Assessment of RBED electron-impact ionization cross sections for - PowerPoint PPT Presentation

October 16, 2017 1 Assessment of RBED electron-impact ionization cross sections for Monte Carlo electron transport Judy Wang 1 , Jan Seuntjens 1 , José M Fernández-Varea 2 , 1 1 McGill University 2 Universitat de Barcelona

• Improving cross sections for MC simulation of electron transport • Accurate track structures in microdosimetry and other applications in medical physics • Need differential and total (integrated) cross sections (DCSs and TCSs) • Present work: ionization of atomic inner shells by electron impact 2 Motivation

• Empirical, semi-empirical, and ab initio (first principles) calculations • Current gold standard is DWBA (Bote and Salvat, 2008): • Projectile wavefunctions distorted by target, not plane waves • Thoroughly validated against experiment (Llovet et al , 2014) • TCS data tabulated in NIST (all atoms; K, L, M shells) • DCS data not tabulated • Much more computationally expensive than PWBA • Focus here on semi-empirical RBED model (Kim and Rudd, 2000) • Yields both DCS and ICS and is very simple Purpose: compare RBED with DWBA and assess limitations of the model 3 Ionization of atoms by electron impact • Valid from low E ( ∼ 50 eV) to relativistic regime

4 1 d f 0 • Binding ( B ) & average kinetic ( U ) energies of the N target electrons • Kinetic energy of the projectile ( T ) Required input: t t ln d w d f 1 Model which combines Møller cross section with Bethe equation 1 N 1 d w RBED 1 Relativistic binary-encounter-dipole (RBED) model ( d σ ) { ( N i / N ) − 2 ( ) 4 π a 2 0 α 4 N 1 + 2 t ′ = w + 1 + ( β 2 t + β 2 u + β 2 b ) 2 b ′ t + 1 ( 1 + t ′ / 2 ) 2 t − w ) [ ] ( b ′ 2 + ( w + 1 ) 2 + ( t − w ) 2 + 2 − N i ( 1 + t ′ / 2 ) 2 [ ]} β 2 t − ln ( 2 b ′ ) + − β 2 N ( w + 1 ) 1 − β 2 • Optical oscillator strength (OOS), d f ( w ) / d w W is the kinetic energy of outgoing electron, w = W / B ∫ ∞ N i is the effective number of electrons in the shell, N i = d w d w

5 d f • If nothing is known about OOS, use the empirical function r d r d w 0 RBEB 2 j N 1. RBEB model: • This choice yields analytical DCS and TCS, hence the popularity of RBEB 2. Hydrogenic OOS: • Fully analytical, non relativistic 2 m e d W 3. Numerical ( ab initio ) OOS: Optical Oscillator Strength models ( d f ) = ( w + 1 ) 2 • Obtained by setting Q = 0 in GOS expressions • Can be applied to any Z by using Z eff according to Slater’s rules ⟨ � � � C ( 1 ) � � 2 j ′ ⟩ 2 ∑ � � � � ℓ ′ 1 � = 3 ℏ 2 ( B + W ) N W ℓ 1 � � κ ′ {∫ ∞ } 2 [ ] P W κ ′ ( r ) P n κ ( r ) + Q W κ ′ ( r ) Q n κ ( r ) ×

• Self-consistent DHFS potential used to calculate numerical OOS • Calculations done for Z spanning periodic table, and K, L, M (sub)shells • Results shown here: OOSs, DCSs and TCSs • Emphasis: comparison of RBED (using the three OOS models) with DWBA 6 Overview of results • Inner shell electrons: B ≳ 200 eV

7 Results: OOSs 10 1 10 2 Ar K Ar L1 10 2 10 3 df/dW [a.u.] df/dW [a.u.] 10 3 10 4 4 10 10 5 DHFS 10 5 BEB 10 6 hydrogenic 10 3 10 2 W [a.u.] W [a.u.] Kr K Kr L3 10 3 10 2 10 4 10 3 df/dW [a.u.] df/dW [a.u.] 10 5 10 4 10 6 10 5 10 7 10 6 10 3 10 4 10 2 10 3 W [a.u.] W [a.u.]

8 (a) Neon 1s, T = 3B (c) Neon 1s, T = 10B Results: DCSs 3.5 2.5 3.0 2.0 2.5 dσ RBED /dσ Moller dσ RBED /dσ Moller 2.0 1.5 1.5 1.0 1.0 RBED DHFS 0.5 0.5 RBED hydrogenic RBEB 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 w w (b) Argon 2p 3 / 2 , T = 3B 4.5 7 4.0 6 3.5 5 3.0 dσ RBED /dσ Moller dσ RBED /dσ Moller 4 2.5 2.0 3 1.5 2 1.0 1 0.5 0.0 0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 w w (d) Argon 2p 3 / 2 , T = 10B

9 Agreement is good for the K-shell of low- Z elements Results: TCSs, K shell Ar K RBED 3.0 RBED H RBEB 2.5 DWBA 2.0 [kb] 1.5 1.0 0.5 0.0 10 4 10 5 10 6 10 7 10 8 10 9 T [eV]

10 But for high Z the relativistic asymptotic behaviour is wrong! Results: TCSs, K shell 0.035 RBED Rn K RBED H 0.030 RBEB DWBA 0.025 0.020 [kb] 0.015 0.010 0.005 0.000 10 5 10 6 10 7 10 8 10 9 T [eV]

11 Results: TCSs, L and M subshells Xe L1 Rn M1 1.0 1.4 1.2 0.8 1.0 0.6 [kb] [kb] 0.8 0.6 0.4 RBED 0.4 RBED H 0.2 RBEB 0.2 DWBA 0.0 0.0 10 4 10 5 10 6 10 7 10 8 10 9 10 4 10 5 10 6 10 7 10 8 10 9 T [eV] T [eV] Kr L3 Xe M5 300 25 250 20 200 [kb] [kb] 15 150 10 100 5 50 0 0 10 4 10 5 10 6 10 7 10 8 10 9 10 3 10 4 10 5 10 6 10 7 10 8 10 9 T [eV] T [eV]

12 t N t t 1 ln b i Relativistic Bethe equation for ionization in the high-energy limit: long t trans RBED high-energy asymptotic limit: 1 t b i ln Asymptotic mismatch [ ] − ln 2 b ′ σ Bethe = 4 π a 2 0 α 4 N − β 2 β 2 1 − β 2 c i β 2 t 2 b ′ � �� � � �� � b i and c i are parameters determined from Fano plots { [ ] ( )} t − ln 2 b ′ 4 π a 2 0 α 4 N σ RBED = − β 2 + 2 − N i ( β 2 t + β 2 u + β 2 b ) 2 b ′ 1 − β 2 β 2 Prefactors are different = ⇒ RBED cannot reproduce the Bethe limit!

13 We can restore the PWBA prefactor to the distant (longitudinal and transverse) part of RBED Highlights limitations of combining two disparate models semi-empirically Can recover correct asymptotic limit, but intermediate region is worse Results: asymptotic behaviour DWBA Au K 0.035 RBED RBED_Bethe 0.030 RBED_Bethe_trans 0.025 RBED_Bethe_long Bethe [kb] 0.020 0.015 0.010 0.005 0.000 10 5 10 6 10 7 10 8 10 9 T [eV]

José would like to acknowledge enlightening discussions with Prof. Francesc Salvat Partial support by the CREATE Medical Physics Research Training Network grant of the Natural Sciences and Engineering Research Council (Grant number: 432290), along with the Fonds de Recherche du Québec - Nature et technologies (FRQNT). Spanish Ministerio de Economía y Competitividad (grant FIS2014-58849-P) 14 Acknowledgements