Dipolar interaction energy

Fig. 11. Model, with the two magnetic dipoles oriented parallel to the static field Bq, used for the calculation of the dipolar interaction energy.

Let us now refer to a set of molecules with their jc, y and z axes iso-oriented in an idealized solid state (Fig. 2.5). If the external magnetic field is aligned with the z axis, the dipolar interaction energy between the nuclear magnetic moment and the electron magnetic moments, according to Eq. (1.4), is... 

From the dipolar interaction energy, the dipolar shift can be obtained by evaluating from Eq. (2.16) A d,p between two states differing by A A// = 1 and dividing it by the nuclear Zeeman energy hyiBo (Appendix IV) ... 

Note that, when x is isotropic, the pseudocontact shift is zero (Eq. (2.18)), but a dipolar shift is observed in the solid state (Eq. (2.17) and Eq. (IV.9)). That the dipolar interaction energy averages zero for isotropic x can be easily verified by averaging Eq. (2.15) over all orientations, as... 

The transition probabilities depend on the mean squared interaction energy relative to the mechanism which causes the transition, times the value of the spectral density at the required frequencies (Eq. (3.14)). The square of the dipolar interaction energy is, as usual (see Eq. (1.4) and Appendix V), proportional to (p, 1 1x2/r3)2, where p and p2 are the magnetic moments of the two spins. The actual equations are... 

It is interesting to note that on rapid (with respect to the anisotropy of the interaction energy in frequency units) rotation, the dipolar interaction energy averages to zero. In contrast, the square of this quantity (which determines the extent of the contribution to relaxation) does not average to zero. The average values obtained by integration of the functions results in... 

Eq. (IV.6) is the general formula for the dipolar interaction energy when the principal axis of x is in a generic k direction. The shift is then obtained by calculating the energy difference between two states differing by AM/ = 1 ... 

Electron spin magnetic moment)-(electron spin magnetic moment) dipolar interaction energy ... 

Udd dipolar interaction energy (per pair of interacting moments)... 

Note that the dimensionless units defined in Table I are used, so that the curvature along the X direction is renormalized to 1. Here Uq is the two-body interaction potential defined in Eq. (2.13). The two terms linear in X are the dipolar interaction energy (with Uj and U2 two unit vectors, respectively, along the z-axis of the fixed frame for the solute and the solvent body, cf. Fig. 1). Finally a quadratic term in X has been added in order to confine the fluctuations of the stochastic field. 

AGS is the strain-energy difference, AV is the dipolar-interaction energy difference, and D is the dielectric constant. If both R j and Rjj are known, AGS and AV can be determined by plotting the left-hand side of equation 2 against l/D. This should be a straight line with slope AV and intercept AGS. If only R is known, the above plot is repeated for various values of R-j-j. The value that gives the best straight line relationship is chosen for R.jj. 

Fig. 2.4 Attractive energy E as a function of the distance r for the NH3 molecule. At large distances, the dipolar interaction energy is predominant at smaller distances, its contribution becomes comparable to that of the dispersive energy. After [M1].

The contact interaction energy is ordinarily eombined with a tensoral energy that represents the dipole-dipole interaction between the magnetic moments of the unpaired eleetron (ps=gePe) and the (loeal) atomie nuelei ( j,7=g P ). The sealar expression for the dipolar interaction energy is... 

The electric-tield-induced phase transition in an ER suspension was found to be different from that in general colloidal suspensions. Tao and Martin [55, 56] predicted theoretically tliat the bet structure has an energy lower than that of the fee (face-centered cubic) and other structures, based on dipolar interaction energy calculations. The dipolar interaction energy per particle for various crystal structures is shown in I able 3. The bet crystal structure is shown in Figure 6. 

Table 3 Dipolar interaction energy per particle tor various structures...

The laser diffraction method [57] was employed to experimentally determine the crystal structure within the flbrillated columns by using a uniform glass microsphere/silicone oil system, and a bet structure was observed as predicted. The diffraction pattern is shown in Figure 7 for monodispersed glass beads of various sizes. The structure constants determined from the laser diffraction experiment were found to agree very well with the theoretical calculation based on dipolar interaction energy. Table 4 lists the experimentally determined and theoretically calculated structure constants for the bet structure formed by the silica spheres. The experimental data are consistent witli the proposed bet structure. 

The presence of neighboring magnetic nuclei alters the local field and therefore the energies of a nucleus. The direct dipolar interaction energy between any two magnetic moments pi and P2 separated by a vector r is... 

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