Electronic stopping cross section for protons incident on biological and biomedical materials within a FSGO quantum chemistry description

Radiation Physics and Chemistry 156 (2019) 150–158

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Radiation Physics and Chemistry journal homepage: www.elsevier.com/locate/radphyschem

Electronic stopping cross section for protons incident on biological and biomedical materials within a FSGO quantum chemistry description

T

⁎

L.N. Trujillo-López , R. Cabrera-Trujillo Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México, Ap. Postal 43-8, Cuernavaca, Morelos 62251, Mexico

A R T I C LE I N FO

A B S T R A C T

Keywords: Stopping cross section FSGO Biomolecules Mean excitation energy Tissue equivalent materials

Radiotherapy and dosimetry are techniques used for the treatment of cancer cells and require the proper description of the energy deposition of swift heavy ions when penetrating a biological material. In this work, we report the electronic stopping cross section, by proton radiation, of several complex organic molecules of biological interest that contain, in particular, F and Ar atoms. Our work implements the Harmonic Oscillator approach for the bound electrons through a Floating Spherical Gaussian Orbital description of the molecule in the intermediate to high collision energy within the first Born approximation. In particular, we study the stopping cross section for the following molecules: DNA, guanine, adenine, alanine, glycine, trimethylamine, tissue equivalent gas based on methane and propane, A-150 plastic equivalent material, tissue equivalent liquid, nylon, and air. For complex compounds, where the structure is given in percent by weight (or volume) of molecular fragments, we use a Bragg's-like rule within our model, to determine the electronic stopping cross section. We calculate the orbital and total mean excitation energies for these molecules, finding that it is the orbital mean excitation energy the principal parameter that characterizes the stopping cross section instead of the total molecular mean excitation energy. Our results are in good to excellent agreement to available experimental and theoretical data reported in the literature for these molecules and have an even better agreement than semiempirical results from approaches like SRIM and PSTAR with the advantage of being analytically simple to implement as well as having a predicting capability for its region of validity.

1. Introduction The energy loss of ions in matter is of importance to understand the interaction between the incident beam ions and the target material. The study of the energy loss process of swift heavy ions has been a research topic since the dawn of quantum mechanics in the last century. The pioneering works of Bohr (1987) and Bethe (1930) opened a door for the investigation of fundamental physics and later, the applications in many fields, e.g., industry (Venturi, 2017), medicine (Sabin and Br, 2007; Besemer et al., 2013), and material science (Limandri et al., 2014). The energy loss, dE , for an ion with initial velocity v is related to the stopping cross-section, Se , by the density of scattering centers, n, that penetrate a target of thickness dx , i.e.,

Se (v ) =

1 dE . n dx

(1)

In the field of medicine the applications span to areas like dosimetry, radiation therapy, nanodosimetry, and radiation protection among others (Sabin and Br, 2007). In particular, for radiotherapy, it is

⁎

very important to have accurate methods to quantify the dose of radiation in cancer treatments (Besemer et al., 2013). To do so, the damage needs to be achieved at a molecular level, i.e., to induce DNA strand breaks and the killing of cancer cells. During the projectile-target collision several processes occur such as excitation, ionization, charge transfer, nuclei displacement, molecular fragmentation, electron cascades, among others, so a complete description of the interaction is an open problem at the forefront of atomic and molecular physics. For large molecules, e.g., DNA or complex like tissue-equivalent materials (TEMs), the problem is even more complicated, as the molecules are surrounded by water and immersed in a cell. Furthermore, TEMs must closely match the volume, density, and chemical composition characteristic of the represented tissue for a proper response at the energy of interest. Although there are many studies of the stopping power for protons in organic compounds like amines, amino acids or nucleobases (Thwaites, 1985; Sauer et al., 1995; Cabrera-Trujillo et al., 1994; Akkerman et al., 2001; Tan et al., 2008, 2010), there are few works for large molecules like DNA or TEMs (Quinto et al., 2017; Tan et al., 2006; Abril and Garc, 2011; Fukuda, 1980; Usta and Tufan, 2017; LaVerne

Corresponding author. E-mail address: [email protected] (L.N. Trujillo-López).

https://doi.org/10.1016/j.radphyschem.2018.10.013 Received 5 July 2018; Received in revised form 18 October 2018; Accepted 19 October 2018 Available online 26 October 2018 0969-806X/ © 2018 Elsevier Ltd. All rights reserved.

Radiation Physics and Chemistry 156 (2019) 150–158

L.N. Trujillo-López, R. Cabrera-Trujillo

assumption of being harmonically bound with an angular frequency ω0i , the target's electronic spectrum is replaced by that of a HO. In this case, the ground state wave-function is given by

and Pimblott, 1995), which are needed in radiation therapy. In Bethe's model, the energy deposition is described through a quantum mechanical approach that predicts a dependence for Se as

ln

( )/v , where I v2 I0

2

0

is the mean excitation energy of the target and

2 2

α 3/4 αi r Ψ0 (r) = 〈r|0〉 = ⎛ i ⎞ e− 2 ⎝π⎠

depends on the dipole oscillator strength (DOS) (Inokuti, 1971). This model is valid for high collision energies where the projectile moves faster than the target electrons. The mean excitation energy parameter carries all the information of the target electronic structure and characterizes how the target absorbs energy from the projectile through excitations and ionization. Consequently, in a quantum mechanical description, the target electronic structure is essential. However, the full determination of the DOS for any projectile-target spectra has not been possible. In this regard, several quantum chemical approaches have been employed in the determination of the mean excitation energy from ab initio theories like Hartree-Fock (Oddershede and Sabin, 1984), or through less sophisticated methods as the local plasma approximation (Cabrera-Trujillo et al., 1994; Meltzer et al., 1990). Most of quantum chemical studies use a Gaussian basis set for the description of the electronic structure. The advantage of the use of Gaussians is that the one- and two-electron integrals are easy to evaluate (Szabo and Ostlund, 1996; Huzinaga, 1965; Boys, 1950). Furthermore, a single Gaussian function is the ground-state wave-function of the quantum Harmonic Oscillator (HO), pondering the question if it is possible to replace the target electronic spectra for an equivalent one of the HO. The answer is yes, as we have already reported (Trujillo-L et al., 2013). The advantages of this approach are an analytical expression for the stopping cross section and that the simplest representation of the target, based on Floating Spherical Gaussian Orbitals (FSGO) (Frost, 1967a), provides a very good description of the stopping cross section (TrujilloL et al., 2013; Cabrera-Trujillo et al., 2016). The FSGO model was proposed by Frost (1967a) in the late 60's. Although it is a very simple model, it has proven to be a powerful tool in the calculation of geometrical properties of molecular structures such as bonds, lengths, and angles, with low computational time (Maggiora et al., 1991). In this work we study the electronic stopping cross section of biomolecules and complex compounds of biological and biomedical importance by implementing the FSGO electronic structure basis set within a simple analytical expression derived from the HO approach. The lay out of this work is as follows. In Section 2, we provide a summary of the theoretical approach by connecting Bethe's stopping power theory to the HO model and to the FSGO molecular structure approach. In Section 3, we present our results for the orbital and total mean excitation energy as well as for the stopping cross sections. We compare the results of our approach with experimental and theoretical data available in the literature. Finally, in Section 4, we give our conclusions and future perspectives of this work.

where = me ω0i / ℏ . One consequence of the HO model is that the DOS takes the value f0i = δi,1 and, consequently, the orbital mean excitation energy is related to the orbital electron angular frequency as

connecting directly the HO angular frequency ω0i and the orbital mean excitation energy, I0i . The GOSs have already been determined for the HO and an analytical expression for the electronic stopping cross section, given in terms of the projectile and target properties, has been found (Cabrera-Trujillo et al., 2016). The expression for the orbital stopping cross section is

ln(1 + 16ϵ 2) I0i Se, i = 2 4 2 πe Zp* ⎛ + ϵ⎞ ⎝π ⎠

N2 ln I0 =

∞ n=1

qmax , i (n)

min, i (n)

Fni 0i (q)

dq q

. Here Zp* is the pro-

(7)

2.2. FSGO molecular target description Consider an electronic system with an even number of electrons 2n , distributed in n localized orbitals represented by the function Ψi , with i = 1, 2, …, n , which are not necessary orthogonal. In this representation, each orbital describes the inner-shell (core), bond, and lone pair (LP) orbitals. Each one of them is represented by a FSGO function given by 2

(r − R i) 3/4 ⎛ ⎞ − 2 2 Ψi ⎜r − R i⎟ = ⎛⎜ 2 ⎞⎟ e ρi , ⎜ ⎟ ⎝ πρi ⎠ ⎝ ⎠

(8)

where ρi is the orbital radius and R i is the position of the Gaussian center. The set of molecular parameters {ρi , R i} is obtained by variationally minimizing the energy. The FSGO approach can predict the electronic structure and geometry of a molecular system reasonably well (Frost, 1967a, 1967b, 1967c; Frost and Rouse, 1968; Rouse and Frost, 1969; Chu and Frost, 1971a). Since ρi represents the radius of a sphere containing about 74% of the electronic charge density, it permits us to ’‘visualize” the molecular orbitals by spheres. In Fig. 1, we show a FSGO visual representation of the orbital decomposition for DNA (C20H27N7O13P2), which consists of an adenine-thymine pair plus two phosphate groups and two sugars (DNA-AT). With 69 atoms and 330 electrons, it is a large and complex biomolecule. The decomposition in terms of FSGO orbitals for this molecule is explained in more detail below. As observed, the fragmentation into FSGO orbitals is very simple and easy. The similarities between Eqs. (4) and (8), allow us to find a relation between the orbital mean excitation energy, the electron angular frequency, and the radius of the FSGO orbital as

(2)

∑ ∫q

Mp I0i

where ni is the occupation number of the i-th orbital, such that N2 = ∑i ni is the total number of electrons in the target.

Here

4πe 4Z12 me v 2

∑ ni ln I0i, i=1

N2

Se, i (v ) =

me Ep

jectile effective charge and Mp is the projectile mass. In this work, we report results for protons, such that Zp* = 1, i.e., the projectile charge is frozen during the collision. With this assumption, our results are valid for collision energies above the maximum of the stopping cross section. Within this approach, the mean excitation energy follows a Bragg'slike rule,

In the independent particle model (Cabrera-Trujillo, 1999), the electronic stopping cross section, within the first Born approximation, is written as

i=1

(6)

where ϵ is the reduced energy given by ϵ =

2.1. Stopping cross section

∑ Se,i (v ).

(5)

I0i = ℏω0i ,

2. Theoretical approach

Se (v ) =

(4)

αi2

(3)

is the contribution of the i-th electron, q goes from a minimum qmin to a maximum qmax momentum transfer, and Fni 0i are the generalized oscillator strengths (GOSs). When a target's electron is described under the 151

Radiation Physics and Chemistry 156 (2019) 150–158

L.N. Trujillo-López, R. Cabrera-Trujillo

Table 1 Values for ρi associated to F, Cl, and Ar atoms used in this work. For each atom, we present the inner-shell orbitals (subindex indicates the K, L, and M shells), bond (- indicates a single bond) and lone pair (LP). Here ρi is in atomic units. Averaged values are giving within “〈 〉”. Exact values are giving within “[ ]” as reported in a)- (Talaty et al., 1976), b)- (Simons and Talaty, 1978), and c)- (Chu and Frost, 1971b). Suggested values are indicated with “( ) ”.

Fig. 1. Visual representation of the chemical structure of a DNA-AT nucleotide (C20H27N7O13P2) in terms of Floating Spherical Gaussian Orbitals.

I0i = ℏω0i =

2ℏ2 . me ρi2

The clear distinction between the three different types of orbitals in the FSGO model allows to subdivide the molecular stopping cross section, according to Eq. (2), into contributions from core, bond, and lonepair electrons, i.e., (10)

By evaluating the contribution of each orbital and summing over the molecule fragments, we obtain the total contribution to Se . For complex compounds, e.g., TEMs, its composition are given in terms of fractions by weight (or volume) of molecular fragments, and one needs to take into account these fractions when calculating Se . Experimentally, there is strong evidence that supports the idea of molecular fragments in the analysis of the stopping cross section (Kreutz et al., 1980; Thwaites, 1992). The data suggests that the molecular stopping cross section can be expressed as a contribution of characteristic molecular groups. For a molecule, the Se per atom, S¯e , is given as

S (molecule) S¯e = Se (10−15eV cm2 / atom ) = e . Nt

(11)

where Nt is the number of atoms that constitutes the molecule. Consequently, for a compound given in terms of a mixture, each fragment contributes with a certain fraction such that the Se per atom is given by Thwaites (1992) Frag

S¯e =

∑j

Orb

f j (∑i

Se, i ) j

∑j f j × Nt j

ρi

Orbital

ρi

Orbital

ρi

FK

〈0.2136〉

ClK

〈0.1075〉

Ar aK

[0.1012]

F-H

ClL

〈0.4210〉

Ar aL

[0.3891]

F-C

〈1.1040〉 (1.7541)

Cl-H

〈1.6632〉

Ar aM

[1.5250]

F-Fc LP(F)

[1.4010] 〈1.0927〉

Cl-Cb LP(Cl)

[1.7541] 〈1.6813〉

lone pairs, and bond orbitals with hydrogen for F and Cl, we take an average of the values for ρi from Frost (1967b, 1967c), Chu and Frost (1971b) and Blustin (1977). However, the F-C bond orbital has not been reported before within the FSGO model. For the missing F-C orbital parameter, we apply the fragment theory to make a replacement. The fragment theory was developed by Christoffersen and Maggiora (1969), Maggiora and Christoffersen (1976) for the calculation of other chemical properties. We have applied this theory previously (Trujillo-L et al., 2013) for the P-O and P=O substitution by C-O and C˭O orbitals, respectively, with great success. Simons and Talaty (1978) reported the values of ρi for the Cl-C bond orbital. As F and Cl have a similar valence orbital configuration, s 2p5 , in the spirit of the model, we substitute the ρi of the F-C bond orbital with that of the Cl-C. In Table 1, we provide the values of ρi to generate the Se , for any molecule involving these atoms. Averaged values are presented by “〈 〉”; exact values are presented with ”[ ]“; and “( ) ” indicates the values of ρi replaced in this work. By using the values of ρi from Table 1 to determinate I0i and using Eq. (7), we obtain the total mean excitation energy, I0 , for the molecules studied in this work. These molecular I0 values are given in Table 2. In the same table, we compare to other available values found in the literature (Tan et al., 2006; Abril and Garc, 2011; Berger and Coursey, 2013; Sauer et al., 2011; Bruun-Ghalbia et al., 2010). Note that I0 is given in terms of the mean values for the orbital mean excitation energy, Eq. (7), and in terms of the mean squared radii as stated in Eq. (9), instead of the excitation spectra of the molecule. Thus, I0 depends on ground state properties which make it smaller than the ones reported by other approaches. This behavior is also present in theoretical approaches dependent on the system density, e.g. the local plasma

(9)

Se (molecule) = Se (core) + Se (bond) + Se (LP).

Orbital

Table 2 Total mean excitation energy of the biological molecules studied in this work, according to our approach. Also shown are the corresponding theoretical calculations from Refs: a)- Tan et al. (2008); b)- Tan et al. (2006); c)- Abril and Garc (2011); d)- LaVerne and Pimblott (1995); e)- Berger and Coursey (2013); f)- Sauer et al. (2012); g)- Akkerman and Akkerman (1999); h)- Bruun-Ghalbia et al. (2010); i)- Ishiwari et al. (1990); j)- Kamakura et al. (2006); All energies are given in eV.

(12) Frag

where f j is the fraction of the j-th fragment with ∑ j f j = 1. Thus, the FSGO provides a self-consistent procedure to determine Se from the molecular configuration of the target together with Eqs. (6) and (10). Here is where the FSGO approach proves very useful. 3. Results 3.1. Orbital and total mean excitation energies Values of ρi were previously compiled for some atoms by Trujillo-L et al. (2013). In this work, we report values of ρi for F, Cl, and Ar atoms as required for this study. The parameters ρi depend on the molecular environment. Nevertheless, the change of these parameters for the same type of orbitals (core, bond, and lone pair) in different molecules is less than 5%, so we propose to use mean average values (Trujillo-L et al., 2013). Values of αi , where αi = 1/ ρi , for inner-shell orbitals of argon are reported by Talaty et al. (1976) and from Simons and Talaty (1978) for the Cl-C bond orbital. The values of ρi for the bond orbital F-F are reported by Chu and Frost (1971b). In the case of values for the core,

Molecule

I0

Others I0

Adenine Guanine DNA (AT) Alanine Glycine† TEGM TEGP TEL Water

56.23 58.37 62.68 56.12 58.88 50.03 55.43 60.05 61.31

71.40e 74.10e

A-150(Mix 1) Nylon 6 Air(Near sea level) †

152

73.90b 75.00b 86.54b 67.50g 71.20g

69.06f 71.58f 81.50c 70.40a 74.00a

71.60e

73. 20 h

72. 0 j

45.50

65.10e

64. 70i

50.46 45.64

63.90e 85.70e

71.90e 61.20e 59.50e

Canonical form of glycine Sauer et al. (2006).

79.00g 77.90d

Radiation Physics and Chemistry 156 (2019) 150–158

L.N. Trujillo-López, R. Cabrera-Trujillo

Table 3 FSGO orbitals, its ρi , and the number of them required to obtain the chemical composition used for the following molecules: A) DNA-AT (C20H27N7O13P2), B) DNA-CG (C19H26N8O13P2), C) Guanine (C5H5N5O), D) Adenine (C5H5N5), E) Trimethylamine (TMA) (C3H9N), F) Alanine (C3H7NO2), G) Glycine (C2H5NO2), H) Methane (CH4), I) Propane (C3H8), J) Carbon dioxide (CO2), K) Nitrogen (N2), L) Ethylene (C2H4), M) Oxygen (O2), N) Propadiene (C3H4), O) 1,3 butadiene (C4H6), P) Acetylene (C2H2), Q) Carbon monoxide (CO), R) Carbon Tetrafluoride (CF4), S) Glycerol (C3H8O3), T) Urea (CH4N2O), U) Water (H2O), and V) Nylon 6 (C6H11NO). Values of ρi are given in atomic units. Orbital

ρi

A

B

C

D

E

F

G

H

I

J

K

L

M

N

O

P

Q

R

S

T

U

V

CK NK OK FK PK PL C-H N-H O-H C-C C˭C C≡C C-N C˭N C-O C˭O C≡O C-F N≡ N O˭O P-O P=O LP(C) LP(N) LP(O) LP(F)

0.3280 0.2770 0.2400 0.2136 0.1230 0.5020 1.6690 1.5540 1.3240 1.6660 1.7950 1.7810 1.5440 1.5440 1.3400 1.3480 1.3480 1.7541 1.4490 1.2690 1.3400 1.3480 1.4890 1.5230 1.3290 1.0927

20 7 13 – 2 8 20 3 4 11 4 – 12 6 8 4 – – – – 6 4 – 7 26 –

19 8 13 – 2 8 17 5 4 11 4 – 12 6 8 4 – – – – 6 4 – 8 26 –

5 5 1 – – – 1 4 – 1 2 – 7 4 – 2 – – – – – – – 5 2 –

5 5 – – – – 2 3 – 1 2 – 6 6 – – – – – – – – – 5 – –

3 1 – – – – 9 – – – – – 3 – – – – – – – – – – 1 – –

3 1 2 – – – 4 2 1 2 – – 1 – 1 2 – – – – – – – 1 4 –

2 1 2 – – – 2 2 1 1 – – 1 – 1 2 – – – – – – – 1 4 –

1 – – – – – 4 – – – – – – – – – – – – – – – – – – –

3 – – – – – 8 – – 2 – – – – – – – – – – – – – – – –

1 – 2 – – – – – – – – – – – – 4 – – – – – – – – 4 –

– 2 – – – – – – – – – – – – – – – – 3 – – – – 2 – –

2 – – – – – 4 – – – 2 – – – – – – – – – – – – – – –

– – 2 – – – – – – – – – – – – – – – – 2 – – – – 4 –

3 – – – – – 4 – – – 4 – – – – – – – – – – – – – – –

4 – – – – – 6 – – 1 4 – – – – – – – – – – – – – – –

2 – – – – – 2 – – – – 3 – – – – – – – – – – – – – –

1 – 1 – – – – – – – – – – – – – 3 – – – – – 1 – 1 –

1 – – 4 – – – – – – – – – – – – – 4 – – – – – – – 12

3 – 3 – – – 5 – 2 3 – – – – 3 – – – – – – – – – 6 –

1 2 1 – – – – 4 – – – – 2 – – 2 – – – – – – – 2 2 –

– – 1 – – – – – 2 – – – – – – – – – – – – – – – 2 –

6 1 1 – – – 10 1 – 5 – – 2 – – 2 – – – – – – – 1 2 –

reported results for solid nucleobases, while our work is for gas phase. Despite this, the agreement is very good at high energies. Note that both nucleobases absorb energy from protons in roughly the same manner which implies that it seems impossible to selectively deposit energy in a specific nucleobase (Trujillo-L et al., 2013; Sauer et al., 2012). Note that the value of Se at the maximum is higher than the ones reported in the literature or in experimental data. This is expected, as our model does not consider charge exchange and thus the projectile has a frozen charge for all the projectile energies. This model is only valid for collision energies above the maximum of Se . DNA: With the nucleobases in a FSGO decomposition, the next logical step is to build a nucleotide, whether it is DNA or RNA. Nucleotides consist on a base pair plus two phosphate groups and two sugars. The most common DNA nucleotides studied in the literature are C20H27N7O13P2 (AT nucleotide) and C19H26N8O13P2 (CG nucleotide) (Quinto et al., 2017; Abril and Garc, 2011; LaVerne and Pimblott, 1995). LaVerne and Pimblott (1995) point out that, the use of either one of those two nucleotides or a combination of them “gives virtually identical results”. In Fig. 2c), we show the molecular electronic stopping cross sections for protons incidents on DNA-AT nucleotide. Similarly to nucleobases, we present the Se for DNA-AT obtained from the sum of the orbital contribution, I0i , (thick line), and from the total mean excitation energy, I0 , (thin line). Since it is the orbital contribution the one that determines the Se , we only consider the orbital contributions further on. In Fig. 2c), we compare with available theoretical data found in the literature. Tan et al. (2006) and Abril and Garc (2011) obtained their results within the dielectric formalism, although Abril et al. use a DNA-AT nucleotide while Tan et al. results are obtained by a combination of DNA-AT and DNA-CG nucleotides. This last combination was recently studied by Quinto et al. (2017), with TILDA program plus a percent of water which we also compare to. Note that for high energies, the agreement with these studies is good. We have to remark that while the molecular properties in these works were obtained through density functionals, Monte Carlo techniques or optical loss functions adjusted to experimental values, our approach takes the

approximation (Meltzer et al., 1990). In order to determine these I0 values, we require to know the chemical composition of the molecules. In Table 3, we provide the chemical FSGO decomposition used in each molecule. It is interesting to note that the values of I0 for all the molecules studied in this work are lower than those reported in the literature. However, the curves of the stopping cross sections have a good to excellent agreement with the theoretical and experimental data available (see below) when the orbital decomposition is used instead of the total I0 . This leads us to emphasize that it is the orbital mean excitation energy, I0i , the principal parameter that determines Se and not the total I0 parameter. In summary, to reproduce our work, one determines the chemical composition from Table 3, use the orbital radius, ρi , from Table 1 as well as the reported ones from Trujillo-L et al. (2013), to determinate the Se, i from Eqs. (6) and (2). This recipe is used to obtain our results shown in the next section. 3.2. Stopping cross sections Nucleobases: Adenine (A), thymine (T), guanine (G), and cytosine (C) are four of the five primary nucleobases that constitute the DNA and RNA nucleic acids, essential to all known forms of life. In Fig. 2, we show the molecular electronic stopping cross sections for protons colliding with guanine [Fig. 2a)] and adenine [Fig. 2b)] nucleobases. For each nucleobase, we present two Se curves: one obtained with the sum of the orbital contribution (I0i , thick line) and other obtained with the total mean excitation energy (I0 , thin line). As we observe, it is the orbital mean excitation that determines the Se of the molecule. We compare to the semi-empirical SRIM (Ziegler, 2013) results, as well as to some available theoretical data obtained within the dielectric formalism from Akkerman et al. (2001) and Tan et al. (2006). Note that below the maximum of Se these theoretical results fall faster while our results have a smooth fall. This is due to the HO lower states contribution to Se (Trujillo-L et al., 2013) that are excited in the low collision energy. We have to point out that Akkerman et al. and Tan et al. 153

Radiation Physics and Chemistry 156 (2019) 150–158

L.N. Trujillo-López, R. Cabrera-Trujillo

Fig. 2. Proton electronic stopping cross section per atom, Se , for a) guanine, b) adenine, and c) DNA-AT. The results obtained by using the orbital decomposition I0i are shown by thick solid line and those obtained by the total I0 are shown by thin line. We compare to the theoretical works of Tan et al. (shortdashed line) (Tan et al., 2006), Abril et al. (long-dash line) (Abril and Garc, 2011), Akkerman et al. (short-long dash line) (Akkerman et al., 2001), Quinto et al. (double-dash line) (Quinto et al., 2017), as well as to SRIM (double dashlong line) (Ziegler, 2013). See text for discussion.

Fig. 3. Proton electronic stopping cross section per atom, Se , for a) trimethylamine (TMA), b) alanine, and c) glycine, as obtained by our approach (thick solid line). For glycine and alanine we compare with Tan et al. (short-dashed line) (Tan et al., 2008). The SRIM results (double dashed-long line) (Ziegler, 2013) are displayed for the three biomolecules for comparison. See text for discussion.

Table 4 Chemical composition for: tissue equivalent gas-methane (TEGM) based, tissue equivalent gas-propane (TEGP) based, tissue equivalent liquid (TEL), A-150 tissue-mixtures, and air (dry, near sea level). For TEGM, TEGP, and air, the composition is given in percent by volume. For A-150 and TEL, the composition is given in percent by weight. All compositions were taken from Refs: a)- White et al., (); b)- Rossi and Zaider (1996); and c)- Awschalom and Attix (1980).

molecular configuration per se without any adjustable parameter. Trimethylamine, alanine, and glycine: Amines are compounds that contain a basic nitrogen atom with a lone pair. Amines are important because they are involved in the creation of amino acids and in nearly every step of RNA, DNA, and protein synthesis (Russell and Snyder, 1968). One of these amines is trimethylamine (TMA), C3H9N, which is an amine that has an ammonia-like odor and it is very volatile. It is found in humans and it is present in digestive processes. When amines are bound to a carboxyl (-COOH) functional group, we obtain amino acids. The most important role of amino acids is the formation of proteins, although they participate in other processes such as neurotransmitter transport and biosynthesis (McDonald and Johnston, 1990). The simplest amino acid is glycine, C2H5NO2, a precursor of proteins containing 40 electrons. The second simplest amino acid is alanine, C3H7NO2, with 48 electrons. All these three biomolecules are of great importance in the study of radiation therapy. Fig. 3 shows the molecular electronic stopping cross sections of protons incidents on TMA [Fig. 3a)], alanine [Fig. 3b)], and glycine [Fig. 3c)] and its comparison to SRIM results (Ziegler, 2013). Note how for TMA the agreement is excellent. We also compare with values from Tan et al. (2008) for alanine and glycine, showing a very good agreement, in spite that the results of Tan et al. are in solid phase. TEM (gas phase): TEMs are one of the most complex biomolecules used in radiation therapy. They were developed originally as substitutes of human tissue or organs, e.g., brain, muscle, skin, lung, or bone to study their properties due to radiation. Examples of TEMs are Nylon, A150, tissue equivalent gas-methane (TEGM) and gas-propane (TEGP) based. The elemental composition of some of these TEMs are reported by NIST (Berger and Coursey, 2013) or ICRU (White et al.,), and they

Tissue Equivalentb

CH4 C3H8 H2O Glycerol Urea Ar CO CO2 N2 O2 C2H4 C3H4 C4H6 C2H2 CF4

A-150 Mixturesc

Aira

TEGM

TEGP

TEL

M1

M2

M3

M4

64.4 – – – – – – 32.5 3.1 – – – – – –

– 55.0 – – – – – 39.6 5.4

– – 65.6 26.8 7.6 – – – – – – – – – –

– – – – – 1.8 10.3 – 3.6 – 44.5 37.8 – – 2.0

– – – – – 1.8 10.3 – 3.6 – 41.3 28.9 12.1 – 2.0

– – – – – 1.8 10.3 – 3.6 – 46.5 32.0 – 3.8 2.0

– – – – – 1.8 10.3 – 3.6 – 43.4 25.3 10.6 3.0 2.0

– – – – – 0.93 – 0.03 78.09 20.95 – – – – –

are given by fraction of weight of chemical elements, usually H, C, N, O, F, P, and Ca atoms. However, to be able to apply the FSGO model and to calculate the Se , we need to put their composition in terms of inner shells, bonds, and lone pair orbitals. In Table 4, we present the chemical composition of TEGM, TEGP (Rossi and Zaider, 1996), and A-150 (gas phase) (Awschalom and Attix, 1980) by providing the percentage 154

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Fig. 5. Proton electronic stopping cross section per atom, Se , for TEL (thick solid line) and water (thin solid line). TEL-Mix1 (thick short-and-long dash line) and TEL-Mix2 (thin short-and-long dash line) indicate the mixtures of TEL proposed by Goodman and Colvett (1977) as obtained from the SRIM code. For watervapor phase, we compare with experimental data available from Mitterschiffthaler and Bauer (1990); Phillips (1953), Reynolds et al. (1953), and Baek et al. (2006). See text for discussion.

Fig. 4. Proton electronic stopping cross section per atom, Se , for (a) TEGM and (b) TEGP. Our results (solid line) are compare to values from SRIM (double dash-long line) (Ziegler, 2013), PSTAR database (long-dash line) (Berger et al., 2005), and Thwaites (filled circle dashed line) (Thwaites, 1985). Experimental values are from Fukuda (open triangle symbols) (Fukuda, 1980). See text for discussion.

even better agreement to the experimental values of Mitterschiffthaler and Bauer (1990); Phillips (1953); Reynolds et al. (1953), and Baek et al. (2006), proving the capability of our approach. A-150: Another wide used TEM is A-150, which is utilized as a radiation energy absorber in microdosimetry techniques. Awschalom and Attix (1980) proposed four A-150 plastic-equivalent gas mixtures in terms of well known molecules for its chemical composition. Table 4, presents the chemical fragmentation of each sample and their percentage by weight. Fig. 6 shows the Se of protons colliding with all the four A-150 mixtures. The results are compared with semi-empirical values from the SRIM program (Ziegler, 2013), PSTAR (Berger et al., 2005), the experimental results of Ishiwari et al. (1990), and the theoretical results reported by Thwaites (1985). From Fig. 6, we see that Mix 1 and Mix 2 have a better agreement with both experimental and theoretical results than Mix 3 and Mix 4 in the high energy region, due to the different composition of propadiene and butadiene. The molecular stopping cross sections obtained by our approach for A-150 mixtures within a FSGO decomposition are extremely good. Nylon: Another organic compound often used in radiation research with applications as biological substitute is nylon to mimic properties of skin (Tsuda et al., 2005; Jones et al., 2003). There are many types of nylon, the two most common ones are for textile and industrial plastics, i.e., Nylon 6 (C6H11NO)n and Nylon 66 [(C12H22N2O2)n], respectively. Note that the percentage of the elemental composition for both compounds is the same (Nylon 66 is made of two monomers of Nylon 6), such that most of the results in the literature do not distinguish between these materials and the mean excitation energy reported by NIST is the same for both of them (see Table 2). In Fig. 7, we present the proton stopping cross section per atom for Nylon-6. The curve is compared with to values of SRIM (Ziegler, 2013) and PSTAR (Berger et al., 2005). For the high energy region the agreement is good even that we are comparing to nylon in solid phase. Air. This compound seems not to be important in the field of biological systems and rather in atmospheric research; however, air has relevance in radiotherapy treatment. For example, the stopping power ratio of an ion colliding with water and air is necessary for accurate ionization chamber dosimetry (Paul et al., 2007). Air is a mixture of

fraction of the chemical fragment that form these molecules. These chemical fragmentation are used in Eq. (6) to determine the stopping cross section. In Fig. 4, we show the stopping cross sections for TEMs. Fig. 4a) shows the Se of TEGM and Fig. 4b) shows for TEGP. We compare both results with the online database PSTAR (Berger et al., 2005) and SRIM (Ziegler, 2013), finding that at high energies, the agreement is good. Note how in Fig. 4a), SRIM and PSTAR results overlap each other. For TEGM, we compare also with the theoretical work of Thwaites (1985) and to the experimental values from Fukuda (1980) between 40 and 200 keV. Interestingly, the agreement of our results with the experimental data above 80 keV is very good. Tissue Equivalent Liquid (TEL): Due to the presence of water in human body, approximately 70%, hydrated tissue equivalent materials are important to study and characterize their properties by ion radiation. The need of these TELs is that dry TEMs cannot mimic the biological reality of organs and human tissue. Consequently, a biological medium composed of water must be considered. In Table 4, we present the fragment chemical composition of TEL, as proposed by Rossi and Zaider (1996) as used in this work. Goodman and Colvett (1977) prepared samples of TEL in terms of their elemental composition, i.e., in percentage by weight of H, C, N, and O atoms. For comparison we have taken samples 1 and 2 from Goodmand and named TEL-Mix1 and TELMix2. Fig. 5 presents the Se for protons on TEL with the composition given in Table 4. In the same figure, we compare with the results from SRIM (Ziegler, 2013) using the weight composition of TEL-Mix1 and TEL-Mix2 (Goodman and Colvett, 1977). Since the TEL is approximately 65% water in its chemical fragment or 75% oxygen in its elemental composition, in Fig. 5, we show the Se for water within our model. The figure inset shows the Se from 800 keV to 3 MeV for better visualization. Note how for this region, the SRIM result presents a “shoulder”, i.e., an abrupt slope change. Instead, our result for TEL within the FSGO has a smooth behavior. For water, our result has a 155

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Fig. 6. Proton electronic stopping cross section per atom, Se , for A-150 plasticequivalent gas mixtures. Our results are given as: Mix-1 (thick solid line), Mix-2 (thin solid line), Mix-3 (thick dash line), and Mix-4 (thin dash line), and are compared to semi-empirical results from SRIM program (double-dash long line) (Ziegler, 2013), PSTAR (long-dash lined) (Berger et al., 2005) and theoretical results by Thwaites (filled-circle dash line) (Thwaites, 1985). The experimental data are from Ishiwari et al. (filled triangle symbols) (Ishiwari et al., 1990). See text for discussion.

Fig. 8. Proton electronic stopping cross section per atom, Se , for air (dry, near sea level). Our result (solid line) is compared to the semi-empirical data from SRIM (double-dash long line) (Ziegler, 2013), PSTAR (long dash line) (Berger et al., 2005), and to the experimental data from Reynolds et al. (1953), Park and Zimmerman (1963), Wolke et al. (1963), Swint et al. (1970), and Bonderup and Hvelplund (1971). See text for discussion.

4. Conclusions In this work we have used an analytical expression for the electronic stopping cross section as derived from the assumption of the target's electrons being harmonically bound and described by FSGO wavefunctions. We reinforce the fact that the FSGO basis set is a suitable approach to characterize and describe the energy deposition into complex biomolecules with biomedical applications. We determine the stopping cross section for DNA, guanine, adenine, alanine, glycine, trimethylamine, tissue equivalent gas (methane and propane based), A150 plastic equivalent (gas), tissue equivalent liquid, and air. We report good to excellent agreement when comparing to other more sophisticated approaches and to available experimental data. We find that the orbital mean excitation energy, I0i , is the principal parameter that determines Se , and not the total molecular I0 parameter. The simplicity and visual appealing of the model allows it to be used in any closed shell system. One consequence of our approach, based on the orbital mean excitation energies, is that it predicts a larger contribution for the stopping cross section, for collision energies around the maximum. This is due to the projectile's frozen charge, such that we expect that the incorporation of charge exchange process might give an even better description of the energy loss process. This is work in progress.

Fig. 7. Proton electronic stopping cross section per atom, Se for Nylon-6 (solid line). We compare our results to the semi-empirical values of SRIM (double dash-long line) (Ziegler, 2013) and PSTAR (long-dash line) (Berger et al., 2005). See text for discussion.

Acknowledgment Work supported by grants UNAM-DGAPA-PAPIIT IN-106-617 and LANCAD-UNAM-DGTIC-228 to RCT and CVU 423798 to LNTL.

many gases, mainly O2 and N2. To perform our calculations, we use the air composition given by White et al., which is reproduced in Table 4. Fig. 8 shows the Se for protons colliding with air as obtained with our model. The symbols are the experimental values reported by Reynolds et al. (1953), Park and Zimmerman (1963), Wolke et al. (1963), Swint et al. (1970), and Bonderup and Hvelplund (1971). Again, our results have a good agreement to the available experimental data showing the suitability of our approach.

References Abril, I., García-Molina, R., Denton, C.D., Kyriakou, I., Emfietzoglou, D., 2011. Energy loss of Hydrogen- and Helium-ion beams in DNA: calculations based on a realistic energy-loss function of the target. Radiat. Res. 175 (2), 247–255. https://doi.org/10. 1667/RR2142.1. Akkerman, A., Akkerman, E., 1999. Characteristics of electron inelastic interactions in organic compounds and water over the energy range 20–10000 eV. J. Appl. Phys. 86 (10), 5809–5816. https://doi.org/10.1063/1.371597. Akkerman, A., Breskin, A., Chechik, R., Lifshitz, Y., 2001. Calculation of proton stopping power in the region of its maximum value for several organic materials and water.

156

Radiation Physics and Chemistry 156 (2019) 150–158

L.N. Trujillo-López, R. Cabrera-Trujillo

of tomographic dosimetry phantoms in pediatric radiology. Med. Phys. 30 (8), 2072–2081. https://doi.org/10.1118/1.1592641.(URL 〈https://www.ncbi.nlm.nih. gov/pubmed/12945973〉). Kamakura, S., Sakamoto, N., Ogawa, H., Tsuchida, H., Inokuti, M., 2006. Mean excitation energies for the stopping power of atoms and molecules evaluated from oscillatorstrength spectra. J. Appl. Phys. 100 (6), 064905. https://doi.org/10.1063/1. 2345478. Kreutz, R., Neuwirth, W., Pietch, P., 1980. Analysis of electronic stopping cross sections of organic molecules. Phys. Rev. A 22 (6), 2606–2616. https://doi.org/10.1103/ PhysRevA.22.2606. LaVerne, J., Pimblott, S., 1995. Electron energy-loss distributions in solid, dry DNA. Radiat. Res. 141 (2), 208–215. https://doi.org/10.2307/3579049. Limandri, S., Fadanelli, R.C., Behar, M., Nagamine, L.C.C.M., Fernandez-Varea, J.M., Abril, I., García-Molina, R., Montanari, C.C., Aguiar, J.C., Mitnik, D., Miraglia, J.E., Arista, N.R., 2014. Stopping cross sections of TiO2 for H and He ions. Eur. Phys. J. D. 68 (194), 1–8. https://doi.org/10.1140/epjd/e2014-40782-6. Maggiora, G.M., Christoffersen, R.E., 1976. Ab initio calculations on large molecules using molecular fragments. Generalization and characteristics of floating spherical gaussian basis sets. J. Am. Chem. Soc. 98 (26), 8325–8332. https://doi.org/10.1021/ ja00442a003. Maggiora, G., Petke, J., Christoffersen, R., 1991. Electronic excited states of biomolecular systems: ab Initio FSGO-based Quantum Mechanical methods with applications to photosynthetic and related systems. In: In: Maksić, Z. (Ed.), Theoretical Treatment of Large Molecules and Their Interactions. Part 4 139. Springer, Berlin, Heidelberg, pp. 65–102. https://doi.org/10.1007/978-3-642-58183-0_3. McDonald, J., Johnston, M., 1990. Physiological and pathophysiological roles of excitatory amino acids during central nervous system development. Brain Res. Rev. 15 (1), 41–70. https://doi.org/10.1016/0165-0173(90)90011-C.(URL 〈https://www. ncbi.nlm.nih.gov/pubmed/2163714〉). Meltzer, D., Sabin, J., Trickey, S., 1990. Calculation of mean excitation energy and stopping cross section in the orbital local plasma approximation. Phys. Rev. A 41 (1), 220–232. https://doi.org/10.1103/PhysRevA.41.220. Mitterschiffthaler, C., Bauer, P., 1990. Stopping cross section of water vapor for hydrogen ions. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 48, 58–60. https://doi.org/10.1016/0168-583X(90)90073-4. Oddershede, J., Sabin, J.R., 1984. Orbital and whole-atom proton stopping power and shell corrections for atoms with Z ≤ 36. At. Data Nucl. Data Tables 31 (2), 275–297. https://doi.org/10.1016/0092-640X(84)90024-X. Park, J.T., Zimmerman, E.J., 1963. Stopping cross sections of some hydrocarbon gases for 40–250 keV protons and helium ions. Phys. Rev. 131, 1611–1618. https://doi.org/ 10.1103/PhysRev.131.1611. Paul, H., Geithner, O., Jäkel, O., 2007. The ratio of stopping powers of water and air for dosimetry applications in tumor therapy. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 256 (1), 561–564. https://doi.org/10.1016/j.nimb.2006. 12.072.(URL 〈http://www.sciencedirect.com/science/article/pii/ S0168583X06013115〉). Phillips, J.A., 1953. The energy loss of low energy protons in some gases. Phys. Rev. 90 (4), 532–537. https://doi.org/10.1103/PhysRev.90.532. Quinto, M., Monti, J., Weck, P., Fojón, O., Hanssen, J., Rivarola, R., Senot, P., Champion, C., 2017. Monte carlo simulation of proton track structure in biological matter. Eur. Phys. J. D. 71, 130. https://doi.org/10.1140/epjd/e2017-70709-6. Reynolds, H.K., Dunbar, D.N.F., Wenzel, W.A., Whaling, W., 1953. The stopping cross section of gases for protons, 30-600 kev. Phys. Rev. 92 (3), 742–748. https://doi.org/ 10.1103/PhysRev.92.742. Rossi, H.H., Zaider, M., 1996. Microdosimetry and Its Applications. Springer, Berlin(URL 〈https://cds.cern.ch/record/1972058〉). Rouse, R.A., Frost, A.A., 1969. Floating spherical gaussian orbital model of molecular structure VI. Double gaussian modification. J. Chem. Phys. 50 (4), 1705–1710. https://doi.org/10.1063/1.1671262. Russell, D., Snyder, S., 1968. Amine synthesis in rapidly growing tissues: ornithine decarboxylase activity in regenerating rat liver, chick embryo, and various tumors. Proc. Natl. Acad. Sci. USA 60 (4), 1420–1427. https://doi.org/10.1073/pnas.60.4.1420. (URL 〈https://www.ncbi.nlm.nih.gov/pmc/articles/PMC224936/〉). Sabin, J.R., Brändas, E., 2007. Advances in Quantum Chemistry 52 Academis Press, USA(URL 〈https://www.sciencedirect.com/bookseries/advances-in-quantumchemistry/vol/52/suppl/C〉). Sauer, S., Sabin, J., Oddershede, J., 1995. Calculated molecular mean excitation energies for some small molecules. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 100 (4), 458–463. https://doi.org/10.1016/0168-583X(95)00370-3. Sauer, S.P.A., Oddershede, J., Sabin, J.R., 2006. Directional dependence of the mean excitation energy and spectral moments of the dipole oscillator strength distribution of glycine and its zwitterion. J. Phys. Chem. A 110 (28), 8811–8817. https://doi.org/ 10.1021/jp061412i. Sauer, S.P., Oddershede, J., Sabin, J.R., 2011. Chapter 5 - Mean excitation energies for biomolecules: glycine to DNA. In: In: Sabin, J.R., Brändas, E. (Eds.), Advances in Quantum Chemistry 62. Academic Press, pp. 215–242. https://doi.org/10.1016/ B978-0-12-386477-2.00011-5. (Chapter 5). Sauer, S., Oddershede, J., Sabin, J., 2012. Radiation damage in biomolecular systems. In: In: Gómez-Tejedor, G.G., Fuss, M.C. (Eds.), Theory and Calculation of Stopping Cross Sections of Nucleobases for Swift Ions Springer Science, Singapore, pp. 191–200. https://doi.org/10.1007/978-94-007-2564-5_12. (Ch. 12). Simons, G., Talaty, E., 1978. A local orbital description of the Cl- . CH3ClSN2 reaction. Chem. Phys. Lett. 56 (3), 554–556. https://doi.org/10.1016/0009-2614(78)89038-1. Swint, J.B., Prior, R., Ramir, J., 1970. Energy loss of protons in gases. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 80 (1), 134–140. https://doi. org/10.1016/0029-554X(70)90308-3.

Radiat. Phys. Chem. 61 (3), 333–335. https://doi.org/10.1016/S0969-806X(01) 00258-4. Awschalom, M., Attix, F., 1980. A-150 plastic-equivalent gas. Phys. Med. Biol. 25 (3), 567. https://doi.org/10.1088/0031-9155/25/3/019/. Baek, W.Y., Grosswendt, B., Willems, G., 2006. Ionization ranges of protons in water vapour in the energy range 1–100 keV. Radiat. Prot. Dosim. 122 (1–4), 32–35. https://doi.org/10.1093/rpd/ncl514. Berger, M., Coursey, J., Zucker, M.A., Chang, J., 2013. NIST stopping power and range tables for electrons, protons and helium atoms., Online, available http://www.nist. gov/pml/data/star/index.cfm. Berger, M., Coursey, J., Zucker, M., Chang, J., 2005. ESTAR, PSTAR, and ASTAR: Computer Programs for Calculating Stopping Power and Range Tables for Electrons, Protons, and Helium ions (version 1.2.3), Online, available: http://physics.nist.gov/ PStar [2018, may]. National Institute of Standards and Technology, Gaithersburg, MD. Besemer, A., Paganetti, H., Bednarz, B., 2013. The clinical impact of uncertainties in the mean excitation energy of human tissues during proton therapy. Phys. Med. Biol. 58 (4), 887–902. https://doi.org/10.1088/0031-9155/58/4/887. Bethe, H.A., 1930. Zur Theorie des Durchgangs schneller Korpuskularstrahlen durch Materie (Theory of the Passage of Fast Corpuscular Rays Through Matter). Ann. hys. 397, 325–400. https://doi.org/10.1002/andp.19303970303. Blustin, P.H., 1977. Simple ab initio calculations using a floating basis. Periodic correlations in second row diatomic hydrides. J. Chem. Phys. 66 (12), 5648. https://doi. org/10.1063/1.433886. Bohr, N., III, 1913. On the theory of the decrease of velocity of moving electrified particles on passing through matter: Phil. Mag. 25, 10. In:: J. Thorsen (Ed.), The Penetration of Charged Particles through Matter (1912–1954), Vol. 8 of Niels Bohr Collected Works, Elsevier, 1987, pp. 47–71. 〈http://dx.doi.org/10.1016/S18760503(08)70140-3〉. Bonderup, E., Hvelplund, P., 1971. Stopping power and energy straggling for swift protons. Phys. Rev. A 4 (2), 562–569. https://doi.org/10.1103/PhysRevA.4.562. Boys, S.F., 1950. Electronic wave functions. I. A general method of calculation for the stationary states of any molecular system. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci. 200 (1063), 542–554. https://doi.org/10.1098/rspa.1950.0036.(URL 〈http://www. jstor.org/stable/98423〉). Bruun-Ghalbia, S., Sauer, S.P.A., Oddershede, J., Sabin, J.R., 2010. Mean excitation energies and energy deposition characteristics of bio-organic molecules. J. Phys. Chem. B 114 (1), 633–637. https://doi.org/10.1021/jp908998s. Cabrera-Trujillo, R., Cruz, S., Soullard, J., 1994. Bond stopping cross sections for protons incident on molecular targets within the OLPA/FSGO implementation of the kinetic theory. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 93 (2), 166–174. https://doi.org/10.1016/0168-583X(94)95683-9. Cabrera-Trujillo, R., Martínez-Flores, C., Trujillo-López, L., Serkovic-Loli, L., 2016. On the universal scaling in the electronic stopping cross section for heavy ion projectiles. Radiat. Eff. Defects Solids 171 (1–2), 146–153. https://doi.org/10.1080/10420150. 2016.1155581. Cabrera-Trujillo, R., 1999. Stopping power in the independent-particle model: harmonic oscillator results. Phys. Rev. A 60 (4), 3044–3052. https://doi.org/10.1103/ PhysRevA.60.3044. Christoffersen, R.E., Maggiora, G.M., 1969. Ab initio calculations on large molecules using molecular fragments. preliminary investigations. Chem. Phys. Lett. 3 (6), 419–423. https://doi.org/10.1016/0009-2614(69)80155-7. Chu, S.Y., Frost, A.A., 1971a. Floating Spherical Gaussian Orbital model of molecular structure. VIII. Second-row atom hydrides. J. Chem. Phys. 54 (2), 760. https://doi. org/10.1063/1.1674908. Chu, S.Y., Frost, A.A., 1971b. Floating spherical gaussian orbital model of molecular structure. IX. diatomic molecules of rst-row and second-rowatoms. J. Chem. Phys. 54 (2), 764–768. https://doi.org/10.1063/1.1674909. Frost, A.A., Rouse, R.A., 1968. Floating spherical Gaussian orbital model of molecular structure. IV. Hydrocarbons. J. Am. Chem. Soc. 90 (8), 1965–1969. https://doi.org/ 10.1021/ja01010a007. Frost, A.A., 1967a. Floating spherical Gaussian Orbital Model of molecular structure. I. Computational procedure. LiH as an example. J. Chem. Phys. 47 (10), 3707. https:// doi.org/10.1063/1.1701524. Frost, A.A., 1967b. Floating Spherical Gaussian Orbital Model of Molecular Structure. II. One and two electron pair system. J. Chem. Phys. 47 (10), 3714. https://doi.org/10. 1063/1.1701525. Frost, A.A., 1967c. Floating Spherical Gaussian Orbital model of molecular structure. III. First-row atom hydrides. J. Chem. Phys. 72 (4), 1289. https://doi.org/10.1021/ j100850a037. Fukuda, A., 1980. Stopping powers of a tissue-equivalent gas for 40–200 keV protons. Phys. Med. Biol. 25 (5), 877(URL 〈http://stacks.iop.org/0031-9155/25/i=5/a= 006〉). Goodman, L.J., Colvett, R.D., 1977. Biophysical studies with high-energy Argon ions: 1. Depth dose measurements in tissue-equivalent liquid and in water. Radiat. Res. 70 (3), 455–468(URL 〈http://www.jstor.org/stable/3574637〉). Huzinaga, S., 1965. Gaussian-type functions for polyatomic systems. i. J. Chem. Phys. 42 (4), 1293–1302. https://doi.org/10.1063/1.1696113. Inokuti, M., 1971. Inelastic collisions of fast charged particles with atoms and molecules–the bethe theory revisited. Rev. Mod. Phys. 43, 297–347. https://doi.org/10. 1103/RevModPhys.43.297. Ishiwari, R., Shiomi-Tsuda, N., Sakamoto, N., Ogawa, H., 1990. Stopping powers of a-150 tissue equivalent plastic for protons from 2.5 to 7.5 MeV. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 47 (2), 111–117. https://doi.org/10. 1016/0168-583X(90)90018-P. Jones, A., Hintenlang, D.E., Bolch, W., 2003. Tissue-equivalent materials for construction

157

Radiation Physics and Chemistry 156 (2019) 150–158

L.N. Trujillo-López, R. Cabrera-Trujillo

Interact. Mater. At. 69 (1), 53–63. https://doi.org/10.1016/0168-583X(92)95738-D. Trujillo-López, L., Martínez-Flores, C., Cabrera-Trujillo, R., 2013. Universal scaling behavior of molecular electronic stopping cross section for protons colliding with small molecules and nucleobases. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 313, 5–13. https://doi.org/10.1016/j.nimb.2013.07.020.(URL 〈http:// www.sciencedirect.com/science/article/pii/S0168583X13008033〉). Tsuda, S., Endo, A., Yamaguchi, Y., 2005. Development of three kinds of tissue substitutes for a physical phantom in neutron dosimetry. J. Nucl. Sci. Technol. 42 (10), 877–887. https://doi.org/10.1080/18811248.2005.9711039. Usta, M., Tufan, M.Ç., 2017. Stopping power and range calculations in human tissues by using the Hartree-Fock-Roothaan wave functions. Radiat. Phys. Chem. 140, 43–50. https://doi.org/10.1016/j.radphyschem.2017.03.005.(URL 〈http://www. sciencedirect.com/science/article/pii/S0969806X1730275X〉). Venturi, M., D’Angelantonio M., 2017. (Eds.), Applications of Radiation Chemistry in the Fields of Industry, Biotechnology and Environment, no. 1 in Topics in Current Chemistry Collections, Springer International Publishing. 〈http://dx.doi.org/10. 1007/978-3-319-54145-7〉. White, D.R., Booz, J., Griffith, R.V., Spokas, J.J., Wilson, I.J., Report 44, Journal of the International Commission on Radiation Units and Measurements os23 (1). 〈http:// dx.doi.org/10.1093/jicru/os23.1〉.Report44. Wolke, R.L., Bishop, W.N., Eichler, E., Johnson, N.R., O’kelley, G.D., 1963. Ranges and stopping cross sections of low-energy tritons. Phys. Rev. 129 (6), 2591–2596. https:// doi.org/10.1103/PhysRev.129.2591. Ziegler, J.F., Online, 〈http://www.srim.org/SRIM〉(2013).

Szabo, A., Ostlund, N.S., 1996. Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory. Dover Publications Inc. Talaty, E.R., Fearey, A.J., Simons, G., 1976. FSGO Calculations of geometries and electronic structures of Argon-core third-row hydrides. Theor. Chim. Acta 41, 133–139. https://doi.org/10.1007/BF01178073. Tan, Z., Xia, Y., Zhao, M., Liu, X., 2006. Proton stopping power in a group of bioorganic compounds over the energy range of 0.05–10 MeV range. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 248 (1), 1–6. https://doi.org/10.1016/j. nimb.2006.04.073. Tan, Z., Xia, Y., Zhao, M., Liu, X., 2008. Electronic stopping power for proton in amino acids and protein in 0.05–10 MeV range. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 266 (9), 1938–1942. https://doi.org/10.1016/j.nimb.2008. 03.043.(URL 〈http://www.sciencedirect.com/science/article/pii/ S0168583X08003121〉). Tan, Z., Xia, Y., Liu, X., Zhao, M., 2010. Proton inelastic mean free path in amino acids and protein over the energy range of 0.05–10 MeV. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 268 (17), 2606–2610. https://doi.org/10.1016/j. nimb.2010.06.014.(URL 〈http://www.sciencedirect.com/science/article/pii/ S0168583X10005884〉). Thwaites, D., 1985. Stopping powers for protons in materials of interest in dosimetry and in medical and biological applications. Radiat. Prot. Dosim. 13 (1–4), 65–69. https:// doi.org/10.1093/oxfordjournals.rpd.a079549. Thwaites, D.I., 1992. Departures from Bragg's rule of stopping power additivity for ions in dosimetric and related materials. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam

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