The Motionally Narrowed NMR Line Shape of Ammonium Selenate
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THE MOTIONALLY NARROWED NMR LINE SHAPE OF AMMONIUM SELENATE*,
CHING YAO, R. HALLSWORTH and I. J. LOWE Physics Department, University of Pittsburgh, Pittsburgh, PA
15260
ABSTRACT The proton free induction decay f(t) of ammonium selenate is measured from temperatures of no apparent effects of 0 lattice motion (-185 C) upon f(t), to substantial motional effects (-145oC) upon f(t). Our measured results are consistent with a classical hindered rotational motion of the ammonium group. The deviation of f(t) from the rigid-lattice fid f (t) can be expanded in a Taylor series as
Sa
Alf(t) = = 1 n=3 From the L1 f(t)
n(T)
n
n!
n
n!
data, a 3 and a 4 are derived,
and they are
well fitted by the simple relations a3=C3M2 v
4w1 2 Tlr 2
a 4 =-
C4M
2
where v is the jump frequency of the rotational motion, T1r is the rotating-frame spin-lattice relaxation time, and w1 the strength of the rotating magnetic field in frequency units.
a, C 3 and C4 are calculable constants
of order unity.
INTRODUCTION The spatial motion of atoms and molecules can affect a nuclear magnetic resonance (nmr) line shape, and the affected line shape can be a useful tool for studying these motions. These motions tend to average and thus reduce the time average of the magnetic interactions among the nuclei. In a continuous wave nmr (cwnmr) experiment, the observed absorption line appears to "narrow". For a transient experiment that produces an observed signal called a free induction decay (fid) (the fourier transform of the cwnmr line) the fid appears to lengthen and last for a longer time. The nmr line shape is said to be strongly motionally narrowed when the width of the observed line shape is much narrower than the observed width when there isn't any motion (the so called rigid lattice). The theory of the strongly motionally narrowed line shape has been well developed, and experimental studies confirm its principal predictions [1]. A rigorous theory for the moderately motionally narrowed line shape, that is the observed nmr line *Supported by NSF Grant DMR 78-15441.
502 shape where lattice motion has influenced the line by only a small amount, is much more difficult and has not been developed nearly as far. For this reason, the measurements of moderately motionally narrowed line shapes is normally used to draw qualitative and not quantitative conclusions about the motion. In this article, we will describe the formulation of a theory of the moderately motionally narrowed line and report on the experimental verifications of some of the predictions of this theory. In Section II, the development of the theory is sketched out, and theoretical results listed. Most of this part can be skimmed by readers not interested in details, except to learn nomenclature. Details of the nmr experiments and data processing are described in Sections III and IV respectively; conclusions are listed in Section V. THEORY Van Vleck, in his classic paper [21 on moments of line shapes showed that the 2nth moment M2n of the "rigid lattice" absorption line (n = 1,2,
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