Synchrotorn Motion. A review:

Transcription

1 A review: H e cm c Synchrotorn Motion ( p ea ( x / ( p x ea x ( p z ea / z The phae pace coordinate are (x,,z with independent coordinate t. In one revolution, the time advance T, called the orbital period. In one orbital period, the particle orbit i equal to the circumference C. All accelerator component repeat in each orbital period. It would be nice to ue a the independent coordinate. How to make thi coordinate tranfer? Thee equation indicate that p become the new Hamiltonian with the (x,p x,z,p z,t,-h and a the independent coordinate. x x H ~ / p( [( px eax ( pz eaz ] ea p ~ ~ ~ ~ ~ H H H H H H ~ x, px, z, pz, t, ( H. px x pz z ( H t Bz Bx x K x ( x, z K z ( z B B Hill equation n n ev (inn in, n n n Synchrotron motion The longitudinal electric field at an rf gap where ω =β c/r i the angular revolution frequency of a reference (ynchronou particle, ε i the amplitude of the electric field, β c and R are repectively the peed and the average radiu of the reference orbiting particle, h i the harmonic number, and i the phae angle for a ynchronou particle. We aume that the reference particle pae through the cavity gap in time t nt +( g/βc, g/βc (n = integer, where g i the rf cavity gap width. The energy gain for the reference particle per paage i The angular ynchrotron frequency i where c i the peed of light and R i the average radiu of the ynchrotron. The ynchrotron tune, defined a the number of ynchrotron ocillation per revolution, i Typically the ynchrotron tune i of the order of 3 for proton ynchrotron and for electron torage ring. The effective voltage een by the orbiting particle i V=ε gt. The acceleration rate for a ynchronou particle i V=V in(ω rf t+φ ω rf =hf

2 Horizontal IPM meaurement Fermilab Booter The ynchrotron tune in a booter cycle. The quare are meaured from turn-by-turn data with ICA method. The croe are meaured from phae ignal with ynchrotron phae detector (SPD. Note that the SPD method ha difficulty in meauring the ynchrotron tune above the tranition energy at around the.5 m. The ynchrotron equation of motion can be derived from the Hamiltonian for phae pace coordinate (, /ω : =5 o =3 o = o Note that the econd term in the Hamiltonian can be viualized a the potential. Stable particle motion i bounded by the potential well. The area of table motion i called bucket. What i wrong with the potential well above? =5 o =3 o = o =5 o = o =35 o

3 h ev (in in η< η> Synchrotron bucket area, eparatrix γ>γ T γ<γ T Left: chematic drawing of the rf potential for = and π/. The dahed line how the maximum energy for table ynchrotron motion. Middle: the correponding eparatrix orbit in (πh η /evβ / x v. The phae u i the turning point of the eparatrix orbit. Right: an example of table rf bucket, called fih diagram, with =π/.

4 In Hamiltonian formalim, the rf electric field i conidered to be uniformly ditributed in an accelerator. In reality, rf cavitie are localized in a hort ection of a ynchrotron, and therefore ynchrotron motion i more realitically decribed by the ymplectic mapping equation The phyic of the mapping equation can be viualized a follow. Firt, the particle gain or loe energy at it nth paage through the rf cavity, then the rf phae n+ depend on the new off-momentum coordinate δ n+. It i no urprie that atifie the ymplectic condition: When the acceleration rate i high, tori of the ynchrotron mapping equation are not cloed curve. The mapping equation for ynchrotron phae-pace coordinate (φ,δ. Figure below how two tori in phae-pace coordinate (φ, /β with parameter V= kv, h=, α c =.3, φ = 3 o at 5 MeV proton kinetic energy. Note that the actual attainable rf voltage V i about - V in a low energy proton ynchrotron. Since the eparatrix i not a cloed curve, the phae-pace tori change from a fih-like to a golf-club-like hape. Thi i equivalent to the adiabatic damping of phae-pace area. Since the acceleration rate for proton (ion beam i normally low, the eparatrix toru i a good approximation. When the acceleration rate i high, e.g. in many electron accelerator, the tori near the eparatrix may reemble that of picture below. The mapping from ( n, δ n to ( n+, δ n+ preerve phae-pace area. The phaepace area encloed by a trajectory (φ, δ obtained from the above mapping equation i independent of energy. It can not be ued in tracking imulation of beam acceleration. During beam acceleration, the phae-pace area in (,/ω i invariant. The phae-pace mapping equation for phae-pace coordinate (,/ω hould be ued. The adiabatic damping of phae-pace area can be obtained by tranforming phae-pace coordinate (,/ω to (, δ. Two tori in phae-pace coordinate (φ, /β obtained from mapping equation with parameter V = kv, h =, α c =.3, and φ = 3 o at 5MeV proton kinetic energy. IUCF Cooler Ring ha typical rf voltage at about kv. Requirement of rf voltage in rapid accelerating accelerator Proton acceleration in IUCF cooler ring from 5 MeV to 5 MeV in econd where ρ. m. Uing R m, we obtain V in φ Volt. Acceleration of proton from 9 GeV to GeV in at the Fermilab Main Injector would require db/dt. Tela/. The circumference i 339. m with ρ=35 m. The voltage requirement become Vin =.MV. For electron torage ring: V in(φ = energy lo per revolution U : proton electron K (GeV (GeV.3 3 p (GeV/c. 3 Brho (T-m.95.7 BL (T-m B (T..3 L(m.97.3 N_dip L_dip (m 3..7 P h f rf (MHz C (m.5 5. L / (m 3.5 DBA nu_x.. nu_z.7.3 A B p [GeV/c/u] Z Bd, Bi i B B /

5 Define in in h ev What happen when the ynchrotron tune i large? (ee HW3.. Summary of Synchrotron quation of Motion A. Uing time t a an independent variable, the equation of motion and the Hamiltonian are lited a follow: Uing (φ, /ω a phae-pace coordinate: Uing (φ, δ a phae-pace coordinate: B. Uing longitudinal ditance a independent variable,

6 3. Adiabatic Synchrotron Motion The ynchrotron Hamiltonian for the phae pace coordinate (,δ i Q h ev ev co (in in ( Small amplitude ynchrotron motion i imple harmonic with tune h co ev co h ev Here ν i the ynchrotron tune at co φ =. The ynchrotron period i T = T /Q, where T i the revolution period. During beam acceleration, the ynchrotron Hamiltonian depend on time. However, if the acceleration rate i low, the Hamiltonian can be conidered a quaitatic. Thi correpond to adiabatic ynchrotron motion, where parameter in the ynchrotron Hamiltonian change lowly o that the particle orbit i a toru of contant Hamiltonian value. The condition for adiabatic ynchrotron motion i where ω i the angular ynchrotron frequency and α ad i called the adiabaticity coefficient. Typically, when α ad.5, the time variation of ynchrotron period i mall and the trajectorie of particle motion can be approximately decribed by tori of contant Hamiltonian value. Fixed point of a Hamiltonian are located at phae pace point with zero velocity field: H H H ( q, p, t : q i, p i. p q i i The Hamiltonian toru that pae through the untable fixed point i called the eparatrix. ev H (, Hx H (, [ co ( in ] ev x [co co ( in ] h The eparatrix ha two turning point, φ u and π φ, where φ u i The fixed point of the ynchrotron Hamiltonian are at phae pace point: (, and (π-,. Small amplitude motion around the table fixed point (, i elliptical with ynchrotron tune Q. Motion near the UFP i hyperbolical. The phae pace area encloed by the eparatrix i called the bucket, where particle motion around the table fixed point i elliptical. The motion around the untable fixed point i hyperbolical. The bucket length i (π φ φ u.

7 The phae-pace area encloed by the eparatrix orbit i called the bucket area. ~ ev AB x ( d b( b( h h Here α b i the moving bucket factor. The maximum momentum deviation of the eparatrix orbit i called the bucket height Φ.5 (rad ~ The bucket area in phae pace (, /ω i AB AB ht The phae pace area meaure the time-width, and energy-pread of the bunch ditribution. Thu the dimenion of the phae pace area i ev-ec. For example, a beam bunch with n bunch length and MeV energy pread have a bunch area of. ev-ec. A beam with MeV energy pread with GeV energy ha a fractional energy pread of 3. II.3 Small-Amplitude Ocillation and Bunch Area The linearized ynchrotron Hamiltonian around the SFP i imple harmonic with where φ= ϕ ϕ, and Q i the ynchrotron tune with ω =Q ω. Bucket area ( b( b h h Bucket height Y ( Y ( h h The phae pace area of the ellipe i

8 A. Gauian beam ditribution The normalized Gauian ditribution i The invariant phae-pace ellipe in (θ, δ phae pace i where σ δ and σ ϕ are rm momentum pread and bunch length repectively. The rm phae-pace area i The phae-pace area that contain 95% of the particle in a Gauian beam ditribution i The ynchrotron phae-pace area A, meaured in ev-, for the phae pace (ϕ/h,/ω for one bunch i The normalized Gauian ditribution in (θ, δ pace become Here σ θ and σ δ are the rm bunch angular width and rm fractional momentum pread. The bunch length i σ =Rσ θ in meter, where R i the average radiu of the accelerator, or σ t =σ θ /ω in econd. We conider N B particle ditributed in a bunch, where N B may vary from to particle. The line ditribution and the peak current (in Ampere of the bunch are where N B e/t i the average current, and π/( πσ θ i the bunching factor. For a given phae-pace area A, the maximum fractional momentum deviation and phae-width are related to the area by Typical parametric dependence of the phae pace amplitude i For a beam with rm momentum and phae pread σ δ and σ, the rm phae pace area i A rm =πσ σ δ =πhσ θ σ δ. Here θ i the orbital angle and i the rf phae angle. A beam bunch with Gauian ditribution i

9 Amplitude dependence of ynchrotron tune: Carrying out canonical tranformation with the generating function: Small amplitude motion around the untable fixed point h ev (in in Near the untable fixed point we et =π +φ. The equation of motion become ev ev co, (in( in h 5 Q tan ( J Q ˆ 3 evh co Thu, The equation of motion around the UFP i hyperbolical. h coht inht inht coht h ~, ~ Define the normalized coordinate: ~ ~ Conider a beam with initial ditribution evolve around the UFP, the phae pace ellipe v time become ~ tanh ~~ ~ t ~ t, ~ coh t Note that the phae pace ellipe become elongated, while the phae pace area i preerved. The UFP can be ued for bunch compreion.

10 We conider the tationary ynchrotron motion above the tranition energy with η>, or φ = π. xperimental Tracking of Synchrotron Motion The fractional off-momentum coordinate of a beam can be derived by meauring the cloed orbit of tranvere diplacement xco at a high diperion function location. The off-momentum coordinate i The ynchrotron phae coordinate can be meaured by comparing the bunch arrival time with the rf cavity wave. Firt, we examine the characteritic of beam current ignal from a beam poition monitor. We aume that the bunch length i much horter than the circumference of an accelerator. With the beam bunch approximated by an ideal δ-function pule, the ignal from a beam poition monitor (BPM or a wall gap monitor (WGM i where NB i the number of particle in a bunch, T i the revolution period, ω =π/t i the angular revolution frequency, and τ=(θ θ /ω i the arrival time relative t o the ynchronou particle. The periodic delta-function pule, in time domain, i equivalent to inuoidal wave at all integer harmonic of the revolution frequency. The ynchrotron tune of the Hamiltonian toru The meaured ynchrotron tune v the maximum phae amplitude of the ynchrotron ocillation i compared with theory( olid line. Requirement of rf ytem:. Acceleration rate: d /dt=f evin with Vin=πRρ(dB/dt. If η<, < <π/, if η>, π/< <π. 3. The bucket area mut be larger than the bunch area, f rf, V, h.. The rf frequency i related to magnetic field by The inet how an example of the ynchrotron phae-pace map meaured at the IUCF Cooler, and the FFT pectrum. The zero amplitude ynchrotron tune wa ν = The bucket area and the bucket height are