Dipole form factor

The latest calculation in [3] with (7.60) and the dipole form factor produced... 

Using the dipole form factor one can connect the third Zemach moment with the proton rms radius, and include the nuclear size correction of order Za) m in (7.62) on par with other contributions in (7.58) and (7.74) depending on the proton radius. Then the total dependence of the Lamb shift on Vp acquires the form [3, 25]... 

The Zemach correction was calculated numerically in a number of papers, see, e.g., [5, 6, 7], The most straightforward approach is to use the phenomenological dipole fit for the Sachs form factors of the proton... 

Since the Zemach and recoil corrections are parametrically of the same order of magnitude only their sum was often considered in the literature. The first calculation of the total proton size correction of order Za)Ep with form factors was done in [12], followed by the calculations in [13, 11]. Separately the Zemach and recoil corrections were calculated in [5, 6]. Results of all these works essentially coincide, but some minor differences are due to the differences in the parameters of the dipole nucleon form factors used for numerical calculations. 

Calculation of the nonlogarithmic part of the polarization operator insertion requires more detailed information on the proton form factors, and using the dipole parametrization one obtains [7]... 

Form factor of the hat-curved model Normalized concentration of molecules Kirkwood correlation factor Steady-state energy (Hamiltonian) of a dipole Dimensionless energy of a dipole Moment of inertia of a molecule Longitudinal and transverse components of the spectral function Complex propagation constant Elasticity constant (in Section IX)... 

Let 0 be angular deflection of a dipole from the symmetry axis of the potential 1/(0), let p be a small angular half-width of the well (p Ci/2), and let (/0 be the well depth its reduced value u Uo/(kgT) is assumed to be 1. Since in any microscopically small volume a dipole moment of a fluid is assumed to be zero, we consider that two such wells with oppositely directed symmetry axes arise in the interval [0 < 0 < 2ji]. For brevity we consider now a quarter-arc of the circle. The bottom of the potential well is flat at 0 < 0 parabolic dependence U on 0. The form factor/is defined as the ratio of this flat-part width to the whole width of the well. Thus, the assumed potential profile is given by... 

We remind the reader that the following free parameters are employed in the HC model (a) the reduced potential will depth u = 6 o / ( /tb 7) (b) the angular half-width (3 of the well (c) the mean lifetime x, during which a near-order state exists in a liquid and (d) the form-factor / defined as follows / = (flat part of the well s bottom)/(total well s width 2p). The SD model is characterized by (e) the inhomogeneity-potential parameter p of our self-consistent well, (f) the lifetime xstr, of restricted rotation, and (g) the fraction rvib of dipoles performing RR with respect to their total concentration N. 

A detailed treatment of the Zemach corrections can be found in [28], Assuming the validity of the dipole approximation, the two form factors can be... 

The form factor / accounts implicitly for an influence on the spectrum of collisions of librating dipoles with the surrounding medium. The stronger the intermolecular interactions in a liquid, the smaller the fitted form factor/—that is, the more the potential profile declines from the rectangular one. 

In view of Table II the main difference of the parameters, fitted for HW, from those, fitted for OW, concerns (i) some increase of the libration amplitude / , (ii) decrease of the form factor /, (iii) decrease of the frequency vq (the center frequency of the T-band) and increase of the moment nq, responsible for this band, and (iv) decrease of the intensity factor gj, which strongly influences the THz band. Comparison of curves 3 in Figs. 4h and 5h shows that the partial dielectric loss peak g"max of HW, located at v near 150 cm-1 and stipulated by harmonic longitudinal vibration of HB molecules, substantially exceeds such a peak of OW, since the elastic dipole moment / (D20) 8.8 D exceeds the moment / (H20) 3.5 D. 

Returning to a rather free libration of a dipole in the hat well, we remark that a curved part of the well s bottom is characterized by the form factor/and by the spread = (1 -f) rH2o/ , where rH2o 1.5 A is the radius of a water molecule. The case of ice, in which0.15 and spread 0.52 A, substantially differs from the case of water, in which0.8 and spread 0.12 A. Hence, short-range interactions of H20 molecules are revealed in ice at longer distances than in water. 

The conversion factors g and g contain the so-called form factors which account for shape anisotropies. In an ellipsoidal molecule the form factors (sometimes called depolarizing factors, which are the components of the depolarizing tensor) of the main polarization axes are Ag = Aa, Ai, A, In line with the vectorial character of the internal and directing fields the g factors of anisotropic molecules are tensors. If the environment of the molecules (which are characterized by the polarizability tensor a and the permanent dipole moment p), can be considered as nonpolar and the overall dielectric permittivity is s, the g factor of the q axis is given by... 

Recently, Koehler and Moon (1972) obtained the form factor of Sm " on both cubic and hexagonal sites of Sm metal from intensity measurements in the ordered phases. Their results are shown in fig. 7.24. Unlike most other form factors, it does not have a maximum at sin 0I = 0. This is partly due to the opposition of the spin and orbital contributions to the moment for this ion. In addition, Koehler and Moon ascribe the very low value of the apparent moment on the Sm ion to compensatory conduction electron polarization. However, they did not fully consider the strong crystal field effects which must be present. De Wijn et al. (1974) have attempted to calculate the form factor including crystal field effects but do not obtain good agreement. However, they use non-relativistic wave functions and also employ the dipole approximation. 

In addition, the Formal Graphs show that all the three effects can be combined into a single dipole formed with only one material that must have a variable Seebeck coefficient. This unique dipole is strictly equivalent to the classical decomposition in several dipoles that separates the Peltier and Seebeck effects with a constant factor on the junctions from the Thomson effect between the junctions and the ends. 

Using the magnetic form factor of the free Tm ion in the dipole approximation, Neuenschwander and Wachter (1990a,b) find that the saturation magnetization B(Tm) at 1.5 K is 1.8 0.4 Ub, much less than the saturation moment of either Tm or Tm. The temperature dependence of the magnetization is shown in fig. 88. The data can be fitted surprisingly well with the power law... 

The full evaluation of eq. (11) is a complex task see, e.g., Marshall and Lovesey (1971) or Squires (1978). Instead of repeating this, we will simply adopt the so-called dipole approximation, which is valid for small Q-values. In principle, this works for all magnetic-structure determinations (sect. 4) and for critical scattering (sect. 5). For an examination of the form factors (sect. 3) it must be reconsidered. 

In the Onsager theory of isotropic dielectrics as well as in its extension to the nematic phase given by Maier and Meier the short range dipole-dipole correlations w e ignored. Therefore the dipole moment p in EQNS (1) - (3) cannot be identified with its value measured in the state [16]. The dipole-dipole correlations were considered in the theory developed by Frdtdich [17] who generalised the former Kirkwood approach [18]. Frdhlich has introduced the dipole-dipole correlation factor (known as the Frdhlich-Kirkwood g-factor) in the form... 

Other factors that can stabili2e such a forming complex are hydrophobic bonding by a variety of mechanisms (Van der Waals, Debye, ion-dipole, charge-transfer, etc). Such forces complement the stronger hydrogen-bonding and electrostatic interactions. 

The positions of substitution, orientation, and configuration of the stable form are determined by a balance between opposing steric and dipole ef-fects. There is less agreement regarding the factors influencing kinetically controlled reaction (see below). Essentially neutral conditions, such as provided by an acetate or pyridine buffer, are required to avoid isomerization. Frequently, however, bromination will not proceed under these conditions, and a compromise has been used in which a small amount of acid is added to start and maintain reaction, while the accumulation of hydrogen bromide is prevented by adding exactly one equivalent of acetate... 

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