Calculation of Coulomb breakup cross sections using a new Coulomb dynamical polarization potential

In Ref. Mar21 we gave the formalism for the scattering of a weakly-bound two-body projectile (p) consisting of a core in its ground state plus a cluster of $n$ valence neutrons from a heavy-ion target and the CDPP was obtained. If the core is in an excited state, this expression for the CDPP can be easily generalized by replacing the binding energy of the valence neutron or neutron cluster with respect to the charged core of the projectile, $\varepsilon_{0}$, with an effective separation energy $\varepsilon^{*}_{0}=\varepsilon_{0}+\varepsilon_{I^{\pi}_{c}}$, where $\varepsilon_{I^{\pi}_{c}}$ is the excitation energy of the core state of spin-parity $I^{\pi}_{c}$. To solve the Schrödinger equations of the system and obtain the CDPP one may use the adiabatic approximation $\Psi({\bf r,R})\approx\psi({\bf R})\phi({\bf r,R})$, where $\psi({\bf R})$ refers to the wave function of the center of mass and $\phi({\bf r,R})$ to that of the relative motion of the projectile; ${\bf R}$ and ${\bf r}$ are the coordinates of the projectile-target and the projectile valence-core systems, respectively. By making the same approximations as in Ref. Mar21 the real and imaginary parts of the CDPP can be given as:

where $F_{0}$ and $G_{0}$ are the regular and irregular Coulomb functions in $\rho=k(R)R$ and $Q(R)=(\mu_{p}/m_{c})(k(R)/\kappa_{0})$ with $\kappa_{0}=\sqrt{-2\mu_{p}\varepsilon_{0}^{*}/\hbar^{2}}$ where $\mu_{p}$ is the core-valence reduced mass, $m_{c}$ the mass of the charged core, and

is the wave number of the charged core in the field of the target that is associated with the wave function of the internal motion of the projectile, $\phi(r,R)$. It depends parametrically on the Coulomb potential between the projectile and target, $V_{C}({R})$ and is different from the wave number of the center-of-mass motion of the system that describes the motion of the projectile along the Rutherford trajectory, $K=\sqrt{2\mu(E-\varepsilon_{0}^{*})/\hbar^{2}}$ where $E$ is the incident energy of the projectile and $\mu$ is the reduced mass of projectile-target system. This CDPP (II.1) depends on the structure of the system but does not depend on the incident energy of the projectile.

Excitations of the projectile to the continuum can be included by adding the continuum energy $\varepsilon$, a continuous variable, to $\varepsilon_{0}^{*}$ so that $k$ now becomes a function of $\varepsilon$:

making the same approximations as before, and we accordingly obtain the Continuum based CDPP, $\delta U(R,\varepsilon)=\delta V(R,\varepsilon)+i\delta W(R,\varepsilon)$:

where $Q$ is now also a function of $\varepsilon$ as well as $R$.

We now consider an $E\lambda$-transition ($E1,E2,...$) from a bound state with angular momentum $l_{0}$ to the $l$-wave continuum, with $l=l_{0}+\lambda$ where ${\lambda}$ is the transition multipolarity. The initial bound state ${J_{0}^{\pi}}$ is described by the wave function of the valence neutron in the $l_{0}j_{0}$ orbital relative to the core $I^{\pi}_{c}$, $\phi^{I^{\pi}_{c}}_{l_{0}j_{0}}({\bf r})$, whereas the final state (continuum) is represented by the wave function $\phi_{\varepsilon lj}({\bf r})$, where $\varepsilon$ in the subscript denotes the dependence on the continuum energy. Here we include the notation $I^{\pi}_{c}$ to indicate that the core is in state $I^{\pi}_{c}$. Then, we define a new CDPP that describes this transition as

where $\delta U^{I^{\pi}_{c}}({\bf R,r},\varepsilon)$ can be represented by expansion in Legendre polynomials as

Let us define the excitation energy distribution of the $E\lambda$-transition ${l_{0}j_{0}\rightarrow\varepsilon lj}$ as

so that we may write

The radial wave function of the bound state is given by $\phi^{I^{\pi}_{c}}_{l_{0}j_{0}}({r})=u_{l_{0}j_{0}}({r})/r$, where the single-particle wave function $u_{l_{0}j_{0}}({r})$ is usually calculated using a single-particle potential model, see for example Ref. Xu94 , by adjusting the Woods-Saxon potential parameters to reproduce the experimental neutron separation energy, taking the excitation energy of the core into account where appropriate. Typical values of the potential radius parameter $r_{0}$ range from 1.15 to 1.25 fm and the diffuseness $a$ from 0.5 to 0.7 fm. It is, however, important to note that the extracted spectroscopic factor can differ by 20$\%$ or more depending on the choice of parameters Pra03 . We choose here the potential parameters ($r_{0}=1.15$ fm, $a=0.5$ fm) suggested by Sauvan et al. Sau00 for light neutron-rich nuclei. Note that the potential includes a Thomas form spin-orbit term with a strength fixed at 7.0 MeV for all the cases studied here.

The final state (continuum) wave function is usually calculated assuming the plane wave approximation (expanded in spherical Bessel functions and Legendre polynomials) with the radial wave function normalized as $\phi_{\varepsilon lj}(r)=\sqrt{2\mu_{p}\kappa/\hbar^{2}\pi}j_{l}(\kappa r)=u_{\varepsilon lj}({r})/r$, where $\kappa=\sqrt{2\mu_{p}\varepsilon}/\hbar$ with relative (continuum) energy $\varepsilon\equiv\varepsilon_{rel}$, and $l$ is the orbital angular momentum of the final state.

Now, by considering the expansions of the bound-state and the continuum plane wave functions, Eq. (7) becomes

Instead of the single-particle wave functions, the asymptotic form for overlap integrals is often used

where $W_{-\eta_{0},l_{0}+1/2}$ is the Whittaker function and $C_{l_{0}j_{0}}$ is the asymptotic normalization coefficient (ANC). The quantity $\kappa_{0}=\sqrt{-2\mu_{p}\varepsilon_{0}^{*}}/\hbar$ where $\varepsilon_{0}^{*}=\varepsilon_{0}+\varepsilon_{I^{\pi}_{c}}$ is the effective separation energy of the valence nucleon(s) with respect to the different core states ${I^{\pi}_{c}}$ of excitation energies $\varepsilon_{I^{\pi}_{c}}$. When the valence cluster is formed exclusively of neutron(s) $\eta_{0}=Z_{c}Z_{v}e^{2}\mu_{p}/\hbar^{2}\kappa_{0}=0$ and $W_{0,{l_{0}+\scriptscriptstyle\frac{1}{2}}}(2\kappa_{0}r)=\frac{2\kappa_{0}r}{\pi}k_{l_{0}}(\kappa_{0}r)$ where $k_{l_{0}}$ is a modified spherical Bessel function and $h^{(1)}_{l_{0}}$ is a spherical Hankel function of the first kind. Then $u_{l_{0}j_{0}}(r)=C_{l_{0}j_{0}}\frac{2\kappa_{0}r}{\pi}k_{l_{0}}(\kappa_{0}r)$. Now by using the Whittaker function for the bound-state wave function, Eq. (9) becomes

In the particular case when the initial bound state has $l_{0}=0$, the radial wave function can be given analytically as the Yukawa form $\phi^{I^{\pi}_{c}}_{l_{0}j_{0}}({r})=\frac{u_{l_{0}j_{0}}({r})}{r}=C_{s_{1/2}}\frac{e^{-\kappa_{0}r}}{r}$, so that for the $E1$ transition to the $p$-wave continuum, we obtain the following analytical function:

which will be especially useful since the valence neutron(s) are in a relative $s$-state in the ground state of many light exotic projectiles of interest.

This new CDPP (8) has many potential applications to the scattering and reactions of weakly-bound neutron-rich nuclei. Here we use it to calculate Coulomb dissociation, keeping other applications for future publications. Coulomb breakup or dissociation can take place when a high energy (several hundred MeV/nucleon) projectile impinges on a heavy target and is excited by absorbing virtual photons from the time-dependent Coulomb field. Under these circumstances the electromagnetic excitation is dominated by dipole excitation Pra03 . The corresponding differential cross section for dipole excitation decomposes into an incoherent sum of components corresponding to the different core states populated by neutron removal. For each core state the cross section further decomposes into an incoherent sum over contributions from the different allowed angular momenta of the valence neutron in its initial state. In these calculations the projectile is assumed to have a core plus valence neutron structure. Choosing the lowest core states there are various ways to couple the spins of the core and the valence neutron to the total angular momentum and parity of the projectile Hei17 . Note that for heavy targets the core is assumed to act as a spectator so that the projectile core remains in its initial state after removal of a neutron Han03 .

We start from the usual formula for the absorption cross section derived from the continuity equation

where $W({R})$ is the imaginary potential, $\psi^{(+)}_{K}$ the usual distorted wave function, $v$ the relative velocity, and $K$ the wavenumber in the center of mass system. By considering the imaginary CDPP (5) that describes the ${E\lambda}$-transitions we can write the total absorption due to Coulomb dissociation and excitation by taking a summation over the ${E\lambda}$-transitions, the different core states $I^{\pi}_{c}$, and contributions from different angular momenta $l_{0}j_{0}$ of the valence neutron in its initial state for each core state:

Then, the differential cross section for the $E\lambda$-transition $l_{0}j_{0}\rightarrow\varepsilon lj$ with the core in state $I^{\pi}_{c}$ is

which is similar to the well-known Hussein–McVoy formula and the similar formulae for the inclusive breakup Hus85 ; Ram21 . Introducing

as the total absorption cross section for the $I^{\pi}_{c}$ core state, the breakup cross section for the core in state $I^{\pi}_{c}$ and the $E\lambda$-transition from ${l_{0}j_{0}}$ is

Equations (15) and (17) can be generalized to account for excitations of the target. An imaginary nuclear potential can be added to account for any other reaction channels. From Equation (17) and using the virtual photon method Ber88 one may obtain the photo-absorption cross section $\sigma_{\gamma n}$, the radiative neutron capture cross section $\sigma_{n\gamma}$, the dipole response function $dB/d\varepsilon$, and the double differential cross section $d^{2}\sigma/d\Omega d\varepsilon$. These calculations can also be generalized to consider transitions between two bound states by replacing the continuum wave function $u_{\varepsilon lj}(r)$ with a bound state wave function $u_{lj}(r)$ similar to $u_{l_{0}j_{0}}(r)$.

Equation (16) can be solved by the method of partial wave expansion and the complete scattering amplitude is obtained by summing over all the partial wave scattering amplitudes. At high energies, i.e. hundreds of MeV/nucleon, the wave function will oscillate rapidly and the calculation of scattering wave functions for each partial wave becomes more complicated. In the eikonal approximation and the optical limit of the Glauber theory Gla59 , the wave function of the projectile-target system can be written in terms of the total potential, $V+iW$, as

where ${\bf R}=(x,y,z)=(\mathbf{b},z)$ and ${\bf b}$ is the impact parameter. Equation (16) may then easily be written as

where $d\mathbf{b}=2\pi bdb$. At this point we should include the correction of the impact parameter due to Coulomb deflection of the particle trajectory: $b^{\prime}=a_{0}+\sqrt{a_{0}^{2}+b^{2}}$, where $a_{0}=Z_{p}Z_{t}e^{2}/mv^{2}$ is half the distance of closest approach in a head-on collision of point charged particles. The relativistic correction to the kinematics can be appropriately made if one replaces the quantity $a_{0}$ with $a_{0}=Z_{p}Z_{t}e^{2}/\gamma mv^{2}$, where $\gamma=1/\sqrt{1-v^{2}/c^{2}}$ is the Lorentz factor Ber03 . Since the phase in Eq. (19) is integrated over $z$ it is invariant under Lorentz transformations, so no further relativistic corrections to the dynamical equations are required Mos19 .