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The Generator Coordinate Method

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Book cover A Microscopic Theory of Fission Dynamics Based on the Generator Coordinate Method

Part of the book series: Lecture Notes in Physics ((LNP,volume 950))

Abstract

This chapter describes in detail the formalism of the Generator Coordinate Method and its use in constructing both static and dynamic collective states of the nucleus, starting from constrained Hartree-Fock-Bogoliubov solutions. The full formalism is then reduced to a Schrödinger-like equation, and the calculation of its inertial tensor and zero-point energies is presented. Next, the calculations are extended beyond the adiabatic approximation and the Schrödinger Collective Intrinsic Model is derived. The multi-\(O\left (4\right )\) schematic model introduced in Chap. 2 is used throughout this chapter to illustrate the formalism.

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Notes

  1. 1.

    If the GC integration domain is bounded or, more generally, if the kernel N(q, q′) is a Hilbert-Schmidt operator (i.e. \(\left |\iint d^{n}qd^{n}q'N(q,q')\right |<\infty \)) the eigenvalues and eigenfunctions of N constitute denumerable sets. We assume here that this is not necessarily the case. Therefore, provided the kernel N(q, q′) obeys appropriate convergence conditions when the q i and \(q^{\prime }_{i}\) go to infinity, its eigenvalues and eigenfunctions are to be labelled by means of a continuous parameter ξ.

  2. 2.

    Let us mention that, when a density-dependent effective interaction is used – which is the case of the applications to fission presented in this book – the nature of the one-body density to insert in the effective interaction when calculating non-diagonal matrix elements \(\langle \varPhi _{q}\vert H\vert \varPhi _{q'}\rangle \), q ≠ q′ is unclear, and various prescriptions have been considered in the literature [12, 13].

  3. 3.

    The denomination “cranking” comes from a model proposed by Inglis to describe rotations in deformed nuclei, the cranking model [54, 60], which gives for the nuclear moment an expression of the same form as the one of the ATDHF collective inertia when the “cranking” approximation is used.

  4. 4.

    As noted in [20], this is not a proper square root since we take the adjoint on the left.

  5. 5.

    The difference in sign from Eq. (C.5) in [20] for \(B_{-1/2}\left (\bar {q}\right )\) and \(B_{1/2}\left (\bar {q}\right )\) is due to the fact that we use \(\left [\hat {P}\bar {N}^{\left (2\right )}\left (\bar {q}\right )\right ]^{2}\) instead of \(\left [\bar {N}^{\left (2\right )\prime }\left (\bar {q}\right )\right ]^{2}\), and there is a minus sign from an i 2 factor that appears when \(\hat {P}\) is written as a differential operator.

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Acknowledgements

This chapter was prepared by a contractor of the U.S. Government under contract number DE-AC52-06NA27344. Accordingly, the U.S. Government retains a nonexclusive, royalty-free license to publish or reproduce the published form of this contribution, or allow others to do so, for U.S. Government purposes.

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Authors and Affiliations

  1. Department of Physics, Lawrence Livermore National Laboratory, Livermore, CA, USA

    Walid Younes

  2. Lawrence Livermore National Laborat, Livermore, CA, USA

    Daniel Marc Gogny

  3. Centre DAM-Ile de France, Retired from Commissariat à l’Energie Atomique, Bruyères-le-Châtel, Arpajon, France

    Jean-François Berger

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Younes, W., Gogny, D.M., Berger, JF. (2019). The Generator Coordinate Method. In: A Microscopic Theory of Fission Dynamics Based on the Generator Coordinate Method. Lecture Notes in Physics, vol 950. Springer, Cham. https://doi.org/10.1007/978-3-030-04424-4_3

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