In both the existing (Opt2 and Opt4) and new (Opt4Rel) electron models of Geant4-DNA, the DRF is used to determine the energy-loss function (ELF) of liquid water through the standard expression for bulk media [34, 35]: ELF ≡ Im [ − 1 ε ( E , q ) ] = ε 2 ( E , q ) | ε ( E , q ) | 2 (1) where E and q are the energy- and momentum-transfer, respectively, and ε ( E , q ) = ε 1 ( E , q ) + i   ε 2 ( E , q ) is the complex-valued DRF with the real and imaginary part being related through the Kramers-Kronig relations. The numerator in Eq. 1 describes the absorption spectrum of the medium and it is partitioned to the individual excitation levels (k) and ionization shells (n) as: ε 2 ( E , q ) = ∑ n i o n i z . ε 2 ( n ) ( E , q ) + ∑ k e x c i t . ε 2 ( k ) ( E , q ) (2)

In both the existing (Opt2 and Opt4) and new (Opt4Rel) models, five excitation levels (A¹B₁, B¹A₁, Ryd A + B, Ryd C + D, db) and four outer shells (1b1, 3a1, 1b2, 2a1) are used and they are analytically represented by a sum of Drude-type functions. The details of the parameterization, however, differ among the models. Specifically, in Opt2 the imaginary part of DRF at the optical limit (q = 0) is represented by: ε 2 ( E , q = 0 ) = ∑ n = 1 4 [ D n ( E ; E n )   Θ ( E − B n ) ] + ∑ k = 1 5 [ D k ∗ ( E ; E k )   Θ ( E − B k ) ] (3) where D n ( E ; E n ) and D k ∗ ( E ; E k ) are the ordinary and derivative Drude functions, E n , k are the Drude coefficients for the transition energies, B n , k are the threshold energies, i.e., shell binding energies and band gap excitation energies, and Θ(…) is the Heaviside function. The step-functions Θ ( E − B k ) eliminate the non-physical contribution of the Drude functions below B n , k . As discussed elsewhere [31], an important shortcoming of Eq. 3 is the partial violation of sum rules and the non-analyticity of ε 1 ( E , 0 ) . To overcome these shortcomings, both Opt4 and Opt4Rel make use of the Emfietzoglou-Kyriakou partitioning algorithm by replacing Eq. 3 with: ε 2 ( E , q = 0 ) = ∑ n = 1 4 { [ D ( E ; E n ) − D ( E ; B n ) exp ( B n − E ) + F n ( E ) ]   Θ ( E − B n ) } + ∑ k = 1 5 { [ D k ∗ ( E ; E k ) + F k ( E ) ]   Θ ( E − B k ) } (4) with D ( E ; B n ) exp ( B n − E ) being an exponential smoothing function for ionizations, and F n , k ( E ) are contributions from higher energy-levels, as determined by the Emfietzoglou-Kyriakou algorithms. The Drude coefficients in both Eqs 3, 4 are obtained through a fit to the experimental optical data [36]. Then, from Eqs 2–4 the ELF of Eq. 1 can be partitioned to the individual excitation levels and ionization shells according to the expression: ELF = EL F excit . + EL F ioniz . = ∑ n i o n i z . ε 2 ( n ) ( E , q ) | ε ( E , q ) | 2 + ∑ k e x c i t . ε 2 ( k ) ( E , q ) | ε ( E , q ) | 2 (5)

It is noteworthy that for each excitation level (k) or ionization shell (n) contribution to the ELF, the denominator of Eq. 5 represents the screening of the entire (outershell) electronic sub-system, i.e., represents the contribution of all k and n. The real part of the DRF, ε 1 ( E , q ) , that enters in the denominator of Eq. 5, is formally obtained by the Kramers-Kronig (KK) relation [35]. In the case of Opt2 (Eq. 3), the KK pair of ε 2 ( E , 0 ) is non-analytic. In contrast, in Opt4 (and Opt4Rel) the real part of the DRF at the optical limit (q = 0) may still be obtained analytically from: ε 1 ( E , q = 0 ) = 1 + ∑ n i o n i z . D n K K ( E ; E n ) + ∑ k e x c i t . D k K K ( E ; E k ) (6) where D n K K ( E ; E n ) and D k K K ( E ; E k ) represent the KK pairs of D n ( E ; E n ) and D k ∗ ( E ; E k ) , respectively [33].

To obtain the ELF for arbitrary values of q, analytic dispersion relations are introduced into the Drude coefficients [33]. These dispersion relations ensure that the ELF has the proper limiting behavior at q = 0 and q = ∞. The dispersion relations used in Geant4-DNA are common to all models (Opt2, Opt4, Opt4Rel).

An important aspect of the dielectric approach is that the fitting of ELF to the experimental data can be tested for internal consistency. In Geant4-DNA models, the following sum-rule constraints have been considered:

f sum-rule: 2 π E p 2 ∫ 0 ∞ E Im [ ε ( E , 0 ) ] d E = 1 (7)

ELF sum-rule: 2 π E p 2 ∫ 0 ∞ E Im [ − 1 / ε ( E , 0 ) ] d E = 1 (8)

KK sum-rule: 2 π ∫ 0 ∞ 1 E Im [ − 1 / ε ( E , 0 ) d E + Re [ 1 / ε ( 0,0 ) ] ] = 1 (9) where E p =21.4 eV for liquid water. In the new Opt4Rel model, the contribution of the 2a1 shell to the ELF is modified in order to better fulfill the sum rules of Eqs 7–9. This is possible since the 2a1 shell with binding energy at ∼32 eV is not within the range of the particular experimental data set used [36], so the Drude coefficients of that shell are at our disposal to improve the fulfillment of the sum rules. This adjustment was deemed necessary to also bring Eq. 4 in better agreement with a more recent experimental data set [37] that extends well beyond the 2a1 energy threshold (not shown here).

In all dielectric models of Geant4-DNA (Opt2, Opt4, Opt4Rel), the ELF is used to describe the excitation and ionization of the outer-shell electrons (8 in number) of the water molecule which have condensed-phase properties, while the ionization of the innermost (K-shell) electrons is described by the BEAX model. Although the latter is of an atomic origin and disregards aggregation effects, it is generally considered a reasonable approximation for inner-shell electrons which are minimally perturbed by the phase of the medium.

In the framework of the PWBA, ELF is the only non-trivial material input to calculate inelastic cross sections due to the proportionality relation: d 2 σ P W B A d E d Q ∝ Im [ − 1 ε ( E , Q ) ] (10) where Q is the free-recoil energy, Q ( q ) = q 2 / 2 m , with m the electron rest mass. By integrating Eq. 10 over Q and E one obtains the differential and total inelastic cross section, respectively. Importantly, by entering Eq. 5 into Eq. 10, one may calculate differential and total inelastic cross sections specific to each excitation level and ionization shell. For example, the singly differential ionization cross section (DICS) which will be presented below is calculated from: d σ P W B A d E = 1 π α 0 N T ∑ n i o n i z . ∫ Q n , − Q n , + ε 2 ( n ) ( W + B n , Q ) | ε ( E , Q ) | 2 d Q Q (11) where α₀ is the Bohr radius, N is the density of water molecules (=3.343 10²² molecules/cm³ in unit density water), T is the non-relativistic incident electron energy and W = E − B n is the kinetic energy of the secondary electron. The limits of integration in Eq. 11 are based on energy-momentum conservation: Q n , ± = [ T 1 / 2 ± ( T − W − B n ) 1 / 2 ]   2 (12)

To calculate inelastic cross sections via Eq. 11, it is necessary to have defined the ELF for arbitrary values of q, i.e., within the limits imposed by Eq. 12. For ionizations, the non-relativistic quadratic (in q) dispersion relation is used and implemented within the Drude functions: E n ( q ) = E n + Q ( q ) (13)

In the framework of the relativistic PWBA (RPWBA), the DICS is obtained as a sum of two terms: d σ RPWBA d E = d σ L d E + d σ T d E (14) where the subscripts “L” and “T” denote the longitudinal and transverse terms, respectively [38]. The longitudinal DICS reads: d σ L d E = 1 π α 0 N m c 2 β 2 ∑ n i o n i z . ∫ Q n , − Q n , + [ ε 2 ( n ) ( W + B n , Q ) | ε ( E , Q ) | 2 ] 1 + Q / m c 2 Q ( 1 + Q / 2 m c 2 ) d Q (15) where and β = υ/c with υ the incident electron velocity and c the speed of light, and the relativistic form of the free-recoil energy is Q ( Q + 2 m c 2 ) = ( c q ) 2 . The limits of integration in Eq. 14 are: Q ± , n = { [ T ( T + 2 m c 2 ) ± ( T − W − B n ) ( T − W − B n + 2 m c 2 ) ] 2 + ( m c 2 ) 2 } 1 / 2 − m c 2 (16)

In Opt4Rel, the following relativistic dispersion relation is used: E n ( q ) = E n + ( m c 2 ) 1 / 2 ( q 2 / m + m c 2 ) 1 / 2 − m c 2 (17)

The transverse DICS term is calculated from: d σ T d E = 1 π α 0 N m c 2 β 2 [ ∑ n i o n i z . ε 2 ( n ) ( W + B n , 0 ) | ε ( E , 0 ) | 2 ]   [ ln ( 1 1 − β 2 ) − β 2 ] (18)

Note that only collisions with zero momentum transfer (q = 0 or Q = 0) contribute to Eq. 18 according to the small-angle approximation [38].

The low-energy corrections implemented in Opt2 and Opt4 are of two kinds, namely, corrections for exchange (EX) in electron-electron interactions and Coulomb (CB) corrections to account for deviations from first-order perturbation theory [39, 40]. As discussed elsewhere, the existing version of Opt4 includes a more consistent implementation of the low-energy (EX and CB) corrections terms [31, 33]. These improvements are also passed onto Opt4Rel. However, all low-energy (EX and CB) correction terms had to be re-evaluated in Opt4Rel using the new Drude coefficients that enter into the ELF.

The application of low-energy and relativistic corrections implemented in Opt4Rel increase the complexity of the calculations. Therefore, corrections are applied only within the energy regime that have a sizeable effect. In the present work, corrections to PWBA are applied only if they change the total electronic stopping power by at least 0.5–1%. Based on this criterion we distinguish between the following three regimes: 1) in the energy range 0.01–1 keV, we use the PWBA expression, Eq. 8, together with EX and CB corrections; 2) in the energy range 1–100 keV, we use the RPWBA expression, Eq. 10 (but setting the transverse term equal to zero), together with EX and CB corrections; and 3) in the energy range 100–1,000 keV, we use the complete RPWBA expression, Eq. 10.