UVa Coure on Phyi of Partile Aelerator B. Norum Univerity of Virginia G. A. Krafft Jefferon Lab 3/7/6 Leture
x dx d () () Peudoharmoni Solution = give β β β () ( o µ + α in µ ) β () () β x dx ( + α() α ) ( () ) in µ, β o µ, in, d α µ + ( α() α ) o µ β (),,, β in µ ( () ( () () ( ) ( )) ) x β x' + α x / β ( ) = x + ( β x' + α x ), ( ) / β ε + Uing the x() equation above and the definition of ε, the olution may be written in the tandard peudoharmoni form () () ( ) β x' + αx x = εβ o µ =, δ where δ tan x The the origin of the terminology phae advane i now obviou.
Hint for Problem
Cae II: K() not periodi In a lina or a reirulating lina there i no loed orbit or natural mahine periodiity. Deigning the tranvere opti onit of arranging a fouing lattie that aure the beam partile oming into the front end of the aelerator are aelerated (and ometime deelerated!) with a mall beam lo a i poible. Therefore, it i imperative to know the initial beam phae pae injeted into the aelerator, in addition to the tranfer matrie of all the element making up the fouing lattie of the mahine. An initial ellipe, or a et of initial ondition that omehow bound the phae pae of the injeted beam, are traked through the aeleration ytem element by element to determine the tranmiion of the beam through the aelerator. The deign are uually made up of wellundertood module that yield known and undertood tranvere beam optial propertie.
Definition of β funtion Now the peudoharmoni olution applie even when K() i not periodi. Suppoe there i an ellipe, the deign injeted ellipe, whih tightly inlude the phae pae of the beam at injetion to the aelerator. Let the ellipe parameter for thi ellipe be α, β, and γ. A funtion β() i imply defined by the ellipe tranformation rule β where () ( ()) = M ( ) ( ) ( ( )) γ M M α + M [( ()) ( ( ) ( )) ] = M + βm αm / β M M ( ) M ( ) M, () M () β
One might think to evaluate the phae advane by integrating the beta-funtion. Generally, it i far eaier to evaluate the phae advane uing the general formula, tan µ ', = β ( M ) ', ()( M ) ()( ) ', α M ', where β() and α() are the ellipe funtion at the entrane of the region deribed by tranport matrix M ',. Applied to the ituation at hand yield tan µ, = β M M ( ) () α M ()
Diperion Calulation Begin with the inhomogeneou Hill equation for the diperion. d D d ( ) + K D = ρ ( ) Write the general olution to the inhomogeneou equation for the diperion a before. ( ) ( ) + ( ) + ( ) D D Ax Bx = p Here D p an be any partiular olution. Suppoe that the diperion and it derivative are known at the loation, and we wih to determine their value at. x and x, beaue they are olution to the homogeneou equation, mut be tranported by the tranfer matrix olution M, already found.
To build up the general olution, hooe that partiular olution of the inhomogeneou equation with boundary ondition ( ) ( ) D D p, = p, = Evaluate A and B by the requirement that the diperion and it derivative have the proper value at (x and x need to be linearly independent!) ( ) ( ) ( ) ( ) ( ) ( ) A x x D = B x x D ( ) = ( ) + ( ) ( ) + ( ) ( ) D D M D M D p,,, ( ) = ( ) + ( ) ( ) + ( ) ( ) D D M D M D p,,,
3 by 3 Matrie for Diperion Traking ( M ) ( ) ( ), M, D p, ( ) ( ) p ( ) D ( ) D( ) D ( ) M ( ), M, D, = D Partiular olution to inhomogeneou equation for ontant K and ontant ρ and vanihing diperion and derivative at = K < K = K > D p, () D' p, () K ρ ( oh( K) ) ( o ) ( K ) ( K) ρ inh in ( K) K ρ ρ K ρ K ρ
Beam Mathing Fundamentally, in irular aelerator beam mathing i applied in order to guarantee that the beam envelope of the real aelerator beam doe not depend on time. Thi requirement i one part of the definition of having a table beam. With periodi boundary ondition, thi mean making beam denity ontour in phae pae align with the invariant ellipe (in partiular at the injetion loation!) given by the ellipe funtion. One the partile are on the invariant ellipe they tay there (in the linear approximation!), and the denity i preerved beaue the ingle partile motion i around the invariant ellipe. In lina and reirulating lina, uually different purpoe are to be ahieved. If there are region with periodi fouing lattie within the lina, mathing a above enure that the beam
envelope doe not grow going down the lattie. Sometime it i advantageou to have peifi value of the ellipe funtion at peifi longitudinal loation. Other time, re/mathing i done to preerve the beam envelope of a good beam olution a hange in the lattie are made to ahieve other purpoe, e.g. hanging the diperion funtion or hanging the hromatiity of region where there are bend (ee the next hapter for definition). At a minimum, there i uually a mathing done in the firt part of the injetor, to take the phae pae that i generated by the partile oure, and hange thi phae pae in a way toward agreement with the nominal tranvere fouing deign of the ret of the aelerator. The ellipe tranformation formula, olved by omputer, are eential for performing thi proe.
Claial Mirotron: Vekler (945) l = 6 Extration l = 5 l = 4 l = 3 l = Magneti Field l = y RF Cavity x µ = ν =
Bai Priniple For the geometry given d v dt x d( γ mv) = e E v B dt + d( γ mv x ) = ev ybz dt d( γ mv y ) = evxbz dt Ω + vx = γ For eah orbit, eparately, and exatly d v y Ω dt + vy = γ v x( t) = vxo( Ωt/ γ ) v y( t) = vx in( Ω t/ γ ) γv x ( t ) t Ω x x x = in ( Ω / γ ) y( t) = o( Ω t / γ ) γv Ω γv Ω
Non-relativiti ylotron frequeny: Ω = πf = ebz / m Relativiti ylotron frequeny: Ω / γ Bend radiu of eah orbit i: ρ = γ v / Ω γ / Ω l l x, l l In a onventional ylotron, the partile move in a irular orbit that grow in ize with energy, but where the relatively heavy partile tay in reonane with the RF, whih drive the aelerating DEE at the non-relativiti ylotron frequeny. By ontrat, a mirotron ue the other ide of the ylotron frequeny formula. The ylotron frequeny dereae, proportional to energy, and the beam orbit radiu inreae in eah orbit by preiely the amount whih lead to arrival of the partile in the ueeding orbit preiely in phae.
Mirotron Reonane Condition Mut have that the bunh pattern repeat in time. Thi ondition i only poible if the time it take to go around eah orbit i preiely an integral number of RF period f γ µ f = RF f γ = ν f RF Firt Orbit For laial mirotron aume an injet o that f γ + ν f Eah Subequent Orbit RF f f µ ν RF
Parameter Choie The energy gain in eah pa mut be idential for thi reonane to be ahieved, beaue one f /f RF i hoen, γ i fixed. Beaue the energy gain of non-relativiti ion from an RF avity IS energy dependent, there i no way (preently!) to make a laial mirotron for ion. For the ame reaon, in eletron mirotron one would like the eletron loe to relativiti after the firt aeleration tep. Conern about injetion ondition whih, a here in the mirotron ae, will be a reurring theme in example! f / f B / B RF = z B = πm λe B =.7 T =.7 kg@m Notie that thi field trength i NOT tate-of-the-art, and that one normally hooe the magneti field to be around thi value. High frequeny RF i expenive too!
Claial Mirotron Poibilitie Aumption: Beam injeted at low energy and energy gain i the ame for eah pa f f RF µ, νγ,, γ / /3 /4 µ, ν, γ, γ µ, νγ,, γ µ, νγ,, γ,,, 3,, 3/, 4,, 4/3, 5,, 5/4, 3,, 3, 4,,, 5,, 5/3, 6,, 3/, 4, 3, 4, 3 5, 3, 5/, 3 6, 3,, 3 7, 3, 7/4, 3 5, 4, 5, 4 6, 4, 3, 4 7, 4, 7/3, 4 8, 4,, 4
For ame mirotron magnet, no advantage to higher n; RF i more expenive beaue energy per pa need to be higher Extration Magneti Field y RF Cavity x µ = 3 ν =
Going along diagonal hange frequeny To deal with lower frequenie, go up the diagonal Extration Magneti Field y RF Cavity x µ = 4 ν =
Phae Stability Invented independently by Vekler (for mirotron!) and MMillan V (t) φ ( µ + l ν ) ( ) / frf φ = π f RF t t / f RF Eletron arriving EARLY get more energy, have a longer path, and arrive later on the next pa. Extremely important diovery in aelerator phyi. MMillan ued ame idea to deign firt eletron ynhrotron.
Phae Stability Condition Synhronou eletron ha Phae = φ E l = E o + lev oφ Differene equation for differene after paing through avity pa l + : φ E l + l + = ev in φ πm λe 56 l φl El Beaue for an eletron paing the avity E after = E before + ev ( o( φ + φ ) oφ )
Phae Stability Condition ρl ( + E / El ) ρ l K ( ( )) i = / ρ Dx, p,= ρi o / ρi πρi πρ l D M 56 = d = ( o / ρ l ) d ρ = i πρ l φ E l + l + ev in φ 4π ρ l λel 4π ρ lev λe l in φ φl El
Phae Stability Condition Have Phae Stability if Tr M < i.e., < νπ tanφ <