217 NSRRC FEL Longitudinal Motion (SYL) 1 Synchrotron Motion RF cavities Charged particles gain and lose energy in electric field via Δ. For DC accelerators such as the Cockcroft-Walton and Van-der- Graaff produce beam depending on the source. DC accelerators are limited by their sustainable maximum electric field. The concept of rf accelerator arises from the efficiency that particles can gain energy from the same electric field multiple times. This leads to the development of cyclotrons; synchrotrons; and RF-linacs. Availability of high power rf sources after the WWII, and the discovery of the phase focusing principle in 1945 provides the foundation for the development of high energy rf accelerators. Synchrotron motion is the result of the phase focusing principle. A collection of SRF cavities developed at Cornell University with frequencies spanning 2 MHz to 3 GHz The longitudinal electric field at an rf gap V=V sin(ω rf t+φ) ω rf =hf Consider a non-synchronous particle with small deviations of rf parameters from the synchronous particle, i.e. where ω =β c/r is the angular revolution frequency of a reference (synchronous) particle, ε is the amplitude of the electric field, β c and R are respectively the speed and the average radius of the reference orbiting particle, h is the harmonic number, and s is the phase angle for a synchronous particle. The reference particle passes through the cavity gap in time t nt +( /2βc, /2βc) (n = integer), where is the rf cavity gap width. The energy gain per passage and the acceleration rate are Here s, θ s, ω, p, E are respectively the rf phase angle, azimuthal orbital angle, angular revolution frequency, momentum, and energy of a synchronous particle, and, θ, ω, p, E are the corresponding parameters for an off-momentum particle. The energy gain per revolution for this nonsynchronous particle is ev sin ϕ, where ϕ is the rf phase angle. The rate of energy gain of a non-synchronous particle is The equation of motion for the energy difference is (1) where the effective voltage seen by the orbiting particle is V=ε T.
217 NSRRC FEL Longitudinal Motion (SYL) 2 The time evolution of the phase angle variable ϕ is 1 Phase slip factor Synchrotron equations of motion Δ Δ Compaction factor TPS: α c =2.4 1 4 TLS: α c =6.78 1 3 Linearized synchrotron equations of motion: cos 2 Stable synchrotron motion requires cos. V Acceleration Lower Energy Synchronous Energy Higher Energy Δ (2) φ s π/2 φ s (1) V η< η> Deceleration (1a) This is the Phase focusing principle π/2 π 3π/2 2π φ Linearized synchrotron equations of motion: cos 2 Stable synchrotron motion requires cos. The angular synchrotron frequency is cos 2 cos 2 where c is the speed of light and R is the average radius of the synchrotron. The synchrotron tune, defined as the number of synchrotron oscillations per revolution, is cos Fermilab Booster cos 2 TPS: Qs=.33 V(MV) The synchrotron tune in a Fermilab booster cycle. The crosses are measured from phase signal with synchrotron phase detector (SPD). The transition energy is crossed at around the 14.5 ms.
217 NSRRC FEL Longitudinal Motion (SYL) 3 Longitudinal Particle Dynamics in a Linac Phase focusing of charged particles by a sinusoidal rf wave also works in a linac. Let t s, ψ s and W s be the time, rf phase, and energy of a synchronous particle, and let t, ψ, and W be the corresponding physical quantities for a non-synchronous particle. Define the synchrotron phase space coordinates as Δt = t t s, Δψ = ψ ψ s = ω(t t s ), ΔW = W W s. The accelerating electric field is E = E sin ωt = E sin(ψ s + ψ), where the coordinate s is chosen to coincide with the proper rf phase coordinate. The change of the phase coordinate is (4) Comparing the phase equation of LINAC and synchrotron, Δ (2) we find that ω/β s c is equivalent to the harmonic number per unit length, ΔW/β s2 E is the fractional momentum spread, and 1/γ s2 is the equivalent phase slip factor, i.e. the momentum compaction in a linac is ZERO, the beam in a linac is always below transition energy. (4) where v = ds/dt and v s = ds/dt s are the velocities of a particle and a synchronous particle, and the subscript s is used for physical quantities associated with a synchronous particle. Similarly, the energy gain from rf accelerating electric fields is (3) The synchrotron equation of motion in LINAC is: (3) (4) η< η> Hereafter, β s and γ s are replaced by β and γ for simplicity. The linearized synchrotron equation of motion is simple harmonic: where k syn is the wave number (2π/λ) of the synchrotron motion. For medium energy proton linacs, k syn is about.1 to.1 m 1. Synchro-beta coupling can be important if k syn is near that of betatron motion. For high energy electrons, k syn 1/γ 3/2 is VERY small. The beam particles move rigidly in synchrotron phase space, and thus one choose, i.e. electron bunches are riding on top of the crest of the rf wave.
217 NSRRC FEL Longitudinal Motion (SYL) 4 h ev (sin sin 2 ) 2 s E Synchrotron bucket area, separatrix Left: schematic drawing of the rf potentials for s = and π/6. The dashed line shows the maximum energy for stable synchrotron motion. Middle: the corresponding separatrix orbits in (πh η /evβ 2 E) 1/2 E sx vs. The phase u is the turning point of the separatrix orbit. Right: an example of stable rf buckets, called fish diagram, with s =π/6. TPS: h=864; above transition; and ϕ s >π/2. γ>γ T γ<γ T
217 NSRRC FEL Longitudinal Motion (SYL) 5 In fact synchrotron motion is more realistically described by the mapping equation: The particle gains or loses energy at its nth passage through the rf cavity, then the rf phase n+1 depends on the new off-momentum coordinate δ n+1. The mapping equation satisfies the condition: The mapping from ( n, δ n ) to ( n+1, δ n+1 ) preserves phase-space area. During beam acceleration, the phase-space area in (,ΔE/ω ) is invariant. The phase-space mapping equation for phase-space coordinates (,ΔE/ω ) should be used. The adiabatic damping of phase-space area can be obtained by transforming phase-space coordinates (,ΔE/ω ) to (, δ). When the acceleration rate is high, tori of the synchrotron mapping equations are not closed curves. The mapping equations for synchrotron phase-space coordinates (φ,δe). Figure below shows two tori in phase-space coordinates (φ, ΔE/β 2 E) with parameters V rf =1 kv, h=1, α c =.434, ϕ s = 3 o at 45 MeV proton kinetic energy. The actual attainable rf voltage V is about 2-1 V in a low energy proton synchrotron. The separatrix is not a closed curve, the phase-space tori change from a fish-like to a golf-club-like shape. This is equivalent to the adiabatic damping of phase-space area. Since the acceleration rate for proton (ion) beams is normally low, the separatrix torus is a good approximation. When the acceleration rate is high, e.g. in many electron accelerators, the tori near the separatrix resemble the picture below. For the Fermilab DTL, ϕ s =58º with ϕ u π-ϕ s 25º 122º The bucket width is 97 degree. If a continuous beam is injected into the Linac, only 25 to 3% (97 /36 ) is accelerated. The acceleration rate is high, and the rf bucket is a golf-shaped contour. Requirement of rf voltage in rapid accelerating accelerators Proton acceleration in IUCF cooler ring from 45 MeV to 5 MeV in 1 second where ρ 2.4 m. Using R 14 m, we obtain V sin φ s 24 Volts. Acceleration of protons from 9 GeV to 12 GeV in 1 s at the Fermilab Main Injector would require db/dt 1.6 Tesla/s. The circumference is 3319.4 m with ρ=235 m. The voltage requirement becomes Vsin s =1.2MV. For electron storage ring: V sin(φ s )= energy loss per revolution U : Ref: Linac Rookie Book by Bruce Worthel, April 24
217 NSRRC FEL Longitudinal Motion (SYL) 6 proton electron A The phase-space area enclosed by the separatrix orbit is called the B 3.33564 p [GeV/c/u] Z bucket area. ~ ev Bd 16 s AB 2, Bi i 2B sx ( ) d 16 b( s) b( s ) 2 2 Eh h B KE (GeV).3 2.999 E (GeV) 1.238 3 p (GeV/c).88 3 Brho (T-m) 2.695 1.7 BL (T-m) 16.936 62.875 B (T) 1.4 1.3 L(m) 12.97 48.366 N_dip 4 48 L_dip (m) 3.24 1.76 P 2 24 h 1 864 f rf (MHz) 1.3-7. 499.654 C (m) 28.5 518.4 L 1/2 (m) 3.5 DBA nu_x 1.68 26.24 nu_z.72 14.3 / Here α b is the moving bucket factor. 1.9.8.7.6.5.4.3.2.1.5 1 1.5 Φ s (rad) γ>γ T γ<γ T Small-Amplitude Oscillations and Bunch Area The linearized synchrotron motion is given by where φ= ϕ ϕ s, and Q s is the synchrotron tune with ω s =Q s ω. The phase space area of the ellipse is Gaussian beam distribution Conclusion: 1. DC voltage is difficult to reach high voltage due to breakdown 2. RF cavities can provide efficient particle acceleration 3. Phase focusing principle is the foundation of all high energy particle accelerators. 4. Development of high power rf source and high gradient cavity has been the central focus of high energy accelerator scientists since 1945. 5. Manipulation of synchrotron phase space in accelerators can provide desired beam properties for all beam experiments and accelerator applications.