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NMR Methods for the Investigation of Structure and Transport pp 5–81Cite as

Fundamentals

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Abstract

The chapter on fundamentals comprises a part on nuclear magnetic resonance (NMR) methods and one on the analysis of NMR images. In the NMR part quantum mechanics are avoided and classical equations are worked out. The basic subjects “macroscopic magnetization,” “equation of motion,” “rotating frame,” “generation of transverse magnetization,” “Fourier imaging,” “pulse sequences,” “resolution,” “slice selection,” and “contrast” including “spectroscopy,” “relaxation,” “diffusion,” and “flow” are covered. In addition less common subjects of relevance for a quantitative analysis such as “pair of continuous and discrete Fourier transform,” “gradient imperfections,” and “online relaxation measurements” are addressed. This part is concluded by a collection of “problems.” Basic points treated in the part on image analysis include “thresholds,” “porosity,” “filters,” and “noise.” Particular subjects including “specific surface,” “segmentation,” “signal variance,” and “image-phase correction” are treated in detail.

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Notes

  1. 1.

    Vectors are set in bold italic face.

  2. 2.

    As common in NMR literature, the \(\vec{B}\)-field is designated as magnetic field. Alternative names are magnetic flux density or magnetic induction. In vacuum the magnetic \(\vec{H}\)-field is proportional to the magnetic induction with the induction constant μ0: \(\vec{B} = {\mu }_{0}\vec{H}\).

  3. 3.

    Sometimes more appropriate as magnetogyric ratio.

  4. 4.

    The relation T 2 ≤ T 1 is obtained in theoretical calculations for several relaxation mechanisms [8]. The condition that the magnitude of the magnetization cannot exceed the magnitude of the equilibrium magnetization through relaxation leads to the weaker condition T 2 ≤ 2T 1.

  5. 5.

    Usually a linear polarized field is irradiated. It can be decomposed into two counter rotating circular polarized fields. For B 1 ≪ B 0 the component rotating with the magnetization acts as described in (2.23). The counter rotating component leads to the Bloch–Siegert Shift, see [7] and Fig. 2.2. Advanced NMR systems allow to generate only the required circular polarized component [18,  27].

  6. 6.

    Experimentally, this corresponds to the signal being mixed with the rf frequency and subsequent low-pass filtering.

  7. 7.

    In contrast to the representation in the form of coordinates in a column vector, denoted as “matrix representation,” that differs between coordinate systems.

  8. 8.

    Usually the coil transmitting \(\vec{{B}_{1}}\) and the receiver coil are identical. Between transmit and receive mode a dead time of some microseconds has to be waited. For the distinction of x and y component of transverse magnetization pairs of data points can be digitized with a delay corresponding to a precession by π ∕ 2. This procedure is denoted as “sequential quadrature detection.”

  9. 9.

    A spatial variation of ⟨μ z ⟩ due to the superposition of \(\vec{{B}_{0}}\) with \(\vec{Gr}\) can be neglected. On the one hand the additional gradient fields are typically at least three orders of magnitude weaker than \(\vec{{B}_{0}}\). On the other hand the gradient fields are usually switched on as pulses with a duration that is short compared to the longitudinal relaxation time T 1.

  10. 10.

    A more general definition with the effective gradient will be given in Sects. 2.1.7 and 2.1.8. The concept of effective gradients includes the effect of rf pulses with ω1τ = π.

  11. 11.

    See also problems 2.5 and 2.6.

  12. 12.

    This corresponds to the claim of unambiguous phase increments k inc r for all positions within the FOV. For magnetization outside the FOV the phase increment between two discretization points calculated from (2.35) and (2.42) is outside the interval [ − π π]. This violation of the sampling theorem leads to a folding of spin density in the calculated image.

  13. 13.

    More precisely the isotropic magnetic shielding, i.e., one-third of the trace of the shielding tensor, see also Sect. 2.1.5.

  14. 14.

    To be replaced by the corresponding tensor in case of anisotropic diffusion, see also Sect. 2.1.7.

  15. 15.

    In the range of 10 ppm (parts per million) for hydrogen nuclei, 200 ppm for the NMR-observable carbon isotope 13C.

  16. 16.

    The angular frequency has been used as variable. In spectroscopy it is common to use the chemical shift δ relative to a reference: \(\delta = 1{0}^{6}({\sigma }_{\mathrm{ref}} - \sigma )/(1 - {\sigma }_{\mathrm{ref}}) \approx 1{0}^{6}({\sigma }_{\mathrm{ref}} - \sigma )\).

  17. 17.

    In contrast e.g. to infrared spectroscopy, where band integrals are influenced by further parameters.

  18. 18.

    There is no closed analytical expression for the convolution.

  19. 19.

    The prefactor of a normalized Lorentzian distribution is 1 ∕ π and not 1 ∕ (2π) as in (2.63). The factor 1/2 is a consequence of the missing signal for negative times. If the distribution is expressed with the linear frequency instead of angular frequency the full width at half maximum amounts to 1 ∕ (πT 2, k  ∗ ).

  20. 20.

    Neglecting translational self diffusion and experimental artifacts.

  21. 21.

    Fourier transformation can be treated as special case of Laplace transformation. The latter is a special case of Fredholm integral equation of the first kind, i.e., a linear integral equation with constant integral limits and the unknown function occurring only in the integral. This general case is treated in [59].

  22. 22.

    Condition for symmetric echoes.

  23. 23.

    That is unintended gradients by eddy currents or remanent changes of the permanent-magnet system that act in the “background.”

  24. 24.

    The magnetization phase angle is given by ϕ = q v w. Thus w max corresponds to an angular frequency.

  25. 25.

    For the case of spins with velocity w and arbitrary profiles of relaxation rate and equilibrium magnetization, the solution of the differential equation \(\dot{{M}_{z}} = w(\partial {M}_{z}/\partial y) = -{R}_{1}(y)({M}_{z} - {M}_{z}^{\mathrm{eq}}(y))\) is \({M}_{z}(y) = (1/w){ \int \nolimits \nolimits }_{-\infty }^{y}{R}_{1}(y\prime){M}_{z}^{\mathrm{eq}}(y\prime)\exp \{ - (1/w){ \int \nolimits \nolimits }_{y\prime}^{y}{R}_{1}(y\prime\prime)\mbox{ d}y\prime\prime\}\mbox{ d}y\prime\). This is obtained by solution of the homogeneous differential equation by separation of the variables and variation of the constant for the particular integral. For a constant relaxation rate R 0 and piecewise constant equilibrium magnetization, the integrals yield \({M}_{z}(y) =\exp \{ -{R}_{0}y/w\}[{\sum \nolimits }_{i=1}^{n}{M}_{i}(\exp \{{R}_{0}{y}_{i}/w\} -\exp \{ {R}_{0}{y}_{i-1}/w\}) + {M}_{n+1}(\exp \{{R}_{0}y/w\} -\exp \{ {R}_{0}{y}_{n}/w\})]\). With a relaxation profile, numerical integration by simple time slicing yields good results.

  26. 26.

    The raw data before transformation requires more memory, as it consists of a real and an imaginary part. The spectrometer used always saves raw data with 32 bits, resulting in 128 megabytes in the present example.

  27. 27.

    There are several common notations for matrices. Here the notation (a) nm is used, see e.g., http://mathworld.wolfram.com/Matrix.html. More detailed notations are (a nm ) nm as well as \({a}_{n=1...N,m=1...M}\). Introduction of a new symbol, such as A or \(underlineunderlinea\) is avoided.

  28. 28.

    Named after Stephen O. Rice.

  29. 29.

    Note that for the Rice or Rayleigh distribution the parameter s 2 is not the variance of the distribution.

  30. 30.

    If the original image is corrected grid point by grid point, it gets modified during the process of correction and the rules would apply to the modified image. Thus it is necessary to apply the corrections to a copy by inspection of the original image.

  31. 31.

    The MathWorksTM, Inc.

  32. 32.

    A free download as file ghist.c is available at http://www.materialography.net/.

  33. 33.

    MAVI – Modular Algorithms for Volume Images, Copyright © 2006 Fraunhofer Institut für Techno- und Wirtschaftsmathematik.

  34. 34.

    Concerning the algorithm, Matlab®; refers to [46], where several watershed algorithms are described.

  35. 35.

    Generation of the distance matrix by a kind of “morphological thinning” is explained in detail, the following steps are only mentioned. However, a straight-forward search for local maxima in the next step leads to an over segmentation. Also in the subsequent association of adjacent grid points to the maxima, care must be taken to avoid distorted pores. Therefore details of the developed procedure are described and in Sect. 6.3 the implementation in Matlab®; is listed.

  36. 36.

    In Zrich, a tablet on the tramway bearing the inscription “Uetliberg hell” (Uetliberg bright) indicates this situation.

  37. 37.

    The distance matrix generated in [26,  55] contains only natural numbers so that a reduction of the levels by less than one is not meaningful in this case.

  38. 38.

    Equivalent and more intuitive is the assignment of new values with the equation \({(z)}_{\Lambda } = {(z <<\ \xi )}_{\Lambda }\). Due to the syntax of Matlab®; this version requires an additional step as the cyclical shifted matrix cannot be indexed with a matrix in the same expression.

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  1. Institut für Mechanische Verfahrenstechnik und Mechanik, Karlsruher Institut für Technologie (KIT), Adenauerring 20b, 76131, Karlsruhe, Germany

    Dr. Edme H. Hardy

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Hardy, E.H. (2012). Fundamentals. In: NMR Methods for the Investigation of Structure and Transport. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21628-2_2

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