Asymptotic normalization coefficients in nuclear astrophysics and structure
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THE EUROPEAN PHYSICAL JOURNAL A
Asymptotic normalization coefficients in nuclear astrophysics and structure C.A. Gagliardi1,a , A. Azhari1 , V. Burjan2 , F. Carstoiu3 , V. Kroha2 , A.M. Mukhamedzhanov1 , A. Sattarov1 , X. Tang1 , L. Trache1 , and R.E. Tribble1 1 2 3
Cyclotron Institute, Texas A&M University, College Station, TX 77843, USA ˇ z, Czech Republic Institute for Nuclear Physics, Czech Academy of Sciences, Prague-Reˇ Institute for Atomic Physics, Bucharest, Romania Received: 21 March 2002 / c Societ` Published online: 31 October 2002 – a Italiana di Fisica / Springer-Verlag 2002 Abstract. The asymptotic normalization coefficient (ANC) for a nuclear system specifies the normalization of the tail of the nuclear overlap function. ANCs may be determined experimentally by measuring peripheral nuclear reactions. They may be used to calculate the astrophysical S-factors for radiative capture reactions at stellar energies and to determine the halo structure of loosely bound nuclei. A brief introduction to ANCs is given and applications to the 7 Be(p, γ)8 B and 11 C(p, γ)12 N reactions and the halo structure of 8 B are described. PACS. 26.30.+k Nucleosynthesis in novae, supernovae and other explosive environments – 26.20.+f Hydrostatic stellar nucleosynthesis – 25.60.Je Transfer reactions – 25.60.Gc Breakup and momentum distributions
1 Introduction to ANCs The asymptotic normalization coefficient (ANC) C for the nuclear system A + p ↔ B specifies the amplitude of the tail of the overlap function of the bound state B in the two-body channel (A p). We show that ANCs can be determined from measurements of peripheral proton transfer reactions or one-nucleon breakup reactions of loosely bound nuclei, and they may be used to calculate direct radiative capture reaction rates of astrophysical interest at stellar energies, to estimate the gamma widths of certain resonances, and to investigate nuclear halo structures. Traditionally, spectroscopic factors have been obtained from DWBA analysis of proton transfer reactions. However, these spectroscopic factors have significant systematic uncertainties associated with the choice of proton single-particle orbitals in the initial and final nuclei. For peripheral transfer reactions, the ANC is better determined and is the more natural quantity to extract. Consider the proton transfer reaction a + A → c + B, where a = c + p and A + p = B. We can write the cross-section in the form [1] σlDW dσ j l j B a = (CApl )2 (Ccpl )2 2 B B a2 a , B jB a ja dΩ b AplB jB bcpla ja j j
(1)
B a
where σlDW is the reduced DWBA cross-section. The B jB la ja a
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C’s are the ANCs at the two vertices, and the b’s are the ANCs of the bound-state proton wave functions in nuclei a and B. If the reaction under consideration is peripheral, the ratio in eq. (1) is independent of bAplB jB and bcpla ja . Thus for surface reactions, the cross section is best parametrized in terms of the product of the squares of the ANCs of the initial and final nuclei, (C B )2 (C a )2 . The important ad
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