Eur. Phys. J. A 13, 227–232 (2002) T HE EUROPEAN P HYSICAL JOURNAL A c  Societ` a Italiana di Fisica Springer-Verlag 2002 Asymptotic normalization coefficients and astrophysical direct capture rates C.A. Gagliardi 1 , A. Azhari 1 , V. Burjan 2 , F. Carstoiu 3 , V. Kroha 2 , A.M. Mukhamedzhanov 1 , A. Sattarov 1 , X. Tang 1 , L. Trache 1 , and R.E. Tribble 1 1 Cyclotron Institute, Texas A&M University, College Station, TX 77843, USA 2 Institute for Nuclear Physics, Czech Academy of Sciences, Prague- ˇ Reˇ z, Czech Republic 3 Institute for Atomic Physics, Bucharest, Romania Received: 1 May 2001 Abstract. Peripheral transfer reactions can be used to determine asymptotic normalization coefficients (ANCs). These coefficients, which specify the normalization of the tail of the nuclear overlap function, determine S-factors for direct capture reactions at astrophysical energies. A variety of proton transfer re- actions involving both stable and radioactive beams have been used to measure ANCs. Tests have demon- strated that ANCs determined from proton transfer reactions can be used to calculate astrophysical direct capture rates to within 9%. The 10 B( 7 Be, 8 B) 9 Be and 14 N( 7 Be, 8 B) 13 C reactions have been used to measure the ANC appropriate for determining the 7 Be(p, γ) 8 B rate, and the 14 N( 11 C, 12 N) 13 C reaction has been used to measure the ANC required to calculate the 11 C(p, γ) 12 N rate. PACS. 26.20.+f Hydrostatic stellar nucleosynthesis – 25.60.Je Transfer reactions – 21.10.Jx Spectroscopic factors 1 Introduction Stellar evolution involves sequences of direct and reso- nant capture reactions, together with beta-decays, whose rates must be known to make reliable predictions. Direct capture reactions of astrophysical interest usually involve systems where the binding energy of the captured pro- ton is low. Hence at stellar energies, the capture proceeds through the tail of the nuclear overlap function. The shape of the overlap function in this tail region is completely de- termined by the Coulomb interaction, so the amplitude of the overlap function alone dictates the rate of the capture reaction. The 7 Be(p, γ ) 8 B reaction, which plays a central role in the solar neutrino problem, is an excellent example of such a direct capture process [1]. The asymptotic normalization coefficient (ANC) C for A + p ↔ B specifies the amplitude of the tail of the over- lap function of the bound state B in the two-body chan- nel (Ap). We have pointed out [2,3] that astrophysical S-factors for peripheral direct radiative capture reactions can be determined through measurements of ANCs us- ing traditional nuclear reactions such as peripheral nu- cleon transfer. Direct capture S-factors derived with this technique are most reliable at the lowest incident ener- gies in the capture reaction, precisely where capture cross- sections are the smallest and most difficult to measure di- rectly. It is extremely important to test the reliability of the ANC technique in order to know the precision with which it can be applied. Below we give a brief description of this technique and discuss three tests that have been performed. We shall then describe our 10 B( 7 Be, 8 B) 9 Be, 14 N( 7 Be, 8 B) 13 C, and 14 N( 11 C, 12 N) 13 C radioactive beam studies and their use to determine the astrophysical S- factors for 7 Be(p, γ ) 8 B and 11 C(p, γ ) 12 N. 2 ANCs from proton transfer reactions Traditionally, spectroscopic factors have been obtained from DWBA analysis of proton transfer reactions. How- ever, it is well known that 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 ex- tract. Consider the proton transfer reaction a +A → c +B, where a = c + p, A + p = B. As was previously shown [3], we can write the DWBA cross-section in the form dσ dΩ =  jBja (C B AplBjB ) 2 (C a cplaja ) 2 σ DW lBjBlaja b 2 AplBjB b 2 cplaja , (1) where σ DW l BjBlaja is the reduced DWBA cross-section and j i ,l i are the total and orbital angular momenta of the transferred proton in nucleus i. The factors b cplaja and