Reorientational dynamics of C60 in the solid state. An avoided level-crossing muon spin resonance study

Chemical Physics ELSEVIER

Chemical Physics 192 (1995) 231-237

Reorientational dynamics of C60in the solid state. An avoided level-crossing muon spin resonance study Emil Roduner a.,, Kosmas Prassides b.,, Roderick M. Macrae b, Ian M. Thomas b, Christof Niedermayer c, Ulrich Binninger e, Christian Bernhard c, Anselm Hofer c, Ivan D. Reid d a Physikalisch-Chemisches lnstitut der Universitilt Ziirich, Winterthurerstrasse 190, CH-8057 Ziirich, Switzerland b School of Chemistry and Molecular Sciences, University of Sussex, Brighton BNI 9QJ, UK c Fakultlltfiir Physik, Universitilt Konstanz, D-78434 Konstanz, Germany d Paul Scherrer Institute, CH-5232 I/illigen PSI, Switzerland

Received29 April 1994

Abstract

The dynamics of the muonium (Mu) adduct to C6o in the crystalline state have been investigated using avoided level crossing muon spin resonance (ALC-/xSR). The motion of C~0 in the orientationally ordered phase ( T < 260 K) is well described by a spherical diffusion or isotropic jump-reorientation motion with an Arrhenius activation energy of 176(1) meV and a frequency factor of 2.95(13) x 10-12 s. This is in striking contrast to the behaviour of MuCT0 adducts whose reorientational motion fits well in a pseudo-static model with a temperature dependent order.parameter.

1. Introduction

Solids of highly symmetric molecules often form plastic phases with nearly freely rotating molecules. These hybrid-type structures which have aspects of the dynamics of a dense gas but at the same time are ordered crystalline materials have always fascinated researchers. The freezing-in of the dynamics can give insight into subtle intermolecular interactions of the order of kT, as a consequence of deviations from spherical symmetry. A comparison of the two most highly symmetric fullerenes C60 and C70 is of particular interest since both molecules consist of carbon atoms only, and they are both built of hexagon and pentagon units. The difference in dynamic behaviour * Correspondingauthors.

originates from the additional belt of hexagons which distinguishes the elongated C70 (D5h symmetry) from the more spherical C60 (Ih symmetry) molecule, thus leading to different steric requirements and to a modified distribution of partial charges which contribute to orienting forces via Coulomb interactions. Based on carbon atom Lennard-Jones potentials and on the interaction between negative partial charges on the electron-rich short bonds with the slightly positive C atoms, and neglecting the quadrupolar character of C70 [ 1 ], the dynamic behaviour of the two fullerenes was predicted in molecular dynamics simulations [ 2,3]. Experimentally, the dynamical properties of C60 have been studied by many different techniques, ineluding nuclear magnetic resonance [4-7], muon spin rotation [ 8 ], thermal conductivity [ 9], quasielastic neutron scattering [ 10], sound velocity [ 11 ], di-

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electric [12] and dilatometric [13] measurements. In brief, C60 forms a high temperature face-centered cubic (fcc) phase with dynamic molecular rotational disorder, characterised by a correlation time of about 12 ps at 300 K. The molecules rapidly tumble through all possible orientations with little evidence of significant correlations between the orientations of neighbouring molecules. On cooling at 260 K, there is a phase transition to a simple cubic phase, accompanied by a drastic change in the reorientational dynamics. Powder diffraction measurements are now consistent with a model in which the molecules shuffle between two nearly-degenerate orientations [ 14] with an energy difference of 11.4(3) meV [ 15,9] in a thermally activated jump motion. The two orientations are characterised by the nesting of hexagon-hexagon fusions (6 : 6) of one molecule over either a pentagonal (major orientation) or a hexagonal (minor orientation) face of neighbouring molecules. At ~ 90 K, an effective freezing of the motion and a transition to an orientational glass occur. Of particular interest are the detailed dynamics in the intermediate "ratchet" phase (90 < T < 260 K) and the specific mechanism by which the C60 molecules can hop between the two energetically favoured orientations. The simplest type of motion that can be envisaged involves 60 ° molecular hops about 3-fold molecular axes, parallel to the unit cell diagonals. Alternatively, a more complicated, essentially quasi-random, sequence of hopping motion, involving ~ 42 ° rotations about the set of (1-10) axes which straddle short C-C bonds, can also take the molecules between the two orientations. These models represent two extreme situations, almost uniaxial and almost isotropic reorientational jump motions. We would like to demonstrate here that the muon spin resonance (/zSR) technique in its variant which takes advantage of avoided level crossings (ALCs) provides a unique means of studying the reorientational mechanism in the intermediate "ratchet" phase of C60. In addition, a comparison with our earlier results on solid C7o [ 16] clearly distinguishes the differing dynamic behaviour of the two fullerenes. In these experiments, energetic spin polarized positive muons which are available at the beam ports of suitable accelerators are injected into the sample of interest. Towards the end of their thermalization track, they neutralize by capture of an electron from the environment, forming the muonium atom (Mu --/z+e - ) which is

in a chemical sense a light isotope of hydrogen. Mu adds chemically across one of the double bonds to form a free radical, leaving the muon as a polarized spin label on the surface of the fullerene. Mu adducts for C60 [ 8,17 ] have been reported prior to the observation of its H adducts [ 18,19]. All carbons of C60 are equivalent, and a single adduct was therefore observed, with an isotropic hyperfine coupling constant Aiso = 325 MHz. The muon-electron hyperfine interaction is anisotropic, but the dynamics average out this anisotropy in specific ways so that its detection provides a direct probe of the dynamical motion.

2. The effect of dynamics on ALC-pSR transitions The Mu adduct to the fullerene can be regarded as a two-spin-I/2 system analogous to Mu, but with a reduced hyperfine interaction. Such systems have four magnetic eigenstates, which in high magnetic fields are designated by [aeo~/~), [aefl/~), ]flea/~), and [fleflg). Muons which are polarized with their spins parallel (ct~) or antiparallel (fl~') to the external field are thus in an eigenstate, and there is no evolution of spin polarization. In the ALC-/zSR experiment the observed signal [20,21], which is a direct measure of this spin polarization, becomes at high longitudinal fields independent of field. In the isotropic case there is a single crossing of energy levels in the above system of four states. It occurs between [aea ~) and [aefl ~) at a field of Bres = Aiso x 36.72 Gauss, where Aisois the isotropic muon-electron hyperfine interaction in units of MHz. Hyperfine anisotropy provides a direct matrix element which mixes these two states. It turns a true crossing into an avoided crossing. As a consequence of mixing, the system oscillates between the two states at a frequency given by the energy splitting, which is of the order of the hyperfine anisotropy. This relaxes the muon polarization, and in the ALC/~SR spectrum leads to a resonance at Bres. Obviously, when reorientational motion is isotropic and its correlation time increases through a critical time scale set by the inverse hyperfine anisotropy, the resonance starts to disappear. On the other hand, when motion is fast, but not isotropic, the hyperfine anisotropy is only partially averaged to a reduced effective value. This leads to a change in the shape and in particular in the width of this resonance which otherwise is not

E. Roduner et al.I Chemical Physics 192 (1995) 231-237

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B (Gauss) Fig. 1. (a) Simulations of static powder patterns for the IAMI = 1 ALC-/xSR resonance of a muon-electron system with Aiso=275 MHz and variable axial hyperfine anisotropies, DII--+6 MHz, +3 MHz, +1.5 MHz, +0.8 MHz, +0.4 MHz, and +0.2 MHz (going from the broad outer to the narrow inner curves), following the theory given in Ref. [24]. (b) Dynamic simulations of powder patterns based on a stochastic Liouville formalism for spherical rotational diffusion [23], calculated for the IAMI = 1 ALC-/~SR resonance for a muon--electron system with Also=326 MHz, DI1=+10.67 MHz, a residual electron relaxation of 1 MHz, and rotational diffusion constants Drot = 1/6~'c of 0.0, 0.3, 1.0, 3.0, 10.0, and 30.0 MHz (going from the inner to the outer curves).

distinguishable from that of a static system. These two types of behaviour are illustrated for axial systems by the simulations in Fig. 1, using parameters which apply to the present work. The upper part shows the pseudo-static case. A large anisotropy leads to the typical asymmetric powder pattern with a cusp, and with the steep slope on the low-field side for a cigar-type anisotropy (Oil = All - Aiso > 0) [22]. With falling anisotropy the resonance becomes n a r r o w e r first, then more symmetric, and finally it loses amplitude and disappears as a small tick when the anisotropy falls below the inverse muon lifetime (T~ 1 = 0.45 MHz). The lower part shows the dynamic behaviour for isotropic rotational diffusion, based on

233

a stochastic Liouville formalism to average the hyperfine anisotropy [ 23 ]. In contrast to the upper case, the line b r o a d e n s when the dynamics set in, and at higher rotational frequencies it flattens out completely. The first experiment which took advantage of the sensitivity of this resonance to study the dynamics of globular molecules was performed on norbornene in its plastic crystalline phase [24]. This system displayed the pseudo-static behaviour, with an anisotropy that decreased monotonically with temperature. A second type of system, which was expected to comply with the rotational diffusion model, was one with cyclohexadienyl radicals adsorbed and diffusing on the surface of spherical silica grains [25]. Quite unexpectedly, this resonance was absent or very weak in early experiments and became more distinct only when lower surface coverages (5% of a monolayer of benzene) were employed [26]. At low coverages the higher differential heat of adsorption obviously serves to suppress an additional averaging process. The resonance discussed thus far is the only one in a two-spin-1/2 system at high fields. Since it affects only the muon spin by flipping it between the states ~ and fl~, it is classified by the selection rule IAMI= 1. This distinguishes it from additional resonances of the type [AM[ = 0,2 [21] which occur in the presence of further magnetic nuclei normally without interference with the IAMI-- 1 line. Roughly half of the fullerene molecules C60 and C70 contain at least one 13C nucleus, which is also of spin-I/2. In the ensemble average, these are distributed over many inequivalent positions so that the intensity of resonances, other than the IAMI-- 1 one, is expected to be at the level of ~<2% of the observed lines.

3. Experimental

Pure C60 (>99%) was prepared and purified using standard procedures [27]. After preliminary heating at 440 K under reduced pressure for a few hours, the sample was sublimed in a high vacuum system at 700 K. Experiments were performed on ~ 700 mg of sublimed C60 at the z'E3 surface-muon beam line of the accelerator at the Paul Scherrer Institute in Villigen, Switzerland, using an experimental set-up described elsewhere [21]. Data were collected in 10 K intervals over the temperature range 100-300 K on heat-

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Fig. 2. (a) ALC-/zSR spectra obtained with C60 at different temperatures. The solid line is a fit of the dynamic stochastic Liouville model for spherical rotational diffusion [23] which treats MuC6o as a system with axial hyperline anisotropy. (b) ALC-/zSR spectra obtained with C7o at different temperatures in the region around 10 kG (D-adduct radical) [ 16,33]. The solid line is a fit of the powder function for static axial systems [24].

ing, with intermediate temperatures collected in a second cycle of measurements which gave no evidence of hysteresis effects.

4. Results ALC-/zSR spectra obtained at different temperatures with C60 in the region of the IAMI = 1 transition are shown in Fig. 2a. A clear resonance is observed which broadens with increasing temperature. At 250 K, it is no longer detectable. Thus the temperature evolution of the experimental spectra appear to conform clearly to the predictions of the model of spherical diffusion (Fig. lb). This qualitative observation can immediately distinguish between the two extreme reorientational models mentioned earlier for the C60 molecules in the "ratchet" phase. It unambiguously favours the occurrence of quasi-isotropic ~ 42 ° dis-

crete hops about (1-10) axes, as this mechanism will allow each molecule to visit all its sixty equivalent orientations in an essentially random fashion [ 14,28 ]. In order to minimize the influence of a non-linear background in the quantitative analysis of the data, the spectrum obtained at 300 K and confirmed to be just baseline was subtracted from all the other spectra. The subtracted data were then analyzed by least squares fitting to the spherical rotational diffusion model for systems of axial symmetry [23]. The hyperfine anisotropy of the muonated C60 radical has been determined accurately by Duty et. al [29] in a zero-field /zSR experiment. The principal components of the hyperfine anisotropy were -t-10.67 MHz, - 4 . 1 3 MHz, and - 6 . 5 3 MHz. Since this is close to axial symmetry, a value of DII = +10.67 MHz ( D ± = -½Dii) was used in the present analysis as a fixed parameter. Moreover, an intrinsic (independent of reorientational dynamics) electron relaxation rate of 1

E. Roduner et al./Chemical Physics 192 (1995) 231-237

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suit the non-axiality of the system, and also because of the fact that the unique axis is not strictly perpendicular to the direction of diffusion as required by the theoretical model used [23 ]. The fit yields an activation energy of 176( 1 ) meV and a frequency factor of 7"0 = 2.95(13) x 10 -12 s.

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5. Discussion

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MHz was also employed. This is the value determined for the three frequencies at low temperature in zero field where muon and electron share a common relaxation. The value is typical for radicals and not crucial for our analysis. It affects mostly the amplitude of the resonance while, for this low value, it changes the shape only slightly by rounding off the cusp. Variable fit parameters were the signal amplitude, the isotropic muon-electron hyperfine coupling, and the rotational diffusion coefficient which is related directly to the correlation time by Vrot = 1/6rc [5], plus a small residual background represented by a polynomial with terms up to cubic. At the lowest temperature, we clearly see that the experimental line, although slightly asymmetric, does not have the characteristic cusp of purely axial systems. This is to be expected from the measured anisotropy [29]. At higher correlation times, the information about the symmetry of the system is lost, and in the Redfield limit the resonance line is always Lorentzian. A fit of a Lorentzian to these lines appears therefore to be of roughly equal quality, on average, with a fit to the dynamic model. The extracted correlation times are shown in the Arrhenius plot of Fig. 3. We have omitted the data points below 170 K that deviate significantly from the straight line behaviour. This principally arises as a re-

The activation energy obtained for C60 agrees with that of other magnetic resonance type determinations [5,6,8] but is lower than that obtained with different techniques [9,11-13]. ~'0 is at the upper end of the literature values, with the exception of ref. [6]. The frequency factor may be interpreted as an attempt frequency for barrier crossing, that is related to the energies of the small-amplitude librations of the C60 molecules in the lattice. Neutron inelastic scattering measurements have given values for the librational energy of ~ 2.5 meV [ 30]. This translates to 20 c m - l or a correlation time of 2.0x 10 -12 s, in excellent agreement with the determination of r0 from the present experimental data. A theoretical estimate of the frequency factor may be also obtained on the asumption of a simple sinusoidal hindrance potential: V(~) = IVn(1 - c o s n $ ) ,

(1)

where (~ is the torsional angle and n = (0hop/360), with 8hop ~ 42 ° in the present case. In the high barrier limit, one can express the force constant as the second derivative, evaluated at the minimum, f = 82V/8~b 2. In the harmonic approximation, the frequency is obtained from 2Try = X / ~ , which using a moment of inertia for C60 o f / = 1.0x 10 -43 kg m 2 [31] and equating Vn with Ea = 176 meV gives for the librational frequency a value of P ~18 cm - l . This is again in excellent agreement with the value measured by neutron scattering [30]. It should be noted that magnetic resonance techniques are not expected to give the same answer for reorientational correlation times as thermal conductivity, sound velocity, dielectric and dilatometric measurements. For diffusion over a periodic potential, there is a certain probability that the diffusing system crosses several barriers once it has sufficient energy to cross one [32]. Such multiple jumps are detected as enhanced averaging of the 13C chemical shift or muon

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hyperfine anisotropy tensor, whereas they are not distinguished from single jumps in the other techniques. In Fig. 2b, we also show for comparison the temperature evolution of the ALC-/zSR spectra obtained for C70 [ 16,33] in the vicinity of 10 kG. The observed resonance has been attributed to the Mu adduct in the D position of C70, in the nomenclature where the carbon atoms are labeled consecutively A-D, from the poles towards the equator. While the lineshape at low temperatures (frozen radical) is quite comparable to, albeit narrower than, that observed for C60, the behaviour as the temperature increases is strikingly different. The line narrows, and then the amplitude decreases, just as it is found in the simulations for the pseudo-static case (Fig. la). Fast rotational dynamics lead to averaging of the hyperfine tensor of the muonated radical. In an anisotropic potential like the one encountered in the low-symmetry phase of solid C70, this results in a partly averaged tensor of axial symmetry with the principal component, DII, proportional to the order parameter S2(T) [34]: Oil = DzzS2(T) (3 cos2 ~b - 1)/2.

(2)

Dzz is the positive component of the static hyperfine anisotropy, ~b is the angle between the rotation axis and the hyperfine principal axis ~, 0 the angle of the motion, i.e. between the fast rotation axis and a preferred crystal direction, and $2 (T) = (( 3 cos2 0 - 1 )/2) gives the temperature dependence of the order parameter. The decrease of DII with temperature reflects a decrease of the order parameter $2 [ 16,33 ]. The fact that the pseudo-static model describes the situation well implies that the correlation times for both the uniaxial rotation and the motion of the unique axis are short compared with the critical time scale given by the inverse hyperfine anisotropy. The temperature dependence of $2 should then be interpreted by invoking a larger amplitude of the motion of the axis at higher T. Whether the motion is of the type of a precession or jump motion or wobbling cannot be specified, but it is plausible that the monotonically increasing unit cell volume [35] allows the ellipsoidal molecules to reorient more easily. It should be noted that the present experiments measured the reorientaion dynamics of MuC60 rather than that of C60 itself. The increase in the moment of inertia by adding Mu to C60 is certainly negligible. Furthermore, there seems to be sufficient space in the 0.30 nm

gap between the fullerene molecules to accommodate the Mu atom in a distance of ~ 0.11 nm to the carbon atom to which it is bound. We therefore believe that the perturbation is negligible and that the motion of MuC60 represents in a good approximation the dynamics of the parent fullerene molecule.

6. Conclusions In the present work, we have demonstrated the unique potential of the ALC-/zSR technique as a sensitive tool for studying molecular reorientational motion in disordered solids. The quasi-random nature of the dynamics of C60 molecules in the intermediate temperature range (90 < T < 260 K) in the "ratchet" phase is unambiguously reflected in the behaviour of the IAMI= 1 resonance line with increasing temperature - the line disappears through broadening. In remarkable contrast, the differing dynamical behaviour of the ellipsoidal C70 molecules is immediately apparent, as the resonance line of the most abundant MuC70 radical disappears with increasing temperature through narrowing. The motion is now consistent with a pseudo-static model with a temperature-dependent order parameter which reflects the temperature variation of the amplitude of the motion of the long molecular axis itself. These dynamic reorientational mechanisms together with the behaviour observed for cyclohexadienyl radicals on the surface of silica or silica-supported Pd catalysts [26] describe three quite different types of motion which are reflected in unique ways in the shapes of the IAMI= 1 transitions and in their temperature dependences.

Acknowledgement Financial support by the Swiss National Science Foundation, the Science and Engineering Research Council, UK and the British Council is gratefully acknowledged.

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