Probing Nuclear Dipole Moments and Magnetic Shielding Constants through 3-Helium NMR Spectroscopy

Abstract

Multinuclear NMR studies of the gaseous mixtures that involve volatile compounds and ³He atoms are featured in this review. The precise analyses of ³He and other nuclei resonance frequencies show linear dependencies on gas density. Extrapolation of the gas phase results to the zero-pressure limit gives the ν₀(³He) and ν₀(ⁿX) resonance frequencies of nuclei in a single 3-helium atom and nuclei in molecules at a given temperature. The NMR frequency comparison method provides an approach for determining different nuclear magnetic moments. The application of quantum chemical shielding calculations, which include a more complete and careful theoretical treatment, allows the shielding of isolated molecules to be achieved with great accuracy and precision. They are used for the evaluation of nuclear moments, without shielding impacts on the bare nuclei, for: ^10/11B, ¹³C, ¹⁴N, ¹⁷O, ¹⁹F, ²¹Ne, ²⁹Si, ³¹P, ³³S, ^35/37Cl, ³³S, ⁸³Kr, ^129/131Xe, and ¹⁸³W. On the other hand, new results of nuclear moments were used for the reevaluation of absolute nuclear magnetic shielding in the molecules under study. Additionally, ³He gas in water solutions of lithium and sodium salts was used for measuring ^6/7Li and ²³Na magnetic moments and reevaluating the shielding parameters of Li⁺ and Na⁺ water-solvated cations. In this paper, guest ³He atoms that play a role in probing the electron density in many host macromolecules are also presented.

1. Introduction

Helium (He), a chemical element, is atomic number 2 located by IUPAC in group 18 (or 8A) of the Periodic Table of Elements [1]. It is a colourless, odourless, tasteless, inert, monoatomic gas. Helium is the second most abundant element in the universe. However, in the Earth’s atmosphere and in natural gas, it is only 5.2 ppm by volume. Seven times lighter than air, it is able to escape from the Earth’s atmosphere to space. Radiogenic helium in natural gas fields occurs in concentrations up to ~7% by volume and can be extracted commercially by a low-temperature fractionated distillation process. For the NMR community, the use of liquid helium as a cooling material for superconducting electromagnets in spectrometers is well known. Two stable helium nuclei are known: ³He and ⁴He.

Both of these helium nuclei play an important role in cryogenics [2]. ⁴He can be a liquid at a low temperature (4.22 K, or −268.93 °C) and can be a solid at a pressure of 25 atm and temperature of 0.95 K (−272.2 °C). Liquid ³He can be used as a refrigerant in the separated phase solution of ³He in ⁴He, where a temperature below 0.1 K is reached. In these very low temperatures, it shows characteristic superfluid properties (Helium II) [3]. The natural abundance of ³He is of the order of 0.000134%, but only this isotope with spin number I = 1/2 can be used in NMR and MRI measurements. The use of artificially enriched material is then obligatory. Most of the rare isotope of helium is produced from the decay of tritium during the maintenance of nuclear weapons in the United States and then accumulated in a stockpile. Tritium (T, ³H) is the radioactive isotope of hydrogen with a half-life of 12.32 years. The ³He nucleus—also known as helion (³He⁺², symbol h)—is composed of two protons and one neutron that is magnetically active. Two simple requirements are obligatory for successful measurements of helium in the NMR spectrometer. It is necessary to have a suitable transmitter with an available helium frequency and a probe with adequate generated resonance tuned to the helium frequency [4]. In fact, many NMR spectrometers do not meet these conditions. Nevertheless, some practical measurements were performed on helium in its gaseous, liquid and solid states. If samples are enriched in the helium-3 isotope, the sensitivity of the NMR analysis is high enough. The resonance signals are sharp when appearing at a spectral range extending from 10 up to −50 ppm. Since helium does not form any “normal” covalent compounds, the NMR experiments have a physical, rather than chemical, character. In recent years, an experiment with helium-3 revealed a very valuable method of investigation for its specific intermolecular-interactions while in the gas phase [5]. Helium atoms are completely chemically inert and do not form any stable compounds. Interestingly, helium-3 atoms can be included into fullerenes and nanotubes and used as probes of magnetic shielding molecular interiors. This aspect of NMR investigations will be discussed at the end of our review. Novel developments of ³He NMR are connected with nuclear hyperpolarisation techniques, carried out in different ways, to receive signals even 10,000 times greater per atom than those routinely produced at thermal equilibrium [6]. A new, general method of NMR spectra measurements expressed in shielding scales, known as the “Alternative Approach to the Standardization of NMR Spectra”, was proposed using the ³He NMR resonance reference [7]. For this purpose, the ³He NMR spectra were measured simultaneously with the residual proton and carbon signals of different deuterated solvents when helium-3 was dissolved. The experimental shielding of deuterated solvents determined in this way is characterised with high accuracy and precision, and will be used in the next steps of this work. The main part of this review is devoted to the NMR method of NMM measurements. The role of ³He NMR spectroscopy in establishing nuclear magnetic moments and magnetic shielding constants is discussed in detail. We utilised the ³He (helium) atom used in mixtures with simple molecules that possessed the nuclei of interest, e.g., ³¹PH₃, ¹³CH₄, ¹⁴NH₃, ¹¹BF₃, ²⁹SiH₄, H³⁷Cl, and some others. Additionally, a few comments on helium-3 NMR achievements and prospects are included. The Scheme 1 of nuclei studied by NMR spectroscopy is shown below:

2. Methodology of Measurements of Nuclear Magnetic Moments

2.1. Methodology of Gas-Phase NMR Experiments

Many physical properties of gases can be conveniently characterised in terms of general virial theorem. In this case, the nuclear magnetic shielding dependence of any nucleus in a pure gas molecule σ(ρ,T) can be described as a virial expansion in the density function (ρ) at a given temperature (T) [8], as follows:

σ(ρ,T) = σ₀ + σ₁(T)ρ + σ₂(T)ρ² + σ₃(T)ρ³ +…

(1)

where σ₀ is the shielding constant of the nucleus in the “isolated” atom or molecule and σ₁, σ₂, and σ₃ are virial coefficients resulting from many body collisions. Below ~40 atm (density 1.64 mol/L at 297 K), the dependences are strictly linear because only the first two coefficients are present, as follows:

σ(ρ,T) = σ₀ + σ₁(T)ρ

(2)

Analogous equations can be treated when the mixtures of gaseous substances are investigated. These systems can be considered as the binary mixtures of gaseous substance A containing the nucleus X, whose shielding σ(X) is measured, and gas B is the solvent. Equation (1) can be formulated as:

σ(X) = σ₀(X) + σ_AA(X)ρ_A + σ_AB(X)ρ_B + …

(3)

where ρ_A and ρ_B are the densities of substances A and B. The coefficients σ_AA(X) and σ_AB(X) contain the bulk susceptibility corrections ((σ_A)_b and (σ_B)_b) and the terms involving intermolecular interactions during the A–A and A–B collisions (σ_A-A(X) and σ_A-B(X)). Again, if the concentration of substance A under investigation is sufficiently low (usually <1%) the σ_AA(X) can be fully omitted, and the final equation is:

σ(X) = σ₀(X) + σ_AB(X)ρ_B

(4)

Nuclear magnetic shielding constants can be measured also in the liquid solutions of different substances. Here, water solutions of inorganic salts are of the most importance. Several water salt solutions were investigated to establish the nuclear magnetic moments of inorganic isotopes. The analysis of chemical shifts measured in some mixtures of different concentrations is much more complicated than that of gases because of nonlinearity dependences. In this case, polynomial functions are more useful:

σ(X) = σ₀(X) + σ_AB(X)ρ_B + σ_AB(X)ρ_B²

(5)

where σ₀(X) means the isotope in the solvated cation or anion under study. The final equation forms (2), (4), and (5) will be exercised for numerous individual cases in detail in the next sections.

2.2. Nuclear Magnetic Dipole Moments and Shielding Constants

The main field of the induction resonance technique was to determine the dipole moment of a given nuclei. For atomic nuclei, the intrinsic angular momentum of particle (p) is directly proportional to its magnetic moment (μ) as μ = γ·p, where the proportionality constant γ is the magnetogyric ratio. The dipole magnetic moment is a vector, but only the projection value that is partially in the direction of the magnetic field can be measured in experimental conditions. The nuclear magnetic moment (NMM) μ_X is related to the nuclear spin number as:

μ_X = ħ·γ_X·I_X = g_X·I_X·μ_N

(6)

where γ_X is the nuclear gyromagnetic ratio, μ_N (which is equal to 5.050783699(31) × 10⁻²⁴J/T) is the nuclear magneton, and g_X is the g factor of nucleus x. The first measurements of NMM were performed by I.I. Rabi [9], who predicted that magnetic moments could flip their magnetic orientation in field B₀ when forced to absorb tiny parts of energy from the additional electromagnetic field B₁. These transitions can be detected when spins go to the lower energy orientations. He called this method atomic or molecular beam magnetic resonance (ABMR). This approach was a prototype of the well-known spectroscopic method NMR (nuclear magnetic resonance) carried out in a static magnetic field on bulk macroscopic samples. The NMM precess (rotate) is in the magnetic field around the z axis at a constant rate ω_L, which can be observed in the NMR spectrometer as a radio resonance frequency (rf). The equations that determine the resonance frequencies of two different nuclei, ν_X and ν_Y, in the resonance condition are as follows:

hν_X = (1 – σ_X)B_z

(7)

hν_Y = (1 – σ_Y)B_z

(8)

where σ_X and σ_Y are the absolute shielding constants. In reality, the shielding (screening) effects in the matter can slightly change the Larmor frequency of any given nuclei in the molecule, depending on the chemical environment of the nucleus within a molecule. It is difficult to measure the strength of the magnetic field with high precision. This problem can be avoided if the ratio of frequencies in Equations (7) and (8) is measured. Dividing both equations, it is possible to eliminate the external magnetic field induction B₀ and receive the very useful relation:

$$\Delta {\mathsf{\mu }}_{\mathrm{X}}^{\mathrm{z}}=\frac{{\mathsf{\nu }}_{\mathrm{X}}}{{\mathsf{\nu }}_{\mathrm{Y}}}\cdot \frac{(1-{\mathsf{\sigma }}_{\mathrm{Y}})}{(1-{\mathsf{\sigma }}_{\mathrm{X}})}\cdot \Delta {\mathsf{\mu }}_{\mathrm{Y}}^{\mathrm{z}}$$

(9)

often used for evaluation of unknown NMM from other taken as references. The nuclear dipole moment of the reference nucleus should be known with the best possible precision and accuracy. Proper measurements of nuclear magnetic moments require corrections for shielding constants (σ) of the given and reference nucleus f = (1 − σ_Y)/(1 − σ_X). These correction factors (f) are usually more or less different than 1.00, which means that shielding effects in atoms and molecules change the proper nuclear magnetic moments. New, and now more accessible, sophisticated theoretical calculations of shielding parameters have provided a new impetus for remeasuring moments with great accuracy and precision.

The most excellent result for the nuclear dipole moment was established very recently for the proton µ(¹H) = µ_p. The new physical method for the direct, high-precision measurement of a single isolated proton in a double Penning trap gives the value μ_p/μ_N = g/2 = 2.79284734462(82)μ_N [10]. The magnetic moment is here expressed in units of nuclear magnetons. The most fruitful method for establishing heavier magnetic dipole moments of stable nuclei is NMR spectroscopy since resonance techniques are able to give fixed values very precisely and accurately. Locating a macroscopic sample in a uniform stable magnetic field, it is possible to measure Larmor frequencies coming from different nuclei present in a sample. These nuclei can originate from the same molecule or different molecules that are simply present in the sample under study. This approach allows the assignment of appropriate frequency relations and, as a consequence, the comparison of two or more NMMs. The basis for further calculations is a good knowledge of one NMM from the pair of chosen nuclei. One can usually use ¹H (proton) as a reference. Other referenced nuclei can be deuteron D(²H), helion ³He, or fluorine ¹⁹F. In this paper, the second example is mostly used. The other form of Equation (9) can be used to check the consistency of nuclear magnetic moments and shielding factors, as follows:

$${\mathsf{\sigma }}_{\mathrm{X}}^{}=1-\frac{{\mathsf{\nu }}_{\mathrm{X}}}{{\mathsf{\nu }}_{\mathrm{Y}}}\cdot \frac{\Delta {\mathsf{\mu }}_{\mathrm{Y}}^{\mathrm{z}}{}_{}}{\Delta {\mathsf{\mu }}_{\mathrm{X}}^{\mathrm{z}}{}_{}}\cdot (1-{\mathsf{\sigma }}_{\mathrm{Y}}^{})$$

(10)

All parameters used in this formula are the same as stated above. It is very convenient to exercise Equation (10) when complex molecules are studied and data for additional nuclei can be applied.

3. Results and Discussion

3.1. ³He Chemical Properties

The helium atom and helion nucleus can be categorised as unusual physical objects. A very small atomic diameter (28 pm, 1 pm = 10⁻¹² m) can cause the glass walls to be penetrated and the glass ampoule to be emptied. This is an indication for fast experimental maintenance of samples containing helium-3 atoms. It is remarkable that the ³He nucleus is the only stable nucleus with a proton number larger than neutrons. The mirror nucleus of helium-3 is the tritium nucleus with two neutrons and one proton (T = ³H). The sign of the ³He nuclear magnetic moment is opposite to that of a proton [11] and negative by convention. Precise knowledge of the helium-3 NMM as a reference is crucial for employing the NMR method for establishing nuclear dipole moments against that of helion. Several attempts have been made to precisely determine the ³He nuclear magnetic moment, and all were undertaken using NMR spectroscopy. The ³He magnetic moment was measured by Anderson and Novick [12], Williams and Hughes (1968) [13], Neronov and Barzakh (1977) [14], Belyi and Shifrin (1986) [15], Hoffman and Becker (2005) [16], Jackowski et al. (2008) [4], Aruev and Neronov (2012) [17], and Makulski (2020) [18]. In our opinion, the best result was achieved by Flowers, Petley, and Richards [19] in pure helium-3 samples against the proton signal in liquid water. The authors measured the ratio of the NMR spin precession frequencies of optically pumped, low pressure helium-3 and protons in water, taking into account the residual inhomogeneities and water diamagnetic corrections accurately. The final results of the nuclear magnetic moment (μ(³He) = −2.127625308(25)μ_N) were corrected by factors that arose from the shielding parameters of helium-3 (σ₀(³He) = 59.96743(10) ppm) [20]. These were included in the latest collections of fundamental physics constants recommended by Stone in the “Table of recommended nuclear magnetic dipole moments” published under the auspices of the INDC (International Nuclear Data Committee) in November 2019 [21]. This value was used for all recalculations of nuclear moments of different nuclei carried out previously and mentioned in this work. It is worth noting that the nuclear moments are rather more experimental than theoretical values. The nuclear moments can be calculated purely theoretically using several nuclear models. The last calculations lead to essentially different results: −1.913042 µ_N (shell model), −2.2905 µ_N (QCD model), and −2.004914 µ_N (Quark model), where QCD refers to the lattice quantum chromodynamics theory [22]. The difference from the best experimental value is rather large (~5% in the last case) and confirms the assumption of nuclear moments’ significance as experimental values. From the quantum mechanical point of view, the ³He nucleus is fermion, while the ⁴He nucleus is a boson. Several chemical properties of the helium atom and helion nucleus are shown in

Table 1. Physical and magnetic properties of ³He atoms and nuclei.

Property		Property
Spin	1/2	Natural abundances	0.000137%
Magnetic moment	−2.127625308	Isotope mass	3.0160293 u
Chemical shift range	58 ppm	Half life	stable
Frequency ratio	76.18%	Boiling point	3.19 K
Reference compound	3He gas	Critical point	3.35 K
Linewidth of reference	0.3 Hz	Heat of vaporization	0.026 kJ/mol
T1 of reference	1000 s	Melting point	below 1 mK
Receptivity rel.to 1H	0.442 when enriched	Covalent radius	32 pm
Magnetic susceptibility	−1.88 cm3/mol	Van der Waals radius	143 pm

. The NMM µ(³He) and spin number (1/2) decide the radiofrequency used for the detection of this nuclei in the stable magnetic field. The Ξ frequency parameter is defined as the ratio of the 3-helium frequency, ν₀ (observed), to that for ¹H of TMS in CDCl₃ in the same magnetic field, expressed as a percentage. For example, in magnetic field B₀ = 11.74 T, the ³He radiofrequency is 381.3575 MHz, just between frequencies for ¹⁹F (470.910 MHz) and ³¹P (202.595 MHz). In this way, the Ξ/MHz is 76.178972 [23]. Fortunately, these measurements are available on Varian INOVA (now Agilent, Santa Clara, CA, USA) spectrometers. The relaxation times of the ³He nuclei in the pure gas phase are very long, up to several thousand of seconds. Therefore, the usage of short pulse duration and long relaxation delays are strongly recommended. The linewidth of recorded signals often does not exceed 1 Hz.

3.2. Other Noble Gases: ²¹Ne, ⁸³Kr, and ^129/131Xe

A simple and effective example for handling Equations (2) and (4) for establishing the nuclear magnetic moments is provided by mixtures of noble gas isotopes that possess the non-zero spin numbers. Mixtures of ³He as reference nucleus with other noble gas atoms, including ²¹Ne [24], ⁸³Kr [25], ¹²⁹Xr and ¹³¹Xe, were carried out using the NMR method [26]. The frequency dependences as functions on the density of the main gaseous ingredients are shown in Figure 1.

The appropriate input data for nuclear magnetic moment calculations—frequency ratios, correction factors, and final nuclear magnetic moments data—are shown in

Table 2. NMR parameters for calculating the µ(ⁿX) nuclear magnetic moments of noble gases.

System	ν0(nX)/ν0(3He)	Correction Factor	μ(nX)	References
3He	1.0000	1.000059965	−2.127625308(10)	[19,20]
			−2.127625307(10) *	[25,26]
21Ne/3He	0.103638	1.000497422	0.66184(7)	[27]
			0.6617774(10) *	[24]
83Kr/3He	0.0505161561(5)	1.003529996	−0.97072965(32)	[25]
			−0.97072965(32) *
129Xe/3He	0.36309748(3)	1.007022726	−0.777961(16)	[26]
			−0.777961(16) *
131Xe/3He	0.107634919	1.007022726	0.691845(7)	[26]
			0.691845(7) *

* Nuclear magnetic moments from ²H(D) parameters.

. Thanks to the high receptivity of ³He NMR signals, the precise evaluation of magnetic moments is mainly a function of shielding correction factors. The correction factor for ²¹Ne is small (~0.05%), but very valuable for xenon isotopes of 0.7%. The results marked with asterisks were calculated against the deuterium ²H(D) resonance used in the lock system. It can be seen that both kinds of results for helium, krypton, and xenon are in very good agreement. This circumstance justifies using the lock system instead of 3-helium for measuring the ²¹Ne nuclear moment.

In our spectrometer, low frequency resonances (like ²¹Ne and ⁸³Kr) should be measured in a broad band probe with test tubes that have a 10 mm external diameter, whereas high frequencies (like ³He) are only available in a special probe with NMR tubes that have a 5 mm external diameter [4]. Both the original lock system and VT facilities were left unchanged in the home-adapted Varian variable temperature probe-head. This probe is notable for its high quality. The duration of the 90° rectangular rf pulse was only 15.5 µs. The lock frequency (²H) was practically constant (76.84640 MHz) for all measurements presented in this work. Our probe generally allows the precise determination of ³He spectra samples with small amounts of helium gas in a short time.

From the final results of NMMs measured against ³He and ²H(D) nuclei (Table 2), it is clear that both are very congruent and can be used interchangeably in these experiments. The accuracy is strictly connected with shielding factors which are large for heavy nuclei and small for light nuclei. The shielding parameters of the lock system (²H,D), for these purposes, show very good quality [4]. It is remarkable that the NMR investigations of other noble gases, mainly argon and radon elements, are not possible. All three stable argon isotopes—³⁶Ar, ³⁸Ar and ⁴⁰Ar—are not magnetically active because of their spin (I = 0). Otherwise, there are known nuclear magnetic moments of short-lived nuclei such as ³⁵Ar, ³⁷Ar, ³⁹Ar, and ⁴¹Ar from physical methods other than typical NMR experiments. Likewise, radon nuclei are all radioactive and short-lived. The electromagnetic properties of radon isotopes, from ²⁰⁹Rn up to ²²⁵Rn, are known from physical experiments [18].

3.3. Simple Hydrides: ¹³CH₄, ¹⁴NH₃, H₂¹⁷O, ²⁹SiH₄, ³¹PH₃, and H^35/37Cl

The investigations of simple, gaseous hydrides are optimal for obtaining the accurate results of shielding and magnetic moments of nuclei. The range of ¹H NMR shifts is very small, practically limited to ~10 ppm, and experimental errors are much less than 0.1 ppm. The shielding constants are very well known from experimental [28] and theoretical [29] studies. The binary hydrides CH₄, NH₃, SiH₄, PH₃, and HCl are gases at normal conditions and can be used directly from lecture bottles for the preparation of gaseous samples for NMR measurements. Furthermore, isotopically enriched substances (e.g., ¹³CH₄ and ¹⁵NH₃) were offered from commercial sources and formerly used in our systematic investigations [30,31,32,33,34]. More problematic were the attempts to prepare water that contained gaseous samples—a 25% enriched preparation was used [35], and next, 90% H₂¹⁷O [36]. In this last case, the samples containing ~5 × 10⁻⁴ mol/L enriched water in the buffer gases CH₃F and CHF₃ were made, and the ¹⁷O spectra were measured with success. All suitable input data for the calculation of the nuclear magnetic moments are shown below in

Table 3. NMR parameters in simple hydrides for the calculation of the nuclear magnetic moments of ¹³C, ¹⁴N, ¹⁷O, ²⁹Si, ³¹P, and ^35/37Cl nuclei from gas-phase experiments.

System	ν0(nX)/ν0(3He)	Correction Factor	μ(nX)	References
13CH4/3He	0.330074361(2)	1.00013507	0.70236944(68)	[30]
			0.70236945(68) *
14NH3/3He	0.094821748(5)	1.00020688	0.40357377(45)	[31]
			0.40357367(40) *
H217O/3He	0.177949095(15)	1.00026852	1.893553(3)	[36]
			1.893553(2) *	[35]
29SiH4/3He	0.260768297(3)	1.00042308	−0.5550520(3)
			−0.5550520(3) *	[32]
31PH3/3He	0.531248246	1.000555132	1.1309247(50)	[33]
			1.1309246(50) *
H35Cl/3He	0.12862043	1.000917130	0.821716(5)	[34]
			0.821721(4) *
H37Cl/3He	0.07063037	1.000917130	−0.683997(5)
			−0.683997(4) *

* indicates values calculated from ¹H NMR data.

.

Certainly, the measurements of ³⁵Cl and ³⁷Cl NMR spectra were the most challenging of all the gaseous systems presented in Table 3. The relatively large quadrupole moment Q of ³⁵Cl (+0.085 b) (1 barn = 10⁻²⁸m²) and ³⁷Cl (−0.064 b) leads to effective quadrupolar relaxation times and broad resonance signals with a linewidth of one-hundred hertz (Hz). It is obvious that analogous experiments with other gaseous hydrides—HBr and HJ—dealing with ⁷⁹Br (Q = +0.51 b), ⁸¹Br (Q = 0.26 b), and ¹²⁹J (Q = −0.696 b) nuclei would be very problematic. Fortunately, water solutions of bromides and iodides with ³He solute can be used in these cases (see, for example, Section 3.5).

3.4. Fluoride Compounds: ^10/11BF₃, ³³SF₆, and ¹⁸³WF₆

Unfortunately, not all elements form simple hydrides which are regular gases or volatile liquids at room temperatures. In such cases, one can use the fluorinated analogs of hydrides, which often are available as commercial chemicals. The benefit of using ¹⁹F nuclei is their high sensitivity, which is almost equal to that of protons and the universal ability of measurements in most spectrometers. In the case of boron, it is suitable for analysing trifluoroborane BF₃ gas rather than diborane B₂H₆, with a specific structure composed of two bridge H atoms between boron atoms and four H terminal atoms. In the end, we used BF₃, which has a trigonal planar structure with D_3h symmetry [37]. Its vapour pressure at 20 °C is more than 50 atm and it can be investigated in a large density range. Analogically, similar experiments were performed with sulfur hexafluoride SF₆ gas in the 5–25 atm pressure range [38]. A big advantage of using this compound is its high octahedral symmetry consisting of six fluorine atoms connected to one central sulfur atom. The electric field gradient (EFG) on sulfur nuclei is then zero and the ³³S signals are very narrow (~1 Hz). Other gaseous sulfur-containing compounds, such as H₂S, COS, SO₃, and SO₂, are notable for their much worse NMR measurement conditions. The gaseous samples containing small amounts of 3-helium atoms dispersed in sulfur hexafluoride were used. Well-resolved sulfur-33 NMR spectra gave the opportunity to recognise the ¹J(³³S,¹⁹F) spin–spin coupling and make use of the INEPT (insensitive enhanced polarisation transfer) sequence to amplify the signal and smooth the baseline in the spectrum. For example, the ³³S NMR INEPT spectrum of 21.8 atm (0.89 mol/L) SF₆ is shown in Figure 2 as coupled to six fluorine-19 nuclei, as well as fully decoupled.

The linear density dependences of the ³He and ³³S radiofrequencies are shown in Figure 3. The observed Larmor frequencies, extrapolated to the zero-pressure limit, belong to the isolated SF₆ molecule and helium atom. The application of Equation (9) gives the ³³S nuclear magnetic moment in terms of ³He μ(³³S) = 0.6432555(10) μ_N. The new result is in good agreement with previously reported values (μ(³³S) = 0.6432474(107) μ_N [39] and μ(³³S) = 0.643251(16) μ_N [40]), but is more accurate by one order of magnitude. The shielding corrections for the sulfur nuclei (σ(³³S) = 392.6 ppm) in SF₆ were taken from the first relativistic calculations performed previously [41]. This value, recalculated from Equation (10) using the fluorine parameters—frequency and shielding constants—of 406.2 ppm is in moderate agreement with the calculated one.

Another important modification of the procedure shown above relies on the measurements of the ¹⁸³WF₆ substance where CF₄ or C₂F₆ gaseous buffers were used [42]. They are necessary for lengthening the relaxation times and narrowing the ¹⁸³W resonance signals. Naturally occurring tungsten (wolfram) contains only one isotope (¹⁸³W) with a non-zero value of I = 1/2 and a natural abundance of 14.31%, with a rather poor receptivity of 1.06 × 10⁻⁵ relative to ¹H. This is where it gets weird: ¹⁸³W is a α-particle-emitter with a long half-life of >1 × 10¹⁷ y. The ¹⁸³W NMR spectra were recorded using the INEPT (insensitive nuclei enhanced by polarisation transfer) sequence for increasing the signal-to-noise ratio and smoothing the baseline distortions. They were decoupled from the ¹⁹F nuclei during acquisition, making use of the known spin–spin coupling (¹J(F,W) = 43.75 Hz). The isotope effect observed in the ¹⁹F NMR spectra connected with the substitution of ¹⁸³W by other tungsten isotopes is negligible. The ¹⁸³W nuclear magnetic moment obtained in this experiment was μ(¹⁸³W) = 0.116953(18) μ_N, which gives the g-factor g_I = μ(¹⁸³W)/I = 0.233901(18) and gyromagnetic ratio γ_I = g_I μ_N/ħ = 1.120249 × 10⁷ rad s⁻¹ T⁻¹. The experimental results were corrected by the shielding constant of tungsten in WF₆ calculated theoretically (σ(¹⁸³W) = 6221.0 ppm [41]). The appropriate NMR parameters for the ³He/fluoride compound systems are collected in

Table 4. NMR parameters for calculations for the nuclear magnetic moments of ^10/11B, ³³S, and ¹⁸³W against that of ³He.

System	ν0(nX)/ν0(3He)	Correction Factor	μ(nX)μN	Reference
10BF3/3He	0.141033238	1.000037916	1.8004636(8)	[37]
			1.80045428 *
11BF3/3He	0.421170045	1.000037916	2.6883781(11)
			2.6883642 *
33SF6/3He	0.100744802	1.0003926	0.6432555(10)	[38]
			0.6432467(16) *
183WF6/3He	0.054630264	1.0061996	0.116953(18)	[42]
		1.0062286	0.116953 *
		1.0061989	0.116950 **

* NMM measured from ¹⁹F parameters. ** NMM measured from ¹⁹F parameters with the theoretical shielding correction factor.

.

It is obvious that the ¹⁹F nuclear magnetic moment can be calculated from the helium-3 results used in the above-mentioned cases (the BF₃/³He, SF₆/³He, and WF₆/³He systems). If we utilise the measured ¹⁹F frequencies and shielding results that come from the best relativistic calculations, the new ¹⁹F NMM values are as follows: μ(¹⁹F) = 2.6283348(26)μ_N, μ(¹⁹F) = 2.6283711(132) μ_N, and μ(¹⁹F) = 2.6283925(263) μ_N, respectively. They are in adequate agreement with those used in previously published results (μ(¹⁹F) = 2.628321(4) μ_N [21] and μ(¹⁹F) = 2.628335(11) μ_N [40]) but in disagreement with the result given previously in the literature tables (μ(¹⁹F) = 2.6288868(8) μ_N [40]). The deviations mainly arise from the shielding uncertainties of the fluorine nuclei. It is important to mention the special position of hydrogen fluoride (HF) in fluorine NMR spectroscopy. This simple molecule has one undesirable feature: it forms very corrosive and penetrating fumes with traces of water when hydrofluoric acid is forming. It attacks the glass vessels and vacuum lines mostly used in laboratories for sample preparation and handling. The intermolecular hydrogen bonds in HF are very strong and obtaining monomers, even in the gaseous phase, is very problematic. Nevertheless, the HF molecule was the basic molecule for establishing the ¹⁹F NMR absolute shielding scale [43]. The following special conditions were ensured: a thin-walled cell from polyethylenes and a Monel metal vacuum line for the transfer of HF and other gases. The results of this investigation have never been repeated and confirmed. For this reason, the secondary reference substances were usually used (SiF₄ and CFCl₃ [44]).

In light of these considerations and taking into account the remarks on the fluorine shielding scale, it is strongly recommended to reevaluate this absolute scale in new experimental and theoretical studies. We have recently started the preparatory work for the revision of ¹⁹F nuclear magnetic moments and shielding constants in gaseous mixtures of ³He/CH₃F and ³He/CF₄. The preliminary findings promise good agreement between older results and those of our new investigations.

3.5. Water Solutions: ^6/7Li and ²³Na Salts

Helium-3 can certainly be used as a reference in NMR measurements of nuclei when the water solutions of different cations are the target of the studies. Neither the lithium nor the sodium elements form any stable gaseous compounds. Li-He, the very weakly interacting Van der Waals chemical compound, is probably only present in low temperatures [45]. The most studied objects in lithium and sodium chemistry are their water solutions, and they were used in the NMM measurements. A total of 1.5 mg of the helium atoms can be dissolved in pure water at 20 °C and 1 bar pressure. Additionally, the shielding correction factors can be known from the theoretical studies of Antušek et al. [46]. These results make use of our new experimental reports on the NMR frequencies of Li⁺ [47] and Na⁺ [48] in water solutions of different salts. The concentration dependencies do not show linear functions, but they can be fitted with a quadratic one. In these cases, the frequency results do not belong to a simple isolated cation, but rather to a few aquatic structures of central ions occurring in different mole fractions. In the case of lithium cation, because of its small radius of 90 pm, as a Li(H₂O)₄⁺ cation, the shielding correction is 91.69 ppm in the infinitely diluted solution. In the octahedral coordinating lithium cation, this value is 90.89 ppm. Both results can give the ^6/7Li nuclear magnetic moments as the lower and upper limits of these numbers. In the case of the ²³Na magnetic moment, the situation is much more complicated. The sodium cation Na⁺ is dynamically transformed between several structures, according to the equation:

Na(H₂O)₃⁺ + H₂O ↔ Na(H₂O)₄⁺ + H₂O ↔ Na(H₂O)₅⁺ + H₂O ↔ Na(H₂O)₆⁺ + H₂O ↔ Na(H₂O)₇⁺

(11)

The molecular dynamics (MD) simulation results suggest that the five-coordinated sodium ions is the most-populated structure (~60%) and the six-coordinated structure is the second-most populated (~30%) [49]. In this case, we prefer to consider the dynamic character of water surrounding the central cation by averaging the results for four-, five-, and six-coordination systems, with their appropriate molar fractions. The shielding correction factor of 580.12 ± 10 ppm for sodium cations in liquid water was finally received.

The helium-3 correction factor in lithium water solutions was measured as 2.7675(25) ppm and 2.747(2) for sodium salts against that of the isolated ³He atom in the gas phase in the chemical shift category. Taking into account the volume magnetic shielding susceptibility effect of pure water (4/3χ_V, where χ_V is volume susceptibility), the shielding corrections are of the order of 0.24 ppm compared with the gas phase in the relative shielding scale. For ^6/7Li and ²³Na nuclear magnetic moments, the NMR results are very consistent with the ABMR results achieved previously [50], where shielding corrections were made for the atomic inert species σ(Li) = 147.2 ppm and σ(Na) = 640.62 ppm. The NMMs of lithium and sodium nuclei measured using the NMR and ABMR methods are shown in

Table 5. NMR parameters for the calculation of ^6/7Li and ²³Na nuclear magnetic moments from water solutions. * NMM calculated from ²H(D) parameters.

System	ν(6Li)/ν(3He)	Correction Factor	μ(6Li)	Reference
Li+ water solution/3He	0.193177745	1.000031563	0.8220456(25)	[47]
	0.95866015	1.000062459	0.8220432(25) *	2H(D)
			0.8220445(10)	[50] ABMR
			0.8220454(25)	[51]
	ν(7Li)/ν(3He)
	0.510164303	1.000031563	3.2564182(98)	[47]
	2.531731524	1.000062459	3.2564085(98) *	2H(D)
			3.2564157(2)	[50] ABMR
			3.2564171(98)	[51]
Na+ water solution/3He	ν(23Na)/ν(3He)
	0.347233571	1.000520714	2.2174997(111)	[48]
	1.723174611	1.000551603	2.2174962(111) *	2H(D)
			2.2175019(133)	ABMR
			2.2175065(55)	[52]

. Additionally, the new results published by Neronov [51,52] are included. They were achieved using an interesting method of the simultaneous observation of the two-spin system with an adapted NMR spectrometer, which reduced the random and systematic errors by an order of magnitude.

3.6. ³He Atoms in Different Chemical Environments

It is worth drawing attention to the new spectacular applications of 3-helium spectroscopic achievements. The ³He atom can be inserted into the internal cavities present in fullerene compounds for the investigation of the magnetic shielding environments inside cavities. The first developments dealt with the endohedral fullerenes ³He@C60 and ³He@C70 [53]. ³He shielding values are 6 and 29 ppm here, respectively, relative to the free atom, which indicates substantial diamagnetic ring currents inside fullerene molecules. It is even possible to incorporate two helium-3 atoms in C70 [54]. Interestingly, the reduction of C60 and C70 fullerenes to their hexaanions ³He2@C60⁻⁶ and ³He2@C70⁻⁶ reduces the magnetic field inside the fullerene, diminishing its aromacity [55]. These observations are in good agreement with the theoretical predictions of [56,57]. ³He NMR spectroscopy was used for probing the small pores in solid substances, e.g., zeolites [58,59], silica aerogels [60], and nanocavities on the crystal surface powders and single crystals [61]. Besides these chemical uses, it is impossible to omit the huge and still growing biological and medicine applications of 3-helium MRI methods. A few remarks about these studies will be added in the conclusion section of this paper.

4. Conclusions

In this review, we have shown the main achievements of the NMR method for establishing the nuclear magnetic moments of different nuclei, which were performed in the gas phase using ³He atoms. It was established that this method is distinguished by its high degree of accuracy and precision. NMR measurements in high magnetic fields, in the spectrometers of advanced technology, and in sophisticated quantum calculation methods leads to unprecedented results. The final nuclear magnetic moment was identified with at least 0.01% accuracy. The interpolation NMR parameters to the zero-point pressure of gases deals with well-defined chemical species, atoms, and molecules. The shielding corrections for these species are often known with great quality from sophisticated theoretical calculations. The experimental conditions in liquid solutions are quite different. The most common water solutions are often complicated systems where cations and anions are solvated by different numbers and time-changing water molecules. The calculation of the shielding values of these different solute compositions is a complicated task and only limited results are available. The references suggested for NMR measurements of nuclear magnetic moments are: ¹H (proton), ²H (deuteron), and ³He (helion). Experiments performed in the gaseous phase are strongly recommended. The helion (³He) nuclear magnetic moment belongs to the more accurately known values among all nuclei in the periodic table, with 1.2 × 10⁻⁸ relative standard uncertainty. Despite this, we hope that newly planned experiments by physicists, using Penning trap methods in combination with laser cooling, will provide even better results [62]. Helium-3 has unique properties that currently makes it very useful in several fields, and its cryogenic properties are useful in low-temperature physics, magnetic resonance imaging (MRI) with hyperpolarisd atomic nuclei in medicine, and in fast neutron high sensitivity for 3-helium detectors. The gaseous helium-3 magnetic resonance method is even more useful in cryogenic environments [63] and as a probe of the nature of the solid-superfluid interface [64]. ³He can be hyperpolarized by spin-exchange optical pumping and other methods. On the other hand, a helium atom with an ³He nucleus can be used as a probe of electron densities inside macromolecule compounds only if NMR spectra can be measured for an encapsulated helium-3 species. Finally, local magnetic fields in liquid crystals utilising ³He NMR spectroscopy are also known from research [65]. More valuable and prospective are MRI investigations with hyperpolarized helium-3 atoms in medical treatment, firstly demonstrated in the lung images of a guinea pig [66].

This unique chemical nucleus and/or atom is definitely a good probe of the matter being investigated. It is truly fascinating to observe the discovery of the new and surprising purposes of helium-3 in environmental research. We are convinced that this small and curious species will find several spectacular applications in chemistry, physics, and biology in the near future.

Upon completion of this paper, we recognize a very recent publication by Pachucki et al. (2021) [67] on QED effects in calculations of the ³He shielding constant. The final result, which gained in accuracy, is: σ(³He) = 59.967029(23) µ_N, which is in good agreement with the previous result mentioned in this work. The new value does not essentially influence the helium-3 nuclear magnetic moment.