Frank Zimmermann, IBS in MAD-X, MAD-X Day, 23.09.2005 IBS in MAD-X Frank Zimmermann Thanks to J....

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


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)


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


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


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


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


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


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


in the limit of zero vertical dispersion new coefficients reduce to CM ones

denominator coefficients (from determinant) for x, z, s


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


in the limit of zero vertical dispersion new coefficients reduce to CM ones

numerator coefficients for l

numerator coefficients for z


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


1st example: LHC – dispersion cont’d

x dispersion y dispersion

Dy [m]

s [m]

Dx [m]

s [m]


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


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]


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


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


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


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


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

Frank Zimmermann, IBS in MAD-X, MAD-X Day, 23.09.2005 IBS in MAD-X Frank Zimmermann Thanks to J....

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