PLAN ANUAL DEL EQUIPO DE ANIMACIÓN LOCAL

Typically, two receiver coils are placed with their axis perpendicular to the direction of B0

and to each other. As M rotates about B0 and simultaneously relaxes back to equilibrium,

voltages are induced in these coils due to Faraday’s Law, which are translated into a NMR signal. The NMR signal is complex, where the real and imaginary parts hold information from the two receiver coils, separately.

The simplest NMR pulse sequence consists of a 90° r.f. pulse followed by signal acquisition, as illustrated in Fig. 2.3. The noise-free NMR signal acquired, known as the Free Induction Decay (FID), is given by:

𝑦̂(𝑡) = 𝑒−𝑡/𝑇2_∫+∞_{𝜌(𝑤)𝑒}−2𝜋𝑤𝑡𝑖_d𝑤

−∞ , (2.7)

where ρ(w)dw is proportional to the number of nuclei with a Larmor frequency in the range

w to w + dw (spin density). It is assumed that a single T2 characterizes the whole system.

If T2 is very large (no relaxation effects), ŷ(t) and ρ(w) are a Fourier pair. Therefore, ρ(w)

can be obtained through a Fourier transform of ŷ(t); peaks in ρ(w) reveal the Larmor frequencies present in the sample and the relative population of the nuclei having that Larmor frequency. If T2 is relatively small, this is manifested in peak broadening of ρ(w),

giving each peak a Lorentzian shape. This is illustrated in Fig. 2.3.

Fig. 2.3. Illustration of the simplest pulse sequence composed of a 90° r.f. pulse, followed

by signal acquisition. Through Fourier transformation of the signal, the spectru m of Larmor frequencies is obtained (in this case illustrated by a single Lorentzian peak).

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2.6. Chemical shift

Not all the nuclei are in the same chemical environment; The most common example is nuclei in different functional groups. The local magnetic field experienced by nuclei in different chemical environments is different. This causes the nuclei to have different Larmor frequencies and it is manifested as multiple peaks in the Fourier transform of the NMR signal (also referred to as the NMR spectrum). NMR spectra are dependent on B0.

As a result, they are typically normalized to give a chemical shift, δ (ppm):

𝛿 = 106(𝑤−𝑤ref

𝑤ref ) , (2.8)

where wref is the Larmor frequency of a reference chemical (most commonly

tetramethylsilane for 1H). Further peak splitting occurs from the magnetic interaction of nuclei in close proximity or connected through covalent bonds, known as J-coupling. The detailed peak splitting leads to complex NMR spectra which are chemical fingerprints. This is the most commonly used NMR method to elucidate chemical structure and composition. The present work is not focused on NMR spectral analysis. However, chemical shift information will be given where necessary.

2.7. Spin echo

The homogeneity of the external magnetic field, B0, is rarely perfect. Corrections can be

made through the application of shimming magnetic field gradients, but some inhomogeneity remains. This is particularly marked when materials with different magnetic susceptibilities are present in the sample, such as air and water, or when traces of paramagnetic material exist, such as in porous media applications. Therefore, the NMR signal decays with a transverse relaxation time T2* < T2. This can be problematic for

several reasons: it leads to a lower signal-to-noise ratio; only short pulse sequences are viable as the signal decays very fast; the T2* decay is more difficult to interpret than the

T2 decay, which is more meaningful because of its chemical and environmental specificity.

The spin echo [9] is one of the basic building blocks of NMR pulse sequences as it retrieves the T2 decay from the T2* decay by refocusing the dephasing that occurs due to B0

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a 180° r.f. pulse after an echo time τ. During the echo time, the spins precessing at different Larmor frequencies due to B0inhomogeneity diphase, leading to a drop in the NMR signal.

After the application of the 180° r.f. pulse, the spins’ direction is inverted and they start rephasing. After an equivalent time of τ, the spins are completely in phase (before starting to dephase again), and this is known as a spin echo. The magnitude of the signal at the spin echo is then related to the magnitude of the signal just after the application of the 90° r.f. pulse through a T2 decay. An illustration of the formation of the spin echo is given in

Fig. 2.4.

Fig. 2.4. (a) The spin echo pulse sequence. (b) Just after the application of the 90 ° r.f.

pulse, all spins are in phase and they start precessing at slightly different Larmor frequencies. Three representative spins with w1 ≠ w2 ≠ w3 ≠ w1 are followed. (c) After an

echo time τ, spins have dephased. When the 180° r.f. pulse is applied, spins are inverted, a process illustrated by the dotted lines. (d) Position of spins just after the application of the 180° r.f. pulse. (e) Spins refocus after an equivalent time of τ; a spin echo is formed. Spins start dephasing again after this point.

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2.8. Stimulated echo

In certain applications, such as when a chemical species interacts strongly with a surface,

T2 << T1, which causes a fast decaying of the signal. A common technique to overcome

this limitation is to store the transverse magnetization in the B0 direction for a time tstore

by the application of a 90° r.f. pulse at a time τ after an initial 90° r.f. pulse, as shown in the pulse sequence in Fig. 2.5. During tstore, the stored magnetization decays with a time

constant T1, which is a slower decay. In order to recall the magnetization back into the

transverse direction, another 90° r.f. pulse is applied and an echo is formed after a time τ. This is known as a stimulated echo sequence and is used in some of the diffusion experiments in the present work. It is noted that other echoes may also form because the second 90° r.f. pulse is, in practice, imperfect and some remnant transverse magnetization persists. These echoes are undesirable and are formed whenever a train of imperfect r.f. pulses is applied [10]. The undesirable echoes are suppressed using spoiler magnetic field gradients, as shown in Fig. 2.5, or phase cycling techniques. These are very important areas in NMR experimental design, but they will not be the focus of the present work.

Fig. 2.5. Schematic of the stimulated echo pulse sequence.

2.9. T

estimation

The most commonly used pulse sequence to measure the relaxation time constant, T1, is

the Inversion Recovery (IR) pulse sequence, shown in Fig. 2.6. An initial 180° r.f. pulse flips the magnetization vector in the opposite direction. The magnetization vector has no component in the transverse direction and, therefore, decays with a relaxation time constant, T1. To make the magnetization vector measurable, the magnetization vector is

rotated into the transverse plane after a time t1 using a 90° r.f. pulse, after which the NMR

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of NMR signals. These are individually Fourier transformed and the area under the NMR spectrum, which will confusingly be known as the NMR signal, is plotted as function of

t1.

Fig. 2.6. Schematic of the Inversion Recovery (IR) pulse sequence. TR refers to the

recovery time which is needed for the longitudinal magnetization vector to return to its equilibrium condition before starting the next loop of the experiment.

For a system characterised by a single T1, the noise-free NMR signal, is given by:

𝑦̂(𝑡1,𝑖) = 𝐴 (1 − 2𝑒− 𝑡1,𝑖

𝑇1

⁄

) , (2.9)

where A = ŷ(t1→∞) is a scaling factor. From the expression in Eq. (2.9), T1 can be retrieved

through a simple fitting from any computational package. If the system is characterised by a distribution of T1, u(T1), where u(T1)dT1 is the contribution to the signal from species

with a relaxation time constant in the range T1 to T1+dT1, the noise-free NMR signal is

given by: 𝑦̂(𝑡1,𝑖) = ∫ 𝑢(𝑇1) (1 − 2𝑒− 𝑡1,𝑖 𝑇1 ⁄ ) d𝑇1 ∞ 0 . (2.10)

The distribution u(T1) needs to be obtained numerically. As a result, Eq. (2.10) is

discretized by seeking a solution only for several relaxation times spaced in some pattern between a minimum and maximum value, where the relaxation times are believed to lie within: 𝑦̂(𝑡1,𝑖) = ∑ 𝑢(𝑇1,𝑗) (1 − 2𝑒 −𝑡1,𝑖 𝑇1,𝑗 ⁄ ) 𝑗 . (2.11)

Eq. (2.11) can be written more compactly in matrix form as:

𝒚

̂ = 𝑲 𝒖 , (2.12)

where ŷ is the signal column vector such that ŷi = ŷ(t1,i), K is a kernel matrix such that

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far, it has been assumed that the NMR signal is not corrupted by noise. However, noise is unavoidable and, therefore, the noisy acquired signal is given by:

𝒚 = 𝑲 𝒖 + 𝒆 , (2.13)

where e is an unknown noise column vector, such that ei is the noise related to yi. u needs

to be estimated from a knowledge of y and K, which is much of the scope of the present work.

2.10. T

estimation

The most commonly used pulse sequence in measuring the transverse relaxation time constant, T2, is the Carr-Purcell Meiboom-Gill (CPMG) [11, 12] pulse sequence,

illustrated in Fig. 2.7(a). The CPMG pulse sequence is composed of a train of equidistant 180° r.f. pulses following an initial 90° r.f. pulse. After each 180° r.f. pulse, a spin echo forms, but only the middle point of the echo is acquired. For a system characterised by a single T2, the noise-free acquired signal follows an exponential decay:

𝑦̂(𝑡_2,𝑖) = 𝐴𝑒

−𝑡2,𝑖

𝑇2

⁄

, (2.14)

where t2,i = 2iτ, i is the echo number; A = ŷ(t2→0) is a scaling factor, and τ is the echo time.

For a system characterised by a distribution of T2, u(T2), a similar argument to the

discussion in Section 2.9 gives the noise-free NMR signal in a discrete setting as:

𝑦̂(𝑡_2,𝑖) = ∑ 𝑢(𝑇_2,𝑗)𝑒−𝑡2,𝑖⁄𝑇2,𝑗

𝑗 . (2.15)

In the presence of noise, the acquired signal is given in a compact matrix form as:

𝒚 = 𝑲 𝒖 + 𝒆 , (2.16)

where y is the signal column vector such that yi = y(t2,i), K is a kernel matrix such that

Kij = exp(-t2,i / T2,j), u is the distribution column vector such that uj = u(T2,j) and e is an

unknown noise column vector, such that ei is the noise related to yi. It is noted that

Eq. (2.16) is the same as Eq. (2.13), which means that the same numerical methods can be used to obtain u from a knowledge of y and K for both T1 and T2 experiments.

In some of the experiments in the present work, a variation of the CPMG sequence, known as Periodic Refocusing of J Evolution by Coherence Transfer (PROJECT) [13] is used. This consists in the addition of a 90° r.f. pulse between every other 180° r.f. pulse

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(illustrated in Fig. 2.7(b)). The PROJECT pulse sequence offers an improvement over the CPMG pulse sequence in cases where the frequency difference between spins coupled through J-coupling is of the order of 1/τ which would lead to signal modulation if the CPMG sequence was used.

Fig. 2.7. (a) Schematic of the CPMG pulse sequence. Only the middle point of each echo

is acquired. (b) Schematic of the PROJECT pulse sequence. Similarly, only the middle point of each echo is acquired.