Abstract
Nuclear magnetic resonance (NMR) is the oldest nuclear method in solid state physics. It is based on the principle, that transitions between nuclear magnetic energy levels corresponding to differently oriented nuclear spins in a static magnetic field should be observable when applying a second, time-dependent magnetic field perpendicular to the static one. The second magnetic field should oscillate at the Larmor frequency of the nuclei. The first NMR measurements were performed contemporaneously by Purcell, Torrey and Pound in Cambridge and by Bloch, Hansen and Packard in Stanford. Purcell and his colleagues observed the radio frequency absorption of protons in solid paraffin at room temperature using a resonant cavity, a sweepable magnet and radio frequency power of about 10-11 W [24]. At the same time Bloch, Hansen and Packard reported in a very short paper (~ 260 words) the observation of radio frequency absorption by protons in water at room temperature using conventional radio frequency techniques [25]. Bloch, who was not limited to a fixed cavity frequency, could already measure at different frequencies and fields and confirmed that the ratio H/v was always the same: the gyromagnetic ratio γ of the protons. In his paper, which was published on Christmas Eve of 1945, Purcell already proposed various applications of the newly established effect, for instance precise measurements of gyromagnetic ratios, investigations of the spin-lattice coupling as well as standardizations of magnetic fields. All these applications (and many more) were realized sooner or later. This was the twofold birth of NMR, which already included the basic principle of NMR based on the isotope-specific gyromagnetic ratios as well as first indications of the broad applicability of NMR in condensed matter and beyond. For completeness and to honor the nowadays unfortunately unpopular practice to publish also negative results, it should be noted that an earlier attempt to detect nuclear magnetic resonance four years before the discoveries of Bloch and Purcell was unsuccessful [26]. Over the years, NMR became a powerful method in condensed matter as well as in in chemistry, biology and medicine, where it is widely used for structural analysis and non-destructive diagnostic imaging.
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Notes
- 1.
In practice, γ is mostly expressed in terms of γ/2π, since this value is used to calculate the radio frequency \( {v_L} = {\omega _L}/2\pi = \gamma {H_0}/2\pi \) with which the system has to be irradiated such that it absorbs the energy \( E = \hbar {\omega _L} = h{v_L} \). For theoretical considerations however, one mainly uses the angular frequency \( {\omega _L} = \gamma {H_o} \) with which the spins precess around the direction of the external field. For 1H γ amounts to 26.7522128x107rad/(s.T) [39].
- 2.
The magnetic field strength H has the units A/m. A correct description would use the magnetic flux density \( B = {\mu _0}H \) in units of T for the description of the magnetic field. However, in order to keep consistency with textbooks [28, 32, 33] and most of the literature, H will be used for the magnetic field in this thesis. See also discussion “B vs H” in [33].
- 3.
Nowadays, specially assembled high resolution NMR can reach much lower limits of detection. For instance, magnetic resonance force microscopy (MRFM) is able to detect down to 106 nuclear spins [40]. This number is already at a stage were statistical spin fluctuations rather than the Boltzmann distribution are dominant.
- 4.
At \( T = 300{\rm{K}} \) for \( I = 1 \) and \( {\mu _0}{H_0} = 1{\rm{T}}\langle {\hat I_z}\rangle \) 〈Î z 〉 amounts only to 10-6.
- 5.
The following deductions assume a single, homogeneous environment of all nuclei. A distribution of different environments would also result in line shifts, line broadening and relaxation effects.
- 6.
RKKY = Ruderman, Kittel, Kasuya and Yosida
- 7.
One arrives at the quantum mechanical description by simply replacing ρ with its quantum mechanical operator \( \hat p(\vec r) = \sum\nolimits_k {{q_k}\rho (\vec r - {{\vec r}_k}} ) \) where the sum runs over the nuclear particles 1,2,…k, …N with charge qk and position \( ({{\vec r}_k}) \).
- 8.
θ is the angle between the principal axis of the EFG, Z, and the direction z of the applied magnetic field, and φ denotes a rotation around z in the xy-plane.
- 9.
This holds only true for NMR spectra on single crystals. In the case of powder spectra the situation is more complicated. (Asymmetric) second-order quadrupole effects (n≠ 0) have to be considered also for the satellites of the powder spectrum [52]. Furthermore, additional satellites due to θ ≠ 90° contributions might appear [53].
- 10.
This is only true for static measurements. The NQR spin-lattice relaxation can still be of magnetic origin.
- 11.
Experimentally, the circularly polarized field \( {{\vec H}_1} \) is realized by applying high frequency pulses generating an alternating magnetic field \( {{\vec H}_{rf}}(t) \) perpendicular to H0. This field can be decomposed into two contra-rotating components with the frequencies ω and -ω. The component rotating with -ω can be neglected, since it is far away from the effective resonance. It is thus sufficient to deal with one magnetic field \( {{\vec H}_1}(t) \) rotating with ω.
- 12.
The receiving and emitting coils are usually one and the same solenoid The separation between in-and out-coming signals is realized by the use of a λ/4-cable, explained in more detail in Chapter 5.
- 13.
Typical values of the induced voltage are of the order of a few microvolts. An amplifier is needed to detect these small voltages (see Chapter 5)
- 14.
BPP= Bloembergen, Purcell and Pound.
- 15.
The relaxation function (2.57) holds for cases of axially-symmetric field gradients (\( \eta = 0 \)). The more complicated situation of \( \eta \ne 0 \) has been considered by Chepin and Ross [73].
- 16.
Note that the spin-lattice relaxation time T1 can be arbitrarily defined with relation to the transition probability W 1 (see App. A.2 for details). In the previous formulas, T1 was chosen according to Eq. (A.29) as \( T_1^{ - 1} = 2{W_1} \). In the case of Eq. (2.57), T1 is defined as \( T_1^{ - 1} = (2/3){W_1} \) [72].
- 17.
Local field inhomogeneities may arise due to inhomogeneities of the external field as well as due to internal dipole fields or inhomogeneities of local fields produced by electronic spins. All these contributions will be eliminated by the Hahn spin echo pulse sequence, as long as they are static over the time scale of the measurement [27], which means as long as τ C ≫T 2 .
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Hammerath, F. (2012). Basic Principles of NMR. In: Magnetism and Superconductivity in Iron-based Superconductors as Probed by Nuclear Magnetic Resonance. Vieweg+Teubner Verlag, Wiesbaden. https://doi.org/10.1007/978-3-8348-2423-3_2
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