Direct field model

E3.15 The direct field model introduced at an early stage [23] provides a pictorial explanation for the occurrence of forbidden (Ami =1) hyperfine lines. The nucleus is exposed to the hyperfine ( Ba) and the external (B) fields. The nuclear spin is oriented in the effective fields (B ) which are not parallel when the hyperfine coupling is anisotropic. A closer examination shows that Am/ = 1 lines then occur in the ESR spectrum, (a) How would the effective fields ( Ba) be oriented in the limits B Ba and B Ba (B Ba applies at X-band for nuclei with small gN, e.g. for N hyperfine structure... 

The intensity ratio between the main and satellite lines has been used to estimate the distance between a paramagnetic centre and protons at surrounding molecules. The method is based on the assumptions that the point dipole approximation applies for the anisotropic hfc, and that this coupling is much smaller than the nuclear Zeeman energy of the proton. It was found advisable to measure this ratio at as high microwave frequency as possible to achieve the latter condition, i.e. Ba < Bn in terms of the direct field model. The procedure was adequate at Q-band but not at X-band to obtain the distance between a proton and the P03 radical in a single crystal of Na2HP03-51120, see [37] for details. 

At X-band microwave frequency traditionally employed in ESEEM studies, the envelope modulation patterns are in certain cases not well developed. This applies particularly for weakly coupled nuclei. The intensity of the forbidden transitions are then suppressed and consequently also the modulation depth. The use of a lower microwave frequency than X-band is accordingly expected to be particularly useful for enhancement of the ESEEM for weakly coupled nuclei. Dramatic enhancement of the ESEEM depth was observed for weakly coupled nuclei in initial studies with a pulsed EPR spectrometer constructed to operate at S-band. Nuclear quadrupole resonance frequencies were observed for N in some nitroaromatic systems under cancellation conditions for the hfc, i.e. when V2A - vn 0 in terms of the direct field model [44]. 

The magnetic fields B+ and B acting on a magnetic nucleus is in the direct field model obtained by vector addition of Bn due to the externally applied magnetic field, and Ba due to the magnetic moment of the electron. 

The field Ba can be oriented two ways depending on the value of the electron spin quantum number (ms = 1/2). The field is according to the direct field model [36] given by (4.2) when the electronic -factor is isotropic ... 

The angle a is calculated by quantum mechanics [38] or in the direct field model by applying the cosine theorem (Fig. 4.25) according to equation (4.7) ... 

By expressing the mean-field interaction of an electron at r with the N- 1 other electrons in temis of a probability density pyy r ) that is independent of the fact that another electron resides at r, the mean-field models ignore spatial correlations among the electrons. In reality, as shown in figure B3.T5 the conditional probability density for finding one ofA - 1 electrons at r, given that one electron is at r depends on r. The absence of a spatial correlation is a direct consequence of the spin-orbital product nature of the mean-field wavefiinctions... 

Data Structures. Inspection of the unit simulation equation (Equation 7) indicates the kinds of input data required by aquatic fate codes. These data can be classified as chemical, environmental, and loading data sets. The chemical data set , which are composed of the chemical reactivity and speciation data, can be developed from laboratory investigations. The environmental data, representing the driving forces that constrain the expression of chemical properties in real systems, can be obtained from site-specific limnological field investigations or as summary data sets developed from literature surveys. Allochthonous chemical loadings can be developed as worst-case estimates, via the outputs of terrestrial models, or, when appropriate, via direct field measurement. 

A simple model, known as the Internal equivalent field model (9) (see "Figure 1") accounts for the different p values of mono-substltuted benzene derivatives R-X, and relates them to the respective donor or acceptor strength of X. It Is assumed In this model that the substituent action on the it electrons of the ring Is equivalent to that of a D-C field EQ with direction and Intensity related to the substituent electronegativity. Identifying dipole expansions of R-X In presence of E and of R In presence of E0 + E yields ... 

Here, the factor A has been introduced to accommodate the fact that L is an effective operator in the ligand field model, and in the same sense A for use in these equations may not have a direct relationship to its use in the interpretation of spectra, say A is generally expected to be somewhat less than unity. There is no similar TIP contribution for AUg) ground terms as there is no higher-lying term of the same multiplicity. 

A 3) and two small negative peaks (-0.6 e A-3) are located in the directions of four C3 axes at 0.40 A from the Co nucleus. This arrangement of positive and negative peaks is just opposite to that observed for the transition metal ions in an octahedral environment. A simple crystal field model can also be applied to this case, because of the highly ionic character of this crystal (Section IV,B). In a tetrahedral crystal field, a fivefold degenerate 3d level splits into a lower doublet e and a higher triplet f2 as follows ... 

The Lorentz-Lorenz equation can be used directly to model the birefringence of a solution of rigid rod molecules subject to an orienting, external field. Figure 7.2 shows a representative molecule, which is modeled as having a uniaxial polarizability of the form... 

Thermal fluctuations can contribute dominantly to the scattering intensity right after the isothermal phase separation starts [70,76], Therefore, conditions 1) and 3) must be fulfilled to ensure that the effect of thermal noise is negligible. The dynamics of phase separation can be adequately described by the mean-field model if condition 2) is satisfied. Condition 2) is a direct consequence of the Landau Ginzburg criterion [75]. Thus, one may establish prerequisites for Eqs. (27) and (33) are the conditions 1) and 3), while Eq. (34) requires conditions 2) and 3). For example, Eq. (27) and as a consequence Eq. (33) cannot be confirmed experimentally not even for small values of q if the quench depth e is too small [70]. Moreover, owing to the effect of thermal fluctuations, Eq. (33) fails at q as qc even if the Landau Ginzburg criterion is fulfilled [70,77]. Thus, in the former case condition 2) is violated whereas in the latter example conditions 1) and 3) are not satisfied. 

When increasing the value of H, the moments gradually rotate towards the applied field direction and when H = Hc, they are parallel to the field. In the molecular field model, the spin-flop is a first-order transition and the spin flop 5 paramagnetic transition a second-order process. Between HSF and Hc magnetization increases linearly with H M/Ms = H (2HE - Ha) where Ms is the saturation magnetization of a sublattice (Ms = A NgMBS). At T = 0 K, the critical field can be written as HSF = [2HE HA/(1 - a)]1/2... 

Grozema, F., Zijlstra R.W.J. and Duijnen RTh. van, Many-body interactions calculated with the direct reaction field model. Chem.Phys. (1999) 246 217—227. 

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