Using the Intramolecular Contribution to the Second Moment of NMR Line Shape to Detect Site Symmetry Breakdown in Molecular Crystals

A new approach to simulating the intramolecular contribution to the anisotropic secondmoment 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 q α , the dynamic weights of irreducible representations of the HMM crystallographic point symmetry group. The application of the theory to describing the intraionic second moment of the proton NMR spectral line in monocrystalline ammonium chloride proves the tetragonal distorted tetrahedral site symmetry of ammonium ions.


Introduction
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 [1].As the magnitude of the second moment depends rigorously on geometrical and physical peculiarities of nuclear motion, measuring the second moment allows one to test various models of HMM taking place in condensed matter.
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) [2,3].Using RDM shown to be very successful by describing HMM happened in liquids [4][5][6][7].FAJM works sufficiently well in isotropic media of powders [8][9][10].At last, the EAJM is appropriated for exploring the HMM of symmetrical molecules in any condensed molecular media: single crystals, powders, and liquids.It was fruitfully applied to discuss the broadened lines of incoherent scattering of neutrons, dielectric and infrared absorption, Raman scattering of light, and the rates of nuclear magnetic relaxation in mono-and polycrystalline molecular media [2,3].This paper is devoted to expanding EAJM approach to simulate the second moment of NMR-absorption line broadened due to the magnetic dipole-dipole interaction in condensed molecular media.It is dedicated also to showing that the intramolecular part of the second moment is sensitive to symmetry breakdown effect in the HMM.As an application, we revise the experimental data of Bersohn and Gutowsky concerning the 2nd moment of the proton spectral line in the single crystal of ammonium chloride [11].A new interpretation of the data follows the tetragonal distorted site symmetry of NH 4 + ions in NH 4 Cl that firstly has been determined by the proton relaxation experiments [12].

Theory
According to Van Vleck, the contribution of identical resonant nuclei to the second moment of NMR spectral line 2 International Journal of Spectroscopy 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 [1]: where the index  labels a target resonant nucleus of a chosen molecule and the index j other nuclei of the substance;   and   are, respectively, the gyromagnetic ratio and the quantum number of the nuclei; ℏ is Planck's constant divided by 2 and   is the polar angle formed by the internuclear vector r  with the induction vector of the stationary magnetic field B 0 determined in the laboratory reference frame (LRF).
For convenience, we shall replace (3cos 2   − 1) 2 by (16/5)| (2)  0 (  )| 2 = (16/5)| (2)  0 (  ,   )| 2 , where  (2)  0 (  ) =  (2)  0 (  ,   ) is the zeroth element of normalized spherical harmonic of the second order, the zeroth component of the unit spherical tensor of the 2nd rank, which depends on the polar angle   , but it has no dependence of the azimuth angle   .Moreover, we shall rewrite (1) in terms of spherical tensor components as By the way, the angles   and   relate LRF whereas the internuclear vectors are fixed in the crystallographic reference frame (CRF).Taking into account the transformation rule of spherical tensors by a rotation to the three-dimensional Euler angles (, , ) from CRF to LRF given by the expression where  (2)  0 (, , ) is an element of Wigner matrix, we can express  () 2 in the form Replacing  (2)  0 (, , ) = √(4/5) (2)   (, ) allows us to reduce (4) to where (, ) and (  ,   ) are, respectively, the spherical angles of the vector B 0 and the internuclear vector r  , determined with respect to CRF.At sufficiently low temperatures, a regime of rigid lattice is valid in molecular crystals.At high temperatures (T > 40 K), the direction of some internuclear vectors changes accidentally due to classical rotational displacement.Consequently, angles (  ,   ) become random functions of time.Such motion changes a spectral distribution law of NMR signals.Therefore, the simulation of a second moment for molecular substances has to be performed by accounting for intramolecular nuclei as well as intermolecular ones: The term  (inter) 2 denoted by the label "(inter)" is the intermolecular part of  () 2 .Its calculation is out of the interest of this paper.
The other term of the sum ( 6) delivered by nuclei inner with respect to the given molecule.It is convenient to present  (intra) 2 in the form [5,13,14] where  is the number of identical resonant nuclear spins in the considered molecule and is the nonnormalized spectral density function of the autocorrelation function: In its turn, the function  (0)  () is the coordinate part of spinspin interaction Hamiltonian of nuclei  and , given by or The integration is prescribed within the limits of the double line width from −] up to +] in (7).By the way, ] = 1/T 2 , where T 2 is the spin-spin relaxation time,  = 2] =    0 is the resonance angular frequency of spins ,   is their gyromagnetic ratio, and  0 is the module of induction vector of the static magnetic field.The outcome of calculus by using ( 7)- (10) depends on the physical model of molecular motion.According to the goal of the present study, we shall apply EAJM approach [2,3] as the model of HMM, within the framework of which an analytical expression of the normalized spectral density function (SDF)  (2)  0 () of zero component of the 2nd rank tensor  (2)  0 [(), ()] can be written for a single crystal by In agreement with the EAJM approach, polar angle   of the internuclear vector r  does not present explicitly in (11).Instead, there are two spherical angles  and , which fix the orientation of main axis of the crystallographic reference frame in the laboratory one.Index  labels irreducible representations (IR) Γ  of the molecule motion point symmetry group G and  labels an intermediate summation.
Taking into consideration (11), nonnormalized SDF takes the form The rated expressions of the factors  0 () are tabulated explicitly as a function of azimuth angle  for all crystallographic point symmetry groups of pure rotation in [2,3].The quantities   are the adjustable parameters of the theory called the dynamic weights of IR Γ  .They characterize the symmetry distortion and have to be experimentally determined.Parameter   is a correlation time adapted to IR Γ  that relates the expression: where   and   are the characters of th and identical classes, respectively,   are probabilities of fundamental acts of the motion appropriate to th class of the group G, and  is a mean time between two consequent steps of motion.The time  is ordered to Arrhenius low: where   is the height of activation energy barrier averaged on the HMM motion symmetry group and  0 is a mean time between two sequential attempts to overlap the barrier.By substituting (12) in (7), we shall obtain Performing just prescribed integration, we shall finally get the second moment expression specified by intramolecular action of spins  to spin  for a given single crystal in an analytical form: For a powder, averaging ( 16) over the angles  (0 ≤  ≤ 2) and  (0 ≤  ≤ ) reduces it to In a regime of the fast thermal molecular motion, the inequality 2 ⋅ ] ⋅   ≪ 1 is valid that follows the equality arctg(2 ⋅ ] ⋅   ) = 0.In this case the dipoledipole contribution to the second moment disappears; that is,  (intra) 2 = 0, and so-called phenomenon of "spectral line narrowing" or "bandwidth narrowing" is observed.
It should be noted that in case the molecular motion is frozen, the alternative inequality 2 ⋅ ] ⋅   ≫ 1 followed by arctg(2 ⋅ ] ⋅   ) = /2 is valid.Furthermore, taking into consideration that the dynamic weights of irreducible representations are normalized to unity, namely, ∑    = 1, (17) reduces down to a permanent value: presented earlier by Abragam and Lösche [15,16].For monocrystalline samples, (16) reduces to an expression maintaining the angular dependence in the second moment: It is noted that in the case of not resonant nuclear spins contribute to the 2nd moment, (19) has to be corrected by the factor 4/9 [17].Labelling not resonant nuclear spins by the index   , this contribution will be expressed in the regime of slow molecular motion as (20)

Application to Monocrystalline Ammonium Chloride
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 bsbl.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  Cl-Cl =  N-N = (3.844± 0.024) ⋅ 10 −10 m, four nearest protons are placed in the vicinity of a nitrogen nucleus at the distance  N-H = (1.038± 0.04) ⋅ 10 −10 m, and the neighboring protons are spaced from each other to  H-H = (1.695± 0.04) ⋅ 10 −10 m [18].
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 NH 4 + tetrahedrons with equal probability in the disordered phase observed at temperatures  ≥ 242.9 K. Hence, any physical quantity obeys the point symmetry group of the octahedron O in the disordered phase of NH 4 Cl.
It was found that both the shape of spectral line [11] and the relaxation times [12] are anisotropic and show temperature dependence in the single crystal of NH 4 Cl.To discuss the continuous wave data the quantum mechanical technique was applied [11].The relaxation data were discussed in the framework of EAJM approach taken as the model of classical HMM of NH 4 + cation [12].To examine the site symmetry of the ion NH 4 + in the regime of its slow motion by using the second moment of the proton NMR spectral line in NH 4 Cl, we have to deal with two contributions:  (intraHH) 2 and  (intraNH)

2
. The first contribution is induced by interaction of an arbitrary chosen proton with other three protons of a chosen cation (19) and the second contribution-with inner nitrogen nucleus (20).
By using   =  H = 1/2 as the value of proton spin, ( − 1) = 3 as the number of adjacent protons,   =  H , and    =  HH , we shall get from (19) the rated expression of Substituting    = 1,    =  N , and     =  NH in (20) follows the rated expression of  (intraNH)

2
, the intraionic contribution to the second moment of proton NMR spectral line in monocrystalline ammonium chloride, prescribed by (26) allows us to calculate the numbering values of dynamic weights  1 and  2 by using experimental values of  (intra) 2 .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 (B 0 ‖ [1, 0, 0]) equal to 36.5 Gs 2 and along the twofold symmetry axis (B 0 ‖ [1, 1, 0])-54.6 Gs 2 [11].The respective high temperature values were equal to 6.6 Gs 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  (intra,exp) 2 equal to 29.9 Gs 2 and 52 Gs 2 for directions of magnetic field induction B 0 ‖ [1, 0, 0] and B 0 ‖ [1, 1, 0], respectively.
By substituting in (26) firstly  (intra,exp) 2 = 29.9Gs 2 ,  = /4, and  = 0 and then  (intra,exp) 2 = 52 Gs 2 ,  = /4, and  = /2, we get the experimental values of dynamic weights  1 and  2 :  1 = 0.254 and  2 = 0.757.These values of dynamic weights of two-dimensional and three-dimensional irreducible representations for tetrahedron point symmetry group of ammonium ion motion in the ordered phase of NH 4 Cl are in good agreement with our data  1 = 0.25 and  2 = 0.73 derived from proton NMR relaxation experiments earlier [12].It results in the tetragonal distorted tetrahedral site symmetry of ammonium ions in the ordered phase of NH 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,  1 and  2 take respective values equal to 0.4 and 0.6.
A surface-plot graph of the intraionic contribution

Conclusion
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.

(intra) 2 to 4 + 2 /
the proton NMR second moment in ammonium chloride single crystal for a slow motion regime of NH cation is shown as a function of spherical angles  and  of the crystal orientation in the static magnetic field for the experimental values of dynamic weights  1 = 0.254 and  2 = 0.757 in Figure1.In Figure2, the graph of  (intra) 2 is shown for  = /4 as a function of the polar angle , which forms the induction vector B 0 of the static magnetic field with respect to the direction [1, 0, 0] of the unit cell main axis.Here, one can see the values of  (intra) 2 for three main orientations of the external magnetic field in the unit cell of the NH 4 Cl crystal:B 0 ‖ [1, 0, 0], B 0 ‖ [1, 1, 1], and B 0 ‖ [1, 1, 0].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 (23) and (24) and accounting for  1 = 0.254 and  2 = 0.757 are shown in Figure3.It is noted that they are spacelike in relation (intraHH) 1,  2 , , ), the function of the dynamic weights  1 and  2 and the angles  and , is equal toΩ ( 1 ,  2 , , ) = { 1 [(1 + 3cos 2 2) − 6 (1 + cos 2 2) cos 2  + 3 (3 + cos 2 2) cos 4 ]