Collective model

The energy levels from the quantum mechanical solution of the rotation of a rigid body have the characteristic feature of increasing separation with angular momentum. The energy levels are given by the expression  

A special class of quantum rotors are the superdeformed nuclei. The moments of inertia, after scaling by A5//3, are all similar due to the fact that the shape of these nuclei is largely independent of mass with an axis ratio of 2 1 due to shell stabilization effects discussed below. 

Another interesting case of nuclear rotation occurs in the spherical nuclei. The observation of equally spaced 7-ray transitions implies collective rotation, but such bands have been observed in near spherical 199Pb. It has been suggested that these bands arise by a new type of nuclear rotation, called the shears mechanism. A few 

It is interesting to note that the vibrational model of the nucleus predicts that each nucleus will be continuously undergoing zero-point motion in all of its modes. This zero-point motion of a quantum mechanical harmonic oscillator is a formal consequence of the Heisenberg uncertainty principle and can also be seen in the fact that the lowest energy state, N = 0, has the finite energy of h to/2. 

From another standpoint, the superposition of all of these shape oscillations can be viewed as a natural basis for the diffuseness of the nuclear surface. 

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In the Introduction the problem of construction of a theoretical model of the metal surface was briefly discussed. If a model that would permit the theoretical description of the chemisorption complex is to be constructed, one must decide which type of the theoretical description of the metal should be used. Two basic approaches exist in the theory of transition metals (48). The first one is based on the assumption that the d-elec-trons are localized either on atoms or in bonds (which is particularly attractive for the discussion of the surface problems). The other is the itinerant approach, based on the collective model of metals (which was particularly successful in explaining the bulk properties of metals). The choice between these two is not easy. Even in contemporary solid state literature the possibility of d-electron localization is still being discussed (49-51). Examples can be found in the literature that discuss the following problems high cohesion energy of transition metals (52), their crystallographic structure (53), magnetic moments of the constituent atoms in alloys (54), optical and photoemission properties (48, 49), and plasma oscillation losses (55). 

Nuclei with certain even numbers of protons and of neutrons are the most stable. One model of the nucleus is the collective model, which pictures nucleons as occupying quantized energy levels and interacting with one another strongly. 

Much attention was devoted to collective models Mottelson reviewed Vibrational Motion in Nuclei Judd, the use of Lie groups Lipkin, the... 

A substantial simplification of the systematics in nuclear phase transition regions is obtained if the data are plotted against the product, Np Nn, of the number of valence protons and neutrons instead of against N, Z, or A as is usually done. Such a scheme leads to a unified view of nuclear transition regions and to a simplified scheme for collective model calculations. 

The other consequence is that Np Nn systematics offer a way to greatly simplify collective model calculations. Normally, such calculations are parameterized for each nucleus individually or for a set of isotopes. The Np Nn curves suggest that an entire region can be treated as a unit in which the collective parameters are taken as smooth functions of Np Nn only. Moreover, Fig. 3 suggests that the same set of parameters... 

Np Nn systematics are very useful for extrapolation far off stability and can be exploited to simplify collective model calculations. 

Calculations are presented and data are reviewed on the properties of the high-j states in the light Au nuclei. Both prolate and oblate structures are observed in this region. It is found that the collective model describes well the band- head and the high-spin properties of the h9/2 and ii3/2 proton states, without resort to an "intruder state" phenomenology. 

In conclusion, our data on high-spin states in 185 186Au and in 184 185Pt are mostly indicative of band structures and alignment processed in prolate nuclei. The oblate hi 1/2 band is observed, but is crossed at intermediate spin by what is probably a prolate hi 1/2 structure. Collective model (Nilsson) calculations of bandhead properties of the Au isotopes agree well with the observed systematics of the "intruding" I19/2 and inn proton states, calling into question the particle-hole interpretation of these states. 

In essence, the data collected, modelled or otherwise generated and reported are what they purport to be. 

Review category Data collection model Typical application Comments... 

Sample Selection, Spectra Collection Modeling, Cross-Validation, Statistical Evaluation ... 

The collective model, in which nucleons are considered to occupy quantized energy levels and to interact with each other by the strong force and the electrostatic (coulomb) force... 

In this chapter we review the recent history of and evidence for collective, moleculelike behavior of valence electrons in atoms and indicate some of the questions that will have to be explored in order to resolve the question of how well the electrons in atoms are described by independent-particle or collective models. We then turn the question around and ask whether atoms in a molecule could, under suitable circumstances, display independent-particle behavior, with their own one-particle angular momenta behaving like nearconstants of the motion. The larger question that emerges is then one of whether few-body systems—the valence electrons of an atom, the atoms that constitute a small polyatomic molecule, and perhaps others such as the nucleons in a nucleus, all of which have heretofore seemed nearly unrelated— share characteristics to the extent that we can devise a unifying picture of the dynamics of few-body systems that will expose their commonalities as well as their obvious differences. 

These problems aside, the present picture is thus one in which many states of atoms, both common and exotic, must henceforth be thought of as collective some states such as the ground and singly excited states of He, clearly fit the traditional independent-particle model and most of the elements of the Periodic Table must now be studied carefully in many states to see how they should be interpreted to fit into a coherent scheme of excitations. Then it may become possible to infer, before doing elaborate calculations, whether any particular atomic state is represented better by an independent-particle model or a collective model, and to test the efficiency of collective basis sets for accurate computations. 

Among collective models which have been developed for special purposes is the alpha particle model. If it is assumed that the four-structure already apparent in the supermultiplet theory of Wigner has some permanence in an actual nucleus, the methods of molecular physics may be applied to deduce a level sequence (Wheeler ). The S5nnmetry of the system imposes special restrictions on the level spins of the lower excited states. The energies of these states are obtained from a decomposition of the motion into rotation and vibration. The treatment is especially suitable for nuclei of mass number An, but it is not restricted to these 3 and the detailed application of the model to a number of nuclei is discussed in references [7] and [itf]. 

Since this article was written a great deal of evidence for the validity of a collective description of the levels of light nuclei has accumulated. It seems likely that the individual particle model, modified by configurational interaction, and the collective model with coupling to single particle motion, are alternative modes of description. 

Inglis [10] has also discussed the possible application of a collective model of the alpha particle type to the Li nucleus, particularly with respect to the broad state at 6.56 MeV which finds no explanation in the LS or // schemes. 

No low level of / = 0 is predicted by these schemes other than the ground state, but it arises naturally as a dilational state in the alpha particle model. This model (Sect. 4) also predicts the 2 state but requires low lying states of J = y and 1" which have not been identified. The 0" state can also be obtained by double nucleon excitation, e.g. in the configuration pfj p i - The transition probability of the O " state for pair emission to the ground state can be estimated from the cross section for excitation of this state with electrons. Schiff claims that the value obtained is too small to be explained by the alpha particle model and too large to be accounted for by a // coupling model with excitation of two -particles he suggests that a collective model intermediate in properties between the two extreme models is necessary. 

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