Multipole order

Fig. 2.1. Dielectric scaling functions for different multipole orders.

Here, r denotes the position vector of the charges qt with respect to the center of the sphere, and r, the distance from the center. By applying the dielectric scaling function for dipoles (Eq. (2.3)), which—as we have seen in Fig. 2.1—is also a good approximation for most other multipole orders, it was immediately clear that the idea of using a scaled conductor instead of the EDBC leads to a considerable simplification of the mathematics of dielectric continuum solvation models, with very small loss of accuracy. It may also help the finding of closed analytic solutions where at present only multipole expansions are available, as in the case of the spherical cavity. Thus the Conductor-like Screening Model (COSMO) was bom. 

A comparison of Figure 3.21 (a) and (b) shows that the behavior of d>44(fc, t) at increasing k values tracks the behavior of C0(t) for increasing multipole order. The similarity in the behavior extends to the fact that, for both functions, there is partial... 

To stress the multipole nature of the binary couphng [68], the singleparticle multipole order is specified as L = 1-30, while the cluster T-matrix... 

Figure 3. Extinction spectra of randomly oriented gold bispheres in water (diameter d = 15 nm, separation distance A / = 0.15 nm). Calculations with different single-particle multipole order...

Transition probabilities for neutron capture y-rays. The multipole orders of the y-rays produced in direct transitions between the capturing state and the ground state can be inferred from the spins and parities of these states when these are known. As pointed out above, the transition probabilities of these y-rays can be deduced when the absolute intensities are known. There are few instances where sufficient data is available some examples are listed in Table 4. It will be seen that the partial radiation widths corresponding to the emission of these y-rays are roughly proportional to the level spacing in the vicinity of the initial state, a phenomenon predicted by Blatt and Weisskopf [4]. In general, the... 

Since the lowest levels of even-even nuclei almost always have the same parity as the ground state and spin 2, it is generally assumed that Coulomb excitation in heavy elements is mainly an electric quadrupole process and that the cross section observed experimentally measures the E 2 transition probability. Similar assumptions are made for odd-A nuclei. In the majority of nuclei this assumption is justified. However, as Bjerregaard and Huus have shown, the multipole order can be determined directly by measurements with bombarding particles with different specific charge, e.g., protons and deuterons (or a-particles). They have verified that the multipole order is indeed two for the excitation of the lowest excited states in the even-even wolfram nuclei. 

The calculation of the Coulomb excitation can be treated by a semi-classical approximation. If v is the velocity, and Zg and Z are the charges on the target nucleus and the bombarding particle respectively, the quantity ri = 2Z Z e l%v is greater than unity, and the cross sections can be calculated from the classical trajectories of the bombarding particles. In general the total cross section for Coulomb excitation for multipole order Z is ... 

The cross section for higher multipole orders falls by factors of the order of 100 to 1000 from one multipole order to another. The cross sections for Ml excitation are generally somewhat less than those of 3 excitation. Recently 3 excitation seems to have been observed by Jones and Phillips. ... 

Examples of decay schemes in deformed nuclei. In this section we shall discuss some examples of decay schemes in nuclei in which more than one rotational family is excited. In some of these nuclei the y-rays are quite numerous and the elucidation of the course of events in such nuclei is possible only if the energies of the y-rays have been measured with some precision and if their multipole order has been determined from measurements of the absolute y-ray intensities and of the intensities of the conversion electrons. 

The full Coulomb interaction energy term can be expressed by combining O Eqs. (25.42) and (O 25.43) as the sum of a number of terms with multipole order 1. The term for / = 0 is the electric monopole term. The / = 2 term gives rise to electric quadrupole interactions and wiU be discussed later. (All the odd / terms vanish for reasons of symmetry. The very small even terms for Z > 2 can also be ignored.)... 

The problem of the nature of the electronic multipole ordering is especially interesting for some praseodymium hexagonrd compounds with the doublet ground state E of the Pr + ion in the crystal field of Csh symmetry (chlorides, bromides, hydroxides, trifluoromethane sulfonates, ethylsulfates). In the effective S=j formalism, neglecting hyperfine interactions, the single-ion Hamiltonian can be written as follows ... 

Table IV. Scaling of Diffusion Times with Multipole Order...

Fig. 1-11. Calculated conversion coefficients for the emission of K electrons for the light elements of the Pt group (Ru, Rh, Pd) versus the transition energy E —EJ [37]. The electric and magnetic multipole orders are denoted as usual by E and (x = 1 to 4), respectively.

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