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 ???