# Formal and Physical -matrix parameters

###### Abstract

Notes from 11 October 2004 lecture presented at the Joint Institute for Nuclear Astrophysics -Matrix School at Notre Dame University.

## I Introduction

The -matrix theory of reactions has proven over the course of time to be very useful in nuclear and atomic physics, both for the fitting of experimental data and as a tool for theoretical calculations. In these notes I will explore the relationship between formal -matrix parameters and “physical” resonance parameters.

## Ii How does one define a resonance energy?

In practice one is likely to encounter many definitions of “the resonance energy” – e.g. the energy where there is a peak of an excitation function, the energy when the phase shift is equal to (for elastic scattering), the complex poles of the scattering matrix,…Many approaches involve the Breit-Wigner resonance formula:

(1) |

where () label the incoming (outgoing) channel, is the incoming wavenumber, is the statistical factor, is the (variable) reaction energy, is the resonance energy, are the observed partial widths of the resonance, and the total observed width is given by sum over all channels of the partial widths: . It should be noted that the Breit-Wigner formula is a very general quantum-mechanical result with applications ranging from atomic physics to particle physics (e.g. the 92-GeV resonance in scattering known as the boson). In some cases the partial widths may be energy dependent. From a time-dependent viewpoint the resonant state decays with time-dependence , where is the mean lifetime and . I will describe below one approach for defining resonance energies and partial widths in the -matrix formalism which is particularly convenient. The differences between various prescriptions for defining resonance energies and partial widths are generally most significant for broad resonances. Let me conclude this section with a caution: since different methods are in use for defining resonance parameters, it is very important to (1) understand what conventions are used by others if you use their results and (2) supply enough information so that others may properly use your results.

## Iii Review of -Matrix theory

I will begin by reviewing some of the notation and results of -matrix theory as described by Lane and Thomas (LT) Lan58 which will be needed later. The matrix is a function of the energy and is defined by

(2) |

where are the level energies, are the reduced width amplitudes, is the level label, and is the channel label. I will assume that the numbers of levels and channels are finite and given by and , respectively. One must also specify the constant(s) , which determine the boundary conditions satisfied by the underlying eigenfunctions. I will also consider here a set of levels which all have the same value (the presence of additional levels with other values would not impact any results discussed here).

When working in the level-matrix framework, it is convenient to define column vectors in level space from the :

(3) |

as well as a diagonal matrix containing the :

(4) |

The level matrix can now be defined by its inverse:

(5) | |||||

(6) |

where is the unit matrix and and are the energy-dependent penetration and shift functions. Note also the notation for any matrix . The collision matrix which determines the cross sections can be calculated from the matrix:

(7) |

where the are defined in terms of Coulomb wavefunctions (see LT, Eq. III.4.5).

## Iv The Single-Level -matrix

In order to interpret the meaning of the -matrix parameters and it is very helpful to consider the case of a single level, i.e. . In this case the matrix is simply a number and the cross section is given by

(8) |

where

(9) | |||||

(10) | |||||

(11) |

The partial and total widths and are known as formal widths. This formula for the cross section is very similar to the Breit-Wigner formula defined above by Eq. (1) (in fact the single-level -matrix formula is sometimes called the Breit-Wigner formula). The relationship between Eqs. (1) and (8) is simplest if the are chosen such that . The energy-dependent quantity can then be expanded in a Taylor series around :

(12) | |||||

(13) |

If we replace with the first non-zero term in the expansion, we have

(14) |

We can now connect the -matrix parameters with the Breit-Wigner resonance parameters. If we identify

(15) | |||||

(16) |

then the one-level -matrix formula [Eq. (8)] and the Breit-Wigner formula [Eq. (1)] are equivalent. The presence of the penetration factor in Eq. (9) gives rise to an energy dependence of the – hence the need for defining and in Eq. (16). Eqs. (15) and (16) can be considered as the definition of the resonance energy and partial widths for the case of a single-level -matrix (i.e. the physical parameters). These relationships are also discussed in Sec. XII.3(a) of LT. The resonance energy defined in this manner may differ from the peak of cross section excitation function due to the energy dependences of the penetration factors and . Note also that the difference between formal and observed partial widths is often on the order of 30% – a large factor compared to typical experimental uncertainties.

## V General definition of physical resonance parameters

When there is more than one level, a one-level approximation to the matrix is very good when is near the -matrix pole energy of a particular level [this approximation becomes exact when ; see Eq. (2)]. Eqs. (15) and (16) from the previous section thus define the physical parameters for that particular level. However, this approach requires defining the boundary condition constants such that . The can only be set once (they are after all “constants”) – therefore we only have a simple relationship between -matrix parameters and physical parameters for one of the levels.

One approach to finding the physical parameters for the remaining levels is to change the to values appropriate for another physical resonance energy. The -matrix parameters and must then be re-determined – either by refitting data or by the transformation procedure discussed in the next section. This procedure must be then be repeated one level at a time until physical parameters have been found for all of the levels. Note also that this procedure is necessarily iterative – we want to set where is a physical resonance energy, but the physical resonance energy is one of the quantities we are seeking to determine. Fortunately an iterative approach starting with the -matrix pole energy converges quickly.

## Vi How to change the

An interesting feature of -matrix theory is that the collision matrix (i.e. the matrix) is invariant under changes in the , provided that the and are suitably adjusted. This result remains true even for the case of finite Bar72 . The transformation is most easily described using matrix equations in level space. Let us consider the transformation , , and . One first constructs the real and symmetric matrix defined by

(17) |

which is diagonalized by the orthogonal matrix such that , with . Note that any real and symmetric matrix can be diagonalized by an orthogonal matrix; an orthogonal matrix satisfies (in other words ). The necessary transformation of the -matrix parameters is then given by Bar72

(18) |

and

(19) |

By using Eq. (6) one can show that these transformations leave and hence [see Eq. (7)] invariant.

The invariance of for changes in has several important consequences. I will highlight two of them: (1) mathematically-equivalent cross section fits should be attainable for any choices and (2) we are free to choose however we desire. We can exploit this freedom by adopting values which are convenient [e.g. ]. This situation is somewhat analogous to that encountered in electromagnetism: transformations of the scalar and vector potentials (known as gauge transformations) can be carried out which leave the physical electric and magnetic fields invariant.

It is instructive to look at the transformation for the case of a single level. The matrix consists of only a single dimension, i.e. a number. The resulting transformation is

(20) | |||||

(21) |

It is straightforward to verify by substitution into the single-level formula [Eq. (8)] that the cross section is invariant.

## Vii Another Approach to Physical Parameters

In this section I will describe another mathematically-equivalent method for extracting physical parameters from -matrix parameters. I begin by defining the real and symmetric matrix :

(22) |

and consider the eigenvalue equation

(23) |

where is the eigenvalue and is the corresponding eigenvector. Note that is implicitly dependent upon through , so the eigenvalue problem is nonlinear. We will assume for convenience that the eigenvectors are normalized so that .

The eigenvalues are in fact just the physical resonance energies. A new set of reduced width parameters is defined via

(24) |

The are equal to the -matrix reduced widths when transformed to . The are sometimes called the “on-resonance” reduced-width amplitudes. The observed (physical) partial widths are thus given by

(25) |

One advantage of this formulation is that the solution of the eigenvalue problem gives immediately all of the physical resonance energies and partial widths.

However one defines the physical resonance parameters, it is desirable that they be independent of the boundary condition constants and channel radii of -matrix theory. The above approach is completely independent of the and approximately independent of the channel radii. Additional details and the justification for this approach are given by Brune Bru02 . The “on-resonance” reduced-width amplitudes defined here are also disused by Barker Bar71 and by Angulo and Descouvemont Ang00 .

I would like to mention in passing another similar eigenvalue equation. If we had instead defined

(26) |

we would have the eigenvalue equation introduced by Hale, Brown, and Jarmie Hal87 . The eigenvalues of this equation are complex and correspond to complex poles of the and matrices (the latter is also known as the scattering matrix). As discussed in Refs. Hal87 ; Til02 this approach to defining resonance energies and widths may be theoretically preferable. However, there are significant complexities associated with calculating the Coulomb wavefunctions for complex momenta. We will not discuss this approach further here.

## Viii Converting Physical parameters to -matrix parameters

Next we will investigate how to translate physical parameters into -matrix parameters Bru02 . It is often desirable to incorporate information from a level compilation or shell model calculation into an -matrix parameterization. Values for must first be converted into reduced widths using Eq. (25). This procedure requires the straightforward solution of a system of linear equations for ; the only complication is that the signs of reduced-width amplitudes cannot be determined. Relative signs can be determined via an -matrix calculation (which automatically includes interference effects between levels) and a comparison to experimental data.

With the physical resonance energies and “on-resonance” reduced-width amplitudes in hand we can determine -matrix parameters. We proceed by defining the matrices and :

(27) |

and

(28) |

where denotes . Note that the construction of requires the adoption of specific values. The eigenvalue equation

(29) |

now holds the key for transforming from the - representation to the standard -matrix parameters and . The eigenvalues are the -matrix pole energies and the eigenvectors yield the -matrix reduced-width amplitudes:

(30) |

The matrices and are real, symmetric, and energy-independent. The solution to this type of eigenvalue problem is discussed in Sec. 8.7.2 of Ref. Gol96 ; I have utilized the LAPACK lapack routine dsygv.

The eigenvectors can be arranged into a square matrix which simultaneously diagonalizes and :

(31) |

and

(32) |

We can also re-write Eq. (30) as

(33) |

## Ix Working Directly With Physical Parameters

It turns to be possible to carry out -matrix calculations directly from the physical parameters Bru02 . This result can be achieved by defining a new (alternative) level matrix from the physical resonance energies and “on-resonance” reduced-width amplitudes :

This matrix is then used with the to calculate the the collision matrix :

(35) |

This method of calculating is mathematically equivalent to first transforming the physical - parameters to -matrix parameters - and then calculating using standard -matrix equations. Note that in this approach the boundary-condition constants never need to be defined at all. One application of this approach is the possibility of using the physical parameters directly as fit parameters.

1 | 2 | 3 | |

(MeV) | |||

(MeV) | |||

(MeV) | |||

(MeV) | |||

(MeV) | |||

(MeV) | |||

(MeV) | |||

(meV) |

## X An Example

At this point it is instructive to introduce a concrete example. The top section of Table 1 shows a simple well-documented set of standard -matrix parameters taken from Azuma et al. Azu94 . This analysis concerns the low-lying states of reacting through channels (, ) and (, multipolarity). The zero of the energy scale is at the threshold. Note that the gamma-ray channel is treated perturbatively and does enter in the calculation of the matrix.

The bottom section of Table 1 shows physical resonance energies and the “on-resonance” reduced-width amplitudes and -decay feeding factors . In addition the physical partial widths are given along with values of needed to related formal and physical widths.

This example is also useful for demonstrating some of the other concepts discussed here. For example one can convert from the - representation back to the standard -matrix - representation. In Table 2 we show the elements of the transformation matrix [see Eqs. (31-33)] needed to carry the transformation back to standard -matrix parameters. One can also verify that the various approaches for calculating the collision matrix [e.g. Eqs. (7) and (35)] yield the same result.

\backslashbox | 1 | 2 | 3 |
---|---|---|---|

1 | |||

2 | |||

3 |

## Xi Summary

Several aspects of the relationship between -matrix parameters and physical resonance parameters have been discussed. First it is necessary to establish how physical resonance energies and partial widths will be defined. By comparing the one-level -matrix formula for the cross section to the Breit-Wigner formula it is possible to define a relationship between -matrix parameters and physical resonance parameters. For the multi-level case the extraction of physical parameters is more complicated but can be achieved by the refitting data with different values, carrying out transformations, or by solving an eigenvalue equation. We next investigated the reverse procedure, i.e. transforming physical resonance parameters into -matrix parameters, which is useful for incorporating information from other sources into an -matrix fit. A method for calculating observables directly from the “on-resonance” -matrix parameters was then discussed. Finally we presented an example case which is useful for demonstrating the ideas which have been presented here.

## References

- (1) A. M. Lane and R. G. Thomas, Rev. Mod. Phys. 30, 257 (1958).
- (2) F. C. Barker, Aust. J. Phys. 25, 341 (1972).
- (3) C. R. Brune, Phys. Rev. C 66, 044611 (2002).
- (4) F. C. Barker, Aust. J. Phys 24, 777 (1971). The quantities and of Ref. Bar71 correspond to and in the present work.
- (5) C. Angulo and P. Descouvemont, Phys. Rev. C 61, 064611 (2000). We have utilized a notation somewhat different from Ref. Ang00 , in order that we could retain the well-established notation of Lane and Thomas Lan58 . The quantities and of Ref. Ang00 correspond to and in the present work. Note also that the procedure of Ref. Ang00 is approximate and only considers the single-channel case.
- (6) G. M. Hale, R. E. Brown, and N. Jarmie, Phys. Rev. Lett. 59, 763 (1987).
- (7) D. R. Tilley, C. M. Cheves, J. L. Godwin, G. M. Hale, H. M. Hofmann, J. H. Kelley, C. G. Sheu, and H. R. Weller, Nucl. Phys. A708, 3 (2002).
- (8) G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd Ed. (Johns Hopkins University Press, Baltimore, 1996).
- (9) E. Anderson Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, D. Sorensen, LAPACK Users’ Guide, 3rd Ed. (Society for Industrial and Applied Mathematics, Philadelphia, 1999).
- (10) R. E. Azuma, L. Buchmann, F. C. Barker, C. A. Barnes, J. M. D’Auria, M. Dombsky, U. Giesen, K. P. Jackson, J. D. King, R. G. Korteling, P. McNeely, J. Powell, G. Roy, J. Vincent, T. R. Wang, S. S. M. Wong, and P. R. Wrean, Phys. Rev. C 50, 1194 (1994).