estimates of intra-beam scattering in abs
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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
OUTLINE1. Motivation (why?)2. Formula for estimating intra-beam scattering (what?)3. Comparison of estimates with measurements (does it
work?)
ABS Intensity
( ) ( )AtfQk
Q inout -124
×××= α
kk number of selected states (1 or 2) number of selected states (1 or 2)
dissociation at nozzle exitdissociation at nozzle exit
QQinin input flux input flux
ff fraction of atoms entering the first magnet fraction of atoms entering the first magnet
tt magnet transmission, calculated with ray-tracing code magnet transmission, calculated with ray-tracing code
AA attenuation factor attenuation factor
Current Situation
Hermes Nov. IUCF ANKE RHIC
Qin (mbar l/s) 1.5 0.6 1.7 1.0 1.0
Bpt (T) 1.5 3.2 1.5 1.7 1.5
dmag (cm) 0.86 1.4 1.04 1.0 1.04
vdrift (m/s) 1953 1750 1494 1778 ~1530
Tbeam (K) 25.0 30.0 16.5 20.3 ~18
length (m) 1.16 1.40 0.99 1.24 1.37
dct (cm) 1.0 2.0 1.0 1.0 1.0
Qout (atoms/s) 6.8x1016 6.7x1016 7.8x1016 7.5x1016 12.4x1016
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
slowfast
Cross Section Definition
(for two intersecting beams)
dv = Number of collisions in time dt and volume dV
n1,n2 = Beam densities
Vrel = Relative velocity of the two beams
dVdtdv relvnnσ 21
21 v-vv =rel
( ) ( )221
2
21 vxv-v-vv
=rel
(Landau and Lifshitz, The Classical Theory of Fields, p. 34, 1975 English edition)
• v1||v2:
• General:
Calculation of IBS losses
dVdtdv relvnσ 2=For scattering within a beam:
Analytical solution, if:
•v1 and v2 co-linear
•constant transverse beam size Am
2kT~v2vΔ beam
rms~
dzdt
dd 2
2mean
Φv
vΔ-2σ-2Φ ==
ν
( )00
1)(
ΦΦ
Φαz
z +=
σα 2meanv
vΔ=
Steffens PST97
Simple Example
Remaining flux for r<= 5 mm Diverging beam from molecular-like starting generator and 2mm nozzlev/v = 0.3
Random point inside nozzle
Isotropic direction (random cos)
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
21 v-vv
=rel
)(),,( znzrn =φ
dzndn 2
meanv
vΔ-2σ=
)-1(' dnnn ii =
rdr2
1
v
1
N
1∝ ∑
π
Calculate n0(z), the nominal beam density without scattering, by counting tracks that remain within the acceptance r<5mm
For each piece dzi, reduce the nominal density by the cumulative loss until that point
Calculate the losses within dz, subtract them from ni to get ni` and add them to the cumulative sum
dzndn ii2
meanv
vΔ-2σ=
ii
ii n
n
nn ,0
1-0,
1-'=
Cumulative loss
Density reduced by scattering
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
Test Bench
nozzle skimmer c. tube QMA
Position (mm) 0 15 800 ~2000Diameter (mm) 4 6 10 -
Molecular Beam Measurements
Velocity Distribution Measurement
TIME OF FLIGHT
To be improved:v and vmean have 10-20% error and one value is used for all fluxes
FITTED PARAMETERS
Attenuation Prediction
MEASURED CALCULATED
2-14H2-H2 cm 2.0x10=effσ
Application to HERMES
Total density
Envelope density
∑
1 1 1
N v 2 rdr
Sum over tracksthat pass through dr
Total number of tracks
Weighted average density:
env all
env
( , ) ( , )( )
( , )
n z r n z r dAn z
n z r dA
Survival Fraction:
)(
)('..
0 zn
znfs =
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 cm2
PRA 60 2188 (1999)(calculation)
What could explain the difference?
Formula assumes that vrel=v1-v2, and this neglects the convergent/divergent nature of the beam Formula assumes that v is constant for the entire length of the beam
( ) ( )
sin)v-(v
vv-)cos-1(
)v-(v
v2v1)v-(v=
vxv-v-vv
22
21
22
21
221
2121
2
21
2
21
θθ ΔΔ+
=
rel
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
Uncertainties
• Starting Generator: E2% on loss– Assumed that tracks with cos>0.1 leave beam
instantly (underestimate losses immediately after nozzle)
• Velocity Distribution (mol. beams): E15% on cross section
• Neglecting RGA ???
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