The hyperfine splitting parameters describe the interactions between unpaired electrons and magnetic nuclei in a paramagnetic center. The 3 × 3 dimensional hyperfine tensor A can be separated into its isotropic and anisotropic (dipolar) components. The isotropic hyperfine splitting A_iso (N) caused by nucleus N is equal to the Fermi contact term and is related to spin densities ρ^α−β (R_N) on the corresponding nuclei by the following equation: (1) A iso ( N ) = 4 π 3 β e   β N   g e   g N   〈 S Z 〉 − 1 ρ N α − βwhere β_e and β_N are the electronic and nuclear Bohr magnetons, respectively, g_e and g_N are the free-electron and nuclear g-values, respectively, 〈S_Z〉 is the expectation value of the z-component of the total electronic spin, and ρ N α − β is the spin density on the nucleus N at the position R_N. The anisotropic hyperfine tensor components T_kl in the Cartesian coordinate axes k and l in the first-order approximation are given by the following equation: (2) T kl   ( N ) = 1 2 β e   β N   g e   g N   〈 S Z 〉 − 1 ∑ μ , ν P μ , ν α − β 〈 φ μ | r N − 5 ( r N 2   δ kl − 3 r N , k   r N , l ) | φ ν 〉where r_N = r − R_N and P μ , ν α − β

represent the spin-density matrix elements. Indices k and l cyclically represent the x, y and z-axes. The T tensor is always traceless and may be brought into diagonal form with diagonal elements T_xx, T_yy and T_zz, where T_xx + T_yy+T_zz = 0. The isotropic and anisotropic (dipolar) components contain information about the spin densities of the unpaired electron on the nuclei in the neighborhood [22].

The g-tensor components g_rs are calculated using the coupled-perturbed density functional theoretical (CP-DFT) formulation of Neese and Estebes et al involving four terms [23,24]: (3) g rs = g e δ rs + Δ g RMC δ rs + Δ g rs GC + Δ g rs OZ / SOC

The first term is the isotropic contribution representing the free-electron g-value. The second term of Equation 3 is a relativistic mass-correction term introduced by Angstl [25] to correct g-values for planar-aromatic complexes, which is calculated with the ground state spin-density and kinetic energy integrals as follows: (4) Δ g RMC = − α 2 S ∑ μ , ν P μ ν α − β 〈 φ μ | T ^ | φ ν 〉where α is the fine-structure constant, S is the total spin of ground state, P μ ν α − β

is the spin-density matrix, φ is the basis set, and T̂ is the kinetic-energy operator. The third term of Equation 3 is a diamagnetic correction introduced by Stone [26] and is calculated depending on the ground-state spin density as follows: (5) Δ g GC = 1 2 S ∑ μ , ν P μ ν α − β 〈 φ μ | ∑ N [ α 2 Z eff 4 2 | r i − R N | 3 ] ( r N   r O − r N , s   r O , s ) |   φ ν 〉where r_N is the position vector of the electron relative to the nucleus N, r_O is the position vector relative to gauge origin, Z_eff is the effective nuclear charge, and the term in square brackets is the effective spin–orbit coupling of the i^th electron at nucleus N. The fourth term of Equation 3 is the dominant correction and comes from crossed terms between the Zeeman orbital (OZ) operator and the spin–orbit coupling. This term is calculated using Neese’s coupled perturbed theory and the DFT methodology (CP-DFT) [23].

From the assumptions made on the experimental spectra, the possible radicals that formed after gamma irradiation were modeled using theoretical computations. The molecular geometry parameters of MSM [(CH₃)₂SO₂] that are shown schematically in Figure 1 were taken from crystal data [27–29]. The geometry optimizations of the model radicals were made with the UMP2 method and the standard 6-311++G(d,p) basis set. The optimizations were performed without any constraints (full optimization). All stationary points were confirmed as local minima by their harmonic vibration frequencies, and the normal-mode calculations were performed at the same level as the geometry optimizations.

The hyperfine and g-values of the modeled radicals were calculated by the UB3LYP method using the TZVP basis set [30] because a successful prediction of the EPR parameters for the sulfur containing radicals has been recently demonstrated using this method/basis set combination by Hermosilla et al. [31]. All geometry optimizations and EPR parameter calculations were performed using the GAUSSIAN 03 program [32].

Figure 2a shows the EPR spectra when the crystalline b axis was parallel to the magnetic field. Four lines with labels I, II, III and IV were detected, and their angular variations in three perpendicular crystalline planes are shown in Figure 3. The angular variations of lines I, II and III are the same where the separation of lines II and III is fixed with the value of 1.2 mT and an axially symmetric g-value averaging 2.0062, which indicated that they belonged to the same radical. However, the intensity of line I was too high and was not comparable to the intensities of lines II and III; therefore, the separation did not arise from hyperfine interactions. Moreover, the width of line I changed slightly with orientation, which indicated some small and irresolvable HFCC values. Lines II and III may correspond to satellites arising from a small HFCC values contained in line I [18] and can be attributed to a sulfur-centered radical. Conversely, the intensity of line IV was too small compared to other lines and behaved differently with an average g-value of 2.0010. This line was completely quenched when heated up to 70 °C or after keeping it at room temperature for approximately two months. This line can be attributed to a SO 2 −

type ionic radical formed as an impurity after irradiation [33,34].

The ³³S isotope has a nuclear spin of I = 3/2 and a natural abundance of 0.75%; therefore, when the spectrometer gain was increased by 200 times or more, four anisotropic lines caused by the ³³S HFCC were clearly seen (Figure 4), and their g-value variation was exactly the same as those of line I in Figure 2a. The parameters measured at room temperature and evaluated with a second-order shift [18,21] are given in Table 1. The ³³S HFCC was axially symmetric with an average value of 7.2 mT, which was greater than the value measured for the radical in DMSO around −50 °C [16] and was determined using ab initio molecular orbital calculations [35]. The unpaired electron occupied the 3s and 3p_z orbitals of the sulfur atom with ratios of 0.06 and 0.53, respectively.

When the temperature was increased from room temperature to +80 °C, the width of line I became narrower, and its amplitude increased due to the tumbling motion of the radical creating that line. Lines II and III showed no appreciable change, but line IV was irreversibly quenched with increasing temperature. When the temperature was increased above +80 °C, the quenching of all lines began because the melting point of MSM (+109 °C) was approached. The EPR parameters did not change significantly compared to the room temperature values.

As the temperature decreased below −50 °C, line I in Figure 2a became weaker, and the ³³S HFCC lines in Figure 4 started to quench; in addition, a new set of lines with an intensity distribution of 1:3:3:1 emerged and reached its highest point at −160 °C (Figure 2b). The HFCC was completely isotropic (1.3 mT), and the g-value was rhombic with average value of 2.0115 (Table 1). Similar spectra were seen in the gamma-irradiated DMSO at low temperatures and were attributed to CH₃SO⁻ ionic radical by Nishikida and Williams [16] and Swarts et al. [35]. The HFCC values for the hydrogen atoms of the bound CH₃ group were supposed to be equivalent, and the group kept rotating even at −160 °C. No ³³S HFCC lines could be detected at low temperatures. In addition, the calculations showed that ³³S lines became smaller and were under the intense spectra, Figure 2b.

When the temperatures decreased to −180 °C and below, the spectra changed further and showed four equally intense lines, Figure 2c. The angular variations of the lines showed that the spectra were created by two nonequivalent methyl protons because there were no other atoms in the structure other than hydrogen that could induce such spectra. The HFCC and g-values were axially symmetric and are given in Table 1. Both values differed from the values measured for the bound CH₃ group at −160 °C.

To assign the observed EPR parameters and determine the possible radical structure, a series of theoretical calculations were performed for several possible model radicals R1, R2, R3, R4, R5 and R6, Figure 5. The molecular structure of MSM obtained from the x-ray crystal data, Figure 1, was taken as the initial geometry. Model R1 was formed by removing the CH₃ fragment from the molecule. Models R2 and R3 were cationic and anionic model radicals formed by removing an oxygen atom from the molecule. Model R4 was formed by removing one of the oxygen atoms and the CH₃ fragment from the molecule. Models R5 and R6 were formed by cationic and anionic form of R4, respectively.

To provide accurate calculations for the hyperfine splitting and g-values, precise geometric structures of possible radicals were needed, and therefore, UMP2/6-311+G(d,p) level geometry optimizations were performed for six modeled radicals; the optimized radical geometries were then used as initial values to find the HFCC and g-values using the UB3LYP/TZVP level density-functional calculations. The theoretically calculated isotropic values of the hyperfine splitting and g-values of six model radicals are given in Table 2 with respect to the atomic numbering scheme shown in Figure 1.

In the calculations, the single and isolated molecule approach was used to perform DFT calculations assuming the radical was in gas phase at 0 K that is the lattice around the radical was not incorporated during EPR parameter calculations or geometry optimization calculations. The usefulness and feasibility of this methodology in the calculations of EPR spectroscopic parameters have previously been extensively demonstrated [22,36]. Moreover, 20% deviations, discussed below, in experimental and theoretical calculations include intrinsically the errors originating from environmental effects.

For most purposes of interpretation and assignment, 20% deviations would be quite acceptable for calculated isotropic hyperfine splitting values of experimentally isolated radicals [36]; the deviations in the calculated ³³S HFCC values from the experimental spectra at room temperature were less than 20% for the R1 model radical. Furthermore, it was difficult to measure the isotropic g-values more accurately than 10⁻³, and for most applications, deviations up to 1000 ppm are considered to be satisfactory [23]. As seen in Table 2, the calculated isotropic g-value deviation of model radical R1 was approximately 100 ppm, which was closer to the room temperature experimental value when compared to other five model radicals. Therefore, it was reasonable to conclude that the theoretically calculated isotropic g-value of the radical and hyperfine splitting value of ³³S for model radical R1 were in good agreement with experimental values at room temperature. Referring to the discussion on the intense line I in Figure 2a, the HFCC caused by the CH₃ protons were less than 0.5 mT and were under the intense line. The CH₃ groups that exist in radical species at higher temperatures freely rotate, and the three protons are assumed to be equivalent [37]. As seen in Table 2, the average HFCC of the three CH₃ protons for the optimized model radical R1 was found to be 0.49 mT and was compatible with the observed radical at room temperature. This structure was not observed by Nishikida and Williams [16] because of the lower melting point of the host DMSO. Meanwhile the principal ³³S HFCC value measured in this work (averaging 7.2 mT) is almost equal to the HFCC measured in x-irradiated MSM by Andersen and is appreciably higher than those of measured in DMSO (averaging 5.9 mT) by Nishikida and Williams [16]. We assumed that this difference originates from that the radical in DMSO contains only one oxygen atom while the radical in MSM contains two oxygen atoms and the spin population is relatively higher on S atom due to polarization effect of oxygen atoms.

As the temperature decreased between −50 °C and −160 °C, the electronic distribution and the structure of the radical changed as discussed above. The parameters obtained after the calculations showed that the model radical R4 fit the observed radical in Figure 5. The average HFCC and g-values for model radical R4 given in Table 2 were 1.08 mT and 2.012, respectively, and the difference in the experimental values (Table 1) was 500 ppm. The radical observed at low temperatures for the gamma-irradiated DMSO provided similar results [16] but no information was found for x-irradiated MSM at 77 K [14]. Taking the ³³S splitting values at room temperature and CH₃ proton splitting values for radicals in DMSO and MSM between −50 down to −160 °C, and also the temperature dependent anti-symmetric stretching mode and hence the elongation of S=O bonds (∼0.1 Å) of MSM molecule [38,39] we assumed that one of the oxygen atoms stayed closer to the S1 atom and the other one moved away in model radical R1; therefore, the radical behaved like model radical R4. The radical is basically a π electron radical, and the unpaired electron may have occupied the 3p_z orbital but not 3s orbital of the sulfur atom [35].

At temperatures of −180 °C and lower, the CH₃ group stops rotating, and the hydrogen atoms become magnetically non-identical. One of the hydrogen atoms displayed an almost zero HFCC value, and the other two hydrogen atoms provided different and axially symmetric HFCC values. The g-values were also different from the radical observed at higher temperatures (Tables 1 and 2). The calculations showed that when the CH₃ group stopped rotating and due to the polarization effect of lone pair electrons of the oxygen atom staying closer to sulfur atom, the unpaired electron density on the CH₃ group shifted towards the two hydrogen atoms; as a result, the observed spectra were produced. According to the calculations which took care of the discussions made above for temperature interval of −50 down to −160 °C, one of the oxygen atoms stayed closer to the S1 atom and the other one went more away in the model radical R1, and therefore, the radical behaved like the model radical R4* below −180 °C. The calculated values (UB3LYP/TZVP) for the R4* structure, which was the modified model radical from R4, are given in Table 2 together with the model radical R4; it was only in this case that the HFCC values of two hydrogen atoms fit the experimental values. In another words, room temperature measurements and calculations indicate two oxygen atoms bound to sulfur atom [14–16] and low temperature measurements and calculations indicate only one oxygen atom. Moreover, S=O bonds are strong and cannot be broken easily. Therefore it seemed the only assumption that could be made to explain behaviors discussed above. The measured and calculated average values of the radical were all in the same order of magnitude and fit the acceptable limits [23,36].