## Abstract

The starting point of all versions of the shell model is the physical idea that the interaction between a given nucleon and all the others resembles that between a nucleon and a fixed field. From this starting point one might attempt to construct a field which is self-consistent but this approach is not followed in most shell-model calculations because of the complications that arise. The more usual approach has been to use the idea of an average field to provide a complete set of sin gle-particle wave functions. Then, if the parameters of the field (e.g. its size) are correctly chosen, we would expect to reach a good approximation to the nuclear-wave function by taking that configuration of single-particle wave functions which has lowest energy in this field. The wave functions could clearly be improved by allowing the mixing of excited configurations but this is rarely done because of the resulting complexity of the problem. Even in the lowest configuration there are in general many independent wave functions for a many-particle system which would all be degenerate in the average field. To find the nuclear energy levels and wave functions we must therefore build up the energy matrix in this
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## Citation Formats

Elliott, J. P.
The Nuclear Shell Model and its Relation with Other Nuclear Models.
IAEA: N. p.,
1963.
Web.

Elliott, J. P.
The Nuclear Shell Model and its Relation with Other Nuclear Models.
IAEA.

Elliott, J. P.
1963.
"The Nuclear Shell Model and its Relation with Other Nuclear Models."
IAEA.

@misc{etde_22095803,

title = {The Nuclear Shell Model and its Relation with Other Nuclear Models}

author = {Elliott, J. P.}

abstractNote = {The starting point of all versions of the shell model is the physical idea that the interaction between a given nucleon and all the others resembles that between a nucleon and a fixed field. From this starting point one might attempt to construct a field which is self-consistent but this approach is not followed in most shell-model calculations because of the complications that arise. The more usual approach has been to use the idea of an average field to provide a complete set of sin gle-particle wave functions. Then, if the parameters of the field (e.g. its size) are correctly chosen, we would expect to reach a good approximation to the nuclear-wave function by taking that configuration of single-particle wave functions which has lowest energy in this field. The wave functions could clearly be improved by allowing the mixing of excited configurations but this is rarely done because of the resulting complexity of the problem. Even in the lowest configuration there are in general many independent wave functions for a many-particle system which would all be degenerate in the average field. To find the nuclear energy levels and wave functions we must therefore build up the energy matrix in this degenerate set, using the inter-nucleon two-body forces, and then diagonalize this matrix. If the detailed form of the nuclear forces was known we might regard such calculations as the first step towards an exact calculation in which higher configurations were included but every indication is that the convergence would be extremely slow. It is more usual to treat an energy calculation in the lowest configuration unashamedly as a model calculation and to attempt to deduce, by comparisons with experimental data in the many-particle nuclei, the nature of the effective nuclear forces required in that configuration. If the model is realistic then we should not expect these effective forces to change very much in going from one nucleus to its neighbour and since there are many more pieces of data than available parameters we may make significant predictions and thus test the model. Even within this class of model calculations there are different philosophies. At one extreme is the Israel group, TALMI, DE-SHALIT and co-workers who keep rigidly to the lowest j{sup k} configuration. This has the great advantage that very few matrix elements of the Hamiltonian are involved and these may be deduced from a fit to the known spectra. It is, however, well known that such simple wave functions give poor agreement with transition rates and moments if the real operators for these processes are used. They must therefore try to extract the matrix elements of model moment operators also from the data. If one takes a more general model, allowing mixing of the lowest configurations, it is no longer possible to deduce all the required matrix elements of the Hamiltonian as there are so many. One must then resort to a definite assumption of a Hamiltonian with possibly a few parameters, such as range and exchange properties, to be chosen. Although such an approach, which is the one I usually take, is not designed to give close fits to the spectra, one finds reasonable agreement and, in addition, the moments and transition data are generally predicted correctly using the real operators, suggesting that the wave functions are a little nearer the truth than in a pure configuration. In these lectures I shall describe some of the group theoretical techniques used in classifying states of a pure configuration and of mixed configurations and in calculating energy matrices. In some cases this will lead to a description of collective behaviour and to a connection with other nuclear models. (author)}

place = {IAEA}

year = {1963}

month = {Jan}

}

title = {The Nuclear Shell Model and its Relation with Other Nuclear Models}

author = {Elliott, J. P.}

abstractNote = {The starting point of all versions of the shell model is the physical idea that the interaction between a given nucleon and all the others resembles that between a nucleon and a fixed field. From this starting point one might attempt to construct a field which is self-consistent but this approach is not followed in most shell-model calculations because of the complications that arise. The more usual approach has been to use the idea of an average field to provide a complete set of sin gle-particle wave functions. Then, if the parameters of the field (e.g. its size) are correctly chosen, we would expect to reach a good approximation to the nuclear-wave function by taking that configuration of single-particle wave functions which has lowest energy in this field. The wave functions could clearly be improved by allowing the mixing of excited configurations but this is rarely done because of the resulting complexity of the problem. Even in the lowest configuration there are in general many independent wave functions for a many-particle system which would all be degenerate in the average field. To find the nuclear energy levels and wave functions we must therefore build up the energy matrix in this degenerate set, using the inter-nucleon two-body forces, and then diagonalize this matrix. If the detailed form of the nuclear forces was known we might regard such calculations as the first step towards an exact calculation in which higher configurations were included but every indication is that the convergence would be extremely slow. It is more usual to treat an energy calculation in the lowest configuration unashamedly as a model calculation and to attempt to deduce, by comparisons with experimental data in the many-particle nuclei, the nature of the effective nuclear forces required in that configuration. If the model is realistic then we should not expect these effective forces to change very much in going from one nucleus to its neighbour and since there are many more pieces of data than available parameters we may make significant predictions and thus test the model. Even within this class of model calculations there are different philosophies. At one extreme is the Israel group, TALMI, DE-SHALIT and co-workers who keep rigidly to the lowest j{sup k} configuration. This has the great advantage that very few matrix elements of the Hamiltonian are involved and these may be deduced from a fit to the known spectra. It is, however, well known that such simple wave functions give poor agreement with transition rates and moments if the real operators for these processes are used. They must therefore try to extract the matrix elements of model moment operators also from the data. If one takes a more general model, allowing mixing of the lowest configurations, it is no longer possible to deduce all the required matrix elements of the Hamiltonian as there are so many. One must then resort to a definite assumption of a Hamiltonian with possibly a few parameters, such as range and exchange properties, to be chosen. Although such an approach, which is the one I usually take, is not designed to give close fits to the spectra, one finds reasonable agreement and, in addition, the moments and transition data are generally predicted correctly using the real operators, suggesting that the wave functions are a little nearer the truth than in a pure configuration. In these lectures I shall describe some of the group theoretical techniques used in classifying states of a pure configuration and of mixed configurations and in calculating energy matrices. In some cases this will lead to a description of collective behaviour and to a connection with other nuclear models. (author)}

place = {IAEA}

year = {1963}

month = {Jan}

}