Pandya theorem
Encyclopedia
Pandya theorem provides a theoretical framework for connecting the energy level
s in jj coupling of a nucleon
-nucleon and nucleon-hole system. It is also referred to as Pandya Transformation or Pandya Relation in literature. It provides a very useful tool for extending shell model
calculations across shells, for systems involving both particles and holes.
The Pandya theorem is a good illustration of the richness of information forthcoming from a judicious use of subtle symmetry principles
connecting vastly different sectors of nuclear systems.
The Pandya transformation, which involves angular momentum
re-coupling coefficients, can be used to deduce one-particle one-hole (ph) matrix
elements.
By assuming the wave function to be ”pure” (no configuration mixing), Pandya transformation could be used to set an upper bound
to the contributions of 3-body forces to the energies of nuclear states.
Nucleon-Hole Interaction in jj Coupling
S.P. Pandya, Phys. Rev. 103, 956 (1956).
Received 9 May 1956
A theorem connecting the energy levels in jj coupling of a nucleon-nucleon and nucleon-hole system is derived, and applied in particular to Cl38 and K40.
that the division into ``particle-hole" and ``particle-particle" channels appears naturally.
(i) It relates diagonal and non-diagonal elements.
(ii) To calculate any particle-hole element, the particle-particle elements for all spins belonging to the orbitals involved are needed; the same holds for the reverse transformation. Because the experimental information is nearly always incomplete, one can only transform from the theoretical particle-particle elements to particle-hole.
(iii) The Pandya transform does not describe the matrix elements that mix one-particle one-hole and two-particle two-hole states. Therefore only states of rather pure one-particle one-hole structure can be treated.
Pandya theorem establishes a relation between particle-particle and particle-hole spectra. Here one considers the energy levels of two nucleons with one in orbit j and another in orbit j and relate them to the energy levels of a nucleon hole in orbit j and a nucleus in j. Assuming pure j-j coupling and two-body interaction, Pandya (1956) derived the following relation:
This was successfully tested in the spectra of
Figure 3 shows the results where the discrepancy between the calculated and observed spectra is less than 25 KeV.
Energy level
A quantum mechanical system or particle that is bound -- that is, confined spatially—can only take on certain discrete values of energy. This contrasts with classical particles, which can have any energy. These discrete values are called energy levels...
s in jj coupling of a nucleon
Nucleon
In physics, a nucleon is a collective name for two particles: the neutron and the proton. These are the two constituents of the atomic nucleus. Until the 1960s, the nucleons were thought to be elementary particles...
-nucleon and nucleon-hole system. It is also referred to as Pandya Transformation or Pandya Relation in literature. It provides a very useful tool for extending shell model
Shell model
In nuclear physics and nuclear chemistry, the nuclear shell model is a model of the atomic nucleus which uses the Pauli exclusion principle to describe the structure of the nucleus in terms of energy levels. The first shell model was proposed by Dmitry Ivanenko in 1932...
calculations across shells, for systems involving both particles and holes.
The Pandya theorem is a good illustration of the richness of information forthcoming from a judicious use of subtle symmetry principles
Symmetry
Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection...
connecting vastly different sectors of nuclear systems.
The Pandya transformation, which involves angular momentum
Angular momentum
In physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...
re-coupling coefficients, can be used to deduce one-particle one-hole (ph) matrix
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
elements.
By assuming the wave function to be ”pure” (no configuration mixing), Pandya transformation could be used to set an upper bound
Upper bound
In mathematics, especially in order theory, an upper bound of a subset S of some partially ordered set is an element of P which is greater than or equal to every element of S. The term lower bound is defined dually as an element of P which is lesser than or equal to every element of S...
to the contributions of 3-body forces to the energies of nuclear states.
History
It was first published in 1956 as follows:Nucleon-Hole Interaction in jj Coupling
S.P. Pandya, Phys. Rev. 103, 956 (1956).
Received 9 May 1956
A theorem connecting the energy levels in jj coupling of a nucleon-nucleon and nucleon-hole system is derived, and applied in particular to Cl38 and K40.
Shell model Monte Carlo approaches to nuclear level densities
Since it is by no means obvious how to extract "pairing correlations" from the realistic shell-model calculations, Pandya transform is applied in such cases. The "pairing Hamiltonian" is an integral part of the residual shell-model interaction. The shell-model Hamiltonian is usually written in the p-p representation, but it also can be transformed to the p-h representation by means of the Pandya transformation. This means that the high-J interaction between pairs can translate into the low-J interaction in the p-h channel. It is only in the mean-field theoryMean field theory
Mean field theory is a method to analyse physical systems with multiple bodies. A many-body system with interactions is generally very difficult to solve exactly, except for extremely simple cases . The n-body system is replaced by a 1-body problem with a chosen good external field...
that the division into ``particle-hole" and ``particle-particle" channels appears naturally.
Features
Some features of the Pandya transformation are as follows:(i) It relates diagonal and non-diagonal elements.
(ii) To calculate any particle-hole element, the particle-particle elements for all spins belonging to the orbitals involved are needed; the same holds for the reverse transformation. Because the experimental information is nearly always incomplete, one can only transform from the theoretical particle-particle elements to particle-hole.
(iii) The Pandya transform does not describe the matrix elements that mix one-particle one-hole and two-particle two-hole states. Therefore only states of rather pure one-particle one-hole structure can be treated.
Pandya theorem establishes a relation between particle-particle and particle-hole spectra. Here one considers the energy levels of two nucleons with one in orbit j and another in orbit j and relate them to the energy levels of a nucleon hole in orbit j and a nucleus in j. Assuming pure j-j coupling and two-body interaction, Pandya (1956) derived the following relation:
This was successfully tested in the spectra of
Figure 3 shows the results where the discrepancy between the calculated and observed spectra is less than 25 KeV.