Shell theory

During recent decades a great amount of knowledge about the properties of atomic nuclei has been gathered. An extensive theory of nucleonic interactions and nuclear structure [liquid-drop theory (7), shell theory (2, 3), unified theory (4), cluster theory (5—7)] has been developed... 

The shell theory has had great success in accounting for many nuclear properties (3). The principal quantum number n for nucleons is usually taken to be n, + 1, where nr, the radial quantum number, is the number of nodes in the radial wave function. (For electrons n is taken to be nr + / +1 / is the azimuthal quantum number.) Strong spin-orbit coupling is assumed,... 

Before concluding this section, it must be pointed out that there are other fields of application of the SRH formalism. Thus, Karwowski et al. have used it in the study of the statistical theory of spectra [30,38]. Also, the techniques used in developing the p-SRH algorithms have proven to be very useful in other areas such as the nuclear shell theory [39,40]. 

Dalton, B. J. (1971), Nonrigid Molecule Effects on the Rovibronic Energy Levels and Spectra of Phosphorous Pentafluoride, J. Chem. Phys. 54,4745. de Shalit, A., and Talmi, 1. (1963), Nuclear Shell Theory, Academic Press, N.Y. 

The overall picture of the atom envisioned by Bohr was a dense nncleus of fixed charge surrounded by rings of electrons. The comphcated optical spectra and the simple x-ray spectra suggested that the ring closest to the nucleus was different than the outer rings. More theory and more observations were necessary to refine this picture, but the shell theory of electronic structure has persisted. 

The approach to analysis of biaxial extension of melts in the simulation of the sleeve inflation process was developed by Pirson and Petrie in 1966-1970 with the use of ideas of the thin shell theory which allows to substitute sleeve film by flat film in analysis. The problem was formulated more accurately and completely and solved in works by Han et al. The author made several conclusion the velocity of material extension changes in the main direction of sleeve motion while effective longitudinal viscosity may increase, decrease, or remain constant depending on the nature of material and the range of strain velocities under consideration longitudinal viscosity of the material at fixed process parameters decreases with temperature rise (the behavior of longitudinal velocity is described more strictly above, in Sect, 2.2.6). 

The use of quarks in atomic shell theory provides an alternative basis to the traditional one. The transformations between these bases can be complicated, but there are many special cases where our quarks can account for unusual selection rules and proportionalities between sets of matrix elements that, when calculated by traditional methods, go beyond what would be predicted from the Wigner-Eckart theorem [4,5], This is particularly true of the atomic f shell. An additional advantage is that fewer phase choices have to be made if the quarks are coupled by the standard methods of angular-momentum theory, for which the phase convention is well established. This is a strong point in favor of quark models when icosahedral systems are considered. A number of different sets of icosahedral Clebsch-Gordan (CG) coefficients have been introduced [6,7], and the implications of the different phases have to be assessed when the CG coefficients are put to use. 

Having put the Coulomb operators (3) in tensorial form, it is not difficult to use the standard techniques of atomic shell theory to calculate the energies of all the terms of the icosahedral g shell. The details have been given elsewhere [10]. Our results agree with those found earlier by more traditional methods by Plakhutin [11] (provided the entry 9 Table 6 of Ref. [10] is corrected to -9). It is interesting to notice that the terms 1 and T2 always occur as degenerate pairs, a result that can be understood by showing that the Coulomb operators (3) are invariant with respect to the automorphism operation [10]. 

Further properties of the g shell can be explored by introducing the notion of quasi-spin <2, in analog to its use in atomic shell theory [12]. We define... 

Four decades ago, Bell [3] introduced a particle-hole conjugation operator CB into nuclear shell theory. Its operator algebra is essentially isomorphic to that of Cq (for example, CB is unitary), the filled Dirac sea now corresponding to systems with half-filled shells. This was later extended to other areas of physics. For example,... 

Our procedure for securing basis independence follows a group-tensor algebraic approach to shell theory, and examines the algebraic interplay of particle-hole conjugation operators with quasispinors and quasispin tensors. The problem with Cs may be remedied while retaining an antilinear transformation only by replacing Z with another antilinear operation which is physical. Apart from an unimportant phase this can be identified as time reversal T, so that C = CT. Hence, the operators to be examined for physical interest are just two in number C and CT. In a later work we will explore the consequences of the work of Ceulemans [7,8,10] from this perspective. 

STa63] de Shalit A and Talmi 1 1963 Nuclear shell theory (Academic Press, New York and London). 

Mechanical Properties of Freestanding Polyelectrolyte Capsules a Quantitative Approach Based on Shell Theory... 

Figure 1 shows a representative force deformation characteristic as obtained from the measurements of a capsule made from PAH/PSS in water. The dried thickness of the capsule was 25 nm and the radius 7.9 microns. For deformations on the order of 1-3 times the shell wall thickness, a linear force deformation characteristic is found. For higher deformations discontinuities in the force deformation characteristic are observed, which are separating quasi-linear sections. The position of these discontinuities as well as their shapes scattered a lot between different shells and the shells showed plasticity in this deformation regime. We avoided this regime in the measurements and obtained the results exclusively from a detailed analysis of the linear regime. Based on classical thin shell theory [20], one would expect a linear force deformation characteristic for deformations up to a few times the wall thickness (fit indicated as dotted line). The onset of buckling should lead to a deviation from the linear dependency, like dis-... 

Niordson FI (1985) Shell Theory, North-Holland Series in Applied... 

This was the state of our knowledge of the structure of the atom when Langmuir, the modern scientific conquistador, attempted to invade the tiny world of the atom. There was an unmistakable conflict between Bohr s theory of the hydrogen atom and the conception of Lewis. Chemists could see but little use in the Bohr atom. They wanted an atom which would explain chemical reactions. The first World War over, Langmuir undertook to reconcile the two theories by publishing his concentric shell theory of atomic structure. 

The concentric shell theory of Langmuir solved other riddles. It explained valence—the tendency of elements to combine with one or more atoms of hydrogen. Valence had baffled chemists ever since Frankland, an English chemist, had introduced the idea in 1852. Valence, according to Langmuir, is the number of electrons which the atom borrows or lends in its effort to complete its outside shell. Thus chlorine, which borrows but one electron, has a valence of one, which means that it combines with but one atom of hydrogen. 

De Shalit, A., Talmi, I. Nuclear shell theory. New York Academic Press 1963. 

Hartree-Fock-Roothaan Closed-Shell Theory. Here [7], the molecular spin-orbitals it where the subscript labels the different MOs, are functions of (af, 2/", z") (where /z stands for the coordinate of the /zth electron) and a spin function. The configurational wave function is represented by a single determinantal antisymmetrized product wave function. The total Hamiltonian operator 2/F is defined by... 

Now, Werner s formula does not admit of the possibility of isomerism of ferrocyanides. This is not the case, however, with the formulae of Deniges, Browning, and of Etard, several rearrangements being possible. According to Friend s 4 shell theory three isomerides, namely, ortho, meta, and para, are possible Thus —... 

Note that commutation of cluster operators holds only when the occupied and virtual orbital spaces are disjoint, as is the case in spin-orbital or spin-restricted closed-shell theories. For spin-restricted open-shell approaches, where singly occupied orbitals contribute terms to both the occupied and virtual orbital subspaces, the commutation relations of cluster operators are significantly more complicated. See Ref. 36 for a discussion of this issue. 

The chemical properties of the atom are determined by the number of valence shell electrons (Z) in an atom, and the way these electrons are arranged in electron shells. The simple Bohr theory quantised the energies of the electrons into discrete K, L, M, N, and O shells (Figure 2.3). This shell theory also allows the prediction of the number of electrons per shell as 2n electrons, namely, 2, 8, 18, 32, etc. electrons, respectively, as shown in Table 2.3, where n is now referred to as the principal quantum number. 

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