Anomalous EPR Intensity Distribution of the Methyl Radical Quartet

Article pubs.acs.org/JPCA

Anomalous EPR Intensity Distribution of the Methyl Radical Quartet Adsorbed on the Surface of Porous Materials. Comparison with Solid Gas Matrix Isolation Nikolas P. Benetis*,† and Yurij Dmitriev‡ †

Department of Pollution Control, Technological Education Institution, TEI, West Macedonia, Kozani 501 00, Greece Ioffe Physico-Technical Institute, 26 Politekhnicheskaya ul., St. Petersburg, Russia

‡

ABSTRACT: The two inner lines of the EPR quartet of methyl radicals trapped in cryogenic gas matrices are superpositions of the inner transitions of an A-proton-spin quartet and an E-proton-spin doublet. Their intensity relative to the outer lines provides information on the population of the methyl-rotation quantum states. The above intensity ratio for the CH3 in solids is a challenging problem of the quantum dynamics and statistical thermodynamics. The influence of the quantum-mechanical/inertial rotation on the intensity distribution of the hf components of methyl radical on the surface of porous materials, e.g., silica gel, is investigated by EPR line shape simulations and compared with spectra of the radical isolated in the bulk of solid gas samples. The experimental part of this study includes the first in literature EPR observation of methyl radical in the bulk of N2O solid and provides new essential information on CH3 in CO2 and Ar matrices, thus, covering both strongly hindered and almost free rotation of the radical. We verify the observation of nonrotating methyl radicals in a N2O matrix, discovered earlier in cold CO2, give a thorough account of their EPR characteristics, and explore their formation at the inner surface of porous materials. Combination of a classical spin-Hamiltonian with employment of quantum effects due to nuclear spin-rotation coupling and the radical symmetry were used to interpret the experimental spectral observations. The cause of experimentally found unexpected contribution of the excited degenerate E-doublets to the EPR spectrum down to 4.2 K and A/E transition amplitude ratios sometimes as high as ca. 1:8 at liquid-N2 temperature is sought. The validity of Bose-Einstein quantum (BEq-) statistics of the spin rotation states in addition to the classical Maxwell-Boltzmann (Boltzmann) statistics was also assessed against experimental population A/E-ratio data. The BEq-statistics were not previously applied to similar systems. Furthermore, detailed consideration of the laboratory/free space rotational degeneracy and the planarity of methyl radical was also included in this work.

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INTRODUCTION The EPR spectra of methyl radicals stabilized in various solid gases at cryogenic temperatures received much attention from numerous investigators and have been studied through several decenniums. Methyl radical is the object of extensive experimental and theoretical treatment, since it possesses very small moment of inertia (tensor) which facilitates characteristic quantum-mechanical effects to become observable at low temperature EPR lineshapes. However, a variety of peculiarities in the EPR spectrum of methyl radicals remain unexplained despite the advances in the field.1 The overall EPR spectrum of the CH3 radical can be considered as a superposition of two kinds of hyperfine (hf) multiplets: a quartet and a doublet attributable to the nuclear (F = Σi Ii) A- and E-symmetry states, respectively, in combination to rotational states (R) that give an overall totally symmetric direct product R ⊗ F state to the D3 group radical figure.2 The lowest rotational state is the nondegenerate, totally symmetric, A irreducible representation of the D3 group, being totally symmetric also with respect to exchange of any pair of © XXXX American Chemical Society

the proton nuclei. The excited states may involve nuclear parts of both A- and E-symmetry types with different overall degeneracy. At the lowest temperatures the population A/E ratio is determined by the superimposed ground quartet and the excited E-doublets (more than one degenerate doublet contributes to the first excited rotational level). One could expect that as the sample temperature increases a delicate interplay between symmetries, degeneracies, and energies of excited states would result in nontrivial behavior of the observed peak intensities. Moreover, the ratio and its temperature dependence should be crucially sensitive to a matrix surrounding and sample preparation conditions. Actually for a higher, usually not exactly prescribed temperature, the quantum effects and the details of the symmetry become less important for the line shape of the EPR spectrum as the Received: November 21, 2012 Revised: April 3, 2013

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dx.doi.org/10.1021/jp311507q | J. Phys. Chem. A XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry A

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found that the E-doublet emerged at liquid helium temperatures in the EPR study of CH3 radicals in solid Kr.5 A close inspection of the spectra led to the conclusion that the lowtemperature doublets originated from CH3 radicals trapped in both disordered regions of the matrix and in some nonequilibrium Kr regular structure with lower symmetry compared to krypton fcc lattice.5,6 In an EPR study of methyl radicals stabilized in molecular CO and CO2 matrices at temperatures 14 K and higher, Kiljunen et al.8 found that the A/E value drops even below the expected minimum theoretical high-temperature limit of 1/2. Furthermore, a remarkable effect of the E-states being populated at relatively low temperatures more efficiently than the A-ones was clearly observed in N2 matrix.8 Namely, the intensity of the central doublet was higher at the lower sample temperature of 16.5 K than at the elevated temperature of 35.4 K. Our experimental results here extend to lower temperatures since our setup provides sample temperature down to 7 K compared to the data of Kiljunen et al.8 with lowest temperature 14 K. In light of the anomalous distribution of the hf transition intensities found by Shiga and Lund9 for the EPR spectra of methyl on silica gel, we consider our most recent experimental part devoted to the EPR of CH3 in N2O and Ar matrices. The unexpectedly low E/A amplitude ratio observed in our system at low temperatures and the results of Eloranta’s group are compared to the extraordinary nonbinomially distributed quartet of Pariiskii et al.10 on silica gel. The anomalous distribution intensity of the methyl quartet has usually been interpreted in the past as a classical spin-Hamiltonian anisotropy problem relevant to a powder spectrum with residual anisotropy.9,10 Alternatively this anomaly was considered as a cross relaxation problem due to the residual anisotropies of the g- and A-tensors, averaged by relatively fast rotation. The latter case originally refers to a more general work of Freed and Fraenkel11 and particularly Fraenkel12 on the extension of the Redfield relaxation theory including offdiagonal elements in the relaxation matrix. That theory applied to the methyl quartet allows variable broadening of the different components of the hf methyl lines and was used by several subsequent EPR investigations on methyl radical; see relevant section Classical, Higher Temperature Results in this work. In Pariiskii et al.10 there are conflicting statements describing the broadening mechanism of the EPR spectrum of methyl radical adsorbed on the surface of silica gel. At first, they state that hindering the thermal motion of the radical leads to broad and anisotropic spectra and intensity of the central components below the magnitude corresponding to the binomial distribution, in spite of the fact that they observe the opposite relation of the central to the outer spectral components. Further they claim that completely frozen radicals should give narrow lines with the binomial intensity distribution, probably referring to the well accepted fact that thermal motion becomes all too slow the closer one comes to zero absolute temperature. In this statement it is not clear what happens with the variation of the magnetic dipole−dipole interaction strength of the different parts of the polyoriented sample and the resulting inhomogeneous broadening. They return to this point later, repeating their initial statement that if the rotations were completely frozen this spectrum would consist of broad lines. This statement is again opposite to the previous one. The idea that the zero rotational level can give sharp spectra due to quantum effects was actually explained earlier by us.2 However, the fast C3-axis rotation of methyl in the first excited

classical thermodynamics and the diffusive processes are dominating. Indeed, earlier and more recent EPR measurements of the A/E amplitude ratio revealed features which although underlined by the authors received no adequate explanation. In studying CH3 radicals in solid methane, CH4, at liquid helium temperatures, Jackel and Gordy3 discovered an unexpectedly large contribution of the E-states to the resulting EPR spectrum. They nearly observed the classical, hightemperature limit methyl quartet with relative intensity ratio 1:3:3:1, for the hf transitions at the unexpectedly low 4.2 K sample temperature. Obviously the methyl radical could not reach the ground rotational state in the cold methane matrix in contrast to the cases of Ar1, Kr4, and Xe3, where the hf components obtain relative intensities 1:1:1:1. It was thus surprising that the hf components could be reduced to nearly equal intensities by the reduction of the temperature from 4.2 to 2.1 K, although this is a rather small interval. The EPR quartet of equal intensities4 of methyl radicals has the simple explanation of the radical at the free, 3D-ground rotational state with angular momentum J = 0.6 The observation of the 1:3:3:1 quartet instead could be explained by the increased radical−matrix interaction and consequently higher hindering potential leading to classical motion, which is unexpected for such a low temperature. Consequently the interpretation of this anomaly in the original article is oriented on different argumentation. In particular, Jackel and Gordy3 explained this peculiarity postulating that the required level populations, changing the intensities from 1:1.9:1.9:1 at 4.2 K to 1:1:1:1 at 2.1 K, could not be achieved by a reduction in temperature alone without, possibly, a marked increase in the effective rotation constant. This was indicating equivalently an increase of the energy spacing between the rotational levels through less restrictive interaction of the CH3 radicals with the CH4 at 2.1 than at 4.2 K. Moreover, based on the observed intensities, Jackel and Gordy3 estimated that at 2.1 K, the N = 1, K = 1 level was about 5 cm−1 (= 7.19 K) or more, above the N = 0 level (N corresponds to our used further quantum number J for the total angular momentum, and K corresponds to our M for its projection on the molecular frame). A simple computation using the classical Maxwell-Boltzmann (from here on simply Boltzmann) distribution for the rotational states shows that, based on the 5 cm−1 energy separation between the levels, the ratio should be 1:1.36:1.36:1 at 4.2 K compared to the experimental one of 1:1.9:1.9:1 at the same temperature, thus implying a rotation constant at 4.2 K smaller than the constant at 2.1 K. The solid CH4 reduces the rotational constant of the trapped radical much more than in any other matrix; on the other hand, with sample temperature decreasing from 4.2 to 2.1 K the constant markedly increases! Of course, other, less complicated explanations without such inconsistencies should be sought. The introduction of a lower effective rotation constant of the methyl radical in solid methane is an empirical way to explain the very unusual behavior not observed for the CH3 in other matrices. This behavior has possibly something to do with the fact that even the matrix CH4 molecules rotate at low temperatures. Large contribution from the first excited rotational E-type hf doublet to the EPR spectrum of CH3 in solid CH4 at 4.2 K was also observed by Wall et al.7 They measured amplitude ratio 1:2.4:2.8:1.1 for the hf transitions. More recently, it was also B



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rotational level already averages the perpendicular components to nearly the value of the parallel dipolar component, diminishing the residual hf anisotropy. Therefore, in literature, methyl EPR is known as a relatively sharp quartet, i.e., a significantly sharper spectrum than the broad powder expected from the rhombic hyperfine tensor of α-protons.13 The high temperature methyl radical EPR 1:3:3:1 quartet in inert gas matrices is well-known and has been discussed extensively over the years and contrasted to the low temperature quartet with lines of equal intensity. Similar results were obtained for methyl radical isolation inside the voids of zeolites.14 In most cases this spectrum is rather isotropic and relatively sharp. The situation becomes very different when the methyl radical is adsorbed on the inner surface of VYCOR glass15−17 or silica gel.9,10 Pariiskii et al.10 are inconsistent in their article by using both the terms “intensity” and “amplitude” for the same ratio of their experimental methyl quartet transitions. They write, namely, that the “intensity” distribution 1:8.5:13:2.5 does not satisfy the binomial law 1:3:3:1, while, in a different part, they refer to the same ratio as unusual “amplitude” ratio.

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NONROTATING METHYL

Of course, it is difficult to register the classical powder spectrum of the expected well-known rhombic hf interaction A = (−30.9, −93.7, −60.3) MHz of the α-proton19 in the presence of the sharp quartet of the methyl radical in the lowest rotation quantum state. However, certain workers have seen remnants of this nonrotating methyl spectrum, appearing as a weak signal emerging a little above the baseline (see, e.g., the CH3 spectrum in CO2 matrix at 14.4 K, Kiljunen et al.8 figure 4). We undertook new EPR experiments on CH3 isolated in solid N2O and found a similar resonance of nonrotating methyl at 9 K. Furthermore it is possible to study the nonrotating methyl spectrum in more detail, also proposing an appropriate experimental adjustment improving its observation. Namely, an increase of the MW power would selectively saturate the narrow stopped rotor quartet of methyl radical in the first rotational level with angular momentum J = 0 and would bring out the nonrotating weak powder signal from the baseline. We observed those lines in solid CO2, see Figure 1a. Our results that are still under further development support the finding by Kiljunen et al.8 The corresponding splitting was also measured to be close to the value reported by Kiljunen et al.,8 while somewhat different from that in N2O. Here, we measured Axx = −1.47 mT, Ayy = −3.83 mT. The third pair of weak lines turned out to be unresolved because it was superimposed on the two much stronger central axially symmetric transitions. Thus, an approximate magnitude of the third principal value Azz = −2.3 mT may be assessed. The obtained parameters match well those reported previously:8 Axx = −1.5 mT, Ayy = −3.8 mT, Azz = −2.3 mT. We found also out that the intensity of the rhombic feature changes reversibly with the sample temperature; see Figure 1a. A very interesting peculiarity of the EPR spectrum of CH3 radicals in solid N2O is the presence of weak transitions clearly seen at low sample temperatures, Figure 1b. Similar transitions were obtained very recently in EPR study of methyl radicals in molecular solids by Kiljunen et al.8 from 2010. The spectrum of the methyl radicals in CO2 matrix, besides a strong axially symmetrical quartet, revealed weak transitions which the authors tentatively assigned to nonrotating methyl. The overall EPR spectrum was then considered as a superposition of two spectra related to radicals trapped in sites of two kinds. For the N2O matrix, the spacing between adjacent y-lines obtained experimentally equals 3.40 mT. While some very weak lines may be discerned between the mF = 1/2 and mF = −1/2 strong axial transitions (see Figure 1b), we are not certain which ones may be assigned to the x component of the E-line pattern. Comparing the Ayy components for CH3 in these two matrices, we notice a difference which is most probably due to the matrix shift of the hyperfine parameters as well as to different averaging of these parameters by the radical motion. Also, a new finding is that both matrices exhibit temperature dependence of the nonrotating CH3 spectrum intensity which decreases with increasing sample temperature. Simultaneously, the E-pattern of the J = 1 rotational state increases. We simulated the EPR spectrum of CH3 in N2O superimposing the A-quartet and a powder-shaped spectrum of nonrotating radicals, Figure 2. Theory predicts a pronounced rhombic anisotropy of the hyperfine interaction of the CH3 unpaired electron with the protons. Kubota and co-workers19 confirmed this prediction,

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EXPERIMENTAL SECTION The experimental technique was described in detail elsewhere.6,13 It will briefly be outlined here for convenience. The solid samples under study were obtained by gas condensation on the thin-walled bottom of a quartz finger located at the center of the evacuated microwave cavity of the EPR spectrometer. The bottom served as the substrate for deposited samples. A Teflon tube of small diameter inserted into the quartz finger supplied liquid He vapor to the finger bottom. The sample temperature controlled by the volume of the He vapor flow could be set in the range 6−80 K and was measured using Ge film on a GaAs resistance thermometer18 supplied by the V. Lashkaryov Institute of Semiconductor Physics and MicroSensor Company, Kiev (V. F. Mitin, MicroSensor available from http://www.microsensor.com.ua). The thermometer was attached to “Triton” temperature gauge (http://terex.kiev.ua). The gases were supplied to the substrate through two channels: the first one directs the gas flow through a glass discharge tube prior the condensation, while the second one allowed the gas flow to be set onto the substrate avoiding the gas discharge zone. An electrode-less pulsed RF gas discharge was excited in the glass tube. The substrate temperature during deposition was about 17 K, for Ar matrix experiments, and in the range 16−25 K, for N2O matrix experiments. A He-CH4 (KLIM Company Ltd., www.klim-spb.ru) mixture was passed through the discharge, while pure gaseous Ar (Linde Gas) or N2O (KLIM Company Ltd.) was fed to the substrate avoiding the gas discharge. Based on the concentration of the gaseous helium−methane mixture (1.5 mol % CH4 for the Ar matrix experiments, and 2.5−10 mol %, for the N2O matrix experiments), the measured gas consumptions from the storage balloons during deposition, and the geometry of the deposition part of the equipment, the impurity content of CH4 was estimated to be 0.14% and 0.2−1% in Ar and N2O matrices, respectively. The He atoms made no contribution to the impurity content because of the very small He adsorption at “such a high” sample temperature. Deposition duration was from half an hour to one hour. C



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Figure 2. Simulated EPR spectrum of CH3 in solid N2O at 9.3 K (see Figure 1b) obtained as a superposition of the A-line axially symmetric pattern, Figure 1b, and the E-line of orthorhombic symmetry (see inset).

for the proton hyperfine coupling tensor of CH3 radicals in CH3COONa·3D2O single crystal. The components of the tensor were determined to be Axxcryst = −31.4 MHz, Ayycryst = −93.7 MHz, Azzcryst = −60.3 MHz, Aisocryst = −61.8 MHz, or, in magnetic field unit, Axxcryst = −1.120 mT, Ayycryst =−3.344 mT, Azzcryst = −2.152 mT, Aisocryst = −2.205 mT. Notice also figure 1 in Shiga and Lund,9 and figures 1a and c in Shiga, Yamaoka, and Lund,20 where weak satellite lines close to the inner CH3 doublet were interpreted as spin-flip transitions due to interaction with protons of the matrix. However, no such satellites are seen in the vicinity of the outer CH3 hyperfine transitions, and in addition, the intensity of these lines is temperature dependent, becoming weaker with increasing sample temperature from 4 to 183 K, in figures 1a and c in Shiga, Yamaoka, and Lund.20 These facts indicate that these are not spin-flip satellites but rather the nonrotating methyl observed by Kiljunen et al. in CO2 matrix,8 and in our study, in N2O and in CO2 matrices; see Figure 1. Moreover, the magnetic field splitting between satellite transitions estimated from the spectra presented by Shiga, Yamaoka, and Lund20 gives Ayy = 3.4 mT in good agreement with the N2O and CO2 results. Two types of the EPR satellite transitions were observed by Brotikovskii et al.21 for CH3 trapped on the silica gel surface at sample temperatures 4.2 and 77 K. The satellites of the first type symmetrically accompanied each of the main quartet transitions and were seen at both low and high sample temperatures. This satellite spectrum was tentatively explained as being due to the hf interaction with the protons of the matrix. The satellites of the second type (two weak pairs of lines) accompanied only the inner transitions of the main quartet and were located nonsymmetrically with respect to each main transition. Moreover, the satellites of this type vanished reversibly with increasing sample temperature. In order to explain these additional satellites, the authors suggested a model of quantized free rotation. The temperature behavior of the satellites of the second type was exactly the same for the satellites in the EPR of CH3 in N2O and CO2 matrices of the present work. Also, the splittings between the satellites, 1.43 mT and 3.64 mT, agreed with those measured, in our

Figure 1. a. Isolated CH3 in CO2 matrix. The figure shows temperature dependence of two pairs of weak transitions attributed to the nonrotating methyl EPR spectrum. b. EPR spectra of CH3 radicals trapped in N2O solid. The spectra were recorded at different sample temperatures. The line positions are recalculated with respect to the EPR resonance frequency f res = 9391.37 MHz. The substrate temperature during deposition was Tsub = 17 K. The indicated components of the anisotropic hyperfine tensor for the proton, Ayy and Azz, are assigned to the nonrotating CH3 radical. Weak transitions of the nonrotating radical which contribute to the Azz spacing overlap with the main axially symmetric pattern and, therefore, cannot be resolved. The figure shows, particularly, a temperature dependence of the amplitudes of the additional weak y-pattern as well as the change in the shape of the mF = 3/2 hyperfine transition of the “A” symmetry quartet. D



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experiments, in solid N2O, Ayy = 3.4 mT, and CO2, Axx = 1.47 mT, Ayy = 3.83 mT. Two types of satellites were found also by Kubota et al.22 in EPR of methyl radicals produced on silica-gel surfaces. By measurements of both the X- and K-band EPR spectra, it was found that one type of the satellite lines was affected by the magnetic field, changing the splitting between transitions, while the other type was not. The first satellite multiplet can thus be due to the spin-flip of the electron of the methyl radical interacting with the protons of the crystallized water. The behavior of the second satellite multiplet is just what one would expect for the splitting of the nonrotating CH3. Thus, the experimental results obtained in solid gases of linear molecules verify that methyl radicals at the surface of the silica gel are adsorbed, in part, at sites where rotation is prohibited. The following new facts were also revealed by our CO2 study: (i) The temperature dependence of the EPR spectrum intensity of the nonrotating CH3 in both the CO2 and N2O matrices is reversible. This is very useful information, which may shed light on the origin of the nonrotating radicals. (ii) By comparing the hf splittings, we found matrix effect on this parameter. This effect of the “nonrotating” spectrum may be attributed to the fact that for methyl radicals in both N2O and possibly CO2, slow rotation around the C3 axes is allowed. This rotation results in partial averaging of the hf tensor components. The rotation is faster in N2O compared to CO2, so that the hf components in these two matrices are averaged to different extent. (iii) We brought attention to the fact that the nonrotating CH3 was probably also present in silica gels, except for matrix isolated methyl radicals.

Such an asymmetry of the individual lines may occur, e.g., because of fast passage condition for the magnetic field sweep rate. Later on, Shiga and Lund9 reported a well resolved axially symmetrical EPR powder spectrum of CH3 radicals on a silica gel surface at 77 K. The experimental line shape was interpreted by assuming rapid radical reorientation about the fixed C3symmetry axes. The least line width in the experiments by Shiga and Lund was half than the relevant value published by Pariiskii et al.10 rendering the residual anisotropy in the latter work not as evident at first sight. To demonstrate this suggestion, we simulated three different EPR spectra as follows. The first simulation in Figure 3 is based

Figure 3. Simulated EPR spectrum: CH3 adsorbed on silica gel at 77 K. The EPR parameters are taken from Shiga and Lund.9

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SPECTRAL SIMULATIONS In order to approach a solution of the E/A ratio problem of the CH3 radical adsorbed at the surface of porous materials (silica gel, VYCOR glass, zeolites) we first consider relevant experimental EPR spectra of the literature. Using the photochemical method for producing methyl radicals adsorbed on a solid surface, Pariiskii et al.10 recorded the EPR spectrum of methyl in a silica gel sample at 77 K and obtained the unusual amplitude ratio, 1:8.5:13:2.5, of the hyperfine components. The linewidths were in the range 0.1− 0.2 mT. The noticeably nonbinomial amplitude sequence was attributed by the authors to incomplete averaging of the radical EPR parameters due to the restricted rotation of the radical bound to the surface. While the assumption seems promising in explaining the ratio of the amplitudes, a question emerges about the relative intensities of the inner to the outer hf transitions. The two inner lines of the methyl quartet are a superposition of the inner transitions of the A-quartet and the E-doublet. Their intensity relative to the outer lines provides information on the rotational states population. Based on the fact that the line intensity is proportional to the line amplitude multiplied by the squared line width and on the ratio of the widths of the third, MF = −1/2, and first, MF = 3/2, hf components taken as 2:1 were obtained for these two components, the intensity ratio close to expected hightemperature limit 3:1. However, the situation is not that simple. The hf components recorded by Pariiskii et al.10 are slightly asymmetrical with respect to the baseline. This asymmetry does not quite resemble an axial one because all hf components have the upper part larger than the lower part.

on the EPR parameters reported by Shiga and Lund9 for temperature 77 K. In particular here, A⊥ = −22.6 G, A|| = −21.9 G, g⊥ = 2.0027, g|| = 2.0024. A line width of 0.5 G and Lorentzian shape of the individual components was assumed. The experimentally obtained amplitude ratio 1.0:4.5:8.3:2.2 by Shiga, Yamaoka, and Lund20 is not far from the one which the simulation in Figure 3 yields, i.e., 1:4.1:6.2:1.8. The EPR parameters for the A-line quartet and E-line doublet were set to be the same. The E-line amplitude was assumed to relate to the A-line amplitude as 2:1 (high-temperature conditions or classical/diffusion limit). The second simulation in Figure 4 aimed at analyzing the spectrum by Pariiskii et al.10 for temperature 77 K is based on the same EPR parameters as in Figure 3 except the individual line width which is set to be 1 G. The simulated relative amplitude ratio is 1:3.68:4.28:1.4. The widths of lines vary between 1.06 and 1.74 G which fairly agrees with the experimental5 1 and 2 G, respectively. Because the simulated amplitude ratio was in poor agreement with the experimental one, we also tested the Gaussian profile of the individual lines, Figure 5. This attempt provided the ratio 1:4.25:5.28:1.7 which does not shift the calculated value much closer to the experimental result, 1:8.5:13:2.5. Moreover, the widths of the calculated lines changed from 1.03 G through 1.44 G, thus, not matching well the targeted experimental ones. The analysis by Pariiskii et al.10 shows that the experimental amplitude ratio may be well fitted if one adopts an E/A amplitude ratio greater than 2/1, which is the classical hightemperature limit. This means identically an increased E



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the symmetry allowed energy spin-rotation levels to be developed.

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CLASSICAL, HIGHER TEMPERATURE RESULTS We start with reviewing and assessing earlier results obtained by theory based on the classical spin Hamiltonian and relaxation due to tumbling of the radical at the relatively high liquid Nitrogen temperature of 77 K, or higher. Mainly the case of the differential broadening of the hf transitions of methyl due to the g- and A-tensor cross relaxation is discussed, in particular the linear dependence on the nuclear spin projection24 ± MI that results in line asymmetry with respect to the center of the multiplet. The details of this issue are given in eqs 1 to 3. Notice here that MI stands for the total nuclear spin projection MF. The cross relaxation can also contribute to the line asymmetry with respect to the baseline as could be anticipated considering the limiting case of no motion (powder spectrum), which is approximately seen in the simulations of our spectra where the effects of the g- and A-tensor anisotropy are cooperating, in particular, in forming the complicated shape of the MI = 3/2 transition. In an EPR study of methyl radicals adsorbed at a silica gel surface at 77 K, Gardner and Casey25 observed a marked dependence of the hf components’ line width on the nuclear spin quantum number. Because of the rather large widths (above 0.08 mT) the A- and E-transitions for CH3 were not resolved. The line width was interpreted in terms of the tumbling of the radicals on the surface of the silica gel, where the anisotropies in the hyperfine and g tensors gave rise to the above-mentioned cross relaxation mechanism dependence on MF.25−27 They also reviewed possible characteristics of the barriers giving rise to the hindering of the radical rotation on the silica gel surface. The authors obtained correlation times of 50 and 79 ns for CH3 and CD3, respectively. These relatively low28 but still sufficiently rapid correlation times to averageat least partiallythe g- and A-tensor anisotropies indicate significantly hindered motion, Friebele et al.29 In particular, Gardner and Casey25 discussed the electrostatic interactions between the methyl radical and the silica surface as a cause of the potential barrier of the radical rotation. These were as follows: the interaction of the methyl radical electric quadrupole moment with the field gradient at the surface, and second-order polarization effects, which could give rise to the radical nonvanishing dipole moment and, hence, to the dipole−surface interaction. However, because of these uncertainties, the exact nature of “tumbling” remained somewhat obscure. It becomes obvious that more elaborate theory is needed to calculate the barrier ab initio such as, e.g., the work of Takada and Tachikawa30 (see further) and the group of Eloranta.8 We agree with Gardner and Casey25 in their estimation that the barrier penetration at 77 K should not be important, but based on the experimental data we propose some other quantum mechanical statistics effects in the following sections. Furthermore, Gardner and Casey reported EPR spectrum of the CH3 radicals which displayed noticeable asymmetry of the transitions with respect to the baseline. Discussing this feature, the authors referred to the incomplete averaging of the g-tensor anisotropy as a possible reason for a part of this asymmetry. However, they excluded, after a brief consideration, their suggestion and named the asymmetry an “unexplained feature”. We should add here that according to the above estimated

Figure 4. Simulated EPR spectrum: CH3 adsorbed on silica gel at temperature 77 K. All EPR parameters are taken from Shiga and Lund9 except for the individual line width which is set to 1 G.

Figure 5. Simulated EPR spectrum: CH3 adsorbed on silica gel. The EPR parameters match those of the spectrum in Figure 4, except that the Gaussian profile substitutes the Lorentzian one.

population of the levels with E-nuclear symmetry due to a higher degeneracy of the excited rotation levels. We notice that there are really significant differences in the present simulation method and the simulation method applied in the paper by Shiga and Lund who used the method developed by Lefebvre and Maruani.23 That method treats powder spectra of systems with significant anisotropy of the magnetic tensors and in addition essentially static systems, i.e., in the absence of motional dynamics. In our case, the residual Zeeman and hf anisotropy from the g- and particularly the Atensor is exceptionally small, due to the efficient rotational averaging of the perpendicular components of the hf tensor rendering the hf parameters almost isotropic, even at the lowest rotational level of the radical2 at ca. 5 K. The present simulation method offers the opportunity to separate the spectral simulation in A and E lines and assigning to them arbitrary relative weight according to the symmetry requirements that quantum mechanics of low temperature impose; see further details in Figure 8. Next, the classical theory of differential, hyperfine-component, nuclear-spin-projection dependent line width will be discussed. Later the rotational energy diagram of methyl will be reviewed and the implication of the population distribution of F



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component, because of the much larger line width of about 0.049 mT of this hf transition. The lines were nonsymmetrical with respect to the baseline and their widths were dependent on the nuclear spin quantum number MF. Furthermore, Friebele et al.29 shortly discussed the observed asymmetry, noting that the powder lineshapes of the methyl radical indicated uniformly averaged g-tensor and hf interactions over random orientation of the radical in the glass. Considering the temperature dependence of the EPR spectrum of the radical in irradiated silica, the authors concluded that the radicals tumbled at a rate which was sufficiently rapid to average at least partially these anisotropies. On the other hand, one can easily see from the spectrum reported in Friebele et al. that the CH3 transitions are characteristic of the EPR pattern for axially symmetric Aand g-tensors. They first consider the results of Kazanskii and Pariiskii33 that the linewidths of a spectrum characterized by incompletely averaged g-tensor anisotropy will depend on MI.2 They conclude that this theory is not adequate since the broadening of the hf components of their quartet spectrum should be symmetric about its center. Finally, they followed the work of Gardner and Casey25 and recognized the full g- and Atensor cross-relaxation mechanism which relates this kind of anisotropy with the linear term in MI; see eqs 1 to 3. Despite the marked asymmetry, Friebele et al. presumed that the tumbling was sufficiently rapid to average the anisotropies completely. We will just discuss some aspects concerning this issue. The above theory is based on the Redfield relaxation theory, requiring the so-called strong narrowing conditions,1 i.e., |Hhf | τR < 1, which is inaccurate in the case of methyl on silica gel surfaces as shown further. Here, we denote the rootmean-square or the standard deviation of the hf interaction by |Hhf | = ⟨|Hhf(t)|2⟩1/2. Friebele et al.,29 for example, estimated rotational frequency νR = 11 MHz for methyl radicals in their system, or equivalently correlation time is τR ≈ 9.1 × 10−8 s, and therefore, assuming hf anisotropy of ca. 30 MHz for planar methyl radicals,

relatively slow reorientation correlation times it is obvious that neither the g-tensor nor the A-tensor can be considered totally averaged. Relatively slow motion is namely a requirement of significant contribution for the cross relaxation mechanism according to Redfield theory.31 This fact is reinforced due to the nature of the transverse relaxation with main contribution from the secular terms of the time-dependent perturbation with spectral densities at zero frequencies, j(0)’s in eq 1, proportional to the rotational correlation time τR. Considering that τR increases with decreasing temperature T, it is obvious why slow motions enhance the cross correlation. We reproduce here the relevant equation from Fraenkel,12 where exponents D and G of the spectral densities j(0)’s stand for Dipole and G-tensor, respectively. T2−1(MI ) = j(D) (0)[I(I + 1) + (5/3)MI2] + −1 (16/3)j(DG) (0)B0 MI + (8/3)j(G) (0)B02 + T2,0

(1)

in which one can recognize the corresponding terms of the following eq 1′. T2−1 = a0 + a1MI + a 2MI 2

(1′)

thus indicating also the final form of the linear term in MI seen in eq 2. a1 = (4/15)bΔg Β0 τR

(2)

where b = 2(A|| − A⊥)/3 is the hyperfine anisotropy, B0 is the external magnetic field, Δg = g|| − g⊥ the g-tensor anisotropy, and τR is the rotational correlation time. The a0 and the corresponding T2,0 terms of eq 1 refer to MI -independent, or cross-relaxation independent, mechanisms. Thus, the cross-relaxation contribution to the broadening Δ(MI ) = [|γe|T2(MI )]−1

(3)

−1

is proportional to T2 ∝ τR as was anticipated above. Consequently, at least for the non deuterated methyl radical and in spite of the fact that the contribution of nonsecular relaxation terms is not important at relatively slow motions, there is still an additional problem with the application of the Redfield theory of relaxation. One simply has entered in the slow motion regime where the Redfield theory is invalid;1 see further. Barnes et al.32 studied both “normal” and “abnormal” methyl radicals produced on a silica gel surfaces by UV photolysis of adsorbed iodomethane. According to their definition, “normal” methyl radicals with hf splitting 2.3 mT were sufficiently stable and were observed on a number of surfaces, while abnormal methyl radicals with hf splitting of 1.9 to 2.1 mT were less stable. The EPR spectrum obtained at 77 K showed no secondorder splitting and was a partly resolved superposition of two quartets of hyperfine lines. As in the previous study by Gardner and Casey,25 the dependence of the line width on the nuclear spin quantum number MF was observed. Moreover, some asymmetry of the hf transitions with respect to the baseline could be discernible in the reported spectrum of the normal methyl radicals. Barnes et al.32 estimated correlation times 93 and 298 ns, for the normal and abnormal methyl radicals, respectively. Friebele et al.29 observed methyl radicals in irradiated highpurity synthetic silicas by EPR. Within the narrow MF = −1/2 hyperfine component, they partly resolved the second-order splitting, but this splitting was obscured for the MF = 1/2

|Hhf |τR ≈ 30 × 106 Hz × 9.1 × 10−8 s = 2.73 > 1

(4)

The higher than unity value of this product invalidates the Redfield theory, actually indicating that a more accurate slow motion theory would actually be needed. In that case the spectrum is expected to display certain anisotropic characteristics of powders. The same can be said for the data of Barnes et al.32 who estimate correlation times 93 to 298 giving values that were too high, ca. 2.79 and 8.94, respectively, for the product |Hhf |τR of eq 4, prohibiting applicability of Redfield’s theory. Even the value of the correlation time τC = 50 ns estimated by Gardner and Casey25 gives the value |Hhf |τR ≈ 1.5 rad to the characteristic product of Redfield theory in eq 4, rendering its applicability questionable. Furthermore, all the above investigators imply nearly full averaging of the anisotropy of the magnetic tensors through a relatively fast motion, i.e., ωSτR < 1, i.e., extreme narrowing conditions. On the contrary for all the above cases within the Xband (ν0S ≈ 9.4 GHz) we find relatively slow motion conditions, as discussed above in connection to the significance of the cross correlation term. Considering, e.g., the largest correlation time τR = 298 ns in Barnes et al.32 we find ωSτR = 1.76 × 104 ≫1. Even the smallest correlation time τR = 50 ns in Gardner and Casey25 gives ωSτR = 2.95 × 103 ≫ 1. G



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In their work, Gardner and Casey25 mentioned in addition the mixing of the real and the imaginary parts of the resonant magnetic susceptibility as a possible mechanism accounting for the asymmetry of the individual transitions with respect to the baseline. The effect was discussed also by Rogers and Pake34 for VO2+ in solution. In the latter work about VO2+, the experimentally observed symmetry ratio (the ratio of the baseline to maximum amplitude derivative heights) does not show any MF dependence. However, the symmetry ratio does vary with MF in the works by Gardner and Casey25 and by Friebele et al.29 and probably depends on MF also in the study by Barnes et al.32 Based on the assumption that the anisotropies were averaged completely by the radical tumbling, Friebele et al.29 calculated correlation time as 92 ns. Referring to the work by Gardner and Casey,25 the authors asserted that the presence of the silica gel network substantially hindered the rotation of the methyl radical, as compared with the radical adsorbed on a silica gel surface. However, the results by Barnes et al.32 suggest that the hindrance of the radical rotation also depends on the formation technique of the radical, giving correlation time equal to the value reported by Friebele et al.29 for CH3 in the bulk of the fused silica gel. The EPR line shape reported by Pariiskii et al.10 was almost isotropic, while slight anisotropy could be discerned in the MF = 3/2 hf component. The width of the lines ranged from 1 to 2 G. The authors attributed the observed nonbinomial hf amplitude ratio to the incomplete neutralization of the anisotropic hyperfine and spin−orbital interactions. The above consideration suggests that though the MF dependent relaxation cannot be excluded as contributing to the nonbinomial amplitude ratio, the major contribution comes from residual EPR anisotropies that are due to the fairly slow radical reorientation about the in-plane symmetry axes, while the radical itself executes fast rotation about the C3-symmetry axis. Thus, in most studies, the CH3 radical is believed to occupy the silanol Si−OH site, model I.30 We will refer next to the energy diagram of Figure 8, displaying the regular free, 3D methyl rotation properties, after symmetry adaptation due to spin−rotation coupling. It will thus be investigated if by specifying detailed rotational level populations using quantum theory can explain the anomalies of the EPR hf quartet relative intensities for methyl radical adsorbed on porous materials, in particular.

comparatively weak intermolecular interaction, enables the C3 axes reorientation with the result of significantly averaged anisotropies. The experimental spectrum of CH3 in N2O in Figure 6 was simulated by superimposing the hf transitions of the unresolved

Figure 6. Experimental EPR spectrum and simulation of CH3 in solid N2O recorded at sample temperature 30.0 K; microwave resonance frequency f res = 9391.37 MHz. (a) Experimental spectrum. (b) Simulated spectrum, sum of the following two: (c) simulated E-line doublet, (d) simulated A-line quartet.

ground A-quartet and the first excited E-doublet35 of the spinrotation states. Based on these line shape simulations, the ratio of the E- to the A-line intensities against the sample temperature are plotted in the Appendix (blue asterisks, Figure A.1), assuming that they follow the population ratios PE/PA of the rotational levels. The simulations were based on the positions in magnetic field of the extrema of the experimental lines and the relative peak intensities of the lines. Gaussian profile and equal widths of the individual lines were suggested for both the quartet and the doublet. For the EPR parameters used in the simulation procedure, see Table 1. The inserts show the hf MF = 3/2 and MF = −1/2 components in detail. They demonstrate how well the simulations (dashed lines) follow the different shapes of the inner and outer hf components. The simulation procedure was very complicated because, generally, the quartet and the doublet are expected to have different EPR parameters. This situation owes the variations in the radical−matrix interaction for the ground and excited rotation states experienced by the trapped radical which is due, in turn, to the difference in the spatial/rotational wavefunctions of the states. Kiljunen et al.8 stressed that “since these (A- and E-) rotational states differ in their spatial characteristics, they should exhibit a varying degree of coupling to the surrounding cage”. Thus, the molecule in these states should be considered as rotating (or oscillating) in slightly different potential wells.

■

EXPERIMENTAL ROTATIONAL LEVEL OCCUPATION Coping with the E/A amplitude ratio problem, we compared literature data, mainly from methyl adsorbed in porous material, with older and new experiments on CH3 radicals in the bulk of solid gases. To cover a variety of the possible radical motions, we tested solid N2O and Ar matrices, corresponding to the two extremes, an overall cylindrical potential, and a relatively inert spherical matrix environment about the radical, respectively. The first methyl-isolation environment is a matrix of partially oriented, linear molecules which keeps the C3 axes of the trapped radical pointing at a specific direction. This cylindrical environment results in an axially symmetrical EPR spectrum with anisotropic line splitting partially obscured by a rather large line width. The major broadening owes to the superhyperfine (shf) interaction between the radical unpaired electron and the magnetic moments of the matrix nuclei. The second one, i.e., the matrix of spherical Ar particles with a H



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Table 1. EPR Parameters for the A- and E-Transitions Which Were Used in EPR Spectra Simulation for the Experimental CH3 Spectrum Obtained at 30.0 K in Solid N2O seen in Figure 6a A-line E-line a

A⊥ G

A|| G

Aiso G

g⊥

g||

giso

ΔH⊥ G

ΔH|| G

23.30 23.60

21.95 21.40

22.85 22.87

2.002672 2.002672

2.002261 2.002261

2.002535 2.002535

1.26 1.26

1.10 1.10

Gaussian profile of the individual lines was assumed.

Table 2. List PE/PA Ratios Based on Experimental Results Available from the Literature for CH3 Radical Adsorbed on the Surface or/and Trapped in the Bulk of Porous Matrices

It turned out that the method of extraction of the experimental E/A ratio intensities from the EPR line shape required special care leading to two possible methods. The details of the assessment of the most appropriate method applied in the following are described in the Appendix. The experimental high-resolution EPR spectrum of CH3 in solid Ar12 permitted separation of the A-quartet and the Edoublet. At low temperatures the MF = 1/2 hf transition of the doublet exhibited anisotropic splitting. On the contrary the MF = −1/2 hf transition of the E-state did not show splitting at any temperature but furthermore exhibited constant peak-to-peak width down to approximately 16 K, the lowest sample temperature accessible for analyzing the intensity ratio with the simulation procedure. Furthermore at the latter temperature the extrema of the MF = −1/2 component of the E-line spectrum superimposing on the relevant A-line are reliably seen. The same holds for the relevant transition of the A-quartet. Based on this finding, and adopting Gaussian profile at all the studied temperatures for the quartet and doublet, we correlated the E/A population ratio to the E/A peak intensity ratio measured for MF = −1/2 hyperfine components. The results recalculated as inner-to-outer intensity ratios are plotted in the Appendix (Figure A.2) together with data by Kiljunen et al.8 Similarly to the simulation procedure that led to the PE/PA ratio data applied to CH3 in solid N2O, the experimental data obtained for Ar matrix were markedly below the calculated gasphase points, suggesting that this, again, may be due to the faint variation in the profile of the E-doublet. Therefore we extracted a corrected inner-to-outer ratio using the double integration procedure. The results, open circles in Figure A.2 in the Appendix, show the ratio shifted to larger values, revealing, thus, the relative intensity variation of the E- to the A-profile. Methyl Radical Adsorbed in Porous Materials. Table 2 accumulates data on PE/PA ratios which are evaluated based on experimental data available from the literature for methyl, CH3, radical trapped on the surface and in the bulk of porous matrices. Since the information is rather scarce, we present not only reported ratios, but also ratios which could be obtained from reported hf peak intensity sequences and/or linewidths and/or published spectra. The experimentally obtained amplitude ratio 1.0:4.5:8.3:2.2 by Shiga, Yamaoka, and Lund20 is close to simulation results, 1:4.1:6.2:1.8, obtained with the 2:1 E/A amplitude ratio, and 1:5.5:8.5:1.8, obtained with the 3:1 E/A amplitude ratio. The double integration procedure performed for the published spectra yields 3.1:1 PE/ PA amplitude ratio. That is why the table lists the results assessed from the paper by Shiga, Yamaoka, and Lund as 2−3. Figure 7 shows PE/PA ratios based on experimental results for CH3 radical adsorbed on the surface or/and trapped in the bulk of porous matrices available from the literature. Notice that most existing literature data were conveniently obtained at the single temperature of 77 K. As for the “porous” data points included in Figure 7 it is the PE/PA ratios of 5−6 that squeeze the curves to the abscissa.

# in Figure

T, K

PE/PA

1 2

Silica gel surface H-ρ zeolite

77 250

2−3 4−5

3 4

77 100

2.2 2

5

Sol−gel silicas Fumed ∼7-nm-sized silica nanoparticles Silica gel surface

77

3

6

Silica gel surface

77

2

7 8

Silica gel surface Metal ion-supported silica-gel, Mg/SiO2 Porous Vycor glass

77 77

6 4−6

77

3.5− 4.5

9

matrix/surface

source Assessed from ref 9 For “B-methyl radical” assessed from ref 14 Reported in ref 36 Reported in ref 37 For Me(n1) assessed from ref 21 For Me(n2) assessed from ref 21 Assessed from ref 10 Abnormal CH3, assessed from ref 38 Abnormal 12CH3, assessed from ref 39

Figure 7. Black open squares: PE/PA ratios based on experimental results available from the literature for CH3 radical adsorbed on the surface or/and trapped in the bulk of porous matrices. Theoretical curves: Boltzmann statistics, solid light blue line for C3-symmetric pyramidal methyl; D3-symmetry planar methyl geometry, magenta dashed-dotted-dotted line. Black open circles: free methyl rotor ratio obtained from calculations published by Kiljunen et al.8 See Table 2 for the data numbering.

This figure shows furthermore the difficulty to explain the PE/ PA ratio anomalies of the existing experimental data, not only by the present theory, but also by other currently known theoretical treatments.

■

SPIN-ROTATION LEVELS OF METHYL There are several facts that should be considered when analyzing the influence of the populations of the rotational states on the EPR spectrum of methyl radical. First of all, it is not sure that for the lowest experimental temperatures the Boltzmann distribution is valid. On the contrary, the quantum statistics of Bosons and Fermions differ from both the wellI



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Figure 8. Model of the symmetric-top free-methyl rotor, i.e., a planar methyl fragment of D3 symmetry6 considered in three dimensions. The rotational C3- and C2-axes are indicated in the figure while the three protons are indexed by −1, 0, and 1. The rotational energy level diagram indicates that the first excited “parallel” rotation level |1 ± 1 L⟩ (molecular frame angular momentum projections M = ±1) has totally sixfold rotational degeneracy due to the laboratory projection L of the angular momentum. As experimental temperature the liquid He temperature is indicated in the energy diagram.6

Table 3. Energy, Nuclear Symmetry, and Degeneracy of the Allowed Spin-Rotation Levels for a Free 3D-Rotor of Pyramidal C3 Symmetrya |J M L⟩ rotation state |0 |1 |1 |2 |2 |2 |3 |3 |3 |3 a

0 0⟩ ±1 L⟩ 0 L⟩ ±2 L⟩ ±1 L⟩ 0 L⟩ ±3 L⟩ ±2 L⟩ ±1 L⟩ 0 L⟩

EM/B

C3 F-nuclear symmetry

symmetry allowed rotation degeneracy

nuclear spin multiplet F

E/K

0 3B 4B 8B 11 B 12 B 15 B 20 B 23 B 24 B

A E A E E A A E E A

1 6 3 10 10 5 14 14 14 7

3/2 1/2 3/2 1/2 1/2 3/2 3/2 1/2 1/2 3/2

0 20.28 27.04 54.08 74.36 81.12 101.4 135.2 155.48 162.24 811.2

The nuclear F-multiplets are included. The computed rotational constant B has the approximate value 6.76 K.

known Boltzmann and from each other at very low temperatures. The distribution according to quantum statistics is not always compatible with the classical view of Boltzmann and the equipartition of energy that always gives the amount kT/2 of thermal energy for each motional degree of freedom, also including harmonic potentials. The classical description is even worse when thermal energy is simply absent from an isolated quantum mechanical system, since the used matrices are considered relatively inert.

This is our case, at experimental temperatures as low as 5 K where the thermal energy kT is too low compared to the energy spacing of methyl radical quantum rotation, rendering the system unable to occupy even the first rotational level. At a little higher temperature, ca. 7 K, the first few rotational levels start to become occupied. Therefore, either Fermi−Dirac or Bose−Einstein statistics become appropriate instead of Maxwell−Boltzmann and should be used depending on the particularly isotope (proton or deuteron) substituted methyl rotor under study.2 The quantum nature of the system at these J



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Table 4. Energy, Nuclear Symmetry, and Degeneracy of the Allowed Spin-Rotation Levels for a Free 3D-Rotor of Planar D3 Symmetrya |J M L⟩ rotation state |0 |1 |1 |2 |2 |2 |3 |3 |3 |3 a

0 0> ±1 L⟩ 0 L⟩ ±2 L⟩ ±1 L⟩ 0 L⟩ ±3 L⟩ ±2 L⟩ ±1 L⟩ 0 L⟩

EM/B

D3 F-nuclear symmetry

symmetry allowed rotation degeneracy

nuclear spin multiplet F

E/K

0 3B 4B 8B 11 B 12 B 15 B 20 B 23 B 24 B

A1 E E E A1 A1 E E -

1 3 5 10 5 7 14 7 -

3/2 1/2 forbidden 1/2 1/2 3/2 3/2 1/2 1/2 forbidden

0 20.28 54.08 74.36 81.12 101.4 135.2 155.48 621.92

The nuclear F-multiplets are included. The computed rotational constant B has the approximate value 6.76 K.

since the carbon mass dominates the total weight allowing the center of mass to remain very close to carbon atom. If a significant amount of the potential energy of the matrix− radical interaction must be added to the kinetic energy, it will increase the total quantum energy of the rotor. However, it will not significantly affect the interval of the energy levels compared to the value of the rotational constant B of methyl rotor, ca. 7 K, seen in the vertical axis of Figure 8. The important issue of the population occupancy here is, however, the additional adjustment by considering the degeneracy of the angular momentum laboratory projection quantum number L of the rotational states; see Figure 8. In that case we totally obtain a sixfold rotational degeneracy in |1, ±1, L⟩, the first excited rotational level, actually all of them E symmetry nuclear spin doublets; see eq 5.

low temperatures goes over to classical random reorientation/ tumbling at higher temperatures, sometimes higher than 28 K.2 In the quantum mechanical limit, one more important factor, the symmetry of the system, has to be taken into account. Namely, only totally symmetric overall spin-rotational wave functions within the symmetry group of the rotor are allowed. The nuclear spin part of the zero-order spin-rotation wavefuntions has thus to comply with the construction of the totally symmetric states overall. As an example, the rotational state |J = 1, M = 0, L⟩ combined with the three-proton quartet is not allowed in the planar methyl within the D3 group, while it is fine in the pyramidal methyl of the C3 group; see further in Figure 8 and Tables 3 and 4. Ogata et al.38 studied methyl radicals obtained by photoactivation of methane on metal-ion supported silica gel. They suggest a pyramidal structure for the radical and indicated that the neat EPR spectrum of methyl radicals they registered, called abnormal, had a significantly smaller hf splitting, 19 G, than regular methyl. The authors provided the amplitude ratio sequence 0.2:3.0:5.4:0.4, i.e., 1:15:27:2, referring to the low field +3/2 hf component. From their figure 1a in that work we obtained the ratio 1:15.4:23.7:1.7, in satisfactory agreement with the former sequence. We also measured linewidths 3.03, 1.74, 1.52, and 2.06 G. The resonance lines were fairly symmetrical with respect to the baseline, allowing thus the estimation of the line intensities as products of the amplitudes with the squared widths. As a result, the intensity sequence 1:5.1:5.9:0.8 obtained, was obviously different from the binomial distribution by any of the above methods or the original estimation of the authors. Another issue is also important in the distinction of classical and quantum cases concerning the potential energy. While the quantum potential modifies the levels by adding energy to the system, probably without a significant change in the level separation, in the classical case it has to be considered as a hindrance of the rotation thus reducing the kinetic energy of the system. This is taken by certain workers as a reduction of the kinetic constant of the radical, which then leads to decreasing level separation and consequently to the classical limit of the system.3,10 Assuming pyramidal shape40 for the CH3 radical due to matrix−carbon interaction, the C3 symmetry instead of a planar D3 rotor could be adopted giving identically a symmetric top rotor. The small deviation from planarity is not able to significantly modify the moment of inertia of the methyl species

⎧ Ψ(A) (1, + 1, L) = |1, + 1, L⟩[ε*|β , α , α⟩ + |α , β , α⟩ ⎪ α ,β ,α ⎪ + ε|α , α , β⟩] ⎨ ⎪ Ψ(A) (1, − 1, L) = |1, − 1, L⟩[ε|β , α , α⟩ + |α , β , α⟩ ⎪ α ,β ,α + ε*|α , α , β⟩] ⎩

(5)

where the imaginary number ε = exp(i 2π/3) signifies the period of the C3 symmetry and L = 0, ±1 the laboratory projections of the rotational angular momentum. The A nuclear spin states Ψ(A) n,n,n(1, ±1, L) for either n = α or β, that should give total nuclear spin Fz = 3/2 are excluded due the requirements of the C3 group for the particular case of the three protons with nuclear spin I = 1/2. On the other hand the nuclear-spin E, overall totally symmetric R ⊗ F states in eq 5, are allowed by symmetry. Thus, going up with the temperature to the thermal energy of the first excited “parallel” |1, ±1, L⟩ rotational levels and before reaching |1, 0, L⟩, the “perpendicular” rotational levels we obtain a quartet ratio 1:7:7:1, comparable to the experimental 1:8.5:13:2.5 quartet in Pariiskii et al.10 When the rotational level |1, 0, L⟩ also becomes populated41 at even higher temperature, a new totally symmetric quartet is available changing the theoretical intensity ratio to 2:3:3:2. This is actually not compatible with either the lowest temperature 1:1:1:1 or the higher temperature 1:3:3:1 quartet, but it is possibly obtainable at certain experimental temperatures. However, with the system already being at the 77 K level, the classical quartet 1:3:3:1 is close to emerge. Our tentative interpretation of the unusual EPR methyl quartet found in silica gel based on a straightforward zeroth order 3D-free quantum rotor wavefunction approximation is under further investigation K



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two allowed R ⊗ F states of eq 5 are only possible for the first exited rotational level comprising only the “parallel” rotation about the z-axis of the symmetric top for the D3 symmetry. For the present problem, quantum effects can only occur to a certain level of temperature. This level seems to be quite high here, as high as 77 K, rather unexpectedly. It is interesting to compare our free-methyl-radical rotation level population to calculated data of Kiljunen et al.8 in their limited temperature range up to 5 to 35 K. Their rotational energies that are computed from parameters based on an older experimental paper by Yamada et al.42 are very close to ours in eq 6 and Tables 3 and 4. Furthermore, at least for low temperatures up to ca. 20 K, their rotational states’ populations very closely followed our Boltzmann computations for planar D3 symmetry, while for the higher temperatures up to 35 K for which they have data, they follow our pyramidal C3 computed points; see Figure 10. Notice that the predictions of the free,

in our laboratory. It constitutes a semiquantitative interpretation to be compared to certain qualitative interpretations of other research groups found further in this text. Furthermore, there are indications that our quantum model explains certain peculiarities of the CH3 spectra in solids and helps in understanding some experimental data presented further. In particular, it will be seen that this model might shed light on similar peculiarities observed by our and Eloranta’s groups’ recent experimental results. Since anisotropy alone cannot bring us closer to the experimental spectrum, the presumed quartet ratio 1:7:7:1 was tested also considering the spectrum anisotropy. The EPR spectrum in Figure 9 shows a simulation with the hf- and gtensor components according to Shiga and Lund9 except for the individual line width which was set equal 1 G.

Figure 9. Simulated EPR spectrum: CH3 adsorbed on silica gel. The EPR parameters match those of the spectrum in Figure 5, except that 6:1 E-to-A amplitude ratio is implied instead of 2:1.

Figure 10. Population ratio PE/PA for freely rotating methyl, including the implications of symmetry restrictions of rotation by the nuclear spin statistics. The figure accounts for the ratio PE/PA of all the A and E states from the ground rotational level |0 0 0⟩ up to the excited rotational level |3, ± 3, L⟩. Solid black curve: Bose-Einstein quantum (BEq) distribution for λ = 0.24 and C3 symmetry. The BEq distribution for D3 planar symmetry is represented by the lower three colored curves for different absolute activities λ. Highest magenta dashed-dotted-dotted line: Boltzmann statistics. From highest to lowest BEq-curves: dashed red λ = 0.2, dotted green λ = 0.5, dashed− dotted blue λ = 0.8.

In addition, the A/E amplitude ratio was changed to 1:6. The simulation yielded the amplitude sequence 1:10.4:12.9:1.7. The widths are between 1.04 and 1.42 G. The calculated amplitude ratio fairly well matches the experimental 1:8.5:13:2.5 one.10 Symmetry Restrictions. The three proton spins (I = 1/2) of methyl radical can combine to a total spin quartet (F = 3/2), and two different doublets (F = 1/2), giving totally (2 I + 1)3 = 8 spin coupled states, irreducible representations within the full rotation group. First, we combine two of the spins I = I1 = I2 = 1/2 to a triplet and a singlet I1 ⊕ I2 ⇒ F12 = 1, 0. The coupling to the third spin I3 = 1/2 results to I3 ⊕ (F12 = 1) ⇒ F = 3/2, 1/2 and I3 ⊕ (F12 = 0) ⇒ F = 1/2. The above three coupled representations F = 3/2, 1/2, 1/2 can be recombined most easily to form totally symmetric states with respect to exchange of any pair among the three spins considered as indistinguishable particles, if they do not already fulfill this symmetry requirement. This is due to the commutation of the rotations to the particle exchange. For higher spin than I = 1/2, even the totally antisymmetric state with respect to exchangeimpossible for three I = 1/2 spins of identical particles can be constructed, e.g., in the case of 3 deuterons in the CD3 radical.1 If one includes the first excited E-type level full twofold nuclear spin-degeneracy, the fourfold nuclear spin degeneracy of the ground A level has also to be used in the computations of the populate rates. However, due to symmetry restrictions the

three-dimensional, planar rotor, within both the classical Boltzmann’s and the BE-q distributions, allow for PE/PA ratios greater than 2:1, as certain experimental data indicate. Theoretically this is explained by the greater restriction of the A-states by the D3 symmetry, which decreases the weight of the numerator in eq 7 and eq 8 in the next section. In addition, our computations and independent experimental data also presented in Kiljunen et al.8 indicate that at temperatures from 30 to 40 K where the classical properties start to emerge the ratio PE/PA = 2:1 will actually prevail. Boltzmann vs Bose−Einstein Populations. To begin with, the energy of the pure rotor levels, irrespective the restrictions due to the coupling to the nuclear spin, will be considered. The energies of these |J M L⟩ rotational states for a symmetric top, with M the molecular and L the lab projection of the angular momentum, are computed using EJM = B[2J(J + 1) − M2] L

(6)



Article

Where the rotational constant is given by B = (2 I)−1, with I the greater moment of inertia about the high symmetry axis of methyl, I = I|| = 2 I⊥, based on the known relation between the parallel I|| and the perpendicular I⊥ valid for an oblate symmetric top. The value of the rotational constant B computed from pure geometric data was approximately 6.76 K.2 Notice also that, in the present simple model of free rotation, the rotational energy is not depending on L, the lab projection of the angular momentum, while on the contrary, this quantum number is crucial for degeneracy; see Tables 3 and 4 and eqs 7 and 8. This relation in eq 6 is quite similar to the equation used by Kiljunen et al.8 in their table 2, based on the experimental spectroscopic work of Yamada et al.42 In Table 3 and Table 4 the energies of the allowed spinrotational states, their nuclear-multiplet-counterpart symmetries within the C3 and D3 groups and their rotational degeneracy, proportional to the possible 2 J + 1 laboratory projections L, as they were illustrated in Figure 8, are listed. On the basis of Tables 3 and 4, a diagram showing the temperature dependence of the population ratio between the first excited states to the ground rotational A-state within both these symmetry groups is given in Figure 10. The degeneracy and the symmetry restrictions of the rotational levels of the total rotation and nuclear spin system were also considered. The open circles in Figure 10 were taken from the paper of Kiljunen et al.8 based on the expected free, 3D rotational levels of the methyl rotor. Best agreement of the latter data extended up to the sample temperature 80 K is found by the theoretical solid black curve of the present work, for BEq-distribution λ = 0.24 and C3 symmetry. If we restrict ourselves to the first seven points from Kiljunen et al.,8 the Bose-Einstein λ = 0.22 statistics applied to the D3 planar symmetry, i.e., the dashed red line in Figure 10, would be the best fit according to the least-squares procedure. On the other hand, one has, surely, to allow for the higher temperature points by Kiljunen et al. which are expected to be close to 2. In that case the Bose-Einstein λ = 0.24 applied to the C3 pyramidal symmetry would be the best, i.e., the solid black curve in Figure 10. The calculation of the populations in this limit does not employ the spin degeneracy rules. The result is obviously not compatible with the well-known experimental high temperature limit 2:1 obtained by the use of the EPR line shape of the methyl group. We continue considering the nuclear spin in the full rotational and nuclear spin space R ⊗ F. According to the Table 3 for the free pyramidal methyl we use the degeneracy of the symmetrized spin-rotation states in order to plot the theoretical Boltzmann and Bose-Einstein population ratio between the A and the E nuclear states. ∑J , M d(J E, M) , Lexp( −E(J E, M) /T ) PE = PA ∑J , M d(J A, M) , Lexp( −E(J A, M) /T )

∑J , M d(J E, M) , L[λ−1 − exp(E(J E, M) /T )]−1 PE = PA ∑J , M d(J A, M) , L[λ−1 − exp(E(J A, M) /T )]−1

(8)

In Figure 10 and Figure 11 it becomes obvious that the smaller the absolute activity λ the more the BEq-distributions

Figure 11. C3 symmetry spin-rotational level population ratio PE/PA including rotational states up to |J = 3, M, L⟩ of the free, 3D pyramidal methyl radical rotor as a function of temperature. The energy and the degeneracy of the spin rotation states are presented in Tables 3 and 4. The Boltzmann curve is seen as the highest (light blue) curve. The Bose-Einstein quantum (BEq) distribution is represented by the lower three curves for different absolute activities λ. From highest to lowest BEq-curves: red dash λ = 0.2, green dot λ = 0.5, blue dash-dot λ = 0.6. Experimental points: Methyl stabilized in N2O matrix black open triangles. In Ar matrix, green open circles obtained by the double integration procedure.

close the classical Boltzmann curve, indicating the smaller quantum nature of the system as it should. It is therefore the curves with the smallest λ that approach first the classical Boltzmann statistics at higher temperatures, here at ca. 60 to 80 K, while the other BE curves with greater λ approach the classical distribution asymptotically at even higher temperatures. Notice also that the classical Boltzmann distribution ratios are systematically highest compared to the corresponding quantum BE distribution. Quantum Statistics Effects. Actually for Bosons, we have the average occupation ⟨nk⟩ of a state k given by the following relation according to the Bose-Einstein (BE) statistics:43 1 ⟨nk ⟩ = −1 λ exp(εk /kT ) − 1 (9) where the absolute activity λ = exp(μ/kT) is an unknown, adjustable parameter. Alternatively instead of λ the chemical potential μ = kT ln λ could be estimated. In contrast to the Boltzmann distribution the partition f unction (statistical sum) Z for the mixed spin-rotational space, given as Z = ∑k∞= 1 exp(−β Ek), is not defined if it follows BEquantum statistics according to the book of McQuarrie.43 An interesting restriction is suggested for the absolute activity parameter λ obtained by using eq 10. Setting the energy of the ground state ε0 = 0 the average population of the lowest spinrotational level ⟨n0⟩ is obtained.

(7)

where dJ,M,L is the degeneracy of the accepted rotational states while EJ,M are the 2-fold M degenerate energies of these symmetric-top-rotor states in units of Kelvin. In addition to the molecular M-projection degeneracy the symbols dJ,M,L also allow for the additional 2 J +1-fold laboratory L-projection degeneracy of the angular momentum; see Tables 3 and 4. Under similar conditions the corresponding BE-distribution gives

⟨n0⟩ = M

λ 1 = −1 1−λ λ −1

(10)



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Figure 12. Fitting of the rotational level populations of methyl radical isolated in solid gas matrices estimated from methyl radical EPR using Boltzmann and Bose-Einstein Statistics, based on rotational constant B = 6.76 K. Triangles in magenta are for the free CH3 radical by Kiljunen et al.8 (a) The solid green curve is obtained by fitting of the absolute activity λ according to Bose-Einstein statistics considering the methyl radical as pyramidal C3-rotor, while the dashed red curve represents a plot of Boltzmann statistics. The best value of the parameter λ for CH3−N2O data (open black circles) was 0.54. (b) Methyl isolated in Ar matrix. Fitting of the data within the BEq-statistics (solid green line) gave best value for the parameter λ = 0.19 considering the methyl radical as pyramidal C3-rotor. Dashed red curve represents a plot of the Boltzmann statistics. (c) Open circles: experimental data of CH3 in solid Ar, present work, and ref 13 Boltzmann fitting curves are shown. Dashed red line: free 3D-rotor of pyramidal C3 symmetry; solid green line: free 3D-rotor of planar D3 symmetry. The planar rotor model that gives better agreement is tested also for BEq-distribution in part d. (d) Standard deviation, χ, of the experimental PE/PA results for CH3 in solid Ar, predicted by the BEq-distribution.

the ratio of the separate contribution of the inner E- to the outer A-lines, i.e., PE/PA = “inner-to-outer” − 1 = Pinner/Pouter − 1. The other ratio, Pinner/Pouter presented in Figure A.1 and Figure A.2, stands for the ratio of the total intensity of the inner doublet (superposed A and E lines) to the outer pure Adoublet. The experimental PE/PA population data of Figure A.1 and Figure A.2 were thus fitted to the Bose-Einstein-quantum (BEq-) statistics using eq 8 and were plotted together with the Boltzmann distribution using eq 7 in Figure 12. Trying to obtain any reasonable conclusion from the value of the absolute activity λ and combining with the above considerations it appears that the BE statistics seem applicable for up to the high temperature data. Actually until temperature ca. 21 K the above average population ⟨nk⟩ of the BE statistics formula should be valid instead for Boltzmann. In that case, as many experimental points as possible obtained at lower temperatures must be included in the fitting procedures as the ones in Figure 12.

Since the average population of any level must be a nonnegative number it is obvious that the absolute activity λ must obey the inequality 0 ≤ λ < 1. This relation allows the population of the ground level to lie inside the limits of zero particles to all the particles N of the system, as it is the appropriate for Bosons to populate any state with any number of particles. There are two main problems, however, in the application of the BE-distribution to our system. The first problem has to do with the classification of methyl radicals, i.e., the quantum rotators combined with the three protons, as Bosons. The other problem to apply BE-statistics is that the absolute activity λ, or alternatively the chemical potential μ, should be assessed after an independent estimation. Both the classical Boltzmann and quantum statistics were used to simulate the experimental temperature profiles of the inner-to-outer hf component intensity ratio of the methyl quartet. They were obtained by the double integration procedure described above. In particular, Figure 12 depicts N



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for models II, III, and IV, respectively. Model I suggests slightly hindered rotation occurring about the C3-symmetry axis while the rotation about the in-plain axes is restricted. The same bound geometry was considered by Pariiskii et al.10 As a result, an axially symmetric EPR line shape was expected. The spectrum was, indeed, observed by Shiga and Lund.9 They reported relatively narrow EPR lines of 0.59 G for the MF = −3/2 hf component and measured the anisotropies of the EPR parameters. In that study, as well as in a number of other experimental studies, a nonbinomial ratio of the hf amplitudes was found. These observations brought about the issue of whether the nonbinomial ratio originated in a residual anisotropy due to incompletely averaged anisotropy by radical rotation, or if the effect was related to the properties of quantum rotation. A classical view of the first optionincomplete averaging for such an amplitude distribution is related to the broadening dependence of the hf components on the nuclear spin projection MF. In that case, the CH3 radical rotation is considered fast enough to average almost completely the EPR anisotropies of the g- and A-tensors, but still slow enough for the cross-relaxation mechanism of these tensors to be significant, actually in the intermediate motional regime. Look in refs 25−27 for a brief account and application of this mechanism. Recall, however, that the original theory was given by Fraenkel and Freed.11,12 In a very recent work by Buscarino et al.45 the correlation time of the CH3 rotation in two types of SiO2 solid was estimated using widths and amplitudes of the hf components, according to the above cross-relaxation model. However, no other attempts were made since the works by Friebele et al.,29 Garbutt et al.,16 and Gardner and Casey25 for better understanding of the delicate interplay between incompletely averaged anisotropy and the quantum effects contributing to the CH3 EPR line width. It seems that this recent paper by Buscarino stresses the need for a new model describing the variation in the CH3-EPR line shape since one could discern quantum effects at the unexpectedly high temperature of 77 K. The peculiar nonbinomial methyl quartet remains actually unchanged from ca. liquid He to liquid N2 temperature as was observed in one of the few reports in the literature of methyl adsorbed on porous material by Lund and Shiga.9,20 An alternative explanation related to the quantum rotation case was recently presented1 revealing the complex nature of the EPR line shape of the trapped CH3 radical in solid gases at low temperatures. This theory was based on the occupation of the degenerate quantum nuclear spin-rotational states of methyl radical. Figure A.1 presents two groups of experimental evaluations for the inner-to-outer hf intensity ratios of CH3 in N2O (i) ratios obtained using the spectrum simulation based on the superposition of the E-doublet and A-quartet; (ii) ratios obtained through the double integration procedure. The superposition method is not sensitive for small E/A ratios. That is why the blue asterisk sequence in Figure A.1 is limited to a set of points above approximately 20 K. On the contrary, the double integration procedure extends to the lowest sample temperatures in the N2O experiments down to 8.5 K in Figure A.1. To understand where the ratio 1:7:7:1 in the simulation of Figure 9 came from, one should start from a two-level system. Let both the lower A-type quartet (F = 3/2) and the first excited E-type levels to be nondegenerate except for the

Another aspect is the adaptation of one of the two alternatives: pyramidal C3 or planar D3 methyl symmetry. From Figure 10 it is obvious that the D3 symmetry can give values that exceed the regularly considered maximum (PE/ PA)max = 2 ratio, which is observed in the data that we present here for methyl isolation in Ar. Considering only pyramidal C3 methyl, in Figure 12a,b the BEq-statistics fitting gave the values 0.54 and 0.19 to the absolute activity λ parameter for methyl isolated in N2O and Ar isolated methyl, respectively. Obviously the experimental data in Figure 12a are best fitted by BEq-statistics rather than the Boltzmann distribution. This cannot be used however as an argument for the correctness of the particular quantum statistics due to the extra adjustable parameter λ. Probably, the difference would be obvious only in the lowest experimental temperatures. In fact, the presence of an extra adjustable parameter was not enough to fit the Ar data in Figure 12c. In that case the planar methyl was a better model in reproducing the methyl in Ar data even using Boltzmann statistics, as was shown in Figure 12c,d. The fitting curves in Figure 12a,b were computed for the free 3D- rotor of pyramidal C3 symmetry according to eqs 7 and 8 and Tables 3 and 4, suggesting that we were able to distinguish the appropriate rotor symmetry. As was discussed above, the planar model and Boltzmann statistics is better for CH3 in solid Ar, Figure 12c, while the pyramidal model and BEq-statistics are better for CH3 in N2O, Figure 12a. The nonplanar nature of the CH3 in N2O may result from strong interaction between the radical and the linear matrix molecules with large eccentricity. Earlier, Misochko et al. reported nonplanar methyl radicals trapped in solid Ar matrix in the reaction cage44 with HF. We note also that methyl radicals in solid Ar may be considered isolated from the host and, therefore, from each other, while those radicals in solid N2O more readily interact with the matrix being thus “immersed” into the phonon bath. Whether these peculiarities have something to do with the above statistics is an open question which is to be resolved in future investigations, both theoretical and experimental. Also, the computed PE/PA ratios using Table 4 data, very surprisingly, indicate that the Boltzmann curve rather than the BE-one fits the experimental Ar points almost exactly. In particular, the fitting of λ Figure 12d suggests the Boltzmann distribution (λ = 0) to be best for the Ar experiment. The same analysis applied to the experimental N2O results confirmed the pyramidal rotor model and Bose distribution with λ = 0.54.

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DISCUSSION An idea about the silica gel structure and possible adsorption sites of CH3 on the silica surfaces may be obtained from DFT and MD study by Takada and Tachikawa on hf coupling constants of methyl radicals adsorbed on model surface of silica gel.30 Four typical binding sites of CH3 on certain proposed cluster models were found: two silanol Si−OH sites (models I and II) and two siloxane Si−O−Si sites (models III and IV). In model I, an OH group orients toward the carbon atom of the • CH3 radical. More specifically, the 2p−spin−orbital of •CH3 with an unpaired electron orients toward the proton Si−OH. The methyl radical in model I of this work has an almost planar structure with an averaged H−C−H bond angle of 119.6°, which is slightly smaller than that of the free methyl radical. Of all four models, model I provides the largest binding energy of 3.2 kcal/mol compared to energies of 0.6, 0.7, and 0.9 kcal/mol O



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rotational left or right sense (the rotational states |J = 1, M = ±1⟩) twofold degeneracy factor 2 for the E-symmetry level. The above two rotational levels also correspond to the two remaining total electron spin doublets with F = 1/2, with which they are by symmetry allowed to combine in the C3 group; see section Symmetry Restrictions. The Boltzmann statistical sum here Z = ∑j 3= 1 exp(−β Ej) where j = 1, 2, 3 stand for the ground level and the above two excited rotational levels, since the third rotation level with J = 1, Jz = 0 has higher energy and is assumed not to be occupied yet at temperature T such that β Ej ≫ 1. Here β = 1/kT, and Ej is for the energy of the jth level. When T → ∞ the probability of the system to occupy the jstate will be Pj = (1/Z) exp(−β Ej) and at least for the above three levels this will be Pj = 1/3. Thus, at a high temperature, the population of the two excited states will be proportional to 2/3, while the population of the ground state will be proportional to 1/3. This brings about a superposition of one quartet and two doublets with the result of 1:3:3:1 amplitude ratio of the hf components in the EPR spectrum. However, with increasing temperature all too many other rotational levels will be occupied and one has to take care of the details of the allowed R ⊗ I states due to identical particle exchange and the symmetry of the radical geometrical shape. This limiting value is also observed approximately in our more general treatment up to the J = 3 level for C3 symmetry; see Tables 3 and 4. The most general problem is however to show theoretically the high temperature ratio 1:3 for the intensity ratio for the outer to the inner doublet in the limiting case of high J, valid for classical thermodynamic conditions. That must be true in spite the low temperature deviation from the binomial distribution. This is however a more involved problem to be considered in a future work together with the details of the general form of the allowed spin-rotation states of CH3 and CD3, as well as more accurate adaptations of these rotors to the Bose-Einstein statistics or Fermi-Dirac statistics at liquid He temperatures.

conclusions for the properties of experimental population E/ A-ratio data. A comparative study of the EPR of methyl radicals isolated in the bulk of solid gases and those trapped at the surfaces of porous substrates suggests that certain surface sites prohibited the radical rotation not only about the C3 axis but about the C2 axes as well. Nuclear quantum statistics restrict the rotational motion of the weakly interacting, almost free methyls in these environments determining the population ratios PE/PA of the E to the A symmetry nuclear states applying from liquid He temperature to the unexpectedly high, liquid nitrogen temperature of 77 K. The pyramidal C3 deviation of the planar D3 symmetry of methyl binding in the environment made of the spherical or cylindrical molecular units in solid gases or adsorbed on the surface of the voids of porous material such as the silica gel, VYCOR glass, and also zeolite may be the critical point for the explanation of the nonbinomial distribution of the methyl quartet. Thus, the planar model and Boltzmann statistics are better for CH3 in solid Ar, while the pyramidal model and BEqstatistics are better for CH3 in N2O.

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APPENDIX

Assessing experimental E/A Ratios

The open circles data points in Figure A.1 were obtained using double integration of the experimental methyl radical spectra in

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CONCLUSIONS Inspecting the existing experimental evidence of methyl adsorbed on the surface of porous material in Table 2 and our results, we conclude that the PE/PA hf intensity ratio for trapped methyl radicals is a complicated issue, as the diversity of the above data indicate. Not only the absolute value but also the ratio dependence on the sample temperature and matrix used needs further experimental and theoretical studies. One can see however a repeated pattern of ratios that systematically exceed the generally accepted high temperature, classical ratio 2:1 of the methyl radical isolated in relatively inactive gases. In order to estimate the relative hf intensities of the wellknown, relatively sharp, methyl radical EPR quartet, the populations of the symmetry allowed spin−rotation states have to be considered within or even outside the classical Boltzmann statistics. Their statistics deviate from the equipartition, requiring discrete quantum level description, in a similar fashion as the vibrational levels. At the lowest liquid He temperatures one has in addition to think if there is really enough thermal energy to populate the relevant rotational states due to the unusually distant rotational energy levels of methyl due to the extremely small moment of inertia of methyl. Two new aspects, not previously investigated, the application of Bose-Einstein quantum (BEq-) statistics and the overall laboratory rotational degeneracy in both the BEq-statistics and the classical Boltzmann statistics seem to signify new

Figure A.1. Inner doublet-to-the outer lines intensity ratio (open black circles) of the methyl quartet isolated in solid N2O as a function of the temperature obtained using double integration. Asterisks in blue: results from the same methyl in N2O sample obtained from spectral simulation where the total spectrum was considered as a superposition of the A-quartet and the E-doublet. Red open triangles: population ratios computed for the gas-phase 3D rotational states of free methyl radical by Kiljunen et al.8 Purple stars: also from that work, CH3 isolated in CO2, see text.

solid N2O gas matrix. Since N2O and CO2 solids are close in a variety of properties, Kiljunen’s data8 of CH3 isolated in CO2 were added to the graph. The low-temperature part of the graph reaches the ratio 1.6 as a limiting value at 10 K. We were unable to extend the ratio data obtained using the simulation procedure below 20 K since the peak intensity contribution from the E-state to the lines is extremely small at temperatures below ca 20 K and cannot be extracted reliably. The high-temperature ratio obtained by P



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ature decreases. It is possible, however, that the contribution to the wings of the inner lines may come from the unresolved Azz components of the EPR spectrum of nonrotating CH3 radicals which contribute markedly to the overall EPR spectrum at low temperatures, while having no effect on the spectrum at high temperatures.

double integration approaches the common high-temperature limiting value three, 3. One can see that the two experimental N2O temperature dependencies in Figure A.1 obtained using different methods differ markedly. The first method of the intensity ratio determination obtained by double integration is closer to the gas-phase results and also the CO2 data.8 Thus, the second method, using simulations at sample temperature 30 K, suggests inner/outer ratio 1.7, while double integration yields, approximately, ratio 2.5. In order to understand the origin of such a discrepancy we examined the shapes of the hf components in detail. The second method, which follows intensity data, measuring directly on the simulated spectrum, shows a large deviation for CH3 radicals in the gas phase at low temperatures, implying thus unreasonably great energy gap between the ground and excited rotational states of the trapped radicals. The data point suites of the different spectra in Figure A.1 demonstrate how the differently obtained PE/PA ratios match

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AUTHOR INFORMATION

Corresponding Author

*Nikolas-Ploutarch Benetis, 13/0501, Kozani, Western Macedonia, Greece. Tel. +30-24610-68290, e-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Y. D. acknowledges a partial financial support from Russian Foundation for Basic Research under grant 13-02-00373-a.

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REFERENCES

(1) Benetis, N. P.; Dmitriev, Yu. A. Dynamical Effects in CW and Pulsed EPR, Part I Trends in Methods, 2nd ed.;Lund, A.; Shiotani, M., Eds.; Springer: Dordrecht, 2012; Chapter 3. (2) Yamada, T.; Komaguchi, K.; Shiotani, M.; Benetis, N. P.; Sørnes, A. R. High Resolution EPR and Quantum Effects on CH3, CH2D, CHD2 and CD3 Radicals under Argon-Matrix Isolation Conditions. J. Phys. Chem. A 1999, 103, 4823−4829. (3) Jackel, G. S.; Gordy, W. Electron Spin Resonance of Free Radicals Formed from Group-IV and Group-V Hydrides in Inert Matrices at Low Temperature. Phys. Rev. 1968, 176, 443−452. (4) In order to estimate the relative populations of the A and the E states one should consider the relative intensities (= amplitudes multiplied by the squared line width) of the lines of the derivative spectrum rather than their amplitudes. The peak amplitudes through the hf components can differ from the expected ratio since the widths also must be considered. The doubly integrated intensity which accounts for both the line peak amplitude and width is indeed 1:1:1:1 for the solid Kr. (5) Dmitriev, Yu. A. Peculiarities of EPR Spectra of Methyl Radicals in Quench-Condensed Krypton Films. Low Temp. Phys. 2008, 34, 75− 77. (6) Benetis, N. P.; Dmitriev, Yu. Inertial Rotation and Matrix Interaction Effects on the EPR Spectra of Methyl Radicals Isolated in “Inert” Cryogenic Matrices. J. Phys.: Condens. Matter 2009, 21, 103201−21. (7) Wall, L. A.; Brown, D. W.; Florin, R. E. Atoms and Free Radicals by γ-Irradiation at 4.2 K. J. Phys. Chem. 1959, 63, 1762−1769. (8) Kiljunen, T.; Popov, E.; Kunttu, H.; Eloranta, J. Rotation of Methyl Radicals in Molecular Solids. J. Phys. Chem. A 2010, 114, 4770−4775. (9) Shiga, T.; Lund, A. g Factor and Hyperfine Coupling Anisotropy in the Electron Spin Resonance Spectra of Methyl-, Ethyl-, and AllylType Radicals Adsorbed on Silica Gel. J. Phys. Chem. 1973, 77, 453− 455. (10) Pariiskii, G. B.; Zhidomirov, G. M.; Kazanskii, V. B. Electron Paramagnetic Resonance Spectrum of the Methyl Radical Adsorbed on a Silica Gel Surface. J. Struct. Chem. 1963, 4, 336−338. (11) Freed, J. H.; Fraenkel, J. K. Theory of Linewidths in Electron Spin Resonance Spectra. J. Chem. Phys. 1963, 39, 326−348. (12) Fraenkel, J. K. Widths and Frequency Shifts in Electron Spin Resonance Specta. J. Phys. Chem. 1967, 71, 139−171. (13) Dmitriev, Yu.A.; Benetis, N. P. EPR Line- Shape Anisotropy and Hyperfine Shift of Methyl Radicals in Solid Ne, Ar, Kr, and p- H2 Gas Matrices. J. Phys. Chem. A 2010, 114, 10732−10741. (14) Danilczuk, M.; Sadło, J.; Lund, A.; Yamada, H.; Michalik, J. EPR Study of Highly Stable Methyl Radicals Trapped in Synthetic H-rho. Nucleonika 2005, 50, S51−S54.

Figure A.2. Open circles: inner doublet-to-the outer lines intensity ratio plotted against sample temperature obtained from experimental EPR data of CH3 radical isolated in solid Ar using double integration. Blue asterisks: experimental results of the present study, obtained for the MF = −1/2 hyperfine components. Red triangles and purple stars: inner-to-outer intensity ratios obtained from peak intensity ratios for CH3 in gas phase and in Ar, respectively, from ref 8.

the inner and the outer hf components using the Gaussian profile of the EPR lines. This comparison verifies the Gaussian shape of the outer lines while suggesting some perceptible Lorentzian contribution to the wings of the inner hf components. This difference in the lineshapes of the hf component can explain the difference in the ratios assessed with the above two different methods. Since the outer components originate from the pure A-quartet one concludes that it is Gaussian, while the deviation of the inner line shape suggests some Lorentzian contribution to the E-doublet. At higher sample temperatures, the blue asterisk points approach the double integration data. The lineshape of the inner hf components at high temperatures were found to be nearly Gaussian. Close inspection of the experimental spectra revealed also that both the perpendicular and the parallel components of the width tensor are temperature dependent. Thus, in the simulations, one has to account for the temperature dependent lineshape of the E-doublet which gains the perceptible Lorentzian contribution as the sample temperQ



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(15) Fujimoto, M.; Gesser, H. D.; Garbutt, B.; Shimizu, M. Paramagnetic Resonance Spectra of Methyl Radicals on Porous Glass Surfaces. Science 1967, 156, 1105−1106. (16) Garbutt, G. B.; Gesser, H. D.; Fujimoto, M. Temperature Dependences of the Hyperfine Interactions, Linewidths, and Line Asymmetries in Methyl Radicals Stabilized on Porous Glass Surfaces. J. Chem. Phys. 1968, 48, 4605−4614. (17) Garbutt, G. B.; Gesser, H. D. Electron spin resonance studies of methyl radicals stabilized on porous VYCOR glass: various surface interactions, second-order splitting, and a linewidth temperature study. Can. J. Chem. 1970, 48, 2685−2694. (18) Mitin, V. F.; McDonald, P. C.; Pavese, F.; Boltovets, N. S.; Kholevchuk, V. V.; Nemish, I. Y.; Basanets, V. V.; Dugaev, V. K.; Sorokin, P. V.; Konakova, R. V.; Venger, V. F.; Mitin, E. V. Ge-onGaAs Film Resistance Thermometers for Cryogenic Applications. Cryogenics 2007, 47, 474−482. (19) Kubota, S.; Iwaizumi, M.; Ikegami, Y.; Shimokoshi, K. Anisotropic Hyperfine Interaction in the Electron Spin Resonance Spectrum of the Methyl Radical Trapped in CH3COONa·3D2O Crystal at Low Temperatures. J. Chem. Phys. 1979, 71, 4771−4776. (20) Shiga, T.; Yamaoka, H.; Lund, A. Conformation and Mobility of Radicals in Heterogeneous Systems ESR Spectra at 4 K of Methyl, Ethyl, n-Propyl and Allyl Radicals Adsorbed on Silica Gel. Z. Naturforsch. 1974, 29, 653−659. (21) Brotikovskii, O. I.; Zhidomirov, G. M.; Kazanskii, V. B. Quantum Effects of Anisotropic Rotation in the ESR of the Methyl Radical. Theor. Exp. Chem. 1972, 8, 653−659. (22) Kubota, S.; Iwaizumi, M.; Isobe, T. ESR Studies of Methyl Radicals Stabilized on Silica −Gel Surfaces. Bull. Chem. Soc. Jpn. 1971, 44, 2684−2688. (23) Lefebvre, R.; Maruani, J. Use of Computer Programs in the Interpretation of Electron Paramagnetic Resonance Spectra of Dilute Radicals in Amorphous Solid Samples. I. High Field Treatment. XBand Spectra of πElectron Unconjugated Hydrocarbon Radicals. J. Chem. Phys. 1965, 42, 1480−1496. (24) The total proton spin projections MI = ±1/2, ±3/2 of the methyl quartet hf components will in this text be exchanged to MF to point out the vector sum F of the spins of the three protons. (25) Gardner, C. L.; Casey, E. J. Tumbling of Methyl Radicals Adsorbed on a Silica Gel Surface Studied by Electron Spin Resonance. Can. J. Chem. 1968, 46, 207−210. (26) Poole, C. P., Jr. Electron Spin Resonance, 2nd ed.; Wiley: New York, 1967. (27) Carrington, A.; McLachlan, A. D. Introduction to Magnetic Resonance with Applications to Chemistry and to Chemical Physics; Harper International: New York, 1967. (28) Fast motions are considered the ones with correlation times about 10−11 s or smaller, even in the order of picoseconds, which occur in liquids, such as water at ambient temperatures. (29) Friebele, E. J.; Griscom, D. L.; Rau, K. Observation of the Methyl Radical CH3• in Irradiated High-Purity Synthetic Fused Silica. J. Non-Cryst. Sol. 1983, 57, 167−175. (30) Takada, T.; Tachikawa, T. DFT and Direct ab-Initio MD Study on Hyperfine Coupling Constants of Methyl Radicals Adsorbed on Model Surface of Silica Gel. J. Mol. Catal. A: Chem. 2009, 311, 54−60. (31) Benetis, N.; Kowalewski, J.; Nordenskiöld, L.; Wennerström, H. P.-O. Westlund Dipole- Dipole Nuclear Spin Relaxation. A Cross Correlation Correction to the Solomon-Bloembergen Equation for T2. Mol. Phys. 1983, 50, 515−530. (32) Barnes, J. D.; Oduwole, D. A.; Trivedi, M. A.; Wiseall, B. On the Catalytic Reactivity of Solid Surfaces. III. Stabilization of Photolytically Produced Methyl Radicals on Silica Surfaces. Appl. Surf. Sci. 1985, 20, 249−256. (33) Kazanskii, V. B.; Pariiskii, G. B. Proc. 3rd Int. Congr. Catal. 1965, 1, 367. (34) Rogers, R. N.; Pake, G. E. Paramagnetic Relaxation in Solutions of VO+2. J. Chem. Phys. 1960, 33, 1107−1111. R


Anomalous EPR Intensity Distribution of the Methyl Radical Quartet

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