Numerical Model of an Internal Pellet Target

Numerical Model of an Internal Pellet Target

O. Bezshyyko*, K. Bezshyyko*, A. Dolinskii†,I. Kadenko*, R. Yermolenko*, V. Ziemann¶

* Nuclear Physics Department, Taras Shevchenko National University†GSI, 64287,Darmstadt, Germany

¶Svedberg Laboratory, Uppsala University, S-75121Uppsala, Sweden

Pellet TargetPellet Target

• Beam of small frozen hydrogen pellets

• Shape of pellet – nearly spheres

• Pellet diameter ~ 30 µm (20 - 70 µm)

• ρH=0.0708 g /cm2

• Mass of 1 pellet ~ 10-9 g (d=30 µm)

• Number of atoms in 1 pellet ~ 5·1014 (d=30 µm)

• Pellet generation rate ~ 50 kHz (20 - 80 kHz)

Pellet TargetPellet Target

• Vertical velocity ~ 50 m/s• Distance between pellets ~ 1

mm (rate 60 kHz, Vv= 50 m/s)• Angle divergence of pellet

beam ~ 0.040

• Distance between injection nozzle and area of beam ~ dozens of cm (real example 241 cm)

• Spread of pellet beam in the point of crossing with antiproton beam ~ ±1 (±1 - ±3)

Pellet beam

Ion beam

Internal target effectsInternal target effects

• Small angle scattering

• Energy loss, energy straggling (relative momentum straggling ∆p/p)

• Moliere theory (with various modifications) – widely used approach

• Main restriction to Moliere theory – number of scatters Ω0≥20

- parameters of Moliere theory,

- critical scattering angle

- atomic electron screening angle

- incident particle charge

- total path length in the scatterer

• Ω02 for pellets with diameter 30 m This value is out of area of Moliere theory application

• 1<Ω0<20 – “Plural Scattering” approach (direct simulation method), used by GEANT toolkit

22

212

2

0

tZb

e inccc

22 , c57721,0

2c2

incZ

t

Coulomb Multiple ScatteringCoulomb Multiple Scattering

Plural Scattering algorithmPlural Scattering algorithm

1. Calculation of scatters number n. Poisson distribution with average

2. Generation of random number - angle of the single scattering

This approximates Rutherford

distribution:

where is a random number uniformly distributed in the interval between 0 and 1

3. Generation of random number (uniformly distributed in the interval between 0

and 2) – to project the scattering angle into the horizontal (or vertical) direction.

4. Calculation of total scattering angle for one hit:

for horizontal direction for vertical direction

n

iii sinˆˆ

0167.1

11ˆ

i

i

2222

)(

2

d

dN

i

n

iii cosˆˆ

Scattering angle distribution for Plural scattering and simple Moliere scatteringRMS scattering angle dependence on diameter of

pellet

Numerical resultsNumerical results

Energy losses and straggling

Energy losses and straggling

Main parameters for choice between theories

1. 2.

- mean energy loss - maximum transferable energy in single collision with an atomic electron

- mean ionization potential of the atom

- the electron mass

- the mass of the incident particle

- charge of the incident particle

- atomic number and weight of the target

- density of the target

- thickness of the target

2

22

max

21

2

x

e

x

e

e

mm

mm

mE

maxE

I

maxEI

KeVxA

ZZ inc ,4.1532

2

em

xm

incZ

AZ ,

x

Area of Gauss distribution

Area of Vavilov distribution5010 and

501001.0 and

5001.0 and

Area of Landau distribution

Pellet target

210 5 and for E=1 GeV, d=30 µm

It is necessary to take into account atomic structure and direct simulation of scattering

Conditions for choice of the modelConditions for choice of the model

Algorithm

1. Calculation of

2. Calculation of ni (Poisson distribution)

3. Calculation of excitation energy loss

4. Calculation of ionisation energy loss

5. Calculation of the total energy loss

6. Calculation of the relative momentum straggling

in

2211 EnEnEex

3

1

max

max1

n

ji

ion

IE

EI

E

ionex EEE

E

E

p

p

1

Urban modelUrban model

ionisationi

levelsenergyofexitationixn ii

3

2,1,

f1+f1=1; f1lnE1+ f2lnE2 =lnI; f1,2 – oscilator strengths

Macroscopic cross-section for exitation (i=1,2):

Macroscopic cross-section for ionisation:

Distribution of ionisation energy loss:

Approximation of g(E) distribution:

is a random number uniformly distributed between 0 and 1

)1()/2ln(

)/2ln(222

222

rImE

Emf

dx

dE

i

iii

r

IIE

IEI

E

dx

dEi

)ln()( maxmax

max

Urban modelUrban model

11

max

max

E

EI

Er

2max

max 1)()(

EE

IIEEg

Subroutine featuresSubroutine features

•Detail 2D (in plane normal to beam axis) geometrical description of particle interaction with pellet is applied

•Spatial distribution of pellet beam in the interaction area is recalculated through spatial and angle distribution at the injection nozzle

•The local (not mean value) thickness of pellet is taken into account

•Main input parameters - x,y,x’,y’, energy of particle, parameters of the pellet beam

•Output data – dp/p (relative momentum straggling ) and (total

scattering angle ) projections into the horizontal and vertical direction

RMS dE distribution Emax dependence on pellet diameter

Numerical test resultsNumerical test results

dp/p dependence on pellet

diameter

Numerical test resultsNumerical test results

ConclusionsConclusions

•Program block for Monte Carlo simulation of the pellet target is developed

•„Plural Scattering“ model for simulation of angle distributions during every turn analysis is used

•Urban model for simulation of energy losses during every turn analysis is used

•Moliere theory and Landau (Vavilov, Gauss) models are preferable only in cases of analysis of target effects after dozens and more turns

•Detailed 2D (in plane normal to beam axis) geometrical description of particle interaction with pellet is applied

•Preliminary numerical results to check the code are obtained, extended tests of code as part of some Monte Carlo codes for analysis of beam parameters are planned in the near future

RMS dE dependence on Ionisation/Excitation ratio

RMS dE dependence on Ionisation/Excitation ratio (log scale)

Ionisation/Excitation ratio, r

Ionisation/Excitation ratio, r