed cooling system The width and power of the fast mode resonances in real structure are the key parameters in the problem of betatron acceleration of circulating beam A possibility of the resonance crossing and beam parameters distortion in the resonance vicinity will determine the choice of acceleration regime In the case of electron beam acceleration with RF accelerator the dynamics investigation will determine the requirements to the momentum spread of injected electron beam The tune of the slow mode of
80. ensity 10 Atoms cm Analogous dependencies for deuteron beam are presented in the Fig 4 4 In the first approximation the cooling time for deuterons is two times longer than for protons but charcteristic time of the emittance growth due to multiple scattering with the target atoms is longer by four times As result the electron beam current required to obtain the equilibrium is about two times lower The values of the electron beam current used in the calculations of the Fig 4 3 4 4 are presented in the Fig 4 5 The equilibrium is reached in the case of deuteron beam during 20 30 minutes and this time is comparable with the particle life time due to single scattering on the target atoms 18 1 4 1 6 1 8 Beam energy GeV Fig 4 4 Equilibrium deuteron beam parameters as a function of the beam energy 1 horizontal emittance in 10 T m rad 2 vertical emittance in 10 T m rad 3 momentum spread in 10 Deuteron number is 10 target density 10 Atoms cm 700 600 500 400 Deuterons 300 Protons 200 Electron beam current mA 100 1 5 2 2 5 wo Beam energy GeV Fig 4 5 The electron beam current required to reach the equilibrium emittance in both planes of about 10 z m rad The stabilization of the ion beam parameters is important in the experiment with cluster beam target The target dimension in t
81. er of the main form Fig 2 1 3 3 File Save as This sub item calls the standard Windows save dialog window Fig 3 2 All the parameters are the same as in chapter 3 1 Save as le Save in I TWAC x aa Betacoal eta Save as lype BetaCool Parameters eta Cancel Za Fig 3 2 Save dialog window 3 4 File Save On Exit If this sub item is checked all the parameters are saved after exit in the file which name is indicated in the left corner of the main form Fig 1 3 5 File Save DeskTop If this sub item is checked the current positions and parameters of all the forms of the program are saved in the file Betacool top Note when the visual form parameters are not correct you need to delete the Betacool top file in the directory where the BetaCool exe is located After that the default parameters will be used for all the forms and after exit the BETACOOL new file Betacool top will be generated automatically 3 6 File Graf as Data When this sub item is checked the parameters of 2D and 3D plots are saved in the files which names coinside with the name of curren eta file after exit of the program At the next run of the program these files will be loaded with corresponding eta file 3 7 File Open Dialog If this sub item is checked the open dialog window chapter 3 1 is called at the beginning of the program work 3 8 File Exit Exit from the program 100 4 Main menu Pa
82. ere Bx is mean beta function in corresponding plane The number of particles inside Tx has the same value as inside the emittance when the kinetic energy of the particles is connected with the beam transverse temperature as E T And we have for transverse temperature the following relation with the beam emittance T E PYMe 4 2 14 u Ba BI aB Summarizing by all degrees of freedom we have Ap p E 3T E M pr 4 2 15 p kin B Yy y 4B 4B Using this definition of the ions and electron temperature in both beams the temperature is averaged over degrees of freedom as an initial values we can solve the system 4 2 10 83 4 2 11 during circulation period Under assumption that circulating period is substantially shorter than the cooling time and that all degrees of freedom are cooled uniformly we can determine the cooling time as follows 1 1 1 1 OH e E Ge de 2 dg long _ p fin p in 4 2 16 e o e ot H ot T Ta T cool circ long where Tpin and Tp sin are initial and final beam temperatures correspondingly In this model the effects of the flattened velocity distribution and magnetization of electron beam are neglected and temperatures of all degrees of freedom of the ion beam are supposed to be near the thermal equilibrium In the case of magnetized electron beam with flattened velocity distribution the cooling time of the system with circulating electron beam can
83. erial gt Calculate button The calculation of the parameters Estr Ost Ap Ptarget Ta is performed only when one pushes the button Calculate in the Target material and geometry form Note If the beam energy or ion kind was changed one need to push the button Calculate to recalculate the target parameters Estr Ost Ap Piarget To 5 6 Effects Injection The beam parameters variation due to repeated injection is calculated in accordance with the following algorithm In the case when new portion of No ions at emittance are injected into the ring and injection repetition period is equal to Ti the characteristic times of emittance variation are given by the expression 124 1 2c 1 e E No 5 25 e N N T inj i inj where N is the particle number in the circulating beam Beam life time at injection has a positive sign which corresponds to increase the particle number 1 N 5 26 Vite inj NT inj Required parameters are entered using the Injection form Fig 5 11 and listed in the Table 5 8 Finjection o Parameters of injected beam Emittances pi m rad Horizontal Vertical Momentum spread Particle number Injection repetition period sec Fig 5 11 The Injection form Table 5 8 Emittances n m rad Horizontal 0 Vertical 0 Momentum spread Ap p Particle number No Injection repetitio
84. es the electron beam acceleration and test of the beam temperature at maximum required energy is z oo s 27 24 6 T eelesten Rl JEE Ao j Ma 2 a ma tt lt ami Q 3 injec tlon kicker isy jutlus imati e iba ATY RA oS Fig 7 1 The assembly drawing of the LEPTA 1 extraction kicker 2 septum 3 injection kicker 4 toroidal sections 5 betatron core 6 electron gun 7 electron collector 8 cooling section 49 Table 7 1 General parameters of the LEPTA Ring parameters Circumference m 18 28 Longitudinal magnetic field G 1000 Major radius of the toroids m 1 45 Bending magnetic field G 1 75 124 Gradient of the quadrupole magnetic field G cm 10 15 Electron beam radius cm 1 0 Residual gas pressure Torr 107 Electron beam parameters At injection After acceleration Energy keV 10 4360 Current A 0 1 0 5 Revolution period nsec 300 60 Acceleration cycle Induction voltage amplitude V 50 Repetition frequency Hz 1 Cycle duration msec 10 7 2 Preliminary program of the experiments General problems which have to be experimentally investigated before design of the full scale electron cooling system with circulating electron beam are the following beam parameter distortion during injection variation of the longitudinal beam temper
85. eters Mean chamber radius b cm Is used for calculation of longitudinal form factor G chapter 5 in ref 1 Pressure P Torr Residual gas pressure Residual gas composition 1 n No It is assumed that residual gas Pn No No composition includes three kinds of o3 n3 No atoms Ai A2 A Z Z2 Z3 Calculated parameters SS density Dss 1 sm Formula 3 2 4 in ref 1 MS density Nms l sm Formula 2 3 2 in ref 1 104 4 2 Parameters gt Beam This sub item calls the form for input the storage ring parameters Fig 4 2 i ox Show Stability characteristics Bunch parameters Particle number C Bunched Coasting Horizontal emittance pi m rad Vertical emittance pi m rad Momentum spread Mean radius Bah Longitudinal form Factor Space charge impedances Longitudinal Ohm Transverse Ohmm Peak current A Fig 4 2 The form for input the storage ring parameters The form contains three tab sheets General parameters Stability characteristics Bunch parameters and menu item Show The menu item Show calls the same form as the main menu sub item Parameters gt Beam evolution and it is described in the chapter 4 3 The tab sheet Bunch parameters is visible only in the case when the option Bunched is chosen The parameters indicated in the edit boxes of the Beam Parameters form are specified in the tables 4 5 4 7 105 Table 4 5
86. f the menu item Effects Calculations of the beam parameter evolution can be started using the button Run in the Dynamics form next chapter after correct loading of the process parameters The 2D plot in the right side of the form is used for output the rates during dynamics simulation After push the bottom Run in the Dynamics form next chapter the absolute values of the rates are plotted versus time in seconds Rates 1 sec fr ECOOL rips 1 TAU_N Decay 1 TAU_Eh RestGas rd 17TAU_E Stochastic a Injection 17TAU_P Target Heating Fig 6 1 The Rates form 6 2 Task Dynamics This form includes the Process control panel in the right part more detail about this object of BOLIDE see in the chapter 8 edit window Step sec for input initial step over time for numerical integration of the system 1 1 ref 1 using Euler method edit window Scale which is used for set the horizontal axis in the plots in the forms Rates chapter 6 1 and Beam parameter evolution chapter 4 3 the minimum of the axis is equal to 0 the maximum is equal to Scale Step sec this parameter is not used in the calculations and plot parameters can be corrected if necessary using Graphics control tools chapter 8 127 check box Beam scrapers if this parameter is checked the particle losses due to acceptance limitation in accordance with chapter 1 2 ref 1 ioi x Step sec Set Stop Sc
87. fe time after reaching the equilibrium It means that at the elevated beam energy and or relatively small target density the electron beam current has to be optimised to obtain maximum integral luminosity at each given experiment duration The possibility to vary the beam emittance during experiment can be investigated by comparison between the electron cooling rate and heating rate due to interaction with the target In the Fig 4 15 the dependencies of cooling and heating rate on the beam emittance at the beam energy of 2 7 GeV are presented At the pellet diameter of 80 um curve 4 in the Fig 4 15 the heating always prevails on the cooling curve 1 equilibrium is impossible and emittance of the beam increases to 27 acceptance of the ring The cooling application can decrease the speed of the emittance growth only At the pellet diameter of 40 um one can see the stable equilibrium point point 1 in the Fig 4 15 and unstable one point 2 When the beam emittance value is less than certain value corresponded to the unstable equilibrium the emittance of the beam decreases under common action of the cooling and target to the stable equilibrium point When the beam emittance is less than its value in the stable equilibrium it increases but the emittance increase is limited by the value corresponded to the stable equilibrium point 0 05 0 045 0 04 0 035 0 03 0 025 0 02 0 015 0 01 Co
88. fect 5 18 is calculated for two different longitudinal distribution of the ion beam density 2 2 m D Gaussian bunch 200 a s 5 20 N coasting beam Cc Another incoherent effect relates to the drift velocity of the electrons in the crossed longitudinal magnetic and bunch electric and magnetic fields For Gaussian bunch the drift velocity can be calculated using the following formula ns eZX S o By B F r 5 21 where B is the longitudinal magnetic field value and the bunch field radial dependence is given by the expression 2 1 exp ears a F r 2 E 5 22 r The drift velocity can be included into calculation of the friction force as an addition the electron transverse velocity 0 0 02 5 23 where 67 is the electron angular spread corresponding to the thermo velocity 1 T Graii fy 5 24 By Ym c For visualization of the deviation of the particle angular spread after passing the cooling section the form Friction is used Fig 5 3 Two tab sheets of the form contain 3D plots for output of transverse and longitudinal component of friction force in the unit angular deviation of the ion after single crossing the cooling section 5 1 2 6 Show force shape button To output the friction force you need to specify co ordinates of the ion in which it crosses the cooling section in the edit windows Distance from axis and Distance from the bunch center both in cm in the tab
89. he horizontal plane is less than the ion beam dimension corresponding to the acceptance limit The numerical model used in these calculations does not permit to solve the problem correctly in the case when the horizontal emittance is varied during the experiment However one can estimate possible gain in the experiment luminosity due to horizontal emittance stabilization as a ratio between maximum beam dimension and 19 target dimension This ratio is about 1 5 2 and taking into account that the characteristic time of emittance growth without cooling is about 10 minutes the gain in the luminosity can be estimated by the value 1 5 The electron beam radius of 1 26 cm required for electron cooling at injection energy is more than two times bigger than ion beam one at experiment with internal target Decrease of the electron beam radius permits to decrease the beam current In the experiments with gas jet and cluster targets this fact does not play a significant role due to small value of the electron beam current density required for effective cooling In the experiment with higher target density and at maximum beam energy the electron beam current density has to be as high as possible and an accurate choice of the electron beam radius is necessary to minimize the electron beam current The optimum electron beam parameters will be discussed below 4 2 Electron cooling in experiments with the pellet target In the experiment with the pellet target
90. he integral luminosity at initial emittance of 0 3 7 mm mrad and 1 5 m mm mrad versus experiment duration beam energy is 2 7 GeV pellet diameter is 80 um cooling is off At maximum experiment energy the luminosity increases with the electron beam current and maximum luminosity corresponds to maximum electron beam current The electron cooling efficiency increases with decrease of the experiment energy Fig 4 12 And at minimum experiment energy it is not necessary to have maximum electron beam current to obtain the maximum luminosity the luminosity dependence on the current has a saturation in the region of 400 mA Fig 4 13 The reason of the saturation can be explained by analysis of the luminosity time dependence Fig 4 14 25 1 8 1 7 P 1 6 1 5 1 4 1 3 1 2 1 1 Gain in integral luminosity 1 5 2 2 5 Beam energy GeV Fig 4 12 Gain in integral luminosity after five minutes of experiment at electron beam current of 0 5 A pellet diameter is 80 um Gain in integral luminosity 7 a oo A a N gt wo oO P O ol Electron beam current A Fig 4 13 The gain in integral luminosity after five minutes of experiment as function of electron beam current beam energy is 1 GeV pellet diameter is 80 um 26 2 5E 34 lt 2 0E 34 oO eb A le 0 1 5E 34 2 le 0 5A A le 0 4A
91. he use of high order Runge Kutt methods meets certain problems related to a long time of the equation right parts calculation 1 2 Particle losses and emittance calculation in the presence of beam scrapers The emmitance growth and the particle survival probability are dependent of the presence of beam scrapers which limit the amplitudes of particle betatron oscillations Maximum admissible betatron amplitude value Y can be calculated from the ring acceptance as follows Y fArcBr gt 1 3 where A are the ring acceptances in the corresponding planes Jx are the beta functions at the scraper position 65 The survival probability of the particle after one revolution in the ring can be estimated by the following expression 2 co PC Daa a a xp A C 1 4 where J is the Bessel function of the first kind and is the k th root of the equation Jo A 0 The dimensionless variable is connected with the mean square amplitude of the betatron oscillations In the case of Rayleigh distribution function of the beam particles over betatron amplitudes 2 PA p y Zro 1 5 where the x 20 g mean square amplitude is so called two sigma beam emittance oy is mean value of Gaussian distribution function in the space of co ordinate one can calculate g 0 25 y 0 25 A 1 6 Then calculating P Q overestimate slightly the particle loss by including a small loss which would have occurred witho
92. hen electron cooling time is rather long That allows to apply an electron cooling system with circulating electron beam 2 Such a system has potentially low cost in comparison with other possibilities At the energy range from 500 keV to 1 5 MeV only longitudinal magnetic field can provide an effective focusing of an intensive electron beam The electron beam acceleration can be produced both by induction acceleration of electrons or using an RF electron LINAC Specific limitations of such a cooling system are discussed Preliminary design of the electron cooling system with circulating electron beam is described in the report This report contains also preliminary program of experiments at LEPTA Low Energy Particle Toroidal Accumulator which are aimed to study the problems of electron cooling system with circulating electron beam Presently the construction of LEPTA ring is in the final stage at JINR and experiments with circulating beam will be started the next year Introduction The existing stochastic cooling system or an electron cooling system which cover total energy range of the proton or deuteron beam at COSY can be used for the following applications at injection energy to increase the intensity of the polarized proton beam with a combined cooling stacking injection at energy range from about GeV to top energy of 2 7 GeV 2 11 GeV for deuteron beam for luminosity preservation in an experiment with internal targets for
93. her particle losses 4 Cooling processes involved into calculations 4 1 Stochastic cooling 4 1 1 Transverse cooling time 4 1 2 Longitudinal cooling time 4 1 3 Power limited cooling 4 2 Electron cooling 4 2 1 Electron cooling with single pass electron beam 4 2 2 Electron cooling with circulating electron beam 5 Calculations of the beam stability characteristics 6 Description of the program References 62 ns m e eme e anank nv enmn V ONNN 16 16 16 17 17 17 17 17 18 18 18 18 22 25 27 31 Abstract The synchrotron and storage ring COSY delivers proton beams with momenta between 300 and 3500 MeV c for internal and external experiments It is operating both with electron and stochastic cooling The electron cooling is used mainly at injection energy with 22 keV electrons to increase the intensity of the polarized proton beam with a combined cooling stacking injection In addition the electron cooled proton beam is used for diagnostic purposes of machine parameters The stochastic cooling for COSY is working in the proton momentum range between 1 0 and 3 4 GeV c energy from 0 4 to 2 5 GeV and is used for experiments with thin internal targets and at a long flat top time The general aim of this work is to estimate a feasibility to extend the energy of electron cooling application up to maximum proton beam energy of 2 6 GeV maximum electron beam energy is about 1 4 MeV and to use electron cooling in c
94. idual gas atoms high order tune resonances of the ion betatron motion due to space charge tune shift The sources of the particle losses nuclear scattering on the target and residual gas atoms single scattering by large angles on the target and residual gas atoms electron capture recombination in the electron cooling section Cooling processes stochastic cooling both transverse and longitudinal electron cooling with single pass and circulating electron beams 1 General description of the physical model 1 1 Beam emittances and particle number evolution The influence of all the processes determining the luminosity variation can be investigated in the frame of the physical model used in BETACOOL program elaborated in JINR 1 The model is based on the following general assumptions 1 the ion beam has Gaussian distribution by all degrees of freedom which does not change during the process 2 algorithm of the problem analysis consists in the solution of the equations for root mean square values of the beam phase space volumes of three degrees of freedom 3 maxima of all distribution functions coincide with equilibrium orbit The evolution of the ion beam parameters during its motion inside the storage ring is described by the following system of four differential equations 64 B Ny T Tife j 1 amp lee es 1 1 1 amp e j Zj 1 Hy Hin Yi J lon j where N is the parti
95. ing cannot effect on the beam parameters effectively The range of the target density from 10 to 2 10 Atoms cm in which cooling can be used for luminosity preservation covers parameters of all types of the ANKE target practically Only in the experiments with solid target the beam life time is so short that luminosity preservation using cooling seems to be unrealistic The difference between the hydrogen and deuterium targets is not substantial because of the general processes which influences on the beam parameters are determined by interaction with the electrons of the target atoms In this report the possibility of the cooling application was investigated in two ranges of the target density long term experiment up to 1 hour at the target density of 10 Atoms cm and short term experiment about 5 minutes at the target density of 10 Atoms cm In the experiment with gas storage cell target when the target dimensions coincide with the aperture of the vacuum chamber the role of the beam cooling is to reduce the particle losses due to aperture limitation by suppression of the emittance growth Maximum achievable gain in the experiment luminosity is determined by the relation between the beam life time and time of the emittance growth up to acceptance For proton beam at the target density of 10 Atoms cm after one hour of experiment maximum possible gain in the beam intensity is more than two times at the energy below 2 GeV
96. ing process The sources of the beam phase volume growth are the multiple scattering on residual gas and the target atoms intrabeam scattering First of all influence of residual gas on the beam life time was investigated under assumption that average pressure in COSY vacuum chamber is 5 10 Torr and gas composition is 95 of hydrogen and 5 of nitrogen At these conditions the particle life time is about 10 seconds and emittance growth rate at minimum beam energy of 1 GeV is about 3 107 2 m rad sec Thus the interaction with the residual gas does not restrict particle life time and slightly influences on the particle losses due to emittance growth 2 1 Interaction with the target Ultimate efficiency of the beam cooling can be estimated in the case of high target density and low intensity of the ion beam in this case beam parameters are determined only by interaction with the target Multiple scattering on the target atoms leads to the linear in time growth of the beam emittance and particle losses takes place when the beam emittance increases up to 0 3 of the ring acceptance 4 The period of time when the emittance growth does not lead to the particle losses can be estimated by the following formula At eter 2 1 T ae 2 1 which gives the time of emittance growth from zero value to acceptance limit Here A is the ring acceptance Te is the characteristic time of the emittance growth calculated at certain value of the beam emitt
97. ion in the space of invariants 1 7 ee ery T Se Si jer Steg aT ay 4 2 8 EEE 0 The cooling time value is equal to 1 _ lde 1 e dt Tu cool i i 4 2 9 T It should be noted that emittance used in Formula 4 2 8 as a parameter of distribution function contains 63 of particles and is twice higher than r m s value and this has to be taken into account in calculations of the standard deviation values Both possibilities can be used in the simulations cooling time can be calculated in accordance with 4 2 9 or to speed up the calculations in accordance with 4 2 7 4 2 2 Electron cooling with circulating electron beam Due to interaction between the particles antiprotons ions and electrons the particle temperature decreases when the electron one increases In accordance with energy conservation low the temperatures of the ion and circulating electron beam are connected together as follows dT dT N N 4 2 10 P dt dt 81 where Ty Te are the particle and electron temperatures in the particle rest frame Np Ne the particle numbers in the rings Thermodynamics of the cooling process in approximation of uniform two component Maxwellian plasma was investigated in 12 The variation of both temperatures in this plasma is described by the equation al _421m n 2 e fe a 4 2 11 3 22 ate dt ymM T L M m where m and M the electron and ion masses Nz is the
98. ion voltage variation kHz 1 Required amplitude of the acceleration voltage V 500 Beam momentum spread caused by voltage variation 2 107 During the electron beam acceleration inside the ring with constant value of the longitudinal magnetic field the working point crosses series of the resonances of the fast mode of oscillation The experiments at Modified Betatron Acceleratator in NRL USA 13 have shown that the crossing of the high order resonances does not lead to the particle losses An increase of the beam transverse temperature in these experiments was not investigated and we plan to perform such investigations at LEPTA ring see chapter 7 46 6 Principles of the electron cooling system design All general elements of electron cooling system with circulating electron beam are presented in the Fig 6 1 If the electron acceleration is performed with RF electron linac the betatron yoke is not necessary and instead of electron gun one needs to install the RF cavity and debunching system if necessary to reduce Ap p see 14 Injection Extraction kicker kicker Injection Extraction septum septum Betatron Helical quadrupole yore winding Electron gun solenoid Collector Fig 6 1 Layout of the electron cooling system with circulating electron beam The electron beam injection into the ring is provided along the longitudinal magnetic field with transverse field of septum coils which displace the electron beam
99. irculating electron beam the time dependence of electron temperature is displayed in graphic format also The view of the form for input of the electron cooling system parameters is presented in the Fig 6 4 All objects describing a beam parameters evolution are developed on the base of the ancestor class Effect which has a virtual function using Beam and Ring objects as input variables and returning array of the rates particle loss two emittances and momentum spread variation In each descendants class corresponding to concrete process this function is reloaded by its individual one All variables of the class effect are automatically included into array of the effects and class effect includes the method which calculates the sum of the rates in the cycle This ancestor class includes also methods for output of the rates in numerical and graphical format see Fig 6 5 89 Electron Cooling Fig 6 4 The form for input of the electron cooling system parameters In order to realize this approach in the program the special class BTemplate for the storage of object pointers was elaborated The declaration and constructor of this template are presented here template lt class B gt class BTemplate public static B BItems static int BCount int BIndex BTemplate j virtual BTemplate template lt class B gt BTemplate lt B gt BTemplate B newItems new B BCount 1 for int i 0 i lt BCount i
100. is one can expect effective application of existing stochastic cooling in the experiment with internal target when the beam intensity is about or below of 2 10 particles and the target density does not exceed 10 Atoms cm 2 Emittance pi mm mrad 0 150 300 Momentum spread 0 1 0 01 0 150 300 Time sec Fig 3 1 Emittance and momentum spread evolution at stochastic cooling Target density is 10 Atoms cm beam energy 1 GeV the beam intensity is 10 protons 1 0E 11 D g 1 0E 10 gt c c 1 0E 09 oO 1 0E 08 T 1 1 0E 14 1 0E 15 1 0E 16 Target density Atoms cm42 Fig 3 2 Proton number corresponding to equilibrium between heating effects and stochastic cooling in the horizontal degree of freedom Beam energy is 2 0 GeV 14 1 0E 11 1 0E 10 Proton number 1 0E 09 1 0E 14 1 0E 15 1 0E 16 Target density Atoms cm42 Fig 3 3 Proton number corresponding to equilibrium between heating effects and stochastic cooling in the horizontal degree of freedom Beam energy is 2 7 GeV 2 t i i k I 2 E E E ES m z Lu z 0 1
101. its of first 7 constants can be chosen from the corresponding lists The numbers from edit windows can be copied to the system buffer using Ctrl Insert combination 130 Physical Constants STIS Plank constant Ay B 2607554E 34 lus Plank constant AB 1 0545726663634 hs x atom mass unt AmuU 6605655627 ka ol neutron mass M a 16749543627 ko al proton mass m p 1 6726485E 27 ka al electrom mass MI aise ka Boltzman const K 1380658126230 0 CO hek x speed of ligh G 239792458 ooo mic electron charge e0289 49 O permittivity E 6 854187817642 0 F m permeability H 1 125663706146 NAS fine structure of 0 00729735307964482 electron radius 28175405238645 m Borh radius an 52517724924641 m gravitational G N E 6725985E41 m ka 2 gray accel q jaso EE Avagadro num Na 6 022136736623 1 mol Fig 7 3 The Physical Constants form Example of the game is presented in the Fig 7 4 Game 1 0 ioj x New Game Opponent Back 131 Fig 7 4 The Game form 132 8 Graphics and process parameters To control the 2D and 3D plot parameters one need to click right mouse when the cursor is over the corresponding object To control the procedure parameters one need to push the button Set in the procedure panel In the BETACOOL only one procedure is used in the Dynamics form 8 1 2D graph The form for controlling the 2D plot parameters includes 5 tab sheets Axis Grid
102. l exe file is needed for operation however it is more convenient to have all the files specified in this chapter 96 2 Main form Main form of the BETACOOL program is presented in Fig 2 1 It contains the name of current file with parameters and Main Menu F CARUSER SAD TWAC Betacool eta F ioj x File Parameters Effects Task Tools POO EEE Fig 2 1 Main form of the BETACOOL program Main menu Menu item File contains the following sub items Open Save Save as Save On Exit Save DeskTop Graf as Data Open Dialog Exit This item of menu is standard for all applications developed with BOLIDE system Menu item Parameters contains the following sub items Ring Beam Beam evolution Load Lattice It is intended for input general parameters of the storage ring Ring Load Lattice ion beam parameters Beam and representation on the screen beam parameters in the graphic format during calculations Beam evolution Menu item Effects contains the following sub items Ecool IBS RestGas Stochastic Target Injection External heating Each sub item is intended for input the general parameters of corresponding cooling or heating effect used in the calculations Menu item Task contains the following sub items Rates Dynamics 97 The sub item Rates is used for specification of the task which is solved in the current calculation and visualisation in the graphic format the particle life time and heati
103. lating electron beam 5 Lattice functions Friction force Calculation parameters Cooler length m o Magnetic field kG o Beam radius cm 0 Beam current 4 a Beam temperature me Transverse ee Longitudinal ee Neutralization factor Fig 5 1 The Electron Cooling form tab sheet Cooler parameters Form contains four tab sheets Cooler parameters Lattice functions Friction force Calculation parameters and menu item Circulating electron beam Main goal of the BETACOOL is calculation of the beam dynamics in the presence of electron cooling and this part of the program gives additional tools for analysis and optimisation of the electron cooling system The calculations can be performed using different models of the cooling process The components of the friction force used in the calculations can be presented as 3D graph For electron cooling system with circulating electron beam the time dependence of the electron beam temperature during the circulation is presented as 2D graph 110 5 1 1 Tab sheets Cooler parameters and Lattice functions General cooler parameters from the tab sheet Cooler parameters Table 5 1 Table 5 1 Cooler parameters Coller length L m Magnetic field B kG Beam radius ae cm Beam current I A Beam temperature meV Transverse T meV Longitudinal T meV Neutralization factor Nn The ring lattice parameters in the cooling section from th
104. left side of the plot window Fig 8 5 If this parameter is equal to zero the legend is not displayed If parameter Keep Image on Redraw is checked the curves calculated in previous run are not redrawn after push the button Run in the Dynamics form The tab sheet Curves Fig 8 6 contains the list of the curves displayed in the plot The example of the curve checked in the list is displayed in the left upper corner of the tab sheet The curve is visible in the plot only in the case when parameter Visible is checked This tab sheet is used for editing of the line and point stile width and colour Note The line style can be edited only in the case when the line width is 0 Each curve of the plot can be saved in the text format using the button Save The button calls Save as dialog window The BOLIDE automatically generates the extension for the file and this file contains two column first one the independent variable values second function The curve in the numerical format is also displayed in the tab sheet Values Fig 8 7 which structure coincides with the structure of cur file Each curve of the plot can be used for displaying numerical data from the file for this aim the button Load is intended 136 450_Graph Curve Curvel Fig 8 6 The Curves tab sheet 450_Graph Curve ot a po m A p T Fig 8 7 The Values tab sheet 8 2 3D graph The form for control of 3D graph par
105. make a choice of the formula which he prefers to use in calculations For instance Fig 4 1 presents a shape of the friction force of the magnetized electron beam given in 10 The dependence of the friction force or drag rate on particle coordinates appears due to space charge effects of the ion and electron beams Calculations are performed taking into account these effects as described in 1 11 To speed up the program run time user can 78 exclude from the calculations the ion beam space charge effects Electron beam neutralization is taken into account under assumption of parabolic profile of the residual gas ions distribution inside the electron beam Influence of electron beam space charge introduces an electron momentum shift and electron drift velocity both are the functions of ion radial position inside the electron beam In presence of dispersion in the cooling section an asymmetry between vertical and horizontal degrees of freedom appears At last in the case of an intense ion bunched beam the friction force depends on the ion distance from the bunch center The variation of the motion invariants after single passing the cooling section are given by the following expressions x 2 SI daft 7 lj tawh lt b es p Oe 6 Nd 2 Pecan Z 2 2 B x o Eag 4 2 3 B p p 2 B p o Ps P p P 2 gt 5 P P 2 l zon PAB n op 4 2 5 p P P where p _s Ap D and D are dispersion and its deriva
106. me with permanent luminosity during the experiment compensating of the particle losses by corresponding increase of the electron beam current and decrease of the beam emittance see formula 4 4 0 2 0 18 0 16 0 14 0 12 0 1 0 08 0 06 0 04 0 02 0 1 00E 08 1 00E 07 1 00E 06 1 00E 05 Emittance pi m rad Fig 4 16 Cooling rate at electron beam current of 0 5 A 1 and heating rate at pellet diameter of 20 um 2 40um 3 and 80um 4 as functions of the beam emittance beam energy is 1 GeV The Fig 4 15 4 17 are plotted without taking into account intrabeam scattering in the proton beam However the intrabeam scattering at the beam intensity up to 10 particles slightly displaces the equilibrium position only 29 0 14 0 12 0 1 0 08 0 06 0 04 0 02 Li lt o D tz o 2 b _ fo z I e e S 1 00E 07 1 00E 06 1 00E 05 Emittance pi m rad Fig 4 17 The equilibrium position depending on the electron beam current Pellet diameter is 80 um the beam energy is 1 GeV In the experiments with the deuteron beam the electron cooling gives a substantial gain in the luminosity even at big pellet diameter due to small deuteron energy But at deuteron beam energy of 2 11 GeV Fig 4 18 the electron cooling completely suppresses the beam heating only at maximum electron beam current
107. momentum due to electron beam space charge is equal to Ap el 1 s 5 11 E Mn 3 3 5 11 e where Nn is neutralisation factor Correspondingly the maximum current of electron beam is limited by the value 39 3 2 B ym c Og 5 12 face Rae However even at zero neutralisation factor this condition practically does not limit a possible value of the beam current up to magnetic field value of several kG Fig 5 5 More serious limitation of the circulating beam current is determined by the threshold of the microwave instability and it is discussed in the next chapter Imax A 50 40 30 20 Tooo 2000 3000 4000 5000 B G Fig 5 5 Upper limit of the circulating beam current determined by momentum shift due to its space charge Beam energy is 500 keV neutralisation factor is equal to zero 5 3 2 Current limitation due to microwave instability At given maximum value of the momentum spread the longitudinal microwave instability limits the intensity of the circulating beam Dynamics of longitudinal motion in the LEPTA is the same as in a standard strong focusing ring and therefore we can use the usual criterion for longitudinal stability of the electron beam which can be written as follows 10 292 mc p Yhl 2 mE a igh oo 5 13 Ofis a spread in momentum deviation half width on half height Z is a longitudinal coupling impedance of the beam for a mode with number n Factor Fjong takes into account the re
108. n heating due to interaction with the target residual gas intrabeam scattering and stochastic cooling was calculated Fig 3 2 3 3 Maximum efficiency of the stochastic cooling corresponds to the maximum beam energy due to dependence of the mixing factor on off momentum factor When the particle number is higher than about 10 the intrabeam scattering IBS begins to play a significant role that one can see from the change of the curve derivatives in the Fig 3 2 3 3 More attractive region of parameters for the existing stochastic cooling system application is the target density below 10 Atoms cm and particle number of a few units of 10 This region corresponds to long term experiment and the Fig 3 4 presents the emittance and momentum spread evolution during one hour under common action of IBS target residual gas and stochastic cooling Beam energy is 2 7 GeV initial intensity is 2 10 particles target density is 10 Atoms cm After the beam relaxation the stochastic cooling stabilises the horizontal emittance and the momentum spread vertical emittance increases during the first 30 minutes to acceptance limit The computer simulation shows that the particle number decreases during 1 hour of experiment to the value of 7 4 10 Without cooling the final particle number is about 5 6 10 Thus the stochastic cooling application gives the gain in the experiment luminosity of about 30 13 As a conclusion from this analys
109. n mass and M the projectile mass is a quantity which is proportional to the areal density px of the target p target density in g cm M ERL Esg Ai py 1 4 P A where Zp and Zr are the charge number of projectile and target atoms I is ionisation potential J Z x16 eV Experimental value of the energy losses follows from Formula 1 1 R AA o str exp 1 5 ET To revolution period Texp 300 sec is the experiment duration Experimental value of the energy loss during one revolution in the ring is 7 3 meV which gives the target areal density of 1 3 10 Atoms cm The maximum designed areal density of the storage cell is expected to be of 10 atoms cm The frozen pellet target provides a total number of atoms per pellet of 1 8 10 1 1 10 Influence of the target on the beam parameters can be estimated by introducing the effective areal density which is proportional to ratio of the atom number per the pellet to the beam cross section We assume that in each moment of time one pellet is inside the beam At the beam emittance of Im mm mrad the effective target density lies between 3 10 and 2 10 Atoms cm Maximum achieved intensity of the proton beam is 7 10 In the future one can expect improvement of the intensity to 10 or even more by upgrade of the injection system Now the maximum beam energy is 2 65 GeV but it is possible in the future to increase this value to
110. n period Tinj sec 5 7 Effects External heating The linear or diffusion increases of the invariants of motion are calculated in accordance with the chapter 2 5 of ref 1 The diffusion coefficients Prans are imputted in the pi m rad and Piong in rad 125 In the External heating form Fig 5 11 one can enter speed of the emittance growth and diffusion coefficients independently for each degree of freedom The linear increase is included into calculation when the parameter Linear increase is checked The diffusional increase is included when the parameter Diffusion is checked External heating loj x I Linear increas Oe ee Emittance deviation pi m tad sec Vertical emittance pi m rad sec Momentum deviation rad tad sec T Diffusion Horizontal f Vertical E Longitudinal Fig 5 11 The External heating form 126 6 Main menu gt Task 6 1 Task gt Rates At the panel Active Effects Fig 6 1 one need to check the effects which are included in the calculations from the list When pushed the button OK the sum of the corresponding rates is displayed into edit windows The edit windows are used only for output If some parameters of process were entered incorrectly BOLIDE system generates the Warning window analogous to that one in the Fig 4 5 In this case one need to correct parameters in the corresponding form called by sub menu items o
111. nce AQy is the resonance width fc is the particle velocity 2 5 External heating 74 During proton beam injection in the beginning of the stacking process the beam in COSY is additionally heated longitudinally with a white noise frequency band around the harmonic by USE kicker and transversely by noise applied to the stripline until for the tune measurement This parallel heating results in a higher intensity of the cooled beam The noise heating can be characterized with the diffusion power P and the corresponding heating rates are calculated as follows 1 2P rans a ets 2 5 1 Tiong H on The heating effects caused by interaction with residual gas and target atoms lead to linear increase with time of the beam emittances The emittance deviation can be calculated in accordance with corresponding formulae or measured experimentally and introduced into calculation as an external linear heating 1 _ de ot te E 1 _ OH 0t 2 5 2 Tiong H pn where de dt and 0H of freedom correspondingly t are the heating power in the transverse and longitudinal degrees lon 3 Particle losses calculation 3 1 Nuclear scattering on residual gas atoms The cross section for nuclear scattering can be approximated by the expression 7 O aimb Are ac 66 3 1 1 Picev ic Introducing the nuclear scattering density nA 3 1 2 we can express the life time as the following ns T e gt 3 1 3 i Bensons where Ons is th
112. nce m Circulating period msec Number of electrons Calculation parameter Number of steps during cire period Show temperature ioj x Fig 5 5 The Electron ring parameters form Table 5 3 Electron ring parameters Electron ring circumference C m Circulating period Teire msec Number of electrons CI output parameter N c Number of steps during circ period n Is used in accordance with algorithm described in formulae 4 2 17 4 2 19 in ref 1 5 1 5 Effects ECool Circulating electron beam Show temperature button With click on Show temperature button one can call the form Electron temperature Fig 5 5 which presents the electron beam longitudinal and transverse temperature time dependencies during taken circulating period The calculation of the dependencies can take a long period of time and they will be plotted only when the calculations finished 10 gt T E T tm T a E o od Time msec Fig 5 5 The Electron temperature form 5 2 Effects IBS The form that dedicated to enter parameters required for Intra Beam Scattering rates calculation is presented in the Fig 5 6 Intra Beam Scattering loj x Calculation model _j i Fiwinski Martini Calculation parameters Nuber of integation step Fig 5 6 The Intra Beam Scattering form In the panel Calculation model you need to make a choice bet
113. ng or cooling rates for beam parameters The sub item Dynamics is used for input parameters of the numerical algorithm for beam parameter dynamics simulation and for control of the calculation process Menu item Tools contains the following sub items Calculator Periodic Table Constants Game The sub items are not used in the calculations directly but can help in the preparation of the initial parameters Calculator is the programming calculator with general algebraic functions Periodic Table contains Mendeleev s Periodic Table of elements Constants the table of numerical values of a number of physical constants Game is included into program especially for our Japanese colleges and presents the version of Ren Dzu game in the field 40x40 98 3 Main menu gt File 3 1 File Open This sub item calls the standard Windows open dialog window Fig 3 1 pen Look in E TWAC T ee Betacool eta a TAl eta aa TC eta a TCo eta a TWAC Leta Files of type BetaCool Parameters eta Cancel Fig 3 1 Open dialog window Za Look in indicates the current directory File Name edit box is used for input the file name or file name can be chosen from the list Files of type default parameter is eta and it can be changed by all types or arbitrary extension 3 2 File Save This sub item is used for saving of current parameters in the file which name is indicated in the left corn
114. nteger and half integer resonances This value can be estimated by the following expressions AQ hast __ Ap z a Q fast j p P 1 so 4 AQ fast 5 9 which gives us 38 Ap 1 5 10 a o E On 5 10 Fig 5 4 presents the dynamic aperture value at the ring circumference of 20 m Width and power of resonances are determined by the errors of the focusing fields Fast crossing of high order resonance leads to increase of the beam transverse temperature only However during a long circulation period the stability conditions 5 8 have to be satisfied to keep a good beam quality even at small instability increment At minimum electron beam energy and magnetic field of 1 1 5 kG the value of the ring dynamic aperture Fig 5 4 is approximately equal to 2 10 and such a value of the electron beam momentum spread has to be provided by the system of the electron beam acceleration At maximum electron beam energy this limitation is not so strong Oo 0 015 0 01 0 005 0 500 1000 1500 2000 B G Fig 5 4 Dynamic aperture on momentum deviation as function of longitudinal magnetic field 1 beam energy is 500 keV 2 beam energy is 1 5 MeV In principle the maximum permitted value of momentum spread limits the circulating beam current because the momentum shift between particles at the axis and at the beam radius produced by the beam space charge has not to exceed Ap Dios The shift in electron
115. of calculations User manual will be included as an addendum into a final report References 1 A Lavrentev I Meshkov The computation of electron cooling process in a storage ring preprint JINR E9 96 347 1996 2 F Hinterberger D Prasuhn Analysis of internal target effects in light ion storage ring NIM A 279 1989 413 422 3 COSY Annual report 1997 4 P Zenkevich et al Modeling of electron cooling by Monte Carlo method to be published 5 A Piwinski Proc 9 Int Conf on High Energy Accelerators p 105 1974 6 M Martini Intrabeam scattering in the ACOOL AA machines CERN PS 84 9 AA Geneva May 1984 7 N Madsen D Moehl at al Equilibrium beam in the Antiproton Decelerator AD NIM A 20000 54 59 8 A Wolf et al Recombination in electron coolers NIM A 441 2000 183 9 I N Meshkov Fiz Elem Chastits At Yadra 25 1994 1487 Engl transl Phys Part Nucl 25 6 1994 631 10 V Parkhomchuk New insights in the theory of electron cooling NIM A 441 2000 9 17 91 11 A Sidorin I Meshkov T Katayama P Zenkevich Ion Bunch Stability in the Double Storage Ring RIKEN AF AC 19 February 2000 12 I Meshkov Electron cooling with circulating electron beam in GeV energy range NIM A 441 2000 p 255 92 Annex II User manual for BETACOOL program A Sidorin A Smirnov G Trubnikov Contents Introduction 1 The BETACOOL files 2 Main form 3 Main menu File 3
116. oling 6 European Particle Accelerator Conference Stockholm 22 26 June 1998 M Chelnokov V Kalinichenko Yu Korotaev I Meshkov S Mironov A Petrov I Sedykh A Sidorin A Smirnov E Syresin I Titkova G Trubnikov Modified Betatron Prototype Dedicated to Electron Cooling with Circulating Electron Beam Proc HEACC 98 Dubna 1999 p 413 I Meshkov Electron cooling with circulating electron beam in GeV energy range NIM A 441 2000 p 255 8 I Meshkov A Sidorin A Smirnov E Syresin The particle dynamics in the low energy storage rings with longitudinal magnetic field 6th European Particle Accelerator Conference Stockholm 1998 p 1067 9 I N Meshkov A O Sidorin A V Smirnov E M Syresin G V Trubnikov Betatron program for simulation of particle dynamics in storage ring with strong coupling of transverse coordinates proceedings of ICAP2000 Darmstadt to be published in Phys Rev Accelerators amp Beams 10 I Meshkov A Smirnov A Sidorin E Syresin E Mustafin P Zenkevich The Stability of the Circulating Electron Beam in the Electron Cooling System Based on the Modified Betatron Proc HEACC 98 Dubna 1999 p 416 11 A V Aleksandrov N S Dikansky N Ch Kot V I Kudelainen V A Lebedev P V Logachev Proc Workshop on Electron Cooling and New Cooling Techniques Legnaro 1990 World Scientufic 1991 p 279 12 JINR electron cooling group The report of the Designing of the Modified Betatron Prototype JINR D
117. oling heating rate sec4 1 0 005 1 0E08 1 0E 07 1 0E 06 1 0E 05 Emittance pi m rad Fig 4 15 Cooling rate at electron beam current of 0 5 A 1 and heating rate at pellet diameter of 20 um 2 40um 3 and 80um 4 as functions of the beam emittance beam energy is 2 7 GeV At the pellet diameter of 20 um the emittance value corresponded to the stable equilibrium point 3 in the Fig 4 15 is less than in the previous case and unstable equilibrium lies outside the ring acceptance At low beam energy the maximum of the cooling rate is displaced to the region of higher emittances this is a kinematical effect the same emittance at low energy corresponds to low particle transverse velocity in the particle rest frame and the maximum value increases due to strong dependence of the cooling time on the particle energy in the laboratory frame At the beam energy of 1 GeV the stable equilibrium points exist and unstable ones are outside the acceptance at all pellet diameters Fig 4 16 In this case the equilibrium beam emittance value can be varied by corresponding variation of the electron beam current The range of the emittance variation is illustrated by the Fig 4 17 At electron beam current of 150 mA the 28 stable equilibrium is reached at the emittance value of about 1 5 m mm mrad and the equilibrium displaces to the value of about 0 8 7 mm mrad at the electron beam current of 500 mA Thus one can obtains the regi
118. oling system Band I Band II Lower frequency GHz 1 1 Upper frequency GHz 1 8 3 Bandwidth GHz 0 8 2 Fig 3 1 shows the emittance and momentum spread evolution of the proton beam at energy of 1 GeV and initial particle number of 10 under common action of the hydrogen target of 10 Atoms cm areal density residual gas intrabeam scattering and stochastic cooling As one can see the emittances in both plane increase to acceptance limit during first 30 seconds and after that cooling does not influence on this value Momentum spread increases during first part of the calculation period but after about 3 minutes cooling begins to prevail on the heating processes and momentum spread decreases To this moment the particle number decreases to the value of about 2 10 and continues to decrease during last minutes Stochastic cooling does not influence on the particle losses at these experiment parameters At target density of 10 Atoms cm the similar situation takes a place at maximum proton beam energy of 2 7 GeV and initial beam intensity of 10 particles stochastic cooling does not compensate emittance growth and the gain in the particle number after 5 minutes of experiment accompanied with cooling is less than 10 In order to estimate a range of the beam parameters in which the stochastic cooling can be effective in the experiment with internal target the particle number corresponding to an equilibrium betwee
119. ombination with stochastic one for luminosity preservation in experiments with internal targets It presumes numerical calculation of the ion beam parameter variation under common action of different heating and cooling processes with and without electron cooling which permits to estimate the gain in luminosity and its life time value due to electron cooling application comparison of efficiency of two medium energy electron cooling methods with single pass electron beam and circulating one preliminary design of the electron cooling system for COSY at energy range required for experiments with internal target elaboration of the experimental program dedicated to investigation of the electron beam dynamics in electron cooling system with circulating electron beam using the LEPTA ring which is under construction in JINR This report includes description of the numerical methods and computer program developed for beam parameter calculations 63 Introduction Luminosity of an experiment with an internal target in an ion storage ring at conventional experiment setting up is limited in time due to the growth of the circulating beam emittance and momentum spread related to different heating effects Other limitation is connected with particle losses during experiment General heating effects are the following interaction of the beam ions with the target atoms intrabeam scattering of the ion beam ion multi scattering on res
120. oscillation can be measured using pick up electrodes The frequency of the slow oscillations at beam energy of 10 keV is about 700 kHz and corresponds to coherent oscillations of circulating beam The tune of the fast mode of oscillations can not to be measured directly characteristic value of the frequency of the fast mode of the betatron oscillations is about 1 GHz and the signals produced by each particle on pick up electrodes are incoherent Its measurements we plan to perform by displacement of the beam energy to the region of the coupling resonance which leads to increase of the amplitude of the slow mode For this aim small RF cavity will be installed in the ring In order to test beam parameters variation after the resonance crossing the beam will be extracted from the ring and directed to the optic diagnostics for the beam temperature measurement The longitudinal temperature of the circulating electron beam will be measured using Schottky diagnostics This permits to measure growth rate of the longitudinal temperature due to transverse longitudinal relaxation which is impossible in the case of single pass electron beam of a traditional electron cooling system Experimental study of the relaxation process allows finding the limit of the maximum value of the electron beam current in the electron cooling devices with circulating electron beam 54 Summary 1 Role of the cooling in experiments with internal targets The beam energy and re
121. perature 0 5 1 eV the magnetic field value of 1 1 5 kG is big enough to provide beam magnetisation More strong limitation of the magnetic field value follows from the requirement of suppression of the transverse longitudinal relaxation in the electron beam during long circulation period This process will be discussed below 5 2 Principles of electron cooling with circulating electron beam The limitations of the electron beam circulating period are determined by the energy exchange between proton and electron beam due to the cooling process The principle scheme 7 of the cooler with circulating electron beam Fig 5 1 includes an injector electron gun storage ring and electron collector electron dump The ring has straight section inserted in the structure of a hadron storage ring where electron beam merges with proton one and cools it Due to interaction between the particles antiprotons ions and electrons the particle temperature decreases when the electron one increases The variation of both temperatures in Maxwellian plasma is described by the equations 7 dT 4V2nn 27e L T T dt y mM T T 3 2 EEA M m 5 4 34 dT dT N T N Pi 5 5 where T T are the particle and electron temperatures in the particle rest frame m and M their masses N Ne the particle numbers in the rings n is the ratio of the cooling section length to the circumference of the particle
122. quired target density in experiments with internal targets is determined by physical problem to be investigated Investigations of nature of ao mesons are to be performed at maximum proton beam energy required level of the luminosity is 10 cm sec and higher This experiment can be provide using a pellet target Measurements of the deuteron break up n K mesons production are performed in the wide energy range from 1 GeV to about 2 GeV and with all the types of the ANKE targets The effective density of the hydrogen cluster beam target achieved in the experiment at January February 2001 is about 2 10 Atoms cm Expected areal density of the gas storage cell is 10 10 Atoms cm Effective density of the frozen pellet target is expected to be between 3 10 and 2 10 Atoms cm at beam emittance of 1 m mm mrad In the experiments with the internal target the upper limit of the circulating beam life time and as a result the experiment duration is determined by single scattering on large angles with the target atoms At the target density below 10 Atoms cm the beam life time and characteristic times of the emittance growth due to interaction with the target are longer than one hour and experiment can be performed without cooling At the effective target density over than approximately 2 10 Atoms cm the beam life time is shorter than expected cooling times at realistic parameters of the cooling system In this case cool
123. rameters 4 1 Parameters Ring This sub item calls the form for input the storage ring parameters Fig 4 1 f Ring parameters oe cx Load lattice lon kind Lattice parameters RF system Vacuum Input parameters Atomic number Charge number Energy MeV amu Life time sec Relativistic factors Fig 4 1 Form for input the storage ring parameters The form contains four tab sheets Ion kind Lattice parameters RF system Vacuum and menu item Load Lattice The menu item calls the same form as the main menu sub item Parameters Load Lattice and it is described in the chapter 4 4 The tab sheet RF system is visible only in the case when in the Beam Parameters form Fig 4 2 the option Bunched is chosen 101 The parameters indicated in the edit boxes of the Ring Parameters form are specified in the tables 4 1 4 4 Table 4 1 Tab sheet Ion kind Input parameters Atomic number A Ion atomic number Charge number Z Ion charge number Energy E Mev Amu Ion energy Life time Tlife Life time of the radioactive ion that is used in the calculation when the process Decay is chosen in the list of the processes in the form Rates chapter 6 1 Relativistic factors gamma The usual relativistic beta T factors output parameters Table 4 2 Tab sheet Lattice parameters Input p
124. rease of magnetic field value due to dependence of Goon the magnetic field see Fig 5 4 I A 100 N 1 400 600 800 1000 1200 1400 1600 Beam energy keV Fig 5 6 Threshold electron beam current of microwave instability 1 magnetic field is 1 kG 2 magnetic field is 2 KG At minimum electron beam energy the threshold beam current is higher than maximum designed value only by two times however even taking into account decrease of the cooling efficiency for the system with circulating electron beam required value of electron beam current at minimum energy does not exceed 0 3 A 41 5 3 3 Transverse longitudinal relaxation of electron beam In the magnetised electron beam intrabeam scattering is substantially suppressed by longitudinal magnetic field Characteristic time of the temperature growth rate can be estimated by the following empirical expression 11 1 1 dT n 2nr n m Le mc e728 on 5 16 T T dt T T relax where re is the electron classical electron radius Le 10 is the Coulomb logarithm ne is the beam density p 4 2mc T eB is the particle Larmor radius in the longitudinal magnetic field B Exponential term in 5 16 leads to strong dependence of the relaxation characteristic time on the temperature of transverse degree of freedom and on the beam current Fig 5 7 At electron beam current of 0 1 A the magnetic field value of 1 5 kG seems to be big enough to suppress intrabeam scattering
125. resented version of the numerical program the particle losses due to longitudinal acceptance limitation are not taken into account correspondingly presented calculations can underestimate the particle losses due to growth of the momentum spread The next chapters of the report are dedicated to calculations of the beam parameter evolution in the presence of stochastic or electron cooling 9 z8 i e z 6 F 5 SA 7 3 c 2 O 1 0 0 1 2 3 Beam energy GeV Fig 2 5 Ultimate gain in the beam intensity which can be obtained using cooling at the experiment duration of 5 minutes 1 proton beam 2 deuteron beam The hydrogen target density is 10 Atoms cm initial beam intensity is 10 particles 11 14 7 T 2 A Tae 33 6 S 9 PE 2 O 0 0 1 2 3 Beam energy GeV Fig 2 6 Ultimate gain in the beam intensity which can be obtained using cooling at the experiment duration of 7 hour 1 proton beam 2 deuteron beam The hydrogen target density is 70 Atoms cm initial beam intensity is 10 particles 12 3 Beam parameter evolution with existing stochastic cooling system The parameters of the stochastic cooling system used in the following calculations are presented in the Table 3 1 The band I is used for horizontal cooling the Band II for longitudinal one Table 3 1 Parameters of stochastic co
126. rget can be estimated as follows 1 Ty are et T a 2 1 4 1 P E T a targ et Tife arg et where 7 is the target parameter which also can be introduced as input parameter for calculations 2 1 3 Target parameters calculation The parameters Ap Piarget Ost Esr can be calculated in accordance with the following formulae 2 0 2NxT Ka alc sap 2 1 5 B 2 CAE a i gt 2 1 6 A E a 75 2 1 7 targ et Here Nx is the number of targets atoms per unit area Zp and Zy are the charge number of projectile and target atoms p and E is the particle momentum and kinetic energy re is the electron classic radius The parametrs 2 and Ab are given by the equations D OR Ay 2 1 8 where D h pis De Broglie wavelength Ar and Ap are the mass numbers of the target and the projectile ro 1 3 fm Z 2 x 1 1307 ss ft zzl 2 1 9 69 where D Q 2 1 10 l 0 8854 Z2 eZee do 0 529 10 cm denotes the Bohr radius 2 2 Ab l In 1130p Sas 2 1 11 Z Zp 1 B 2 where uin is a constant determined by the electron configuration of the target atom from the Thomas Fermi model one finds uin 5 8 for the H atom exact calculation yields tutin 3 6 for Li and O atoms the values of uin are 4 6 and 5 0 respectively Emax is the maximum energy loss in a head on collision of the projectile with a target electron 2 DD ao me BY 2 1 12 max 2 1 2y da EEA Mie
127. rgets used in the different internal experiments at COSY Target COSY 13 EDDA COSY 11 ANKE strip or filament strip filament strip cluster X x pellet x atomic beam X X storage cell X x At the first stage of numerical calculations we plan to investigate the beam parameters evolution in the ANKE experiments The results for the cluster target will be compared with the COSY 11 experiment The cluster beam of the COSY 11 cluster target has always the maximal density of 10 atoms cm The cluster beam is round with a diameter of 9 mm ANKE is the experiment where all types of targets will be applied in time Up to now solid strip targets Table 2 and the cluster target H D were used Table 2 The ANKE solid targets Type Density ug cm CH2 200 C in form of polycristalline 500 1000 maximum density is not clear 1 7 or 3 5 diamond g cm or in between Cu 200 Au 400 The solid targets were flat stripes of 18 mm length and shaped in form of a triangle 2 mm width above with the tip at the lower end in order to minimize the beam target overlap as few as possible The targets were positioned above the beam axis and the p beam was moved into the target with beam orbit deformation Typical cycle times were of 1 minute The experiments were also run with a constant target rate by applying a feedback loop The ANKE cluster beam apparatus is very similiar to COSY 11 ho
128. ring Lc Coulomb logarithm ne the electron density t current time in Laboratory reference frame The particle temperature in a cooler with single pass electron beam and with Maxwellian velocity distribution decreases in accordance with the 1st equation where T const Therefore electron temperature in cooling process increase very fast and electron beam is to be renewed after electrons have got a significant temperature For typical parameters of the cooler the number of circulating electrons N is comparable with that one of the particles N circulating in COSY WLM ELECTRON GUN Injector Fig 5 1 Principle Scheme of Electron Cooler with Circulating Electron Beam For electron beam with flattened velocity distribution electron beam parameter evolution during cooling process can be calculated as described in 1 An example of the electron temperature variation during circulation is presented for COSY parameters in the Fig 5 2 35 4000 Temperature me 2000 o 2500 5000 Time msec Fig 5 2 Evolution of the transverse 1 and longitudinal 2 electron beam temperature during circulation Proton beam energy is 1 GeV proton number is 10 initial transverse electron temperature is 300 meV longitudinal 50 meV electron beam radius is 0 5 cm current 0 5 A electron ring circumference is 20 m magnetic field is 1 2 kG cooling section length 1 4 m O NWA ON W
129. ring experiment can give additional gain in the luminosity of about 50 In the experiment with the pellet target the luminosity is inversely proportional to the circulating beam cross section because of the beam cross section always larger than pellet one In this case the beam cooling can provide increase the luminosity by several times When the cooling prevails on the heating effects one can realise the regime with constant luminosity during the time of experiment via compensation of the particle losses by corresponding decrease of the beam emittance Thus in the experiments with the gas jet and cluster beam targets the experiment duration can be about one hour without substantial loss of the luminosity and cooling application can give gain in the luminosity from 50 to 5 10 times at energy from 1 to 2 7 GeV In the experiment with the pellet target the experiment duration can be about 5 minutes and cooling application can give a gain in the luminosity of about 2 3 times in the total range of the beam energy 2 The possibility of the existing stochastic cooling system application The effectiveness of the stochastic cooling strongly depends on the ion beam intensity In this report the possibilities of the stochastic cooling were investigated under assumption that one of the chains is used for cooling of the horizontal degree of freedom other one for longitudinal In this case the stochastic cooling system can be used for luminosity preserv
130. ron cooling system with circulating electron beam has substantially low cost but in such a system the electron beam quality is determined by a method of the beam acceleration stability of electron motion in the electron storage ring number of electrons and the ring circumference The required electron beam energy lies in the range from 0 5 to 1 5 MeV Stable motion of the intensive electron beam at such relatively small energies can be provided only in the storage ring with longitudinal magnetic field Electron beam acceleration can be provided using linear 31 RF accelerator which periodically fills up the ring with new portion of electrons or directly in the electron storage ring using induction acceleration In the first case longitudinal temperature of the electron beam can be even higher than transverse one and one of the general problems of this method is to adjust the momentum spread of the electron beam with the ring dynamic aperture on momentum deviation Induction acceleration of the electron beam permits to keep the flattened velocity distribution of the electrons and in principle provides the same longitudinal temperature as HV accelerator However at constant value of the guiding magnetic field the working point crosses resonance regions of the Larmor oscillations Due to this fact the transverse temperature of the electron beam can be higher than one in traditional cooling system by several orders of magnitude The design of th
131. s Other Curves Values which are presented in the Fig 8 1 8 3 8 6 In the upper string of the form the Graph number and name of the current curve are indicated 3000_Graph Curvel Axis Grids Other Curves Values Temperature me ak 4 100 Sl II Min Time msec 4 gt Min 4 gt Max Digits l Log a Cancel Fig 8 1 The Axis tab sheet The Axis tab sheet is used for modifying the scales and type linear or logarithmic of x and y axis In the case when the Maximum or Minimum value is determined incorrectly BOLIDE system generates Warning window Fig 8 2 133 1 BOLIDE Warnings iof x TGrat has negative or zero value of the logarithmic scale Graph 10 Fig 8 2 The BOLIDE Warning window The warning includes short description of the mistake and number of the Graph To prolonge the work after the Warning message one need to correct the mistake and push the button Accept Note Depending on Windows version the control of the plot can be loosed after mistake in the plot parameters In this case one need to input correct values of the plot parameters push the button Accept save current file in necessary exit BETACOOL and start it again 450_Graph Curve AVIS Grids Other Curves Values Rates 1 sec Major style and widt Tick Number kotia Major 2 M Minor f gt E jo Color tim TE i Minor style and widt ick Number
132. tically indicated If you save a file without specification of file extension the program automatically generates the extension eta When program starts the file Betacool eta is automatically loaded if exists in the current directory When the file loaded all the editing windows in all the form of the program are filled with the data from input file Modifying the existing file and saving it with new name one can generate new input file The files Betacool grf Betacool srf contain the parameters of all the plots in the program scale of the axis intervals and so on and are automatically loaded when program starts If these files do not exist in the current directory the plot parameters are generated automatically and when the program has been closed these files are generated and saved in the current directory The file Betacool top contains the status of all the forms of the program during the previous session The files with extension Its lf are used in the case of intrabeam rates calculation File Its contains lattice parameters of the storage ring in text format and can be generated by different programs of particle dynamics simulation for instance MAD File llf contains specification of the Its file and must be generated by user in accordance with the format of using Its file 95 All the files required by BETACOOL program except the Its file can be generated during the program work and only BetaCoo
133. tion has a lower cost in comparison with RF acceleration but general problem of this system is beam quality degradation after crossing of number of resonances This problem as well as optimisation of injection system can be experimentally investigated at LEPTA ring 48 7 Program of experiments at LEPTA 7 1 LEPTA parameters The general aim of the LEPTA ring construction Fig 7 1 is to provide storing low energy positrons electron cooling of positrons and generation of the directed flux of positronium atoms However at design study of the ring all the parameters were chosen such that we have a possibility to test beam parameters required for electron cooling system with circulating electron beam The parameters of the ring in the mode with circulating electron beam Table 7 1 are closed to the parameters of COSY cooling system and designed parameters of the betatron yoke cover the total energy range of the circulating beam required for COSY Adjustment of the injection and extraction system investigations of an influence of the helical quadrupole winding will be performed during ring commissioning using optic method of the beam temperature measurement 3 At the first stage of the LEPTA operation the study of the beam dynamics will be performed at the beam energy of 10 keV and variation of the beam energy will be provided using small RF cavity Final test of the system will be performed after installation of the betatron yoke and it presum
134. tive in the cooling section P P D D D The expressions 4 2 3 4 2 5 include the diffusion terms proportional to the square of momentum deviations These terms determine equilibrium beam parameters in absence of other heating effects In presence of more powerful heating processes like IBS or scattering on residual gas atoms we can ignore the diffusion effects and speed up the program runtime excluding them from the calculations 79 1 5E 5 Force mradsurn 7 5E 6 o 0 4 0 2 angle mrad Force mradturn o 0 1 0 2 angle mrad Fig 4 1 An example of the dependencies of the transverse upper picture and longitudinal lower picture components of the friction force on the ion momentum and angular shiftcalculated with friction force formulae from Ref 10 Under assumption that the ion distribution over betatron and synchrotron phases is uniform in an interval 0 27 a stationary beam we can calculate average invariant deviation for ions having the same invariants of motion 80 20 0 These values calculated with invariants of motion are equal to beam emittances and can be used for evaluation of so called single particle cooling rate 1 _ 1 8 aN fie cool sp rev 4 2 7 T where T ev is the particle revolution period in the storage ring The cooling rates for ion beam with Gaussian distribution are calculated in BETACOOL by averaging 4 2 6 over distribution funct
135. to the vicinity of the equilibrium orbit at the distance of about diameter of the gun solenoid Than the kicker plates displace the beam to the equilibrium orbit Inside the septum coil the electron beam drifts in the longitudinal magnetic field of the septum solenoid and transverse magnetic field of the septum coil The field of the septum coil has a zero value at the orbit of circulating beam due to special coil design Possible design of the septum coil is presented in the next chapter The septum coils are operated in the DC mode The field of the kicker displaces the electron beam to the equilibrium orbit and after the beam occupies the total ring circumference the field of the kicker has to be switched off during the time substantially shorter than revolution period Drift of the electrons inside the kicker can be performed by transverse magnetic field or by electric field When electrostatic kicker is 47 applied the cooling system can be used also in the single pass mode of operation In this case both kickers injection and extraction are used in the DC mode and electron beam is displaced by injection kicker to the orbit inside the cooling section and after moving through the cooling section is displaced to extraction septum and to the collector by extraction kicker Such a possibility looks like very attractive for COSY it permits to use electron cooling system at injection energy in the single pass regime with existing electron gun an
136. tracted to the possibility of the electron cooling system with circulating electron beam In the electron energy range from 0 5 to 1 4 MeV only a longitudinal magnetic field can provide strong focusing of an intensive electron beam General parameters of the electron ring with longitudinal magnetic field can be the following magnetic field value is 1 1 5 kG the ring circumference is about 20 m electron beam radius is 0 5 mm maximum current value is 0 5 A The electron beam acceleration can be provided using RF electron linac or by induction acceleration of the electrons in the ring The induction acceleration is more attractive from the side of the momentum spread of accelerated beam and installation cost but it did not tested experimentally The proposed design of the injection system of the electron ring permits to perform two modes of operation with the single pass electron beam at injection energy and with the circulating electron beam at the energy of experiment Some elements of the existing COSY electron cooling system can be used in the new cooling device General problems of the proposed cooling system are the following distortion of the electron beam parameters during its injection into the ring motion stability of the electron beam during circulation heating of the electron beam due to its interaction with the cooled ion one coherent energy losses of both beams ion and electron due to interaction of the ion beam
137. ubna 1997 13 C Kapetanakos et al The Naval Research Laboratory Modified Betatron Accelerator and Assessment of its Results Phys Fluids B 5 7 1993 p 2295 14 K Balewski et al Preliminary Study of Electron Cooling Possibility of Hadronic Beam at PETRA 6 European Particle Accelerator Conference Stockholm 22 26 June 1998 p 1079 60 Annex I Electron cooling application for luminosity preservation in an experiment with internal targets at COSY Interim report JINR I N Meshkov Yu V Korotaev A L Petrov A O Sidorin A V Smirnov E M Syresin G V Trubnikov S V Yakovenko COSY H J Stein Dubna 2001 61 CONTENTS Abstract Introduction 1 General description of the physical model 1 1 Beam emittances and particle number evolution 1 2 Particle losses and emittance calculation in the presence of beam scrapers 2 Heating processes involved into calculations 2 1 The interaction with the target atoms 2 1 1 Description of the COSY internal targets 2 1 2 Emittance growth and beam lifetime 2 1 3 Target parameters calculation 2 2 Intrabeam scattering of the ion beam 2 2 1 Piwinski model 2 2 2 Martini model 2 3 Ion multi scattering on residual gas atoms 2 4 Tune resonances of the ion betatron motion 2 5 External heating 3 Particle losses calculation 3 1 Nuclear scattering on residual gas atoms 3 2 Single scattering on large angles 3 3 Recombination in the electron cooling section 3 4 Ot
138. ulations of formula 2 1 1 Alpha Qhor ref 1 Dispersion D m Dispersion derivative D Vertical Beta Bvert m Alpha vert When the internal target is used for multiple charge exchange injection the parameter Stripper foil is to be checked and parameters of the injection system are to be entered in the tab sheet Injection Table 5 6 Table 5 6 Injection Repetition period Tia rget sec Number of crossings Ntarget are used for calculations of formula 2 1 2 ref 1 5 5 2 Target material form 123 The target parameters used for invariant variation calculation can be calculated using characteristics of the target material density and ion beam parameters using the menu item Target material of the Target form The target parameters are entered in the Target material and geometry form Fig 5 10 They are listed in the Table 5 7 The calculations are performed in accordance with chapter 2 1 3 ref 1 Mass number ee Charge number y Density atoms cm 3 ee Density gfem 3 ee Length cm H Electron config constant o Calculate Fig 5 10 The Target material and geometry form Table 5 7 Target material and geometry Mass number AT Charge number ZT Density N atom cm input parameter Density p g cm is output parameter calculated from N Length x cm Electron cooling constant Uin output parameter Effects Target Target mat
139. umber decreases from 10 to 3 6 10 The particle number after five minutes of experiment calculated without taking into account aperture limitation is equal 10 to 4 85 10 The ratio between final particle number calculated without and with the aperture limitation can be used as a parameter which characterises the gain in the experiment luminosity when the cooling completely suppresses the emittance growth The ultimate possible cooling effect on the experiment ratio of the beam intensity with cooling to that one without cooling at high target density is presented in the Fig 2 5 The possible gain in the particle number after one hour of experiment at the target density of 10 Atoms cm is presented in the Fig 2 6 One can see that the beam cooling can provide maximum gain in the beam intensity at minimum beam energy and for the proton beam where this value is up to ten times The gain for deuteron beam is about two times less due to intrabeam scattering dependence on the particle mass At maximum energy the gain decreases to the value of about 50 It is explained by the strong dependence of the intrabeam scattering on energy as B y As result at maximum beam energy the emittance growth due to this process does not play a significant role The difference between hydrogen and deuterium targets is negligible because of the main processes are determined by the interaction of the beam with electrons of the target atoms In the p
140. used for Schnell Zotter criterion calculation formula 5 2 in ref 1 Factor of distribution F Is used for Schnell Zotter criterion function Ft calculation formula 5 2 in ref 1 for standard criterion it is equal to 1 Transverse coupling Zi Ohm m This value is added to space charge impedance transverse impedance Dipole instability Formula 5 4 in ref 1 106 Table 4 7 Tab sheet Bunch parameters Number of bunches Number of bunches in the bunched beam Calculated parameters RMS bunch length O cm Is calculated in accordance with formula 5 8 in ref 1 without taking into account the synchrotron tune depression Maximum particle number No Formula 5 6 in ref 1 Synchrotron tune Q Formula 5 5 in ref 1 Bunching factor Br 4 3 Parameters Beam evolution This sub item calls the form for output the beam parameters in the graphic format Fig 4 3 This form can be called also by the menu item Show in the Beam Parameters form Beam Parameters a Zio T E E a a a G E E wi Fig 4 3 The form for output the beam parameters in the graphic format The form includes 7 tab sheets Beam parameters Emittance Beam parameters Momentum Beam parameters Bunch length Beam parameters Betatron tune Beam parameters Synchrotron tune Beam parameters Stability Beam parameters Particle number 107 The tab sheets Bunch length and
141. ut any emmitance growth However this overestimation is negligible as long as y lt 0 1Y If particle loss occur in both transverse planes the total survival probability P is the product of two one dimensional probabilities Po PG P G 1 7 The particle loss on scrapers can be included into calculations at emittance growth amp gt 0 by introduction of the lifetime 1 Jae P a _ tot 1 8 T site T rev where Tev is the revolution period In the case of particle loss on scrapers the emittance growth has to be corrected in accordance with The function f A for lt 0 05 and tends rapidly to the constant asymptotic value of 0 3084 The asymptotic value is in accordance with the theoretical expectation for a homogeneous density distribution with a smooth cutoff near the amplitude limit 2 2 Heating processes involved into calculations 66 2 1 The interaction with the target atoms 2 1 1 Description of the COSY internal targets COSY experiments are performed in the energy range from 0 5 to 2 6 GeV polarized or unpolarized protons There are various heating effects caused by the various targets solid strip targets cluster and pellet targets and a storage cell used especially for polarized atoms There are four experimental areas with internal targets ANKE COSY 11 COSY 13 and EDDA 3 The types of the targets being used in the experiments are listed in the Table 1 Table 1 The type of ta
142. ween two possibilities Piwinski calculations are performed in accordance with chapter 2 2 ref 1 Martini calculations are performed in accordance with chapter 2 2 2 ref 1 120 Number of integration step is calculation parameter which specify the algorithm of numerical calculation of corresponding integrals 5 3 Effects RestGas The sub menu item RestGas calls the form Scattering on gas Fig 5 7 where one can include inside the calculation following processes Single scattering chapter 3 2 in ref 1 Nuclear scattering chapter 3 1 in ref 1 Multiple scattering chapter 2 3 in ref 1 Electron capture For calculation of the single scattering process the ring acceptance is to be entered in the form Ring parameters Scattering on gas Ioj x Nuclear Scattering Multiple scattering Electron capture Fig 5 7 The form Scattering on gas 5 4 Effects Stochastic For each degree of freedom of ion oscillation there is a corresponding tab sheet Fig 5 8 where one need to enter Lower frequency Upper frequency both in GHz Parameter used is to be checked in the case when corresponding chain of the stochastic cooling system is used in the calculations The cooling rates are calculated in accordance with chapter 4 1 1 ref 1 121 Fig 5 8 The Stochastic form 5 5 Effects Target 5 5 1 Target form Stochastic E used Lower frequency GHz Upper frequency GHz ii i
143. wever it has different dimensions It is of rectangular shape 5 x 10 mm 10 mm longitudinally We take in calculations the maximum value of the areal density equal to 5 x 10 atoms cm The pellet target is planned to have parameters listed in the Table 3 Table 3 Parameters of frozen pellet target 67 Diameter of pellet um 20 80 mass g 3 10 _ 2 10 total number of atoms per pellet 1 8 10 1 1 10 Speed m s 40 100 Pellet flux pellet sec 10 The storage cell at ANKE will be fed by an atomic beam source 10 atoms cm The maximum areal density of 10 atoms cm is expected The cell cross section is 30 x 10 mm 10 mm horizontal the length along the beam is 400 mm 2 1 2 Emittance growth and beam lifetime The parameters of the circulating ion beam vary at single pass through the target in accordance with the following expression 4 P targ et 2 A AE joy Bror i T H ror Ae vert Ben str gt Z 1 x 1 fi 2 E AE ing at 2 Ap B YAE P target where Ap Purgets Ost Esr are the target parameters Bror Byer are horizontal and vertical beta functions in the target position 2 H 1 aL D 20 hor n DD Bpo D gt Qhon D are the alpha function and the dispersion hor in this point The target parameters Ap Piarget Ostr Esr are calculated for the given width and material of the target parameters and particle energy either can

Annex I Electron cooling application for luminosity preservation in an

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