Fluka advanced calculations on stopping power and multiple Coulomb scattering

Transcription

1 Fluka advanced calculations on stopping power and multiple Coulomb scattering Andrea Fontana INFN Sezione di Pavia

2 Outline Stopping power Ionization potential Range calculation: which range? Fluka options Multiple Coulomb scattering Molière theory Nuclear interactions Fluka options Tools for analysis USRBIN readout and plot with ROOT USRBIN dynamic scan

3 Case study This study is performed in a simplified geometry, consisting of a water phantom in water (xx30 cm 3 ), hit by a monochromatic beam of protons with different energies. Geometry setup Andrea Fontana INFN Sezione di Pavia 3/45

4 Case study This study is performed in a simplified geometry, consisting of a water phantom in water (xx30 cm 3 ), hit by a monochromatic beam of protons with different energies. Gaussian (σ=0.6 cm) beam profile (lin) Andrea Fontana INFN Sezione di Pavia 4/45

5 Case study This study is performed in a simplified geometry, consisting of a water phantom in water (xx30 cm 3 ), hit by a monochromatic beam of protons with different energies. Gaussian (σ=0.6 cm) beam profile (log) Andrea Fontana INFN Sezione di Pavia 5/45

6 Case study This study is performed in a simplified geometry, consisting of a water phantom in water (xx30 cm 3 ), hit by a monochromatic beam of protons with different energies. Pencil beam profile (log) Andrea Fontana INFN Sezione di Pavia 6/45

7 Stopping power de [ = Kz 2 Z dx A ϱ 1 1 β 2 2 ln 2mec2 β 2 γ 2 T max β 2 δ(βγ) ] I 2 2 Andrea Fontana INFN Sezione di Pavia 7/45

8 Ionization potential A material is completely described by a single number, the mean excitation potential I. Felix Bloch approximation (1933): I = (ev )Z whit Z atomic number of the atoms of the material. Modern tables (e.g. ICRU Report 37 and 49) yield better results than this formula. For compounds: ( ) Z Mj ZM lni = Σw j lni j / A Mj A M with ZM A M = Σw j Z Mj A Mj Z M and A M are charge number and mass number of the target. Andrea Fontana INFN Sezione di Pavia 8/45

9 "What is the ionization potential of water?" FLUKA default value is I=75 ev, but it is usually redefined. Its exact knowledge is difficult, both theoretically and experimentally. MAT-PROP 77. WATER Andrea Fontana INFN Sezione di Pavia 9/45

10 Ionization potential Bragg peak and water ionization potential for 200 MeV protons energy (a.u.) I=75 ev z (cm) Andrea Fontana INFN Sezione di Pavia /45

11 Ionization potential Bragg peak and water ionization potential for 200 MeV protons energy (a.u.) I=75 ev I=77 ev z (cm) Andrea Fontana INFN Sezione di Pavia 11/45

12 Ionization potential Bragg peak and water ionization potential for 200 MeV protons energy (a.u.) I=75 ev I=77 ev I=79 ev z (cm) Andrea Fontana INFN Sezione di Pavia 12/45

13 Range calculation: which range? Mean range Projected range CSDA range 90% distal fall-off range analytical range Andrea Fontana INFN Sezione di Pavia 13/45

14 CSDA range The CSDA (Continuous Slowing Down Approximation) ranges for protons of different energies in water is investigated. The CSDA range is calculated by integrating the inverse of the linear stopping power of the proton from zero to the initial energy: no energy straggling R CSDA = no delta ray production no nuclear interactions L no Coulomb multiple scattering 0 dx = 0 E 1 de/dx de Energy (MeV) Ranges CSDA (mm) FLUKA NIST This corresponds essentially to follow the primary particle ionization energy loss only. Andrea Fontana INFN Sezione di Pavia 14/45

15 Fluka options EMFCUT: transport and production thrs. for e +, e and γ EMFCUT -3E-05 3.E-6 1. PROD-CUT EMFCUT -3E-05 3.E PART-THR: all partice thrs. kev, neutron thr MeV PART-THR -1E PART-THR NEUTRON 0.0 THRESHOL: infinite hadron elastic scattering and inelastic interactions THRESHOL Andrea Fontana INFN Sezione di Pavia 15/45

16 DELTARAY: infinite delta ray thr. DELTARAY -0.E PRINT IONFLUCT: switch off hadrons and leptons ionization straggling IONFLUCT FLUKAFIX: fraction of the kinetic energy to be lost in a step FLUKAFIX STEPSIZE: set maximum step size STEPSIZE 0.01 Andrea Fontana INFN Sezione di Pavia 16/45

17 CSDA range Andrea Fontana INFN Sezione di Pavia 17/45

18 Analytical stopping power? Øverås has found a simple empirical relation between p(x) 2 β(x) 2 and normalized residual range (see also Gottschalk papers): ) k ( p(x) 2 β(x) 2 = p(0) 2 β(0) 2 1 x R ( p(x) 2 β(x) 2 p(0) 2 β(0) 2 1 x ) R Valid for many materials of density ρ (g/cm 3 ) and radiation length X 0 (cm): k = exp( 0.09ρX 0) Andrea Fontana INFN Sezione di Pavia 18/45

19 Analytical range calculation? For an incident proton of kinetic energy E(MeV), Ulmer has published a very accurate formula for a medium of density ρ (g/cm 3 ) is: R(cm) = 1 A M ρ Z M N α ni p i E n n=1 order α i p i e e e e where I is the mean ionization potential of the medium (in ev) and Z M and A M are the effective atomic and mass numbers. For water and protons of incident kinetic energy E < 300 MeV, a sum with N = 1..4 gives results accurate to a more than 0.5%. Andrea Fontana INFN Sezione di Pavia 19/45

20 Multiple Coulomb scattering Molière has found a probability distribution function for θ that is a solution of the transport equation and that well describes the experimental data. The expression of this solution reads as: f(θ)θdθ = f M (θ)d(cosθ)dφ/2π and, by using the approximation d(cosθ) = sinθ dθ θ dθ, can be written as the sum of three terms: f M (θ) = 1 [ f 0 (θ ) + f 1 (θ ) 2θM 2 B ) + f 2 (θ ] ) B 2 where θ M is the characteristic multiple scattering angle of the target, θ = θ/( 2θ M ) is the reduced angle and B is related to the logarithm of the effective number of collisions in the target. Andrea Fontana INFN Sezione di Pavia 20/45

21 Molière theory Molière theory depends on two parameters: χ 2 c and χ 2 α. The parameter χ 2 c is related to the RMS of the scattering angle: χ 2 c = Z 2 z 2 x 1 A p 2 β, 2 The parameter χ 2 α is related to the screening of the Coulomb potential: ( ) µ 2 = z2 Z 2 χ 2 α = µ 2 χ β 2 0 ( χ 2 ħ Z 1/3 0 = p (cm) ) 2 Andrea Fontana INFN Sezione di Pavia 21/45

22 Multi-gaussian approximation f 0 (x) = 2e x f 1 (x) = 2e x (x 1)[Ēi(x) lnx] 2(1 2e x ) f 2 (x) = e ([ψ x 2 (2) + ψ(2)](x 2 4x + 2) + 4 y 3 dy[lny/(1 y) ψ(2)] 0 ) [(1 y 2 )e xy 1 (x 2)y (x 2 /2 2x + 1)y 2 ] whit x = θ 2, ψ(n) = d{lnγ(n + 1)}/ dn the Digamma function and Ei(x) the Exponential Integral function. The first term is a gaussian function that constitutes the "core" of the distribution, while the extra terms account for the tails of the distribution that are non-gaussian: they can be evaluated by numerical integration or by using mathematical tables. 1 Andrea Fontana INFN Sezione di Pavia 22/45

23 Nuclear interactions Andrea Fontana INFN Sezione di Pavia 23/45

24 Nuclear interactions In the case of protons in water, the total nuclear cross-section for proton-nucleus interactions is often calculated only taking into account the p 16 O reactions: p O n F p O p + n O p O p + p N p O α N p O d O n O p N n O p + n N In general terms: t(x) f(x) = W pf M (x)+(1 W p) + t(x)dx f M (x) = Molière electromagnetic theory t(x) = parametrization of nuclear tails W p = fraction of events without nuclear interactions Andrea Fontana INFN Sezione di Pavia 24/45

25 Analytical description of nuclear interactions Ulmer formula for the determination of the percentage W p of events without nuclear interactions (primary protons), as a function of the traversed thickness, for protons of incident kinetic energy E and range R in water, at a certain water thickness x: [ W p = 1 ( ) E f ] [ ( )] Eth x R x erf, (1) 2 m R τ where erf is the error function, f = 1.032, m is the proton mass in MeV, E th = 7 MeV is the 16 O threshold energy of the Coulomb barrier. The parameter τ takes into account the range variation due to the straggling along the beam path and can be parametrized as: τ = R t, where t = { if R 1cm if R < 1cm (2) Andrea Fontana INFN Sezione di Pavia 25/45

26 Decrease of the fluence of primary protons in water Andrea Fontana INFN Sezione di Pavia 26/45

27 Secondaries production protons Andrea Fontana INFN Sezione di Pavia 27/45

28 Secondaries production neutrons Andrea Fontana INFN Sezione di Pavia 28/45

29 Secondaries production deuterons Andrea Fontana INFN Sezione di Pavia 29/45

30 Secondaries production alphas Andrea Fontana INFN Sezione di Pavia 30/45

31 Fluka options Fluka is the only MC code firmly based on the full Molière theory for MCS. MULSOPT: to define tracking conditions with many sub-options (Fano correction, Molière screening angle, spin relativistic corrections, finite nuclear size correction, single scattering at boundaries or very low energy). To completely suppress MCS: MULSOPT MCSTHRESh to specify accuracy for MCS of heavy charged particles, depending on DEFAULTS. Andrea Fontana INFN Sezione di Pavia 31/45

32 Fluka options Lateral profile for 200 MeV protons at z=25 cm energy (a.u.) -3 default z (cm) Andrea Fontana INFN Sezione di Pavia 32/45

33 Fluka options Lateral profile for 200 MeV protons at z=25 cm energy (a.u.) -3 default precision z (cm) Andrea Fontana INFN Sezione di Pavia 33/45

34 Fluka options Lateral profile for 200 MeV protons at z=25 cm energy (a.u.) -3 default precision optimize z (cm) Andrea Fontana INFN Sezione di Pavia 34/45

35 Fluka options Lateral profile for 200 MeV protons at z=25 cm energy (a.u.) -3 default precision optimize fano off z (cm) Andrea Fontana INFN Sezione di Pavia 35/45

36 Fluka options Lateral profile for 200 MeV protons at z=25 cm energy (a.u.) -3 default precision optimize fano off mlsh off z (cm) Andrea Fontana INFN Sezione di Pavia 36/45

37 Fluka options Lateral profile for 200 MeV protons at z=25 cm energy (a.u.) -3 default precision optimize fano off mlsh off born 1st z (cm) Andrea Fontana INFN Sezione di Pavia 37/45

38 Fluka options Lateral profile for 200 MeV protons at z=25 cm energy (a.u.) -3 default precision optimize fano off mlsh off born 1st born 2nd z (cm) Andrea Fontana INFN Sezione di Pavia 38/45

39 Fluka options Lateral profile for 200 MeV protons at z=25 cm energy (a.u.) -3 default precision optimize fano off mlsh off born 1st born 2nd size 1st z (cm) Andrea Fontana INFN Sezione di Pavia 39/45

40 Fluka options Lateral profile for 200 MeV protons at z=25 cm energy (a.u.) -3 default precision optimize fano off mlsh off born 1st born 2nd size 1st size 2nd z (cm) Andrea Fontana INFN Sezione di Pavia 40/45

41 Fluka options Lateral profile for 200 MeV protons at z=25 cm energy (a.u.) -3 default precision optimize fano off mlsh off born 1st born 2nd size 1st size 2nd off z (cm) Andrea Fontana INFN Sezione di Pavia 41/45

42 Tools for analysis USRBIN readout and plot with ROOT macro plotbrg.c : to plot 2D USRBIN ASCII output and project longitudinal profile. macro plotmcs.c : to plot 2D USRBIN ASCII output and project trasversal profile. USRBIN dynamic scan macro scan.c : to quickly inspect 2D USRBIN ASCII output and perform fits. Other files Fluka input brg.inp and brgcsda.inp : for Bragg peak and range studies. Fluka input mcs.inp : for Multiple Coulomb scattering studies. Andrea Fontana INFN Sezione di Pavia 42/45

43 References [1] J. R. Sabin et al., Advances in Quantum Chemistry, Vol. 65, 2013 [2] H. Øverås, CERN Yellow Report No , [3] B. Gottschalk, Med. Phys. 37(20)352 [4] R. Fruhwirth et al., NIM A 456(2000)369 [5] V.L. Highland, NIM 129(1975)497 [6] G.B. Arfken, Mathematical Methods for Physicists, Academic Press 2005 [7] W. Ulmer, Rad. Phys. and Chem. 76(2007)89 [8] W. Ulmer et al., Rad. Phys. and Chem. 80(2011)378 Andrea Fontana INFN Sezione di Pavia 43/45

44 Questions? Contact:

45 Acknowledgements Fluka team: A. Ferrari, P. Sala, A. Mairani et al. Pavia Fluka group: A. Rotondi, G. Magro, M. Mori, E. Bellinzona and A. Embriaco