Owned and operated as a joint venture by a consortium of Canadian universities via a contribution through the National Research Council Canada Propriété.

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1 Owned and operated as a joint venture by a consortium of Canadian universities via a contribution through the National Research Council Canada Propriété d’un consortium d’universités canadiennes, géré en co-entreprise à partir d’une contribution administrée par le Conseil national de recherches Canada Canada’s national laboratory for particle and nuclear physics Laboratoire national canadien pour la recherche en physique nucléaire et en physique des particules Accelerating Science for Canada Un accélérateur de la démarche scientifique canadienne Low energy reaction theory: The forsaken realm Lothar Buchmann TRIUMF

2 Owned and operated as a joint venture by a consortium of Canadian universities via a contribution through the National Research Council Canada Propriété d’un consortium d’universités canadiennes, géré en co-entreprise à partir d’une contribution administrée par le Conseil national de recherches Canada Canada’s national laboratory for particle and nuclear physics Laboratoire national canadien pour la recherche en physique nucléaire et en physique des particules Accelerating Science for Canada Un accélérateur de la démarche scientifique canadienne What kind of nuclear reactions do we encounter? Direct measurements: Low energy scattering Continuous cross section Narrow resonances Wide resonances Direct reactions Extrapolation Low energy reaction theory No extrapolation Nuclear structure theory

3 Owned and operated as a joint venture by a consortium of Canadian universities via a contribution through the National Research Council Canada Propriété d’un consortium d’universités canadiennes, géré en co-entreprise à partir d’une contribution administrée par le Conseil national de recherches Canada Canada’s national laboratory for particle and nuclear physics Laboratoire national canadien pour la recherche en physique nucléaire et en physique des particules Accelerating Science for Canada Un accélérateur de la démarche scientifique canadienne Endangered species act While with the resurgence of low energy physics in the context of radioactive beams facilities the experimental data are growing, there are very few low energy reaction theorists left. In 10-20 years there may be none left. Experi- menters muddle through

4 Owned and operated as a joint venture by a consortium of Canadian universities via a contribution through the National Research Council Canada Propriété d’un consortium d’universités canadiennes, géré en co-entreprise à partir d’une contribution administrée par le Conseil national de recherches Canada Canada’s national laboratory for particle and nuclear physics Laboratoire national canadien pour la recherche en physique nucléaire et en physique des particules Accelerating Science for Canada Un accélérateur de la démarche scientifique canadienne Why do we need low energy scattering theory: 1.Data need to be interpreted. While models may fail the test of being strictly reality, it always has been the goal of physics, to unify observations by simple, all encompassing models. Typically, a first reduction of scattering data is to obtain nuclear potentials and eigenstates of those potentials. 2.The need for predictions, or in nuclear astrophysics, extrapolations. Frequently nuclear cross sections are needed where they cannot be measured. Sometimes other quantities, even at zero energy are also important to characterize a reaction. The task is to make extrapolations as reliable as possible. While there are many approaches to it no king’s way has been determined yet for this problem. Worse, errors on extrapolations can only be derived within a certain approximation used for extrapolation, but the possible errors of the model can not be computed.

5 Owned and operated as a joint venture by a consortium of Canadian universities via a contribution through the National Research Council Canada Propriété d’un consortium d’universités canadiennes, géré en co-entreprise à partir d’une contribution administrée par le Conseil national de recherches Canada Canada’s national laboratory for particle and nuclear physics Laboratoire national canadien pour la recherche en physique nucléaire et en physique des particules Accelerating Science for Canada Un accélérateur de la démarche scientifique canadienne An amateur view on nuclear scattering Most things we do in nuclear physics is scattering: the view with the microscope.` Nuclei are quantum systems: typically very few (low) angular moments are selected in nuclear reactions  phaseshifts. Radial Schrödinger equation

6 Owned and operated as a joint venture by a consortium of Canadian universities via a contribution through the National Research Council Canada Propriété d’un consortium d’universités canadiennes, géré en co-entreprise à partir d’une contribution administrée par le Conseil national de recherches Canada Canada’s national laboratory for particle and nuclear physics Laboratoire national canadien pour la recherche en physique nucléaire et en physique des particules Accelerating Science for Canada Un accélérateur de la démarche scientifique canadienne Introducing the jargon: Phaseshifts For elastic scattering: energy (in cm) is conserved for scattered particle, thus the (de Broglie) wavelength is conserved. The only effect of elastic scattering is a shift in phase in the periodic wave function. Phaseshifts are intimately connected to nuclear potentials and eigenstates. For non charged particles is (spin 0): This sum breaks down typically to a few or even one angular momentum. Thus maximum cross section at 90 deg  resonance  eigenstate. Typical 1 - resonances in 12 C(α, ɣ) 16 O. Nota bene: Phaseshifts for charged particles contain Coulomb dependent parts.

7 Owned and operated as a joint venture by a consortium of Canadian universities via a contribution through the National Research Council Canada Propriété d’un consortium d’universités canadiennes, géré en co-entreprise à partir d’une contribution administrée par le Conseil national de recherches Canada Canada’s national laboratory for particle and nuclear physics Laboratoire national canadien pour la recherche en physique nucléaire et en physique des particules Accelerating Science for Canada Un accélérateur de la démarche scientifique canadienne The scattering length The concept of the scattering length comes from neutron physics, where phaseshifts can be easily derived at zero energy. For non charged particles is (for the s wave): With a 0 being the scattering length and r 0 being the interaction radius. A positive value of a 0 corresponds to an infinite (attractive) square potential of the same radius, a negative a 0 corresponds to an infinite (repulsive) square potential of the same radius. For charged particles a Coulomb correction is necessary: Thus l is a linear function of the energy. Sommerfeld parameter

8 Owned and operated as a joint venture by a consortium of Canadian universities via a contribution through the National Research Council Canada Propriété d’un consortium d’universités canadiennes, géré en co-entreprise à partir d’une contribution administrée par le Conseil national de recherches Canada Canada’s national laboratory for particle and nuclear physics Laboratoire national canadien pour la recherche en physique nucléaire et en physique des particules Accelerating Science for Canada Un accélérateur de la démarche scientifique canadienne The transition matrix: Typically, there are many channels open to a reaction, even if only elastic scattering is allowed as reaction channels. This is due to the fact that in many cases non zero spins are present, and each spin combination presents its own separate channel. However, as processes are coherent, mixed terms between different channel combinations are also possible. Mixing can be described e.g. by the concept of mixing angles. Due to different angular moments, the resulting cross sections typically show an angular dependence beyond Rutherford scattering. So far scattering matrices are an abstract concept and need to be brought in a form that allows relatively simple terms for input. These are (here): The phaseshift approach The R-matrix approximation Example: 7 Be(3/2 - )+p(1/2 + )  Channel spin: s=1 and s=2 (π=-1).

9 Owned and operated as a joint venture by a consortium of Canadian universities via a contribution through the National Research Council Canada Propriété d’un consortium d’universités canadiennes, géré en co-entreprise à partir d’une contribution administrée par le Conseil national de recherches Canada Canada’s national laboratory for particle and nuclear physics Laboratoire national canadien pour la recherche en physique nucléaire et en physique des particules Accelerating Science for Canada Un accélérateur de la démarche scientifique canadienne Nuclear part of the collision matrix: with the mixing coefficients ( 7 Be+p case): Formalism by R.G. Seyler: c, p order parameters. Phaseshift analysis: 7 Be case: Number of angular moments included. Phases are eigenphases for each angular momentum and channel spin combination.

10 Owned and operated as a joint venture by a consortium of Canadian universities via a contribution through the National Research Council Canada Propriété d’un consortium d’universités canadiennes, géré en co-entreprise à partir d’une contribution administrée par le Conseil national de recherches Canada Canada’s national laboratory for particle and nuclear physics Laboratoire national canadien pour la recherche en physique nucléaire et en physique des particules Accelerating Science for Canada Un accélérateur de la démarche scientifique canadienne The collision matrix can be expressed in R-matrix terms as: The summation runs over states. It is And the inverse of the state (pole) matrix: hardsphere phaseshift R-matrix approach: Reduced width (state) amplitudes Energy independent.

11 Owned and operated as a joint venture by a consortium of Canadian universities via a contribution through the National Research Council Canada Propriété d’un consortium d’universités canadiennes, géré en co-entreprise à partir d’une contribution administrée par le Conseil national de recherches Canada Canada’s national laboratory for particle and nuclear physics Laboratoire national canadien pour la recherche en physique nucléaire et en physique des particules Accelerating Science for Canada Un accélérateur de la démarche scientifique canadienne Example: 12 C(α,α) 12 C Excitation ratio: 89 o to 58 o. 12 C+α leads to the tightly bound 16 O. It shows a series of narrow or medium width resonances. The scattering is well described by the R-matrix formalism (see fit) including a hard sphere potential. The s wave follows essentially hard sphere. However, the use of a hard sphere potential is not always justified.

12 Owned and operated as a joint venture by a consortium of Canadian universities via a contribution through the National Research Council Canada Propriété d’un consortium d’universités canadiennes, géré en co-entreprise à partir d’une contribution administrée par le Conseil national de recherches Canada Canada’s national laboratory for particle and nuclear physics Laboratoire national canadien pour la recherche en physique nucléaire et en physique des particules Accelerating Science for Canada Un accélérateur de la démarche scientifique canadienne Example: 7 Be(p,p) 7 Be: Theoretical s-wave prediction: Descouvement Navratil (2010) In general, these s waves cannot be fitted by R-matrix expressions. 8 B is a very weakly bound nucleus! Hard sphere phaseshift

13 Owned and operated as a joint venture by a consortium of Canadian universities via a contribution through the National Research Council Canada Propriété d’un consortium d’universités canadiennes, géré en co-entreprise à partir d’une contribution administrée par le Conseil national de recherches Canada Canada’s national laboratory for particle and nuclear physics Laboratoire national canadien pour la recherche en physique nucléaire et en physique des particules Accelerating Science for Canada Un accélérateur de la démarche scientifique canadienne Derivation of phaseshifts in 7 Be(p,p) 7 Be: A phaseshift analysis using all 41 parameters resulting from above expressions will result in a very good fit to the data, if done for individual points. However, neighbouring points in energy and angle can deviate considerably in phases and mixing angles. As both of them are expected to show a smooth angle and angular dependence, a global fit to all data with less parameters is required. However, it can be shown in single fits that the mixing angle parameters and the imaginary phaseshifts are largely irrelevant to describe the cross section. For our global fits, three forms of the phaseshift are chosen: Polynomial approach. For the s wave it can be shown that b 0 ~a 0. Fits Navratil s-wave perfectly well. Or Resonance approach. (i) ɣ λ, E λ real; (ii) ɣ λ, E λ complex.

14 Owned and operated as a joint venture by a consortium of Canadian universities via a contribution through the National Research Council Canada Propriété d’un consortium d’universités canadiennes, géré en co-entreprise à partir d’une contribution administrée par le Conseil national de recherches Canada Canada’s national laboratory for particle and nuclear physics Laboratoire national canadien pour la recherche en physique nucléaire et en physique des particules Accelerating Science for Canada Un accélérateur de la démarche scientifique canadienne Poles and Resonances The R-matrix approach produces poles of the function for In contrast, e.g. to the K matrix, where Describes a resonance as i.e. the definitions are not equal. J. Humblet et al. (1998) also find that in neither theory poles necessarily correspond to resonances, even with ɣ 2 >0 for R-matrix terms.

15 Owned and operated as a joint venture by a consortium of Canadian universities via a contribution through the National Research Council Canada Propriété d’un consortium d’universités canadiennes, géré en co-entreprise à partir d’une contribution administrée par le Conseil national de recherches Canada Canada’s national laboratory for particle and nuclear physics Laboratoire national canadien pour la recherche en physique nucléaire et en physique des particules Accelerating Science for Canada Un accélérateur de la démarche scientifique canadienne Phaseshift for 7 Be(p,p) 7 Be for Oak Ridge data:

16 Owned and operated as a joint venture by a consortium of Canadian universities via a contribution through the National Research Council Canada Propriété d’un consortium d’universités canadiennes, géré en co-entreprise à partir d’une contribution administrée par le Conseil national de recherches Canada Canada’s national laboratory for particle and nuclear physics Laboratoire national canadien pour la recherche en physique nucléaire et en physique des particules Accelerating Science for Canada Un accélérateur de la démarche scientifique canadienne Scattering lengths in 7 Be(p,p) 7 Be (discussion): The l-function for the previous (s wave) phaseshifts. Correlation between a 0 and b 0 for a penetration-polynomial fit.

17 Owned and operated as a joint venture by a consortium of Canadian universities via a contribution through the National Research Council Canada Propriété d’un consortium d’universités canadiennes, géré en co-entreprise à partir d’une contribution administrée par le Conseil national de recherches Canada Canada’s national laboratory for particle and nuclear physics Laboratoire national canadien pour la recherche en physique nucléaire et en physique des particules Accelerating Science for Canada Un accélérateur de la démarche scientifique canadienne Scattering lengths in 7 Be(p,p) 7 Be (results): Least squares dependence of a 2-,0. a 2- and a 1- correlation.

18 Owned and operated as a joint venture by a consortium of Canadian universities via a contribution through the National Research Council Canada Propriété d’un consortium d’universités canadiennes, géré en co-entreprise à partir d’une contribution administrée par le Conseil national de recherches Canada Canada’s national laboratory for particle and nuclear physics Laboratoire national canadien pour la recherche en physique nucléaire et en physique des particules Accelerating Science for Canada Un accélérateur de la démarche scientifique canadienne Experimental considerations: Elastic scattering data are in general folded by the finite experimental conditions: These include target thickness, angular resolution, detector energy and angle resolution, finite beam resolution in space and energy, and straggling effects throughout the target and in the detectors. While those effects can be modeled or treated by Monte Carlo simulations, the experimental effects are best studied at narrow resonances. There these effects are most pronounced which typical leads to several iterations in the folding model between modeled cross section and experimental yields. One other interesting point is, how many parameters to be used, i.e., how many partial waves and mixing ratios. The fits shown are restricted in this respect as there are typically more parameters than necessary for fitting.

19 Owned and operated as a joint venture by a consortium of Canadian universities via a contribution through the National Research Council Canada Propriété d’un consortium d’universités canadiennes, géré en co-entreprise à partir d’une contribution administrée par le Conseil national de recherches Canada Canada’s national laboratory for particle and nuclear physics Laboratoire national canadien pour la recherche en physique nucléaire et en physique des particules Accelerating Science for Canada Un accélérateur de la démarche scientifique canadienne Conclusions: 1.A phaseshift analysis of typical scattering data is always possible in a general approach. 2.R-matrix analysis with real poles is frequently possible. However, description of background, i.e., non pole terms is sometimes insufficient. 3.With introducing as many poles as necessary to fit data in R-matrix, it is not guaranteed that such poles correspond to eigenstates of the compound nucleus. On the hand, R-matrix analysis is relatively robust against fluctuations in the data. 4.A phaseshift analysis shows the general physics trends of the data. However, it is not robust against fluctuations in the data and the influence of other partial waves. The result is that frequently poles with no justification are produced in the fit. Therefore dense and very precise data are required. 5.A K-matrix analysis is as robust as the R-matrix one. Background terms are more flexible than in R-matrix, for better or for worse. The analysis of scattering data in (light) nuclei is a fascinating area requiring specialized knowledge. This knowledge is leaving us now, because no faculty is hired to replace outgoing specialists in the field. Books and other publications are just not good enough to guarantee a thorough and valid analysis of data,