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LEONIDAS PETRAKIS
Quadrupolar Relaxation of Aluminum-27 Nuclear Magnetic Resonance in Aluminum Alkyls1
by Leonidas Petrakis Gulf Research & Development Company, Pittsburgh, Pennsylvania
16990
(Received M a y 20, 1968)
The width of the 27Alnmr line has been recorded for many aluminum alkyls, neat and in solution in hydrocarbons, over wide temperature and concentration ranges. The main nmr line-width-determining relaxation mechanism is quadrupolar relaxation. The data are treated in terms of the equation appropriate in the limit of extreme motional narrowing, and the various parameters that are important in that treatment are examined critically. The observed line widths varied from 3 to 46 G at 11.09 MHa and correspond to nuclear quadrupole coupling constants which are reasonable for the aluminum nucleus. The correlation times are calculated variously including the use of the Debye equation and the microviscosity models. The temperature dependence of the line width led to nondrrhenius plots, with activation energies for nuclear relaxation AE,,, between 1 and 10 kcal/mol. The previously reported discrepancies between AE,,, and AE,,,, (the latter variously reported greater or smaller than the former) are resolved.
I. Introduction Nuclear magnetic resonance of the 27Al nucleus in aluminum alkyls has proved a sensitive and useful probe of the environment of these systems.2 The electron deficiency of the aluminum atom in compounds with a coordination number of 3 results in oligomer formation and well-defined complexes with electron donors such as tertiary amines and ethers.2b-6 In addition, an intramolecular alkyl exchange6 and dimermonomer equilibria in these systems are well recogni~ed.~& Asl ~a result, large and greatly variable electric field gradients are being set up about the aluminum nucleus, which couple with the large nuclear electric quadrupole moment to produce an efficient relaxation mechanism. This, as is normal with quadrupolar nuclei, dominates all other relaxation mechanisms, 2&,8-12 and evidence has been adduced which indeed supports this contention. However, in quadrupolar relaxation in particular and in nuclear-spin relaxation in general, several problems remain, among which is the widespread and continuing use of the Stokes theory to calculate the correlation tirne 7,. I n this paper we report the results of the experimental study of the relaxation of 27Al in aluminum alkyls. We have reexamined certain systems which were reported previously, and we have made a systematic extension to other pertinent systems and ranges of the factors that enter in quadrupolar relaxation. Our emphasis is on the critical evaluation of these factors and especially on the applicability of the Stokes theory to these systems.
IT. Quadrupolar Relaxation In nuclear spin relaxation we are interested in the magnitude and the nature of the energy of interaction between the nuclear spins and the other degrees of The Journal of Physical Chemistry
freedom in the lattice such as rotations, vibrations, and translations. These random motions of the molecules, which are the carriers of the nuclear spins, produce fluctuations in the local electric and magnetic fields with components at appropriate frequencies that induce transitions between the nuclear Zeeman levels and produce the relaxation of the spins. These time-dependent fluctuations are expressed in terms of a correlation function G ( r ) that shows the probability that a function of the coordinates of the molecule will not have changed after a time 7. There are a number of mechanisms through which nuclear spins could be coupled to the lattice motion. For quadrupolar relaxation the electric field gradients set up at the nucleus are assumed to fluctuate with the random molecular motions.13 (1) Presented in part at the 155th National Meeting of the American Chemical Society, San Francisco, Calif., April 1968. (2) (a) C. P. Poole, Jr., H. E. Swift, and J. F. Itzel, Jr., J . Chem. Phys., 42, 2576 (1965); (b) L. Petrakis and H. E. Swift, J . Phys. Chem., 72, 646 (1968), and references therein. (3) G. E. Coates, “Organo-Metallic Compounds,” John Wiley & Sons, Ino., New York, N. Y., 1956. (4) H. Zeiss, Monogram Series, No. 147, American Chemical Society, Washington, D. C., 1960. (5) H. E. Swift, C. P. Poole, Jr., and J. P. Itzel, Jr., J . Phys. Chem., 68, 2509 (1964). (6) N. Muller and D. E. Pritchard, J. A m . Chem. Soc., 8 2 , 248 (1960). (7) K. S. Pitzer and H. S. Gutowsky, ibid., 68, 2204 (1946). (8) A. Abragam, “The Principles of Nuclear Magnetism,” Clarendon Press, London, 1961. (9) D. E. O’Reilly and G. C. Schacher, J. Chem. Phw.,3 9 , 1768 (1963). (10) J. A. Pople, Discussions Faradau SOC.,34, 192 (1967). (11) W. B. Moniz and H. S. Gutowsky, J . Chem. Phys., 38, 1155 (1963). (12) G. Bonera and Rigamonti, ibid., 4 2 , 175 (1965). (13) A. Abragam and R. V. Pound, Phys. Rev., 9 2 , 943 (1953).
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QUADRUPOLAR RELAXATION OF 27AlNMRIN ALUMINUM ALKYLS The interaction Hamiltonian X is given by 2
X = Q.VE =
p = -2
(-l)pQp(VE)-p
(1)
where Q,the nuclear electric quadrupole moment, and VE, the electric field gradient, are second-rank tensors and transform under coordinate rotations as do the spherical harmonics with 1 = 2. This time-dependent perturbation is used to calculate the transition probability per unit time between two states with the perturbing field fluctuating in a random stochastic manner, from which, in turn, we may obtain the expression for Ti 1
Ti
-
-+
3n2 21 3 X 20 12(21 - 1) (1
+ f>(T g)2 J(w)
(2)
where J ( w ) is the Fourier transform of the correlation function G ( r ) evaluated at the Larmor frequency wo, B is the asymmetry factor, Q is the electric quadrupole moment, and b2V/bZ2is the gradient of the electric field at the nucleus. The other symbols have their usual significance. In order to evaluate J ( w ) , a model of the random motion is required. The most commonly used model for random reorientational motion in liquids is that of Brownian rotational diffusion. Debye14 first applied this model to dielectric relaxation. It assumes exponential decay functions of the type G(r) = exp( -./ 7,) showing that the probability of change of the coordinate functions decreases exponentially with a characteristic time constant 7,. The Fourier transform J ( w ) of this function gives the intensity of fluctuations of w,, and it has the particularly simple form of
2r0 (3) at extreme narrowing conditions (w0.,
