Van Vleck second moment

The earliest work employed the method of moments, and Fraissard et al (192) were the first to consider in detail the second moment of the proton NMR resonance in nonspinning solids. In general, when the line shape is determined solely by the I-I and I-S magnetic dipole-dipole interaction, the Van Vleck second moment, M2, is given by (193)... 

Very recently, an interesting modification of HETCOR correlation was published by Chan and co-workers.69 In this experiment, a DQ filter is incorporated into the pulse sequence of HETCOR spectroscopy so that a DQ excitation profile can be obtained by measuring a series of 2D spectra (Figure 14). This method offers a simple experimental approach to extract the van Vleck second moment of a multiple-spin system under high-resolution condition. Hydroxyapatite (Ca10(PO4)6(OH)2) and brushite (CaHP04 2H20) were used as reference samples. 

For bone, the P spin-lattice relaxation time Tf is ca 100 s [36,39] and decreases by approximately 15% from the dried to the fully hydrated sample [36]. In contrast, for synthetic HA, Tf is considerably shorter and varies from 1 to 22 s [31, 36]. The cited results are for differently prepared samples, different magnetic fields and different MAS rates. Homonuclear 3ip.3ip spin-spin relaxation in the sohd state can be conveniently characterized by van Vleck second moments M [46,47].Wuetal. [41] measured Mf in bone, dental enamel and synthetic apatites, HA and CHA-B, using the P Hahn spin echo under proton decouphng. The initial P magnetization was prepared either by a nl2 pulse or... 

For more precise discussion we use the Van Vleck second moment equation. 

Residual van Vleck Second Moments of Cross-Linked Series of Natural Rubber. Proton residual second and fourth van Vleck moments were measured by AIMS for the samples of cross-hnked series of natural rubber (NR). The residual van Vleck moments M2 and M4 can be evaluated by the best fits of the composite signals with equation 21 and these results are shown in Figures 8a and 8b, respectively, for the whole NR series. 

Van Vleck 8) has shown how the second moment and fourth moment interactions between the magnetic nuclei. 

The first attempts to rationalize the magnetic properties of rare earth compounds date back to Hund [10], who analysed the magnetic moment observed at room temperature in the framework of the old quantum theory, finding a remarkable agreement with predictions, except for Eu3+ and Sm3+ compounds. The inclusion by Laporte [11] of the contribution of excited multiplets for these ions did not provide the correct estimate of the magnetic properties at room temperature, and it was not until Van Vleck [12] introduced second-order effects that agreement could be obtained also for these two ions. 

As was discussed qualitatively in Section II,A,2, the local magnetic fields produced at a nucleus in a solid by the magnetic dipole moments of nuclei around it are often responsible for the observed line widths. Van Vleck (73) has derived, in a rigorous manner, an expression for the second moment of the absorption curve of the nuclei in terms of the magnetic moments, spins, and internuclear distances of the nuclei. The second moment ((AH )) of the shape function g(H — Ho) normalized to unit area is... 

Van Vleck s second moment formula has been experimentally proven to be correct by Fake and Purcell (74)- It has become a valuable tool in structure determination of the solid state. 

An instructive illustration of the effect of molecular motion in solids is the proton resonance from solid cyclohexane, studied by Andrew and Eades 101). Figure 10 illustrates their results on the variation of the second moment of the resonance with temperature. The second moment below 150°K is consistent with a Dsi molecular symmetry, tetrahedral bond angles, a C—C bond distance of 1.54 A and C—H bond distance of 1.10 A. This is ascertained by application of Van Vleck s formula, Equation (17), to calculate the inter- and intramolecular contribution to the second moment. Calculation of the intermolecular contribution was made on the basis of the x-ray determined structure of the solid. 

The 7Li resonance in zeolites is also difficult to interpret, even though the quadrupole moment is much lower. Lechert et al. (227) believe that the 7Li linewidth is controlled by the dipole-dipole interaction with 27A1 nuclei in the aluminosilicate framework. According to Herden et al. (232) the increase of 7Li frequency from 9 to 21 MHz does not affect the second moment of the spectra in zeolites Li-X and Li-Y, which means that the quadrupolar interaction is small. The second moment was also independent of the Si/Al ratio. The mean Li-Al distance calculated from the van Vleck formula was 2.35 A. Small amounts of divalent cations reduce the movement of Li + considerably, with the activation energy for this process increasing from 30 to 60 kJ/mol. 

Diffusional behavior of sorbed species is studied by NMR using one of three approaches the van Vleck method of moments, relaxation measurements, and the pulsed-field-gradient method. An example of the use of the method of moments is the work of Stevenson (194) on H resonances in zeolite H-Y (see Section III,K). Another is the study by Lechert and Wittem (284) of C6H6 and C6H3D3 adsorbed on zeolite Na-X. Analysis of second moments of H resonances allowed the intra- and intermolecular contributions to the spectra to be extracted. Similarly, second moments of H and 19F spectra of cyclohexane, benzene, fluorobenzene, and dioxane on Na-X provided information about orientation of molecules within zeolitic cavities (284-287). 

The second high-frequency term involves a sum over all discrete states and an integration over the continuum states the difficulties involved have been outlined before. Little is known about the continuum states, but what few calculations there are for simple systems92 suggest that they may be at least as important as the discrete states. For this reason early calculations were done in the closure approximation, notably by Van Vleck in the 1930 s. The difficulties of calculating xHF have been reviewed by Weltner.93 Experimentally xHF may be obtained from rotational magnetic moments. For linear molecules these can be obtained from molecular-beam experiments, which also measure the anisotropy x Xi- directly. The anisotropies may also be derived from crystal data, the Cotton-Mouton effect and, recently, Zeeman microwave studies principally by Flygare et al.9i... 

Figure 3 shows a typical pulse sequence developed for DQ excitation. A systematic variation in the DQ excitation and reconversion periods can be used to probe the van Vleck s second moment of the coupled spins,57 whose magnitude depends on both the number of interacting spins and the internuclear distances ... 

To determine the molecular process which causes a change in the second moment of 13.6 G2 for PC and 10.4 G2 for PMST, it is necessary to calculate the second moment. According to Van Vleck (22) the second... 

In this section, we shall use the degenerate perturbation theory approach to derive the form of the effective Hamiltonian for a diatomic molecule in a given electronic state. Exactly the same result can be obtained by use of the Van Vleck or contact transformations [12, 13]. The general expression for the operator up to fourth order in perturbation theory is given in equation (7.43). Fourth order can be considered as the practical limit to this type of approach. Indeed, even its implementation is very laborious and has only been used to investigate the form of certain special terms in the effective Hamiltonian. We shall consider some of these terms later in this chapter. For the moment we confine our attention to first- and second-order effects only. 

For an isolated pair of nuclei in a polycrystalline system the band shape shows a Pake doublet. When the number of interacting nuclei becomes larger and a polycrystalline material is analyzed, the resulting spectrum is much more complex and much less informative. Nevertheless an expression for the dipolar second moment (or mean square width) of the NMR spectrum in a polycrystalline sample has been suggested by Van Vleck [12] ... 

In spherically symmetric systems the induced diamagnetism depends primarily on the mean square radius of the valence electrons as the small contribution from the inner-shell electron core can usually be neglected 1 ). In the case of molecules with symmetry lower than cubic, the quantum mechanical treatment by Van Vleck 23> indicates that another term must be added to the Larmor-Langevin expression in order to calculate correctly diamagnetic susceptibilities. This second term arises because the electrons now suffer a resistance to precession in certain directions due to the deviations of the atomic potential from centric symmetry. The induced moment will now be dependent on the orientation of the molecule in the applied magnetic field and thus in general the diamagnetic susceptibility will not be an isotropic quantity 19-a8>. 

The terms and in the Van Vleck equation are the first- and second-order Zeeman coefficients obtained from the expansion of the magnetic moment in terms of a power series of the magnetic field. 

The reduction of the inhomogeneous linewidth follows from Van Vleck s calculation of the second moment (AB ). It states that (AB > is proportional to N is here the number of nuclear spins. In an A - B pair, the number of nuclear spins is doubled in comparison to an isolated molecule. Therefore, the inhomogeneous linewidth is reduced by the factor 1 /-/z. The experimental values in naphthalene confirm this interpretation in the immediate neighbourhood of the A-M - B crossing points (a = 120° in Fig. 7.4), the experimental value for the ratio of the linewidths of the mini-exitons to those of the isolated molecules is ABm/ABab = 1/1.6. The width AB is thus an additional confirmation of the model for a mini-exciton it consists of the two molecules in a unit cell. At higher N-hg concentrations x = 10%), A-A-B and A-B-B mini-excitons have also been observed [4]. 

Since f(H) is a symmetric function, all odd numbered moments are zero. Van Vleck has also shown how the second moment and fourth... 

Inserting this into the Van Vleck equation for the second moment gives ... 

As remarked above. Van Vleck also showed how to calculate the NMR fourth moment. Whereas the second moment data relate to the values of P (cosA) with n = Q, 2,4, the fourth moments relate to n = 6,8 as... 

In the case of a dense system of randomly oriented moments, the field distribution can be assumed to have Gaussian shape. Truly random orientation is certainly fidfilled for a nuclear moment system (except at extremely low temperatures, which are out of the reach of jxSR). For electronic moments it is strictly true only for a free paramagnet. This field distribution is added to Bapp and in summary Bp is distributed. The width of this field distribution can be characterized by its second moment, the so called polycrystalline Van Vleck moment, originally derived for nuclear moments ... 

This series converges slowly, which implies that the moments of the system must be known to high order if the FID and lineshape is to be completely determined. Van Vleck gives the method for calculating the moments from the structure of a rigid solid, using traces of commutators that reduce to a series of sums over internuclear vectors. The second moment involves summing the square of an interaction term for every pair of spins. The fourth moment involves the square of that sum, the sum of the fourth power of the interaction... 

The M —H distances are approximately equal to the sum of covalent radii as shown by neutron and X-ray studies. In order to find out the precise structure of metal hydrides, both neutron and X-ray studies are needed. The distances obtained from NMR studies in the solid state based on the second moment calculated from the line shape are too small. This results from the inaccuracy of the Van Vleck equation. The Mn-H distance in [MnH(CO)5] calculated from the second moment of the Van Vleck equation is 128 pm. The longer distance (144 pm) was calculated from NMR data based on the modified Van Vleck equation. " A similar distance was calculated from electron diffraction studies in the gas phase. All these distances are lower than the sum of covalent radii which equals 157 pm. The Mn-H distance in [MnH(CO)5] obtained from neutron diffraction studies equals 160.1 pm. Similar distances were found for other hydride complexes. 

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