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Marianne Dufour. Microscopic Cluster Models. Université de Strasbourg - IPHC. f S. f B. R. H Y = E Y = A f B f S g (r). Pierre Descouvemont. Université libre de Bruxelles. This seminar is devoted to:

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microscopic cluster models

Marianne Dufour

Microscopic Cluster Models

Université de Strasbourg - IPHC




  • HY= E Y
  • = A fB fSg(r)

Pierre Descouvemont

Université libre de Bruxelles


This seminar is devoted to:

Microscopic Cluster Models based on the combination of the Generator-Coordinate-Method and of the Microscopic R-matrix method.

They are very efficient tools to study the nuclear many-body problem.

  • They are particularly well-adapted to the study of:
  • Nuclear reactions at very low energy where antisymmetrization effects between nucleons are expected to be important.
  • Important application: Reactions of astrophysical interest
  • Structure of (exotic) light nuclei, in particular for states which are known to be strongly deformed, and which presents a marked cluster structure.
  • Typically:halo nuclei, Molecular states, condensate states.
  • Reactions of astrophysical interest and the nuclear many-body problem
  • Overview of the Theoretical Framework

Generator Coordinate and Microscopic R-Matrix Methods

  • Recent applications
reactions of astrophysical interest and the nuclear many body problem
Reactions of astrophysical interest and the nuclear many-body problem

Nowadays, it is commonly accepted that stellar energy is due to nuclear reactions occurring in the core (Eddington 1920) and that stellar nucleosynthesis can explain all the nuclei with A>=12 and one part with A<12.

At human scale, stellar temperatures are very high

At nuclear scale, a star is a very cold medium

Supernova :

nuclear reactions of astrophysical interest
Nuclear Reactions of Astrophysical Interest

Besides, deeper investigations of the stellar plasma show that:

  • At low density, electrons of the plasma can be neglected.
  • The ensemble of nuclei forms an ideal gas.
  • The reactions are considered as between bare nuclei (no atom).
  • Nuclear reactions occur at (very) low energies.

Reactions can be studied in principle in accelerators on earth.

BUT, energies are (very) low …

interactions between 2 nuclei at low energy neutron special case
Interactions between 2 nuclei at low energy(neutron : special case)

Repulsive Coulomb interaction

Short range nuclear attractive interaction

Resonant/non resonant reaction


Charge induced nuclear reactions in the center of the stars can only proceed because the nuclei penetrate the repulsive Coulomb barrier that separates them. Since the stellar energies are significantly lower, the cross sections drop rapidly to very small values.

nuclear astrophysics difficulties
Nuclear astrophysics difficulties
  • Direct measurements of the cross sections in the range of astrophysical energies are impossible in most cases.
  • Additional experimental problems:

Needs for radioactive nuclei, very exotic nuclei, etc …

Theoretical investigations are necessary

resolution of the many body problem theoretical difficulties
Resolution of the many-body problemTheoretical difficulties
  • Very low energies.
  • The de Broglie wavelength associated to the relative motion is greater than the typical scale of the nuclear system in interaction.

The system of the two nuclei must be treated as a system of nucleons in interaction.

quantum many body problem
Quantum Many-Body Problem

Nuclear reaction:

Reaction between Nucleons: 12N + 4N



16 Nucleons in interaction

a a1 a2 fermions in interaction
A = A1+A2 fermions in interaction

The Pauli Principle must be exactly treated.

The system of A nucleons must be antisymmetrized.

  • Rigorous treatment of the channels of reaction.
  • Interactions at the Nucleon level (NN, NNN, …).
  • Necessary to compute bound states and scattering states.

In such a context, microscopic cluster models appear to be very efficient tools to handle the A nucleon problem.

  • Historically, the observation of clustering starts with the a particle which presents a large binding energy and tends to keep its own identity in light nuclei.
  • The description of states based on a cluster structure was first suggested by Wheeler (1937) and Margenau (1941) and then extended by D. Brink.
  • Since, many other developments ...

D. Brink, Proc. Int. School, E. Fermi 36, Varenna, Academic Press NY 1966.

Lecture Notes In Physics 818, Clusters in Nuclei, Editor: C. Beck, Springer (Vol.1 (2010), Vol.2, Vol.3)



Microscopic Cluster Models – Basic idea

Cluster= Harmonic Oscillator Potential

8 Nucleons = a + a


Pauli Principle


R=Generator Coordinate

Localizes the HO orbitals

Two cluster model

Generalization possible



D. Brink, Proc. Int. School. E. Fermi 36, Varenna, Academic Press NY 1966


Antisymmetrized cluster configurations for Na nuclei

Here only s clusters

R = set of generator coordinates

  • Investigations of different values of R
  • In the limit where R goes to 0, we get SM configurations
  • Binding Energy minima for cluster configurations (~1960)
overview of the mcm theoretical framework
Overview of the MCM Theoretical Framework
  • The Schrödinger equation of the A-Nucleon system is approximatively solved with the Generator Coordinate Method (GCM) combined with the Microscopic R-Matrix method (MRM).
  • In particular, this framework ensures a good asymptotic behaviour of the wave functions.

P. Descouvemont, D. Baye, Rep. Prog., Phys. 73 (2010) 036301.

P. Descouvemont, M. Dufour, Microscopic Cluster Models, Lecture Notes

in Physics, Springer T2 (2011) (Ed. C. Beck).


Determination of the total Wave Function

  • The specificity of microscopic cluster models is that the WF of the A-nucleon system is described at the cluster approximation.
  • The A nucleons are assumed to be divided in clusters described by shell-model wave-functions.
  • The total WF is fully antisymmetric.

Here, we consider a reaction between two nuclei (1) and (2) with respectively A1 and A2 nucleons.

  • The (1,2) system is called the unified nucleus.

Schematic representation of a Two-Cluster GCM-Basis State

All the quantum numbers are exactly treated

microscopic r matrix method
Microscopic R-Matrix Method

External Region

r = a = Channel Radius


Internal Region



Coulomb functions


Theoretical Framework Summary

  • Unified description of bound, resonant and scattering states
  • Exact treatment of antisymmetrization: the Pauli principle is exactly treated.
  • Rigorous center of mass separation.
  • The quantum numbers associated with the colliding nuclei are restored.
  • Exact asymptotic behaviour of the WF’s.
  • Once the interaction is fixed, the results are parameter free.
  • Cluster approximation – The GCM variational basis is finite
  • Effective interactions.
  • No systematic, heavy framework.
how to improve the cluster wfs
How to improve the Cluster WFs ?

To increase the number of cluster: multicluster model

(Technical difficulties also increased: Projections, implementation of the interactions, …)


Extended Two Cluster Model (ETCM)

Increase the number of major shells of the HO


M. Dufour et P. Descouvemont, Physics Letters B 696 (2011) 237


Kalpachieva et al. 2000

Lecouey et al. 2009



  • Microscopic Cluster Models well adapted to
  • Reactions at low energy when antisymmetrization is necessary (Nuclear astrophysics).
  • Physics of light nuclei (molecular states, etc …)
  • Work in progress:
  • Study of Condensate states in 12C and 16O