Capture cross-section and rate of the ¹⁴C(n, γ) ¹⁵C reaction from the Coulomb dissociation of ¹⁵C

Abstract

We calculate the Coulomb dissociation of ¹⁵C on a Pb target at 68 MeV/u incident beam energy within the fully quantum mechanical distorted wave Born approximation formalism of breakup reactions. The capture cross-section and the subsequently rate of the ¹⁴C(n, γ) ¹⁵C reaction are calculated from the photodisintegration of ¹⁵C, using the principle of detailed balance. Our theoretical model is free from the uncertainties associated with the multipole strength distributions of the projectile.

Appendix A. Validity of the local momentum approximation

The factorization of the breakup amplitude (eq. (2)) is affected with the help of the local momentum approximation [20–22], on the outgoing charged fragment b. Essentially, one uses the Taylor expansion of $\chi _{b}^{(-)*}({\mathbf {q}}_{b},{\mathbf {r}})$ about r _i, which is exact and helps in the separation of variables r _i and r ₁,

$$\begin{array}{@{}rcl@{}} \chi^{(-)}_{b}(\mathbf{q}_{b},\mathbf{r}) = \mathrm{e}^{-\alpha\mathbf{r}_{1}.\nabla_{r_{i}}}\chi^{(-)}_{b}(\mathbf{q}_{b},\mathbf{r}_{\mathrm{i}}). \end{array} $$

(A.1)

The approximation involves the replacement of del-operator by an effective local momentum, K( = −i∇ _{r i} ), whose magnitude is given by

$$\begin{array}{@{}rcl@{}} \mathbf{K} = \sqrt{\frac{2\mu_{bt}}{\hbar^{2}}(E_{bt}-V(R))}, \end{array} $$

(A.2)

where μ _bt is the reduced mass of the b–t system, E _bt is the energy of particle b relative to the target in the c.m. system and V(R) is the Coulomb potential between b and the target at a distance R. Thus, the local momentum K is evaluated at some distance R (10 fm, in our case) and its magnitude is held fixed for all the values of r. The direction of K is taken as that of the outgoing fragment b. The condition of validity (see eg. [20]) is that the quantity

$$\begin{array}{@{}rcl@{}} \eta(r) = \frac{{1\over 2} K(r)}{\left| {\mathrm{d} K(r)}/{\mathrm{d}r} \right|}, \end{array} $$

(A.3)

calculated at some representative distance R should be more than the projectile radius, r _a .

To check the validity of the approximation, in figure 5 we show the variation of η(r) (upper half) and K(r) (the magnitude of the local momentum) (lower half) as a function of r, for the Coulomb breakup reaction ¹⁵C + Pb → ¹⁴C + n + Pb at the beam energy of 68 MeV/u. At r = 10 fm, η(r)≫r _a = 3.12 fm, the projectile r.m.s. radius.

Figure 5

Variation of η(r) (upper half) and K(r) (lower half) with r for the Coulomb breakup of ¹⁵C.