  Nuclear magnetic resonance (NMR) is a widely used nondestructive method to
study random motion of spin-bearing particles in different systems. In the
long-time limit the theoretical description of the NMR experiments is well
developed and allows proper interpretation of measurements of normal and
anomalous diffusion. The traditional description becomes, however, insufficient
for the shorter-time dynamics of the particles. In the present paper, the
all-time attenuation function of the NMR signal in a magnetic-field gradient
due to the Brownian motion (BM) of particles in incompressible liquids is
calculated by using the method of accumulation of phases by a precessing
magnetic moment, without reference to a concrete model of the stochastic
dynamics. The obtained expressions are then used to evaluate the attenuation
within the hydrodynamic theory of the BM. It is shown that the well-known time
behavior of the formulas corresponding to the Einstein theory of diffusion in
the case of steady gradient and Hahn's echo experiments is reached at times
much larger than the characteristic time of the loss of memory in the particle
dynamics. At shorter times the attenuation function significantly differs from
the classical formulas used to interpret these experiments.
