SpinâLattice Relaxation and 13C NMR Line Shape at Multiaxial

J. Phys. Chem. B 2002, 106, 5335-5345

Spin-Lattice Relaxation and in Fullerite C60

13C

5335

NMR Line Shape at Multiaxial Reorientation of Molecules

Dmitry E. Izotov Max Planck Institute for the Physics of Complex Systems, Noethnitzer str. 38, Dresden, 01187 Germany

Valerii P. Tarasov* KurnakoV Institute of General and Inorganic Chemistry, Russian Academy of Sciences, Leninskii pr. 31, Moscow, 119991 Russia ReceiVed: May 17, 2001; In Final Form: October 1, 2001

Molecular dynamics in the fcc-sc phases of fullerite C60 (13C NMR line shape and spin-lattice relaxation time T1) is described in terms of discrete orientational states of molecules. Molecular reorientations in the low temperature orientationally ordered sc phase are assumed to take place about molecular symmetry axes. Transition into the high-temperature disordered fcc phase is modeled by additional molecular reorientations about C4 symmetry axes of the cubic unit cell. At low temperatures, the NMR line shape functions are demonstrated to depend significantly on the model of molecular reorientations in contrast with T1. In the vicinity of the phase transition at 260 K, T1 changes dramatically for all models, unlike the NMR line shape. The model of relatively fast molecular jumps about a single C3 symmetry axis along with slow reorientations about the other axes is found to adequately reproduce the T1 experiment and qualitatively reflects the temperature changes in NMR line shape.

1. Introduction Fullerite C60 forms soft molecular crystals with statically or dynamically disordered molecules.1 Three crystal modifications of fullerite C60sface-centered cubic (fcc), simple cubic (sc),2,3 and hexagonal close-packed (hcp)4-6 modificationssare currently known. Different crystal modifications of fullerite are considered to be connected with different orientational states of C60 molecules in the unit cell.7 Changes in these orientational states are supposed to be responsible for the fcc-sc structural phase transition of the first order at 260 K.2,3 According to XRD and neutron scattering data,2,8,9 at roomtemperature, C60 molecules form a fcc lattice with orientationally disordered molecules (high-temperature (HT) phase); below 260 K, C60 molecules form a sc lattice with definite molecular orientations (low-temperature (LT) phase), the centroids of molecules slightly displacing (∼0.3%, ref 8) against their positions in the HT lattice. Orientational order implies that one of the C3 axes of each C60 molecule is directed along one of the body diagonals of a unit cubic cell and that no other symmetry elements, except inversion and identity elements, coincide.3,7 This orientation can be derived from the so-called standard orientation (when a C3 axis is aligned with a body diagonal and three mutually orthogonal C2 axes are aligned with the edges of a unit cubic cell) by rotation through ∼22° about the 〈111〉 direction10,11 (Figure 1). Several models of molecular orientation assumed for the HT phase are in agreement with diffraction experiments. One of the models implies that C60 molecules can occupy eight inequivalent positions, with the orientation of molecules in the LT phase being one of these orientations.7 These orientations can be derived, for example, from one fixed orientation by * To whom correspondence should be addressed. E-mail: tarasov@ igic.ras.ru (V.P. Tarasov).

Figure 1. Two standard orientations of a C60 molecule in a cubic unit cell related by a rotation of π/2 about any Cartesian axes in a cubic crystal which pass through three orthogonal two-fold axes of the molecule.

rotation about C4 axes of the lattice cube. The second model implies a continuum distribution of orientations of the C60 molecule in the HT phase.12 Note that none of the models for HT phase is preferable over the other model from the standpoint of diffraction experiments. NMR13-18 and inelastic neutron scattering19,20 evidences show that C60 molecules rotate. The 13C NMR line shape and T1 for C60 were measured in different magnetic fields: 1.4, 4.75, 6.3, 7.04, and 9.4 T13-17 at temperatures 80-400 K. The most important conclusion drawn from those experiments is that, first, the relaxation rate linearly depends on squared polarizing field intensity and, second, the NMR line shape is asymmetric at low temperatures (80-120 K). These observations provide convincing evidence that both the NMR line shape and T1 in fullerite C60 are dominated by the only type of magnetic interaction, namely, carbon chemical shift anisotropy; this is rather uncommon. The effect of direct dipole-dipole interaction on 13C nuclei is negligible, because its magnitude for adjacent 13C nuclei, about 4 kHz,21 is considerably smaller than the chemical shift anisotropy in fields higher than 7 T, about 20 kHz. In addition, at p ) 1.1% natural abundance, the probability of a particular

10.1021/jp011904p CCC: $22.00 © 2002 American Chemical Society Published on Web 05/04/2002

5336 J. Phys. Chem. B, Vol. 106, No. 21, 2002

Izotov and Tarasov

Figure 2. Experimental 13C NMR spectra (ppm from TMS) of C60 at (a) 9.4 T (reproduced with author’s permission from ref 13. Copyright 1991 Am. Phys. Soc.) and (b) 1.4 T (reproduced with permission from ref 15. Copyright 1991 Am. Chem. Soc.) and (c) calculated spectrum at 7.04 T (this work).

nucleus having directly bonded 13C nucleus equals 1 - (1 - p)3 = 0.033. Strictly speaking, spin-rotational interaction expected at high temperatures for symmetrical C60 molecules should not be ruled out. This type of interaction significantly influences the spinlattice relaxation time but not the NMR line shape. No experimental manifestations of spin-rotational interaction have been found thus far. NMR data on C60 have some specific features. T1, unlike NMR line shape, dramatically changes (by 2 orders of magnitude) in the vicinity of the phase transition. On the other hand, the NMR line shape at low temperatures contains a narrow signal at 143 ppm, with the temperature of its emergence depending on the resonance frequency (Figure 2). The interpretation of published data is of a qualitative and conflicting character. Until now, temperature dependence of T1 has been considered only in terms of the isotropic diffusion rotation model (also called a Debye-type model), both above and below the phase transition point,13-17 which disagrees with diffraction experiments. A high potential barrier in the LT region (about 20 kJ/mol13,14,17,19) also testifies in favor of the jump model. The 13C NMR line shape has not been simulated, and the narrow signal at 143 ppm in the sc phase has been attributed to the presence of noncrystalline phase. The absence of adequate theoretical concepts of molecular reorientations in fullerite C60 stimulated our investigation aimed at developing the model that describes temperature dependence of T1 and NMR line shape in C60 with allowance for diffraction data. In sections 2 and 3, we present the formalism describing spin-lattice relaxation and NMR line shape caused by chemical shift anisotropy at multiaxial reorientations. The T1 and NMR line shape calculated by the formulas obtained are reported in sections 4 and 5, respectively. Our findings are discussed in section 6.

13C

2. Spin-Lattice Relaxation Time Following Abragam,22 the relaxation of the longitudinal component of magnetization in the case of chemical shift

anisotropy in the laboratory frame (Hx ) Hy ) 0, Hz ) H0) can be described by the differential equation

d 1 〈I 〉 ) - (〈Iz〉 - 〈Iz〉0) dt z T1 where

1 ) 3H02J(ω0), J(ω0) ) 2 T1

∫0∞G(1)(t) cos(ω0t) dt,

G(1)(t) ) 〈F(1)(0)F(1)*(t)〉 - 〈F(1)(0)〉〈F(1)*(t)〉 (1) J(ω0) is the spectral function, G(1)(t) is the correlation function, F(1) is the lattice function, Iz is the spin operator, ω0 is Larmor frequency, 〈‚〉 stands for ensemble average over lattice coordinates, and the asterisk denotes complex conjugation. The lattice functions F(m) upon rotation behave as secondorder spherical harmonics. In the principal axis system (PAS) of the CSA tensor σ, F(m) are determined as follows:22

F(0) )

γδ γδη , F((1) ) 0, F((2) ) 2 2x6

where γ is the gyromagnetic ratio, δ ≡ σ33 - σii/3 is the chemical shift parameter, η ≡ (σ11 - σ22)/δ is the asymmetry parameter, and σii are the principal components of the shielding tensor σ. To specify the correlation function G(1)(t) in the laboratory frame (LF,1), let us introduce additional coordinate systems: the crystal coordinate system (CCS,2), the PAS of the CSA tensor at the nucleus position at the moment t ) 0 (PAS0,3), and the PAS of the CSA tensor at the nucleus position at the moment t (PASt,4). The transition between the systems can be

Reorientation of Molecules in Fullerite C60 specified through the Wigner rotational matrixes D

J. Phys. Chem. B, Vol. 106, No. 21, 2002 5337 D matrixes are unitary, we have that

Dnk(Ωi, Ωj) ) We have

G(1)(t) )

F(1)(0, LF) )

/ D1m (1, 2)D/mn(2, 3)F(n)(0, PAS0) ∑ m,n

/ / / D1m′ (1, 2)Dm′n′ (2, 3)Dn′k (3, 4)F(k)(t, PASt) ∑ m′,n′,k

F(1)(t, LF) )

1

Wij(t)D/km(Ω0, Ωi)Dnm(Ω0, Ωj)F(n)F(k)* ∑ ∑ 5 i,j m,n,k (2)

Hereafter the symbol Ω0 is omitted for brevity. If the CSA tensor has axial symmetry, η ) 0

Because

G(1)(t) ) 1

/ D1m′ dΩ ) δmm′, ∑D/mnDmn′ ) δnn′, ∫Dmn dΩ ) 0 ∫D1m 5 m

where Ω are the Euler angles and δij is the Kronecker delta, and then taking into account powder averaging, we have

G(1)(t) )

1

∑〈Dnk(3, 4)〉 F(n)F(k)*

5 n,k

Here the functions F(n) and F(k) are specified in the PAS of the CSA tensor. Let us define the ensemble average as

〈Dnk(3, 4)〉 )

Wij(t)Dnk(Ωi,Ωj) ∑ i,j

where Wij(t) is the probability of finding the nucleus in the position where the orientation of the PAS of the CSA tensor is determined by the Euler angles Ωi at the initial moment and by Ωj at the moment t. The same matrix D(Ωi, Ωj) describes the rotation of the coordinate system turned through an arbitrary angle Ω with respect to the PAS0. If the CSA tensor is strictly bound to the molecule, the molecular coordinate system can be chosen as the system PAS0, and hence, the D(Ωi, Ωj) matrix is easy to determine. In this particular case, there is no need to know how the principal axes of the CSA tensor are disposed with respect to the molecule. This at first glance surprising conclusion results from the way of powder averaging of G(1)(t). In fact, we approximated the averaged macroscopic magnetization 〈M〉 ) M0(1 - 〈exp(-t/T1)〉) by the leading term 〈M〉 = M0(1 - exp(-t〈1/T1〉)). Strictly speaking, the higher terms of the expansion should give nonexponential decay of 〈M〉, as well as the dependence of T1 on the orientation of the CSA tensor with respect to the molecule. The effects of analogous terms were considered in ref 23 for tetrahedrally coordinated four-spin-1/2 system. For such a system, the correction was found to be negligible. Precise calculation of these terms in the case of C60 is a challenge. However, because τ , T1 in fullerite over the entire temperature range known, the 13C nucleus changes sites many times during T1, and the initial relaxation of the powder magnetization is roughly exponential.24 It is convenient for calculations to define rotations from one fixed orientation Ω0 into the other orientations Ωi rather than between all ΩiΩj pairs (in the first case, N rotational matrixes are to be specified, and in the second, N2). Because the

D/km(Ω0, Ωi)Dnm(Ω0, Ωj) ∑ m

1 20

Wij(t)D/0k(Ωi)D0k(Ωj) ∑ i,j,k

γ 2δ 2

Note that D0k depends only on two of the three Euler angles, R and β, which coincide with the polar angles φ and θ of the Oz′ vector of the CSA tensor principal axis system. In the formula for the conditional probability, Wij ) wijwi, wij is the probability of the nucleus being in the position j at the moment t provided that it was in the orientation i at the initial moment, and wi is the a priori probability of the nucleus being in the orientation i at the initial instant. Let us assume that wi ) 1/N, where N is the number of different nucleus positions. If the motion of the molecule is assumed to represent the Markovian process with discrete states and continuous time, the time dependence of probabilities wij is determined by the system of differential equations

dw ) wH, w(0) ) E dt

(3)

where H is the kinetic matrix and E is the unit matrix. The matrix element Hij is the sum of the frequencies of the jumps about each of the rotation axes Cn that transfer the molecule from the ith orientation to the jth one. If the axis multiplicity is higher than two, the jump frequency is multiplied by a factor of 1/2, because rotation about the axis in two directions is possible. The matrix H is usually assumed to be symmetrical. The solution of system (3) for the symmetrical matrix H is

wij )

∑k CkiCkjeλ t, k

i, j, k ) 1,...N,

(4)

where λk are the eigenvalues and C ) |Cij| is the matrix of the eigenvectors of H. Hence, wij ) wji at any moment t g 0. Substituting (2) and (4) into (1) gives

1 T1

)

3 10N

γ2H02δ2

∑i

|λi|

∑CijCik ×

λi2 + ω02 j,k

fl fm ∑D/ln(Ωj)Dmn(Ωk) ∑ l,m n

(5)

where f0 ) 1, f(1 ) 0, f(2 ) η/x6, and γH0 ) ω0. Let us consider the right-hand side of (5). Because the


Izotov and Tarasov

D matrixes are unitary and the C matrix is orthogonal, we have

CijCik ∑ fl fm ∑D/ln(Ωj)Dmn(Ωk) ) ∑ j,k l,m n C fl fm∑D/ln(Ωj)Dmn(Ωj) + ∑j ij2∑ l,m n / / C C f f D (Ω ∑ ij ik ∑ l m ∑ ln j)Dmn(Ωk) + Dln(Ωk)Dmn(Ωj) ) j Tc. This implies that jumps about the C4 axes are added to the reorientations about the molecular symmetry axes when the C60 molecule is passing to the HT phase.

(1) (1) Figure 3. Mutual arrangement of C(1) 2 , C3 , and C5 rotation axes in C60 molecule.

Figure 4. Schematic drawing of the CSA tensor layout in a C60 molecule.

Figure 5. Theoretical T1 temperature dependence for models A-C of molecular reorientations in fcc-sc phases of fullerite C60 (B0 ) 4.7 T). Kinetic parameters are listed in Table 1.

The following parameters were used for T1(T) computations:17

ω0 ) 2π × 50.288 MHz, σ11 ) 214 ppm, σ22 ) 187 ppm, σ33 ) 34 ppm hence it follows that σ ) 145.0 ppm, δ ) -111.0 ppm, and η ) 0.2432. Because the number of the discrete rotations models for the C60 molecule is great, those were chosen that served either as indicators or could manifest themselves in the experiment. The kinetic parameters of reorientations about the C4 axes in the high-temperature phase were taken identical for all models

ν04 ) 2.1 × 1010 Hz, Ea4 ) 0.8 kJ/mol.


Figure 6. Theoretical T1 temperature dependence for models D-G of molecular reorientations in fcc-sc phases of fullerite C60 (B0 ) 4.7 T). Kinetic parameters are listed in Table 1.

Figure 7. Experimental (°, ref 17) and theoretical (--, -‚-) T1 temperature dependence for models G-I of molecular reorientations in fcc-sc phases of fullerite C60 (B0 ) 4.7 T). Kinetic parameters are listed in Table 1.

Molecular reorientation axes for all of the models considered are listed in Table 1. The same kinetic parameters were assumed for the molecular axes of the same type. The mutual arrangement (1) (1) of C(1) 2 , C3 , and C5 molecular axes is shown in Figure 3. The results of the calculations are shown in Figures 5-7. 5.

13C

NMR Line Shape Calculation for C60

The same models and kinetic parameters were used to calculate the NMR line shape function (Table 1). The magne-

Izotov and Tarasov tization function was computed according to formula 11, and powder averaging taken into account, with subsequent Fourier transformation. This scheme appeared to be robust relative to computational roundoff in comparison with direct calculation of the line shape function by formula 12, because a very high accuracy was required for computing ϑk, ωk, ak, and bk. Because S(t) is invariant relative to the substitution β′ ) π - β and γ′ ) γ - π, it is sufficient to integrate S(t) over 0 e β e π and 0 e γ e π or over 0 e β e π/2 and 0 e γ e 2π. Furthermore, S(t) analytically depends on the variables β and γ; hence, the algorithms for highly smooth functions can be applied. Three integration schemes have been sampled: the rectangular formula and Gaussian quadratures with two and three integration points at every elementary interval for each variable.31 These integration schemes appeared to give the results of almost the same quality, with the more sophisticated integration scheme resulting in slight improvement. The number of the integration intervals was varied in a wide range to provide convergence to the smooth spectrum. In the LT region, the mesh nβ × nγ ) 100 × 200 was used in most instances. In the intermediate and HT regions when the jump frequencies are comparable with or much higher than the rigid spectrum width, respectively, the following feature was pointed out: the higher the number of reorientation axes and the higher the multiplicity of those axes, the lower the number of integration intervals that had to be used to provide a satisfactory smoothness of the spectra. For instance, at high temperatures for models A (C2 single axis) and B (C3 single axis), as much as 400 × 800 integration intervals were required, whereas for multiaxial rotation models, 30 × 60 or even 15 × 30 intervals were used. All of those mentioned above deal with the computations without convolution of the line shape with the Gaussian or Lorentzian function. Once convolution has been used, the smooth spectrum that changes slightly with increasing the number of integration nodes can be derived even with the use of a small number of intervals. For all models, a residual broadening was taken into account by convolution with the Gaussian function with the width at half-maximum equal to 5 ppm. It is worth noting that the Gaussian integration scheme uses irregular intervals on β variable. The regular integration intervals scheme was found to be ineffective for computing NMR powder spectra.32,33 The so-called ASD-algorithm32 makes it possible

Figure 8. 13C NMR C60 theoretical spectra for three different positions of the CSA tensor (a,b,c) upon molecular rotation about a single C2 symmetry axis with different jump frequencies (model A), B0 ) 7.04 T.

Reorientation of Molecules in Fullerite C60

J. Phys. Chem. B, Vol. 106, No. 21, 2002 5341

Figure 9. 13C NMR C60 theoretical spectra for three different positions of the CSA tensor (a,b,c) upon molecular rotation about a single C3 symmetry axis with different jump frequencies (model B), B0 ) 7.04 T.

Figure 10. 13C NMR C60 theoretical spectra for three different positions of the CSA tensor (a,b,c) upon molecular rotation about a single C5 symmetry axis with different jump frequencies (model C), B0 ) 7.04 T.

to decrease computation time at the expense of irregular integration intervals over the Euler angles. However, this algorithm can be applied only to rigid spectra computation, which is of little interest to us. The magnetization function was computed by the following scheme. Time intervals were taken to be ∆t ) 26.5 µs (the maximum frequency is 18.87 kHz), where the number of intervals n was equal to 1024 or 2048. The values of Sk(t) ) eλkt(ak + ibk) were computed iteratively by the following formulas N

Sk(t + ∆t) ) eλk∆tSk(t), Sk(0) ) ak + ibk, S(t) ) Re

∑Sk(t)

k)1

where N is the number of different sites of a 13C nucleus. The parameters used for line shape were the same as those used to calculate T1, except ω0 ) 2π × 75.5 MHz. From the symmetry of the C60 molecule, we can conclude that two of the three principal axes of the CSA tensor are located in the symmetry plane passing through the edges shared by hexagons

(Figure 4). Thus, there is some arbitrariness in deciding between the CSA orientations in the molecule. To estimate the influence of CSA orientation on the NMR line shape function, the computations were carried out for three cases: (a) The Ox′ axis is directed along the edge connecting hexagons. (b) The Oz′ axis is directed along the molecular radius (Figure 4). This orientation can be derived from the first one by rotation about Oy′ through ∼11° clockwise. (c) The Ox′y′ plane coincides with a pentagon plane. This orientation can be derived from the first one by rotation about Oy′ through ∼30° clockwise. The results of computations are shown in Figures 8-16. For models A-C, the jump frequencies about the single axis are listed in kHz; for multiaxial rotation models (D-I), the temperatures are given. A change to jump frequencies can be made by the Arrhenius equation (13) with parameters listed in Table 1. 6. Results and Discussion Reorientations of the C60 molecule in the LT phase are considered to be instantaneous jumps about the axes of


Izotov and Tarasov

(1) Figure 11. 13C NMR C60 theoretical spectra for three different positions of the CSA tensor (a,b,c) upon molecular rotation about C(1) 2 and C3 symmetry axes (model D), B0 ) 7.04 T. Kinetic parameters are listed in Table 1.

(1) Figure 12. 13C NMR C60 theoretical spectra for three different positions of the CSA tensor (a,b,c) upon molecular rotation about C(1) 2 and C5 symmetry axes (model E), B0 ) 7.04 T. Kinetic parameters are listed in Table 1.

symmetry that belong to the point symmetry group Ih. In particular, the axis set for model I coincides with that of group Ih, and the group theory formalism can be applied.34 A 60-dimensional linear space of probability vectors defines representation Γ of group Ih. In the case of model I, this representation can be reduced as follows:

Γ ) A1 + F1g + 2F1u + F2g + 2F2u + 2Gg + 2Gu + 3Hg + 2Hu The correlation times τ of corresponding irreducible representations are determined by the formula

τµ-1 )

∑q nqνq(1 - χµq/χµE)

where nq is the number of axes of the class q, χµq is the character of the irreducible representation µ for the rotation axes of the class q, and E denotes the identity operation. The values

of τµ for model I are listed in Table 2. Thus, there are four different correlation times for model I. In other models, the reorientation axis sets are the subsets of group I ⊂ Ih. These specific cases require particular symmetry analysis that differs from the one in ref 34, and the symmetry group of eq 3 has to be considered.35 However, in these cases, τµ-1 is hard to be expressed in the explicit form by means of the group theory. Physically, preexponential τ0n corresponds to the period of torsional librations of the molecule in the lattice. To estimate it in the LT region, inelastic neutron scattering data were used:20 Elib = 25 kJ/mol, whence ν0n = 1013 Hz. As to reorientations about the C4 axis at high temperatures, no relevant data are available. However, estimations from the diffusion rotation model can be assumed (ν04 = 1010 Hz), because these models are similar for the HT phase. The computations of temperature dependence of spin-lattice relaxation time in C60 reveals the following peculiarities. 1. All models exhibit the discontinuity in the T1(T) curves at

Reorientation of Molecules in Fullerite C60

J. Phys. Chem. B, Vol. 106, No. 21, 2002 5343

(1) Figure 13. 13C NMR C60 theoretical spectra for three different positions of the CSA tensor (a,b,c) upon molecular rotation about C(1) 3 and C5 symmetry axes with identical kinetic parameters (model F), B0 ) 7.04 T.

Figure 14. 13C NMR C60 theoretical spectra for three different positions of the CSA tensor (a,b,c) upon molecular rotation about single C2 and all C5 symmetry axes (model G), B0 ) 7.04 T. Kinetic parameters are listed in Table 1.

the point of the phase transition (260 K) because of switching the jumps about C4 axes of the lattice cube. 2. The HT portions of the T1(T) curves are identical for all models, because the jump frequencies about molecular symmetry axes are much less than the ones about the C4 axes of the unit cell. 3. For models with a single reorientation axis (A-C), the minimum T1 value is higher than for models of multiaxial reorientations. Furthermore, the minimum points are observed at different temperatures, with other parameters being the same. 5. For models of multiaxial reorientations, the minimum T1 values are identical and agree with the experiment better than in the case of models of single reorientations. The minimum positions on the temperature scale significantly depend on the kinetic parameters. 6. Fitting the calculated data to the experimental ones results in almost identical T1(T) curves for different models of multiaxial reorientations.

For models A-H, the kinetic parameters were slightly varied to fit the calculated NMR line shape function to the experimental line shape. For model I, we tried to fit T1(T) to experimental dependence. The simulated 13C NMR spectra of C60 much more strongly depend on the reorientation model used, and have the following specific features. 1. In the LT region (when jump frequencies are much less than the line width for the rigid lattice), all models exhibit identical spectra, characteristic of CSA interaction. 2. For models D-I of multiaxial rotations, the temperature evolution of the NMR consists in a progressive shift of the basic signal from 187 to 143 ppm accompanied by a decrease in spectrum width. 3. In the extreme cases of anisotropic distribution of rotation axes (single axis models A-C), the spectra consist of a combination rigid spectra with different principal components of the CSA tensor averaged by rotation.


Izotov and Tarasov

Figure 15. 13C NMR C60 theoretical spectra for three different positions of the CSA tensor (a,b,c) upon molecular rotation about single C3 and all C5 symmetry axes (model H), B0 ) 7.04 T. Kinetic parameters are listed in Table 1.

Figure 16. 13C NMR C60 theoretical spectra for three different positions of the CSA tensor (a,b,c) upon molecular rotation about all C2, C3, and C5 symmetry axes (model I), B0 ) 7.04 T. Kinetic parameters are listed in Table 1.

TABLE 2: Correlation Times τµ of Molecular Reorientations in the LT Phase of Fullerite C60 for Model I µ

τµ-1

A1 F1g, F1u F2g, F2u Gg, Gu Hg, Hu

0 20ν2 + 20ν3 + (10 - 2x5)ν5 20ν2 + 20ν3 + (10 + 2x5)ν5 15ν2 + 15ν3 + 15ν5 12ν2 + 24ν3 + 12ν5

4. For models with slow and fast reorientation axes, the LT spectra change because of jumps about the fast axis. Reorientation about the slow axis manifests itself as smoothing and narrowing the spectra with an increase in temperature. 5. In the vicinity of the phase transition at 260 K, models D-I of multiaxial reorientations exhibit no distinguishable spectral changes because of high jump frequencies (significantly exceeding the line width of the rigid spectrum). In the HT region, all models demonstrate a slightly higher slope of the T1 curve in comparison with the experimental one. As was pointed out in ref 17 for some C60 fullerite samples (with defects), T1(T) curves have inverse temperature depen-

dence in the HT phase. One of the reasons behind this observation can be the spin-rotational (SR) contribution to T1. The feature of this contribution is the increase of SR correlation time τJ with an increase in temperature, because τJ depends not on the molecular angular rotation rate but on change of that rate in magnitude and direction

1 ) (8π2kT/p2)IC2τJ T1sr where I is the moment of inertia and C is the SR constant. For a spherical molecule, correlation times τJ and τθ are related with each other by the following equation:36

τθτJ )

I at τJ , τθ, 6kT

hence T1sr and T1csa have opposite temperature slopes. We may suppose that the slight decrease in T1(T) magnitude at high temperatures is caused by SR interaction. Preliminary estimations show that SR interaction might appear noticeable at

Reorientation of Molecules in Fullerite C60 temperatures higher than 400 K. To gain deeper insight into the role of this contribution, HT relaxation experiments are required. 7. Conclusions The suggested models of multiaxial discrete reorientations satisfactorily describe the T1(13C) temperature dependence with allowance for singularity at the phase transition point, as well as the evolution of NMR line shape as a function of kinetic parameters (jump frequencies and potential barrier about selected axes). The discontinuity in the T1 temperature dependence curve at the phase transition point results from additional molecular jumps with significantly lower barrier between inequivalent orientations about unit cell axes. We emphasize that the model of discrete molecular jumps used above meets the symmetry requirements for both HT and LT phases of fullerite C60 in contrast with the diffusion rotation model. However, we cannot rule out the diffusion model for the HT phase on the basis of present experimental data. When considering the models, we tried to reproduce the signal at 143 ppm in the line shape curve. In the framework of our models, one could expect that agreement with the experiment can be obtained by introducing slow axes to single-axis models (A-C). These additional axes smooth the spectrum and decrease its width to the single line at 143 ppm with an increase in temperature. Model H is composed with allowance made for that requirement. The model agrees with the experiment better than the other models; that is, it reproduces the features of the NMR spectrum. It is worth noting also that model H agrees with the idea that C60 fullerite has a special crystal direction, a body cube diagonal, along which C3 molecular axes are directed and about which rotations take place with lower potential barrier.3 The narrow signal at 143 ppm in the LT phase of C60 has previously been assigned to a small amount (∼5%) of imperfect or noncrystalline domains in the samples;13,15 that is, the latter were assumed to be heterogeneous. It is believed that C60 molecules in these imperfect domains rotate rapidly at low temperatures, presumably, because of relatively low barriers to reorientations.13 Modeling such a two-phase system is a separate, hitherto untouched problem, which we are planning to consider in the future. Our present model refers to the homogeneous system. Our findings show that the presence of the noncrystalline phase is not the only factor that can account for the narrow signal at 143 ppm. Even for perfect crystalline C60 samples, the simulated NMR line shape has a sharp maximum at 143 ppm, depending on the rotation frequency and activation energy and the model employed. Currently, it is impossible to distinguish between the contributions of the noncrystalline phase and the molecular reorientation anisotropy to the experimentally observed narrow signal. There are some grounds to believe that single-crystal experiments will shed light onto this issue. Note that the models that reproduce well the NMR line shape do not reproduce T1 temperature dependence with the same quality, kinetic parameters being identical, and vice versa. One of the possible reasons for this discrepancy might be the necessity of accurately solving the inverse problem, which is a challenge. Our basic guideline in deciding which type of reorientation model to employ was the same trend in the line shape of experimental and simulated spectra observed with a change in temperature and model kinetic parameters. Fitting the calculated values to experimental data was beyond the scope of this work.

J. Phys. Chem. B, Vol. 106, No. 21, 2002 5345 It is worth noting also that considered models may be applied to the interpretation of inelastic neutron scattering experiments. Furthermore, the formalism developed is general enough and can be brought into use for the analysis of molecular rotation dynamics in hcp phase of fullerite C60. Acknowledgment. This work was supported by the Russian Foundation for Basic Research (Project 98-03-32513). The authors are grateful to prof. T.N. Khazanovich (N.N. Semenov Institute for Chemical Physics, Moscow, Russia) for useful discussions. References and Notes (1) Parsonage, N.; Staveley, L. Disorder in crystals; Clarendon: Oxford, U.K., 1978; Vols. 1-2. (2) Heiney, P.; Fischer, J.; McGhie, A.; Romanow, W.; Denenstein, A.; McCauley, J.; Smith, A. Phys. ReV. Lett. 1991, 66, 2911-2914. (3) Dresselhouse, M.; Dresselhouse, G.; Eldund, P. Science of Fullerenes and Carbon Nanotubes; Academic Press: San Diego, CA, 1996. (4) Kra¨tschmer, W.; Lamb, L.; Fostiropoulos, K.; Huffman, D. Nature 1990, 347, 354-358. (5) Arkhngel’skii, I.; Skokan, E.; Velikodnyi, Y.; Chernyshov, V.; Sidorov, L. Dokl. Akad. Nauk 1998, 363, 494-496. (6) Ce´olin, R.; Tamarit, J.; Lo´pez, D.; Barrio, M.; Agafonov, V.; Allouchi, H.; Moussa, F.; Szwarc, H. Chem. Phys. Lett. 1999, 314, 21-26. (7) Aksenov, V.; Shakhmatov, V.; Osip’yan, Y. JETP 1998, 86, 591597. (8) David, W.; Ibberson, R.; Dennis, T.; Hare, J.; Prassides, K. Europhys. Lett. 1992, 18, 219-225. (9) Moret, R.; Albouy, P.; Agafonov, V.; Ceolin, R.; Andre´, D.; Dworkin, A.; Szwarc, H.; Fabre, C.; Rassat, A.; Zahab, A.; Bernier, P. J. Phys. I France 1992, 2, 511-515. (10) Wochner, P.; Xiong, X.; Chow, P.; Moss, S. Phys. ReV. B 1997, 55, 5678-5683. (11) Blaschko, O.; Krexner, G.; Maier, C.; Karawatzki, R. Phys. ReV. B 1997, 56, 2288-2291. (12) Heiney, P. J. Phys. Chem. Solids 1992, 53, 1333-1352. (13) Tycko, R.; Dabbagh, G.; Fleming, R.; Haddon, R.; Makhija, A.; Zahurak, S. Phys. ReV. Lett. 1991, 67, 1886-1889. (14) Mizoguchi, K. J. Phys. Chem. Solids 1993, 54, 1693-1698. (15) Yannoni, C.; Johnson, R.; Meijer, G.; Bethune, D.; Salem, J. J. Phys. Chem. 1991, 95, 9-10. (16) Tycko, R.; Haddon, R.; Dabbagh, G.; Glarum, S.; Douglass, D.; Mujsce, A. J. Phys. Chem. 1991, 95, 518-520. (17) Privalov, V.; Muravlev, Y.; Arkhangel’skii, I.; Skokan, E.; Sidorov, L. Zh. Neorg. Khim. 1997, 42, 1031-1038. (18) Pennington, C.; Stenger, V. ReV. Mod. Phys. 1996, 68, 855-910. (19) Neumann, D.; Copley, J.; Cappelletti, R.; Kamitakahara, W.; Lindstrom, R.; Creegan, K.; Cox, D.; Romanow, W.; Coustel, N.; McCauley, J.; Maliszewskyj, N.; Fischer, J.; A. B. Smith, I. Phys. ReV. Lett. 1991, 67, 3808-3811. (20) Neumann, D.; Copley, J.; Reznik, D.; Kamitakahara, W.; Rush, J.; Paul, R.; Lindstrom, R. J. Phys. Chem. Sol. 1993, 54, 1699-1712. (21) Yannoni, C.; Bernier, P.; Bethune, D.; Meijer, G. J. Am. Chem. Soc. 1991, 113, 3190-3162. (22) Abragam, A. The principles of nuclear magnetism; Clarendon: Oxford, U.K., 1961. (23) Peternelj, J.; Pintar, M. Phys. ReV. B 1977, 15, 5097-5106. (24) Lalowicz, Z.; Punkkinen, M.; Vuorima¨ki, A.; Ylinen, E.; Detken, A.; Ingman, L. Solid State Nucl. Magn. Reson. 1997, 8, 89-107. (25) Kaplan, J.; Fraenkel, G. NMR of chemichally exchanging systems; Academic Press: New York, 1980. (26) Lalowicz, Z.; Sagnowski, S. Z. Naturforsch. 1991, 46a, 829-840. (27) Keller, A.; Benz, S.; Haeberlen, U. J. Magn. Res. 1987, 72, 434448. (28) Haeberlen, U. High-resolution NMR in solids. SelectiVe aVeraging; Academic Press: New York, San Francisco, London, 1976. (29) Feldman, E. Teor. Eksp. Khim. 1974, 10, 787-793. (30) Michel, K. J. Chem. Phys. 1973, 58, 142-152. (31) Kalitkin, N. Numerical methods; Nauka: Moscow, 1978. (32) Alderman, D.; Solum, M.; Grant, D. J. Chem. Phys. 1986, 84, 3717-3725. (33) Greenfield, M.; Ronemus, A.; Vold, R.; Vold, R.; Ellis, P.; Raidy, T. J. Magn. Res. 1987, 72, 89-107. (34) Bee, M. Quasielastic Neutron Scattering; Hilger: Bristol, U.K., 1988. (35) Izotov, D.; Tarasov, V. Chem. Phys. 2000, 261, 399-410. (36) Hubbard, P. Phys. ReV. 1963, 131, 1155-1165.

SpinâLattice Relaxation and 13C NMR Line Shape at Multiaxial

Recommend Documents

SpinâLattice Relaxation and 13C NMR Line Shape at Multiaxial