The evolution of the phase space density of particle beams in external fields

Transcription

1 The evolution of the phase space density of paticle beams in extenal fields E.G.Bessonov Lebedev Phys. Inst. RAS, Moscow, Russia, COOL 09 Wokshop on Beam Cooling and Related Topics August 31 Septembe 4, 2009 IMP, Lanzhou, China

2 INTRODUCTION In 1958 K.W.Robinson poved the theoem in the field of cicula paticle acceleatos. He deived at once the sum of damping ates (decements) of thee paticle oscillation modes in acceleatos. In ode to deive damping incement, he evaluated the deteminant of the tansfe matix of the infinitesimal element of length of a paticle obit. The deteminant detemines the evolution of a 6D phase space volume of the beam o its density along the tajectoy. In the poof K.W.Robinson did an expansion the enegy loss pe tun (aveage powe) taken fo the abitay fictional foces and then took the deivative of the enegy loss

3 ove the paticle enegy fo the pivate case of adiative eaction foce. That is why his final fomulae lost geneal fom and do not include the tem with the deivative. Late (1965), A.A.Kolomensky deived the fomulae in the geneal fom fo the elativistic case and applied it to the ionization cooling. He calculated sepaately damping ates fo thee diections in the cuvilinea coodinate system and then took thei sum. P.L.Csonka in 1992 fo this pupose evaluated the infinitesimal 6D phase space volume as well and used some additional conditions. H.Wiedemann in his textbook pesented the poof of the theoem following Robinson s idea but keeping the deivative of the powe losses ove the enegy.

4 Now the theoem in the paticle acceleato community is named by Robinson theoem o Robinson s damping citeion.

5 In this pape the evolution of the phase space density of paticle beams in extenal fields is found poceeding fom the continuity equation in the sixdimensional phase space. The Robinson theoem, which includes the Liouville theoem as a special case, was poved in a moe simple and consistent altenative way valid fo abitay extenal fields, aveaged fields of the beam (self-geneated electo-magnetic fields except intabeam scatteing) and abitay fictional foces (linea, nonlinea). Limits of the applicability of the Robinson theoem in case of cooling of excited ions having a finite living time ae pesented.

6 EVOLUTION OF PARTICLE BEAM DEN SITY IN THE EXTERNAL E.M. FIELDS Let us poceed fom the continuity (Liouville s) equation in the 6D phase space coodinate-momentum (, p) u d ρ dt + ρ div v = Hee components of the 6D velocity v = ( v, vp) ae x, y, z, px, py, pz, whee i = di / dt, pi = dpi / dt. This equation o the equivalent equation ρ / t+ div ( ρv) = 0 expesses the numbe of paticles consevation law. 0 (1)

7 The solution of the Eq. (1) can be pesented in the integal fom ρ = ρ0 exp[ divvdt ( ) ]. The divegence of uu uu uu the 6D-velocity is div v = divv + divpv p, divv = 0 u as the velocity v = cp/ p + m c does not depend uu u u u on spatial coodinates. The value vp = p = ph+ pf = u u u F = FH + FF is the foce acting upon the paticle. u u uu The consevative foce FH = ee(, t) + ( e/ c)[ v H(, t)] is detemined by extenal fields and the fields of the paticle beam ( div F H = 0), while F F is the fictional u u foce. p

8 That is why div v = divpff. Finaly we have ρ = ρ0 exp[ divpffdt]. The fictional foce u can be pesented in the fom FF = χf (, p, t) n, whee p= p, u n= p/ p, χf (, pt, ) is the fictional coefficient. divf p F = χfdivpn n gad pχf = 2 χf / p χf / p. The fictional powe is PF = FF v = χf (, pt,) n v = cβ χf (, pt,), whee β = v / c. It follows that χf = PF(, pt, ) / cβ and the above equation become ρ = ρ0exp[ α6d( pt,, ) dt], whee α6d( pt,, ) = div FF = 2 χf / p + χf / p o 1 PF (, ε,) t PF (, ε,) t α6d(, ε, t) = (1 + ) +. 2 β ε ε

9 The 6D ate of the beam density change is detemined by the fictional powe and its deivative with espect to the paticle enegy. The expession is valid fo the abitay systems (linea, nonlinea, coupled). We did not use a cuvilinea coodinate system, the Jacobee s fomula fo the system of linea diffeential equations, matices, any additional conditions. In ou case the expession is valid fo the nonelativistic case as well. If the aveage enegy of paticles is kept constant, powe loss of paticles of the beam in the limits of its phase space volume is a linea function of the enegy, then, the ate of the beam density change in these machines is detemined by the 6D-damping incement

10 whee 1 PF ( ε) PF ( ε) = div F = (1 + ) ε= ε +, s ε= εs β ε ε 6D F 2 P F ( ε ) is the aveage ate of the paticle enegy loss due to fiction. If the enegy of paticles is maintained at constant level, P α ( p) 0, and P ( p) / ε ~ P ( p) / ε then, accoding to (3), the F F F density of the paticle beam will incease by e = 2.7 times afte paticles of the beam will lose the enegy Δε ~ ε. If P ( p) / ε ~ P F ( p) / Δε ( Δ ε << ε ) then the the same incease of the density b b will be afte paticles of the beam lose the enegy Δε b is the initial enegy spead of the beam. F Δ ε =Δε, whee To sepaate the longitudinal component of the momentum fom the tansvese components we can pass on to a cuvilinea coodinate system. b

11 In this system the 6D incement is the sum of two tansvese (adial, vetical) and longitudinal incements: α = 2α + 2α + 2 α. 6D x z s The incements fo longitudinal and uncoupled vetical oscillations ae found without a poblem by diect calculations and the adial one is detemined fom the Robinson theoem. In the elativistic case: 1 Ps P dp 1 Ps 1 dp αx = + s s, αz =, αs = s. 2 ε ε dε 2 ε 2 dε s s

12 Robinson and Liouville theoems ae valid fo identical paticles (electons, potons, muons and so on). The Robinson theoem is valid if fictional foces exist only at the moment of thei inteaction with media o extenal fields and thee is no time delay between the inteaction time and the fictional foce. Excited ions have highe est mass then unexcited ones and have finite lifetime. It means that in geneal case the above theoems ae not valid fo ion cooling (excited ions ae not identical to unexcited ones and have finite lifetime). The theoems woks well if the lifetime of excited ions is less then some chaacteistic time fo the pocesses of the beam evolution detemined by concete conditions. E.g., in the case of paticle acceleatos the delay time between the moment of the ion excitation by a lase beam and following photon eemission must be less then the peiod of betaton oscillations. Othewise, the additional cooling o heating of the ion beam is possible. Limits of the applicability of the Robinson theoem in case of cooling of excited ions having a finite living time ae pesented.

13 Fig. 2. Motion of a paticle in longitudinal-adial plane. is a jump of the closed obit. Δ x η

14 CONCLUSION In this pape the evolution of the phase space density of paticle beams in extenal fields is found poceeding fom the continuity equation in the sixdimensional phase space. The Robinson theoem, which includes the Liouville theoem as a special case, was poved in a moe simple and consistent altenative way valid fo abitay extenal fields, aveaged fields of the beam (self-geneated electomagnetic fields except intabeam scatteing) and abitay fictional foces (linea, nonlinea). Limits of the applicability of the Robinson theoem in case of cooling of excited ions having a finite living time ae pesented. Cooling of excited ions having a finite living time in stoage ings equies additional investigations.