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26ISSN 2G72-59B1
Volume ll, No. 2, pages 33-37, 2009
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August 2, 1996 First Issue was appeared at July 25, 1997
© Kazan State University (KSU)
”Magnetic Resonance in Solids. Electronic Journal' (MRSey) is a
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Editors-in-Chief Jean Jeener (Universite Libre de Bruxelles, Brussels) Boris Kochelaev (KSU, Kazan) Raymond Orbach (University of California, Riverside)
Executive Editor
Yurii Proshin (KSU, Kazan) Editor@mrsej.ksu.ru
Editors
Vadim Atsarkin (Institute of Radio Engineering and Electronics, Moscow) Detlef Brinkmann (University of Zurich, Zurich) Yurij Bunkov (CNRS, Grenoble) John Drumheller (Montana State University, Bozeman) Mikhail Eremin (KSU, Kazan) Yoshio Kitaoka (Osaka University,
Osaka)
Boris Malkin (KSU, Kazan) Haruhiko Suzuki (Kanazawa University, Kanazava) Murat Tagirov (KSU, Kazan)
*
In Kazan State University the Electron Paramagnetic Resonance (EPR) was discovered by Zavoisky E.K. in 1944.
Gazizulin R.R.*, Klochkov A.V., Kuzmin V.V., Safiullin K.R., Tagirov M.S., Yudin A.N.
Kazan State University, Kremlevskaya, 18, Kazan 420008, Russia
* E-mail: g-rasul@yandex.ru
(Received November 19, 2009; revised December 24, 2009; accepted December 26, 2009)
For the first time the inverse Laplace transform was applied for analysis of 3He relaxation in porous media. It was shown that inverse Laplace transform gives new information about these systems. The uniform-penalty algorithm has been performed to obtain the 3He relaxation times distribution in pores of clay sample. It is possible to obtain pores’ sizes distribution by using applicable model.
PACS: 67.30.E-, 67.30.er, 67.30.-n, 67.30.ej, 76.60.Es, 89.20.Bb.
Keywords: inverse Laplace transform, uniform-penalty algorithm, liquid 3He, pulse nuclear magnetic resonance, clay, porous media
1. Introduction
The use of inverse Laplace transform to investigate some relaxation curves is well-known problem in many fields of material research [1-2]. The nuclear magnetic resonance (NMR) in porous media and crystal powder samples is one of the most informative methods for investigating of such systems [3-9]. The measuring longitudinal magnetization relaxation time T1 is important in this investigation. Usually, there is a distribution of the relaxation times T1 in most of the samples. Obtaining the relaxation times distribution from the longitudinal magnetization recovery curve is important for investigating properties of the systems. Various algorithms have been used to analyze experimental relaxation data. Fundamental requirement for such algorithms is the obtained distribution accordance to the properties of the real systems. In our work we have researched the applicability of the uniform-penalty inversion algorithm [10] for investigating properties of the systems by measuring NMR 3He relaxation in the clay sample.
The pulse NMR measurements are a very useful tool for understanding various properties of different systems. One of the key parameters that are widely used in pulse NMR is the longitudinal relaxation time T1. This time constant characterizes the process of longitudinal magnetization recovery. This process is commonly described by the following equation
A (t ) = A
1 - exp
(1)
1 JJ
In most porous systems, for example in clay samples, the diffusion isn’t fast enough and we can assume that the molecules are located in the same pore during experiment. In this case the relaxation of whole spin system is the superposition of contributions from the pores with different sizes, which results in T1 distribution. But the experimental data usually can be described by following equation in most cases
A (t ) = A
1 - exp
T
K11 J
(2)
J J
Such equation contains all information about T1 distribution. To have a clear understanding of sample properties through NMR measurements it is necessary to obtain reliable inversion of experimental data.
The experimental data represent sums of exponentially decaying components
A (t, (~ A0 +I fk (Tl (exp
f .л л
T
V k J
where fk(Ti) is the weighting factor.
The uniform-penalty inversion (UPEN) algorithm is one of the most succesfull regularization processes applied to NMR relaxation data. The uniform-penalty algorithm can give sharp lines, not broadened more than conditioned by noise, and in the same distribution it can show wide lines, not broken into several peaks [10]. In general, this algorithm represents a least square minimization routine with adding a penalty factor. A coefficient Ck is multiplied to the penalty factor. The coefficient Ck is iteratively adjusted to ensure a unifromity of the penalty factor. Thus, the function to be minimized is
f t. л
T
V J1k J
I A +I fk (Tl (exp --f- - A (tt ( +I Ck ( (Tl (-2fk (Tl ( + fk+l (Tl (( ,
(4)
where the first component is the least minimization routine component and the second component is the penalty factor.
Using this algorithm we have created a program which contains the inverse Laplace transform. For the first time it has been applied to the magnetic relaxation of liquid 3He.
k
2. Methods
The simulated data were created for single component T1 process with T1 = 11.9 s. Then real experimental noise pattern was added to simulated mono-exponential curve so that signal-to-noise ratio was the same order of magnitude which we usually obtain in our experiments (see Fig. 1).
UPEN from simulated mono-exponential recovery data is shown on Fig. 2, where the pick with finite width can be seen. As the noise added, the broadening of the computed peak is observed.
To investigate the applicability of the uniform-penalty algorithm for describing properties of real systems by the NMR relaxation measurements of 3He, the specially prepared sample of clay filled by 3He was used. In this system T1 relaxation data of 3He contain information about porosity of the sample and the time distribution probably corresponds to pores’ sizes distribution. The clay sample
•Є
Я
C
з
§
й
t, s
T1, s
Figure 1 Simulated mono-exponential recovery Figure 2 UPEN from simulated mono-exponential curve with parameter T1 = 11.9 s with noise data without noise (green curve) and
(dark circles) and without noise (open UPEN from simulated mono-exponential
circles). data with noise (red curve).
was cut out of clay mineral in the shape of tablet with diameter of 5 mm and height of 3 mm.
A hole (diameter of 1.7 mm) has been drilled out in the center, along the tablet axis, forming a calibration cavity (see Fig. 3). The relaxation in the calibration cavity has to be mono-exponential and has much bigger contribution to relaxation curve than contribution from 3He relaxation in each and any pore. The idea is to obtain the contribution from calibration cavity into the 3He relaxation time distribution. By comparison this contribution with simulated mono-exponential distributions obtained by the uniform-penalty algorithm we can make a conclusion about its applicability.
The sample was placed in the pyrex tube which was sealed leak tight to the 3He gas handling system. On the outer surface of the pyrex tube a NMR coil was mounted. The hand made pulse NMR spectrometer has been used (frequency range 3 - 50 MHz). The pulse NMR spectrometer is equipped by resistive electrical magnet with magnetic field strength up to 10 kOe. The sample has been filled by 3He under saturation vapor pressure at the temperature 1.5 K (see Fig. 3).
The longitudinal magnetization relaxation time T1 of 3He was measured by the saturation recovery method using spin-echo signal. The observation of the spin-echo signal was possible due to specially induced inhomogeneity of the external magnetic field H0 (NMR probe has been decentered in the magnet).
Figure 3 Geometry of the NMR experiments.
3. Results and discussion
The nuclear spin relaxation time T1 of liquid 3He in clay sample was measured on frequency 12 MHz at the temperature 1.5 K. The recovery of the spin echo signal is non single-exponential, what can be seen in Fig. 4. The liquid 3He nuclei spin echo signal recovery curve can be described by the following equation
Aecho = A + A
1 - exp
(5)
//
Here the fitting parameters are A0 = 0.014 ± 0.004 arb.units, A = 1.004 ± 0.007 arb.units, T1 = 7.609 ± 0.130 s, n = 0.626 ± 0.008. The power n = 0.626 ± 0.008 in the equation (5) indicates the presence of distribution of relaxation times Ti. The curve contains information about all components but it is necessary to inverse data to obtain it.
The distribution of relaxation times computed from real experimental data by UPEN are shown in Fig. 5. We see narrow peak (in semi-log scale) and wide peak with shorter T1 values. The narrow peak corresponds to 3He in the calibration cavity. Compared this distribution with mono-exponential distributions (see Fig. 2) we can assume that the relaxation in the calibration cavity is monoexponential, as well. The wide peak corresponds to 3He relaxation in the pores of clay sample.
Thus, inverse Laplace transform revealed unique information about our system, which was not shown on the relaxation curve (see Fig. 4). According to analysis of relaxation curve our system should have distribution of T1, which was indicated by power n ~ 0.6 in the equation (5). While, inverse Laplace transform unambiguously shows the presence of two different areas of our system (calibration cavity and clay pores), where relaxation differs significantly.
I I I I Mil-1—I I I I I III---------1—I I I I MM------------1—I I I I MM
1.0 -^echo = A0 + ^-exp^T/)) jyi -
- A0 = 0.014 ± 0.004 arb.units jp -
0.8 ~A = 1.004 ± 0.007 arb.units // -
- T = 7.609 ± 0.130 s _
e 3 0.6 1 // -n = 0.626 ± 0.008 f/ -
•e - / -
0.4 - jF -
- ^ -
0.2 - -
0.0 ■ ■
0.1
10
100
t, s
Tv s
Figure 4 The recovery of liquid 3He nuclei spin Figure 5 Distribution of liquid 3He T1 relaxation
echo signal in clay sample on frequency 12 MHz at temperature 1.5 K (■ - experimental data).
times for a clay sample obtained by regularized inverse Laplace transform UPEN algorithm.
The distribution of relaxation times, obtained by inverse Laplace transform, could be converted into pore size distribution, using specific models of relaxation. For instance, in case of very small pore’s sizes, when the molecular diffusion of 3He during the time of an experiment is longer than the size of pores the longitudinal relaxation of liquid 3He would be single-exponential in each pore. The T1 relaxation mechanism in this case would be surface relaxation, and direct knowledge of surface relaxation time T1s, would allow to obtain pore’s size distribution. The surface relaxation itself could have different mechanisms, specially complicated by using geological sample, like in our case. Certainly, in some specific cases, the NMR 3He relaxation in each pore could be non singleexponential and in these cases inverse Laplace transform would not allow to obtain pore’s size distribution.
4. Summary
We have demonstrated that the UPEN algorithm of the inverse Laplace transform of 3He relaxation curves in porous media can give information about properties of media. The distribution of relaxation times could be converted into pore size distribution, using specific models of relaxation.
Acknowledgments
We wish to acknowledge A.V. Egorov for enlightening discussions.
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