Numerical approach for simulation of Palmer cooling

In this paper the time domain algorithm is described and applied for the Palmer Cooling method. The possibility of using the Palmer system in such case for simultaneous longitudinal and transverse cooling by a suitable choice of the pickup to kicker distance was described by Hereward [1, 2]. Using...

Ausführliche Beschreibung

Gespeichert in:

Bibliographische Detailangaben
Datum:	2018
Hauptverfasser:	Dolinska, M.E., Doroshko, N.L.
Format:	Artikel
Sprache:	English
Veröffentlicht:	Національний науковий центр «Харківський фізико-технічний інститут» НАН України 2018
Schriftenreihe:	Вопросы атомной науки и техники
Schlagworte:	Применение ядерных методов
Online Zugang:	https://nasplib.isofts.kiev.ua/handle/123456789/147307
Tags:	Tag hinzufügen Keine Tags, Fügen Sie den ersten Tag hinzu!
Назва журналу:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Zitieren:	Numerical approach for simulation of Palmer cooling / M.E. Dolinska, N.L. Doroshko // Вопросы атомной науки и техники. — 2018. — № 3. — С. 150-154. — Бібліогр.: 7 назв. — англ.

Institution

Digital Library of Periodicals of National Academy of Sciences of Ukraine

id	nasplib_isofts_kiev_ua-123456789-147307
record_format	dspace
spelling	nasplib_isofts_kiev_ua-123456789-1473072025-02-09T11:57:34Z Numerical approach for simulation of Palmer cooling Чисельний підхід для симуляції охолодження методом Palmer Численный подход для симуляции охлаждения методом Palmer Dolinska, M.E. Doroshko, N.L. Применение ядерных методов In this paper the time domain algorithm is described and applied for the Palmer Cooling method. The possibility of using the Palmer system in such case for simultaneous longitudinal and transverse cooling by a suitable choice of the pickup to kicker distance was described by Hereward [1, 2]. Using his method the special computer code has been developed to calculate beam cooling in time domain approach. Описується алгоритм «timedomain», який застосовується для охолодження Palmer-методом. Можливості застосування Palmer-системи для поздовжнього і поперечного охолоджень шляхом підбору відстані між pick-up і кікер-магнітами були описані в роботах Херевальда [1, 2]. Використовуючи даний підхід, була розроблена комп'ютерна програма для розрахунків параметрів охолодження із застосуванням «timedomain»- методики. Описывается алгоритм «timedomain», который применяется для охлаждения Palmer-методом. Возможности применения Palmer-системы для продольного и поперечного охлаждений путем подбора расстояния между pick-upи кикер-магнитами были описаны в работах Херевальда [1, 2]. Используя данный подход, была разработана компьютерная программа для расчетов параметров охлаждения с применением «timedomain»-методики. 2018 Article Numerical approach for simulation of Palmer cooling / M.E. Dolinska, N.L. Doroshko // Вопросы атомной науки и техники. — 2018. — № 3. — С. 150-154. — Бібліогр.: 7 назв. — англ. 1562-6016 PACS: 02.60.Cb, 29.20.−c https://nasplib.isofts.kiev.ua/handle/123456789/147307 en Вопросы атомной науки и техники application/pdf Національний науковий центр «Харківський фізико-технічний інститут» НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Применение ядерных методов Применение ядерных методов
spellingShingle	Применение ядерных методов Применение ядерных методов Dolinska, M.E. Doroshko, N.L. Numerical approach for simulation of Palmer cooling Вопросы атомной науки и техники
description	In this paper the time domain algorithm is described and applied for the Palmer Cooling method. The possibility of using the Palmer system in such case for simultaneous longitudinal and transverse cooling by a suitable choice of the pickup to kicker distance was described by Hereward [1, 2]. Using his method the special computer code has been developed to calculate beam cooling in time domain approach.
format	Article
author	Dolinska, M.E. Doroshko, N.L.
author_facet	Dolinska, M.E. Doroshko, N.L.
author_sort	Dolinska, M.E.
title	Numerical approach for simulation of Palmer cooling
title_short	Numerical approach for simulation of Palmer cooling
title_full	Numerical approach for simulation of Palmer cooling
title_fullStr	Numerical approach for simulation of Palmer cooling
title_full_unstemmed	Numerical approach for simulation of Palmer cooling
title_sort	numerical approach for simulation of palmer cooling
publisher	Національний науковий центр «Харківський фізико-технічний інститут» НАН України
publishDate	2018
topic_facet	Применение ядерных методов
url	https://nasplib.isofts.kiev.ua/handle/123456789/147307
citation_txt	Numerical approach for simulation of Palmer cooling / M.E. Dolinska, N.L. Doroshko // Вопросы атомной науки и техники. — 2018. — № 3. — С. 150-154. — Бібліогр.: 7 назв. — англ.
series	Вопросы атомной науки и техники
work_keys_str_mv	AT dolinskame numericalapproachforsimulationofpalmercooling AT doroshkonl numericalapproachforsimulationofpalmercooling AT dolinskame čiselʹnijpídhíddlâsimulâcííoholodžennâmetodompalmer AT doroshkonl čiselʹnijpídhíddlâsimulâcííoholodžennâmetodompalmer AT dolinskame čislennyjpodhoddlâsimulâciiohlaždeniâmetodompalmer AT doroshkonl čislennyjpodhoddlâsimulâciiohlaždeniâmetodompalmer
first_indexed	2025-11-25T22:46:48Z
last_indexed	2025-11-25T22:46:48Z
_version_	1849804255771230208
fulltext	ISSN 1562-6016. ВАНТ. 2018. №3(115) 150 NUMERICAL APPROACH FOR SIMULATION OF PALMER COOLING M.E. Dolinska, N.L. Doroshko Institute for Nuclear Research, National Academy of Sciences of Ukraine, Kyiv, Ukraine E-mail: kinr@kinr.kiev.ua In this paper the time domain algorithm is described and applied for the Palmer Cooling method. The possibility of using the Palmer system in such case for simultaneous longitudinal and transverse cooling by a suitable choice of the pickup to kicker distance was described by Hereward [1, 2]. Using his method the special computer code has been developed to calculate beam cooling in time domain approach. PACS: 02.60.Cb, 29.20.−c INTRODUCTION For improving beam quality in storage rings is used Stochastic cooling (SC). It is also used for accumulation of high intensity ion beams. In our previous papers, the main principles of stochastic cooling were described. Many theoretical results for the stochastic cooling sys- tem, which is planned to be used in the Collector Ring (CR) of the FAIR complex (Darmstadt, Germany), were presented [3 - 7]. The momentum cooling and antiproton accumula- tion are usually simulated by numerically solving the Fokker-Planck (FP) equation [1]. This approach howev- er can not be easily extended to account for many real- life complications, particularly the two-way transverse- longitudinal coupling due to the finite betatron size at the pickup. An alternative approach to stochastic cooling simu- lation can be described as the macroparticle simulation using the discrete particles and the time- domain re- sponse functions of the pickup (PU) – kicker (KK) cir- cuits. The beam dynamic under influence of the stochas- tic cooling forces can be studied by a particle by particle and turn by turn in the time-domain treatment. This treatment escapes the involvement of complicated, un- certain and changing frequency spectra, which anyhow are likely to be incomplete by considering Fokker- Planck equation and its solution. To keep the computa- tion times within reasonable limits, the scaling law that cooling times are proportional to the number of particles (for zero preamplifier noise and all other parameters remaining unchanged, except the gain) has been applied throughout. A typical simulation super-particle number is about (1…10)10 4 . 1. THE NUMERICAL CODE FOR CALCULATION OF COOLING BY PALMER METHOD The Collector Ring (CR) at FAIR [3] is going to be used for fast cooling of hot ions coming from separa- tors. This ring will be equipped with stochastic cooling systems, which can allow to cool beam by different methods: TOF (time of flight), notch filter and Palmer. The Palmer cooling system will be used as a pre-cooling of radioactive ion beams (RIBs), since this system al- lows to avoid the Schottky band overlap of ions, for which the η – accelerator slip factor is rather large 0.178. The Palmer cooling will be useful in the first stage of stochastic cooling process at the CR. It serves to detect signals in all 3 phase space planes. After the rms Δp/p (root mean square of momentum spread) decreases below 0.1%, it is possible to switch off the signals from the Palmer Pick up and turn to Notch filter cooling. The initial normalized particle coordinates xc,i and x’c,i are generated and transformed to the normalized coordinated Xn,i and X’n,I c PU PU cPUn PU c n xxX x X      ; . (1) Here βPU and αPU are Twiss parameters of a ring at Pick up (beta and alfa functions). The normalized coor- dinates are used because of the simple modeling of betatron oscillation in the ring. The particle amplitude is defined 222 nn XXA  . The maximum radius of A 2 is interpreted as emit- tance ε. The particle coordinate is generated with a Gaussian distribution with the σ = εrms, where 2 rmsrms A . 1. The initial momentum deviation Δp/pi is assigned to each particle. The initial particle ensemble has the Gaussian momentum distribution with σ = Δp/prms=δrms. 2. The time length ts of one sample is defined using the bandwidth characteristic of cooling system W W ts 2 1  . For a certain number of samples ns the total time length of beam is calculated: Tb=nsts. For each particle the initial time ti is generated within the time domain from 0 up to Tb with a homogenous probability. Now one can say that each particle belongs to one of the samples (from 1 to ns). This is important issue, which characterizes the particle ensemble of each sample. In the code each sample is analyzed turn by turn. Now the cooling process together with beam dynamic in the ring is calculated. It should be noted that Palmer cooling is characterized by a fact that a test particle has a horizon- tal orbit displacement i PUi p p Dx   proportional to its momentum error Δp/p, where DPU is the value of the “orbit dispersion function” at the pickup as determined by the focusing properties of the storage ring. This dis- placement is detected by horizontal position pickup. It is assumed that the particles momentum contribution dom- inates at the pickup. 3. The single particle displacement at the PU is cal- culated by i PUninin p p DXX   , , where Dn,PU is the normalized dispersion function at PU. mailto:kinr@kinr.kiev.ua ISSN 1562-6016. ВАНТ. 2018. №3(115) 151 4. At the PU the accessory of a particle to the certain sample s is defined and saved. Belonging to the certain sample depends on its time value ti. The average value of <Xn>s is calculated for each sample s by    sN i ni s sn X N X 1 1 , where Ns is a number of parti- cles in sample s. 5. On the way from PU to KK the particles migrate from sample to sample, which means the time of each particle at KK has new value and calculated by iii ttt  , i PKPKi p p Tt    , Δti is a time change of the particle. For each sample the creation of new en- semble with a new time characteristic is called as “mix- ing”. 6. The particle coordinates at the KK have also mixed and new values due to betatron oscillation recal- culated by                         Pn Pn PKPK PKPK Kn Kn X X CS SC X X , , , , . (2)    PKPKPKPK SC   sin;cos ; ΔμPK is a phase advance fro PU to KK. One can see that the beta- tron oscillation is considered as simple rotation in nor- malized coordinates with radius A. 7. At the KK the accessory of particle to certain sample s is defined and saved. Belonging to a certain sample s depends on its time value ti. Depending on the sample number s the single particle correction is calcu- lated by )( tsX D g p p p p psn PUii      , (3) )( tsXgXX tsnnn   . (4) The g is a normalized gain, αp,t is damping factor, which reduce the gain efficiency due to the noise. In the code this factor is calculated by formulae given in chap- ter 3. The s(Δt) is a time profile of the signal, which is calculated by the formula given in chapter 2. The Eqs. (3) and (4) describe the cooling effect in the time domain approximation. One can see that for the Palmer method the momentum error of particles is corrected proportionally to the center gravity of sample, which characterized by average value of coordinate Xn. 8. On the way from KK to PU the particles migrate again from sample to sample, which means the time of each particle at PU is changed and calculated by iii ttt  , i KPKPi p p Tt    . 9. The particle coordinates at the PU are rearranged and recalculated by                         Kn Kn KPKP KPKP Pn Pn X X CS SC X X , , , , .    KPKPkPKP SC   sincos , ΔμKP is a phase advance from KK to PU. 10. At the position of PU the analyses of the rms values of Δp/prms and emittance εrms are performed and saved. 11. The next turn is calculated. Go to step 4. 2. THEORETICAL DESCRIPTION OF THE MAIN PARAMETERS OF STOCHASTIC COOLING 2.1. GAIN FACTOR For momentum cooling the gain factor g can be ex- pressed [1] as 2 2 2 1 ............(5) ( / ) n rev n n rev av p e f R N e f R N g EE W E e E     Here the sum goes over all harmonics in the pass- band, i.e. from fmin=n1frev to fmax=n2frev and average im- pedance can be approximately calculated by    nav R nn R )( 1 12 . For the transverse cooling the gain factor g can be expressed as [4] kpaoKoPUkp ic i t G x S ZZnn EAm ceNZ g    2 2 0 224          . (6) Z0PU and Z0K transverse impedance of Pick-up and Kick- er; RPU and RK input impedance of network (shunt im- pedance); np and nk are number of Pick-up and Kicker units; βp and βk are betatron functions at Pick-up and Kicker; fgPU and fgKU are geometry factors of electrodes; Ga is an electrical gain; Zi and Ai are the ion charge and mass; f0 is a revolution frequency; e is the electron charge; β=v/c; γ is a relativistic factor; N is the number of particles; mc is the middle harmonic. 2.2. NOISE The gain damping factor αp can be calculated by  pp U g  1 2 1 for momentum cooling and  xx U g  1 2 1 for betatron cooling. Here Up and Ux are the noise-to-signal ratio. For Palmer cooling this value is defined by [10] 2 2 22 2 rms n rmsPU rms p D U      and 2 2 2 22 rms n rms rmsPU x AA D U   . In this equation the noise-to-signal ratio 22 / rmsn  can be calculated according the formula    RnfeNZ kT pi s rms n 0 22 10 2 2 2 10  . The ν is noise of the amplifier (usually 1.5…3.0 dB); R is input resistance of the amplifier; λ is a sensitivity factor; N – number of particles in the beam; e – electron charge; f0 – revolution frequency; np – number of pick- ups; k – Boltzmann constant; Te – temperature (K). The possible increase of Up as the beam shrinks, and many construction details, are hidden in λ. As first approxima- tion, in the code it is assumed that the sensitivity factor calculated by 2 0 2 0)(    rms rmsp  and 0 0)(    rms rmsx  . The parameter λ0 is geometrical factor, which char- acterizes the PU. In simulation it is assumed λ0=0.5. δ0=(Δp/prms)0 is initial rms momentum spread of beam. ε0=(εrms)0 is the initial rms emittance. These equations reveal that the noise limits the cooling rate. ISSN 1562-6016. ВАНТ. 2018. №3(115) 152 2.3. KICKER ACTION Synchronism between particles and their correcting pulse on their way from pick-up to kicker must be properly calculated. In the time domain approach the incoherent heating effect is calculated in a simpler way compare to that is done by Focker-Planck Equation. It is assumed that the action at the kicker produces the time- pulse curves. These curves can be calculated by inverse Laplace transformation of a single passage of sample for a certain bandwidth cooling system. As a result of such transformation the signal shape similar to that shown in Fig. 1, can be obtained. Fig. 1. Approximated correcting pulse in the kicker In Fig. 1 Δt is the time error of the particle p2 with respect to the p1, which is located in the middle of the sample and synchronized with an ideal test particle p1. The particle p2, which arrives at KK with a time delay Δt, gets a partial kick. Tc is the useful width of the cor- rection pulse and usually equal to the sample length ts for low pass system. But Tc is shorter than ts for a high frequency band-pass system. (this is subject is still un- der study). For simplification, in this work a parabolic response model of the form ,1)( 2           cT t ts (7) for Palmer and TOF signal approximation is used. In Fig. 2 (left) the signal shape calculated by Eq. (7) is shown. Here is assumed that Ts=ts/2.3, ts is the time length of sample. Fig. 2. Test signals s(t) reproducing a signal shape (left) for Palmer and TOF (right) for the Notch filter method For the notch filter the signal shape is approximated by ,1)( 2           c c T Tt ts (8) where Tc=ts/4.3. The signal shape calculated by Eq. (8) is shown in Fig. 2 (right). 3. ANALYTICAL APPROACHES FOR STOCHASTIC COOLING In this chapter the analytical formulae are presented in order to cross check the results of simulation obtained with numerical method given in this work. Using these formulae the beam evolution is calculated and results are compared with time domain approach described above. A simple, but, useful, calculation of the stochas- tic cooling by analytical formulae can be done using cooling rates 1/τ. The set of equation are solved to cal- culated rms emittance and rms momentum spread a each time step i with a time step of Δt:           h i rms i rms t   exp1 ,               p i rms i rms t p p p p  exp 1 . (9) Here the cooling rates 1/τ are calculated by formulae [5]   )()()(2 1 2 tUtMgtBg N W ppp p   and   )()()cos()(2 1 2 tUtMgtBg N W hhPKh h    . The parameters B, M, U are time dependent func- tions. The average mixing factor M(t) for the Gaussian distribution is expressed as         1 2 ln )(22 1 n n ntp M rms . The δp (t) is a rms momentum spread, which de- pends on time. The noise-to-signal rations Up and Uh for Palmer method are calculated by 2 2 2 ,            i rms PU PUnPU i rms p p p D x U  , i rmsptPUp ls h ZnfNe ZkT U  2 0 2 2  , s v PUn kTx 10 , 10   , δv = 1.5 dB  noise. (10) For Palmer or TOF method  rmspkPK tpntB )(2cos)( 2  . For Notch filter cooling  rmseff tpntB )(5.0cos)( 2 . Zl is char- acteristic impedance of the electrode; N – number of particles in the beam; e – electron charge; f0 – revolution frequency; np – number of pick-ups, k – Boltzmann constant; Te – temperature (K); βp – beta function at the pickup. ΔμPK – phase advance between pick-up and kicker. The optimal value of the gt can be calculated [4] as )()( )cos()( , tUtM tB g PK optt     . Some useful analytical formulae formula, which can be used for test simulations. The noise-to-signal ration can be expressed through the noise electron current In and Schottky currect at PU IPU. 2 2 PU n I I U  , h D I rms PU   , 2 SCrms rms Ix D  . ISSN 1562-6016. ВАНТ. 2018. №3(115) 153 The λ is sensitivity factor; h – half of gap of electrodes; In  noise current due to temperature of electronic R WkT I e n v 1010  and WNfeISC 0 22  Schottky current. 4. NUMERICAL SIMULATIONS 4.1. TEST CALCULATION: COMPARISON NUMERICAL WITH ANALYTICAL CALCULATIONS The special tracking code, where the Palmer cooling is modelled in the time domain approach, has been writ- ten in FORTRAN language. As an example a set of parameters for a beam and CR cooling system [5] is used. The beam cooling with the Palmer method has been performed over the time of 5 s. In the Fig. 3 one can observe the Probability Density Function (PDF) evolution of dp/p and emittance during cooling time at each 0.4 s. In Fig. 4 the evolution of rms values of dp/p and emittance calculated by tracking code and by the analytical formulae Eq. (9, 10) are shown. Here the gain factor g is 0.5. Fig. 3. Probability Density Functions (PDF) calculated by tracking code in time domain approach. Cooling of an Ur beam: horizontal beam profiles (left) and momentum spread distribution recorded every 0.4 s Fig. 4. Evolution of the emittance (left) and momentum spread (right) calculated with the Palmer method in the time domain approach and by the analytical formulae for the CR machine. Parameters used in calculation are given in [5]. The gain factor g = 0.5 4.2. PALMER COOLING AT THE CR In this chapter the preliminary simulations of beam cooling with the Palmer method are presented and com- pared with an analytical model. In the code the Palmer cooling is modeled by following steps. Fig. 5 shows the results of simulation for uranium beam cooling at the CR for the different gain factors. The parameters of the CR ring and beam are given in [5]. One can see that the optimal fast cooling in both planes can be performed if the gain g is 0.4…0.5 that corresponds to the amplification factor of 150 dB. Here the ideal signal is assumed at the Kicker (see chapter 2). The realistic signal shape must be obtained from inverse Laplace transformation and included in simulations. Under given above conditions the rms Δp/p below 0.1% can be achieved in 0.5 s, while the rms emittance is cooled down to about 8 mm mrad as shown in Fig. 5. Simulations obtained with analytical formulae Eqs. (9), (10) show approximately the similar results, which are shown in Fig. 6. Here the one can see that the optimal gain factor is 0.3…0.4. Analytical model pre- dicts that the rms emittance after 0.5 s is above 10 mmmrad, while the rms Δp/p is 0.1%. ISSN 1562-6016. ВАНТ. 2018. №3(115) 154 Fig. 5. Evolution of the rms emittance and the momentum spread of the U beam for different gains and obtained in numerical calculation by a time domain method Fig. 6. Evolution of the rms emittance and the momentum spread of the U beam during cooling for different gains and obtained in calculation by analytical formula CONCLUSIONS The precise method for the higher dimensions Palm- er cooling calculations has been presented. The special code has been developed to investigate the beam dy- namic in storage ring, where the stochastic cooling process is used. Presented method can be used for TOF and Notch filter cooling systems. Presented method can be used for the wide range of tasks and will be applied to the realistic parameters for studying the dynamics of particles in storage rings. REFERENCES 1. D. Möhl. Stochastic cooling of particle beams. Ce- neva: “Springer-Verlag”, 2013. 2. H.G. Herewald. Statistical phenomena-theory // Proc. 1 st Course of International School of Particle Accelerators, Geneva, CERN. 1976, №77/13, p. 281. 3. M. Dolinska, C. Dimopoulou, A. Dolinskii, et al. Simulations of Antiproton Stochastic Cooling in the CR // GSI Scientific report. Darmtadt, 2011, p. 327. 4. C. Dimopoulou et al. Simulations of stochastic cool- ing of antiprotonsin the Collector Ring CR // Pro- ceedings of COOL’11, Alushta, Ukraine. 5. M. Dolinska. Macroparticle simulation of stochastic cooling: GSI internal report, 2012. 6. M. Dolinska. Fokker-Planck equation solver for study stochastic cooling in storages rings // Nuclear Physics and Atomic Energy. 2011, v. 12, № 4, p. 407-413. 7. M. Dolinska. Numerical algorithm based on the PDE method for solution of the Fokker-Planck equation // Problems of Atomic Science and Technology. Series “Nuclear Physics Investigations”. 2011, № 5(75), p. 63-66. Article received 23.02.2018 ЧИСЛЕННЫЙ ПОДХОД ДЛЯ СИМУЛЯЦИИ ОХЛАЖДЕНИЯ МЕТОДОМ PALMER М.Е. Долинская, Н.Л. Дорошко Описывается алгоритм «timedomain», который применяется для охлаждения Palmer-методом. Возможно- сти применения Palmer-системы для продольного и поперечного охлаждений путем подбора расстояния между pick-upи кикер-магнитами были описаны в работах Херевальда [1, 2]. Используя данный подход, была разработана компьютерная программа для расчетов параметров охлаждения с применением «timedo- main»-методики. ЧИСЕЛЬНИЙ ПІДХІД ДЛЯ СИМУЛЯЦІЇ ОХОЛОДЖЕННЯ МЕТОДОМ PALMER М.Е. Долінська, Н.Л. Дорошко Описується алгоритм «timedomain», який застосовується для охолодження Palmer-методом. Можливості застосування Palmer-системи для поздовжнього і поперечного охолоджень шляхом підбору відстані між pick-up і кікер-магнітами були описані в роботах Херевальда [1, 2]. Використовуючи даний підхід, була розроблена комп'ютерна програма для розрахунків параметрів охолодження із застосуванням «timedomain»- методики.

Numerical approach for simulation of Palmer cooling

Institution

Ähnliche Einträge