Resonance Widths and Unitarity

1972 ◽  
Vol 50 (2) ◽  
pp. 84-92 ◽  
Author(s):  
C. T. Tindle
Keyword(s):  
Energy Dependence ◽  
Cross Section ◽  
Matrix Theory ◽  
Neutron Cross Section ◽  
Matrix Formalism ◽  
R Matrix ◽  
Level Width ◽  
S Matrix ◽  
The Difference ◽  
Energy Dependent

The low energy neutron cross section of 135Xe is analyzed using both the R-matrix theory of Wigner and Eisenbud and the S-matrix theory of Humblet and Rosenfeld. Particular attention is given to the role played by the total resonance level width for it is well known that the R-matrix widths are energy dependent but the S-matrix widths are not. This different energy dependence leads to different analytic forms for the cross section and the n + 135Xe reaction offers what may be the simplest and best physical example for comparing these two forms. To the accuracy of the present data the difference is not detectable. The different energy dependence of the resonance widths is shown to be related to unitarity. A general proof that the R-matrix formalism is always unitary is given. The difficulty of satisfying unitarity in the S-matrix formalism is discussed and it is shown for the n + 135Xe reactions that this can lead to physically unacceptable solutions. This "lack of unitarity" does not, however, lead to any difficulties in fitting the present experimental data.

LOW ENERGY NEUTRON CROSS SECTION OF MANGANESE

1955 ◽  
Vol 33 (11) ◽  
pp. 622-634 ◽  
Author(s):  
R. Krotkov
Keyword(s):  
Cross Section ◽  
Resonance Energy ◽  
Neutron Cross Section ◽  
Coherent Scattering ◽  
High Energy ◽  
Positive Energy ◽  
Matrix Formalism ◽  
Perfect Agreement ◽  
R Matrix ◽  
Section Curve

The total neutron cross section of Mn55 has three (predominantly scattering) resonances in the energy range 300–2500 ev. In the present paper these resonances are examined in detail from the point of view of the R matrix formalism. It is found that quite good (but not perfect) agreement with the experimental cross section curve is obtained by assigning (in order of increasing resonance energy) spins (2, 3, 3) and neutron widths (23, 11, 395 ev.) to the resonances. The addition of a constant increment ΔR11 = 0.16 × 10−12 cm. representing the contribution of distant, positive energy, J = 3 resonances to the R matrix element for s-neutron scattering improves agreement with experiment on the high energy side of the highest energy resonance. The (n, γ) cross section at 0.025 ev. is fitted well by a constant radiation width of 0.5 ev. However, with no set of parameters was it possible to fit both the experimental total cross section curve, and the observed coherent scattering cross section at 0 ev., although the sign of the latter was accounted for correctly. The discrepancy is 2.0 barns (calculated) compared to 1.7 barns (observed).

scholarly journals Application of Reaction Matrix Theory to Meson Processes

1968 ◽  
Vol 21 (6) ◽  
pp. 769
Author(s):  
JL Cook
Keyword(s):  
Boundary Conditions ◽  
Matrix Theory ◽  
Matrix Formalism ◽  
Pion Cloud ◽  
R Matrix ◽  
Reaction Matrix ◽  
Pion Nucleon ◽  
Full Analysis

The possible application of reaction matrix theory to pion-nucleon processes is discussed. It is found that suitable boundary conditions can remove the effects of the nucleon-pion cloud, leading to the conventional R-matrix formalism. A simple dynamical channel.coupling scheme is proposed which permits full analysis of experiments.

scholarly journals Interference Between Levels in Nuclear Reactions

1967 ◽  
Vol 20 (3) ◽  
pp. 341 ◽  
Author(s):  
FC Barker
Keyword(s):  
Nuclear Reaction ◽  
Cross Section ◽  
Nuclear Reactions ◽  
Matrix Theory ◽  
R Matrix ◽  
Single Level ◽  
The Cross

In this note we consider the cross section for a nuclear reaction in which one of the product nuclei is unstable, with two or more levels contributing to its decay. Previously a formula had been derived from R-matrix theory for the case where contributions come from only a single level of the nucleus with a given spin and parity.

scholarly journals Study of the (n,γf) process on 239Pu

2019 ◽  
Vol 211 ◽  
pp. 02004 ◽  
Author(s):  
Esther Leal-Cidoncha ◽  
Gilles Noguere ◽  
Olivier Bouland ◽  
Olivier Serot
Keyword(s):  
Cross Section ◽  
Matrix Theory ◽  
Fission Cross Section ◽  
Neutron Multiplicity ◽  
Prompt Neutron ◽  
Neutron Resonance ◽  
Fission Reactions ◽  
R Matrix

In the neutron resonance range, fission cross section of 239Pu can be seen as a sum of the immediate (n,f) and the two-step (n,γf) fission reactions. In that case, five channel widths should be considered for a proper evaluation, those are: two opened fission channels for Jπ = 0+, one opened fission channel for Jπ = 1+ and two J-dependent for the (n,γf) reaction. The sizeable contribution of the (n,γf) process should have an impact on the determination of the capture and fission widths involved in the Reich-Moore approximation of the R-matrix theory. The present work aims to investigate this impact by using the CONRAD code and the Γγf available from literature. Prompt neutron multiplicity (νp) has been also reproduced including the contributions of the (n,γf) process.

On threshold resonances

1970 ◽  
Vol 48 (15) ◽  
pp. 1747-1758 ◽  
Author(s):  
C. T. Tindle
Keyword(s):  
Cross Sections ◽  
Matrix Theory ◽  
Threshold Level ◽  
R Matrix ◽  
Neutral Particles ◽  
S Matrix ◽  
Energy Neutron ◽  
Square Well ◽  
Neutron Cross Sections ◽  
Threshold Resonances

A comparison is made of the way R-matrix and S-matrix theories analyze the threshold resonances which occur in the scattering of neutral particles by a square well. Both approaches are found to provide very good approximate formulas. However, modifications of the usual S-matrix expansions must first be made. The behavior of the energy of the threshold level is quite different in the two alternatives. By comparing the two theories in their interpretation of the low-energy neutron cross sections of 1H, 16O, and 208Pb it is concluded that.R-matrix theory provides a better interpretation for unbound levels and S-matrix theory is preferable when the threshold level is bound.

THEORETICAL INVESTIGATIONS OF THE 12C(α,γ)16OE2 CROSS SECTION

2008 ◽  
Vol 17 (10) ◽  
pp. 2176-2181 ◽  
Author(s):  
MARIANNE DUFOUR ◽  
PIERRE DESCOUVEMONT
Keyword(s):  
Cross Section ◽  
Matrix Theory ◽  
Cluster State ◽  
Matrix Analysis ◽  
State Transitions ◽  
R Matrix ◽  
S Factor ◽  
Theoretical Investigations ◽  
Cascade Transitions ◽  
Asymptotic Normalization Constant

The E2 component of the 12 C (α,γ)16 O cross section is investigated in three ways: by a microscopic cluster model, by R-matrix fits and by a combination of both. The microscopic calculation leads to an estimate of the S-factor at a typical energy of 300 keV of SE2(300 keV )≈ 50 keV-b for ground-state transitions. Cascade transitions to the [Formula: see text] and [Formula: see text] excited states of 16 O are also studied. Then the S-factor is analyzed in the phenomenological R-matrix theory. We show that the background term plays a crucial role, and cannot be determined without ambiguity. Consequently only an upper limit on the extrapolated S-factor can be obtained [SE2(300 keV )< 190 keV-b ]. Finally, we use the microscopic Asymptotic Normalization Constant (ANC) of the [Formula: see text] level, well known to be a cluster state to constrain the R-matrix analysis. This procedure strongly reduces the uncertainties on the R-matrix fits, and we end up with a recommended value of SE2(300 keV ) = 42 ± 2 keV-b .

S-matrix theory and the density matrix formalism

1979 ◽  
Vol 38 (2) ◽  
pp. 553-560 ◽  
Author(s):  
M. Roberts ◽  
W.E. Hagston
Keyword(s):  
Density Matrix ◽  
Matrix Theory ◽  
Matrix Formalism ◽  
Density Matrix Formalism ◽  
S Matrix

scholarly journals A Method for Inverting the Channel Matrix in R-Matrix Theory

1972 ◽  
Vol 25 (3) ◽  
pp. 253
Author(s):  
WK Bertram
Keyword(s):  
Recurrence Relation ◽  
Matrix Theory ◽  
Matrix Formalism ◽  
R Matrix ◽  
Channel Matrix

A technique is presented for inverting the channel matrix which occurs in R-matrix theory. The collision matrix is expressed in a form which does not involve matrix inversions; it is evaluated by the application of a recurrence relation. The connection between this method and the level matrix formalism is examined.

scholarly journals The low-energy 9Be(g, n)8Be cross section.

2000 ◽  
Vol 53 (2) ◽  
pp. 247 ◽  
Author(s):  
F. C. Barker
Keyword(s):  
Cross Section ◽  
Radioactive Isotope ◽  
Low Energy ◽  
Inelastic Electron ◽  
R Matrix ◽  
Section Data ◽  
Isotope Data ◽  
Particle Potential ◽  
The Difference ◽  
Parameter Values

Fits are made to low-energy 9 Be(g;n) 8 Be cross-section data using one-level R-matrix formulae including channel contributions. Fits with reasonable parameter values are found for the newer radioactive-isotope data, and also for data obtained from inelastic electron scattering on 9 Be, but not for older radioactive-isotope data. This differs from the result of recent fits using a semi-microscopic model, which supported the older data. The difference is attributed to the use in the latter calculation of a single-particle potential description of the continuum wave function.

scholarly journals ELASTIC BACKSCATTERED CROSS-SECTIONS OF 14N(P,P0)14N AT HIGH ANGLES

2020 ◽  
Vol 20 (3) ◽  
pp. 749-754
Author(s):  
MOHAMED ELTAYEB M. EISA ◽  
JOHAN ANDRE MARS ◽  
MUSTAFA J. ABUALREISH ◽  
MARWA L. WAREGH
Keyword(s):  
Cross Section ◽  
Cross Sections ◽  
Ion Beam ◽  
Region Of Interest ◽  
Laboratory Frame ◽  
Matrix Formalism ◽  
Cross Section Data ◽  
R Matrix ◽  
Section Data ◽  
Elastic Backscattering

The importance and present needs of proton cross section data of nitrogen needed by the Ion Beam Analysis (IBA) community are briefly reviewed. Previous experimental data presently used for the theoretical determination of the proton cross-sections are discussed. The Azure code based on the R-matrix formalism was then used to evaluate the data and to determine the nitrogen cross section in the previous and presently desired angular domain and energy region of interest. The experimental elastic backscattering cross section data, as spectra, for back-scattering analysis determined at angles in the laboratory frame of reference, θi,lab, of 165o, 170o and 176o are presented.

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