1. Frank Zimmermann, IBS in MAD-X, MAD-X Day, 23.09.2005 IBS in MAD-X Frank Zimmermann Thanks to J. Jowett, M. Korostelev, M. Martini, F. Schmidt

2. Frank Zimmermann, IBS in MAD-X, MAD-X Day, 23.09.2005 Motivations (1) CERN experiments at low or moderate energy are said to disagree with MAD predictions (J.-Y. Hemery); Michel Martini recommended the implementation of the Conte-Martini formulae, which are a non-ultrarelativistic generalization based on Bjorken-Mtingwa (2) check of algorithm implemented in MAD (3) extend formalism to include vertical dispersion which is important for damping rings and for the LHC (neglecting vertical dispersion often gives shrinkage of  y )

3. Frank Zimmermann, IBS in MAD-X, MAD-X Day, 23.09.2005 References: J.D. Bjorken, S.K. Mtingwa, “Intrabeam Scattering,” Part. Acc. Vol. 13, pp. 115-143 (1983) general theory and ultrarelativistic limit M. Conte, M. Martini, “Intrabeam Scattering in the CERN Antiproton Accumulator,” Part. Acc. Vol. 17, p.1-10 (1985). non-ultrarelativistic formulae M. Zisman, S. Chattopadhyay, J. Bisognano, “ZAP User’s Manual,” LBL-21270, ESG-15 (1986). possible origin of MAD-8 IBS formulae? K. Kubo, K. Oide, “Intrabeam Scattering in Electron Storage Rings,” PRST-AB 4, 124401 (2001) factor 2 correction for bunched beam There is an alternative earlier theory by Piwinski as well as a “modified Piwinski” algorithm by Bane – however we stayed with the BM approach, since it was already implemented in MAD-8 and should give the same answer

4. Frank Zimmermann, IBS in MAD-X, MAD-X Day, 23.09.2005 Outline re-derive general formulae including vertical dispersion in the limit of zero vertical dispersion compare with Conte-Martini expressions; find a slightly different result in x example 1: LHC example 2: LHC upgrade example 3: CLIC

5. Frank Zimmermann, IBS in MAD-X, MAD-X Day, 23.09.2005 IBS growth rates in general Bjorken-Mtingwa theory where a=x,l, or y, r  the classical particle radius, m the particle mass, N bunch population, log=ln(r max /r min ) --- with r max the smaller of  x and Debye length, and r min the larger of classical closest approach and quantum diffraction limit from nuclear radius, typically log~15-20 ---,  Lorentz factor,  =(2  ) 3 (  ) 3 m 3  x  y    z the 6-D invariant volume vertical dispersion enters here note: above formulae refer to bunched beams

6. Frank Zimmermann, IBS in MAD-X, MAD-X Day, 23.09.2005 Bjorken-Mtingwa gave solution for zero vertical dispersion in ultrarelativistic limit, neglecting  x /  x and  y /  y relative to (  D x ) 2 /(  x  x ), (  x /  x )  2  x 2 and  2 /   2 Conte-Martini kept the terms neglected by B-M, which are important for  <10

7. Frank Zimmermann, IBS in MAD-X, MAD-X Day, 23.09.2005 big surprise! contrary to the prevailing belief in the AB/ABP group, it was found that MAD8 & the previous MAD-X version had already implemented the Conte-Martini fomulae and not the original ultra-relativistic ones from Bjorken-Mtingwa

8. Frank Zimmermann, IBS in MAD-X, MAD-X Day, 23.09.2005 The general form of all solutions is with 9 coefficients in the integral to be determined

9. Frank Zimmermann, IBS in MAD-X, MAD-X Day, 23.09.2005 in the limit of zero vertical dispersion new coefficients reduce to CM ones denominator coefficients (from determinant) for x, z, s

10. Frank Zimmermann, IBS in MAD-X, MAD-X Day, 23.09.2005 numerator coefficients for x in the limit of zero vertical dispersion, CM does not agree with our derivation, namely the two red terms are absent on the right for the example applications, which follow, the contribution from these two terms turns out to be negligible

11. Frank Zimmermann, IBS in MAD-X, MAD-X Day, 23.09.2005 in the limit of zero vertical dispersion new coefficients reduce to CM ones numerator coefficients for l numerator coefficients for z

12. Frank Zimmermann, IBS in MAD-X, MAD-X Day, 23.09.2005 1st example: LHC - dispersion vertical dispersion is generated by the crossing angles at IP1 and 2, as well as by the detector fields at ALICE and LHC-B; the peak vertical dispersion is close to 0.2 m x & y dispersion

13. Frank Zimmermann, IBS in MAD-X, MAD-X Day, 23.09.2005 1st example: LHC – dispersion cont’d x dispersion y dispersion D y [m] s [m] D x [m] s [m]

14. Frank Zimmermann, IBS in MAD-X, MAD-X Day, 23.09.2005 without crossing angles and detector fields with crossing angles (285  rad at IP1 and 5) and detector fields old MAD-X new MAD-X old MAD-X new MAD-X  l [h] 57.5 58.6  x [h] 103.3 102.5104.2  y [h] -2.9x10 6 436.1 1st example: LHC – IBS growth rates

15. Frank Zimmermann, IBS in MAD-X, MAD-X Day, 23.09.2005 local vertical IBS growth rate around the LHC with nominal crossing angles at all 4 IPs, zero separation, and ALICE & LHC-B detector fields on, as computed by the new MAD-X version; the highest growth rates are found in the IRs 1 and 5 1st example: LHC – local y IBS growth rate 1/  y [1/s] s [m]

16. Frank Zimmermann, IBS in MAD-X, MAD-X Day, 23.09.2005 2nd example: LHC upgrade higher bunch charge, possibly larger transverse emittance, possibly smaller longitudinal emittance, higher harmonic rf, larger crossing angles, etc. IBS tends to get worse  l [h]  x [h]  y [h] nominal N b =1.15x10 11 58.6104.2436.1 N b =1.7x10 11, 0.7x longit. emit,  z =3.8 cm,   =1.55x10 -4, 7.5xV rf =120MV,  c =445  rad 46.442.577.3 2x charge N b =2.3x10 11 29.251.9217.5 N b =2.3x10 11 & 2x transv. emittance (  ) x,y =7.5  m 72.5254.21075. N b =2.3x10 11, 1/2 longit. emit,  z =5.2 cm,   =7.86x10 -5 9.332.8138.3 N b =2.3x10 11, 1/2 longit. emit,  z =3.7 cm,   =1.11x10 -4, 4xV rf 14.626.0108.6

17. Frank Zimmermann, IBS in MAD-X, MAD-X Day, 23.09.2005 3rd example: CLIC damping ring - dispersion IBS is dominant effect determining equilibrium emittance; field errors creating vertical dispersion have a profound effect on the vertical IBS growth rate and, thereby, on the emittance example: CLIC-DR dispersion functions obtained with random quadrupole tilt angles of 200  rad, cut off at 3  wiggler arc

18. Frank Zimmermann, IBS in MAD-X, MAD-X Day, 23.09.2005 no errorsquadrupole tilt errors old MAD-X new MAD-X old MAD-X new MAD-X  l [ms] 2.2  x [ms] 2.2 2.1  y [ms] 12.6 2.0 3rd example: CLIC DR – IBS growth rates in new MAD-X, y growth time decreases by factor 6 when errors are included → in tuning studies for CLIC DR, dependence of IBS y growth rate on residual vertical dispersion must be taken into account

19. Frank Zimmermann, IBS in MAD-X, MAD-X Day, 23.09.2005 3rd example: CLIC-DR – local IBS growth rates 1/  x [1/s] 1/  y [1/s] s [m] horizontal vertical 1/  l [1/s] s [m] longitudinal with quadrupole random tilt angles, computed by new MAD-X arc wiggler

20. Frank Zimmermann, IBS in MAD-X, MAD-X Day, 23.09.2005 conclusions:  applying B-M recipe, generalized expressions for the three IBS growth rates were derived  the new formulae are valid also if the beam energy is non-ultrarelativistic, or if vertical dispersion is present either by design or due to errors  in the limit of zero vertical dispersion, we recover the Conte-Martini result, except for a small difference in the horizontal growth rate  3 examples illustrate that the effect of vertical dispersion is significant  the new formulae have been committed to MAD-X