Electron Beam Cooling Calculations Based on Plasma Relaxation Formalism; Is Single Pass Cooling Possible?

Ady Hershcovitch

January 23rd, 2009 Prevailing plasma physics relaxation time formalism is applied to electron beam cooling, with higher density electrons than in presently used electron beams, in contrast to currently used electron beam cooling computations. As an example electron beam cooling is examined, using the test particle model, as an option to reduce momentum of gold ions exiting the EBIS LINAC before injection into the booster. Electron beam parameters are based on experimental data (obtained at BNL) of electron beams extracted from a plasma cathode. Preliminary calculations indicate that single pass cooling might be feasible; momentum spread could be reduced by an order of magnitude in about one meter. The two approaches are not contradictory, since each is applicable to different plasma regimes.

Due to seemingly still debatable issue of the Booster acceptance and due to the need for further needed electron gun R&D, the presentation is focused on differences and applicability of different models to cooling computations, and validation of a previously used approach.

Although the subject matter might still be academic at present, “in- principle” feasibility must be ascertained before proceeding further. Friction Calculations Models (Analogies? Shielding is Missing) Friction force due to multiple small- angle binary collisions = ball moving through a field of billiard balls. Slowing down (dynamic friction) under test-particle plasma relaxation might be analogous to a ball moving through and interacting with many billiard balls simultaneously (Vlasov is wave- particle interaction). Fokker-Planck equation is basically a model for treating collisional relaxation due to binary collisions. With it changes in plasma distribution are calculated due to binary collisions in terms of the cross section for two-body scattering. But in dense plasma, a particle is not seen as bare. It carries its shielding cloud when undergoing scattering. The test particle model includes shielding in collision calculations. Basic difference: Fokker-Planck type collisional relaxation due to binary collisions describes a process that occurs in vacuum, while collisional relaxation based on the test particle model are essentially collisions that occur in a dielectric medium. Ion Beam Parameters Expected ion beam parameters at the exit of the EBIS-RFQ-LINAC system are: energy 2 ∆p || = 10−3 MeV/u, momentum spread p , and

∆p −3 ⊥ = 5x10 p , beam diameter 1 cm, and gold ion charge state Au+32, with ion density2 at the 7 -3 LINAC exit ni=8x10 cm . For electrons to match ion velocity, their energy U must be about 1 KeV. At these energies, ion and electron velocities are about 2x107 meter/second, hence β = 0.0667 and γ = 1.0022. Electron Beam Parameters Previously developed electron gun with plasma cathode, from which 9 A were extracted at 1 KeV through a 6 mm aperture, is considered. Based on these parameters the electron density n n = I can be computed from Aev , to be about n ≈1011 cm −3 . Balk electrons energy spread before extraction was about 0.1 eV. Due to kinematic compression, energy spread of the 1 T = T ∗0.5()T 2 accelerated electrons Te is e U , where T is thermal spread of unaccelerated electrons. For T = 0.1 eV and U = 1 KeV, Te = 5x10-4 eV. Presentation Layout

• Ion and E-beam parameter; embodiments. • Conventional cooling formulas & their origin (very Short). • Cooling based on Parkhomchuk’s empirical formula. Parkhomchuk claims that an empirical formula is in much better agreement with computer simulations of fully magnetized cooling for magnetic fields of up to 4kG. Additionally, experiments and computer simulations showed reasonable agreement with Parkhomchuk. • Test Particle Model: derivation and cooling/heating calculations. Derivation of Parkhomchuk’s empirical formula. Heating calculations. New issue to be resolved. • Other pertinent physics issues • Justification for using the Test Particle Model: when to use which model Possible Embodiments

A Couple of Electron Gun Options Origin of “Conventional” Cooling Formulas Boltzmann works well in gases: short-range forces. Fokker-Planck equation was originally derived to treat Brownian motion (short-range forces). Later was used to evaluate collision terms in the Boltzmann equation for cases of multiple small angles binary collisions due to long-range forces: Chadrasekhar multiple long-range binary collisions between stars, and Spitzer’s long-range binary Coulomb collisions. Finally there is the elegant mathematical treatment by Rosenbluth, MacDonald, and Judd for multiple long-range binary Coulomb collisions in magnetized plasma and deriving the Fokker-Planck coefficients like friction and diffusion. Belyaev and Budker derived energy transfer time, due to friction force for multiple small angle binary Coulomb collisions,. Based on that formalism, Budker proceeds to evaluate electron beam cooling for protons and antiprotons. Finally, theory for computing magnetized friction forces & cooling times was derived by Derbenev and Skrinsky 4 2 2 4 2 2 2 2π e Z n λ 3 v i ⊥ 2π e Z n λ 2 v i ⊥ − 2 v i || F || = − • 5 v i || F ⊥ = • 5 v i ⊥ Cooling rate, M v i M v i which is proportional to ion energy E loss rate to electrons,

dE Mv i = F • V τ = is defined as dt . H ence, F Consequence of “Conventional” Cooling Formulas

For Maxwellian distribution ion cooling due to the friction force is larger than ion heating due to velocity space diffusion by a factor of 2 only!

Based on Derbenev and Skrinsky, Particle Accelerators, 8, 235 (1978). And unpublished note by Alexei Fedotov January 8, 2004. Preceding Theory is Characterized by Particle Discreteness (binary collisions) Parkhomchuk’s Empirical Formula Parkhomchuk claims that an empirical formula is in much better agreement with computer simulations of fully magnetized cooling for magnetic fields of up to 4kG. Additionally, experiments and computer simulations showed reasonable agreement with Parkhomchuk. (2.4 Tesla is used in this case; based on booklet)

Magnetized friction force and cooling time based on the empirical formula in cgs units, are (for the case of hot ions 4Z 2e4nλ F = v and cold electron) i 3 and cooling time mvthi 3 Mvi Mvimvthi   v    thi  -8 τ c = =   2 4  ω pe   λ = 3.4; τc = 6x10 sec F 4Z e nλvi λ = ln   ρe      and a cooling length of 1.2 meter is needed. Based on Derbenev and Skrinsky, Parkhomchuk, as well as a booklet by Dikansky, Kudelainen, Lebedev, Meshkov, Parkhomchuk, Sery, Skrinsky, and Sukhina, “Ultimate Possibilities of Electron Cooling,” INP, Novosibirsk, USSR Report, Preprint 88-61 (1988), resultant cooling, electron energy spread, ion loss, needed magnetic field (except for ion heating due to velocity space diffusion) etc. are computed 1/3  2  2 1/3 τ0Ωe Thermal equilibrium Ti⊥ is given by   in cgs units, where τ0 is time Ti⊥ =5Ze n 4π  ω   pe  an ion spends in the electron beam, ωpe and Ωe are electron plasma and cyclotron -8 frequencies respectively. For a magnetic field of 2.4 Tesla, and τ0 of 6x10 sec, the perpendicular ion total velocity spread si reduced to 7.9x103 m/s, and the transverse

∆p −4 momentum (full) spread to ⊥ ≈3.9x10 , & the parallel momentum spread is reduced p by an order of magnitude as well. Electron beam thermal spread due to the electrostatic space charge 2 1/3 potential e n in cgs units yields an energy spread of 6.68x10-4 eV, which is orders of magnitude lower than the ion beam thermal spread. Inelastic interactions are: electron capture and ionization. But due to the relative low energy differential, the only ion loss mechanism is due to recombination, 2 1/3 Z  11.32Z T   α =3.02x10−13 ln +0.14 e⊥   2   T  T   Z  e⊥   e⊥  

Rate coefficient for recombination γ in cgs units except for T, which is in eV. For our parameters τrec is about τrec = nα 1.5 msec, which is orders of magnitude longer than any computed cooling time. Additionally, electron capture is suppressed in such a large magnetic field. Hence, electron recombination is not an issue in this process. Test Particle Model L i o u ville e q u a t i o n (ba s e d o n L i o uville t h e o re m c o n s e r v a t i o n o f p r oba b ilit y i n p h ase space) f o r p h ase space den s i t y D ( X 1 , X 2 , X 3 ,… ..X N ; t) ∂ N ∂ q v v ∂ {[+•∑ vv i −(Et(X i ,)+vv i ×B)• ]}D =0 ∂ t i = 1 ∂ X i m ∂ vv i wh e r e B i s c o n s ta n t m a gn e t i c fi e l d a p p l i e d e x te r n a lly. H e re Xxii= (,vvvi) is th e 6 d i m e n s i o n a l p h as e s p a ce ( pos it i o n a n d v e loc i t y ) coor d i na te o f t h e i th p ar t i c l e. Onl y Co ul o m b f o r ces ar e con s i d er ed, h e n c e v N ′ ∂ 1 Et(,xr i )= q∑ , t h e pr i m e ind i c a t e s th a t t h e te r m i = j i s o m it te d. j = 1 ∂ xxvvi ||ij− xv B y ta k i ng m o m e n t s o f t h e Lio u ville e q ua ti o n , a hie r a r c h y s i m ila r to t h e B B G K Y (B og o l i u b o v - B o ron - G r e e n - K i rk w o o d - Y v o n ) h i e r a r c h y w a s obta i n e d b y Ros t oke r a n d Ros e nb lu t h . B a s i c a lly , Ros t oker a n d Ros e nb lu t h inte g r a t e d o u t t h e c o o r d i n a te s o f a l l pa r t ic l e s b u t o n e , but tw o, e t c . of t h e L i o u ville e q u a t i o n to obta i n a c h a i n o f e q u a t i o n s f o r o n e - bod y , tw o - b od y , e t c . , d i s t ri bu ti on f u n c ti on s .

s s N e x t re d u c e d proba b ilit y d i s t r i b u t i o n s a r e d e fine d a s f s /V , wh er e V is t h e conf igur a t i o n s p ac e v o lu m e ( f s a. k. a . s - body s f unct i o n ), f s ( X 1 , X 2 ,..... X s ; t ) ≡ V ∫ DdX s + 1 ...... dX N ;a n d m o m e nts of th e L i o uville e q u a t i o n a r e ta ke n . A f te r th a t c l u s te r e x pa n s i o n , s i m ila r to M a y e r c l u s te r e x p a n s i o n [ f (X )= f (X ); f (X X )= f (X )f (X )+ P ( X X ); f (X X X )= f (X )f (X )f (X1 )+1 f (X1 )P 1 (X 2X )+1 f 2 (X 1)P ( X1 1X )+2 f (X )P1 (X2 X ) +T3 ( X1 X2 X3 ) , 1wh e1 r e1 P an2 d1 T ar3 e co1 r r1e l at i o2n f3u n c 1t i o n2 s] i s p3 e rf1 o r m1 e d 3. Bu t , 2 1 Ros t oker1 2 an3 d Ros e n b lut h pe r f o r m e d t h e e xpan s i o n to o r de r o f t h e di s c r e te n e s s para m e te r , whi c h i s pr opo r t i o nal to t h e p l as m a 3 − 1 p ar a m et er g = ( n λ D ) . In de n s e pl a s m a g i s a v e r y s m a l l num b e r, th e r e R o s t oke r a n d R o s e n b lut h tre a te d T a s highe r t h a n firs t orde r, P a s firs t or der a n d f 1 a s z e ro th o r d e r. To m o v e a w a y f r o m (d e c re a s in g ) p a rt i c l e di s c re te n e s s , th e li m its , wh e r e q → 0 , m → 0 , n → ∞ su ch t h at q/ m a n d q n re m a in c o n s ta nt a r e ta ke n . T h e s e a r e a l s o th e li m its ta ke n to de r i v e t h e V l aso v eq u a t i o n . Th e d i f f er e n ce bet w ee n t h e u s e o f t h e V l as o v eq u a t i o n an d t h e t e st par t i c l e m o d e l i s t h at i n t h e f i r s t i n t e r act i o n s ar e v i a wa v e s, while in t h e l a tte r s h i e lde d pa rt i c l e s a r e inte ra c t ing . Test Particle Model (continued) After lots of math, Rostoker and Rosenbluth were able to derive a Fokker-Planck like equation with expressions for (dynamical) friction force, diffusion, etc. on a test particle. However, since the shield cloud of the test particle had a very complex form, it was very difficult to even determine the maximum impact parameter. Rostoker continued to develop the shielding aspect of the theory (“dressed test particles”), which behave like statistically independent quasiparticles. However, applying the resultant equations to experimental setups is rather challenging even for 21st century computation capabilities (let alone the early 1960’s). Finally exact relaxation rates expressions for a Maxwellian field particle distribution were derived. Commonly used in fusion! Pertinent (to this case) relaxation rates νi/e in sec-1 (ion test particle in a background of field electrons) are given in the following equations (without relativistic corrections!) Test Particle Model (continued) d v = − ν i / e v dt i s i ( 1) d ( v − v ) 2 = ν i / e v 2 dt i i ⊥ ⊥ i ( 2) d ( v − v ) 2 = ν i / e v 2 dt i i || || i ( 3) Vel o c i t i es ar e de n o t e d b y v w h i l e r a t e s ar e in d i cated b y ν . S u bs c r ip ts ( s , ⊥ ,& ||) den o t e sl ow i n g dow n , t r a n s v er se d i f f u s i o n in v e l o c i t y space an d p ar a l l e l d i f f us i o n in v e l o c i t y space r e spect i v e ly . Av er a g es ar e per f or m e d o v er an en s e m b l e of t e s t p a r t i c l e d i s t r i bu t i on s f o r a M a x w el l i an f i el d p a r t i c l e di s t r i b u t i o n . E x a c t f o r m ula s e x i s t f o r re l a x a tio n ra te s th a t c a n be w r i t te n a s , ν i / e =  1 + M ψ ()x ν i / e ; ν i / e = 2 [( 1 − 1 x )ψ ( x ) + ψ ' ( x )] ν i / e ; ν i / e = [ψ ( x ) / x ]ν i / e s  m  0 ⊥ 2 0 || 0 i / e 2 4 2 3 wh e r e ν 0 = 4 π Z e λ n / M v i ; x i s e s s e n t i a l l y th e r a ti o of th e t e s t p a r t i c l e (i on ) en er gy t o t h e f i e l d par t i c l e ( e l e c t r o n) te m p er at u r e. Z i s i o n c h ar g e state, e e l em en t a r y ch a r g e an d λ is t h e C o u l o m b lo ga r i t h m. x ψ ( x ) = 2 t 1 / 2 e − t dt an d ψ ' ()x = d ψ , ( h er e e i s not t h e el e m e n tar y c h ar g e ) π ∫0 dx In c a se s w h e r e x > > 1 o r x < < 1 , (i .e . f o r v e ry f a st o r v e ry s l o w t e st p a r t i c l e s ) sim p l e r li m i t i ng f o r m s o f t h e r e l a x a t i o n r a t e s e x i t . T h ese eq u a t i o n s a r e u t i l i z ed i n t h e n e xt su bsect i o n s f o r i o n coo l i n g a n d i o n h eat i n g co m p ut at i o n s . C o m p ar ing eq u a t i o n s 2 a nd 3 to eq u a t i o n 1 , i t i s ob vio u s t h at o n a lo ng t i m e scal e i o n co o l i n g d o m i n a t e s, si n ce i o n v e l o c i t y sl o w s- do w n w i t h t i m e t , while io n h e at ing de v e l ops a s t , a s i s t h e cas e i n m u lt i- pass cool ing. Test Particle Model (continued) Experimental Verification The test particle model has been experimentally verified: J. Bowles, R. McWilliams, and N. Rynn, Phys. Rev. Letters 68, 1144 (1992); J. Bowles, R. McWilliams, and N. Rynn, Physics of Plasmas 1, 3418 (1994); J.J. Curry, F. Skiff, M. Sarfaty, and T.N. Good, Phys. Rev. Letters 74, 1767 (1995). Performed in Q machines by optical tagging and laser induced fluorescence (LIF): Ba, Ar ions are pumped from ground state to a long lived metastable state (tagged ions). LIF leads to photon emissions from tagged ions, from which velocity space distributions and their evolutions are determined (measurements in quiescent plasmas only without turbulence). Test Particle Model (continued) Fully justified to apply the test particle model plasma -4 physics formalism to this case: λD=7.43x10 cm⇒1346 Debye lengths in a beam diameter & 182 electrons in a Debye sphere. Electron gyro-radius is 3.13x10-5 cm⇒ almost 32,000 electron gyro-radii in a beam diameter. Electron gyro-frequency is 6.72x1010 Hz. During an interaction time of τ = 6x10-8 sec, an electron completes 4032 gyrations. Ion gyro-period over a factor of 5 larger than longest cooling time⇒ions are not magnetized. Ion inter-particle distance is 2.3x10-3 cm, i.e. larger than 3 Debye lengths⇒ions are totally shielded; relaxation rates of a single ion (“dressed” test particle), whose energy equals ion beam thermal spread, streaming through cold electrons (field particles), is a reasonable representation for the ion beam velocity space relaxation rates. Ion Cooling C o m puta t i o ns a r e pe r f or m e d i n th e be am r e s t f r am e , s i nc e γ = 1.0 0 22, c o rre c ti o n s to t i m e dila tio n s a r e m i nus c u le . P e rti n e n t re la x a tion ra te s ν i/ e in s e c -1 (ion te s t pa rtic le s l ow ing dow n in a ba c k g r ound of f i e l d e l e c tr ons ) a r e : i / e − 4 1 / 2 2 − 3 / 2 ν s = 1 . 7 x 10 µ nZ λε ( 4) i / e − 7 − 1 / 2 2 − 3 / 2 ν ⊥ = 1 . 8 x 10 µ nZ λε (5 ) i / e − 4 1 / 2 2 − 5 / 2 ν || = 1 . 7 x 10 µ nZ λ T e ε (6 ) Un i t s are cgs an d eV . T e is the e l e c tron, µ io n to pro t o n m a s s ra tio. Sin c e Ma x w e llia n dis t ribut io n is a s sum e d, a n d t h e sp read i n t h e p e r p en d i c u l a r d i rect i o n i s m u ch l a rger 1 2 ε ⊥ = 2 Mv th i ⊥ It i s th e e n erg y o f “t est ” rep r es en tat i v e i o n . E l ect ro n are m a gn et ized ⇒ the Coulom b log a rithm (b is the s m a lle s t i m pa c t pa ra m e te r) is  ρ e  λ = ln  b  ≈ 3 . 2 From e q u a tio n 4 c ool ing tim e − 8 τ c = τ s ≈ 1 i / e ≈ 5 . 35 x 10 ν s s e c , ( 7 ) w h ic h im plie s a c o oli n g l e ng th of 1. 07 m e te r ! T h e dif f e r e n c e be t w e e n this v a lue a n d tha t obta i ne d f r om P a r k h o m c huk ’ s em pir i c a l f o r m ula is a bou t 1 2 % . Ion Heating (velocity space diffusion) Slowing down (cooling) transverse velocity space diffusion ration,

ν i / e 1.7x10−4 µ1/ 2nZ 2λε −3/ 2 s = ≅ 1.7x105 i / e −7 −1/ 2 2 −3/ 2 ν ⊥ 1.8x10 µ nZ λε

5 −8 −2 τ ⊥ ≈1.7x10 • 6x10 ≈1x10 sec, i.e., about 10 msec

Slowing down (cooling) parallel velocity space diffusion ration,

ν i / e 1.7x10−4 µ1/ 2nZ 2λε −3/ 2 ε s = = ≅ 1.87x105 i / e −4 1/ 2 2 −5/ 2 ν || 1.7x10 µ nZ λTeε Te

5 −8 −2 τ || ≈ 2x10 • 6x10 ≈ 1.1x10 sec, i.e. about 11 msec

Thus, velocity space diffusion relaxation times are 5 orders of magnitude longer than the computed cooling times. Ion Heating (mobility due to electric field) i / e i / e Interesting to note that as ε → Te ,ν s →ν || In ca se of a n el ectr ic fie ld (ex ternal, due to char ge separ ation, or sp ac e ch ar ge im bal ance) t here is als o a mo bil ity term , w hich can res ult i n m odif icatio n of both v elocity and con fig urati on spa ce dis tribu tio n funct ion s. I n al l c om puta tion s so far sp ati ally homo gen eou s pl as ma s ar e as sum ed; not n ecessari ly corre ct.

M odificati on (p erturb ati on) t o the di stri but ion f unctio n

q ~ vr • ∇r f + ( Er + vr × Br ) • ∇r f = −ν f r 0 m v 0 m

U nreso lved i ssue; can lea d to radia l io n c urrent (s ligh t radia l reduct ion); hav e n ot h ad a ch an ce to ev alu ate Comparison of Parkhomchuk’s empirical formula to Test Particle Model Plasma Physics Formalism C o o l in g t i me c o mp u t e d fr o m P a r k h o mc h u k ’ s e m p i r i c a l fo r m u l a , 3 Mv i Mv i mv th i τ c = F = 4 Z 2 e 4 n λ v Subs ti tuti ng n u m e r i c ia l v a l u e s f o r t h e c o ns ta nt s m a n d e 3 9 Mv th i τ c = 4 . 28 x 10 Z 2 n λ ( 8)

T e st P a rticle Mo del P l as m a P h y s i c s f o rm alis m c o oling tim e τ c c a n be w r itte n a s 3 / 2 3 ε τ c = 5 . 88 x 10 Z 2 n λµ 1 / 2 i n m i x e d ( f us i on pl a s m a ; m o s t l y M 1 2 µ ≡ ε ≡ Mv th i cg s and e V ) u n its. Bu t m p and 2 . H e nc e , 3 / 2  1  1 / 2 3 / 2 3   m p M v th i − 3 / 2 3  2  3  − 12 er g  τ c = 5 . 88 x 10 Z 2 n λ M 1 / 2 ()2  1 . 6 x 10 eV  2 9 Mv thi τ c = 3 . 76 x 10 2 (9) Z n λ Co m p arin g equ a tio ns 8 & 9 y i e ld s v e r y g o od a g ree m ent b e t w e e n c ool i n g ti m e c o m put e d f r om P a r k hom c huk ’ s em piric a l f o r m ula a n d c o oli n g ti m e c o m p ute d f r om pla s m a r e la x a t i on ba s e d o n t h e t e s t pa r t i c l e m ode l . Crux-of-the-Matter Dressed ions with shielding clouds (test particle model) versus bare ions in “conventional cooling theories”. According to “conventional theory” rates for ion cooling and ion heating are comparable. But, cooling evolves as t while ion heating develops as t . Need long cooling time! In plasma relaxation theories based on the test particle model, ion dynamic friction (cooling, slowing-down) rate is much faster than ion velocity space diffusion (heating). Other Pertinent Physics Issues Electron beam stability and adverse effects the electron beam (or electron gun) might have on the gold ions. Magnetic Field Required for Electron Beam Equilibrium and Stability: to prevent electron beam expansion. Square electron density profile, electric I field Ee at the outer beam radius R is given in mks units by Ee = − 2πε 0vR 2 2 B 2 2 Ee 6 needed magnetic field 0.5ε 0 Ee = ; B = 2 Ee=2.7x10 V/m. B = 90 2µ0 c Gauss. A more stringent magnetic field requirement is imposed by plasma 2 ω pe 3 stability 2 ≤ 1 which necessitates a magnetic 2x10 Gauss or 0.2 Tesla. Ωe These magnetic fields are small compared to the 2.4 Tesla magnetic field, which maximizes the cooling decrement. Other channels of recombination like three body collisional recombination or dielectronic recombination have extremely low probability Cross section for the latter is usually of the order of 10-19 cm2 or less. The low cross sections combined with 10’s nsec interaction time and a mean free path of 108 cm render these processes unimportant. Other Pertinent Physics Issues (continued) C harge E xchange

Pressure in H C plasm a cathode electron gun 10-5 Torr of argon gas. H igh charge exchange cross sections 10-1 4 cm 2 : m ean free path is 336 cm . Inside the hollow cathode the p ressure is about 10-2 T o rr: M F P < 0 .3 c m . B u t w ith c o n c e n tric H C sy ste m io n b e a m in je c tio n th ro u g h h o llo w cathode m ight be an option. In the extractor pressure is under 10-5 Torr; outside the extractor pressure is 10-7 Torr, w here charge exchange is no longer an issue. Electron guns w ith carbon fiber cathodes have generated close to 1 M A of electron current. W ith this cathode currents of up to 2 kA at 2 kV w ere obtained in m icroseconds long pulses. D epending on the current generated, pressure during the electron beam pulse can be betw een 10-3 to 1 0 -6 T orr (or even low er w here large p um ping capability is available). Since the needed electron beam currents are w ell below 100 A , pressures below 10-6 T orr, w here charge exchange is not an issue, are expected.

O th e r P la sm a In sta b ilitie s

The electron beam should be stable for an axial m agnetic field of 2.4 T esla. If the electron beam is stable, there should, in principle, be no other instabilities. The only p ossible plasm a instability m ight be to the ions (like a ro ta tin g tw o stre a m in sta b ility ). L ik e a ll b e a m in sta b ilitie s, it has a density threshold. Since the ion density is m ore than three orders of m agnitude low er than the electron density, there should be no beam instabilitie s. Personal Chronology • 1994 – suggested using intense electron diode as final stage (single pass) cooling of muons for the muon collider. • 1996 – results included in major muon collider documents; Palmer,…Hershcovitch… et al, “Muon Collider Design”, Nuclear Physics B 51A, 61 (1996).

• 1997- PAC97 was told wrong equations, should use: N.S. Dikansky, V.I. Kudelainen, V.A. Lebedev, I.N. Meshkov, V.V. Parkhomchuk, A.A. Sery, A.N. Skrinsky, and B.N. Sukhina, “Ultimate Possibilities of Electron Cooling,” Institute of Nuclear Physics, Novosibirsk, USSR Report, Preprint 88-61 (1988). • Shortly after told muon collider colleague: re-evaluate previous results. 3 9 • 2008 – old approach is probably valid nλD ≈ 10 Discussion: justification & limitations 3 Plasma parameter , n λ D which is inversely proportional to the discreteness parameter must be a very large number. Here it is 182. In hollow cathode arcs it is over 106; in the Plasma Window it is over 108; and, in fusion grade plasma (Tokamak) the plasma parameter is over 109! Nevertheless, ion are well shielded from each other, since intra ion distance is more than three Debye lengths. Additionally simplified (useful) equations assume Maxwellian distributions. As ions cool off the asymptotic expressions lose validity. Discussion: space charge affect on ions In quasi-neutral dense plasmas, interactions can be separated into two classes: strong close-range (binary or multi) and collective long-range through average electric and magnetic fields, which in absence of turbulence, are much weaker. Here the plasma in non-neutral! Radial electric field effect on peripheral ion in the presence of a strong magnetic field has yet to be analyzed including centrifugal forces. Work-in-progress If space charge is a problem, explore space-charge mitigation (like variation on plasma flood guns [2.777x10-6 Torr CX MFP over 12 m] or proton plasma); must be done below instability density threshold or using stabilizing mechanisms…haven’t thought it through yet