F. Zimmermann and M.-P. Zorzano, CERN, Geneva, Switzerland

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1 LHC Project Note Touschek Scattering in HERA and LHC F. Zimmermann and M.-P. Zorzano, CERN, Geneva, Switzerland Keywords: Touschek effect, beam loss, coasting beam Summary We estimate the rate at which coasting beam is generated via the Touschek effect by evaluating analytic expressions for flat and round beams. The Touschek scattering rate computed for HERA at 920 GeV is of the order of one percent over 8 hours. In the LHC at injection the coasting beam component is about after one hour. At top energy the scattering rate is two times smaller, and, taking into account particle loss due to synchrotron radiation, it will give rise to a steady-state coasting beam component of about Introduction In HERA a significant portion of the proton beam is coasting, i.e., not trapped in the rf bucket, potentially impairing the HERA-B experiment [1]. We here explore whether the generation of the coasting beam component can be explained by the Touschek effect, and we estimate the corresponding coasting-beam component at the LHC. 2 Formulae The name Touschek effect [2] refers to a particle-particle collision within a bunch, by which so much energy is transferred from transverse into longitudinal phase space, that the scattered particles leave the stable rf bucket. Since it is caused by a particle-particle collision, the loss rate due to Touschek scattering is quadratic in the bunch population, namely dn b dt = αn 2 b. (1) The number of particles outside the rf bucket increases as N coast = αn 0t 1+αN 0 t N 0 (2) This is an internal CERN publication and does not necessarily reflect the views of the LHC project management. 1

2 where N 0 = N b (0) denotes the initial bunch population. The traditional expressions for the Touschek effect in an (electron) storage ring refer mainly to flat beams. For such a flat beam (with ɛ x /β x ɛ y /β y ), the Touschek coefficient, α = α fl,isgiven by [3, 4] α fl = 4πr2 pc J(η, δq) (3) γ 2 η 2 V where r p (= m) is the classical proton radius, V =8π 3/2 σ x σ y σ z the bunch volume, c the speed of light, γ the energy divided by the rest mass, η =( E/E) max the maximum relative energy deviation accepted by the rf system, δq = γσ x /β x,andjafunction of η and δq, which, for δq < 0.1 and η less than 10 3, can be approximated as [3] J(η, δq) ( ) 2 1 η 4 ln (4) πδq δq For round beams, a Touschek lifetime formula was derived by Miyahara [5]. However, it appears that there is a factor of one half missing in his result [6]. The correced formula for a round beam is α rd = πr2 0 c ( ) β x β y δq γ 4 σ x σ y Vη D η (5) with D(ɛ) = e u ( u ɛ ɛ u 3/2 ɛ ln u ) du ɛ (6) Piwinski has derived the Touschek lifetime for arbitrary aspect ratio [7, 8]. Ignoring the contribution from dispersion his expression reads α piw = r0c 2 8πγ 2 F(τm,B 1,B 2 ) σ z σ x σ y (7) where the brackets denote the average over the whole circumference, and ( ( ) π/2 F (τ m,b 1,B 2 )=2 π(b1 2 B2) 2 (2τ +1) 2 τ/τm κ m 1+τ 1 /τ + τ ( 2+ 1 ) ln τ/τ ) m 2τ 1+τ ττ m (1 + τ) e B 1τ I 0 (B 2 τ) 1+τdκ (8) with I 0 the modified Bessel function, τ = tan 2 κ, κ m = arctan τ m, τ m = β 2 η 2 (β is the relativistic factor), B 1 = 1 ( ) β 2 x + β2 y 2β 2 γ 4 σx 2 σy 2 (9) and [ ] 1/2 B 2 = B1 2 β2 x β2 y. β 4 γ 4 σx 2σ2 y (10) 2

3 variable symbol value rms horizontal beam size σ x 450 µm rms vertical beam size σ y 450 µm rms bunch length σ z 60 mm average beta function β x,y 50 m momentum compaction factor α c beam energy E 920 GeV number of proton bunches n b 174 number of protons per bunch N b revolution time T 0 21 µs transverse emittance (1σ) ɛ x,y 4.1 nm relativistic factor γ 981 bunch volume V 541 mm 3 rms uncorrel. trans. momentum in units of m 0 c δq st rf voltage ˆVrf,1 100 kv 2nd rf voltage ˆVrf,2 540 kv 1st harmonic number h nd harmonic number h energy acceptance η Estimates 3.1 HERA Table 1: HERA proton-beam parameters Table 1 compiles relevant HERA parameters. HERA has two rf systems operating at the harmonic numbers h 1 (=1100) and h 2 (=4400) with voltages ˆV 1 and ˆV 2. The rf energy acceptance η provided by this double rf system is computed as η ( ) ( E 2e = E max πα c E 0 [ ˆVrf,1 h 1 + ˆV ]) 1/2 rf,2, (11) h 2 where α c denotes the momentum compaction factor. We remark that, roughly, η V rf and σ z 1/ V rf so that α fl 1/ V rf, ignoring the weak dependence of J on the rf voltage. Using the numbers of Table 1, we find J 71 from Eq. (4), and the flat-beam Touschek coefficient, Eq. (3), evaluates to α fl s 1. For a two times higher rf voltage, we find J 60 and α s 1. Evaluating the more adequate round-beam expression, Eq. (5), for the HERA parameters of Table 1, we obtain α rd s, and with a two times higher rf voltage α rd s. Thus, for HERA the round-beam formula predicts a factor 2 longer Touschek lifetime than the flat-beam formula. Note that the ratio of the flat and round Touschek rates, ( ln ( ) ) η 2 δq α fl α rd γσ y πηβy 3 D(η 2 /δq 2 ), (12)

4 rf voltages 100 kv and 540 kv rf voltages 200 kv and 1080 kv Figure 1: Coasting-beam component in HERA, N coast /N 0 due to Touschek scattering as a function of time in hours, calculated from Eqs. (2) and (7). The two curves refer to two different settings of the rf voltages. could in principle assume any value. Evaluating Piwinski s formula, Eq. (7), we find for the nominal rf voltage α piw s, and with the two times higher rf voltage α piw s, in excellent agreement with the round-beam estimate of Eq. (5). Figure 1 shows the increase of the coasting-beam component in HERA over an 8 hour period for two different rf voltages, calculated using Eq. (2) and the Piwinski formula, Eq. (7). After an 8-hour store the expected fraction of particles outside the rf bucket, due to Touschek scattering, is of the order of 1%. 3.2 LHC We can also estimate the Touschek effect for the LHC. The corresponding parameters at injection and at top energy are listed in Table 2. As in HERA, two rf systems are employed. The rf parameters are taken from Ref. [9]. In case of the LHC, the round beam estimate, Eq. (5), evaluates to α rd s 1 at injection, and α rd s 1 at 7 TeV. From Piwinski s formula, Eq. (7), we obtain similar numbers: α piw s 1 and α piw s 1, respectively. Thus, using N 0 = and Eq. (2), coasting beam is produced at a rate per proton of hr 1 during injection and hr 1 at top energy. Once the protons are outside of the bucket, they lose energy due to synchrotron radiation [10]. This energy loss amounts to dδ/dt s 1 at 450 GeV and dδ/dt s 1 4

5 at 7 TeV [11]. If the collimators provide an energy aperture of [12], a scattered proton is lost after about τ loss 390 hours at injection or after τ loss 6.5 minutes at top energy, respectively. While the energy drift due to synchrotron radiation is unimportant at injection, in collision it gives rise to a steady-state coasting beam fraction of α piw N 0 τ loss variable symbol value (inj.) value (top) rms horizontal beam size σ x 883 µm 220 µm rms vertical beam size σ y 883 µm 220 µm rms bunch length σ z 130 mm 77 mm average beta function β x,y 100 m 100 m momentum compaction factor α c beam energy E 450 GeV 7000 GeV number of proton bunches n b number of protons per bunch N b revolution time T µs 800 µs transverse emittance (1σ) ɛ x,y 7.8 nm 7.8 nm relativistic factor γ bunch volume V 4515 mm mm 3 rms uncorrel. trans. momentum in units of m 0 c δq st rf voltage ˆVrf,1 750 kv kv 2nd rf voltage ˆVrf, kv 0kV 1st harmonic number h nd harmonic number h energy acceptance η Table 2: LHC parameters at injection and top energy. 4 Conclusion In HERA, the coasting beam component generated by Touschek scattering is estimated to be about 1% over 8 hours. This may be compatible with observations, but it is probably not the dominant effect, since recent studies [13] suggest that about 4% coasting beam is generated in 3 hours and that the dominant process exhibits diffusion characteristics. In the LHC, after 1 hour at injection energy a coasting beam fraction of has been created by the Touschek effect. At top energy, protons are scattered outside of the rf bucket at a rate of 10 4 per proton and hour, i.e., two times smaller than at injection. Taking into account the loss of these protons within 6 7 minutes due to synchrotron radiation and energy collimation, the steady-state coasting-beam component at top energy is about A more refined calculation of the Touschek scattering would take into account the contribution from dispersion and the variation of lattice parameters around the ring. 5

6 Acknowledgements We thank D. Kaltchev, A. Piwinski, F. Ruggiero, A. Verdier, and R. Wanzenberg for useful comments and inspiring discussions. References [1] K. Ehret et al., Observation of coasting beam at the HERA Proton Ring, DESY (2000). [2] C. Bernadini et al., Phys. Rev. Letters, vol. 10, p. 407 (1963). [3] R.P. Walker, Calculation of the Touschek Lifetime in Electron Storage Rings, PAC [4] U. Voelkel, DESY 67/5, [5] Y. Miyahara, Jap. J. Appl. Phys., vol. 24, p. L742 (1985). [6] A. Piwinski, private communication (2000). [7] A. Piwinski, The Touschek Effect in Strong Focusing Storage Rings, DESY (1998). [8] A. Piwinski, Touschek Effect and Intrabeam Scattering, in A. Chao and M. Tigner (eds.), Handbook of Accelerator Physics and Engineering, World Scientific (1999). [9] E. Shaposhnikova, Longitudinal Beam Parameters during Acceleration in the LHC, SL Note in preparation (2000). [10] A. Verdier reminded us of this effect. [11] The LHC Study Group, The Large Hadron Collider - Conceptual Design, CERN/AC/95-05 (1995). [12] D. Kaltchev, private communication (2000). [13] M.-P. Zorzano and R. Wanzenberg, Intrabeam Scattering and the Coasting Beam in the HERA Proton Ring, CERN-SL AP, DESY-HERA (2000). 6