... Particle acceleration whithout emittance or beta function

Transcription

1 Introduction to Tranvere Beam Otic Bernhard Holzer II. ε &... don't worry: it' till the "ideal world" Hitorical note:... Particle acceleration whithout emittance or beta function 4 N ntz e i N θ * 4 8 πε r K in θ / Nθ Rutherford Scattering, 9 Uing radioactive article ource: -article of ome MeV energy θ

2 Reminder of Part I Equation of fmotion: Solution of Trajectory Equation K K K ρ k k hor. lane: vert. Plane: M * M drift l M foc co K l in K l K K in K l co K l M defoc coh K l K inh K l inh K l K coh K l

3 Tranformation through a ytem of lattice element combine the ingle element olution by multilication of the matrice M M M M M M total QF * D * QD * Bend * D*...,* M focuing len diole magnet defocuing len court. K. Wille tyical value in a trong foc. machine: mm, mrad

4 Quetion: what will haen, if the article erform a econd turn?... or a third one or... turn

5 Atronomer Hill: differential equation for motion with eriodic focuing roertie Hill equation Eamle: article motion with eriodic coefficient equation of motion: k retoring force cont, we eect a kind of quai harmonic k deending on the oition ocillation: amlitude & hae will deend kl k, eriodic function on the oition in the ring.

6 The Beta Function General olution of Hill equation: i ε co φ ε, Φ integration contant determined by initial condition eriodic function given by focuing roertie of the lattice quadruole L Inerting i into the equation of motion d Ψ hae advance of the ocillation between oint and in the lattice. For one comlete revolution: number of ocillation er turn Tune Q y d π o

7 9. Beam Emittance and Phae Sace Ellie 9. Beam Emittance and Phae Sace Ellie Liouville: in reaonable torage ring γ ε Liouville: in reaonable torage ring area in hae ace i contant. A π*εcont ε beam emittance woozilycity of the article enemble, intrinic beam arameter, cannot be changed by the foc. roertie. Scientifiquely oken: area covered in tranvere, hae ace and it i contant!!!

8 Phae Sace Ellie ti l t j t ε co { φ } articel trajectory: ma. Amlitude: ˆ ε at that oition? ut ˆ into ε γ and olve for ε γ ε ε ε / * * A high -function mean a large beam ize and a mall beam divergence. et vice vera!!! In the middle of a quadruole maimum, zero! and the ellie i flat

9 Phae Sace Ellie ε γ ε γ olve for, ± ε ε γ d and determine ˆ via: d εγ ε ˆ εγ ˆ ± ε γ ε hae and orientation of the hae ace ellie deend on the Twi arameter γ

10 Emittance of the Particle Enemble: ε co Ψ φ ˆ ε Gauß Particle Ditribution: ρ N e σ e πσ article at ditance σ from centre 68.3 % of all beam article ingle article trajectorie, N er bunch LHC: σ ε * 5* m*8m. 3 mm aerture requirement: r * σ

11 Emittance of the Particle Enemble: article bunch Z, X, Y Eamle: HERA beam arameter in the arc 8 m ε 7* 9 rad m σ σ ε.75 mm

12 . Tranfer Matri M ye we had the toic already general olution of Hill equation { } ε co φ ε co{ } in { } φ φ remember the trigonometrical gymnatic: ina b etc ε co coφ in inφ ε [ co co φ in in φ in co φ co in φ ] tarting at oint, where we ut Ψ co φ, ε in φ ε inerting above

13 co in in { } { } { co in} { co in} which can be ereed... for convenience... in matri form M M co in in co in co in * we can calculate the ingle article trajectorie between two location in the ring, if we know the γ at thee oition. * and nothing but the γat thee oition. *! * Äquivalenz der Matrizen

14 . Periodic Lattice co in in M co in co in Thi rather formidable looking matri imlifie coniderably if we conider one comlete revolution ELSA Electron Storage Ring co turn in M γ in turn turn co in turn in turn turn turn L d turn hae advance er eriod Tune: Phae advance er turn in unit of π Q π d

15 Stability Criterion: Quetion: what will haen, if we do not make too many mitake and your article erform one comlete turn? Matri for turn: co turn in turn in turn M γ inturn co turn inturn co in γ Matri for N turn: N M co J in co N J inn N J The motion for N turn remain bounded, if the element of M N remain bounded real co Tr M

16 tability criterion. roof for the dibelieving collegue!! tability criterion. roof for the dibelieving collegue!! Matri for turn: co in in turn turn turn M co in f in co in turn turn turn M γ γ I J Matri for turn: in co in co J I J I M in in co in in co co co J JI IJ I now I I * J I γ γ * J I γ γ * I J I J J I J I * γ γ γ γ γ γ in co J I M in co J I M

17 . Tranformation of. Tranformation of,, γ conider two oition in the torage ring:, g g, ' * ' M S C S C M Betafunction in a torage ring ince ε cont Liouville:... remember W CS -SC γ ε γ ε inerting into ε M ' * ' C C S S C C S S M C C ort via, and comare the coefficient to get... S S C C S S C C γ ε

18 in matri notation: C SC S γ CC SC SC SS γ γ C SC S γ C SC S CC SC CS SS γ C SC S γ!. thi ereion i imortant. given the twi arameter,, γ at any oint in the lattice we can tranform them and calculate their value at any other oint in the ring. 3. the tranfer matri i given by the focuing roertie of the lattice element, the element of M are jut thoe that we ued to calculate l ingle article trajectorie. 4. go back to oint.

19 3. Lattice Deign: how to build a torage ring ρ B ρ / q d Circular Orbit: diole magnet to define the geometry d ρ dl ρ Bdl Bρ field ma of a torage ring diole magnet The angle run out in one revolution mut be π, o for a full circle Bdl π Bdl π B ρ q define the integrated diole field around the machine. Nota bene: ΔB B 4 i uually required!!

20 Eamle HERA: 9 GeV Proton torage ring diole magnet N 46 l 8.8m qe B dl N l B π / q 9 π 9 ev B 5. 5 Tela 8 m m 3 e

21 The FoDo-Lattice A magnet tructure coniting of focuing and defocuing quadruole lene in alternating order with nothing in between. Nothing element that can be neglected on firt ight: drift, bending magnet, RF tructure... and eecially eeriment... Starting oint for the calculation: in the middle of a focuing quadruole Starting oint for the calculation: in the middle of a focuing quadruole Phae advance er cell μ 45, calculate the twi arameter for a eriodic olution

22 Periodic olution of a FoDo Cell QF QD QF y L Outut of the otic rogram: Nr Tye Length Strength y y y m /m m /π m /π IP,,,6,, 5,95,, QFH,5 -,54,8,54,4 5,488 -,78,7 QD 3,5,54 5,488 -,78,7,8,54,66 3 QFH 6, -,54,6,,5 5,95,,5 4 IP 6,,,6,,5 5,95,,5 Q X,5 Q Y,5. 5* π 45

23 Can we undertand, what the otic code i doing? matrice M foc co K lq in K l K K in K lq co K lq q M drift ld trength and length of the FoDo element K /-.54 m - lq.5 m ld.5 m The matri for the comlete cell i obtained by multilication li of the element matrice M M * M * M * M * M FoDo qfh ld qd ld qfh Putting the number in and multilying out... M FoDo

24 The tranfer matri for one eriod give u all the information that we need!. i the motion table? trace M FoDo.45 <. Phae advance er cell co in M γ in in co in co Trace M.77 arc co Trace M hor -function 4. hor -function M, co.6 m, M in in

25 III. Acceleration and Momentum Sread The not o ideal world

26 Remember: Beam Emittance and Phae Sace Ellie: equation of motion: k general olution of Hill equation: beam ize: ε co ϕ σ ε "mm" ε γ * ε i a contant of the motion it i indeendent of * arametric rereentation of an ellie in the ace * hae and orientation of ellie are given by,, γ γ ε γ εγ????? ε? ε? ε γ εγ ε ε? ε / ε

27 4. Liouville during Acceleration ε γ εγ ε γ ε Beam Emittance correond to the area covered in the, Phae Sace Ellie ε Liouville: Area in hae ace i contant. But o orry... ε cont t! Claical Mechanic: hae ace diagram of the two canonical variable oition & momentum L j ; L T V kin. Energy ot. Energy q & j

28 According to Hamiltonian mechanic: hae ace diagram relate the variable q and q oition γ momentum γmv mcγ v c ; & c Liouville Theorem: dq cont for convenience i.e. becaue we are lazy bone we ue in accelerator theory: d d dt d dt d where v /c dq mc γ d dq mcγ d ε the beam emittance ε d hrink during γ acceleration ε ~ / γ

29 Nota bene:. A roton machine or an electron linac need the highet aerture at injection energy!!! a oon a we tart to accelerate the beam ize hrink a γ -/ in both lane. σ ε. At lowet energy the machine will have the major aerture roblem, here we have to minimie ˆ 3. we need different beam otic adoted to the energy: A Mini Beta concet will only be adequate at flat to.,8km 3 m HERA roton otic at 9 GeV HERA roton otic at 4 GeV

30 Eamle: HERA roton ring injection energy: 4 GeV γ 43 flat to energy: 9 GeV γ 98 emittance ε 4GeV. * -7 ε 9GeV 5. * -9 7 σ beam enveloe at E 4 GeV and at E 9 GeV

31 The not o ideal world 5. The / Problem ideal accelerator: all article will ee the ame accelerating voltage. Δ / nearly ideal accelerator: Cockroft Walton or van de Graaf Δ / -5 Vivitron, Straßbourg, inner tructure of the acc. ection MP Tandem van de Graaf Accelerator at MPI for Nucl. Phy. Heidelberg

32 Linear Accelerator 98, Wideroe chematic Layout: Energy Gain er Ga : W q U inω RF t drift tube tructure at a roton linac 5 MHz cavitie in an electron torage ring * RF Acceleration: multile alication of the ame acceleration voltage; brillant idea to gain higher energie

33 Problem: anta rhei!!! Heraklit: v. Chr. Z, X, Y Eamle: HERA RF: Bunch length of Electron cm U ν 5 MHz t c λ ν λ 6 cm λ 6 cm o in9 in84 o.994 Δ U U 6. 3 tyical momentum read of an electron bunch: Δ. 3

34 6. Dierion: trajectorie for Δ / Quetion: do you remember lat eion, age? ure you do ŷ Force acting on the article d F m dt mv ρ e By v y ρ ρ remember: mm, ρ m develo for mall d mv m eb v y dt ρ ρ conider only linear field, and change indeendent variable: t B y B B y e B ρ ρ mv e mv g Δ but now take a mall momentum error into account!!!

35 Dierion: Dierion: develo for mall momentum error Δ << Δ Δ eg eg eb B e Δ Δ ρ k g ρ ρ k k eb * * * * Δ Δ ρ ρ ρ ρ ρ k Δ ρ ρ k Δ Momentum read of the beam add a term on the r.h.. of the equation of motion. inhomogeneou differential equation.

36 Dierion: Δ k ρ ρ general olution: h i K h Δ i K i ρ h Normalie with reect to Δ/: i D Δ Dierion function D * i that ecial orbit, an ideal article would have for Δ/ * the orbit of any article i the um of the well known and the dierion * a D i jut another orbit it will be ubject to the focuing roertie of the lattice

37 Dierion Eamle: homogeneou diole field Cloed orbit for Δ/ >. ρ Δ i D Matri formalim: Δ D C S Δ D ΔΔ C S D C S D

38 Reume : beam emittance ε γ beta function in a drift γ and for article trajectory for Δ/ inhomogeniou equation ρ k Δ ρ and it olution Δ D