Van Vleck moment

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 ... 

Fig. 9. Magnetic moment components of the Tm ion in a TmES crystal the external magnetic field forms an angle 0=1 with the c-axis of the crystal the effective (enhanced) field at the nucleus H and the rmclear monrent m, form an angle ff=4A° the electron (Van Vleck) moment Jlf , an angle of 74" the electron-nuclear irronrent Af, an angle of 89 .

At low temperatures, when only the ground state of the lanthanide ion in the crystal field is populated, the total magnetic moment of the ion is the sum of the induced (Van Vleck) moment and the intrinsic moment (the latter differs from zero only in the degenerate state). The contributions to the magnetostriction and the elastic constants due to changes in the intrinsic magnetic moment of the lanthanide ion with lattice strain can be written explicitly when considering the effective spin Hamiltonian. The latter contains a smaller number of independent parameters (constants of spin-phonon interaction) than the Hamiltonian of the electron-deformation interaction (18) and is more suitable in the description of experimental data. 

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. 

Fig. 8. Proton residual van Vleck moments M2 (a), M4 (b), and the ratio MJM2 (c) measured by AIMS for the samples of a series of cross-linked natural rubber versus the shear modulus G. Reproduced from Ref 70, with permission from Elsevier.

Parameters of the Dipolar Correlation Function for a Cross-Linked Elastomer Series. The mixed echo decays were measured for a series of cross-linked natural rubber (64). The model involving the distribution of correlation times (see eq. 29) was used to fit the mixed echo decay. The residual second van Vleck moments obtained from fits as a function of shear modulus G. 

The development of NMR methods that allow accurate measurements of residual second van Vleck moments and the distribution of correlation times yields new tools to test polymer network theories that are more sophisticated than the theory of the freely jointed chain corresponding to a Markov chain. 

The domain thicknesses and the heterogeneity of the chain dynamics of PS-b-PMMA diblock copolymers were studied by and solid-state NMR. The chain dynamics heterogeneity of different components and inside the interfacial region was investigated by different NMR methods, including residual second van Vleck moments of PS and Ti relaxation. 

Fig. 326. Cu(C44Hj8N4) sc, p. Dependence of Xm versus 1/T. Magnetic temperatures were measured using powdered cerrous magnesium nitrate in which 90% of the Ce sites were replaced by La. Full lines indicate zero-field susceptibilities xii and calculated using the spin Hamiltonian parameters g = 2.179, gx=2.033,. 4=212.2 10 cm", 5 = 30.10" cm", S = i, I=, where r is parallel to c axis. Also shown is the Curie law behavior of Xn obtained for 5 = 0. Broken lines give first-order correction to Xn and Xj. obtained by the Van Vleck moment-expansion method [7413].

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

Consider the orbital angular momentum of a free-ion term. Here L = 3 and the orbital degeneracy is 7. Application of Van Vleck s formula (5.8) predicts an effective magnetic moment. 

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. 

For discussion of experimental methods and more detailed discussion of the theory see C. P. Smyth, Dielectric Constant and Molecular Structure, McGraw-Hill Book Co., New York, 1955 or J. W. Smith, Electric Dipole Moments, Butterworths, London, 1955. A more detailed theoretical treatment is given by J. H. Van Vleck, The Theory of Electric and Magnetic Susceptibilities, Oxford University PresB, 1932. 

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)... 

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 readers will undoubtedly agree that the elucidations and derivations of the equations 13—15 are beyond the scope of this book, but chemists with some physical inclination may find some satisfaction in reading through Van Vleck s [207] Theory of Electric and Magnetic Susceptibilities to quench their mathemetical thirst. Instead of tabulating the measured magnetic moments of all europium compounds with variations we will mention the magnetic behaviour of the individual compounds as we deal with them. 

Then the van Vleck formula applies for the magnetization of a paramagnetic system in the absence of a permanent magnetic moment ... 

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... 

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