Continuum damping effects in nuclear collisions associated with twisted boundary conditions

TL;DR

This study uses time-dependent Skyrme Hartree-Fock calculations with twisted boundary conditions to investigate continuum damping effects and rotational damping in nuclear collisions, revealing persistent damping phenomena and angular momentum retention in the evolved compound nucleus.

Contribution

It introduces the implementation of twisted boundary conditions in TDHF calculations to effectively study continuum damping effects in nuclear collisions.

Findings

Continuum damping effects persist after fusion.

Rotational damping widths are clearly extractable.

The compound nucleus evolves towards sphericity while retaining angular momentum.

Abstract

The time-dependent Skyrme Hartree-Fock calculations have been performed to study $^{24}$Mg +$^{24}$Mg collisions. The twisted boundary conditions, which can avoid finite box-size effects of the employed 3D coordinate space, have been implemented. The prolate deformed $^{24}$Mg has been set to different orientations to study vibrations and rotations of the compound nucleus $^{48}$Cr. Our time evolution results show continuum damping effects associated with the twist-averaged boundary condition play a persistent role after the fusion stage. In particular, a rotational damping in continuum is presented in calculations of both twist-averaged and absorbing boundary conditions, in which damping widths can be clearly extracted. It is unusual that the rotating compound nucleus in continuum evolves towards spherical but still has a considerable angular momentum.

Equations

$$\psi (\text{r}+\mathbf{n}L)=e^{i\theta \cdot \mathbf{n}}\psi (\text{r})\text{,}$$

$${\hat{h}}_{\theta }{\psi }_{\alpha \theta }(\mathbf{r})={ϵ}_{\alpha \theta }{\psi }_{\alpha \theta }(\mathbf{r})$$

$$⟨\hat{O}(t)⟩=\frac{1}{8{\pi }^3}{\iiint }_0^{2\pi }{d}^3\theta ⟨{\Psi }_{\theta }(t)|\hat{O}|{\Psi }_{\theta }(t)⟩$$

$$k_{i,m}=\frac{2\pi m+{\theta }_i}{L_i},m=0,\pm 1,\pm 2,\cdots ,\pm m_{max}.$$

Full text

Introduction.— The real-time nuclear dynamics such as collective responses, large amplitude collisions and fissions have been studied extensively to probe effective interactions, many-body correlations and transport properties negele ; tddft ; tdhf . The basic theoretical framework for quantum many-body dynamics is the time-dependent Schrödinger equation with various approximations. In this respect, the microscopic time-dependent Hartree-Fock (TDHF) (or time-dependent density functional theory) was very successful for studies of nuclear dynamics, particularly the large amplitude dynamics negele ; tddft ; tdhf ; umar ; guo ; scamps ; paul . The improved time-dependent-Hartree-Fock-Bogoliubov calculations have also been developed for superfluid systems, relying on tremendous computing capabilities tdhfb1 ; tdhfb2 . Besides, the quantum molecular dynamics calculations have been widely used for heavy ion collisions at higher energies with two-body dissipations qmd . In the small amplitude limit of collective motions, TDHF results can match the random phase approximation (RPA), or linear resonance theory tddft ; umar05 . For nuclear collisions involving considerable excitation energies, the TDHF without pairing is a reasonable approximation. For too high excitation energies, the TDHF is not applicable when two-body collisions become prominent.

TDHF calculations are usually performed in 3D coordinate spaces with periodic boundary conditions sky3d . Due to the limits of computational resources, the calculations suffer from a systematic error related to finite box-sizes that were employed. The finite box sizes lead to tinny wave reflections at boundaries or wave interferences between periodic images abc1 . Furthermore, the continuum can not be accurately discretized within small coordinate spaces abc2 . This is not a serious problem for descriptions of bulk properties but it could be not negligible after long-time evolutions. The important role of continuum in nuclear reactions has also been demonstrated by the widely adopted continuum-discretized coupled-channels calculations cdcc . For highly excited compound nuclei produced by fusion reactions, the surface pressures are equilibrized by thermalized continuum gases which allow for particle evaporations zhu . In weakly bound nuclei, the accurate treatment of continuum couplings is important for halo structures and associated dynamics matsuo ; zhang ; sun .

To avoid the box-size dependence in the treatment of continuum, the twisted averaged boundary condition (TABC) has been applied to TDHF calculations of giant resonances schuer . The twisted boundary condition (TBC) is a generalized periodic boundary condition (PBC) for Bloch waves with non-zero twisted angles. In condensed-matter physics, TABC results by averaging twisted angles can significantly cancel finite box-size effects lin . Partial TBC has also been found to be useful in Lattice QCD calculations of few body systems luu . In nuclear physics, the quasiparticle RPA (QRPA) with outgoing boundary conditions has been realized for spherical nuclei matsuo but it is extremely difficult for deformed nuclei. It was known that QRPA calculations with not-well discretized continuum can cause false resonance peaks abc2 ; wang . An effective way to smooth resonances of deformed nuclei is to use the absorbing boundary condition (ABC) abc1 ; abc2 . It has been demonstrated that the TABC and ABC behave similarly in damping effects to smooth giant resonances schuer . The TDHF calculations can in principle take into account Landau damping, escaping damping and and the damping due to complex configuration couplings tdhf . The continuum treatment is essential in all of these damping mechanisms. The ABC calculation has to adjust the imaginary potential to absorb waves exactly at boundaries, and this is tedious. On the other hand, TABC can be easily implemented for complex systems. The TABC is very successful in studying giant resonances which are considered as small amplitude collective motions. Therefore, it is desirable to explore the influences of TABC calculations of large amplitude nuclear collisions with time evolutions.

In this work, we intend to study the 24Mg +24Mg collisions by TDHF calculations with TABC, PBC and ABC boundary conditions. The compound nucleus 48Cr can have hyperdeformed states at high spins from cranking calculations high-spin , indicating multi-$\alpha$ clustering structures. Such $\alpha$-conjugate compound nuclei are expected to be favorable for searching collective molecular motions. Indeed, several experiments have been performed for 24Mg +24Mg collisions and narrow resonances in inelastic cross sections have been reported wuossma ; nitto ; zurmuhle . However, some experiments didn’t find resonance structures in 24Mg +24Mg fusion cross sections Jachcinski . Note that the full picture of clustering structures in compound nuclei should take into account the dynamical nonlocalized clustering zhou . In the precompound nuclei, the time-dependent nucleon localization indicates that clustering vibrations are important in the initial stage of fusion schuer3 . The prolate 24Mg allows studies of collisions with different orientations. To characterize different collision reactions, the time evolutions of total kinetic energies and deformations have been studied using the Fourier transformation.

Method.— We utilize the 3D Skyrme-TDHF solver Sky3D sky3d , which solves the self-consistent HF equation and the TDHF equation. Calculations are performed in the 3D uniform coordinate space, and there is no symmetry restrictions on the wavefunctions. The full Skyrme energy functional adopts the SV-bas svbas force, in which the spin-orbit and time-odd terms have been included.

The grid spacing is set to be 1 fm and the time step of dynamical evolution takes 0.2 fm/c. In Sky3D, the time propagator is evaluated by the Taylor series expansion up to the sixth order. Computations with these settings have been demonstrated to be good enough for descriptions of the essence of dynamical properties dai . The static calculations of 24Mg+24Mg are firstly carried out to obtain the ground-state wave functions, which are inputs for time-evolution calculations. The 3D box sizes (along x, y, z-axis) in static and dynamical calculations are taken as 24$\times$24$\times$24 fm and 48$\times$24$\times$48 fm, respectively. Note that the static wave functions can be transformed into larger coordinate spaces by using the Fourier and inverse Fourier transformations. The energies and density distributions as a function of time are the main outputs.

Boundary conditions.— PBC is a natural choice for plane wave representation and is efficient for computations in the uniform 3D grids sky3d . TBC is a generalized Bloch boundary condition as written as schuer ,

[TABLE]

where $\mathbf{r}$ denotes the 3D coordinates, $L$ denotes the box size, and $\mathbf{n}$ is the unit vector in 3D Cartesian coordinates. The twisted angle $\theta$ changes from zero to $\pi$. Eq.1 can go back to PBC when the twisted angle $\theta$ is zero. The single-particle HF equation can be written as,

[TABLE]

where $\alpha$ is the discrete label of the single-particle wave functions.

In the TABC method, the expectation value of an observable Ô can be obtained by averaging over the twist angles schuer ,

[TABLE]

where $\Psi_{\theta}(t)$ is the HF Slater wave function at time $t$. The twisted angle is discretized in practical calculations. In this work, the 3D integration over $\theta$ is performed using a four-point Gauss-Legendre quadrature between 0 and $\pi$. This means that total 64 TDHF calculations are carried out for each case. The momentum k is modified accordingly as,

[TABLE]

In principle, we can recover a continuous spectrum of k with varying twisted angles. Calculations with different twist angles will give rise to different finite-volume corrections luu . It has been demonstrated that averaging results over $\theta$ can significantly cancel finite-volume effects lin ; schuer ; schuer2 .

For comparison, we also implemented ABC using the mask function method abc1 ; schuer . It was known that the mask function is effective as the imaginary absorbing potential abc1 . The mask function applies to wavefunctions and is given as ${\rm cos}(\frac{\pi}{2}\frac{r-L/2+l_{abs}}{l_{abs}})^{p}$ for L/2-l_{abs}$$<r$$\leqslant$$L/2. In ABC calculations, the box sizes are taken as 56$\times$56$\times$56 fm. In this case, the absorbing thickness $l_{abs}$ is 12 fm and $p$ is taken as 0.04. These values are dependent on the specific box sizes and time steps.

Results.— We have performed 3D TDHF calculations using the Sky3D solver for 24Mg+24Mg collisions. The ground state of 24Mg has a large prolate deformation of $\beta_{2}$=0.49 (axis ratio is 1.7:1) in our Skyrme Hartree-Fock+BCS calculations. The dimensionless quadrupole deformation is defined as $\beta_{2}$=$\frac{4\pi}{3AR_{0}^{2}}<r^{2}Y_{20}>$  bender . The reaction threshold energy is -14.93 MeV since the binding energies of 24Mg and 48Cr are 198.26 MeV and 411.45 MeV wangm , respectively. The fusion barriers in this case are from 22 MeV (head to head) to 24 MeV (side to side) depending on the collision orientation.

Fig.1 shows the time evolution of the total kinetic energy for head-to-head collisions of 24Mg+ 24Mg, with collision energies at $E_{c.m}$=29, 49 and 69 MeV, respectively. The corresponding excitation energies of the compound 48Cr are 43.93, 63.93, 83.93 MeV. In the fusion stage, there are strongly damped oscillations in kinetic energies related to the bulk dissipation. In this stage, there are negligible differences between PBC and TABC calculations before $t<1000$ fm/c. In PBC calculations, small amplitude oscillations are persistent after $2000$ fm/c. These small amplitude oscillations behave like molecular vibrating states and are presented at three different collision energies. In TABC calculations, however, the small amplitude oscillations are quickly damped and the compound nucleus at equilibrium is obtained. This damping effect has been demonstrated in the TDHF calculations of strengthes of giant resonances with TABC and ABC schuer . The particles are not really escaped in TABC, however, the box-size dependence of continuum treatment is actually diminished in TABC. For PBC calculations with not sufficiently large box sizes, the continuum is not precisely discretized. The continuum damping plays a persistent role after the fusion stage. This means that “molecular vibration states” obtained in TDHF calculations with PBC are questionable.

Fig.2 displays the Fourier analysis of the evolution of kinetic energies from TDHF-TABC calculations as shown in Fig.1. It can be seen that for the three collisions at energies of 29, 49 and 69 MeV, the main peaks are at 5.8, 5.0 and 6.0 MeV, which are lower than that of typical giant resonances pring . The damping widths for $E_{c.m}$=29, 49, 69 MeV are about 2.2 MeV, 1.4 MeV and 2.3 MeV, respectively. The head-to-head fusion is a typical underdamping process with damping widths much smaller than oscillation frequencies, in contrast to the widely recognized overdamped fission process overdamp ; overdamp2 . The main dampings are more or less similar in three cases. It can be seen that the damping time of the 49 MeV collision is longer than other two cases, which is related to its narrower damping width. For PBC calculations, the small amplitude oscillations correspond to frequencies of 7.4, 7.0, 7.6 MeV, respectively. These frequencies are much smaller than the Ikeda clustering threshold energies and are not likely to be physical molecular vibrations.

The time evolutions of quadrupole deformations from TDHF-TABC calculations of the above-mentioned collisions are shown in Fig.3. At the fusion stage, the large-amplitude oscillations are strongly damped. It can be seen that the amplitudes are larger with higher collision energies. However, the minimum deformations ($\approx$260 fm2) in the three fusion processes are close. The oscillations of deformations are not as symmetric as that in kinetic energies. The fusion is not a simple damped oscillator regarding the quadrupole deformations, due to the density dependence of incompressibility. At the equilibrium stage, the quadrupole deformations are 448, 485, and 510 fm2, respectively. These prolate deformations of the compound 48Cr are extremely large with axis ratios are 3.9:1, 3.8:1, and 3.2:1, respectively. Therefore larger quadrupole deformations at higher collision energies are not necessarily related to larger axis ratios, and volume expansions of compound nuclei play a role. The corresponding final kinetic energies of the three cases in Fig.1 are 783.5, 776, and 768 MeV respectively, which are lower than the initial kinetic energies at 809, 829, and 849 MeV respectively. The differences indicate that the total potential energies increase significantly as deformation energies play a role.

Fig.4 shows the time evolutions of the 24Mg+ 24Mg side-to-side collision at 49 MeV. Similar to Fig.1, the PBC and TABC calculations are close at the fusion stage before 1000 fm/c. This process is complex and doesn’t like a damped oscillator as in the head-to-head fusion. Small amplitude oscillations in quadrupole deformations and kinetic energies are also persistent in PBC calculations, but they are being damped in TABC calculations. This is the same continuum damping as demonstrated in the head-to-head collisions. The final kinetic energy is about 824 MeV, which is larger than 776 MeV of the head-to-head collision with the same collision energy. The final quadrupole deformation is about 220 fm2 which is much smaller than that of the head-to-head collision, showing the role of deformation energy and dependence of collision orientations.

The last but most interesting part of this work is the fusion-rotation reaction. In this case, the collision energy is taken as 40 MeV and the impact parameter is 2 fm for the side-to-side collision. In our calculations, the collision direction is along z-axis and the compound nucleus rotates in the x-z plane. To study the rotation evolution, the expectation values of $<z^{2}>$ are given in Fig.5(a). It is striking to see that the rotation amplitudes are slowly damped in TABC and ABC calculations, while the rotation is almost a perfect cosine function in PBC calculations. This rotational damping is not a surprise considering the pervious damping of small amplitude vibrations. It is more interesting because it illustrated a very clear damping picture compared to previous vibrational cases. The vibrational damping has been studied extensively bertsch , while the rotational damping has rarely been discussed rotdamp . To further study the damping effects, the spectral analysis by Fourier transformation are also shown in Fig.5(b). Note that the $<z^{2}>$ frequency (period is about 1300 fm/c) is two times the rotation frequency. The resulted rotation frequencies $\omega$ are 0.43, 0.46 and 0.49 MeV for TABC, PBC and ABC calculations, respectively. ABC calculations resulted in a larger rotational frequency (in Fig.5(a)) and angular momentum (in Fig.5(c)). Note that the damped rotation amplitudes can be written as $A(t)=A_{0}e^{-\Gamma t/2}\cos(\omega_{D}t+\phi)$, where the damped frequency $\omega_{D}=\sqrt{\omega^{2}-\Gamma^{2}/4}$. Therefore $\omega_{D}$ of TABC is slightly smaller than that of PBC due to damping effects. The damping width $\Gamma$ can be estimated to be about 0.16 MeV for TABC and ABC calculations, while $\Gamma$ is about 0.08 MeV for PBC calculations. The obtained damping widths are comparable with results from cranking shell model calculations of compound nuclei rotdamp , which can be measured by experiments rotdamp2 ; rotdamp3 .

Fig.6 shows the evolutions of density distributions in the x-z plane, corresponding to rotations in Fig.5. With time increases, we see that the surface density distributions becomes more and more uniform in TABC and ABC calculations. Note that surface gases in TDHF calculations are dynamic rather than static. The gases in ABC calculations are being absorbed at outer boundaries as well as being produced simultaneously. Consequently, the particle numbers with ABC are not conserved. The gas density is about $10^{-5}$ fm*-3*, to which protons and neutrons have similar contributions and $\alpha$ emission is possible. The uniform gas is similar to the thermal gas (dominated by neutrons) in heavy compound nuclei from finite temperature Hartree-Fock-Bogoliubov calculations zhu . The thermal neutron escaping width is proportional to the gas density which provides an equilibrium pressure. The density of thermal neutron gas is dependent on the temperature and independent of box sizes. The surface gas corresponds to emitted particles and in principle it should be removed. We see PBC, TABC and ABC all lead to a low-energy bump at $\hbar\omega$=0.2 MeV in Fig.5(b), due to the floating continuum gases, as discussed in schuer . In contrast, the density distributions at surfaces in PBC calculations have non-uniform structures. The non-uniform surface density is related to finite box-sizes, which doesn’t cause damping effects in Fig.5. The quadrupole deformation of PBC calculations is about 227 fm2 with an axis ratio of 2.2:1. In TABC and ABC calculations, however, the averaged bulk density distributions evolve towards spherical. This is consistent with the damped rotation amplitude in Fig.5. In head-to-head collisions, differently, large equilibrium deformations are maintained in compound nuclei.

TABC calculations show that the near-spherical compound nucleus 48Cr is still rotating at each twisted angles. For the rotation in x-z plane, the calculated averaged angular momentum <$$J_{y}$$> is about 6$\hbar$ at 7000 fm/c, which decays slowly and smoothly. The calculated <$$J_{y}$$> with PBC is decaying slightly faster associated with oscillations. It is known that hot nuclei at equilibrium become spherical at high temperatures in the mean-field framework. In our case, the angular momentum <$$J_{y}$$> from TABC is persistent although density distributions become spherical. It is understandable that spherical compound nuclei at equilibrium can rotate due to the fading of quantum effects, although spherical quantum systems don’t rotate. In Fig.5(c), we see that the total angular momentum is not conserved with different boundary conditions. In TABC and PBC, the angular momentum is not conserved whenever the emitted particles from highly-excited compound nuclei encounter boundaries, while this conservation is preserved for TDHF cranking calculations guo . In this respect, ABC is more reasonable since its angular momentum <$$J_{y}$$> decreases slowly due to real particle emissions and <$$J_{y}$$> is larger than TABC results.

Summary.— We implemented the twisted boundary condition in time-dependent Skyrme Hartree-Fock calculations of 24Mg +24Mg collisions, which are performed in 3D coordinate spaces. In head-to-head and side-to-side collisions, small amplitude vibrations are persistent in calculations with periodic boundary conditions, but they are damped with twist-averaged and absorbing boundary conditions. In TABC, this kind of damping mechanism is related to the cancelation of box size dependence in continuum treatment. By studying the side-to-side collision with an impact parameter of 2 fm, we found that the rotation amplitude is damped as well, in which the continuum damping width can be clearly extracted. The density distributions show that in TABC and ABC calculations the compound nucleus becomes spherical surrounded by a uniform gas. The surface density distributions in PBC calculations are non-uniform, due to finite box-size effects. The angular momentum decreases slowly due to particle emissions. The near-spherical compound nucleus are still rotating with a considerable angular momentum. These results are inspiring and provide a better understanding of rotating compound nuclei. In principle, ABC can be applied to a finite system with continuum by avoiding wave reflections. PBC descriptions are insufficient for a system when continuum is not negligible. TABC can remedy the spurious effects due to finite box sizes by describing the periodicity of a system with continuum correctly. With a very large box, different boundary conditions should give consistent results abc1 . We demonstrated that consequences of continuum damping after long-time evolutions could be significant. Further applications of twisted boundary conditions in nuclear reactions and weakly bound nuclei will be valuable.