# THEORETICAL MODELS OF ATOMIC NUCLEI

## Abstract

7 ott Model and the P/sub 2/ Force. A number of problems are considered which arise when one tries to understand the relation of the shell model of the nucleus to the collective (rotational) model of the nucleus. The Elliott model is a version of the shell model which assumes a central spherical oscillator potential and mixes the various degenerate LS-coupling configurations to obtain a set of wave functions which have certain properties characteristic of the wave functions of the Bohr-Mottelson rotational model. (Elliott, 1958). It is also known that when a residual momentum-dependent quadrupole-quadrupole interaction is added to this model an L(L + I) rotational spectrum results (Bargmann and Moshinsky, 1960). This development is reviewed and an equivalent momentumindependent residual interaction ( P/sub 2/' interaction) is obtained which differs somewhat from the P//sub 2/ interaction commonly assumed to give rotational spectra in the model. The model Hamiltonian is separated into a rotational Hamiltonian, a deformed intrinsic Hamiltonian, and a perturbation term. The eigenfunctions and eigenvalues of the intrinsic Hamiltonian are found and used in Inglis' cranking model' formula to calculate the moment of inertia. This calculation is of interest because it is not known whether the cranking model calculationmore »

- Authors:

- Publication Date:

- Research Org.:
- Stanford Univ., Calif.

- OSTI Identifier:
- 4819445

- NSA Number:
- NSA-16-024825

- Resource Type:
- Journal Article

- Journal Name:
- Dissertation Abstr.

- Additional Journal Information:
- Journal Volume: Vol: 22; Other Information: Orig. Receipt Date: 31-DEC-62

- Country of Publication:
- Country unknown/Code not available

- Language:
- English

- Subject:
- PHYSICS; BOHR-MOTTELSON MODEL; BORN APPROXIMATION; COLLECTIVE MODEL; CROSS SECTIONS; DIFFERENTIAL EQUATIONS; EIGENVALUES; ELECTRONS; ELEMENTARY PARTICLES; EXCITATION; HAMILTONIAN; INERTIAL FORCE; INTERACTIONS; MATRICES; MOMENT OF INERTIA; NUCLEAR MODELS; NUCLEI; NUMERICALS; OSCILLATIONS; PERTURBATION THEORY; QUANTUM MECHANICS; RELATIVITY THEORY; ROTATION; SCATTERING; SHELL MODELS; SPECTRA; TRANSIENTS

### Citation Formats

```
Willey, R S.
```*THEORETICAL MODELS OF ATOMIC NUCLEI*. Country unknown/Code not available: N. p., 1962.
Web.

```
Willey, R S.
```*THEORETICAL MODELS OF ATOMIC NUCLEI*. Country unknown/Code not available.

```
Willey, R S. Fri .
"THEORETICAL MODELS OF ATOMIC NUCLEI". Country unknown/Code not available.
```

```
@article{osti_4819445,
```

title = {THEORETICAL MODELS OF ATOMIC NUCLEI},

author = {Willey, R S},

abstractNote = {7 ott Model and the P/sub 2/ Force. A number of problems are considered which arise when one tries to understand the relation of the shell model of the nucleus to the collective (rotational) model of the nucleus. The Elliott model is a version of the shell model which assumes a central spherical oscillator potential and mixes the various degenerate LS-coupling configurations to obtain a set of wave functions which have certain properties characteristic of the wave functions of the Bohr-Mottelson rotational model. (Elliott, 1958). It is also known that when a residual momentum-dependent quadrupole-quadrupole interaction is added to this model an L(L + I) rotational spectrum results (Bargmann and Moshinsky, 1960). This development is reviewed and an equivalent momentumindependent residual interaction ( P/sub 2/' interaction) is obtained which differs somewhat from the P//sub 2/ interaction commonly assumed to give rotational spectra in the model. The model Hamiltonian is separated into a rotational Hamiltonian, a deformed intrinsic Hamiltonian, and a perturbation term. The eigenfunctions and eigenvalues of the intrinsic Hamiltonian are found and used in Inglis' cranking model' formula to calculate the moment of inertia. This calculation is of interest because it is not known whether the cranking model calculation correctly gives the moment of inertia of a freely rotating system. It is found that in the limit of very many particles in a shell the correct moment of inertia is obtained. For only a few nucleons in a shell, the values obtained from the formula fluctuate about the correct value. The conse quences, for a simple configuration, of modifying the model by taking a mixture of the long range P/sub 2/' interaction with the short range sigma -function interaction are considered. For an intermediate mixture, spectra are found which resemble the spectra predicted by the collective vibrational model. Finally, the implications of a P/sub 2/ residual interaction for direct interaction inelastic scattering processes are considered. There are two points of interest here. One is to try and see if one can actually see a P/sub 2/ residual interaction in rotational nuclei. The other is to see if the strength of the P/sub 2/ interaction obtained from such scattering experiments is consistent with the strength determined from the observed rotational spectrum. Within the approximations made, the few experimental results available seem to be roughly consistent with the calculation. Single-Particle Excitations in Inelastic Scatterinff of Electrons frora Nuclei. Recent experiments of Kendall and collaborators (Cranell, et al., 1961) make it desirable to have more detailed calculations of inelastic electron scattering to resolved final nuclear states than the previous estiraates (Schiff, 1954). Walecka (1961) has studied excitations of collective states of the nuclei. A detailed analysis of single- particle excitations is given. The cross section is calculated in relativistic gorn approxiraation in terms of the general longitudinal electric and transverse electric and magnetic multipole . operators (Alder, et al., 1956). Using the sums of singleparticle operators for the nuclear charge, current, and magnetization densities and single-particle/1 s j) wave functions with harmonic oscillator radial functions, the matrix elements and differential cross sections are calculated for some representative cases. For a p/sub 3/2, yields /sub 1/ 2, excitation (corresponding magnetic radiative transition) the Ml contribution is found to be dominant in the sharply forward region and E2 in the sharply backward region. In the experiraentally observed range of 40 to 90 deg , all three (Coulomb, transverse electric, and transverse magnetic) contributions to the cross section are the same order of magnitude. For a d/sub 5/2 yields s/ sub 1/2 excitation (corresponding electric radiative transition) the Coulomb contribution dominates in the entire forward direction. In the backward},

doi = {},

journal = {Dissertation Abstr.},

number = ,

volume = Vol: 22,

place = {Country unknown/Code not available},

year = {1962},

month = {6}

}