Weak-field approximation

Complexes containing weak field ligands such as H O are more likely to absorb visible light, making them coloured. The wavelength range for the visible region Is approximately 400-700 nm. 

Spodumen is a monoclinic pyroxene, space group C C2 c), with two not equivalent metal cation sites Ml and M2. The aluminum occupies the smaller Ml site, which is approximately octahedral (actual symmetry C2) with an average metal-oxygen distance of 1.92 A. The M2 site, occupied by Li, is also six-fold coordinated with an average metal-oxygen distance of 2.23 A. Both A1 and Li sites may be substitutionally replaced by ions of the transitional metals in various proportions. Both Mn " " and Cr " centers have been identified in luminescence spectra by steady-state spectroscopy (Tarashchan 1978 Walker et al. 1997). At room and lower temperatures only one emission band of Mn + occurs and the excitation spectra taken for the different wavelengths of the luminescence bands are always the same. So it is very probable that Mn + ions in the spodumen matrix present only in one site. The calculated values of 10D,j and B are consistent with the occupation of larger M2(Li) weak-field site. Mn + is mainly in Li-sites rather than Al-sites. 

The lines at 686 and 693 nm with a long decay time of approximately 1 ms in the titanite emission spectrum are not correlated with any other lines and bands (Fig. 4.34). Such lines are very typical for Cr in a high field coordination and may be connected with such a center. The broad luminescence band appears peaking at 765, which may be ascribed to Cr + in a weak field coordination. The band at 765 nm has distinct dips at 749, 762, 793, 798, 804 and 820 nm. Comparison with the titanite absorption spectrum (Fig. 5.19) demonstrates that those lines exactly coincide with the absorption spectrum of Nd (Bakhtin and Gorobets 1992). Cr is a good energy sensitizer, because it has broad, allowed absorption bands with a broad emission spectrum, which overlaps the absorption bands of the lasing ion (Nd " ", Ho " ). 

A weakly bound state is necessarily nonrelativistic, v Za (see discussion of the electron in the field of a Coulomb center above). Hence, there are two small parameters in a weakly bound state, namely, the fine structure constant a. and nonrelativistic velocity v Za. In the leading approximation weakly bound states are essentially quantum mechanical systems, and do not require quantum field theory for their description. But a nonrelativistic quantum mechanical description does not provide an unambiguous way for calculation of higher order corrections, when recoil and many particle effects become important. On the other hand the Bethe-Salpeter equation provides an explicit quantum field theory framework for discussion of bound states, both weakly and strongly bound. Just due to generality of the Bethe-Salpeter formalism separation of the basic nonrelativistic dynamics for weakly bound states becomes difficult, and systematic extraction of high order corrections over a and V Za becomes prohibitively complicated. 

The eigenvalue problem for the simple cos y potential of Eq. (4) can be solved easily by matrix diagonalization using a basis of free-rotor wave functions. For practical purposes, however, it is also useful to have approximate analytical expressions for the channel potentials V,(r). The latter can be constructed by suitable interpolation between perturbed free-rotor and perturbed harmonic oscillator eigenvalues in the anisotropic potential for large and small distances r, respectively. Analogous to the weak-field limit of the Stark effect, for linear closed-shell dipoles at large r, one has [7]... 

The starting point for most of the redox chemistry considered in this review is the nickel(II) ion. The nickel(II) ion has a d8 electronic configuration and, with weak-field ligands such as H20, it forms a six-coordinate ion with approximately octahedral symmetry and a paramagnetic (two unpaired electrons) 3A2 ground state. The characteristic solution chemistry of six-coordinate nickel(II) is well documented and, in particular, the substitution behavior has been extensively studied and is the subject of recent reviews (11, 12). It is a labile ion with solvent exchange rates around 104 sec-1 at 25°C and activation parameters are consistent with dissociatively activated interchange behavior (13). 

In the experiments with Rydberg atoms it is very difficult to observe radiatively assisted collisions with cross sections more than a factor of 10 smaller than the resonant collision cross sections, so the deviations from Eq. (15.29) are not apparent. However, in other contexts, such as laser assisted collisions, this limitation does not apply, and it is interesting to consider how the above description passes over into the weak field regime, in which Jm(KEmv//oj) is small. If we restrict the integration in Eq. (15.27) to the large r region of space, in which the approximations we have used are valid, we can rewrite Eq. (15.27) as... 

Although Cl- is a weak-field ligand and CN- is a strong-field ligand, [CrClg]3- and [Cr(CN)g]3- exhibit approximately the same amount of paramagnetism. Explain. 

For the weak field case, we have the situation where the crystal field interaction is much weaker than the electronic repulsion. In this approximation, the Russell-Saunders terms 3F, 3P, 1G, lD, and 5 for the d2 configuration are good basis functions. When the crystal field is turned on, these terms split according to the results given in Table 8.4.2 ... 

These are the states arising from the weak field approximation. 

Of course, these are the same states obtained in the weak field approximation. The way these two sets of states is correlated is shown in Fig. 8.6.1. In drawing the connection lines, we minimize crossing, even for species with different symmetry. Note that the four lines connecting the triplet states constitute the Orgel diagram for d2 complexes (Fig. 8.5.4 or Fig. 8.5.8). 

The case of intermediate coupling of momenta (between Hund s cases (a) and (6)), as well as that of breaking weak field approximation axe discussed in [294]. The molecular (/-factors for Hund s case (c) coupling are discussed in [92, 364]. 

Enokida et al. (1991) measured hole mdbilities of PMPS before and after ultraviolet exposures. The exposures were of the order of 1 erg/s-cm2. Prior to the exposures, the mobilities were approximately 10-4 cm2/Vs and weakly field dependent. Following the exposures, a decrease in the mobility was observed. Under vacuum exposure conditions, a decrease of approximately 40% was observed for a 1 h exposure. Under atmospheric conditions, however, the decrease was approximately a factor of 4. Enokida et al. attributed the decrease in mobility to the formation of Si-O-Si bonds in the Si backbone chain. A similar study of PMPS was described by Naito et al. (1991). While the field and temperature dependencies of the mobility were not affected by the ultraviolet exposures, the dispersion in transit times increased significantly. The change in dispersion could be removed by subsequent annealing. The authors attributed the increase in transit time dispersion to a reduction in the hole lifetime, induced by Si dangling bonds created by the ultraviolet radiation. 

This form, as it stands, is only an approximate form of solution of the Bloch equation at Eq. (6) valid for small P and/or weak field strength F. Following... 

We consider further the case in which the system is subject to the action of an external homogeneous field and spatial quantisation exists (which is approximately true for weak fields). The alterations of m and the polarisation of the light are then subject to the rules derived above. It is easy to see that the transition possibilities Aj = —1, 0, +1, which are valid for a free system, remain true for j. 

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