Cooling and diffusion rates in coherent electron cooling concepts
Sergei Nagaitsev, Valeri Lebedev, Gennady Stupakov, Erdong Wang, William Bergan
CCooling and diffusion rates in coherent electron cooling concepts
Sergei Nagaitsev ∗ and Valeri Lebedev Fermi National Accelerator Laboratory, Batavia, IL 60510, USA
Gennady Stupakov
SLAC National Accelerator Laboratory,Stanford University, Menlo Park CA 94025, USA
Erdong Wang and William Bergan
Brookhaven National Laboratory, Upton NY 11973, USA (Dated: February 23, 2021) a r X i v : . [ phy s i c s . acc - ph ] F e b bstract We present analytic cooling and diffusion rates for a simplified model of coherent electron cooling(CEC), based on a proton energy kick at each turn. This model also allows to estimate analyticallythe rms value of electron beam density fluctuations in the ”kicker” section. Having such analyticexpressions should allow for better understanding of the CEC mechanism, and for a quicker analysisand optimization of main system parameters. Our analysis is applicable to any CEC amplificationmechanism, as long as the wake (kick) function is available.
I. INTRODUCTION
Let us consider a 1D longitudinal coherent electron cooling (CEC) scheme as proposedin Ref. [1–5]. Figure 1 presents a simplified schematic of CEC. The electron bunch picks updensity modulations from co-propagating protons in the ”Modulator” section. These densitymodulations are then amplified by some mechanism in the ”Amplifier” section (blue).The proton beam line (red) is arranged in such a way that when protons arrive at the”Kicker” section, faster (slower) protons overcome (lag behind) a reference on-energy parti-cle.In our simplified model we will assume that at the end of the ”Kicker” section, the protonenergy experiences a kick as shown in Fig. 2. For convenience, we will call the proton energychange dependence versus z the wake function —apart from a different normalization, it isthe same as the conventional longitudinal wake in accelerator physics. For simplicity, we willassume that the proton’s longitudinal position, z , in the ”Kicker” section does not change andis equal to z = R δ , where δ = δpp is the proton’s relative momentum deviation and R is FIG. 1. A simplified schematic of CEC. ∗ [email protected]; Also at the University of Chicago, Chicago, Illinois 60637, USA z depends only on the proton momentum deviation. One can now seefrom Fig. 2 that faster (slower) protons would lose (gain) energy after the ”Kicker” sectionpassage. The wake function, introduced above, is the main element in various modifications FIG. 2. The electron beam density modulation due to a single proton (arb. units) and a corre-sponding energy kick (in eV) after the ”Kicker” section as a function of the proton’s longitudinalposition ( µ m). of coherent electron cooling. For the microbunched electron cooling (MBEC) concept it wascalculated in Refs. [5]; for the plasma-cascade (PCA) cooling concept, the wake functioncan be found in Ref. [6]. In what follows, we will use the wake function calculated for anMBEC cooler currently being designed for the electron-ion collider (EIC) (for details seeRef. [7]) and shown in Fig 2. Table I gives an example of system parameters, used in ourcalculations. We will discuss both the cooling rate and the diffusion rate due to neighboringprotons producing random kicks and, thus, creating a heating mechanism. Other diffusionmechanisms will also be considered. II. ENERGY KICK
To allow for analytical treatment of the problem, we will use the following model expres-sion for the proton energy kick in the ”Kicker” section, w ( z ) = − V sin (cid:18) π zz (cid:19) exp (cid:18) − z σ (cid:19) , (1)where we introduced three adjustable parameters: V , the amplitude of the kick, z , thecharacteristic wavelength, and σ , the characteristic width. The negative sign reflects the3 ABLE I. CEC system parameters (example)
Parameter Symbol Value Unit
Proton energy E
275 GeVLorentz factor γ C f N p Proton rms momentum spread δ p − Proton rms bunch length σ pz N e Electron rms bunch length σ ez σ ey σ ex L k
40 m fact that the leading particle ( z >
0) loses its energy after the kick. For example, the energykick, calculated using the system parameters in Table I and shown in Fig. 2, is presentedin Figure 3 (red curve) together with our model, Eq. (1) (blue curve). One can see fromFigure 3 that the proposed approximation slightly underestimates the far tales of the wake.This does not affect the cooling rate but slightly underestimates the diffusion rate.For the calculated energy kick, the following model parameters provide the best fit: V = 28 eV, z = 6 . µ m, and σ = 3.0 µ m. One can notice that at | z | > z the energy kickchanges its sign and cooling becomes anti-cooling. This determines the so-called coolingrange, the number of ”sigmas” n such that nR δ p = z / III. FOKKER-PLANCK EQUATION
To describe the evolution of the proton momentum distribution function, we will use theFokker-Planck equation in the following form: ∂ψ∂t + ˙ s ∂ψ∂s + ˙ δ ∂ψ∂δ = − ∂∂δ ( F ( δ, s ) ψ ) + 12 ∂∂δ (cid:18) D ( δ, s ) ∂ψ∂δ (cid:19) , (2)4 IG. 3. The energy kick (eV) after the ”Kicker” section as a function of the proton’s longitudinalposition z = R x ( µ m) with respect to the reference on-energy proton. The red curve is acalculated wake, based on Ref. [8, Eq. C7]. The blue curve is the proposed approximation, Eq.(1). where ψ ( δ, s, t ) is the proton distribution function, δ is the relative momentum deviation, s is the longitudinal coordinate in the lab frame (with respect to the bunch center), F ( δ, s ) = f w ( R δ, s ) /E is the cooling force and D ( δ, s ) is the diffusion coefficient. The diffusion caninclude various contributions, such as heating due to near-by protons, electron beam noise,intra-beam scattering, etc. Eq. (2) corresponds to a bunched-beam case. We will transformEq. (2) to unperturbed longitudinal action-angle variables ( J, φ ) in order to analyse thecooling and diffusion processes in terms of the longitudinal bunch emittance [9, 10], δ = (cid:115) Jβ sin φ, s = (cid:112) J β cos φ, (3)where β is the so-called longitudinal beta function, β = σ pz /δ p ≈
88 m for the parametersin Table I. If the characteristic cooling and diffusion times are longer than the synchrotronoscillation period, it is reasonable to assume that the bunch distribution is continuouslymatched to the shape of the trajectories in the (
J, φ )-phase plane and that the distributionfunction depends explicitly only on J and not on φ , that is ψ = ψ ( J, t ). This simplifiesconsiderably the left-hand side of Eq. (2), ∂ψ∂t = − (cid:112) β ∂∂J (cid:16) √ J ˜ F ( J ) ψ (cid:17) + β ∂∂J (cid:18) J ˜ D ( J ) ∂ψ∂J (cid:19) , (4)where the cooling force ˜ F is given by˜ F ( J ) = 12 π (cid:90) π F ( δ, s ) sin φ dφ (5)5nd the diffusion term ˜ D is given by˜ D ( J ) = 12 π (cid:90) π D ( δ, s ) sin ( φ ) dφ (6)with δ and s given by Eq. (3). For a detailed derivation of Eqs. (5) and (6), see Ref. [10].In its simplest form, the cooling force can be presented as F ( δ, s ) = − λδ , while thediffusion as a constant D ( δ, s ) = D . The Fokker-Planck equation becomes ∂ψ∂t − λ ∂∂J ( J ψ ) = β D ∂∂J (cid:18) J ∂ψ∂J (cid:19) . (7)We will now multiply both sides of Eq. (7) by J and integrate in order to obtain theevolution of the rms longitudinal emittance, (cid:15) L = (cid:82) ∞ ψ J dJ (here we assume that ψ isnormalized by unity, (cid:82) ∞ ψ dJ = 1), d(cid:15) L dt + λ (cid:15) L = β D . (8)In a steady state (that is for d/dt = 0), the equilibrium rms emittance is (cid:15) L = D β λ . (9)For a more realistic cooling force, we notice (see Table I) that the electron bunch is muchshorter than the proton bunch, σ ez (cid:28) σ pz . We will therefore use the following cooling forceapproximation: F ( δ, s ) = − f V E sin (cid:18) π R δz (cid:19) exp (cid:18) − R δ σ (cid:19) exp (cid:18) − s σ ez (cid:19) (10)with δ and s given by Eq. (3). This equation assumes that the electron bunch is placed at thecenter of the proton bunch and the interaction happens only for protons with s ≈
0, because σ ez (cid:28) σ pz . Therefore, we can use the following approximation: exp (cid:16) − s σ ez (cid:17) ≈ √ πσ ez δ ( s ),where δ ( s ) is the Dirac delta function. Using Eq. (5), one can obtain the cooling force as afunction of action J :˜ F ( J ) = − f V E σ ez √ πβJ sin (cid:32) π R z (cid:115) Jβ (cid:33) exp (cid:18) − R σ Jβ (cid:19) . (11) IV. COOLING RATE
To obtain the cooling rate, τ c , from the fokker-Plank equation, Eq. (4), we will evaluatethe following integral: 1 τ c = √ β(cid:15) L (cid:90) + ∞ J ∂∂J (cid:16) √ J ˜ F ( J ) ψ (cid:17) dJ, (12)6here ˜ F ( J ) is given by Eq. (11) and the distribution function ψ = 1 (cid:15) L exp (cid:18) − J(cid:15) L (cid:19) . (13)The resulting cooling rate is1 τ c = πf V δ p nE σ ez √ σ pz (cid:18) z n σ (cid:19) − / exp (cid:18) − π n + z /σ (cid:19) , (14)where n is the cooling range, such that nR δ p = z /
2. Figure 4 shows the cooling time, τ c , as a function of the cooling range, n , for the CEC system parameters in Table I and V = 28 eV, z = 6 . µ m, and σ = 3.0 µ m. One can see that there is a shallow minimum FIG. 4. Cooling time (in minutes) as a function of the cooling range n defined as nR δ p = z . of about 60 minutes for n in the range 3.5 to 4.5. For example, choosing n = 3.7 results in R = Z / (2 nδ p ) ≈ R element. If the proton path length between the ”Modulator” and the ”Kicker” sectionis L ≈
100 m, the kinematic portion of the R element is L/γ ≈ R matrix element. It alsomeans that the Kicker section cannot be too long as its length increases the effective valueof the R element. For n → ∞ , the cooling time increases linearly with n and the Eq. (14)becomes: τ c − ≈ λ , as expected. There are additional constraints on the Kicker section length, due to plasma oscillations in the electronbeam, for example. . DIFFUSION RATE The diffusion coefficient, D ( δ, s ), is usually a function of the momentum deviation, δ .However, it was shown in Ref. [11] that in the case of stochastic cooling with a strongSchottky band overlap, the diffusion coefficient due to random kicks from neighbouringprotons is a constant, i.e. independent of δ . The CEC method, having the typical frequenciesof c/z ≈
45 THz, is in the regime of a strong Schottky band overlap. In this regime, thediffusion coefficient at the center of the electron bunch can be written as D = (cid:104) ( w ( z ) /E ) (cid:105) T , (15)where the angular brackets (cid:104) ... (cid:105) indicate averaging of random energy kicks from neighboringions, and T = 1 /f is the revolution period in the ring. Taking into account that the numberof ions per unit length at the center of a Gaussian bunch is N p / ( √ πσ pz ), we obtain D = f N p √ πσ pz V E (cid:90) + ∞−∞ (cid:18) sin (cid:18) π zz (cid:19) exp (cid:18) − z σ (cid:19)(cid:19) dz. (16)We can finally write D = N p f V E σ σ pz (cid:18) − exp (cid:18) − π σ z (cid:19)(cid:19) ≈ N p f V E σ σ pz . (17)As expected, the diffusion rate is independent of the cooling range n and is proportionalto the width of the kick, σ , which can be viewed as the inverse band-width of the system.Recalling that the electron bunch is much shorter than the proton bunch, Eq. (10), we canwrite the diffusion coefficient for any longitudinal position, s , within the proton bunch as D ( s ) = D exp (cid:18) − s σ ez (cid:19) ≈ D (cid:114) π σ ez δ ( s ) . (18)After averaging over angle φ by using Eq. (6) we obtain˜ D = D σ ez σ pz . (19)From Eq. (4) the evolution of the longitudinal rms emittance, (cid:15) L is determined by1 (cid:15) L d(cid:15) L dt = − τ c + ˜ Dβ(cid:15) L , (20)with τ c from Eq. (14) and ˜ D from Eq. (19). For the CEC system parameters in Table Iand for V = 28 eV, z = 6 . µ m, and σ = 3.0 µ m, the diffusion time is ( ˜ Dβ/(cid:15) L ) − ≈ n = 3.7, τ c ≈
62 minutes. This indicates that, in theory, the overall sum of cooling and diffusion ratesin Eq. (20) can still be increased by increasing the kick amplitude, V . For the so-called”optimal gain” [12] condition we have: 1 τ c = 2 ˜ Dβ(cid:15) L . (21)From this we can obtain the optimal kick amplitude, V opt , for maximum cooling: V opt = 4 √ πE δ p nN p σ pz σ (cid:18) z n σ (cid:19) − / exp (cid:18) − π n + z /σ (cid:19) . (22)This yields V opt ≈
150 eV. With this optimal kick amplitude, the achievable cooling timebecomes 2 τ c ≈
24 minutes (the factor of 2 is due to Eq. (21)). We note that only thediffusion due to neighboring protons is taken into account in Eq. (22). Other diffusionmechanisms can be added to analyze the effective cooling rate in Eq. (20).
VI. ELECTRON BEAM DENSITY FLUCTUATIONS DUE TO PROTONS
First, we can estimate the rms value of random energy kicks per turn due to diffusionEq. (17) for a proton at the center of the electron bunch. For the CEC parameters Table I,the diffusion coefficient D ≈ . × − sec − . The rms energy kick per one turn due todiffusion can be written as δE rms = E (cid:112) D T ≈
26 keV . (23)This random kick is due to other protons in the vicinity of a reference proton. This shouldbe compared to a cooling wake (kick) value of about 20 V (max) per turn as can be seen inFig. (3).Let us estimate the rms electron beam density fluctuations, resulting from the superposi-tion of incoherently-added wakes from neighbouring protons. First, we will use a simplifiedsingle-wavelength density modulation model for a single proton with k = 2 π/z , a wavevector of this modulation. In the electron rest-frame, we can use the Poisson’s electrostaticsequation (in Gaussian units): dE z dz (cid:48) = 4 πen e ( z (cid:48) ) , (24)9here e is the electron charge and z (cid:48) is the transformed rest-frame z coordinate (note that E z = E (cid:48) z ). Using Eq. (24) for the calculation of the electron density perturbation we actuallyreplace electrons by uniformly charged thin slices and assume that the distance between theslices is much smaller than their transverse size. A more accurate model of Gaussian sliceswith elliptic cross-section and an arbitrary transverse size was used in Ref. [8].From Eq. (24) we find the following electron density modulation amplitude (in the beamrest frame) due to a single proton: n (cid:48) e = kE z πγe . (25)The rest-frame electric field E (cid:48) z can be estimated as E (cid:48) z = E z ≈ V / ( eL k ). Thus, we obtainfor a single proton, the amplitude of the density modulation, n (cid:48) e ≈ kV πγe L k . (26)The average over z (cid:48) of this density modulation is zero, (cid:104) n (cid:48) e (cid:105) = 0, but the rms value isnon-zero: (cid:113) (cid:104) n (cid:48) e (cid:105) ≈ kV πγe L k (cid:112) ∆ N p , (27)where ∆ N p is the number of protons in a sample, near a reference proton:∆ N p = N p σ √ πσ pz . (28)We finally arrive at an estimate for the rms electron beam density modulation: (cid:113) (cid:104) n (cid:48) e (cid:105) ≈ kV πγe L k (cid:115) N p σ √ πσ pz . (29)The maximum electron bunch density in the rest-frame is n (cid:48) e = N e (cid:112) (2 π ) γσ ez σ ex σ ey . (30)For the parameters of Table I we obtain, (cid:112) (cid:104) n (cid:48) e (cid:105) n (cid:48) e ≈ kV σ ez σ ex σ ey e L k (cid:112) N p N e (cid:115) √ πσ σ pz ≈ . . (31)Thus, a simplified estimate gives a 15% relative rms density fluctuations value in the electronbunch, needed to support cooling in the presence of other protons . Obviously, it’s not a Eq. (24) treats the particles as charged sheets and thus overestimates their interaction at large distance.A more accurate model of elliptic slices from Ref. [8] takes into account that the interaction is localizedat the distance ∼ σ ex /γ, σ ey /γ ; it gives a larger value for the estimate of (cid:113) (cid:104) n (cid:48) e (cid:105) /n (cid:48) e that depends on thecross section of the electron and proton beams. V ≈
150 eV may not be possible for the parameters of Table I. A moredetailed analysis, obtained by performing averaging of n e ( z (cid:48) ) with n e ( z (cid:48) ) = 14 πe L k dw ( z (cid:48) ) dz (cid:48) , (32)yields (cid:113) (cid:104) n (cid:48) e (cid:105) = (cid:115) N p √ πγσ pz (cid:90) + ∞−∞ n e ( z (cid:48) ) dz (cid:48) = V πγe L k (cid:115) N p σ σ pz (cid:115) k σ + 1 − exp (cid:18) − k σ (cid:19) . (33)Using Eq. (33) yields the relative rms density fluctuation of 13%. VII. ELECTRON BEAM NOISE
In this section we estimate the electron beam shot-noise contribution to the diffusioncoefficient. Since the electron longitudinal charge density is similar to that of protons (seeTable I), we can estimate the electron shot-noise contribution to the diffusion to be similar tothe proton beam contribution, Eq. (17). This doubles the effective diffusion coefficient andgives the effective diffusion time of ( ˜
Dβ/(cid:15) L ) − ≈
330 minutes, still much greater than thecooling time, ≈
60 minutes. One can see that exceeding the shot-noise value in the electronbeam by a factor of 2-3 is possible for the chosen parameters, before the electron beam noisebecomes a dominant diffusion factor. One has to remember, however, that doubling thediffusion coefficient increases the rms electron beam density fluctuations by a factor of √ III. CONCLUSIONS
In this note, we considered a simplified cooling wake (kick) model, given by Eq. (1). Thismodel allows to derive analytic expressions for cooling and diffusion rates, as well as for theelectron beam rms density fluctuations. Having such analytic expressions should allow forbetter understanding of the CEC mechanism, and for a quicker analysis and optimizationof main system parameters.We would like to emphasize that even though we have used the wake calculated for theMBEC amplification scheme, our analysis can be easily applied to other coherent coolingtechniques, for example, to the PCA concept, as long as the wake (kick) function, similar tothe one shown in Fig. 3, is available.
IX. ACKNOWLEDGEMENTS
This manuscript has been authored by Fermi Research Alliance, LLC under ContractNo. DE-AC02-07CH11359 with the U.S. Department of Energy, Office of Science, Office ofHigh Energy Physics. This work was also supported by the Department of Energy, ContractNo. DE-AC02-76SF00515 and by Brookhaven Science Associates, LLC under Contract No.DE-SC0012704. [1] Y. S. Derbenev, On possibilities of fast cooling of heavy particle beams, AIP ConferenceProceedings , 103 (1992), https://aip.scitation.org/doi/pdf/10.1063/1.42152.[2] V. N. Litvinenko and Y. S. Derbenev, Coherent electron cooling, Phys. Rev. Lett. , 114801(2009).[3] D. Ratner, Microbunched electron cooling for high-energy hadron beams, Phys. Rev. Lett. , 084802 (2013).[4] G. Stupakov, Cooling rate for microbunched electron cooling without amplification, Phys.Rev. Accel. Beams , 114402 (2018).[5] G. Stupakov and P. Baxevanis, Microbunched electron cooling with amplification cascades,Phys. Rev. Accel. Beams , 034401 (2019).
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