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
in-homogeneous 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
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 Ám)
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)