Coulomb excitation of the deuteron in peripheral collisions with a heavy ion

We develop an ab initio, nonperturbative, time-dependent basis function (tBF) method to solve the nuclear structure and scattering problems in a unified manner. We apply this method to a test problem: the Coulomb excitation of a trapped deuteron by an impinging heavy ion. The states of the deuteron system are obtained by the ab initio nuclear structure calculation implementing a realistic internucleon interaction with a weak external trap to localize the center of mass and to discretize the continuum. The evolution of the internal state of the deuteron system is directly solved using the equation of motion for the scattering. We analyze the excitation mechanism of the deuteron system by investigating its internal transition probabilities and observables as functions of the exposure time and the incident speed. In this investigation, the dynamics of the Coulomb excitation are revealed by the time evolution of the system's internal charge distribution.

Figure 1

Setup of the peripheral scattering (adopted from Ref. [35]). See the text for the details.

Figure 2

The eigenbasis vector of the target deuteron confined in an external HO trap of strength 5 MeV. This vector makes explicit the basis representation for our model and lists the channel quantum numbers, the angular momentum projection, and the eigenenergies. For the ab initio structure calculation, the 3DHO bases are adopted, for which the basis strength is set to be $\omega =5$ MeV and the truncation parameter $N_{\text{max}}$ to be 60. The lowest-lying seven states are chosen to construct the basis representation for the target. Note there are the expected degeneracies with respect to the target system's magnetic projection $M$.

Figure 3

The Coulomb excitation (only the $E1$ multipole component is included) of the target illustrated as a function of the time and the incident speed of the HI in the middle of the scattering. The target is characterized by seven basis states. It is initially prepared to be in the state $({}^{3}S_1,{}^{3}D_1),M=-1$. The HI projectile is taken to be a fully stripped uranium nucleus, ${\mathrm{U}}^{92+}$, and the incident speed is taken as 0.1, 0.2, and 0.4. The transition probabilities of each basis state of the target are calculated via the nonperturbative tBF method and compared with results from first-order perturbation theory (curves labeled by “Pert” in the legend).

Figure 4

Illustration of the state populations changing with time during the scattering. Note the transition probabilities are all scaled. The $E1$-allowed state ${}^{3}P_1,M=-1$ is populated initially. Then one observes the transport of population from the state ${}^{3}P_1,M=-1$ to the first-order forbidden state $({}^{3}S_1,{}^{3}D_1),M=0$. Later, the transition network populates the second-order forbidden state ${}^{3}P_1,M=1$ with that population fed from the state $({}^{3}S_1,{}^{3}D_1),M=0$. The forbidden states also receive population from other states, in which cases relative phases can lead to interference.

Figure 5

Illustration of the geometric factor [panel (a)] $\frac{Y_{1\mu }^{*}\left({\mathrm{\Omega }}_R\right)}{{\left|R\left(t\right)\right|}^2}$ ($\mu =\pm 1,0$) and its time variation [panel (b)] during the scattering. The incident speed is taken as $v=0.1$. Since the scattering plane is the $xz$ plane, the azimuth angle vanishes and $Y_{1\mu }^{*}\left({\mathrm{\Omega }}_R\right)$ is real. The results related to $Y_{1-1}\left({\mathrm{\Omega }}_R\right)$ are omitted owing to the fact that $Y_{11}^{*}\left({\mathrm{\Omega }}_R\right)=-Y_{1-1}\left({\mathrm{\Omega }}_R\right)$.

Figure 6

Selected observables of the $np$ target as functions of the exposure time and the incident speed of the HI in the middle of the scattering. Panels (a)–(d) show the evolutions of r.m.s. charge radius, r.m.s. momentum, r.m.s. orbital angular momentum, and intrinsic energy, respectively.

Figure 7

Expectation values of the $\widehat{z}$ projection of the total angular momentum as functions of the exposure time and the incident speed in the middle of the scattering. These values are calculated in the same manner as the observables in Fig. 6.

Figure 8

The internal charge distribution (in ${\text{fm}}^{-3}$) of the $np$ target before scattering. The initial target is prepared in the state $({}^{3}S_1,{}^{3}D_1),M=-1$, in which the polarization is antiparallel to the $\widehat{z}$ axis. The $xz$ plane is the scattering plane (see Fig. 1), the $xy$ and $yz$ planes are respectively perpendicular and parallel to the impinging HI.

Figure 9

The overview of the evolution of the internal charge distribution (in ${\text{fm}}^{-3}$) of the $np$ target during the scattering. The simulation conditions are the same as those in Fig. 3, except that the incident speed is chosen as 0.1. These graphs show the difference in the internal charge distributions between the scattered targets at $T=-1.975,-0.255,0,0.255,1.975{\mathrm{MeV}}^{-1}$ and the initial target ($T=-5{\mathrm{MeV}}^{-1}$) in three orthogonal coordinate planes, where the transition amplitudes of each basis state of the target at the selected moments are calculated via the nonperturbative tBF method. See the text for the details.

Figure 10

Stabilization of the target after the scattering. These graphs show the difference in the internal charge distributions (in ${\text{fm}}^{-3}$) between the initial target ($T=-5{\mathrm{MeV}}^{-1}$) and the scattered targets at $T=0.36{\mathrm{MeV}}^{-1}$ (top row), $0.435{\mathrm{MeV}}^{-1}$ (second row), $0.565{\mathrm{MeV}}^{-1}$ (third row), $0.695{\mathrm{MeV}}^{-1}$ (bottom row). The other simulation conditions are the same with Fig. 9 and the transition amplitudes of each basis state are calculated via the tBF method. See the text for the details.