Improving Nuclear Magnetic Dipole Moments: Gas Phase NMR Spectroscopy Research

Abstract

High-resolution NMR spectroscopy is the leading method for determining nuclear magnetic moments. It is designed to measure stable nuclei, which can be investigated in macroscopic samples. In this work, we discuss the progress in research into light nuclei from the first three periods of the Periodic Table and several selected heavy nuclides. The ¹H and ³He nuclear magnetic moments, established using the new double Penning trap facility, are also considered. Both nuclei can be used as references in gaseous mixtures. Gas-phase NMR spectroscopy enables precise measurements of the frequencies and shielding constants of isolated single molecules. They can be used to determine new, accurate nuclear magnetic moments of nuclides in stable, gaseous substances. Particular attention is paid to the importance of diamagnetic corrections for obtaining accurate results. Finding precise diamagnetic corrections—shielding factors —even for light nuclei in molecules is a significant challenge. To date, nuclear moments have been obtained primarily from experimental data. The theoretical approach is mostly unable to predict these values accurately. Some remarks are also made on pure theoretical treatments of nuclear moments.

1. Introduction

Nuclear magnetic moments are related to a nucleus with nuclear spin, and they are expressed as a vector sum of the magnetic moments of the nucleons inside it. This is a valuable physical variable (μ), and it can be used in tests of different calculation theories. These are “ab initio”, semiempirical, or density functional theory calculations. Unfortunately, progress in theoretical computations is still limited to light nuclei, and nuclear magnetic moments are generally treated as experimental quantities. Tables of Nuclear Magnetic Dipole Moments have been published periodically by N.J. Stone for several years. One of the most effective methods for establishing these values is NMR spectroscopy. In this technique, the most critical information is the specific radiofrequency at the peak response in the absorption spectrum. This frequency is proportional to the static B₀ field, the nuclear magnetic moment of the observed nucleus, and the shielding effects. Besides the frequencies being observed (ν), a crucial role in the procedure for measuring the nuclear magnetic moments is attributed to knowledge of minor but valuable corrections for shielding effects (σ), previously known as diamagnetic effects. This paper describes recent developments in the determination of nuclear magnetic moments using

NMR spectroscopy. The nuclides of interest are shown in Figure 1 below.

In the Laboratory of NMR Spectroscopy at the Department of Chemistry at Warsaw University, we have performed several studies in this field in the gaseous state, where diamagnetic corrections and appropriate frequencies can be incorporated throughout the procedure. Gas-phase physical properties can be written as linear functions of the pressure/density of the prepared samples. Linear regression is used to solve linear dependencies analytically. The best-fit line can yield extrapolated NMR parameters for a single “isolated” molecule. The paper covers the experimental details and final results for nuclides from the first three rows of the Periodic Table of Elements, obtained from gaseous experiments, all of which are highly precise and accurate.

2. Methodology

2.1. NMR Method of µ(X) Measurements

To obtain an NMR signal, the conditions require inducing flipping between different nuclear magnetic orientations in the static magnetic field B₀. In spectroscopy machines, the B₀ field is produced by superconducting magnets, with fields ranging from 1 T to ~22 T (1 T = Tesla). The appropriate electromagnetic waves are in the radio-frequency (RF) region from 100 MHz to ~1000 MHz. To absorb a portion of energy in the NMR regime, the simple relations known as the Larmor equations need to be fulfilled:

$$\mathrm{h}{\mathsf{\nu }}_{\mathrm{Y}}=\mathsf{\Delta }{\mathsf{\mu }}_{\mathrm{Y}}^{\mathrm{z}}(1-{\mathsf{\sigma }}_{\mathrm{Y}}){\mathrm{B}}_{\mathrm{z}}$$

(1)

$$\mathrm{h}{\mathsf{\nu }}_{\mathrm{X}}=\mathsf{\Delta }{\mathsf{\mu }}_{\mathrm{X}}^{\mathrm{z}}(1-{\mathsf{\sigma }}_{\mathrm{X}}){\mathrm{B}}_{\mathrm{z}},$$

(2)

where B_z represents the z component of the firm, stable, and homogeneous magnetic field B₀.

The precession (rotation) of the nuclear magnetic moments (NMMs) occurs around the z-axis of the magnetic field (for two different nuclei: ν_X and ν_Y) at a constant rate ω_L, which can be observed in the NMR spectrometer as a radio resonance frequency, where σ_X and σ_Y are the absolute shielding constants. The conventional relationship between two nuclear magnetic moments and two observed frequencies (ν_X, ν_Y) measured in the given sample in the same magnetic field B₀ can be formulated as [1]:

$$∆{\mathsf{\mu }}_{\mathrm{Y}}^{\mathrm{z}}={\displaystyle \frac{{\mathsf{\nu }}_{\mathrm{Y}}}{{\mathsf{\nu }}_{\mathrm{X}}}}·{\displaystyle \frac{\left(1-{\mathsf{\sigma }}_{\mathrm{X}}\right)}{\left(1-{\mathsf{\sigma }}_{\mathrm{Y}}\right)}}·{\displaystyle \frac{{\mathrm{I}}_{\mathrm{Y}}}{{\mathrm{I}}_{\mathrm{X}}}}·∆{\mathsf{\mu }}_{\mathrm{X}}^{\mathrm{z}}$$

(3)

where µ_X and µ_Y are two different magnetic moments, ν_X and ν_Y are NMR frequencies from the spectra, and σ_X and σ_Y are shielding constants often called diamagnetic corrections. Measurements of nuclear magnetic dipole moments using NMR spectroscopy are exact and accurate. These measurements are well suited to studies of stable isotopes and to those involving slow decay (see, for example, ³H(T)). Finally, we can consider Equation (3) converted to the form:

$${\mathsf{\sigma }}_{\mathrm{X}}=1-{\displaystyle \frac{{\mathsf{\nu }}_{\mathrm{X}}}{{\mathsf{\nu }}_{\mathrm{Y}}}}·{\displaystyle \frac{\mathsf{\Delta }{\mathsf{\mu }}_{\mathrm{X}}}{\mathsf{\Delta }{\mathsf{\mu }}_{\mathrm{Y}}}}·{\displaystyle \frac{{\mathrm{I}}_{\mathrm{X}}}{{\mathrm{I}}_{\mathrm{Y}}}}·(1-{\mathsf{\sigma }}_{\mathrm{Y}})$$

(4)

This will be used to check the consistency of the new shielding results and nuclear magnetic moments. It is possible to use other pairs of nuclei than those applied in Equation (3). In this case, experiments with the addition of helium-3 gas are beneficial.

2.2. Gas Phase NMR Measurements

The best-known extension of the gas property to higher powers of density is the so-called “virial expansion”. The dependence of density and pressure for any molecular property in a pure gas can be expressed in this straightforward manner. Several physical properties of gases can be easily understood using the general virial theorem. In our case, the more important property—the nuclear magnetic shielding of a given nucleus in a pure gas σ(ρ, T)can be described as a virial expansion in the density function (ρ) at a specific temperature (T), as shown here [1,2]:

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

(5)

where σ₀ is the shielding constant of the nucleus in the “isolated” atom or molecule, and σ₁, σ₂, and σ₃ are virial coefficients arising from two, three, or more-body collisions. At low pressures (density below ~1.64 mol/L at 297 K), the dependences are strictly linear because only the first two coefficients are essential, in this manner:

σ(ρ,T) = σ₀ + σ₁(T)ρ

(6)

Similar equations can be used when exploring mixtures of gaseous substances. These systems can be studied as binary mixtures of a gaseous substance A containing the nucleus X, whose shielding σ(X) is measured, and gas B, the solvent. Equation (1) can be written as:

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

(7)

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_AA(X) and σ_AB(X)). Again, if the concentration of substance A under investigation is sufficiently low (usually <1%), the σ_AA(X) can be entirely omitted, and the final equation is:

σ(X) = σ₀(X) + σ_AB(X)ρ_B

(8)

Equation (6) was used for pure simple gases such as CH₄, SiH₄, and PH₃. Sometimes the additional ingredients were maintained by small amounts of gaseous ³He. In this case, Equation (7) is valid because the amount of helium is small. Contributions to σ₁ or σ_AB include the Van der Waals field, the electric field, and the molecular anisotropy of neighboring molecules, as well as the bulk susceptibility. The bulk susceptibility (4πχ_M/3 in B‖ external field) can generally be removed from σ₁ or σ_AB, so that only the proper intermolecular effects are present. Each σ₁ parameter is described by a complex function of the intermolecular separation and orientation between two interacting molecules (an intermolecular shielding surface). Ab initio methods can calculate this function, but the appropriate procedure is computationally intensive and, to date, has been obtained only for the simplest molecular systems.

2.3. State-of-the-Art Quantum Mechanical Computations

Gas-phase NMR measurements are usually straightforward and can yield precise frequency measurements when relaxation times are sufficiently long and the spectral resolution is sufficiently high. On the other hand, diamagnetic corrections (shielding constants) are more demanding and require advanced theoretical methods. Nuclear shielding in molecules is, in principle, the function of their electronic structure. Usually, a large number of electrons increases the shielding effect; however, several deshielding effects have also been observed in diamagnetic molecules [1,3]. Gas-phase experiments allow us to eliminate all intermolecular interactions by extrapolating shielding measurements to the zero-pressure limit. In this context, the experiment is associated with a significant portion of quantum chemical calculations, specifically those involving the so-called “isolated” single molecules. More popular methods include perturbative treatments. In a perturbative theoretical approach, iterative corrections yield approximate solutions [4]. The gauge-including atomic orbitals (GIAO) approach is employed chiefly here, incorporating magnetic-field-dependent gauge factors into the atomic basis sets. Several approximate perturbation theories with specific variants are used: the HF (Hartree-Fock), MCSCF (Multi-Configuration Self-Consistent-Field), the CC (Coupled Cluster), such as CCS, CC2, CCSD, CCSDT, etc., and the MP (Möller-Plesset) model up to FCI (Full Configuration Interaction). In larger molecules, the same specific methods were used: DFT (Density Functional Theory) or the ONIOM method, with great success [3]. NMR spectral parameters are first computed for the molecule’s equilibrium geometry, which can be experimental or fully optimized to minimize electron repulsion forces. These shielding values σ_e(X) are then corrected for vibrations and rotations, e.g., by applying the zero-point vibration correction σ(X)ZPV and by averaging over the Boltzmann distribution up to room temperature, σ(X)_T (usually at 300 K). Sometimes, using gas-phase NMR data, one can separate these corrections by studying isotope effects and comparing them with quantum-chemical computations (e.g., for the H₂O molecule [5]). Three factors influence calculations of NMR parameters: the size of the basis set, electron correlation, and relativistic effects. Electron-correlation effects were incorporated into the primary methods mentioned above. Even for light nuclei, it is essential to account for relativistic corrections. For example, within the so-called four-component relativistic framework, the absolute shielding of the isolated NH₃ molecule was established [6]. For the BF₃ molecule, the shielding constant was obtained as the sum of a nonrelativistic electron-correlated contribution and a relativistic correction computed at a lower level of theory [7]. The spin-rotation and nuclear magnetic shielding constants are analyzed for both nuclei in the HCl molecule [8]. Nonrelativistic ab initio calculations at the CCSD(T) level of theory show that relativistic effects are essential to obtain spin-rotation constants consistent with accurate experimental data. For small molecules, experimentally determined nuclear magnetic shielding should be consistent with that computed using the more sophisticated methods. It is worth noting that fully relativistic coupled-cluster (RCC) methods, often using four-component Dirac-Hartree-Fock (DHF) or two-component (2C) Hamiltonians (e.g., ZORA), are crucial for calculations of nuclear magnetic shielding constants for heavy nuclei (for example, ⁷⁵As, ¹²¹Sb, ¹²³Sb, ¹²³Te, ¹²⁵Te, and others), which significantly correct non-relativistic results. They improve agreement with experiments and resolve discrepancies in absolute shielding results. These new treatments, which incorporate spin–orbit coupling and complex calculations, are essential for heavy elements, clarify large relativistic corrections, and complement the collapse of nonrelativistic relationships between nuclear spin-rotation constants (C_I) and shielding constants (σ).

2.4. Absolute Shielding Scales

Standard NMR spectra make it possible to measure the chemical shifts (δ), values expressed in ppm units against the reference, which is the liquid sample taken arbitrarily during the NMR development. Nevertheless, the more fundamental measure of electron screening effects is nuclear magnetic shielding (σ). The absolute shielding scale measures the local magnetic field at a nucleus, influenced by the nearby electron clouds. It comprises several results for a series of molecules with well-characterized shielding constants at room temperature (preferably 300 K) and for a single liquid reference substance. The absolute shielding scales utilize the magnetic shielding of at least one simple compound, for which the shielding constant is well documented and widely accepted [2,3,4]. For example, ¹H [9] and ¹³C [10] NMR shielding is well established for many small “isolated” molecules, and the CH₄ molecule is a primary reference. For other nuclei, new, usually complicated and bothersome theoretical studies are necessary, at least for a simple molecule considered as a reference. They should distinguish between intramolecular and intermolecular effects on the shielding tensor, depending on the external and induced fields at a given nucleus. They can often be compared with experimental results [2,11]. The liquid reference substance, e.g., TMS (tetramethylsilane), can be used in ¹H, ¹³C, and ²⁹Si resonance studies to evaluate results obtained in both gaseous and liquid states. For the determination of the nuclear shielding in the molecule on which the scale is based, it is possible to use different measurable quantities: spin precession frequencies, spin rotation constants, and/or nuclear relaxation times [3]. Gas-phase experiments are preferred because they eliminate strong intermolecular interactions that are always present in condensed phases. They are omitted entirely when gas-phase results are extrapolated to the zero-pressure limit. Using these methods, absolute shielding scales for ¹H, ¹³C, ¹⁵N, and ²⁹Si, as well as for other nuclei, were established. An alternative procedure for determining absolute shielding scales is based on the Ramsey–Flygare relationship [10], which, for heavy nuclei, requires relativistic corrections. The Ramsey–Flygare relationship enables the determination of nuclear shielding from the spin-rotation constant. Recent developments in absolute shielding scales for NMR spectroscopy and relativistic effects on nuclear magnetic shielding are discussed in Refs. [12,13].

3. Results

The previously studied gaseous systems are listed in

Table 1. Gas-phase NMR measurement systems for nuclear magnetic moments (NMMs) have been developed over the past 20 years.

NMR Spectra	N.M.M. Transfer	Molecules	References
1H and 2H(D)	1H → 2H(D)	HD	[14]
1H and 3H(T)	1H → 3H(T)	HT	[14]
19F and 10B	19F → 10B	BF3	[7]
19F and 3He	3He → 10B	BF3/3He
19F and 11B	19F → 11B	BF3
19F and 3He	3He → 11B	BF3/3He
1H and 13C	1H → 13C	13CH4	[15]
3He and 13C	1H → 13C	13CH4/3He
1H and 14N	1H → 14N	NH3	[6]
3He and 14N	3He → 14N	NH3/3He
1H and 15N	1H → 15N	NH3
3He and 15N	3He → 15N	NH3/3He
1H and 17O	1H → 17O	H2O	[5]
1H and 19F	1H → 19F	CH3F	[1]
1H and 29Si	1H → 29Si	SiH4	[16]
1H and 31P	1H → 31P	PH3	[17]
3He and 31P	3He → 31P	PH3/3He
3He and 33S	3He → 33S	SF6/3He	[18]
1H and 35/37Cl	1H → 35/37Cl	HCl	[8]
3He and 35/37Cl	3He → 35/37Cl	HCl/3He

.

Experimental values of nuclear moments for light nuclei, from the first rows of the Periodic Table, can be found in

Table 2. Nuclear magnetic dipole moments of light nuclei. The quoted uncertainty reflects the experimental and theoretical uncertainties.

Isotope	I(X)	µ(X)µN Ref. [19]	µ(X)µN	Reference
1H	1/2+	+2.792847351(9)	+2.79284734462(82)	Schneider et al./2017 [20]
2H	1+	+0.857438231(5)	+0.8574382335(22)	Puchalski et al./2015 [14]
3H	1/2+	+2.978962460(14)	+2.978962471(10)	Puchalski et al./2015 [14]
3He	1/2+	−2.12762531(3)	−2.1276253498(15)	Schneider et al./2022 [21]
10B	3+	+1.8004636(8)	+1.8004636(8)	Jackowski et al./2009 [7]
11B	3/2+	+2.688378(1)	+2.6883781(11)	Jackowski et al./2009 [7]
13C	1/2−	+0.702369(4)	+0.70236944(68)	Makulski et al./2011 [15]
14N	1+	+0.403573(2)	+0.40357368(7)	Makulski et al./2022 [6]
15N	1/2−	−0.2830569(14)	−0.28305791(3)	Makulski et al./2022 [6]
17O	5/2+	−1.893543(10)	−1.893553(2)	Makulski et al./2018 [5]
19F	1/2+	+2.628321(4)	+2.628321(13)	Jaszuński et al./2012 [1]
21Ne	3/2+	−0.66170(3)	−0.6617774(10)	Makulski et al./2020 [22]
29Si	1/2+	−0.555052(3)	−0.5550516(31)	Makulski et al./2006 [16]
31P	1/2+	+1.130925(5)	+1.1309246(50)	Lantto et al./2011 [17]
33S	3/2+	+0.64325(1)	+0.6432555(10)	Makulski/2022 [18]
35Cl	3/2+	+0.82170(2)	+0.821721(5)	Jaszuński et al./2013 [8]
37Cl	3/2+	+0.68400(1)	+0.683997(4)	Jaszuński et al./2013 [8]

. They belong to particular nuclei: hydrogen isotopes ^1/2/3H, ³He, ¹³C, ¹⁴N, ¹⁵N, ¹⁷O, ¹⁹F, ²¹Ne, ²⁹Si, ³¹P, ³³S and ^35/37Cl. All values are expressed in the units of the nuclear magneton, i.e., µ_N = 5.0507837393(16) × 10⁻²⁷ JT⁻¹. The nuclear magneton is a physical constant calculated as µ_N = eħ/2cm_p where p refers to the proton. The previously published results are in the third column of the table and are cited as Ref. [19]. The preferred results, with the highest accuracy and precision, are listed in the fourth column, along with their corresponding sources. The signs of nuclear magnetic moments (plus or minus) cannot be obtained from the usual NMR measurements. They were established using molecular beam magnetic resonance (the ABMR method), in which asymmetry in the absorption resonance curves upon introduction of an oscillating field can identify the quantum numbers and the sign of the nuclear moment. The spin quantum numbers, along with their parity, are also included. Hydrogen isotopes are notable for their excellent accuracy and high precision. ¹H and ²H isotopes can serve as useful reference points, as they are found in many volatile hydride compounds (HD, HT, CH₄, NH₃, H₂O, CH₃F, SiH₄, PH₃, HCl) and as lock channels in NMR apparatuses. Therefore, it is possible to move ¹H or ²H frequencies and diamagnetic correction factors (shielding constants) to other nuclides using the list of molecules shown above (Table 1).

Remember that the nuclear magnetic moments μ(X)μ_N (written also as µ_X) shown in Table 2 are the projection value of the full vector onto the measurement axis. In fact the total length of the nuclear magnetic moment vector is ${\mathsf{\mu }}_{\mathrm{X}}^{\mathrm{length}}=(\sqrt{\left({\mathsf{{\rm I}}}_{\mathsf{{\rm X}}}\right({\mathsf{{\rm I}}}_{\mathsf{{\rm X}}}+1)}/{\mathrm{I}}_{\mathrm{X}})/{\mathrm{\mu }}_{\mathrm{X}}$ larger than this projection, despite this µ_X is commonly referred to in the literature as the nuclear magnetic moment. Sometimes, physicists make use of the g factor of nuclei instead of nuclear magnetic moments expressed in nuclear magnetons. The g factor (g value) is a dimensionless quantity which characterizes the magnetic properties of nuclei and can be ciphered as g_X = µ_X/µ_NI_X. For proton (¹H) the g factor equals +2.79284734463(82)/1/2 = +5.5856946893(16). On the other hand, the gyromagnetic ratio represented by the γ symbol, which describes the ratio of a nuclear magnetic moment to its angular momentum, is closely related to the following g factor: γ_X = gnµ_N/ħ, where ħ is the h-bar. It can be expressed in SI units s⁻¹T⁻¹ or equivalently in MHzT⁻¹. The proton gyromagnetic ratio is γ(¹H) is 2.6752218708(11)·10⁸ s⁻¹T⁻¹ or 42.577478461(18) MHz/T. The gyromagnetic ratio of a given nucleus plays an important role in both NMR and MRI spectroscopic methods. These three constants µ(X), g(X) and γ(X), which characterize the magnetic properties of nuclei are utilized alternatively in different branches of chemistry and physics (see Figure 2). The appropriate analogs of nuclear moments one can find in Appendix A.

4. Discussion

The dipole magnetic moment of a proton (¹H) is a fundamental constant in nuclear physics, with far-reaching consequences for chemistry, physics, astrophysics, and quantum theory. It has been measured many times before. Recently, a direct, precise measurement was performed using a double Penning trap system by physicists at Mainz University [20]. The experiment was conducted in two parts: the first step included isolation of a single proton in the magnetic trap, and the second step involved measurements of two frequencies—the spin-precession Larmor frequency and the cyclotron frequency of the proton in the magnetic field. Three individual protons were used, and the experiment lasted approximately 4 months, including preparatory work. The μ(¹H) result published in Science in 2017 was measured at 2.79284734463(82) µ_N, with an accuracy of 0.3 parts per billion. Deuteron (²H/D) and triton (³H/T) nuclear magnetic moments were determined from measurements of frequency ratios in HD and HT molecules, supplemented by precise calculations of the shielding effects of H, D, and T nuclei [14]. The shielding corrections for deuterated and tritiated nuclei in HD and HT molecules are minimal: σ(D) = 26.372801(1) ppm (parts per million) and σ(T) = 26.391456(2) ppm. The operation of ³H(T) is more demanding due to its radioactivity. Triton has a long half-life of 12.32 years and can be taken as gaseous HT. The source of gaseous tritium gas (HT) can be titrated water (diluted HTO in excess of H₂O). Fortunately, tritium atoms decay by beta emission of low-energy electrons (average 5.7 keV), which can penetrate only 6 cm of air and cannot escape from NMR test tubes. Careful manipulation of samples remains necessary.

Helium-3 was investigated in a Penning trap to directly measure the nuclear g-factor of the ³He+ ion, shielded by only a single electron. The nuclear magnetic moment of the ³He bare nucleus can be obtained by diamagnetic correction of 35.50738(3) ppm in He⁺ ion [21]. The final result is µ(³He) = −2.1276253498(15) µ_N, with an accuracy of 0.7 parts per billion. The terrestrial ³He gas can be supplied by some chemical trading companies (e.g., Chemgas, Isotec) at low pressures up to ~2.5 atm. It is beneficial that helium-3 can be mixed with appropriate gaseous substances and measured in conventional NMR spectrometers (see, for example: ³He/H₂, ³He/NH₃, ³He/CO, and ³He/CH₄ and other mixtures) [18]. Analysis of these systems yields nuclear magnetic moments for ¹⁰B, ¹¹B, ¹³C, ¹⁴N, ¹⁵N, ³¹P, and ³³S.

The ¹³C nuclear magnetic moment was calculated in the enriched ¹³CH₄ molecule against ¹H parameters where ν₀(¹³C)/ν₀(¹H) is 0.2514473143(66) using shielding correction σ₀(¹³C) = 195.01(90) ppm [15]. Confirmation of this result was achieved using helium-3 parameters, where ν₀(¹³C)/ν₀(³He) in the zero-pressure limit is 0.3300743617(61). The final results are in excellent agreement with each other. Analogously, the appropriate frequency ratios in the NH₃ molecule are as follows: ν₀(¹⁴N)/ν₀(¹H) = 0.0722342561(4) and ν₀(¹⁵N)/ν₀(¹H) = 0.101327121(1). These numbers were corrected using new relativistic shielding constants for the nitrogen nucleus in ammonia, σ₀(^14/15N) = 266.78 ppm [6] with an error of less than 1 ppm. ³He spectroscopic data confirm these final results for the magnetic moments. Similarly, accurate calculations were made in H₂O (σ₀(¹⁷O) = 328.4 ± 0.3) ppm) and CH₃F (σ₀(¹⁹F) = 470.85 ± 5 ppm) molecules to achieve ¹⁷O [5] and ¹⁹F [1] nuclear magnetic moments. ¹⁷O-enriched water was dispersed in pure gases (Kr, Xe, CH₃F, and CF₃H), and ¹H and ¹⁷O NMR spectra were recorded. The most accurate calculations of magnetic shielding were performed in H₂O, the smallest oxygen-containing molecule suitable for an NMR reference procedure. On the other hand, the ¹H and ¹⁹F NMR spectra were measured in a pure CH₃F molecule.

In the case of the ²⁹Si nucleus, the stable silane, the silicon analog of methane, was employed. The measured frequency ratio is ν₀(²⁹Si)/ν₀(¹H) = 0.198650063(1) [16]. The main error in the magnetic moment arises from the diamagnetic correction applied to the ²⁹Si shielding value. It can be as large as 2 ppm, depending on the precision of the theoretical calculations; σ₀(²⁹SiH₄) = 482.85 ± 2 ppm. Phosphine was the simplest phosphorus hydride compound used for gaseous experiments and NMR measurements of ¹H and ³¹P NMR spectra. The glass samples containing ³He/phosphine mixtures were prepared with great care due to PH₃’s extremely poisonous, flammable, and explosive properties. The frequency ratios here are as follows: ν₀(³¹P/ν₀(¹H) = 0.404698969 and ν₀(³¹P)/ν₀(³He) = 0.5312482456. The diamagnetic corrections were computed theoretically in absolute shielding units: σ₀(³¹P) = 614.758 ± 4 ppm and σ₀(¹H) = 29.305 ± 0.2 ppm [17].

³³S is a quadrupolar nucleus, and consequently, its NMR signal widths are often large, depending on the chemical symmetry occupied by the nucleus in molecules. Short relaxation times in the gas phase also broadened signals for simple compounds such as SO₂, SO₃, H₂S, and CS₂. Fortunately, sulfur hexafluoride (SF₆), with a central sulfur atom bonded to six fluorine atoms, has an octahedral geometry. SF₆ is a very stable gaseous substance. Several samples of gaseous ³He/SF₆ mixtures were prepared and analyzed by ³He, ¹⁹F, and ³³S NMR spectroscopy. The ratio of frequencies ν₀(³³S)/ν₀(³He) = 0.10074480(17) were measured in the zero-pressure limit. The new nuclear magnetic moment of ³³S is generally consistent with previous results but is substantially more accurate. Recalculated shielding constant σ₀(³³S) in the SF₆ molecule using Equation (4) against ¹⁹F parameters is 396.3 ± 5 ppm [18] in good agreement with σ₀(³³S) = 392.6 ppm from theoretical calculations [23].

The last case of hydride compounds discussed here is hydrogen chloride, a gaseous substance in a mixture with helium-3, which was studied using ¹H, ³He, ³⁵Cl, and ³⁷Cl NMR measurements [8]. The appropriate frequency ratios at the zero-pressure limit are as follows: ν₀(³⁵Cl)/ν₀(¹H) = 0.0979818(4), ν₀(³⁵Cl)/ν₀(³He) = 0.1286204(5), ν₀(³⁷Cl)/ν₀(¹H) = 0.0815596(9), and ν₀(³⁷Cl)/ν₀(³He) = 0.1070630(1). The best ab initio calculations of shielding constants in an isolated HCl molecule give σ₀(^35/37Cl) = 976.20 ± 5 ppm, irrespective of chlorine isotope, and σ₀(¹H) = 31.403 ± 0.5 ppm. These were used to calculate µ(³⁵Cl) and µ(³⁷Cl) of bare nuclei.

Unfortunately, boron forms only one stable gaseous hydride, diborane (B₂H₆), which is very reactive with moisture and oxygen and highly toxic. We therefore decided to use the fluorinated gas BF₃, which is safer to handle and commercially available [7]. A reference nucleus is now the ¹⁹F nuclear magnetic moment. The experimental frequency ratios here are: ν₀(¹¹B)/ν(³He) = 0.42117005(4), ν₀(¹⁰B)/ν₀(³He) = 0.141033238(4), ν₀(¹¹B)/ν₀(¹⁹F) = 0.341028031(5), and ν₀(¹⁰B)/ν₀(¹⁹F) = 0.114196831(4). The diamagnetic corrections of boron nuclei are as follows: σ(¹¹B) = 97.882 ± 3 ppm and σ(¹⁰B) = 97.879 ± 3 ppm, suggesting a small isotope effect. The ¹⁹F diamagnetic corrections are here even larger: σ(¹⁰B¹⁹F₃) = 331.95 ± 4 ppm and σ(¹⁰B¹⁹F₃) = 332.01 ± 4 ppm. The accuracy of the nonrelativistic results for the equilibrium geometry largely determines the accuracy of these values and the estimated error bars.

It is known that neon does not form stable compounds at standard conditions. Because of it, the ²¹Ne moment was established from the ¹H moment and the frequency ratio ν(²¹Ne)/ν(¹H) = 0.07894287214(35) measured in gaseous neon against the proton resonance frequency of the residual signal of benzene-d₆ used in the lock system [22]. Diamagnetic correction applied for the neon nucleus in the neon atom is assumed as σ₀(²¹Ne) = 557.11 ± 1.5 ppm [24].

The most abundant isotopes of argon, consisting of 18 protons and a different number of neutrons, are ⁴⁰Ar, ³⁸Ar, and ³⁶Ar. None of them possesses a magnetic moment and a non-zero spin number. In fact, argon is a unique case in the Periodic Table where contemporary NMR spectrometers have recorded no NMR spectra. However, spectroscopic measurements were carried out for ³⁹Ar nuclei [25], with nuclear spin I = 7/2 and magnetic moment µ(³⁹Ar) = −1.590(5)µ_N, employing optical spectroscopy and a Fabry-Perrot interferometer with photoelectric detection. The ³⁹Ar nucleus is long-lived, with a half-life of 268(8) years; its samples are radioactive. They can be produced in the ³⁹K(n,p)³⁹Ar reaction. The quantity ratio of the ³⁹Ar/Ar was established as ≤4·10⁻¹⁷. The nuclear magnetic moments of other short-lived argon isotopes were measured in fast-beam neutral-atom experiments, with reference to the NMR measurement of ³⁷Ar [26]. The internal structure of even-even argon isotopes was studied using nuclear magnetic moments within the shell model [27].

Interestingly, specific gas phase experiments were performed for heavy nuclei: ⁸³Kr (in gaseous krypton) [28], ¹²⁹Xe and ¹³¹Xe (in gaseous xenon) [29], ¹⁸³W (for WF₆ in gaseous CF₄) [30], ¹¹⁷Sn and ¹¹⁹Sn (for SnMe₄ in gaseous CO₂ and N₂O) [31] and ²⁰⁷Pb (for PbMe₄ in gaseous Xe and SF₆) [32]. Obviously, the NMR method can be applied to light and heavy nuclides, which are observed primarily in liquid solutions. Good examples are here: ⁶Li and ⁷Li (LiCl and LiNO₃ water solutions) [33,34], ²³Na (NaCl, NaNO₃, NaClO₄ water solutions) [35], ³⁹K (KCl and KNO₃ water solutions) [36], ⁷⁷Se (SeMe₂ molecule) [37], ⁷⁹Br and ⁸¹Br (KBr · Kryptofix 222 complex) [38], ⁷⁵As (Na₃AsO₄) [39], ⁸⁵Rb and ⁸⁷Rb (RbCl and RbNO₃ water solutions) [40], ¹²¹Sb, ¹²³Sb (KSBF₆) [39], ¹²³Te and ¹²⁵Te (TeMe₂ molecule) [37], ¹³³Cs (CsF, CsCl and CsNO₃ water solutions) [41] and finally ²⁰⁹Bi (Bi(NO₃)₃ and Bi(ClO₄)₃ water solutions) [42,43]. Most of the papers listed above were published in the past ten years, underscoring the importance and status of investigations of nuclear magnetic moments.

All experimental results mentioned above and preferred today are included in Table 2. The frequency ratios discussed above can be helpful for future recalculations if only better diamagnetic corrections become available. When shielding corrections are denoted with a subscript 0 (σ₀), they refer to a parameter for a single isolated atom or molecule. For convenience, the relevant nuclear magnetic moments μ(X) = hγ_XI_X = µ_Ng_XI_X in nuclear magnetons (μ_N) were recalculated to full-length analogs μ(X)^length, g(X) factors, and gyromagnetic factors γ(X), which can be found in Appendix A 

Table A1. Different values of magnetic properties for light nuclei from ¹H up to ³⁷Cl.

Isotope	μlength/μN	gX	γ(X) × 107 s−1 T−1	γ(X)/2π MHz T−1
1H	+4.837353498(1)	+5.585694689(2)	+26.752218759(8)	42.57747854(1)
2H	+1.212600779(3)	+0.857438234(2)	+4.10662889(1)	6.53590288(2)
3H	+5.159714352(2)	+5.95792494(2)	+28.53498451(1)	45.4148384(2)
3He	+3.685155205(3)	+4.255250700(3)	+20.38016826(1)	+32.43604519(2)
10B	+2.0789963(9)	+0.6001545(3)	+2.8743899(1)	+4.574734(2)
11B	+3.470681(1)	+1.7922521(7)	+8.583842(4)	+13.661609(6)
13C	+1.216540(1)	+1.404739(1)	+6.727879(7)	+10.70775(1)
14N	+0.5707394(1)	+0.40357368(7)	+1.9328824(3)	+3.076278(1)
15N	−0.49027068(5)	−0.56611582(6)	−2.711364(3)	−4.3152706(5)
17O	−2.240482(2)	−0.7574212(8)	−3.627605(4)	−5.773512(6)
19F	+4.55241(2)	+5.25667(2)	+25.1764(1)	+40.0695(2)
21Ne	−0.854351(1)	−0.4411849(7)	−2.113018(3)	−3.362973(5)
29Si	−0.961378(2)	−1.110103(2)	−5.31675(1)	−8.46187(2)
31P	+1.958819(8)	+2.26185(1)	+10.83294(5)	+17.241156(8)
33S	+0.830439(2)	+0.428837(1)	+2.053879(3)	+0.32688500(5)
35Cl	+1.060837(7)	+0.547814(3)	+2.62371(2)	+4.17576(3)
37Cl	+0.883036(5)	+0.455998(3)	+2.18396(1)	+3.47589(2)

. The appropriate correction factors for μ(X)^length can be found in Appendix A,

Table A2. Multipliers for the calculation of full length for the nuclear magnetic vector used in Appendix A Table A1.

IX	Nucleus Example	$\sqrt{\left({\mathsf{{\rm I}}}_{\mathsf{{\rm X}}}\right({\mathsf{{\rm I}}}_{\mathsf{{\rm X}}}+1)}/{\mathbf{I}}_{\mathbf{X}}$
1/2	1H, 3H, 3He, 13C, 15N, 19F, 29Si, 31P	1.7320508076
1	2H, 14N	1.4142135624
3/2	11B, 21Ne, 33S, 35Cl, 37Cl	1.2909944487
2	204Tl	1.2247448714
5/2	17O	1.1832159566
3	10B	1.1547005384
7/2	39Ar, 45Sc, 49Ti, 51V	1.1338934190
4	40K	1.1180339887
9/2	73Ge, 83Kr	1.1055415968

.

5. Conclusions

NMR experiments conducted over the past 20 years in the gaseous state have successfully probed various nuclear magnetic moments. Gases are well suited to this procedure because they enable the acquisition of NMR parameters from “isolated” molecules. Such molecules are not burdened by intermolecular interactions, which can be calculated relatively easily and accurately using quantum-mechanical methods. Experimental frequencies can be obtained by measuring several gaseous samples at different pressures and extrapolating to the zero-pressure limit. For low-density gases, the use of enriched material is strongly recommended. This method is not limited to the gas phase but can, under certain conditions, be extended to liquids and solutions. Measurements of nuclear magnetic moments are among the most accurate in physics and chemistry. The precision of final results depends on so-called “diamagnetic corrections,” which are more critical for heavier isotopes than for lighter ones. Therefore, theoretical advances in this field are crucial for establishing accurate nuclear magnetic moments. For the light nuclides discussed in this work, discrepancies between results (¹H–²¹Ne) corrected using different shielding constants are typically below 0.0005%. The diamagnetic corrections rise in the series of nuclides ¹H, ³He, ^10/11B, ¹³C, ^14/15N, ¹⁷O, ¹⁹Ne, ²¹Si, ³³S, up to ^35/37Cl. The same systematic increase in errors is observed in their accuracy. In this light, the hydride substances or gaseous helium-3 are more valuable reference substances than fluorine compounds. In the case of ¹²³Sb, this error can be in the order of 0.01% of the magnetic moment and even higher for very heavy nuclei.

The best-known nuclear magnetic moment is that of the proton (¹H) nucleus, +2.79284734463(82)μ_N. The analogous magnetic moment was estimated for the short-lived antiproton particle. A unique apparatus, using the two-particle spectroscopy method in an advanced cryogenic multi-Penning trap system, employed at CERN, was also used for the measurement of the antiproton magnetic moment with a fractional precision of 1.5 ppb (parts per billion). In the experiment, the traps are cooled to very low temperatures using cryocoolers, thereby reducing thermal noise and surface outgassing, improving the low vacuum level. Multiple traps work together to form a reservoir trap to prepare, cool, and measure ions efficiently. Superconducting magnets generate strong, uniform magnetic fields, which are essential for confining charged particles in Penning traps. The apparatus is equipped with highly sensitive cryogenic detectors, such as silicon photomultipliers, integrated directly into the cold-trap system for efficient fluorescence detection. Finally, experiments indicate that the magnetic moments of the proton and antiproton are nearly equal [44,45]. The CPT (Charge-Parity Time) theorem is then satisfied, meaning that symmetry holds between particles and their antiparticles, which have identical masses, lifetimes, and magnetic moments. It is worth remembering that magnetic dipole moments are only one of several nuclear parameters, including size, shape, charge, electric quadrupole moment, and, for unstable nuclei, decay mode and half-life. All of them serve as central points in the interpretation of nuclear structure in physical theories of matter. The history of research on nuclear magnetic moments in the light of Gamow-Teller transitions was summarized [46], where several nuclear phenomena—configuration mixing, meson exchange currents, and relativistic effects are examined.

A universal, effective calculation method for predicting nuclear magnetic moments does not exist. Precise theoretical calculations of nuclear moments in light nuclei using lattice QCD (Quantum Chromodynamics) are subject to several uncertainties arising from lattice artifacts and contamination from excited states [47]. The best results show at least a slight discrepancy between theory and experiment. Recent advances in theoretical studies of nuclear magnetic moments are reviewed in Refs. [48,49]. We can say that modern chemical procedures play an essential role in the numerical determination of nuclear moments [50,51]. Still, the benefits accrue exclusively to the fields of nuclear physics, radiochemistry, and astrophysics. Finally, currently available data indicate that nuclear magnetic moments are more experimentally determined than theoretically predicted, and even recently developed theoretical models remain approximate. Nevertheless, accurate nuclear magnetic moments can serve as good references for quantum-mechanical methods that provide the magnetic properties of particular nuclei. For specific nuclear cases, we recommend an online database of nuclear electromagnetic moments accessible via a web browser [52,53]. Recently, N.J. Stone published a short paper describing developments in the determination of nuclear magnetic and electric quadrupole moments not only of stable nuclei but also of short-lived states (<1ms) [54]. In summary, he noted that over the past thirty years, interest in the subject referred to in the title of this paper (“Nuclear moments: recent developments”) is still great and continuous. We refer the reader to this interesting publication.