EPR study of methyl radical in van-der-Waals solids

Physica B 440 (2014) 104–112

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EPR study of methyl radical in van-der-Waals solids Yu.A. Dmitriev a,n, V.D. Melnikov a, K.G. Styrov b, M.A. Tumanova c a

A.F. Ioffe Physico-Technical Institute, 26 Politekhnicheskaya ul., 194021 St. Petersburg, Russia Institute of Computing and Control, St. Petersburg State Polytechnical University, 26 Politekhnicheskaya ul., 195251 St. Petersburg, Russia Faculty of Information, Measurement and Biotechnical Systems, Saint Petersburg Electrotechnical University, 5 Prof. Popova ul., 197376 St. Petersburg, Russia b c

art ic l e i nf o

a b s t r a c t

Article history: Received 22 November 2013 Received in revised form 22 December 2013 Accepted 29 January 2014 Available online 5 February 2014

High-resolution EPR spectra of CH3 radicals trapped in solid N2 matrix were obtained in a temperature range 7–28 K. The analysis of the spectrum suggests that, in the whole temperature range, the radical performs both fast rotational motion around the C3-symmetry axis and libration motion about the C2-symmetry axes. A comparison study of the methyl libration motion in gas matrices of linear molecules, N2, CO, N2O and CO2, was carried out using a model of the infinite potential well with flat bottom. As a result, averaged angular deviations of the radical orientation and dimensions of the potential wells were estimated. We assessed contributions of both the classical motion and the quantummechanical correction to the radical reorientation. A possibility of utilizing the trapped CH3 radical as a spin probe to study order–disorder structural phase transition at the microscopic level was discussed. An empirical formula obtained earlier for the matrix shift of the isotropic hyperfine coupling (hfc) constant of CH3 in matrices of spherical particles was shown to hold true for matrices of linear molecules as well. The experimental results and their treatment made it possible to report an estimate of the hfc tensor anisotropy for the free methyl radical. This information is of special interest to the theory of atoms and molecules. & 2014 Elsevier B.V. All rights reserved.

Keywords: CW EPR Matrix isolation Methyl radical Classical and quantum rotation Matrix shifts

1. Introduction A number of research studies have shown that the methyl radical EPR spectrum lineshape depends on the radical surroundings which impose restriction on CH3 rotation [1]. A proton hyperfine coupling (hfc) tensor of the radical has three different principal components [2]. In case of the free rotation, the components average out leaving an isotropic EPR spectrum. While the hfc anisotropy was observed experimentally for the matrix-isolated radical and confirmed by several theoretical studies, the “ab initio” calculation of the methyl radical EPR parameters is still a challenging problem highlighted by appreciable discrepancies between the theoretical and experimental results. Up to now, there is only one publication reported measurement on the EPR of the free methyl radical [3]. The authors studied only the Fermi contact interaction. That is why theoretical approaches have commonly been evaluated by linking the predicted values to those measured in matrix isolation experiments. Such a comparative studies both gave hints about the hfc anisotropy and provided much smaller error bars for the Fermi contact term. On the other hand, the

n

Corresponding author. Tel.: þ 7 812 292 7130; fax: þ 7 297 1017. E-mail address: [email protected] (Yu.A. Dmitriev).

http://dx.doi.org/10.1016/j.physb.2014.01.039 0921-4526 & 2014 Elsevier B.V. All rights reserved.

matrix shifts of the measured parameters are inherent limitations to the accuracy that can be obtained when these data are used to estimate the free radical values. However, only few investigations are found in the literature concerning systematic study of the effect which a matrix has on the methyl radical isotropic hfc constant, and no investigations were carried out to elucidate contribution of a matrix surrounding to a shift in the components of the anisotropic hfc tensor. While the anisotropic part of the tensor is relatively easier to calculate than the isotropic hfc constant, the exact value of the hfc anisotropy of the free radical is of special importance. This is because of the fact that the residual anisotropy is a fingerprint of the interaction between the trapped radical the matrix surrounding which hinders the radical rotation. With the use of hybrid DFT, Takada and Tachikawa [4] computed ab initio hydrogen hfc constants of the methyl radical in solid H2, Ar and CH4 matrices. The absolute values of hfc constants for the matrix isolated CH3 turned out larger than that of the free radical [3]. This result is in disagreement with the experimental observations [5,6]. Based on the experimental data of hfc constants for matrix isolated CH3 available from the literature, McKenzie et al. [5] reported these constants to be proportional to αM =R6 , where αM is the matrix particle polarizability volume and R is the distance between the methyl radical and the neighboring molecules. This coefficient contributes

Yu.A. Dmitriev et al. / Physica B 440 (2014) 104–112

majorly to the sensitivity of the radical/matrix particle van-derWaals (vdW) interaction to the nature of the matrix. However, on the basis of their experimental results for CH3 in para-H2, Ne, Ar and Kr matrices, Dmitriev and Benetis [6] have shown that the hyperfine splitting plotted versus the vdW radical/matrix-particle interaction energy, EV, cannot be cannot be satisfactorily fitted by the linear dependence. They found out that the situation may be improved remarkably if one changes EV for EΣ ¼(EV þ1.63EP), where EP is the energy of Pauli repulsion between the CH3 radical and the matrix particle; the coefficient of 1.63 was obtained in the fitting procedure. Moreover, it turned out that not only the splittings between the hf EPR transitions for the matrix isolated methyl radical closely follow the linear dependence on EΣ, but the free methyl radical experimental value [3] hits this linear regression line almost exactly. In the present study, we verify an assumption that the empirical dependence obtained in Ref. [6] is common to CH3 matrix-isolated in solid gases. Another object of the present work was to study the EPR of the CH3 radical in solid N2. Earlier, having analyzed EPR spectra of CH3 trapped in H2, Ne, Ar, Kr and CO matrices at liquid helium temperatures, Dmitriev and Zhitnikov [7] and Dmitriev [8] came to the conclusion that the radical is almost free to rotate in solid hydrogen and inert gas matrices while only torsional rotation around the three-fold axes fixed at a specific direction is allowed in solid CO, a matrix with orientational ordering. Also, it was suggested that such a torsional rotation is common to the radical in other matrices with orientational ordering. Previously, the free rotation was reported by Yamada et al. [9] based on the highresolution EPR spectra of methyl radical isotopomers trapped in Ar matrix. The above model of the CH3 rotation was subsequently verified and developed in high-resolution experiments by Popov et al. [10], for solid Ar, Kiljunen et al. [11], for solid Kr, Kiljunen et al. [12] and Benetis and Dmitriev [13], for solid CO2, Kiljunen et al. [12], for N2 and CO solids, Benetis and Dmitriev [13], for solid N2O. Of all molecular gas solids with orientational ordering, N2, CO, N2O, CO2, solid nitrogen is composed of molecules with the smallest excentricity. In that respect the solid N2 may be considered as nearly an intermediate matrix between those composed of spherical symmetric particles allowing almost free rotation of the methyl radical and those composed of linear molecules fixing the radical three-fold axes at a specific direction. Indeed, Kiljunen et al. [12] found peculiarities in the EPR spectrum of CH3 in N2 which were not observed in other molecular solids. Considering both A and E states (the total nuclear spin of the radical being I ¼3/2 with symmetric spin functions, in the former states, or 1/2 with antisymmetric spin functions, in the later states, respectively) contributing to the overlapped EPR spectrum, the authors found that the A/E intensity ratio was very small at all temperatures despite of the fact that the rotational ground A state should be dominant at low temperatures. Moreover, the E states was populated more efficiently at 16.8 than at 35.4 K sample temperature. Thus, the spectral lines were surprisingly inconsistent with the gas phase level structure and the Boltzmann distribution. This behavior was opposite to all the other matrices that had been studied earlier by the authors. They stressed that, in N2 matrix, the spectral lines were broader and more difficult to analyze compared to CO and CO2 solids, to say nothing of solid Ar and Kr. The spectrum in N2 matrix was axial but appeared nearly isotropic. Earlier, having analyzed the literature data, Benetis and Dmitriev [14] concluded that the methyl radical EPR lineshape is to some extent subject to the experimental procedure applied of the radical stabilization. In experiments by Kiljunen et al., the radicals were produced by the laser driven electrical breakdown (high temperature plasma) few millimeters away from the cooled substrate. In that event, the radicals with large kinetic energy may be formed in the gas phase prior to condensation as well as in situ in the

105

deposited sample under the action of the UV radiation and fast particle bombardment from the hot plasma. The abundance of the energy allows the radicals overcome potential barriers and reach a variety of trapping sites or even distort the nearest regular surroundings. On the contrary, our experimental procedure utilizes deposition from the remote gas discharge cooled with liquid nitrogen vapor, thus providing trapping cold molecular radicals with no excess kinetic energy in the substitutional site of the undistorted matrix lattice. It is worth noting that chemical reactions in N2:CH4 ices are of astrophysical interest. Indeed, the surface ices of trans-Neptunian objects (existing beyond the orbit of Neptune in the solar system and also known as a Kuiper Belt), such as Triton, Pluto and Eris, are found to contain predominantly N2 ices with diluted CO, CO2, CH4, H2O, the latter two making up the largest percentage of the impurity content. The Kupier Belt is proposed to be main reservoir of short-periodic comets that are thought to be carriers of organic molecules to Earth [15]. Because of wide interest in this topic, the N2:CH4 icy mixtures were subjected to various kinds of radiation and the IR spectra of photoproducts were studied by several research groups. Among other products, methyl radicals were found by Milligan and Jacox [16] and Hodyss et al. [17] with the use of UV light from a hydrogen discharge lamp, by Wu et al. [18] using VUV light generated with a synchrotron, and by Wu et al. [15] utilizing electron bombardment. The photolyzed experiments revealed that new products were formed by dissociation of CH4 in N2 matrix, including formation of ethane molecules produced in the reaction of two methyl radicals CH3 þCH3-C2H6 This effect testifies that the radicals produced in situ are energetic enough to migrate in the matrix and could occupy (or form) sites unreachable by slow thermal molecules from a cooled gas discharge, thus supporting the idea by Benetis and Dmitriev [14] about sensitivity of the CH3 EPR spectrum shape to the method of the radical trapping.

2. Experimental The experimental technique was described in detail elsewhere [6,14]. It will briefly be outlined in the following 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 outward surface of the bottom facing the vacuum 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 thermometer [19] 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, while the second one allowed the gas flow to be set onto the substrate avoiding the gas discharge zone. An electrodeless pulsed RF gas discharge was excited in the glass tube. We did not study systematically the influence of the deposition temperature, usually about 13 K, on the shape of the EPR spectrum. However, a change of the temperature in a rather narrow range, 12–16 K, had no effect on the spectrum. As well no effect of the sample annealing was noticed. The deposition rate was 5–9 mmol/h.

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A He–CH4 (KLIM Company Ltd, www.klim-spb.ru) mixture was passed through the discharge, while pure gaseous N2 (KLIM Company Ltd) was fed to the substrate avoiding the gas discharge. Based on the concentration of the gaseous helium–methane mixture in various runs (3–8 mol%), 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 0.14–0.31 mol%. 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 40 min to 1 h.

3. Results and discussion Fig. 1 shows EPR spectrum of CH3 radicals recorded at the sample temperature of 27.3 K. The substrate temperature during deposition was about 10 K. The spin Hamiltonian parameters used in the simulations in Fig. 1 are listed in Table 1. The experimental EPR spectra presented in Figs. 1 and 2 are of some higher resolution and signal-to-noise ratio compared to those published by Kiljunen et al. [12]. Our results evidence that the axial anisotropy still emerges in the case of CH3 in N2 at all available sample temperatures despite the smallest N2 molecular excentricity. However, the anisotropy turned out to be the smallest among matrices of linear molecules under consideration, Table 4. The fact that the extent of the anisotropy depends on the matrix used and shows no change with temperature variation is a fingerprint of both: the large contribution from the tunneling to the rotation which the trapped methyl radical performs around its two-fold axis and a very large barrier to such a rotation. Our results, Fig. 2, suggest also no non-monotonic temperature dependence of the A/E relative intensity ratio for the CH3 radicals trapped in solid N2 from the gas phase.

Fig. 1. EPR spectrum of CH3 in N2 solid. The spectrum was taken at the sample temperature of 27.3 K; microwave resonance frequency, fres ¼ 9400.30 MHz. (a) Experimental spectrum, (b) simulated spectrum, (c) simulated E-line doublet, (d) simulated A-line quartet. For the EPR parameters used in the simulation procedure see the text.

3.1. Isotropic hyperfine coupling matrix shift The hyperfine interaction of the unpaired electron of the trapped methyl radical with the proton magnetic moments of the molecule is influenced by the radical neighbors making a matrix cage. In a previous study [6], we have shown that the magnitude of the hfc constant can be linked to a certain combination of the attractive van-der-Waals (vdW) and repulsive Pauli exclusion (RPE) forces between the CH3 molecule and matrix particles of the noble-gas or molecular hydrogen matrix. If the matrix shift of the hfc constant is defined as difference between the free-molecule hfc magnitude measured in the gas state and the magnitude obtained for a matrix isolated radical, then the vdW interaction contributes with a negative value to the shift, while Pauli interaction contributes with the positive one. It turned out that the hfc constant varied almost linearly with |EV þ 1.63 EP |, where EV is for the vdW attractive energy, and EP is for the Pauli repulsive energy. A coefficient of 1.63 measures a weight of the Pauli repulsion in the hfc matrix shift. The coefficient was obtained in the least square fit to the experimental data. It is interesting to learn whether the hfc constant shifts for the CH3 radicals trapped in matrices of linear molecules, N2, CO, N2O, and CO2, check with the results predicted by the above dependence. Based on the interactions between various instantaneous multipole moments, the EV has the familiar form obtained by perturbation theory [20], EV ¼  C 6 =R6 C 8 =R8  C 10 =R10  …

ð1Þ

where the first term is the dipole–dipole interaction. Here, R is the separation between the interacting particles. Let us neglect all terms except the first one being asymptotically the largest at

Table 1 The spin Hamiltonian parameters used in the simulations in Fig. 1 for the particular conditions indicated in the text. The principal axes of the A-, and g-tensors are assumed to coincide. The hyperfine splitting and linewidth are measured in mT. Gaussian shape of the individual line is suggested; ΔΗ is for the width of this line. Parameters for the A- and E-line transitions are set equal. Parameter

CH3/N2 matrix

A|| A? Aiso ΔH g|| g? giso

 2.251  2.349  2.316 0.043 2.00222 2.00262 2.00249

distances encountered in the present study, EV   C 6 =R6

ð2Þ

here [20,5] C6 ¼

3 ECH3 EM  αCH3 αM 2 ECH:3 þ EM

ð3Þ

where ECH3 and EM denote the ionization energies of the CH3 radical species and a matrix particle, respectively; the quantities αCH3 and αM are their polarizability volumes. The ECH3 was found [21] to be 9.838 eV. In calculating EV it will be assumed that the unknown value of αCH3 is equal to that of methane: αCH4 ¼ 17:3a30 .

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Table 3 The literature data [22,23] and the computed values of intermediate parameters leading to the Repulsive Pauli Exclusion (RPE) forces, see in the caption of Table 2 for details. The last column in the table presents the value of the sum of the interaction energies,  (EV þ 1.63 EP), which is linked to the isotropic part of the hfc-tensor matrix shifts. Matrix

RM (a0)

 EV (10  4 ε0)

εM (10  4 ε0)

E (10  4 ε0)

 (EV þ 1.63 EP) (10  4 ε0)

Ne Ar Kr H2 N2 CO N2O CO2

5.95 7.09 7.59 7.14 7.548 7.544 7.538 7.421

3.891 5.039 4.824 2.239 3.708 3.989 5.881 5.819

1.32 4.52 6.32 1.10 3.01 3.17 7.46 6.93

2.264 2.054 1.592 0.821 1.362 1.537 2.178 2.212

0.201 1.691 2.229 0.901 1.488 1.483 2.33 2.213

The negative (long range) term gives attractive potential, EV ¼ 2ε (Rmin/R)6, while the positive (short range) one—the repulsive potential energy, EP ¼ε(Rmin/R)12. By considering the balance of these terms, the unknown repulsive potential energy can be estimated by the following relation, EP ¼

Fig. 2. EPR spectra of CH3 in N2 solid. The spectra were taken at different sample temperatures indicated in the figure. The figure testifies the temperature dependent contribution of the rotational E-symmetry state to the intensity of two central transitions. Note that the spectrum anisotropy does not change noticeably with temperature.

Table 2 List of experimental literature data for the CH3 radical trapped substitutionally in solid gas matrices: nearest neighbor distances, RM; polarizability volumes of the matrix particles, αΜ; and their ionization energies [22,23], ΕΜ, versus the computed quantities EV. Hartree atomic units are utilized for relevant quantities: ε0 ¼27.212 eV unit of energy, a0 ¼0.52918 Å unit of length. Matrix

RM (a0)

αM (a30)

EM (ε0)

 EV (10  4 ε0)

Ne Ar Kr H2 CO N2O CO2 N2

5.95 7.09 7.59 7.14 7.553 7.538 7.421 7.548

2.68 11.08 16.74 5.178 13.341 20.245 17.748 11.924

0.792 0.579 0.514 0.567 0.515 0.474 0.507 0.573

3.891 5.039 4.824 2.239 3.961 5.872 5.818 3.708

Table 2 presents results of the EV calculation as well as the involved parameters for matrices under consideration. For matrices of linear molecules, the distance R was taken as the distance between the centers of mass of the radical and the matrix molecule located at the site of the fcc crystal lattice. To account for the repulsion forces the total radical–matrix interaction in the form of Lennard–Jones (LJ) potential will be used [20,24]
V ¼ε

Rmin R

12 2 


 ! Rmin 6 R

ð4Þ

here ε is the depth of the potential well and Rmin is the equilibrium distance at the minimum of the typical 6–12 LJ potential.


2 EV 1  ε 2

ð5Þ

and the previously obtained value of EV. In addition, the wellknown empirical mixing rule by Lorentz–Berthelot, see e.g. Layer et al. [25], gives the depth of the potential well, εAB, for two different interacting particles, A and B, in the form involving the geometric mean of the depths of the involved pure substances, i.e. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi εAB  εA  εB ð6Þ where εA and εB are the depths for A–A and B–B potentials. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Therefore, ε can be expressed as ε  εCH3  εM , where εM is the depth of the potential between the matrix, M, particles. Instead of the unknown εCH3 , the value 2:11  10  4 ε0 for εCH4 estimated from the He–CH4 and He–He interaction potential wells was used here. The results are collected in Table 3. Table 4 lists our experimental data on the A-tensor components of CH3 trapped in different matrices. Nearly isotropic A-constants for CH3 in the noble-gas and molecular hydrogen matrices were taken as averaged over the three quantities for the spacing between four adjacent hf-components measured in magnetic field units. The spacing increases going from the low-field to the highfield EPR transition, owing to the second order effect. The components of the axially symmetric A-tensor are also listed for matrices of molecules with eccentricity. For reference, components of the g-tensor are also collected for CH3 trapped in matrices under consideration in the present study. Fig. 3 shows isotropic part of the hfc tensor, Aiso, plotted versus  (EV þ 1.63 EP) for CH3 in various matrices. Dashed fitting line is taken from the previous study [6] devoted to the EPR in solid noble gases and molecular hydrogen. When only data for CH3 in noble gas matrices are accounted for, the correlation coefficient of the linear regression is 0.99. After taking all experimental results into consideration, Fig. 3, the correlation coefficient drops to 0.75. Indeed, it is seen from the figure that while the overall Aiso tendency checks fairly well with the linear regression, the results for the molecular matrices with large eccentricity of molecules are left-shifted. This feature suggests some additional attraction forces between the CH3 and the non-spherical molecules contributing to the negative hfc shift. We believe that this additional attraction in the anisotropic non-central interaction between the radical and a linear molecule. In order to achieve this goal one has to account for

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Table 4 The spin Hamiltonian parameters for the A-line transition of the CH3 radical trapped in low temperature matrices. The principal axes of the A- and g-tensors are assumed to coincide. The hyperfine splitting is measured in mT; ΔA ¼ A ?  A||. Each hfc component is represented by its modulus. The values for noble gas matrices and solid H2 are taken from Ref. [6], while the values for matrices of linear molecules are obtained in the present study. Matrix

Parameters A?

Ne Ar Kr H2 N2 CO N2O CO2

2.350 2.343 2.333 2.339

(2) (5) (4) (6)

A||

Aiso

2.252 2.233 2.198 2.197

2.333 2.313 2.300 2.324 2.317 2.306 2.288 2.292

(4) (5) (4) (2)

ΔA (5) (5) (5) (7) (5) (7) (4) (4)

g?

0.099 0.110 0.135 0.142

(3) (6) (3) (6)

g||

2.00262 2.00263 2.00267 2.00264

(12) (12) (12) (12)

2.00225 2.00220 2.00229 2.00215

Δg,10  4

giso

(12) (12) (12) (12)

2.00250 2.00249 2.00254 2.00248

(12) (12) (12) (12)

3.70 4.14 3.84 4.93

(25) (25) (42) (30)

neglect the methyl anisotropic dipole polarizability assuming α ? CH3 ¼ α== CH3 ¼ αCH3

ð9Þ

The dispersion dipole–dipole interaction accounting for the anisotropy of the matrix molecule polarizability may be written in a form Edd V ¼ 

ECH3 EM αCH M  6 3  ð4αM == þ2α ? Þ 4ðECH3 þ EM Þ R

ð10Þ

here we adopted a pair interaction potential derived earlier [26] for the para-H2-radical particle interaction. Let us set 1 kM ¼ γ M 3 Fig. 3. hfc Constants of the methyl radical matrix isolated in low temperature matrices plotted against an empirical coefficient accounting for the contribution of the vdW attractive and Pauli repulsive interactions to the hfc matrix shift. Dashed fitting line is taken from Ref. [6] where it was obtained as a result of the least squared procedure of fitting the hfc data for CH3 in solid noble gases and molecular hydrogen.

ð11Þ

an anisotropy parameter which measures an extent of the anisotropy of the matrix molecule polarizability. Then αM == ¼ ð1 þ 2kM Þ  αM αM ? ¼ ð1  kM Þ  αM

ð12Þ

From Eqs. (12) and (10) may be expressed as the anisotropy of the polarizability volumes of matrix molecules which contribute to the dispersion forces [23]. In treating the pair interaction between the methyl radical and a linear matrix molecule, we will take into account the lack of the dipole moment of the CH3, N2 and CO2 molecules and smallness of the dipole moments of non-symmetrical CO and N2O molecules. In general, it is well known that induction interactions in systems, which we are considering here, are much smaller than the dispersion forces [23]. Let us consider CH3–M dipole–dipole and dipole–quadrupole dispersion interactions, where M is for a matrix molecule. The average dipole, α, and anisotropic dipole, γ, polarizabilities are defined by 1 α ¼ ðα== þ2α ? Þ 3

ð7Þ

and γ ¼ ðα ?  α== Þ

ð8Þ

where α== and α ? are parallel and perpendicular components, respectively. Let us estimate the anisotropy of the methyl polarizability, basing on the ab-initio calculation [26] which gave γ CD3 ¼ 0:0811  10  41 ðC  m2 =VÞ or 0.049 a30 in atomic units. Obviously, γ CH3 is of the same order of magnitude and, thus, far below 17.3 a30 which we set as a measure of the CH3 average dipole polarizability. We will

Edd V ¼ 

2ECH3 EM αCH  6 3  ð1 þkM Þ 3ðECH3 þ EM Þ R

ð13Þ

Thus, the contribution of the anisotropy of the matrix molecule polarizability to the dispersion energy is proportional to the anisotropy parameter. This anisotropy increases an absolute value of the vdW interaction. We suppose that Eq. (13) overestimates the anisotropy contribution because we neglected the dispersion dipole–quadrupole interaction between methyl radical and the matrix molecule. An expression for the vdW energy accounting for both interactions [26] may be written as follows  ECH3 EM 1 M M Edd;dq ¼ αCH3 ð4αM zz þ αyy þ αxx Þ V 4ðECH3 þ EM Þ R6  6 M M M M ð14Þ  7 αCH3 ð  AM x;xz  Ay;yz þ Az;xx þ Az;yy  2Az;zz Þ ; R where A(M) is the quadrupole polarizability tensor of the matrix molecule. To simplify our calculation, we will use Eq. (13), changing kM to = an effective dipole anisotropy parameter, kM , which accounts for the both dispersion interactions. Then

EV  

2ECH3 EM αCH =  6 3  ð1 þ kM Þ 3ðECH3 þ EM Þ R

ð15Þ

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Based on experimental data for the attractive interaction energy between matrix molecules, we will obtain an approximate = estimation of the kM parameter. In the case of two interacting matrix particles, Eq. (10) may be substituted by Edd V ¼ 

EM EM 1 M M M   ð4αM ==  α== þ 2α ?  α ? Þ 4ðEM þ EM Þ R6

3.2. Anisotropic hyperfine coupling matrix shift

2EM EM 1 2 Edd   ð1 þ 2kM þ 3kM Þ V ¼  3ðEM þ EM Þ R6

ð17Þ

=

For a small kM , Eq. (17) becomes as 2EM EM 1   ð1 þ 2kM Þ 3ðEM þ EM Þ R6

ð18Þ

Then an approximate expression for the vdW energy interaction between two matrix molecules which accounts for contribution of the dispersion dipole–dipole and dipole–quadrupole forces takes form EV  

2EM EM 1 =   ð1 þ 2kM Þ 3ðEM þ EM Þ R6

ð19Þ

Table 5 lists central, Ec, and non-central, Enc, including electrical quadrupole, Eq, pair interactions between matrix molecules of solid N2, CO, N2O and CO2. To obtain the part of the non-central energy owing to the dispersion forces, one have to calculate = Edisp ¼ Enc Eq . The effective anisotropy parameter, kM is taken as =

kM 

lower hfc constant shift for CH3 in the solid N2 matrix as compared to the solid CO was also reported earlier [12]. Thus, the empirical dependence obtained in our previous study [6] for the noble-gas and H2 matrices turned out to be surprisingly satisfactory for much wider set of low-temperature vdW solids.

ð16Þ

From Eqs. (12) and (16) may be obtained in a form as follows

EV  

109

Edisp Ec

As opposed to the solid noble gases and H2 where the CH3 anisotropy emerges only through different amplitudes and widths of the different hf components [1,14], the matrices of linear molecules provide EPR spectra of the CH3 with the anisotropy of the hf- and g-tensors evident via the splittings of the hf-transitions. This distinction owes to the different character of the radical rotation in the two groups of matrices: rapid rotation around the C3 axis accompanied by a slightly hindered reorientation around the C2 axis, for radicals in the first group of matrices, and rapid rotation around the C3 axis with forbidden reorientation of this axis, in the second group of matrices [6–8]. For the radical executing fast rotation around the symmetry axis, the principal Ax and Ay components average out yielding A== ¼ ð1=2ÞðAx þ Ay Þ which nearly coincides with A ? ¼ Az [27]. As a result, the spectrum reveals comparatively slight anisotropy [28]. This anisotropy fades away for the spectra of broad lines. One can see from Table 4 that a measure of the hf-anisotropy, ΔA¼A ?  A||, increases in value going from N2 to CO2 matrix. If one supposed that only the radical rotation contributes to ΔA then the hf coupling anisotropy corrected to the matrix shift would be the same in different matrices. The simplest possible correction to the matrix shift of

ð20Þ

We believe that the valences forces contribution to the anisotropic part of the intermolecular potential is small compared to the isotopic interaction. The indirect evidence follows from the data available for the energy interaction between matrix particles themselves [23]. Fig. 4 shows experimental hf constants for CH3 radicals against pair interaction energies with matrix molecules corrected to the anisotropic forces. In the figure, the linear regression is taken from the previous study [6] and is a result of the fitting procedure for CH3 in noble gas and molecular hydrogen matrices. The correlation coefficient of the linear regression in Fig. 4 is 0.98. Though our very simple model linking the radical–matrix molecule interaction to the hfc shift is in good agreement with experimental data, some features of the empirical dependence in Fig. 4 suggest that more thorough theoretical treatment is still needed. Indeed, solid N2 and CO matrices should bring about, in our model, nearly equal hfc constant shifts of the CH3 radical, but experimental results do not support this conclusion. Interestingly

Fig. 4. hfc Constants of the methyl radical matrix isolated in low temperature matrices plotted against an empirical coefficient accounting for the contribution of the vdW attractive and Pauli repulsive interactions to the hfc matrix shift. Fitting line is taken from Ref. [6] where it was obtained as a result of the least squared procedure of fitting hfc data for CH3 in solid noble gases and molecular hydrogen. The pair interaction energies between the methyl radical and the linear matrix molecules were corrected to the anisotropic forces (see Section 3.2. for detail).

Table 5 Contributions of the different components of the attractive intermolecular interaction to the ground state energy at T ¼ 0 K (kcal/mole). Ec, Enc, Eq are taken from Ref. [23]. The =

last row of the table gives an effective anisotropy parameter for the molecule polarizability, kM , according to Eq. (20). Type of interaction/energy

N2

CO

N2O

CO2

Central attraction/Ec Non-central attraction/Enc Quadrupole interaction contributing to the Enc/Eq Dispersion attraction/Edisp

 3495.6  343.0  252.9  90.1 0.013

 4066.4  726.8  445.1  281.7 0.035

 10143.6  6230.0  3980.0  2250 0.111

 10786.6  7725.8  4819.7  2906.1 0.135

=

Effective anisotropy parameter for the molecule polarizability/kM

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~ which accounts for the the anisotropy is through a coefficient α, ratio of the isotropic hf coupling of the matrix isolated radical to the hfc of the free methyl molecule: α~ M ¼

AM iso

ð21Þ

ree Afiso

where M denotes a specific matrix. Than the correction to the matrix shift is ΔAM corr ¼

ΔAM α~ M

ð22Þ

where ΔAM is taken from Table 4. The results of the calculation are listed in Table 6. One can see from the table, columns 2 and 3, that the above correction procedure for the matrix shifts does not bring the anisotropies, ΔAcorr, closer to each other. This suggests some mechanism contributing to ΔA in matrices with prohibited reorientation of the symmetry axis of the methyl radical. We suppose that different radical librations in different matrices are responsible for the ΔA discrepancy. The averaged hfc-tensor elements resulting from the combined motion—the rapid reorientation about the C3 axis and the hindered oscillation about the C2 axis, may be obtained using expressions for the hfc-tensor averaged over the anisotropic oscillation motion of a spin label [29,30]

the tensor from the axial symmetry which suggests the methyl radical oscillation about only one of two “in plane” perpendicular axes. The finding owes to the low, orthorhombic Cmmm space group, symmetry of the radical surrounding. In our study, we observed no deviation from the axial symmetry of the hf-tensor. This result is expected for the fcc crystal lattice. In the study of CH3 radicals trapped in solid noble gases and para-H2 [6], the anisotropy of the EPR spectrum was shown to be linked to the Pauli repulsive forces between the radical and the nearest matrix particles. No contribution from the vdW interaction was necessary for improving linear regression which fitted an experimentally obtained anisotropy measure to the energy interaction between the radical and a matrix. We suppose that the same conclusion holds true in the case of the matrices of linear molecules. The valence forces are the short-range interactions. Therefore, we will consider the librating CH3 as a molecule executing orientational oscillation in the field with potential UðθÞ ¼ 0 at jθjo θ0 and UðθÞ ¼ 1 at jθj 4 θ0 . Then the mean-squared sine averaged over the energy levels of the particle in this well and with respect to possible motion for a particle at the particular energy level is determined by the expression [32]
 1 sin ð2θ0 Þ sin ð2θ0 Þ 1 Wn  〈 sin 2 ðθÞ〉 ¼ ð26Þ þ ∑ 2 4θ0 4θ0 n ¼ 1 1  ðπ  n=2θ0 Þ2

〈Ax 〉osc ¼ Ax þ ðAz  Ax Þ〈 sin θy 〉osc þ ðAy  Ax Þ〈 sin θz 〉osc

where n ¼1, 2, 3, …, and the probability of finding the system in a state with number n at temperature T has a form

〈Ay 〉osc ¼ Ay þ ðAz  Ay Þ〈 sin 2 θx 〉osc þ ðAx  Ay Þ〈 sin 2 θz 〉osc

W n ¼ exp½  ðn2 1Þβ  expf  ½ðn þ 1Þ2  1βg;

2

osc

〈Az 〉

2

2

osc

¼ Az þ ðAx  Az Þ〈 sin θy 〉

2

þ ðAy  Az Þ〈 sin θx 〉

osc

ð23Þ

where 〈 sin 2 θi 〉 is the mean-squared sine of the displacement angle averaged over all paramagnetic particles, i ¼x, y, z. Tensor elements averaged over rotation about z axis directed along the radical symmetry axis are expressed by 1 〈A ? 〉r ¼ ðAx þ Ay Þ; 〈A== 〉r ¼ Az ; 2

ð24Þ

Combined motions yield ~ == 〉r  〈A ? 〉r Þ〈 sin 2 θ〉 〈A ? 〉 ¼ 〈A ? 〉r α~ þ αð〈A 2 r r ~ 〈A== 〉 ¼ 〈A== 〉r α~ þ 2αð〈A ? 〉  〈A== 〉 Þ〈 sin θ〉

ð25Þ

where 〈A== 〉r and 〈A ? 〉r are to be calculated using Eq. (24) with Ax and Ay taken for the free radical; 〈A== 〉 and 〈A ? 〉 are the EPR parameters obtained in the experiment; α~ is the previously introduced coefficient, Eq. (21), accounting for the matrix shift of the hf-tensor components. Earlier, the procedure of the rotation–oscillation averaging was applied for estimation of the dipolar tensor elements of the α-proton coupling for CH3 in single crystal of CH3COOLi  2H2O [31]. The authors observed slight deviation of Table 6 Hyperfine coupling anisotropy for methyl radicals trapped in low-temperature matrices, ΔA; correction to the matrix shift of the anisotropy, ΔAcorr; correction coefficient, α~ M ; averaged angular deviation of the CH3 symmetry axis orientation,
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin  1 〈 sin 2 θ〉 ; dimension of the potential well for rotation motion of trapped pffiffiffiffiffiffiffiffi 〈θ2 〉. methyl radical, θ0 ; averaged angular deviations of the matrix molecules, Sample temperature is 20 K. Matrix ΔA (mT)

ΔAcorr (mT)

α~ M

sin

1


qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 〈 sin θ〉

θ0 (grad)

pffiffiffiffiffiffiffiffi 〈θ2 〉 (grad)

55.8 47.9 25.3 14.4

19.49 14.67 5.57 5.17

(grad) N2 CO N2O CO2

 0.099  0.110  0.135  0.142

 0.1  0.111  0.138  0.145

0.991 19.63 0.987 16.92 0.979 9.06 0.981 5.20

ð27Þ

with 2

β¼

h 32Iθ0 2 kT

ð28Þ

The perpendicular component of the inertia tensor of the CH3 radical for the rotation about the C2 axis is I ? ¼ 2:972 10  47 kg/m2. Eq. (26) suggests that, for light molecules and low temperatures, the averaged squared sine slightly depends on temperature while being rather sensitive to the size of a potential well. This conclusion agrees with experimental results. We found no appreciable temperature dependence of the hfc-tensor anisotropy for every matrix under study while the anisotropy magnitude differs for different matrices. In Eq. (26), the first term included in parentheses coincides with the result of averaging during classical motion in a potential well and does not depend on temperature, while the second term is a quantum-mechanical correction. This correction is significant in the case of light particles at large dimensions of the potential well. For low-amplitude vibration, 〈 sin 2 ðθÞ〉 nearly equals to 〈θ2 〉. Eqs. (25)–(28) enables us to obtain the dimension,θ0 , of the potential well. To carry out the calculation we need a reliable estimate of the components of the free radical hf-coupling tensor. Since no gas phase state data are available with the exception of Afree iso , we used data obtained in ENDOR experiments carried out with matrices of low symmetry [31,33]. Toriyama et al. [31] found B3, which is the principal element along the spin orbital axis, to be of þ0.90 G. Based on the isotropic hf-coupling value of 21.16 G [31], the α~ correction, Eq. (21), is 0.905. Then 〈A== 〉r ¼  23:373 þ ð0:90= 0:905Þ ¼  22:379 G is obtained. In the studies of x-irradiated DL-serine single crystal, Lee and Box [33] measured αproton dipolar tensor elements of þ12.04 G,  12.96 G, þ0.89 G, with an isotropic coupling value of  20.96 G. These data yields 〈A== 〉r ¼ 22:381 G in fine agreement with the previous value. Thus, we based our calculation of the averaged angle deviation on 〈A== 〉r ¼  2:238 mT. Table 6 presents averaged angle displacement

Yu.A. Dmitriev et al. / Physica B 440 (2014) 104–112

111

Table 7 Ground state energy of the methyl radical libration motion at 20 K, E1, and barriers to the rotation of the matrix molecules at 0 K [23], Eb. Matrix

E1 (K)

Eb (K)

N2 CO N2O CO2

347.9 472.2 1692.4 5224.3

325.6 688.2 5844.5 7293.8

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi obtained as inverse sine of the 〈 sin 2 θ〉 and the dimension of the potential well, θ0 . The sample temperature was set equal 20 K. The pffiffiffiffiffiffiffiffi mean deviation angle 〈θ2 〉 for the matrix molecules at T ¼20 K [23] was listed as a reference. Mean deviation angles for the methyl radical librations, column 5 of the table, and that of the matrix molecule, column 7, turned out surprisingly close to each other. The quantum contribution to the CH3 angle displacement, Eq. 26, was found to be relatively large: in CO2, 61.9%, in N2O, 60.3%, in CO, 58.4%, in N2, 56.7%. The suitability of our approximation can be also judged from comparing the ground state libration energy of the methyl radical, E1, and barriers to the rotation of the matrix molecules, Eb, Table 7. The methyl radical energy levels are given by Ref. [32] 2

En ¼

n2 h 32 Iθ0 2

ð29Þ

The table testifies the suitability of the applied model. Indeed, E1 values obtained from the EPR measurement are of the same order of magnitude with Eb barriers known from the literature. Moreover, both E1 and Eb increase with increasing the matrix molecule eccentricity. In three matrices, CO, N2O and CO2, of larger molecule eccentricity, E1 is smaller than Eb. This seems to happen because of the fact that, in these matrices, the “matrix molecule– matrix molecule” anisotropic interaction is larger than the “methyl radical–matrix molecule” anisotropic interaction. The reverse situation, Table 6, is observed in solid N2 composed of linear molecules of the smallest eccentricity. This may be explained as follows. If the methyl radical reorientation around the C2 axis is governed by the repulsive interactions with the nearby matrix molecules than the barrier to the CH3 rotation may be believed somewhat larger compared to the N2 rotation barrier in N2 solid. Qualitatively, this presumption follows from the sizes of two molecules. The lengths of the N–N and C–H bonds are nearly equal: 110 pm and 109 pm, respectively. If one takes the H–H distance as a measure of the size of the methyl molecule, then the size will be 189 pm. It is of interest that a 230 cm  1 (331 K) rotation barrier between 〈1 0 0〉 and 〈1 1 0〉 directions was estimated based on the crystal field model for CH3 radical in solid Kr [11]. Solid Kr and N2 have very close distances between nearest molecules: 7.59 and 7.55 a0, respectively. One could suppose that the barrier would be larger in solid N2 due to the additional noncentral interactions. As mentioned above, Eq. (26) suggests slight temperature dependence of the hf-coupling tensor parameters through a temperature dependence of the average squared sine, 〈 sin 2 θ〉. In order to give a hint about a magnitude of possible variation of the observed EPR parameters with temperature, Fig. 5 shows 2 2 D ¼ ðð〈 sin ðθÞ〉T =〈 sin ðθÞ〉0 Þ 1Þ  102 , where T is for a given temperature. Thus, the value of D gives us, in percentage, a deviation 2 of 〈 sin ðθÞ〉T from the value it takes at T ¼0. One can readily see from the figure that the temperature shift of the recorded EPR parameters increases with increasing temperature, and reaches 2.5%, at maximum, for CH3 in N2 matrix

Fig. 5. Calculated effect of the sample temperature on the averaged (over the libration motion) deviations of the C3 symmetry axis from the equilibrium direction for CH3 radicals in matrices of linear molecules. See the text for D definition.

at 28 K, the upper limit of the temperature range in our N2 experiments. Because of small values of D, we observed no appreciable temperature effect on the components of the g- and A-tensors.

4. Conclusions Despite the expected superhyperfine broadening, the CH3 radicals in solid N2 matrix show rather well resolved EPR spectrum with distinct powder axially symmetric anisotropy of both g- and A-tensors. The highly resolved experimental spectra made possible accurate measurement of the EPR parameters and show no overcome of the potential barrier to the radical rotation around the C2 axes when the sample temperature increases up to a point at which the radicals start to decay in combination reactions. The value of the anisotropy, however, turned out to be the smallest compared to that of CH3 in other solid gases of linear molecules, while the libration amplitude is the largest and mostly temperature sensitive. The CH3 in N2 matrix is thus believed to be mostly sensitive to changes in the matrix surroundings which would enable more free rotation of the radical reducing the spectrum anisotropy. Based on the finding, one may think of a new

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technique of studying the structural phase transition, orientational order–disorder, in solid nitrogen films at a variety of conditions and on a variety of surfaces using the trapped CH3 radical as a probe. This technique may provide insight into the structure dynamics at the microscopic level which is poorly understood so far. In order to demonstrate potentials of the idea, we launch a study on N2–Ar solid mixture based on tracking the EPR spectrum shape of the trapped CH3 radicals. The first results clearly show the order–disorder transition depending on the Ar impurity content and sample temperature. Earlier Kiljunen et al. [12] first undertook an attempt of recording the α–β phase transition in solid N2 at 36 K by observing possible changes in the trapped methyl radical EPR line shape. Unfortunately, the measurements did not reveal any special features around this temperature because possibly of the rather large linewidths or the transition temperature being affected by the methyl radical probe [12]. Our previous [6] and present study of the CH3 isotropic hyperfine splitting, Aiso, testifies that in a wide range of molecular matrices the matrix effect fits nicely well the linear dependence, Fig. 4, of the Aiso on the certain combination of the attractive and repulsive pair interactions between the methyl radical and a matrix particle. Surprisingly, the mean value of the measured isotropic free-radical hfc constant [3] locates exactly on the fitting line at radical–matrix interaction equal zero, Fig. 4. Thus, our finding suggests that Afree iso ¼  2.337 mT measured in Ref. [3] is an exact magnitude of the hfc constant of the free radical despite very large error bars, Fig. 4, which the authors indicated. Based on that value of Afree iso we suggested, in the present study, an estimate of the parallel component, 〈A== 〉r ¼  2:238 mT, for the free radical. The perpendicular component is, thus, 〈A ? 〉r ¼  2:387 mT, and the extent of the hfc tensor anisotropy, ΔAfree ¼  0.149 mT. The estimate is supported by our experimental results. Indeed, in solid CO2 which is a tough matrix with almost prohibited reorientation of the C3 axes of the methyl molecule, Table 6, the hfc tensor anisotropy recalculated to the free radical value gives  0.145 mT, Table 6. This result is in fine agreement with the estimate of ΔAfree. Thus, one may consider the estimate as the first experimental approach to the hfc anisotropy of the free methyl radical. The obtained value will help in checking correctness of theoretical models used to calculate parameters of this basic quantum rotator. Acknowledgments The reported study was supported, in part, by the Russian Foundation for Basic Research (RFBR), Research project no 13-02-00373a.

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