Orientation dependence of the resonance frequency

From the Hamilton operator (3.1.22), the general form of dependence of the resonance frequency on the orientation of the magnetic field in the principal axes system of the coupling tensor is calculated to be 

Here i2 denotes the angular-dependent resonance frequency centred at the isotropic mean value or the Larmor frequency tat- 

In powders and polycrystalline materials all orientations arise with equal probability. For p = 0 the powder average of the resonance frequency (3.1.23) leads to a spectrum (Fig. 3.1.3(a)) with the lineshape function 

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If the orientation dependence of the resonance frequency of a spin 5 is determined by just one interaction, it can be exploited for use as a protractor to measure angles of molecular orientation. In powders and materials with partial molecular orientation, the orientation angles and, therefore, the resonance frequencies are distributed over a range of values. This leads to the so-called wideline spectra. From the lineshape, the orientational distribution function of the molecules can be obtained. These lineshapes need to be discriminated from temperature-dependent changes of the lineshape which result from slow molecular reorientation on the timescale of the inverse width of the wideline spectrum. The lineshapes of wideline spectra, therefore, provide information about molecular order as well as about the type and the timescale of slow molecular motion in solids [Sch9, Spil]. 

Anisotropic Interactions The Orientation Dependence of the Resonance Frequency [4]... 

Spin interactions that can be exploited to obtain information on liquid crystals are chemical shifts, magnetic dipole-dipole interactions between spins, and, for nuclei with spins /> 1, quadrupole interactions between the electric quadrupole moment of the nucleus and the electric field gradient at the site of the nucleus. The anisotropy of all these spin interactions leads to an orientation dependence of the resonance frequency v(or the corresponding angular frequency =271 v), given by... 

The NMR lineshape of solids is determined by the quadrupolar interaction, which can be described by two parameters the quadrupole frequency, o Q, and the asymmetry parameter, t/ (19,20). The parameter q)q is determined by the electric quadrupole moment of the deuteron and the zz component of the electric field gradient at the deuteron site. For deuterons bonded to carbon atoms, the asymmetry parameter is approximately zero and the z axis is along the C—D bond. In this case, the dependence of the resonance frequency, m, from the orientation of the molecule with respect to the magnetic field applied is given by a relation similar to Eq. (18) (19). 

The orientational distribution fimction P (cos 0) enters the shape of the wideline spectrum 5(f2) in a slightly hidden way. The angular dependence of the resonance frequency is given by (3.1.23) via the orientation of the magnetic field in the principal axes system XYZ of the coupling tensor (cf. Fig. 3.1.2), while the orientational distribution function specifies the distribution of the preferential direction n in a molecule-fixed coordinate frame (Fig. 3.2.2(a)). Figure 3.2.3 shows the relationship between the different coordinate frames and the definition of the relative orientation angles. 

Fig. 4 Mn NMR spectrum of an oriented pofycrystalline sampfe of Mnl2 at 1.4 K in zero applied magnetic field. The three peaks correspond to the three different Mn environments in Mnl2 PI is Mn(IV). Inset applied magnetic field dependence of the resonance frequencies for PI to P3. The Mn(IV) peak shifts in the opposite sense to the two Mn(III) peaks, indicating opposite signs of the internal magnetic fields. Figiu e from [63]...

Fig. 9.1 The dependence of the resonance frequency upon orientation for an anisotropic interaction, namely the CSA of a nucleus in a carboxyl group. The orientations illustrated...

An important development in recent years has been the advent of quartz crystal thermometry. When a quartz crystal is cut at a certain orientation to the axes of the crystal lattice the temperature dependence of the resonant frequency is large and nearly linear, and the Hewlett Packard Company has... 

Since the applied field is fixed in the laboratory, the components in the crystal coordinate system will be Hz = Ho cos 0, H = Ho sin 0 cos <(>, and H, = Ho sin 0 sin , where 0 and are the usual spherical coordinate angles. Thus, for NMR in a single crystal of a metallic solid for which the Knight shift tensor has axial symmetry, the dependence of the resonance frequency v on crystal orientation with respect to the laboratory field Ho will be given by [Abragam (I960)] ... 

Kumagai et al. (1975) bear on the question of the coexistence of superconductivity and ferromagnetism and are referred to again in section 4.2.2. in connection with ESR studies of these compounds. The observation of the NMR in a wall-free single crystal sphere of GdAl2 has been achieved by Fekete et al. (1975). In this case, the quadrupole splitting of the Al resonance was observed as well as the dependence of the resonance frequencies on the orientation of the crystal with respect to an applied field. We refer to this work again in connection with the analysis of transferred hyperfine interactions in section 2.2.3.3. 

This expression shows that the chemical shielding anisotropy (CSA) leads to a variation of the resonance frequency with orientation according to the familiar (3 cos2 0—1) dependence and thus produces typical powder patterns, as illustrated in Fig. 7.8 for axial and nonaxial symmetries. 

The interaction of the spin magnetic moment with the orbital magnetic moment of the unpaired electron leads to an orientationally dependent shift of the resonance frequency. This effect is normally described by an effective spin operator S and an anisotropic g matrix. The quantimi-mechanical Hamilton operator for the interaction of the electron spin with the external magnetic field Zeeman interaction) can, therefore, be described by... 

In order to interpret these NOEs in terms of the bound conformation, the contributions from the free state have to be negligible. The NOE effect vanishes for molecular orientation correlation times in the order of the inverse of the resonance frequency (i.e., a few nanoseconds for 300-600 MHz H frequency) [15]. This is the case for molecules with molecular masses of ca. 100-1000 Da (depending on temperature, solvent viscosity and spectrometer frequency). 

In solid-state NMR, a very important concept is that the resonance frequency of a given nucleus within a particular crystallite depends on the orientation of the crystallite [3—5]. Considering the example of the CSA of a nucleus in a carboxyl group, Fig. 9.1 illustrates how the resonance frequency varies for three particular orientations of the molecule with respect to the static magnetic field, Bq. At this point, we note that the orientation dependence of the CSA, dipolar, and first-order quadrupo-lar interactions can all be represented by what are referred to as second-rank tensors. This simply means that the interaction can be described mathematically in Cartesian space by a 3 X 3 matrix (this is to be compared with scalar and vector quantities, which are actually zero- and first- rank tensors, and are specified by a single element and a 3 X 1 matrix, respectively). For such a second-rank tensor, there exists a principal axes system (PAS) in which only the diagonal elements of the matrix are non-zero. Indeed, the orientations illustrated in Fig. 9.1 correspond to the orientation of the three principal axes of the chemical shift tensor with respect to the axis defined by Bq. 

NMR presents a convenient local probe for the temperature dependence of the spontaneous magnetization. Evidently, the measurement of the temperature dependence of the zero-field resonance frequency is much easier than the careful analysis of the variation of the resonance frequency with orientation and strength of an external field and with temperature in a single crystal. It has, however, always to be kept in mind that the zero-field NMR analysis of a multidomain powder sample can only yield information about the phenomena in the Bloch wall if the zero-field signal originates from nuclei in the Bloch wall. This variation should not be compared with the temperature dependence of the magnetization in the domain. Only a careful NMR analysis is thus worthwhile for a discussion of the eventual differences in the variations of h T) = ... 

The quantity is a part of the line width, which is due to the fluctuational mechanism and is designated as T in eq. (168). Its value does not depend on the resonance frequency o>o at parallel orientation at other orientations that dependence has an approximate form Tj =c +c"v. The measurements in TmES at high frequencies (v>20MHz) confirm the expected dependence (fig. 19). At low frequencies the temperature-dependent broadening is small, and the lineshape is strongly deflected from the Lorentzian. 

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