ψ←ψ Coherent Synchrotron Radiation in the Isochronous Muon Collider Ring∗ Juan C. Gallardo Center for Accelerator Physics, Brookhaven National Laboratory, Upton, NY 11973 USA ABSTRACT Table I: High energy-high luminosity µ+ µ− collider We show that the coherent (shielded) synchrotron radiation is negligible in a 4TeVc-of-m muon collider. Maximum c-m Energy [TeV] 4 Luminosity L[1035cm−2s−1] 1.0 I INTRODUCTION Circumference [km] 8.08 − − − Time Between Collisions [µs] 12 To achieve the luminosity of L =1035 cm 2s 1 in a µ+µ −3 Energy Spread σE[units 10 ] 2 collider [1], two bunches per sign of N =2×1012 particles ∗ Pulse length σz[mm] 3 each and a betatron function of β =3mmat the interaction ∗ Free space at the IP [m] 6.25 point (IP) are required. This small β at the IP constrains the Luminosity lifetime [No.turns] 900 ∼ ∗ size of the bunch to be σz β . To maintain this rather short Horizontal betatron tune, ν 55.79 bunch without excessive rf power consumption, an isochronous x Vertical betatron tune, νy 38.82 lattice has been chosen for the final collider ring. [2] One of −6 rms emittance, x,y [10 π m-rad] 0.0026 the important advantages of muons as opposed to electrons is − rms normalized emittance, γ [10 6π m-rad] 50.0 that at up to at least 2 TeV energy it is possible to accelerate x,y Beta-function values at IP, β∗ [mm] 3 muons in circular machines as their synchrotron radiation is re- x,y − rms Beam size at IP [µm] 2.8 duced by a factor of ( me )2 ∼ 23 × 10 6 with respect to elec- mµ Quadrupole pole fields near IP [T] 6.0 trons. Nevertheless, the large number of muons in a short bunch Peak beta-function, β [km] 284 suggests the possibility of strong shielded coherent synchrotron xmax Peak beta-function, [km] 373 radiation. First, we use the well known formulae[3][4] to evalu- βymax ate the power of shielded coherent synchrotron radiation in the Magnet Aperture closest to IP [cm] 12 isochronous muon collider ring. Finally, following the results Beam-Beam tune shift per crossing 0.05 obtained by Kheifets and Zotter [5] for a bunch with a Gaus- Repetition Rate [Hz] 15 sian longitudinal charge distribution we show that the coherent rf frequency [GHz] 1.3 synchrotron radiation in the isochronous µ+µ− collider ring is rf voltage [MeV] 130 12 negligible if the rms bunch length is larger than ≈ 0.3 mm. Particles per Bunch [units 10 ] 2 No. of Bunches of each√ sign 2 I II INCOHERENT SYNCHROTRON Peak current = eNc/ 2πσz [kA] 12.8 I RADIATION Average current = eNc/Circum [A] 0.032 Bending Field [T] 8.5 The main parameters of the present design of a muon collider ring are shown in Tb. I. To calculate the incoherent radiation power we start by recalling the Larmor formula for a circular machine [5]; 4 4 GeV −5 c E [GeV ] me Pγ =8.8575 × 10 (1) s 2π ρ [m] mµ and electron synchrotron radiation); therefore the shielding of the super-conducting magnets is determined by the products of is the instantaneous radiation power per particle. For a beam of muon decay (µ → e + νe + νµ). Although the energy is Ne particles the total synchrotron (incoherent) radiation power deposited mainly at 90 and 270 azimuthal angle, the simplest per turn is solution to bring the energy deposition on the super-conducting 4 4 coils to safe levels is to include a 6 cm radius of Tungsten liner −2 E [GeV ] me Pγ [MW]=8.8575 × 10 I [A] (2) in the beam pipe [7]. Any increase in the synchrotron radia- ρ [m] m µ tion would thus be rather expensive and/or unmanageable from cβ ≈ a cooling power point of view or realistic pole-tip fields. There- where the current I[A]=eNµ 2πρ 13.75 mA. For the 4T in center-of-mass muon collider ring, Pγ ' fore, it is natural to study the possibilityof coherent synchrotron 10kW which it corresponds to ∼ 8.5W/m. This has to be com- radiation. pared with 1 KW/m produced by muon decay [6] (electrons ∗ Work supported by the US Department of Energy under contract DE- Before discussing coherent radiation we recall some useful ACO2-76-CH00016 and DE-AC03-76SF00515. expressions from Jackson [8]. First, the critical frequency is 215 given by: and after an elementary integration 2 2 2 − n σs 3 2 3 c 2 me = ρ (9) ωc = γ ,c[keV ]=0.665E [GeV ]B[T] fn e 2 ρ mµ (3) From this expression we can derive the coherent condition in 3 and harmonic number[9] nc =3γ . Eq.3. The second term in Eq.5 gives the power of coherent ra- − ' 2 Second, assuming a Gaussian angular distribution of the radi- diation Nµ(Nµ 1) Nµ, i.e. proportional to the square of the ation (θ is the angle between the observation direction and the number of particles or equivalently to the square of the bunch tangent to the trajectory of the particle at the moment of emis- current In at the n-th harmonic. This dependence suggests to as- sion) then the rms divergence is: sign an effective impedance to the coherent radiation following 2 Ohm’s law Pn = ZnIn. 1/3 1.07 ωc > γ ω ω ωc In our case, λ∼1.8cm, hence is comparable to the transverse ≈ 0.64 ω = ω (4) size of the beam pipe. Under this condition the coherent ra- σθ γ c 1/2 0.58 ωc diation and its characteristics are substantially changed by the ω ωc γ ω presence of metallic walls. The induced currents tend to de- crease the radiation fields created by the bunch (destructive in- Notice that at low frequencies (ω ωc) the radiation is ∼ 1 terference). This fact is denoted as shielding[11],[12] much wider than the average σθ γ and that at high frequency (ω ωc), the angular range is much narrower than the average. This concludes this brief summary of very well known results B Metallic boundaries of incoherent synchrotron radiation. In the presence of metallic boundaries the image charges and currents created by the beam itself (self induced fields) have III COHERENT SYNCHROTRON RADIATION the potential to destructively interfere with the radiation field emitted by the beam; this effect is clearly more effective the A Free space closer the beam to the wall, i.e. the smaller is the dimension of the beam pipe. Coherent radiation at wavelength λ is obtained if the relative The condition for coherent synchrotron radiation can be made distance between particles is approximately of order λ.Inthis plausible followingthe arguments from Kheifets and Zotter[11]. case the radiation fields emitted by these particles interfere con- structively and the resulting field is the radiation of a represen- Assume a beam moving between 2 parallel conductive plates; tative particle (charge e) multiplied by the total change of the the tangential radiation field must vanish at the conductor, hence P | | | |2 | |2 | |2 Nµ 2 k⊥ = k σθ with k = k⊥ + kk . At the surface ensemble ( j=1 e = eNµ). This leads to the characteristic Nµ 2 kk =0then | k | =| k⊥ |S . The cutoff wave vector due to the behavior of coherence. π π ⊥ waveguide is h , therefore k > h or If the longitudinal length of the bunch of muons is σµ, then the condition for coherence in free space is πρ σ > (10) θ nh < λ c σ ∼ ≡ (5) ≈ µ 2 At the end of the previous section, Eq.5 we recalled that σθ π ω 1/3 1.07 3 , therefore To understand this fact we refer the reader to Jackson[10]. As- 2n r sumeasetofNµ muons moving in a circle with fixed relative 1 2 πρ 3/2 positions around the circle, they produce a power n> ≡ nth (11) 1.07 3 h 2 XNµ hence the possible coherent radiation harmonic number is lim- −inφi P = Pn e = PnNµ + Nµ(Nµ − 1)Pnfn (6) ited from below. Now from Eq.8 the condition for coherence i=1 limits the harmonic number from above, i.e. ρ where Pn is the incoherent power emitted by one particle in the n<ncoh = (12) σs frequency ω = nω0 and fn is a form factor Combining Eqs.11 and 12 we get approximately (up to small Z 2 numeric factors) fn = dφ cos (nφ)S(φ) (7) r 2 πρ 3/2 ρ nth ≡ <n< ≡ncoh (13) where S(φ) is the longitudinal distribution of the bunch and 3 h σs ω0 = c/ρ is the revolution frequency. For a Gaussian distribu- tion with rms bunch length σ or in terms of wavelength s s 2 2 − φ ρ h 6h 1 ρ 2 2 ≡ S(φ)dφ = √ e σs (8) σs <λ< λth (14) 2π σs π πρ 216 Notice that the upper limit λth is smaller than the waveguide [7] I. Stumer, private communication. ≈ h cutoff λc.o. h( ρ 1). Consequently, there will be coherent [8] J. D. Jackson, Classical Electrodynamics, John Wiley & Sons, synchrotron radiation only if New York, 1967. r 3/2 [9] Kwang-Je Kim, Characteristic of Synchrotron Radiation,AIP 3 h Conference Proceedings 184, pp. 582, Eds. M. Month and σs < 2πρ (15) 2 πρ M. Dienes, AIP 1989. Notice that ωc is one half the critical frequency defined by Jackson in Before applying Eq.15 to the collider ring to show that there Ref.9.