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 OUTLINE 1. Motivation (why?) 2. Formula for estimating intra-beam scattering (what?) 3. Comparison of estimates with measurements (does it work?)

ABS Intensity

Q out

=

k

4 ( 2

α Q in

) ×

f

×

t

× ( 1 -

A

)

k

a number of selected states (1 or 2) dissociation at nozzle exit

f Q

in input flux fraction of atoms entering the first magnet

t A

magnet transmission, calculated with ray-tracing code attenuation factor

Current Situation

Q in (mbar l/s) B pt (T) d mag (cm) v drift (m/s) T beam (K) length (m) d ct (cm) Q out (atoms/s) Hermes 1.5

1.5

0.86

1953 25.0

1.16

1.0

6.8x10

16 Nov.

0.6

3.2

1.4

1750 30.0

1.40

2.0

6.7x10

16 IUCF 1.7

1.5

1.04

1494 16.5

0.99

1.0

7.8x10

16 ANKE 1.0

RHIC 1.0

1.7

1.0

1778 20.3

1.24

1.0

7.5x10

16 1.5

1.04

~1530 ~18 1.37

1.0

12.4x10

16

HYPOTHESIS:

the beam density has an upper limit due to intra-beam scattering(IBS)

Parallel beam fast slow 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.

Cross Section Definition

dv

 σ n 1 n 2 v

rel dVdt

(for two intersecting beams)

dv =

Number of collisions in time

dt

and volume

dV

n 1 ,n 2 = Beam densities V

rel

= Relative velocity of the two beams •

v 1 || v 2 :

•

General :

v

rel

= v 1 v 2 v

rel

= (  v 1  v 2 2 ) (  v 1  x v 2 ) 2 (Landau and Lifshitz, The Classical Theory of Fields, p. 34, 1975 English edition)

Calculation of IBS losses

For scattering within a beam: Analytical solution , if: •v 1 and v 2 co-linear •constant transverse beam size A

dv

= σ n 2 v

rel dVdt d

Φ = -2

d ν dt

= Δ v -2σ v mean 2 Φ 2

dz

Δ v ~

α

= 2 v rms ~ 2kT beam m Δ v v mean 2

σ

Φ (

z

) = Φ 0 ( 1 +

z α

Φ 0 )

Steffens PST97

Simple Example

 Remaining flux for r<= 5 mm  Diverging beam from molecular-like starting generator and 2mm nozzle  D v/v = 0.3

q Isotropic direction (random cos q,f ) Random point inside nozzle

Fast Numerical Solution

dn

= Δ v -2σ v mean

n

2

dz

• 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 s is temperature (relative-velocity) independent

n

(

r

,

φ

, v

rel

=

z

)  v 1 = -

n

 v ( 2

z

)

Calculate n 0 (z), the nominal beam density without scattering, by counting tracks that remain within the acceptance r<5mm ∝ 1 N ∑ 1 v 1 2

π

rdr For each piece dz i , reduce the nominal density by the cumulative loss until that point Calculate the losses within dz, subtract them from n i to get n i ` and add them to the cumulative sum

n i

=

n i

' 1

n

0,

i

1

n

0 ,

i

Cumulative loss

dn i

=

n i

' = Δ v -2σ v mean

n i

2

dz n i

( 1 -

dn

) 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

Position (mm) Diameter (mm) nozzle 0 4 skimmer 15 6 c. tube 800 10 QMA ~2000 -

Molecular Beam Measurements

Velocity Distribution Measurement

TIME OF FLIGHT FITTED PARAMETERS To be improved:

D v and v mean have 10-20% error and one value is used for all fluxes

Attenuation Prediction

MEASURED CALCULATED

σ eff

H2 H2 = 2.0x10

-14 cm 2

Total density  1 N  1 1 Sum over tracks that pass through dr Total number of tracks Envelope density

Application to HERMES

∑

Weighted average density:  

n

env ( , ) all ( , ) 

n

env Survival Fraction:

s

.

f

.

=

n

' (

z

)

n

0 (

z

)

Comparison with measurements

 Reasonable agreement!

 s(H 1 -H 1 )  3x10 -14 cm 2 PRA 60 2188 (1999) (calculation) Note that IBS measurement is in a region of converging beam, while calculation assumes co-linear velocity vectors.

What could explain the difference ?

  Formula assumes that v Formula assumes that D rel =v 1 -v 2 , and this neglects the convergent/divergent nature of the beam v is constant for the entire length of the beam v

rel

= (  v 1  v 2 2  v 1  x v 2 ) 2 = (v 1 v 2 ) 1 + (v 2v 1 v 2 1 v 2 ) 2 ( 1 cos Δ

θ

) (v v 1 2 v 2 2 1 v 2 ) 2 sin 2 Δ

θ

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 D 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: E 2% on loss – Assumed that tracks with cos q >0.1 leave beam instantly (underestimate losses immediately after nozzle) • Velocity Distribution (mol. beams): E 15% on cross section • Neglecting RGA ???