1. Estimates of Intra-Beam Scattering in ABS M. Stancari, S. Atutov, L. Barion, M. Capiluppi, M. Contalbrigo, G. Ciullo, P.F. Dalpiaz, F.Giordano, P. Lenisa, M. Statera, M. Wang University of Ferrara and INFN Ferrara OUTLINE 1.Motivation (why?) 2.Formula for estimating intra-beam scattering (what?) 3.Comparison of estimates with measurements (does it work?)

2. ABS Intensity k number of selected states (1 or 2)  dissociation at nozzle exit Q in input flux f fraction of atoms entering the first magnet t magnet transmission, calculated with ray-tracing code A attenuation factor

3. Current Situation HermesNov.IUCFANKERHIC Q in (mbar l/s) 1.50.61.71.0 B pt (T) 1.53.21.51.71.5 d mag (cm) 0.861.41.041.01.04 v drift (m/s) 1953175014941778~1530 T beam (K) 25.030.016.520.3~18 length (m)1.161.400.991.241.37 d ct (cm) 1.02.01.0 Q out (atoms/s)6.8x10 16 6.7x10 16 7.8x10 16 7.5x10 16 12.4x10 16

4. HYPOTHESIS: the beam density has an upper limit due to intra-beam scattering(IBS) POSSIBLE WAY TO INCREASE INTENSITY: increase the transverse beam size while keeping the density constant A method for estimating IBS is essential to work near this limit. Parallel beam slow fast

5. Cross Section Definition (for two intersecting beams) dv = Number of collisions in time dt and volume dV n 1,n 2 = Beam densities V rel = Relative velocity of the two beams (Landau and Lifshitz, The Classical Theory of Fields, p. 34, 1975 English edition) v 1 || v 2 : General :

6. Calculation of IBS losses For scattering within a beam: Analytical solution, if: v 1 and v 2 co-linear constant transverse beam size A Steffens PST97

7. Simple Example  Remaining flux for r<= 5 mm  Diverging beam from molecular-like starting generator and 2mm nozzle   v/v = 0.3 Random point inside nozzle  Isotropic direction (random cos  )

8. Fast Numerical Solution Begin with a starting generator. Use tracks to calculate the beam density in the absence of collisions. Calculate the losses progressively in z. Approximations: Uniform transverse beam density Co-linear velocities  is temperature (relative-velocity) independent

9. Calculate n 0 (z), the nominal beam density without scattering, by counting tracks that remain within the acceptance r<5mm For each piece dz i, reduce the nominal density by the cumulative loss until that point Calculate the losses within dz, subtract them from n i to get n i ` and add them to the cumulative sum Cumulative loss Density reduced by scattering

10. Experimental Tests 1.Dedicated test bench measurements with molecular beams and no magnets 2.Compare with HERMES measurement of IBS in the second magnet chamber 3.Calculate HERMES ABS intensity including the attenuation and compare with measurement

11. Test Bench nozzleskimmerc. tubeQMA Position (mm) 015800~2000 Diameter (mm) 4610-

12. Molecular Beam Measurements

13. Velocity Distribution Measurement TIME OF FLIGHT To be improved:  v and v mean have 10-20% error and one value is used for all fluxes FITTED PARAMETERS

14. Attenuation Prediction MEASUREDCALCULATED

15. Application to HERMES Total density Envelope density Sum over tracks that pass through dr Total number of tracks

16. Weighted average density: Survival Fraction:

17. Comparison with measurements Note that IBS measurement is in a region of converging beam, while calculation assumes co-linear velocity vectors.  Reasonable agreement!       3x10 -14 cm 2 PRA 60 2188 (1999) (calculation)

18. What could explain the difference ?  Formula assumes that v rel =v 1 -v 2, and this neglects the convergent/divergent nature of the beam  Formula assumes that  v is constant for the entire length of the beam

19. Conclusions The parallel beam equation has been freed from the assumption of constant transverse beam size. The new equation reproduces molecular beam measurements reasonably well. Some more work remains to be done on velocity measurements and RGA corrections. The large losses from IBS measured by HERMES in the second half of the ABS are incompatible with the overall losses in the system, given the current assumptions. Predictions can be improved by introducing a z dependence into  v to account for changing velocity distribution and/or convergence angle. This attenuation calculation can be done in 1-2 minutes after the average density is obtained, and is thus suitable for magnet parameter optimization

20. Uncertainties Starting Generator: E 2% on loss –Assumed that tracks with cos  >0.1 leave beam instantly (underestimate losses immediately after nozzle) Velocity Distribution (mol. beams): E 15% on cross section Neglecting RGA ???