1. 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

2. Pellet Target Beam of small frozen hydrogen pellets Shape of pellet – nearly spheres Pellet diameter ~ 30 µm (20 - 70 µm) ρ H =0.0708 g /cm 2 Mass of 1 pellet ~ 10 -9 g (d=30 µm) Number of atoms in 1 pellet ~ 5·10 14 (d=30 µm) Pellet generation rate ~ 50 kHz (20 - 80 kHz)

3. Pellet Target Vertical velocity ~ 50 m/s Distance between pellets ~ 1 mm (rate 60 kHz, V v = 50 m/s) Angle divergence of pellet beam ~ 0.04 0 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

4. Internal target effects Small angle scattering Energy loss, energy straggling (relative momentum straggling ∆p/p)

5. Moliere theory (with various modifications) – widely used approach Main restriction to Moliere theory – number of scatters Ω 0 ≥20 - parameters of Moliere theory, - critical scattering angle - atomic electron screening angle - incident particle charge - total path length in the scatterer Ω 0  2 for pellets with diameter  30  m This value is out of area of Moliere theory application 1<Ω 0 <20 – “Plural Scattering” approach (direct simulation method), used by GEANT toolkit Coulomb Multiple Scattering

6. Plural 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

7. Scattering angle distribution for Plural scattering and simple Moliere scattering RMS scattering angle dependence on diameter of pellet Numerical results

8. 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

9. Area of Gauss distribution Area of Vavilov distribution Area of Landau distribution Pellet target 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 model

10. Algorithm 1.Calculation of 2.Calculation of n i (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 Urban model

11. f 1 +f 1 =1; f 1 lnE 1 + f 2 lnE 2 =lnI; f 1,2 – 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 Urban model

12. Subroutine 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

13. RMS dE distributionEmax dependence on pellet diameter Numerical test results

14. dp/p dependence on pellet diameter Numerical test results

15. Conclusions 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

16. RMS dE dependence on Ionisation/Excitation ratio RMS dE dependence on Ionisation/Excitation ratio (log scale) Ionisation/Excitation ratio, r