J. Phys. Chem. 1994, 98, 12133-12141
12133
Reorientational Dynamics of Solid C70 Probed by Positive Muons Roderick M. Macrae, Kosmas bassides,* and Ian M. Thomas School of Chemistry and Molecular Sciences, University of Sussex, Brighton BNI 9QJ, U.K.
Emil Roduner* Physikalisch-Chemisches Institut der Universitat Zurich, Winterthurerstrasse 190, CH-8057 Zurich, Switzerland
Christof Niedermayer, Ulrich Binninger, Christian Bernhard, and Anselm Hofer Fakultat f i r Physik, Universitat Konstanz, 0-78434 Konstanz, Germany
Ivan D. Reid Paul Scherrer Institute, CH-5232 Villigen PSI, Switzerland Received: July 25, 1994; In Final Form: September 19, 1994@
The dynamics of muonium (Mu) adducts to c 7 0 in the crystalline state have been studied using zero-field muon spin relaxation (ZF-pSR) and avoided level crossing muon spin resonance (ALC-pSR) spectroscopy. The two techniques are sensitive to complementary ranges of temperature and together give a complete and consistent picture of the molecular motion. The behavior of the MuC7o radicals which are characterized by a prolate hyperfine matrix with components D, = -6.8(2) MHz, D, = -3.7(2) MHz, D, = 10.5(2) MHz is well-described by a pseudostatic model with a temperature-dependent order parameter. In contrast, M U G O shows a spherically isotropic diffusive or jump reorientation mechanism. C70 is frozen on a time scale of around 30 ns up to 170 K. On further heating, the motion is found to be complex, consisting of a uniaxial rotation part together with a nutational or jumping motion of the unique axis. Anisotropy on the 30 ns time scale persists up to 370 K, well into the face-centered cubic phase of solid C70. Above 390 K, the reorientational motion is essentially isotropic and the avoided level crossing resonance line disappears.
1. Introduction Bulk crystalline fullerenes such as c 6 0 and c 7 0 exhibit dynamical behavior characteristic of plastic phases with a lowtemperature frozen solid form and a freely reorienting, but still crystallographicallyordered, high-temperature This is a consequence of the extremely high symmetry of the molecules and the comparability of intermolecular orientational ordering energies with kT in the vicinity of room temperature. The orientational ordering energies are subtly related to deviations from spherical symmetry in the molecular framework and the electronic charge distribution. Studies of the changes in the type and degree of rotational disorder as the temperature is varied are of great utility in elucidating the detailed nature of the orienting forces present. In particular, it is of interest to compare CSOand C70, since any difference in dynamical behavior originates in the “belt” of 10 additional carbon atoms: which introduces a “unique axis” into the molecular geometry, changes the molecular point group from Ih to D5h, modifies the symmetry and degeneracy of HOMO and LUMO, correspondingly, and creates an electric quadrupole m ~ m e n t . ~ Many experimentaltechniques have been applied to the study of fullerene structure and dynamics, and for c 6 0 the various crystalline phases and the motional degrees of freedom present therein are well-e~tablished.~~~ Recent ALC-pSR work on the orientationally ordered simple cubic phase (T < 260 K) has indicated that the behavior in this region is well-described by an isotropic diffusive or jumping motion with an Arrhenius activation energy of 176(1) meV and a pre-exponential cor-
* To whom correspondence should be addressed. @
Abstract published in Advance ACS Abstracts, November 1, 1994.
0022-365419412098-12133$04.5010
relation time factor, TO, of 2.95(13) x s.* These parameters are in good agreement with those obtained by NMR for c 7 0 is more and other t e c h n i q u e ~ . ~ ~The ~ - ~situation ~ ~ o m p l e x . ~Diffraction ~J~ reveal two structural transitions on cooling; the high-temperatureface-centered cubic (fcc) phase transforms to rhombohedral, followed by a transition to a monoclinic phase. The fcc to rhombohedral transition appears on cooling at ~ 2 7 0 - 2 8 0 K but persists on heating to ~ 3 4 0K. Experimental information on the reorientational dynamics has thus far come from muon spin rotation,20-2113CNMR,22-24 and neutron inelastic s ~ a t t e r i n g . At ~ ~ first pSRZ1 and then also NMR% and neutrons25detected some residual anisotropy above 270 K. Several studies discuss the dynamics at T < 270 K in terms of uniaxial motion, but pSRZ1and NMRZ3 give evidence of a more complicated behavior which is described by means of a temperature-dependent order parameter. No clear agreement has been reached about the transition to a completely frozen state, which has been assigned to temperatures from below 100 K up to 250 K. In this article, we show that positive-muon spectroscopy provides a unique means of studying the dynamical behavior of C70 and of distinguishing it from that of Ca. The two variants of the technique used here are (i) that which employs the characteristic frequencies of muon-electron hyperfine transitions in the absence of an applied magnetic field (ZF-pSR) and (ii) that which takes advantage of avoided level crossings under a strong magnetic field, applied parallel to the initial muon spin direction (ALC-pSR). In both cases, energetic spin-polarized positive muons, generated at suitable accelerators from the parity-violating two-body decay of positive pions produced by proton irradiation of a light elemental target, are injected into 0 1994 American Chemical Society
12134 J. Phys. Chem., Vol. 98, No. 47, 1994
Macrae et al.
TABLE 1: Eigenvalues, Eigenvectors, Transition the sample of interest. Late in the thermalization process, Frequencies and Intensities for a Fully Anisotropic toward the end of their radiation track they neutralize by Muon-Electron Hamiltoniana capturing an electron from the environment to form the eigenvalues eigenvectors muonium atom (Mu ,de-), which is, in a chemical sense, a light isotope of hydrogen (mMu/mH = U9). In fullerenes, a large fraction of the Mu formed then adds chemically across one of the double bonds to form a free radical, leaving the muon as a polarized spin label on the molecule’s surface. The muonelectron hyperfine interaction is anisotropic, but the reorientafrequencies intensities tional motion averages out this anisotropy in specific ways so that its detection through hyperfine transitions provides a direct (P/2)COSZ p’ vi2 = (112164yy - Am) (P/2)sinzp’ sinZa’ probe of the motional dynamics. The remainder of the muonium vi3 = (1/2)(& - A d (P/2)sin2p’ cos2a’ vZ3 = (1/2)(Azz - A,) is trapped inside the fullerene spheroids to form “endohedral” (P/2)sin2p’ cosza’ vi4 = (1/2)(Az +A,) muonium.20s26 In c70 at low temperatures, this species too (P/2)sin2p’ sin2a’ vz4 = (1/2)(&2 +A,) possesses an anisotropic hyperfine interaction which serves as (P/2)cos2p’ ~ 3= 4 (112)(Ayy + An) a measure of the dynamics.20 The angle dependence of the intensities is expressed in terms of Mu adducts have been reported for both C ~ Oand~C7027.28 ~ , ~ ~ the total polarization P and the Euler angles (a’, F , y’) relating the prior to the observation of the H All carbons of principal axis of the hyperfine system to the polarization/observation Cm are equivalent, and a single adduct was therefore observed, axis. with an isotropic hyperfine coupling constant Ais0 = 325 MHz. In C70, there are five inequivalent carbons, but until recently, and three large (“singlet-hiplet”) transition frequencies, denoted only four of the five possible isomers of the adduct had been (~12,Y13, Y23) and (Y14, v24, v34), respectively. These frequencies detected.28 We have now succeeded in observing the “missing” are given in the second part of Table 1 in terms of A,, Ayy,and fifth radical;32the values for Aiso of the five radicals at 324 K A,. They are independent of the orientation of the hyperfine are 275.42(4), 339.96(5), 354.4(3), 361.5(5), and 365.7(8) system with respect to either the initial muon spin direction or M H z . ~Semiempirical ~ molecular orbital calculations are in the detection direction. In zero field, the initially spin-polarized moderate agreement with experiment and qualitatively support p+ are not in eigenstates and there is evolution of the muon the assignments suggested previously, based on other arguspin due to transitions at these frequencies. This leads to a set ments .2 1,27,28,33 of oscillations superimposed on the radioactive decay curve of the muon (lifetime q, = 2.197 x s) in the time-resolved 2. The Effect of Dynamics on pSR Transitions number spectrum of decay positrons, measured along a given direction with respect to the initial orientation of the muon spin Radicals formed by Mu addition to unsaturated organic vector. The amplitudes oscillating at these frequencies have compounds possess an unpaired electron which, if localized, an orientation-dependence, expressed in Table 1 in terms of the resides in a p-orbital centered on the carbon nucleus B to the initial polarization P and the Euler angles, a‘, ,&, and y‘, muon position or else, if delocalized through an extended connecting the hyperfine system with the laboratory system. n-system, contributes to spin density spread over P, y , and more On averaging over the sphere, however, the lines are all of equal distant centers.34 The muon-electron hyperfine interaction is intensity, each taking 1/6 of the total muon polarization. There reduced in strength from that in free Mu and possesses an is no nonoscillating component. Note that the isotropic part of anisotropic pseudodipolar component due to the lack of spherical the coupling cancels from the intratriplet frequencies and that symmetry in the electron distribution around the muon position. these therefore give a direct indication of the anisotropy. A In general, due to the presence of protons, such radicals must fully anisotropic matrix leads to three signals obeying the sum be regarded as multispin systems, with hyperfine spectra which 3 ~ 1 3while , an axial matrix (with e.g. 0, = rule ~ 1 2 ~ 2 = are complex, except under certain limiting experimental condi-20, = -20,) leads to only a single intratriplet line. In the tions. In fullerenes, the situation is simplified by the absence, case of a matrix averaged by molecular motion to complete if the contribution from the natural abundance of 13C is isotropy, only the singlet-triplet line remains. discounted, of magnetic nuclei other than the muon and becomes In the strong magnetic fields of order 1 T characteristic of describable by the hyperfine Hamiltonian of a two-spin- 1/2 ALC-pSR experiments, the uncoupled Zeeman products laeab), system. In the absence of an external magnetic field, the laep), Ipab),and Ipp)are eigenstates of the system. Muons Hamiltonian is very simple, with no Zeeman terms, containing polarized with their spins parallel (ab) or antiparallel @) to only the scalar “Fermi contact”, As0,and traceless matrix the external field are thus in an eigenstate, and there is no dipolar components, D, of the hypenrfine interaction. It may be evolution of spin polarization. The observed ~ i g n a l ,which ~~,~~ expressed in terms of the electron S,and muon spin operators, directly relates to this polarization, becomes field-independent. Ib, with the hyperfine matrix in frequency units, as In the isotropic p+e- system, there is a single field point at which the energies of two of the four states cross. The crossing is between laeab)and laep)and occuts at a field of B,,,, = 36.72AisOG , where Aiso is expressed in units of megahertz. In the presence of an anisotropic hyperfine interaction, off-diagonal terms in the Hamiltonian mix these two states. If the anisotropy is axial, these terms take the form This can be solved in the basis of Cartesian spin functions for s = 1/2. In the uncoupled representation, the basis functions for the p+e- system are Zeeman products, which may be written in the form (memb)with m taking values a = 1/2 and B = -1/ 2. The eigenvalues and eigenvectors are displayed in the first part of Table 1 . With (Aii - A,jl ( 54.7" leads to Dll < 0, which is in agreement with expectation for the feature C. Because Mu addition to C70 induces only a minor distortion, we expect the order parameter to be the same for all MuC70 isomers. Since C is so prominent in Figure 8 and B is nearly absent, we conclude that V(B) is close to 54" and that B is therefore undergoing microscopic magic angle spinning. We are not aware of a previous report of such a situation. The decrease of Dll with temperature is obvious for the D adduct but less clear for the others. (These latter are weak signals and fitted near the limit of the tolerance of the data.) It reflects a decrease of the order parameter S2. Old21and new data for radical D are combined in Figure 9. The fact that the pseudostatic model describes the situation well implies that the correlation times for the uniaxial rotation and the motion of the unique axis, both short compared with the critical time scale given by the inverse hyperfine anisotropy, are sufficiently different that the two motions can be considered separately. When the motion "switches on" at 175-200 K, it is initially almost entirely uniaxial. The subsequent temperature dependence of S2 has to be interpreted by an amplitude of the motion of the axis which increases monotonically with T as the molecules traverse increasingly large sections of the available potential energy surface. Whether the motion is of the type of a precession or jump motion or "wobbling" cannot be specified without ambiguity, but it is plausible that the monotonically increasing unit cell volume18 allows the ellipsoidal molecules to reorient more easily, that is to say, gradually flattens the maxima in the potential energy surface. The nonzero value for S2 shows clearly that there is a directing force even in the fcc phase at 370 K, possibly along the unit cell diagonals.I8 This would imply that below this temperature jumps between the $J
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Macrae et al.
12140 J. Phys. Chem., Vol. 98, No. 47, 1994
different cell diagonal orientations are slow on the critical time scale. Above 370 K the directing force becomes sufficiently small that it does not lead to any favored molecular orientations with respect to the crystal axes on the time scale of the muon lifetime. The disappearance of the lines seems to be fairly sharp, with little difference between the 350 and 370 K spectra and no signal at all in the 390 K spectrum. The high temperature at which the upper phase transition occurs is consistent with diffracti~n~ and ~ . ~ a~l o r i m e t r i cdata ~ ~ in terms of the known chemical and crystallographic purity of the sample. The lack of exact correspondence between the ZF intertriplet frequencies and the D radical ALC resonance at low temperatures implies that the ZF signal contain information from all four radicals and that these radicals have differing anisotropies. While it is difficult to link the results quantitatively to a complex potential temperature-dependence in both symmetry and barrier size, some simpler models of the motion have been proposed with which crude comparison is possible. Blinc et aZ.ll successfully fit their NMR data to a model based on MD calculation^^^ in which three motional modes are distinguishable: uniaxial reorientation only, uniaxial reorientation together with jumping between three (110) directions, and uniaxial reorientation together with tumbling between six (1 10) directions. This hinges on the idea that C~O’S electronic charge distribution privileges the (1IO) directions over the (1 11) unit cell body diagonal. (In fact, a 1:l ratio of the (111):(110) distribution at low temperature implies a (1 11) orientation more stable than (110) by an amount on the order of kT In 3, where T is the temperature below which (1 11)-( 110)jumping ceases to occur. Consistency of the present ALC data with such a model was tested using a simple thermodynamic approach based on that used by Saito, Dresselhaus, and Dresselhaus (SDD) in their study of the specific heat anomaly in C60.4~In the case of c70, there are three modes (and thus four separate populations) to be considered, and the self-consistent equation for the frozen population X1 (=N1/N) is
where
+
a = 1 p2 exp(-TdT)
m,)= exp(-(a
15,30), implying that, even in the high-temperature cubic phase, on short time scales intermolecular forces cause a preference for alignment along (1 10) axes adjacent to one particular (1 11) axis. Application of the above to the Dll values for the D radical is possible by assuming that all motions are fast on the ALC time scale, noting that (1 lo) A (1 11) = $JD = n/5 and writing
where 0:’ is the anisotropy of an axial hyperfine Hamiltonian approximating the true Hamiltonian. An attempt was made to fit the Dll values to eq 5 by combining the simplex search algorithm as implemented in the NAG routine E04CCF in (T,, Tb, Tc) space with a golden section48search to the solution of eq 4. The values of (p2, p3, p4) chosen by Blinc et al. did not lead to a realistic solution for (T,, T b , Tc) for any reasonable 0;’. With ( p 2 , p3, p4) = (5,20, 100) (implying that both (1 11) and (1 10) sites are available and that all (110) and (1 11) sites are sampled in the high-temperature phase) and DE’ = 7.5 MHz the best fit to the data was given by the dotted curve in Figure 9, with parameters T, = 270(10) K, T b = 500(10) K, and Tc = 680(10) K. The subtler points of the experimentally observed T-dependence of 41highlighted by the solid curve in Figure 9 are, not surprisingly, not well-reproduced by the simple model, which does not take into account factors such as abrupt changes in lattice constant accompanying the structural phase transitions.18 Indeed, the values of (T,, Tb, Tc) obtained seem rather low compared to those used by SDD in their study of C a and the entropy per mole as a function of temperature extracted from this model according to the formula
is rather flat over the region 300-400 K, preventing the calculation of a value for the enthalpy change (AH) at the hightemperature phase transition which might be compared with calorimetric measurements. These weaknesses suggest that the assumption underlying ( 5 ) is too strong and that a more involved model of the motion based on a temperature-dependent potential is required to fully account for experimental phenomenology on the sub-microsecond time scale. 5. Conclusion
+BY(X,)YJT)
with (T,, Tb, Tc)the temperature associated with the barriers to each type of motion, and (p2, p3, p4) the number of orientations available to each mobile population (p1 = 1). In particular, T, = ~ S - ~ J where ,, J, is the barrier to uniaxial motion, while Tb,c = k s - ’ d b , c relate to intermolecular energies Jb,c associated with motion of the long axis among a restricted subset of the available orientations and the full set, respectively. The model considers only nearest neighbor interactions, with z (=12) the number of nearest neighbors. The mobile populations are given by
X , = ( a - 1)X,
x, = Br(x,)x, x, = 1 - x,- x, - x, The values of (p2, p3, p4) depend on the detailed nature of the motion. In the model of Blinc et al., they take the values (5,
The anisotropic hyperfine interaction of the free radical species formed by muonium addition reactions to c70 prove to be a sensitive probe of the reorientational motion. Studies at zero field clearly show the temperature range over which the molecule is orientationally frozen ( T 5 170 K) and demonstrate the uniaxial nature of the motion when it commences. Studies of the avoided level crossings at high field illustrate the way in which the motion gradually becomes isotropic at the hightemperature end of the plastic phase. The contrast with the behavior in C a is striking.8 The results demonstrate the capacity of muon spectroscopy techniques to afford acute insights into reorientational motion. The effect of the perturbation introduced by muonium addition to the fullerene molecule is expected to be small. The moment of inertia for uniaxial reorientation differs between the bare fullerene molecule and the D adduct by only one-forthieth of one percent. Further, the C-Mu bond length of approximately 1.1 8, is considerably shorter that the 3 8, distance separating fullerene molecules in the lattice.
Rotational Dynamics of Solid C70 The muon techniques taken in combination offer a picture of C70 in which the system is frozen on a time scale of around 30 ns up to about 170 K, whereupon motion begins in a manner which is anisotropic on a similar time scale up to at least 370 K. This latter temperature is higher than those at which any other techniques have so far succeeded in detecting anisotropy.
Acknowledgment. Financial support by the Engineering and Physical Sciences Research Council, U.K., the Swiss National Science Foundation, and the British Council is gratefully acknowledged. We also thank T. J. S. Dennis and C. Christides for help with the sample preparation and the staff at PSI for their technical support. We thank R. Saito for drawing our attention to ref 46. References and Notes (1) Prassides, K.: Kroto, H. W.: Tavlor. R.: Walton. D. R. M.: David. W. I. F.; Tomkinson, J.; Haddon, R. C.; Rosseinsky, M. J.; Murphy, D. W. Carbon 1992, 30, 1277. (2) Heiney, P. A. J. Phys. Chem. Solids 1992, 53, 1333. (3) Johnson, R. D.; Bethune, D. S.; Yannoni, C. S. Acc. Chem. Res. 1992, 25, 169. (4) Kroto, H. W.; Heath, J. R.; O’Brien, S. C.; Curl, R. F.; Smalley, R. E. Nature 1985, 318, 162. ( 5 ) Roduner, E.; Reid, I. D. Chem. Phys. Lett. 1994, 223, 149. (6) David, W. I. F.; Ibberson, R. M.; Dennis, T. I. S.; Hare, J. P.; Prassides, K. Europhys. Lett. 1992, 18, 219. (7) Yu, R. C.; Tea, N.; Salamon, M. B.; Lorents, D.; Malhotra, R. Phys. Rev. Lett. 1992, 68, 2050. (8) Roduner, E.; Prassides, K.; Macrae, R. M.; Thomas, I. M.; Niedermeyer, C. ; Binninger, U.; Bemhard, C.; Hofer, A,; Reid, I. D. Chem. Phys., in press. (9) Tycko, R.; Dabbagh, G.; Fleming, R. M.; Haddon, R. C.; Makhija, A. V.; Zahurak, S. M. Phys. Rev. Lett. 1991, 67, 1886. (10) Johnson, R. D.; Yannoni, C. S.; Dom, H. C.; Salem, J. R.; Bethune, D. S. Science 1992, 255, 1235. (11) Blinc, R.; Seliger, J.; DolinSek, J.; Arfon, D. Phys. Rev. B 1994, 49, 4993; Europhys. Lett. 1993, 23, 355. (12) Kiefl, R. F.; Schneider, J. W.; MacFarlane, A,; Chow, K.; Duty, T. L.; Estle, T. L.; Hitti, B.; Lichti, R. L.; Ansaldo, E. J.; Schwab, C.; Percival, P. W.; Wei, G.; Wlodek, S.; Kojima, K.; Romanow, W. J.; McCauley, J. P.; Coustel, N.; Fischer, J. E.; Smith, A. B. Phys. Rev. Lett. 1992, 68, 2708. (13) Shi, X. D.; Kortan, A. R.; Williams, J. M.; Kini, A. M.; Savall, B. M.; Chaikin, P. M. Phys. Rev. Lett. 1992, 68, 827. (14) Alers, G. B.; Golding, B.; Kortan, A. R.; Haddon, R. C.; Theil, F. A. Science 1992, 257, 51 1. (15) Gugenberger, F.; Heid, R.; Meingast, C.; Adelmann, P.; Braun, M.; Wiihl, H.; Haluska, M.; Kuzmany, H. Phys. Rev. Lett. 1992, 69, 3774. (16) Prassides, K. In Physics and Chemistry of the Fullerenes; Prassides, K., Ed.; Kluwer Academic Publ.: Dordrecht, 1994; p 203. Prassides, K. Phys. Scr. 1993, T49, 735. (17) Fischer, J. E.; Heiney, P. A. J . Phys. Chem. Solids 1993,54, 1725. (18) Christides, C.; Thomas, I. M.; Dennis, T. J. S.; Prassides, K. Europhys. Lett. 1993, 22, 611. (19) Vaughan, G.; Heiney, P. A.; Cox, D. E.; Fischer, J. E.; McGhie, A. R.; Smith, A. L.; Strongin, R. M.; Cichy, M. A,; Smith, A. B. Chem.
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