IBS in MAD-X
description
Transcript of IBS in MAD-X
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
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-Martiniformulae, which are a non-ultrarelativisticgeneralization 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 verticaldispersion often gives shrinkage of y)
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
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
Frank Zimmermann, IBS in MAD-X, MAD-X Day, 23.09.2005
ylx LLLL
000
0
012
xxx
x
x
xx HL
000
010
000
2
2
lL
yx
yxyxyxyx
DD
,
,,,, 2
''
10
0
0002
y
yyyy
yy HL
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(rmax/rmin) --- with rmax the smaller of x and Debye length, and rmin the larger of classical closest approach and quantum diffraction limit from nuclear radius, typically log~15-20 ---, Lorentz factor, =(2)3 ()3 m3 xyz the 6-D invariant volume
vertical dispersion enters hereyx
yxyxyxyx
DH
,
2,
2,
2,
,
0 21
2132
02 1
Tr31
TrTrILdet
log1
ILL
ILL
dNcmr aa
a
note: above formulae refer to bunched beams
Frank Zimmermann, IBS in MAD-X, MAD-X Day, 23.09.2005
Bjorken-Mtingwa gave solution for zero vertical dispersion in ultrarelativistic limit, neglectingx/x and y/y relative to (Dx)2/(xx), (x/x)2x
2 and 2/2
Conte-Martini kept the terms neglected by B-M, which are important for <10
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
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
0
2/323
212320
2 log1
cba
badHNcmr xx
x
x
x
0
2/323
21
2
2320
2 log1
cba
badNcmr ll
l
02/323
21320
2 log1
cba
badNcmr yy
y
y
y
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
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
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
Frank Zimmermann, IBS in MAD-X, MAD-X Day, 23.09.2005
1st example: LHC - dispersionvertical dispersion is generated by the crossing angles at IP1 and 2, as wellas by the detector fields at ALICE and LHC-B; the peak vertical dispersionis close to 0.2 m
x & ydispersion
Frank Zimmermann, IBS in MAD-X, MAD-X Day, 23.09.2005
1st example: LHC – dispersion cont’d
x dispersion y dispersion
Dy [m]
s [m]
Dx [m]
s [m]
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 57.5 57.5 58.6
x [h] 103.3 103.3 102.5 104.2
y [h] -2.9x106 -2.9x106 -2.9x106 436.1
1st example: LHC – IBS growth rates
Frank Zimmermann, IBS in MAD-X, MAD-X Day, 23.09.2005
local vertical IBS growth rate around the LHC with nominal crossing anglesat all 4 IPs, zero separation, and ALICE & LHC-B detector fields on, as computedby 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]
Frank Zimmermann, IBS in MAD-X, MAD-X Day, 23.09.2005
2nd example: LHC upgradehigher 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 Nb=1.15x1011 58.6 104.2 436.1Nb=1.7x1011, 0.7x longit. emit, z=3.8 cm, =1.55x10-4, 7.5xVrf=120MV,c=445 rad
46.4 42.5 77.3
2x charge Nb=2.3x1011 29.2 51.9 217.5Nb=2.3x1011 & 2x transv. emittance ()x,y=7.5 m
72.5 254.2 1075.
Nb=2.3x1011, 1/2 longit. emit, z=5.2 cm, =7.86x10-5
9.3 32.8 138.3
Nb=2.3x1011, 1/2 longit. emit, z=3.7 cm, =1.11x10-4, 4xVrf
14.6 26.0 108.6
Frank Zimmermann, IBS in MAD-X, MAD-X Day, 23.09.2005
3rd example: CLIC damping ring - dispersionIBS 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
Frank Zimmermann, IBS in MAD-X, MAD-X Day, 23.09.2005
no errors quadrupole tilt errors
old MAD-X
new MAD-X
old MAD-X
new MAD-X
l [ms] 2.2 2.2 2.2 2.2
x [ms] 2.2 2.2 2.2 2.1
y [ms] 12.6 12.6 12.6 2.0
3rd example: CLIC DR – IBS growth rates
in new MAD-X, y growth timedecreases by factor 6when errors are included
→ in tuning studies for CLIC DR, dependence of IBS y growth rate on residual vertical dispersion must be taken into account
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]s [m]
horizontal vertical
1/l [1/s]
s [m]
longitudinal with quadrupolerandom tilt angles,computed by newMAD-X
arc wiggler
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 recoverthe Conte-Martini result, except for a smalldifference 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