Evaluación de la Escala de Ansiedad de Zung.

In modern NMR spectrometers, the static magnetic field is produced by a supercon- ducting coil24 _{submerged in a tank filled with liquid helium, surrounded by another} tank filled with liquid nitrogen to limit the fusion of the liquid helium, as illustrated in Figure 1.8a.

The RF pulse is produced with a coil L included in a RLC25 _{passband circuit (Figure} 1.8b), whose resonance frequency corresponds to the RF frequencyωRF.[74,75] As all nuclei have different gyromagnetic ratios, the passband circuit will have the effect of selecting a limited range of frequencies and thus only the nucleus of interest. This can be achieved by adjusting the variable capacitances of the capacitorsCM and CT

23_{Adiabatic pulses are described in Appendix F using the vector model.}

24_{There are alternative experimental set-ups, for example electromagnets (often for relaxometry), perma-}

nent magnets [72] or electromagnet placed inside a superconducting coil to reach very high fields.[73]

b

a

c

Rotor (containing the sample)

Figure 1.8:(a): Schematic of a modern NMR spectrometer equipped with a superconducting coil that creat-

ed the static field_B⃗_{0.(b): Equivalent circuit diagram of a simple probe, withCMandCTbeing}

the matching and tuning capacitors, respectively, andLis the RF/recording coil.(c): Schematic

representation of the rotor containing the sample inside the coilLpictured in(b), showing the

direction of the RF field_B⃗_{1and the unit magnetic field⃗b.}

illustrated Figure 1.8b. The probed can be qualified as ”tuned” for a given nucleus if the following three conditions are fulfilled:

• The resonance frequency must be at (or close to) the Larmor frequency of the desired nucleus with an applied static magneticB0.

• The total impedance must be as near as possible to 50 Ω to limit dissipated or reflected power.

• The quality factor must be high in order to select only the Larmor frequency of the nucleus of interest.

The electronics in Figure 1.8b are incorporated into a probe, placed at the center of the superconducting coil (Figure 1.8a). According to Faraday’s law of induction, the voltage observed at the terminals of the RF coil Us(t) (equivalently called the ”NMR signal”)

generated by the bulk magnetisationM⃗ of the nuclei placed in the RF coil (Figure 1.8c), can be expressed as being in the following form

Us(t) =− ∂ΦB ∂t = ∂ ∂t ∫ ⃗b(⃗r). ⃗M(⃗r, t).dr . (1.3.15) whereΦB is the magnetic flux through the recording coil. As the observed signal de-

pends on the characteristics of the RF coil, such as position, structure, gain,etc.,⃗b(⃗r)

is defined as the magnetic field produced at a point at⃗r from the origin by a unit cur- rent, as illustrated in Figure 1.8c, so Equation (1.3.15) is always valid, and illustrates the electromagnetic reciprocity,i.e., that the coil can be exploited both to produce the RF fiend and to record the NMR signal. As an illustration,⃗bwould be in the center of a ”long” coil composed ofN loops of radiusR,

⃗b=µ0N I0

2R ⃗ub , (1.3.16)

whereI0corresponds to a unit electric current of 1 A and⃗ubis a unit vector along the

axis of symmetry of the coil. It can directly be deduced by combining Equations (1.1.13) and (1.3.15) that the NMR signal oscillates at the Larmor frequency,e.g,

Us(t)∝ d∥M⃗T(t)∥ dt ∝ d dt ∥e iω0t∥ _{. (1.3.17)}

From Equations (1.3.10) and (1.3.15), it can be shown that the amplitude of the NMR signal is proportional to

Us∝

γ3_B2 0

T . (1.3.18)

In solid-state NMR, the sample is generally packed in a rotor as illustrated Figure 1.8c, so it can be rotated (see Section 2.2.3). The RF fieldB⃗1 as shown on Figure 1.8c goes along the axis of the coil, which at first sight does not seems consistent with previously written Equations (1.1.14) and (1.2.57), describing the RF magnetic fieldB⃗1as describ- ing a circle. However, the oscillating magnetic fieldB1 produced along an axis can be expressed as the sum of two oscillating components atωRFand−ωRFsuch that

B1.exp(iθRF)cos(ωRFt) =

1

2B1.exp(iθRF) (exp(+iωRFt) +exp(−iωRFt)) . (1.3.19)

If+ωRF approaches the Larmor frequency, then−ωRF component is far off resonance and has no effect on the nuclear spins, therefore can be omitted. With only the+ωRF component, we find an expression for_B⃗_{1similar to that given in Equation (1.1.14).} A NMR spectrum spectrum is usual presented processing with FT26_{(see Appendix G) of} the raw NMR signal, or FID27_{.[76] In general, relaxation processes are sufficiently long} to be ignored during pulses.28 _SinceB⃗_{1and⃗bare perpendicular with respect toB}⃗_{0, the}

26_{FT:Fourier Transform}

27_{FID:Free Induction Decay}

28_{This may no be applicable if long pulses are applied (see examples in Section 3.1.3 and Appendix F), or if}

signal is maximum whenM⃗ is moved perpendicular toB⃗0,i.e., in the transverse plane. This is accomplished when the bulk magnetisationM⃗ is moved by an angle of 90° from its equilibrium value,e.g, whenB⃗1is applied for a durationτpfor which the quantity

β=ω1τp , (1.3.20)

denoted the flip angle, is equal to π/2. Acquiring a basic NMR spectrum consists of applying a 90° pulse, to record the FID, and to apply a FT. When a system is perturbed from equilibrium, the system is submitted to the Zeeman interaction and returns to equilibrium by a photon-driven process. The photon having an angular momentumI= 1, the rule of spin conservation imposes that a transition can take place only between adjacent energy levels,i.e., that satisfy the selection rule

∆mI =±1 . (1.3.21)

This results in onlyp = ±1 coherences (1Q) being directly observable in NMR experi- ments. By convention, only the coherence orderp=−1is observed, which consists in the detection of ⟨_Iˆ_{+⟩(see Equation (3.1.9) for the implication of this for the NMR sig-}

Chapter 2