NMR Hahn and CPMG spin-echo measurements showed such a complex
spin-echo
decay, that it was decided to develop a numerical model. In this model,
the magnetic moments of nuclei accumulate a certain phase, depending on
their random trajectory through an inhomogeneous magnetic field. The
results
of this numerical model for classical situations (e.g., free diffusion
in a uniform magnetic-field gradient) are in perfect agreement with the
well-known analytical solutions. Figure 1., e.g., shows the simulated
magnetization
as a function of time for a Hahn spin-echo.
Figure 1: The simulated magnetization as a function of time for
a Hahn spin-echo.
We continued with simulations of the dephasing behavior of nuclear
spins
in a spherical pore. First, a uniform gradient was assumed in this
pore.
Using these simulations the existing theory, which predicts two
asymptotical
dephasing regimes (`motional averaging and `localization'), has been
verified
and extended with an `intermediate regime'. Next, in the model a
dipolar
magnetic field was created inside the pore, by putting a magnetic
point-dipole
in the solid matrix surrounding the pore. The decay of the simulated
spin-echoes
can be predicted with scaling laws. For the `motional averaging regime'
with a homogeneously distributed nuclear magnetization in the pore,
this
relation could be derived directly from existing theories. However, for
the `localization regime' with an inhomogeneously distributed
magnetization,
these scaling laws were derived from basic principles by ourselves.
Figure 2: The iso-magnetization surface of half the equilibrium magnetization for four succesive spin-echo times.
The magnetic dipole (strength=10E-14 Am^2) is situated at the top of the pore with a radius of 3 µm)
In figure 2 all spins with a magnetization of half the equilibrium
magnetization is shown for four successive spin-echo times. In this
simulation,
the magnetic dipole is situated at the top of the pore. For a detailed
description of the magnetization decay and the explanation of figure 2,
the reader is reffered to J.Chem. Phys. 2001.
Figure 3: Hahn spin-echo intensities as a function of the spin-echo time for a 4.3% weight Fe2O3 doped sample. The solid squares denote experimental data. The open squares reflect the corresponding model prediction. The solid curve represents a fit to the experimental data of a sum of a mono-exponential decay and an appropriate noise level (dashed line).
The NMR spin-echo measurements on a series of magnetically doped
clay samples were done. In this paper, the dephasing effects are
explained
without the use of the model. However, further analysis of Hahn and
CPMG
spin-echo measurements with a much better signal-to-noise ratio
revealed
behavior that can only be explained with the random-walk model. The
measured
spin-echo decay agrees with the results of the model of a dipolar
magnetic
field inside a pore. Therefore, in the model, the magnetic impurities
are
described as additional magnetic point-dipoles with a strength
depending
on the doping fraction. The experimental results agree well with the
predictions
from this model, as can be seen in figure 3 for a Hahn spin-echo
measurement.
This holds also for the CPMG measurements, up till spin-echo times for
which the diffusion length becomes comparable to the pore size.
R.M.E. Valckenborg, H.P. Huinink, J.J. v.d. Sande, and K. Kopinga, Random walk simulations of NMR dephasing effects due to uniform magnetic field gradients in a pore, Phys. Rev. E. 65, 021306 (2002)
R.M.E. Valckenborg, L. Pel, and K. Kopinga, NMR relaxation and diffusion measurements on iron(III) doped Kaolin clay, J. Mag. Res. 151, 291-297 (2001).
R.M.E. Valckenborg, NMR on technological porous materials, Ph.D. thesis, Eindhoven University of Technology (2001).
R.M.E. Valckenborg, H.P. Huinink and K. Kopinga; NMR Dephasing
Effects
due to a magnetic dipole in a spherical pore, Journal of Chemical
Physics
118,
3243-3251 (2003)