New Method for the Determination of Diffusion Constants from Partially Narrowed NMR Lines
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NEW METHOD FOR THE DETERMINATION OF DIFFUSION CONSTANTS FROM PARTIALLY NARROWED NMR LINES*
D. WOLF Materials Science Division,
Argonne National Laboratory,
Argonne,
IL 60439
ABSTRACT The effect of atomic and molecular motions on the NMR free-induction decay (FID) and lineshape is investigated theoretically in the intermediate temperature range in which the NMR line is only partially narrowed. It is shown that the FID may be decomposed into the weighted sum of a rigid-lattice (background) contribution and an exponentially decaying part containing all the information on the diffusive or reorientational motions in the crystal in terms of the spin-spin relaxation time T 2 . INTRODUCTION In contrast to spin-lattice relaxation processes, spin-spin relaxation is most effective in the absence of atomic or molecular motions. The result is a background ("rigid-lattice") free-induction decay (FID) or linewidth which prohibits the investigation of internal motions in crystals if the mean time between consecutive jumps of an atom or molecule, r, is longer than the inverse of the rigid-lattice second moment, LWRL. Owing to the complexity of NMR lineshape theories in solids, it has been difficult in the past to extract quantitative information (such as the diffusion coefficient) from the relatively simple FID or lineshape measurements in the intermediate temperature range in which the "N-R line is Lorentzian (as in the motionally-narrowed region) nor practically
neither Gaussian
(as
in the rigid-lattice region). In this article it is shown that the rigid-lattice (RL) and motional FID or linewidth contributions are simply additive and that, therefore, the information on the atomic motions is extracted rather easily by subtracting the experimentally measurable RL background contribution from the actual FID or linewidth in the intermediate region. For simplicity, the following discussion is coiched in terms of the FID. As is well known, the lineshape is obtained by a simple Fourier transform of the FID. BASIC THEORY As discussed in some detail in Ref. I, for NMR purposes the Hamiltonian H of a crystal is subdivided into a "lattice" Hamiltonian, HL, a Hamiltonian, HS, of the completely isolated spin system embedded in the crystal, and the spin-lattice coupling Hamiltonian, HSL, according to H = HS + HSL + HL
*Work supported by the U.S.
-
Department of Energy.
(1)
530 By definition, [Hs,HL] = 0. In general, HS includes internal (rigid-lattice) spin-spin interactions inside the isolated spin system (such as the direct dipolar interaction Hamiltonian, HDRL), as well as Zeeman interaction Hamiltonians with externally applied time-independent (11Z)and time-dependent [H4(t)] magnetic fields, ih and h It), respectively; hence, considering only internal dipole-dipole interactions,
Hs = HRL +
If
+ I11(t)
(2)
The "lattice"-induced fluctuations of HSL are governed by the expression
i/A liLt }{SL(t)
where it
= e
was observed that,
According to Eqs.
e HR
= e1
-i/M
lLt
RL =
HD(t)
by definition,
D11)
-
[H1,1IL
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