Universal scaling behavior of molecular electronic stopping cross section for protons colliding with small molecules and nucleobases

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Abstract

The electronic stopping cross section and mean excitation energy for molecules and 5 nucleobases have been calculated within the first Born approximation in terms of an orbital decomposition to take into account the molecular structure. The harmonic oscillator (HO) description of the stopping cross section together with a Floating Spherical Gaussian Orbital (FSGO) model is implemented to account for the chemical composition of the target. This approach allows us to use bonds, cores, and lone pairs as HO basis to describe the ground state molecular structure. In the HO model, the orbital angular frequency is the only parameter that connects naturally with the mean excitation energy. As a result, we obtain a simple expression for the equivalent mean excitation energy in terms of the orbital radius parameter, as well as an analytical expression of the stopping cross section. For gas phase molecular targets, we provide HO based orbital mean excitation energies to describe any molecule containing C, N, O, H, and P atoms. We present results for protons colliding with H2, N2, O2, H2O, CO2, propylene (C3H6), methane (CH4), ethylene (C2H4) and the nucleobases – guanine (C5H5N5O), cytosine (C4H5N2O2), thymine (C5H6N2O2), adenine (C5H5N5) and uracil (C4H4N2O2). The results for the stopping cross section are compared with available experimental and theoretical data showing good to excellent agreement in the region of validity of the model. The HO approach allows us to obtain a universal stopping cross section formula to describe a universal scaling behavior for the energy loss process. The universal scaled curve is confirmed by the experimental data.

Introduction

This year marks the centennial of Bohr’s pioneering study of the interaction of swift, heavy particles with matter and whose applications extend to many fields of science, e.g. astrophysics, metallurgy, electronic industries, nuclear physics, materials science, radiotherapy, and dosimetry among others. In particular, radiation therapy consists of destroying the cancerous tumor cells by ionizing radiation, a process known as radiation damage. Radiation damage is produced in the material in two ways: directly, in which the radiation destroys the molecules, and indirectly, in which electrons and low energy photons are absorbed by the material, ionizing it. In living matter, the most significant damage from radiation is due to red water molecule fragments and secondary electrons or by a direct hit that ruptures or fragments a DNA molecules. That is why finding the slowing down characteristic curve for biomolecules, as well as the study of the mechanism by which the ion losses energy as it penetrates matter is of paramount importance.

As mentioned above, Niels Bohr [1], proposed the first model to analyze the energy loss for a projectile while penetrating a target using only classical ideas. Bohr treated the collision system by assuming the target electrons were harmonically bound and excited by the electric field of the incoming projectile. Later, Hans Bethe [2] proposed the first quantum mechanical treatment of the problem using perturbation theory for the projectile-target interaction by means of the first Born approximation. The fundamental idea behind Bethe’s treatment is the use of momentum transfer from the projectile to the electrons in the target as the dynamic factor in the collision.

According to Bethe’s theory, the electronic stopping cross section, Se, for a swift, heavy ion with charge Z1e colliding with a target with N2 bound electrons is given by:Se(v)=4πe4Z12N2mev2ln2mev2I0,valid for projectile velocities, v, higher than the target electron velocity. Here, I0 is the mean excitation energy of the target which is defined in terms of the dipole oscillator strength (DOS), fn0, as:lnI0=nfn0ln(En-E0)nfn0.Here, E0 and En denote the energies of the system in the initial and final state, respectively. However, to obtain I0 requires knowledge of the full excitation spectrum of the target, a difficult task even in these days. So, it may be necessary to resort to quantum chemistry methods [3], [4] or to fits to experimental data [5]. For molecular targets, the situation is even more complicated due to the larger N-body problem [6].

In the field of radiotherapy, accurate values of the mean excitation energy for biological materials, such as DNA, are necessary [7]. A good characterization of I0 as well as the penetrating range and stopping curve of therapeutic ion beams is biologically relevant for the accuracy of the energy deposition in nanometer volumes [8], [7].

The study of the mean excitation energy for several small biomolecules has been the focus of research of the Danish group [9] based on the work of Oddershede and Sabin (OS) [10]. OS have treated shell corrections and mean excitation energies by using the propagator method within the Bethe approximation to obtain the excitation spectra of molecules containing C–H, C–C, Cdouble bondC, O–H, C–O and Cdouble bondO bonds. Once I0 has been obtained, use of the kinetic theory [11] was made to report molecular stopping cross sections. In this context, Lindhard’s theory [12] has been also used in the study of several biomolecules, finding good agreement with other similar treatments if the proper optical energy loss function is used [8], [13].

Experimentally, there exists very strong evidence that supports the use of molecular fragments in the analysis of stopping cross section [14], [15], [16]. Recently, the atomic Bragg rule has been under scrutiny, reaching the conclusion that it fails in the low energy region where molecular bonds start to become important [17], [18]. The data suggest that the molecular stopping cross section can be expressed as a sum of contributions of characteristic molecular groups. Thus, OS implemented a core and bonds description of a molecular target which allowed a fragment treatment of the target. Use of these ideas was made by Cabrera-Trujillo et al. [19] within the molecular floating spherical Gaussian orbital (FSGO) model [20], [21], [22], [23], [24], [25], the local plasma approximation (LPA) [26], and the kinetic theory to describe the molecular stopping cross section, confirming the main idea of OS.

These previous treatments were based mainly on reporting mean excitation energies needed in the Bethe approximation, Eq. (1). However, the full analytical evaluation of the stopping cross section within the first Born expression, to obtain an analytical formula outside of the Bethe approximation, is necessary. Such a formula should account, in a simple way, for the full target excitation spectra, as well as to take into account the bond contribution to correct Bragg’s rule [27], [17], [18] which states that “the total stopping cross section is the weighted sum of the atomic stopping cross sections Se.i, where the weight factors are the numbers of atoms of type i”.

Several analytical expressions for the stopping cross section, based on functional fits [28] or universal curves [29], have been reported in the literature. In this regard, an analytical expression for the stopping cross section based on the harmonic oscillator (HO) model has been also reported [30]. The justification for the use of a HO scheme is that to first order any bound electron can be described by a harmonic potential, as originally considered by Bohr. A further advantage of the HO model is that its exact ground state is a Gaussian wave function which connects naturally with quantum theoretical wave-functions used in Quantum Chemistry models, e.g. the FSGO model.

In this work we propose to study the stopping cross section of small molecules and nucleobases by knowing only the ground state electronic structure, i.e. by implementing the HO analytical description of Se in a self contained and parameter free approach, providing a simple analytical procedure to describe any molecular target. As a consequence of the HO model, we obtain an universal scaling behavior of the electronic stopping cross section.

The work is presented with the following structure. In Section 2 we start with a summary of Bethe’s stopping power theory, within the independent particle model and the harmonic oscillator approach in conjunction with the FSGO model. In Section 3 we discuss and compare the results of our approach with experimental and theoretical available data. We present the result in two subsections: small molecules and nucleobases. Finally in Section 4, we present the conclusions and future prospectives for this work.

Section snippets

Stopping cross section

In this section we summarize the harmonic oscillator description within the independent particle model for Se as proposed in Ref. [30]. In contrast to Ref. [19], we calculate all the physical parameters within the same model in a self contained way. The only approximations to our approach are the first Born approximation and the independent particle model.

Mean excitation energy

In Table 1, we provide the orbital radius ρi for a set of molecules found in the literature [20], [21], [22], [23], [24], [25] and the corresponding orbital mean excitation energy. In the same table, we give the mean excitation energies as reported by the LPA [19], I0iLPA and the OS [10], I0iOS, approaches.

The values for the core (1s2) orbital mean excitation energies are larger in our model than those reported by the LPA approximation for almost a factor of two. Their values are only shown in

Conclusions

In this work, we have shown that a Bragg sum rule for molecular core, bond, and lone-pair orbitals, provides a good description for the electronic stopping cross section if a proper model of the molecular structure is used. The molecular structure FSGO model gives, in a natural way, the core, bond, and lone-pair orbital description parameters of a molecule. Using a harmonic oscillator approach and the FSGO model, we found analytical expressions for the angular frequency (or the equivalent

Acknowledgement

This work was supported by DGAPA-UNAM through grant PAPIIT-IN-101–611. RCT would like to thank Prof. S. A. Cruz for helpful discussions during sabbatical stay at UAM-I.

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