A derivation is given of the effect of a time‐dependent magnetic field gradient on the spin‐echo experiment, particularly in the presence of spin diffusion. There are several reasons for preferring certain kinds of time‐dependent magnetic field gradients to the more usual steady gradient. If the gradient is reduced during the rf pulses, H1 need not be particularly large; if the gradient is small at the time of the echo, the echo will be broad and its amplitude easy to measure. Both of these relaxations of restrictions on the measurement of diffusion coefficients by the spin‐echo technique serve to extend its range of applicability. Furthermore, a pulsed gradient can be recommended when it is critical to define the precise time period over which diffusion is being measured.

The theoretical expression derived has been verified experimentally for several choices of time dependent magnetic field gradient. An apparatus is described suitable for the production of pulsed gradients with amplitudes as large as 100 G cm−1. The diffusion coefficient of dry glycerol at 26°±1°C has been found to be (2.5±0.2)×10−8 cm2 sec−1, a value smaller than can ordinarily be measured by the steady gradient method.

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and, for example,
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8.
The method described by Carr and Purcell (Ref. 2) for the calculation of the echo amplitude has been suitably modified and carried through for this particular G(t). The result is the same as that shown in Eq. (6), although more cumbersome to obtain. We have not succeeded in obtaining Eq. (5) by this method.
9.
Strictly speaking, if g0 vanishes completely, there is no echo identifiable as such since the nuclei will not lose phase coherence after the 90° pulse until the first gradient pulse appears and will regain complete phase coherence immediately after the end of the second gradient pulse. The expression for ln[A(2τ)/A(0)] is still valid, however.
10.
Torrey (Ref. 6) mentions “drift” terms which arise from the space dependence of the equilibrium magnetization M0 when the gradient is very large and which can complicate the steady gradient experiment. In the pulsed gradient experiment, if δ≪T1, the “drift” terms will not have time to develop any importance regardless of the magnitude of g.
11.
The 180° pulse could be dispensed with if the second gradient pulse were applied with reverse polarity. Abragam (Ref. 7, p. 63) explicitly points out the analogous phenomenon for the steady gradient experiment; Anderson et al. (Ref. 5, p. 1333) imply as much in their discussion of electron spin echoes.
12.
A discussion of the effect on the spin‐echo experiment, particularly the pulsed gradient experiment, of a position dependent, anisotropic diffusion coefficient, molecular forces which modify the diffusional motion, and hydrodynamic flow will be shortly submitted for publication by one of us (E.O.S.).
13.
Depending upon g1 and g2 there may be echoes observed at other times—whenever ψ = A, as indicated by Eq. (3). It is also possible that echoes may be observed for which ψ≠A.
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16.
A more detailed discussion will soon be submitted for publication by one of us (J.E.T.);
see also the Ph.D. thesis of J. E. Tanner (in preparation).
17.
If allowed to flow steadily, such a current would burn up the gradient coils. Keeping the duty cycle for the pulsed gradient current flow minimizes heating of the gradient coils.
18.
Motorola Power Transistor Handbook, edited by R. Greenburg (Motorola, Inc., Phoenix, Arizona, 1960), 1st ed., p. 146ff.
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see also
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D. W.
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1965
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