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A new approach to simulating the intramolecular contribution to the anisotropic second moment of NMR spectral lines broadened by magnetic dipole-dipole interaction of nuclei is suggested. The extended angular jump model is used by approximating the local hindered molecular motion (HMM). The theoretical result allow describing the site symmetry distortion by new experimental parameters

The second moment of the NMR spectral line broadened due to the magnetic dipole-dipole interaction has been yielded by Van Vleck as one of the main probes of hindered molecular motion (HMM) in crystalline substances [

At present, three principal model diversities of local HMM exist: the rotational diffusion model (RDM), the fixed angular jump model (FAJM), and the extended angular jump model (EAJM) [_{4}Cl that firstly has been determined by the proton relaxation experiments [

According to Van Vleck, the contribution of identical resonant nuclei to the second moment of NMR spectral line broadened owing to their magnetic dipole-dipole interaction with resonant and with nonresonant nuclei in a crystal lattice, can be calculated by using a well-known expression [

For convenience, we shall replace

At sufficiently low temperatures, a regime of rigid lattice is valid in molecular crystals. At high temperatures (

The term

The other term of the sum (_{2}, where_{2} is the spin-spin relaxation time, _{0} is the module of induction vector of the static magnetic field.

The outcome of calculus by using (

In agreement with the EAJM approach, polar angle

Taking into consideration (

The rated expressions of the factors

By substituting (

In a regime of the fast thermal molecular motion, the inequality

It should be noted that in case the molecular motion is frozen, the alternative inequality

It is noted that in the case of not resonant nuclear spins contribute to the 2nd moment, (

Ammonium chloride NH_{4}Cl, being one of the most studied substances, is frequently used as a touchstone of the validity of various theories on the structure and physical properties of crystals. It is an ionic crystal, at which ammonium ions exhibit random reorientation and consequently they have no orientation ordering. Unit cell of NH_{4}Cl is a body-centered cube of type СsСl. At the center of a cubic cell, a tetrahedron of ammonium cation is placed and its corners are occupied by anions of chlorine. The lattice parameter is

Below 242.9 K, the crystal is in its ordered phase. The random local motion of ammonium ions does not change the ordered structure of the crystal. It means that the reorientation symmetry group of any ammonium ion vector is the point symmetry group of tetrahedron T.

There are two equilibrium dispositions of _{4}Cl.

It was found that both the shape of spectral line [_{4}Cl. To discuss the continuous wave data the quantum mechanical technique was applied [

To examine the site symmetry of the ion _{4}Cl, we have to deal with two contributions:

By using

Using the table data, ^{−1}Gs^{−1}, ^{−1}Gs^{−1}, and

The theoretical dependence of

The mean value of the second moment at low temperature (−195°C) that is in the ordered phase has been determined for direction of the induction vector of the external static magnetic field chosen along the fourfold symmetry axis in the unit cell of NH_{4}Cl (^{2} and along the twofold symmetry axis (^{2} [^{2} and 2.6 Gs^{2}. Because of the fact that the intraionic part of the second moment decreases down to zero at high temperatures, we suggest that the respective values of the second moment consist of the interionic parts only. Taking into account just presented point of view about the experimental second moments we can assume that the difference between low and high temperature values of the total second moment can be considered as the pure intraionic parts for low temperature second moments. Performing the appropriate calculations, we get the experimental values of the intraionic second moments ^{2} and 52 Gs^{2} for directions of magnetic field induction

By substituting in (^{2}, ^{2}, _{4}Cl are in good agreement with our data _{4}Cl. It is noted that no anisotropy is predicted neither for the intraionic second moment no for the spin-lattice relaxation rates in the ideal cubic symmetry position. In this case,

A surface-plot graph of the intraionic contribution _{4}Cl crystal:

The angular dependence of intraionic contribution to the proton NMR spectral line second moment in ammonium chloride single crystal drawn according to (

Angular dependence of the intraionic contribution to the proton NMR spectral line second moment in ammonium chloride single crystal drawn according to (^{2} (^{2} (

Estimation of comparative contributions of the hydrogen spins versus nitrogen one to the intraionic 2nd moment is represented as interesting. The graphs of those drawn according to formulas (

Surface-plot graphs of

The presented approach of simulating the intramolecular contribution to the second moment of NMR spectral line allows one to perform an accurate analysis of dynamics and geometry of internal motion. It requires using the extended angular jump model by approximating the hindered molecular motion, knowledge of structure data for the studied crystal, and point symmetry property of the molecule motion as well as that of its surroundings, inclusively. Alternatively, by examining the anisotropic behavior of the second moment, this theory allows one to obtain the precise structure knowledge.

The authors declare that there is no conflict of interests regarding the publication of this paper.

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