Frank Zimmermann, material for LTC coherent tune shift due to collimator impedance - its dependence on gap size, bunch length, chromaticity, beta function, conductivity, beam energy, #bunches - thanks to Elias Metral & Javier Resta Lopez

Frank Zimmermann, material for LTC Questions by Francesco to Elias, Giovanni and me “Understand/clarify the scaling of the effective LHC collimator impedance as a function of the collimator gap. Does a simple scaling law exist?” → Impact of Running LHC with non-nominal settings → Rating of ILC IR upgrade options Questions by Jean-Pierre to Ralph and me 1)Is the Cu collimator an option for the upgrade? It was the nominal system, rejected for lack of robustness. Can it be considered again for the upgrade with much higher beam power? If yes, impedance problem disappears. 2) What is dominant and should be used for scaling: single bunch or coupled bunch? Shall I take the tune shift to be proportional to the bunch charge or to the product of the bunch charge by the number of bunches?

Frank Zimmermann, material for LTC C Cu C  C =10 5  -1 m -1  Cu =5.9x10 7  -1 m -1  =70 m b=6   m d=3 cm 7 TeV B-L theory impedance of single flat collimator

Frank Zimmermann, material for LTC carbon factor 2 3 factor 2 2 factor 2 1 factor 2 2 impedance increase for 2x smaller gap

Frank Zimmermann, material for LTC m=0 m=1 Q’=10 Q’=0 Q’=10 Gaussian weight functions

Frank Zimmermann, material for LTC coherent coupled-bunch head-tail tune shifts F. Sacherer, 1974 A. Chao, 1993

Frank Zimmermann, material for LTC coh.tune shift –  =70 m, b=6  7TeV

Frank Zimmermann, material for LTC coh.tune shift –  =70 m, b=6  7 TeV half of the modes unstable all modes damped all modes unstable “Ruggiero graph”

Frank Zimmermann, material for LTC coh.tune shift –  =70 m, 7 TeV opening the collimators from 6 to 8  reduces tune shift more than 2 times

Frank Zimmermann, material for LTC critical modes: largest  Q or Im  Q  z (cm) mat.bmQ’mode# max(  Q) mode# max(Im  Q) 7.55C  C  C  C  Cu  C  always 0 or 3564changes by a few 100 carbon Q’=0, m=0  =70 m, 7 TeV injection

Frank Zimmermann, material for LTC maximum growth rate Im  Q vs. gap for m=0, Q’=0 fitted curve where b is the half gap size nearly inversely linear dependence on gap size! 7 TeV ~1/b! ~1/b 2 ! variation with conductivity

Frank Zimmermann, material for LTC maximum growth rate Im  Q vs. gap for m=0, Q’=0 fitted curve where b is the half gap size computed growth rates fit result nearly inversely quadratic dependence on gap size! carbon injection ~1/b 2 !

Frank Zimmermann, material for LTC maximum tune shift |  Q| vs. gap, m=0, Q’=0,  =70 m Q’=0, m=0 carbon copper fitted curves variation with conductivity ~1/b 2.7 ~1/b 3 ~1/b 2.3 ~1/b 2 7 TeV

Frank Zimmermann, material for LTC maximum tune shift |  Q| vs. gap for m=0, Q’=0 & 10 carbon m=0 Q’=0 Q’=10 fitted curves almost ~1/b 3 ! variation with chromaticity 7 TeV

Frank Zimmermann, material for LTC maximum tune shift |  Q| vs. gap for m=0, Q’=0 computed tune shifts fitted curves carbon m=0 Q’=0 fitted curves almost ~1/b 3 ! injection

Frank Zimmermann, material for LTC maximum tune shift |  Q| vs. gap for m=0 & 1, Q’=5 carbon Q’=5 m=0 m=1 fitted curves almost ~1/b 3 ! variation with head-tail mode

Frank Zimmermann, material for LTC maximum tune shift |  Q| vs. gap, m=0, Q’=0,  varying computed tune shifts fitted curves carbon Q’=0, m=0  =70 m  =700 m fitted curves almost ~1/b 3 ! variation with beta function

Frank Zimmermann, material for LTC tune shift decrease for 10x larger  about factor  2 carbon, Q’=0, m=0

Frank Zimmermann, material for LTC maximum tune shift |  Q| vs. gap, m=0, Q’=0,  varying carbon Q’=0, m=0 gap=6  variation with beta function cont’d computed tune shifts fitted curves

Frank Zimmermann, material for LTC maximum tune shift |  Q| vs. gap, m=0, Q’=0,  =70 m computed tune shifts fitted curves Q’=0, m=0 carbon  z =3.77 cm fitted curves  z =7.55 cm variation with bunch length ~1/b 3 !

Frank Zimmermann, material for LTC maximum tune shift |  Q| vs. n b, m=0, Q’=0,  =70 m fitted curves variation with # bunches ~nb!~nb! ~const.

Frank Zimmermann, material for LTC maximum growth rate vs. n b, m=0, Q’=0,  =70 m fitted curves variation with # bunches ~n b 1.2 ~n b 1.4

Frank Zimmermann, material for LTC maximum tune shift |  Q| vs. gap, m=0, Q’=0,  =70 m computed tune shifts fitted curves Q’=0, m=0 carbon, 2808 bunches,  z =7.55 cm, N b =1.15x10 11 fitted curves variation with bunch length, #bunches, conductivity, bunch charge ~1/b 2.2 copper, 5616 bunches,  z =3.77 cm, N b =1.7x10 11 ~1/b 2.7

Frank Zimmermann, material for LTC correction factor from nonlinear wake components derived from A. Piwinski’s wake field, in “Impedance of Elliptical Vacuum Chambers,” DESY , Eq. (52)

Frank Zimmermann, material for LTC nonlinear correction vs. gap b/  NLC round beam

Frank Zimmermann, material for LTC can we detect the inductive bypass effect with single bunches in the SPS?  for larger opening nonlinear correction is small, but tune shift is small too  if we go very close to integer resonance, classical formula diverges b=4   m 270 GeV 1 bunch

Frank Zimmermann, material for LTC conclusions for carbon jaw  Q ~1/b 2.75, for Cu jaw ~1/b 2.5 value for Cu almost 10 times smaller weak dependence on  :  Q ~1/  0.25 halving bunch length increases  Q by ~50% LHC upgrade with half  z, 1.7x10 11 ppb, 5616 bunches, and Cu collimators → 1/3 tune shift of nominal LHC with C jaws correction from nonlinear wake field a few percent for half gaps of 6  or larger