Physics 598ACC Accelerators: Theory and Applications

Transcription

1 Physics 598ACC Accelerators: Theory and Instructors: Fred Mills, Deborah Errede Lecture 6: Collective Effects 1

2 Summary A. Transverse space charge defocusing effects B. Longitudinal space charge effects C. Transverse collective instability 2

3 Collective Effects The effects of particles upon each other, either directly or through fields from charges and currents induced in the walls, lead to alterations in the orbits of the particles, to collective oscillations of beams, occasionally to instability and beam loss. We will consider several of these phenomena which are day to day problems in accelerator operation. 1. Transverse space charge defocusing effects Consider a charged particle beam of radius a = β L ε with uniform line density λ particles/m. According to Gausses law and Ampere's law, the electric and magnetic fields in the beam are (ignoring walls) r 10.1 E r = 2eλ, 4πε 0 a 2 B θ = 2eλv μ 0r 4πa 2 3

4 The radial force on a particle is ( ) ( 2 ) 2e λ F = e E vbθ = ee 1 β = r πε γ r r r a and the force can be represented by an equivalent defocusing (in both x and z) quadrupole of strength K ( I = evλ, I 0 is often called the "Budker Current") B 10.3 K =, for protons Bρ = 2I π 10 7 A I 0 a 2 βγ This causes a tune shift Δν in each plane ( ) 3 I 0 = 4πε 0mc 3 e I R Δν = β Kds = 4π o I ( ) L 3 0 ε βγ 4

5 This is often written in terms of the number N of particles, the "invariant emittance" ε n = βγε and the "classical radius" r = e 2 /(4πε 0 mc 2) ~ m for protons. Δν = Nr p 2πε n βγ using λ = N/(2πR) If the beam is bunched the local density is higher by a factor 1/"B" where "B" is called the "bunching factor". The tune shift is not the same for all particles, because in general the beam density is not uniform. Thus there is a tune spread due to the space charge. The effects on the beam generally cannot be calculated precisely, because they depend on many things not well known, such as the strengths of error fields, and thus the strengths of resonances. Thus the tune shift is just an indicator of the importance of a complicated phenomenon, like the Reynolds number in fluid flow. 5

6 For rapid acceleration, such as in a rapid cycling synchrotron, tune shifts of about 0.25 can be tolerated, while in cases of long beam storage, smaller tune shifts are needed. In electron-positron collider rings with beam-beam effects, radiation damping seems to help keep stability for Δν below about 0.05, whereas hadron colliders can tolerate less tune shift. Tune shifts up to two have been seen without beam loss for several revolutions. The beam also reacts to fields induced for currents and charges induced in the walls of the beam environment. From the point of view of tune shift, the largest effect is to spoil the 1-β 2 cancellation in the tune shift (magnetic field canceling the electric field) due to DC magnetic fields penetrating conducting walls. On the other hand this part of the tune shift can be tuned away with ring correction elements, while the direct part cannot be, since the sources of the fields are in the beam, and not the walls. 6

7 2. Longitudinal space charge effects The same fields seen above can lead to longitudinal fields, if the beam current I depends on longitudinal position in the beam. As an example, consider a round beam in a smooth round pipe of inner radius b. Inside the beam of radius a the fields are given by 10.1, while out side the beam 10.6 E r = 2eλ, B θ = 2eλvμ 0 4πε 0 r 4πr We can apply Faraday's law to the rectangle in (s,r) composed of (s,0) >(s,b) >(s+ds,b) >(s+ds,0) >(s,0). With a field E wall at r = b, what results for the field E s on the axis is 10.7 eλ(s) 4πε ln b a + E eλ(s + ds) wallds 4πε ln b a E s e Ý ds = e λ vμ 0 4π4 π 1+ 2ln b a ds 7

8 The line density λ and the current I = λv change because the beam is moving at velocity v dλ dt = dλ ds ds dt = dλ ds v e dλ 2 b Es = Ewall ( 1 β ) 1+ 2ln 4πε ds a 0 The current including a current at the nth harmonic is written as 10.9 I = eλv = I ave + I n e i ns R nω 0t reflecting the fact that the beam is moving. The energy gain U n per turn is 2πRe times the electric field ( ) U = 2π ReE = ii Z + Z, n s n i wall Z 0 = μ 0 ε 0 8

9 The term Z wall would be zero for a perfectly conducting smooth wall. Wall resistivity, or added inductance, as from bellows, cracks, RF cavities, etc. yield added impedance to the chamber in addition to the capacitive space charge term Z i. For example, if there is an inductance L Henry/m, Z wall = -nω 0 L. The skin effect in a conducting wall adds both inductance and resistance to the wall impedance These fields can cause problems in several ways. In the first place, the fields act in many ways like those in RF cavities, in that they can accelerate or decelerate parts of the beam, and hence distort the equivalent potential characterizing the bucket. The bucket shape can be modified leading to loss of beam or other problems. 9

10 In the second place, the fields can lead to instability. The way this happens is as follows. A current at frequency nω 0 can lead, via its impedance, to collective fields which accelerate particles. This causes the frequency associated with this current to shift to nω 0 - Δω r -Δω i. It develops that usually, only those modes which are slower than the beam (travel backwards in the beam frame) can extract energy from the beam longitudinal motion and grow, i.e. be unstable. The beam has a spread of frequencies, leading to a spread of frequencies in the mode n. For example, here, if the fractional momentum spread of the beam is δ, the spread in revolution frequencies is ηω 0 δ, and the spread of frequencies in mode n is nηω 0 δ. Those modes which can be unstable become unstable when the real collective frequency shift Δω r is larger than the beam frequency spread nηω 0 δ. The growth rate of the instability above threshhold is equal to the imaginary frequency shift Δω i. ( here η = (α 1/γ 2 ) is the slip factor ) 10

11 In the case of longitudinal instability, the real shifts come from reactive impedances such as the capacitive space charge impedance or from inductance in the wall, while the imaginary shift comes from resistive impedance such as wall resistance etc. A general "rule of thumb" estimate of instability in a typical case is given by the "Keil-Schnell" criterion mc 2 γη ΔP Z n P ei ave 2 It is clear that the important impedances for a relativistic beam are the wall impedances, while the space charge capacitive impedance usually dominates for nonrelativistic beams. 11

12 3. Transverse collective instability Consider again the beam in a round pipe. If the beam moves off axis in the z (or x) direction a distance ζ < b, the field pattern changes. According to the theory of images, the electric fields due to the wall charges are now given by the direct (free space) field of the moved beam (fields 10.1 evaluated at -ζ) plus the field of an opposite sign beam located at x =R, where R =b 2 /ζ. From 10.1 and 10.6 we can infer that for small ζ, the beam feels an electric and magnetic deflection (assuming AC boundary conditions at the wall) E r = 2eλζ 1 1, 4πε 0 b 2 a 2 B θ = 2eλvμ 0ζ 4π 1 b 1 2 a 2 12

13 The force on the beam is equivalent to that from an electric field E 10.13, Z = RZ g 0 2π RE, g βγ 2 = 1 a 1 = Z Iζ 2 b 2 For the transverse motion, the beam position at any point s depends on time because the beam is revolving at frequency ω 0 and oscillating at frequency νω 0. Then the frequencies of oscillation are (n ±ν)ω 0. Modes n in the unbunched beam are described, by coherent betatron oscillations in the beam whose phases complete 2πn in a snapshot of the whole beam circumference at one given time. The modes which travel slower than the beam, so are potentially unstable, are those where n > ν. The spread of frequencies in the mode are due to revolution frequency spread and chromaticity. 13

14 Again, instability occurs when the frequency shift Δν exceeds the frequency spread δν. These are 10.14, δν = η(n ν) ξν δp p fwhm Δν = ( ) Re IZ n 4πνβΕ This yields the stability limit ; eiz < 4πβEν R η(n ν) ξν δp p fwhm 14

15 If the wall is resistive, there is an additional impedance Z wall (1+ j) ρrα = n δbn 2ρ δ= μω α= 2(1 j)d 1+ exp δ 2(1 j)d 1 exp δ d = Wall thickness

16 and the added transverse impedance is 2cZ wall Z = 2 ωb Bunched beams exhibit similar instabilities, but modified from the ones described above. An example is the "head-tail" instability. Due to chromaticity, the betatron oscillation phase of a particle when in the tail of the beam is out of phase with those in the head of the beam. Then the wake field of those in front can cause the betatron oscillations of those in the tail to grow and then vice versa. Many complicated modes are possible, including the "m = 0" or rigid bunch mode in which all particles are in phase. If the chromaticity is made positive, all modes except m = 0 become stable. The m = 0 mode can be stabilized by a relatively wide band feedback system which measures the position of each bunch at some location and kicks the beam at a location 90 away in betatron phase. 16

17 End of Lecture 17

18 18