The effective-range theory application to study the nuclear vertex constants for bound and resonant states of the

lightest nuclei up to 8Ве.

Orlov Yu.V.

Skobeltsyn Institute of Nuclear PhysicsLomonosov Moscow State University

orlov@srd.sinp.msu.ru

FBS-Dubna - 2012

Resume

• The effective-range theory taking into account the Coulomb interaction has been developed and applied to computing the nuclear vertex constants (NVC) for a decay (fuse) АВ+С and for relating the asymptotic normalization coefficients (ANC) of the radial wave functions for a nucleus A. We study the 3Не s-wave bound state and the resonant s-wave states of 2Не, 3Не, 8Ве, the p-wave states of 5Не and 5Li, and the d-wave state of 8Ве. Experimental phase shifts and pole positions are used as input data.

Literature

1. V. O. Eremenko, L. I. Nikitina, and Yu. V. Orlov. The Vertex Constant forthe Virtual Decay of a Nucleus into Two Charged Particles within theEffective Range Theory. Bulletin of the Russian Academy of Sciences: Physics, 2007, Vol. 71, No. 6, pp. 791–797.

2. Yu. V. Orlov, V. O. Yeremenko, B. F. Irgaziev, and L. I. Nikitina. The Resonant Pole and the Residue of the Scattering Amplitude While the Coulomb Forces Gradually Decrease. Bulletin of the Russian Academy of Sciences: Physics, 2009, Vol. 73, No. 6, pp. 778–784.

3. Yu. V. Orlov, B. F. Irgaziev, and L. I. Nikitina. Effect of Coulomb Forces onthe Position of the Pole in the Scattering Amplitude and on Its Residue. Physics of Atomic Nuclei, 2010, Vol. 73, No. 5, pp. 757–772.

4. Yu. V. Orlov and L. I. Nikitina. Nuclear Vertex Constants for the VirtualDecay 7Be → 3He + 4He of the 7Be Bound States from a Phase-ShiftAnalysis in the Effective-Range Theory. Physics of Atomic Nuclei, 2011, Vol. 74, No. 11, pp. 1636–1640.

5. Yu. V. Orlov and L. I. Nikitina The Characteristics of the 8Be GroundState in the Effective-Range Approximation. Bulletin of the RussianAcademy of Sciences. Physics, 2012, Vol. 76, No. 4, pp. 446–449.

The effective-range theory

The dynamics of the spin-doublet nucleon–deuteron system at low energies isdetermined by the Feynman diagram of one-nucleon exchange in the elastic-scatteringamplitude. In the partial-wave amplitude f(k), this diagram generates a singularityclosest to the threshold. The corresponding logarithmic branch point occurs at anenergy of Ecm = −Ed = −epsilond/3 ≈ −0.74 MeV, , which is equivalent to the region|Elab| < 1.1 MeV in the laboratory frame (epsilond is the deuteron binding energy). The position of this singularity determines the radius of convergence of the numeratoron the right-hand side of (1) in powers of k2 in the complex plane of momentum.Strictly speaking, it is the boundary that specifies the region of lower energies fromthe point of view of convergence of the expansion in (1). In the literature, the term“low energies” was treated rather widely. For example, the energy range 1 ≤ Elab ≤46.3 MeV was considered in [J. Arvieux, Nucl. Phys. A 221, 253 (1974)] inperforming a partial-wave analysis of proton–deuteron scattering.

On the convergence of an effective-range function expansionfor the Nd scattering

K(k2) = (−1/a + C2k2 + C4k4)/(1 + k2/κ02), (1)

The phase-shift-analysis data for the 3He4He scattering are analyzed using the effective-range theory, with the Coulomb interaction taken into account. We find both the renormalized nuclear vertexconstants for the vertex 7Be → 3He + 4He in the ground (3/2−) and in the first excited (1/2−) boundstates of 7Be, and the corresponding asymptotic normalization constants of the radial wave functionsin these states. The results obtained can be used in an astrophysical S-factor calculation for radiativecapture reaction 4He(3He, γ)7Be.

We consider it necessary to obtain more precise measurements within the low-energy area. Inparticular, solving the problem of the narrow 7/2− resonance influence on the phase shift in otherstates including the states studied in the present work needs to be done. On the whole, the resultsobtained for the vertex constants are smaller in absolute value in comparison with those of modelcalculations in the literature. The fact that the ANC for the excited state 1/2− are larger than thosefor the ground state 3/2− is unexpected in some ways.

We find the vertex constant for the synthesis α + α → 8Be and the corresponding asymptotic nor-malization coefficient of the Gamov wave function for 8Be in the ground state. We use modern dataon the position and width of the narrow resonance of this state as well as the energy dependence ofthe α–α scattering phase shift in the s–wave known from the literature. The effective-range theorywas applied with the Coulomb interaction taken into account. The parameters of the standardeffective-range function expansion up to the member with k4 (k being the relative momentum) werefound.

The first excited state of 8Be (2+)

The effective-range parameters (a2, r2, P2), renormalized NVC and the abs. value of ANC

22Im renG

1.0621.081

0.01540.0151

0.01130.0125

19.845.9

0.2750.196

32.743.1

1.311.24

2.872.77

56

1.1361.1461.1341.143

0.01210.01270.01130.0119

0.01880.01900.01920.0194

2.080.833.102.64

0.2330.2200.2830.267

172188107116

1.471.511.471.51

3.023.023.043.04

1234

C2fm 1/2

fmfm

P2, fm 8

r2, fm 3

a2, fm5

2, MeV

E2, MeV

№ 22Re renG

Note. In variants 1-4 the values E2 and 2 are input data from [Tilley, D.R., Kelley, J.H., Godwin, J.L., et al., Nucl.Phys. A, 2004, vol. 745, p. 155].In variants 5-6 values we calculated E2 and 2 using the found effective-range function

parameters. While fitting a2, r2 and P2 we use the minimal values of 2 in var. 5 and maximum values of 2 in var. 6 (see Afzal S.A., Ahmad A.A.Z., Ali S., Rev. Mod. Phys. 1969. V. 41. P. 247; Warburton E.K., Phys. Rev. C, 1986. V. 33, P. 303).

The points no (1-4) on the complex energy plane are taken from the review [Tilley, D.R., Kelley, J.H., Godwin, J.L., et al., Nucl.Phys. A, 2004, vol. 745, p. 155].One can see that the fitting the phase shift in the full energy region leads to poles shift to the smaller ReE2 and ImE2.