Scaling laws for b * in the LHC interaction regions

1. Valencia, 17th October 2006 CARE workshop LUMI-06 Scaling laws for b* in the LHC interaction regions E. Todesco, J.-P. Koutchouk CERN, Accelerator Technology Department Magnets, Cryostats and Superconductors Group Acknowledgements: B. Bellesia, R. De Maria, P. Fessia, E. Laface, F. Regis, L. Rossi, W. Scandale

2. Contents • The pieces of the puzzle: • An analytical model for beta functions in a triplet lay-out (NEW) • The requirement for quadrupole aperture (well known) • What Nb-Ti and Nb3Sn can give as aperture/gradient (just published) • Solutions • Lay-out for either materials in terms of triplet length, aperture, gradient, b* (NEW) • Large apertures are found: • Stress limits for large aperture quads (just published) • Field errors and aberrations dependence on aperture (NEW) • Gain in b* comparing for the same chromaticity • For each material, distance to IP, the lay-out with minimum bm is found • Setting a maximum tolerable chromaticity, the corresponding b* is found E. Todesco, Accelerator Technology Dept., CERN

3. A triplet structure for parametric analysis • Magnet features • Q1, Q3 with the same length • All quads with the same aperture • All quads with same gradient • Lay-out features • Distance to IP: l*=23 to 13 m • b* free parameter • Triplet length lt=2(l1+l2) • Optics constraints • bmax identical in both planes • Approximate matching in Q4 • negative derivative of b • No minima between Q3 and Q4 • Q4 shifted towards IP to find solutions for l*<23m E. Todesco, Accelerator Technology Dept., CERN

4. Scaling laws for b functions and gradients Empirical fits • bm • Linear with triplet length with al* • Gradient Plot for nominal b*=55 cm E. Todesco, Accelerator Technology Dept., CERN

5. Aperture requirements Budget for the aperture • d: beam separation • r: beam radius • A: alignment tolerances • B: beam screen • C: closed orbit • D: dispersion Using the previous scaling for bm and scaling for crossing angle [F. Zimmermann et al.] E. Todesco, Accelerator Technology Dept., CERN

6. Gradient versus aperture for Nb-Ti and Nb3Sn • Order zero: G/2  critical field • 13T for Nb-Ti, 25T for Nb3Sn • This is a bad approximation ! • Results relative to a sector coil for   100 mm • Nb-Ti: G/2  10 T • Nb3Sn: G/2  15 T • Nb3Sn: 50% more than Nb-Ti • Some dependence on aperture • Better for large apertures • A safety margin of 20% is then subtracted on both cases [L. Rossi, E. Todesco, Phys. Rev. STAB, October 2006, and IEEE Trans. Appl. Supercond., ASC06] E. Todesco, Accelerator Technology Dept., CERN

7. Triplet aperture and length vs b*, technology, l* Ex.:l*=23 mb*=0.28 cm • Nb-Ti: aperture 94 mm, triplet length 30 m, gradient 160 T/m • Nb3Sn: aperture 81 mm, triplet length 20 m, gradient 275 T/m • Solutions can be found for both materials • Large apertures: is this possible? • Stresses, aberrations ? E. Todesco, Accelerator Technology Dept., CERN

8. Limits on aperture due to Lorentz forces • Stress due to Lorentz forces • peak on coil mid-plane • proportional to current density • increases with aperture • Analytic approximation of maximum stress in the coil • For Nb-Ti values below 150 MPa up to =240 mm: ok • For Nb3Sn at 140-240 mm one reaches 150200 MPa • Apertures larger than 120 mm may have too large stresses • Possible cure: reduce current density [P. Fessia, F. Regis, E. Todesco, IEEE Trans. Appl. Supercond., ASC 2006] E. Todesco, Accelerator Technology Dept., CERN

9. Limits on aperture due to field errors • Average (systematic) field errors can be set to zero within the random component • n=8 for one triplet • Estimate of the random component • Dominated by cable positioning (geometric) • A rescaling of aperture, cable width, reference radius and cable positioning gives the same random errors • Estimate of cable positioning for RHIC and LHC based on measurements • Between 0.015 and 0.030 mm • Best guess: positioning independent of aperture • Proposed scaling [B. Bellesia, E. Todesco, to be published] E. Todesco, Accelerator Technology Dept., CERN

10. Aberrations due to field errors • Does a large aperture triplet leads to large aberrations ? • Example: detuning induced by octupolar term in the triplet • Rescaling of aperture: • Rescaling of reference radius: • Rescaling of beta functions: • Rescaling of multipoles: • Constant integrated gradient: • Detuning scales linearly with the aperture • Terms  kn scale with  • Terms  (kn)k scale with k • Not catastrophic, but to be checked (see De Maria talk) E. Todesco, Accelerator Technology Dept., CERN

11. Gain in b* versus technology and l* (distance to IP) • Comparison of lay-outs giving the same linear chromaticity • For each technology, apertures and triplet length optimized • Both technologies used at the limit • Aperture set at the minimum requirement (energy deposition ?) • For the same linear chromaticity, • Nb3Sn gives 25% to 30% more • reducing l* to 13 m gives 20% to 25% more E. Todesco, Accelerator Technology Dept., CERN

12. A simple argument to understand the gain in Nb3Sn • Nb3Sn gives improvement in G of a factor =1.5 • Constant integrated gradient: triplet length decreases with  • Linear chromaticity proportional to bm • Equal chromaticity, constant int. G  equal bm • Using the empirical fit for bm • We obtain the gain in 1/b* • For +50% in G, +35% in 1/b* E. Todesco, Accelerator Technology Dept., CERN

13. Conclusions • Main results of the study • Nb3Sn gives 50% more gradient than Nb-Ti for the same aperture, but 25-30% more in terms of b*, assuming the same linear chromaticity • Material and distance to IP determine apertures, triplet length and structure ! • Reducing distance to IPto 13 m gives 20% to 25% more than the nominal • Going for large apertures  : are there bottlenecks ? • Field errors are likely to scale with 1/ (scaling the reference radius) • First order (in gradient) aberrations are likely to scale linearly with  • Stresses can be an issue for Nb3Sn at large apertures 120 mm E. Todesco, Accelerator Technology Dept., CERN