Fission barriers from multidimensionally-constrained covariant density functional theories

In recent years, we have developed the multidimensionally-constrained covariant density functional theories (MDC-CDFTs) in which both axial and spatial reflection symmetries are broken and all shape degrees of freedom described by βλμ with even μ, such as β20, β22, β30, β32, β40, etc., are included self-consistently. The MDC-CDFTs have been applied to the investigation of potential energy surfaces and fission barriers of actinide nuclei, third minima in potential energy surfaces of light actinides, shapes and potential energy surfaces of superheavy nuclei, octupole correlations between multiple chiral doublet bands in 78Br, octupole correlations in Ba isotopes, the Y32 correlations in N = 150 isotones and Zr isotopes, the spontaneous fission of Fm isotopes, and shapes of hypernuclei. In this contribution we present the formalism of MDC-CDFTs and the application of these theories to the study of fission barriers and potential energy surfaces of actinide nuclei.


Introduction
Many intrinsic nuclear shapes which break certain spatial symmetries play important roles in determining nuclear structure and fission properties [1][2][3][4][5].The nuclear shape can be described by parametrizing the nuclear surface or the nucleon density distribution with the multipole expansion method.The deformation is then characterized by the expansion coefficient β λµ .
Since the discovery of the fission phenomena [26], a quantitative description of the whole fission process has been very challenging.It was already known that the fission dynamics are mostly determined by the barriers prohibiting the dissolving of the nucleus [27].Thus for solving the fission problem, one needs very accurate information on fission barriers, e.g., the heights.e-mail: sgzhou@itp.ac.cnDue to the shell effects, many actinide nuclei are found to be characterized by the two-humped fission barrier [28][29][30][31][32][33][34][35][36][37].Consequently any models concerning the fission phenomena should be able to reproduce the barrier heights with certain assumptions on the nuclear shapes.For example, in early calculations of the fission barriers, the nuclei are usually assumed to be axially symmetric.Later it was found that the occurrence of the triaxial and the octupole deformations has non-negligible consequences in determining the heights of the inner and outer fission barriers, respectively.It has been known from the macroscopicmicroscopic model calculations that the inner fission barrier is lowered when the triaxial deformation is allowed, so is the outer one by the reflection asymmetric distortion [30,31].Such lowering effects were also revealed in the non-relativistic [35] and relativistic [38,39] density functional calculations, respectively.Recently, it is shown in Ref. [40] that the outer barriers are further lowered by the triaxial deformation compared with axially symmetric results.This lowering effect for the reflection-asymmetric outer barrier is of 0.5~1 MeV, accounting for 10%~20% of the barrier height.It is thus necessary to include these shape degrees of freedom properly in any systematic calculations.a model, the so called multidimensionally-constrained covariant density functional theories (MDC-CDFTs), by breaking the reflection and axial symmetries simultaneously.
In MDC-CDFTs, the nuclear shape is assumed to be invariant under the reversion of x and y axes, i.e., the intrinsic symmetry group is V 4 and all shape degrees of freedom β λµ with even µ (β 20 , β 22 , β 30 , β 32 , β 40 , • • • ) are included self-consistently.The MDC-CDFTs consist of two types of models: the multidimensionally-constrained relativistic mean field (MDC-RMF) model and the multidimensionallyconstrained relativistic Hartree-Bogoliubov (MDC-RHB) model.
In the MDC-RHB model, pairing correlations are treated by making the Bogoliubov transformation which generalizes the BCS quasi-particle concept and provides a unified description of particle-hole (ph) and pp correlations in a mean-field level.A separable pairing force of finite range [53][54][55][56][57] is adopted.The MDC-RHB model has been used to study the spontaneous fission of fermium isotopes [58], octupole correlations between multiple chiral doublet (MχD) bands in 78 Br [59], octupole correlations in Ba isotopes [15] and the Y 32 correlations in neutron-rich Zr nuclei [60].
In this contribution, we present briefly the formalism of the MDC-CDFTs and some results of fission barriers.The formulae of the MDC-CDFTs is given in Sec. 2. The results and discussions are presented in Sec. 3. Finally we summarize in Sec. 4.
The starting point of the relativistic NL-PC density functional is the following Lagrangian: where M B is the nucleon mass and L lin , L nl , L der and L Cou are the linear coupling, non-linear coupling, derivative coupling and the Coulomb part, respectively, see Refs.[50][51][52] for details.Starting from the Lagrangian, we can deduce the relativistic mean field equations with the variational principle: where S (r) and V(r) are the scalar and vector potentials, respectively.As usual we suppose that the states are invariant under the time-reversal operation, which means that all time-odd components of the currents and the potentials vanish.
The pairing correlations play an important role in the fission process [79,80].We have implemented the BCS method or the Bogoliubov transformation for pairing in the MDC-RMF or MDC-RHB models.
The axially deformed harmonic oscillator (ADHO) basis [81,82] was adopted in the development of MDC-CDFTs.In contrast to the traditional ADHO code, in this work the components with different quantum numbers are mixed, allowing us to treat triaxial and octupole nuclear shapes simultaneously.
The ADHO basis are defined as the eigensolutions of the Schrödinger equation with an ADHO potential, where ω z and ω ρ are the oscillator frequencies along and perpendicular to the symmetry axis, respectively.The solution of Eq. ( 3) reads where φ n z (z) and R m l n ρ (ρ) are the HO wave functions, χ s z is a two component spinor and C α is a complex number introduced for convenience.These bases are also eigenfunctions of the z component of the angular momentum j z with eigenvalues K = m l + m s .
These bases form a complete set for expanding any two-component spinors.For a Dirac spinor with four components, the following expansion can be made, where the sum runs over all the possible combination of the quantum numbers α = {n z , n r , m l , m s }, f α i and g α i are the expansion coefficients.
In the general case that the axial symmetry as well as the space reflection symmetry are broken, different components with different m j and π are mixed with each other.In this case it is convenient to choose the operator Ŝ = ie −iπ j z with S 2 = 1 as the conserved quantity to classify the energy levels.S is a good quantum number for the basis with Ŝ Φ α = S Φ α = (−1) K α − 1 2 Φ α , which means that the bases Φ α with K α = 1/2, −3/2, 5/2, −7/2, • • • span the subspace with S = 1, while their time-reversed states span the one with S = −1.Note that now the block with K = 1/2 should be mixed with that with K = −3/2 instead of that with K = 3/2.
For deformed nuclei with the V 4 symmetry, we expand the potentials V(r) and S (r) and the densities in terms of the Fourier series, The matrix elements of the potentials can then be calculated from the expansion coefficients [42,60].
In addition to the mean field energy, the centre of mass correction is calculated either phenomenologically or microscopically, depending on which effective interaction is used.The intrinsic multipole moments are calculated from the densities by where Y λµ (Ω) is the spherical harmonics and τ refers to the proton, neutron or nucleon.The deformation parameter β λµ is obtained from the corresponding multipole moment by where R = 1.2 × A 1/3 fm and N is the number of proton, neutron or nucleons.

Results
The double-humped fission barriers of actinide nuclei are often used to benchmark theoretical models for fission barriers.As the nucleus evolves from the ground state to the fission configurations, various shape degrees of freedom play important and different roles in determining the heights of the inner and outer barriers.As mentioned in the introduction, the inner barrier is lowered when the triaxial deformation is allowed, while for the outer barrier, the reflection asymmetric shape is favored.In Ref. [40] we have shown that, the reflection asymmetric outer barrier may be further lowered by the triaxial distortion with the MDC-RMF model.  24Pu with various self-consistent symmetries imposed.The solid black curve represents the calculated fission path with V 4 symmetry imposed: the red dashed curve that with axial symmetry (AS) imposed, the green dotted curve that with reflection symmetry (RS) imposed, the violet dot-dashed line that with both symmetries (AS & RS) imposed.The empirical inner (outer) barrier height [83] is denoted by the grey square (circle).The energy is normalized with respect to the binding energy of the ground state.
In Fig. 1 we show the one-dimensional potential energy curves (PEC) from an oblate shape with β 20 about −0.2 to the fission configuration with β 20 beyond 2.0 which are obtained from calculations with different selfconsistent symmetries imposed: the axial (AS) or triaxial (TS) symmetries combined with reflection symmetric (RS) or asymmetric (RA) cases.The importance of the triaxial deformation on the inner barrier and that of the octupole deformation on the outer barrier are clearly seen: The triaxial deformation reduces the inner barrier height by more than 2 MeV and results in a better agreement with the empirical value [83]; the RA shape is favored beyond the fission isomer and lowers very much the outer fission barrier.Besides these features, it was found in Ref. [40] that the outer barrier is also considerably lowered by about 1 MeV when the triaxial deformation is allowed.In addition, a better reproduction of the empirical barrier height can be seen for the outer barrier.It has been stressed that this feature can be revealed only when the axial and reflection symmetries are simultaneously broken [40].The solid black curve represents the calculated fission path with V 4 symmetry imposed, the blue dotted curve represents that with axial symmetry (AS) imposed.The empirical inner barrier height [85] is denoted by the dots.The energy is normalised with respect to the binding energy of the ground state.The parameter set used is PC-PK1 [84].
In Fig. 2 we show the calculated potential energy curves for 240 U.The results with and without non-axial deformations are presented.Near the second barrier both the axial and non-axial results are reflection asymmetric.The empirical values of the fission barriers are taken from Ref. [85] and depicted by red dots.The calculated inner and outer fission barrier heights are 4.96 and 5.43 MeV, respectively.This means that in 240 U, the outer fission barrier is higher than the inner one, which is in contrast to the empirical values.It's worthwhile to mention that the outer fission barrier height is very close to 5.5 MeV, a value which was recently obtained in JAEA [86].It is clearly shown that the inclusion of non-axial shapes improve the agreement between the calculation and the experiment.
In Ref. [42] we applied the MDC-RMF model to a systematical study of fission barriers of the actinide nuclei with both axial and reflection symmetries broken.In Fig. 3 we show the calculated potential energy curves near  [83,85].Taken from Ref. [42].
the outer barriers of the actinide nuclei.For comparison the results with and without non-axial deformations are both presented.Note that near the second barrier all the results are reflection asymmetric.Because only the β 20 deformation is constrained, the total binding energy is automatically minimized against other possible deformation parameters.The empirical values of the fission barriers are depicted by red dots.Clearly, all the axial symmetric calculations overestimate the barrier height.After including the triaxial deformation, the agreement between the calculated values and the empirical ones becomes better.
Besides the double-humped barriers of the PES's in the actinide region, the occurrence of a third barrier at large deformations beyond the second one was predicted by macroscopic-microscopic model calculations in the 1970s.But contradictory results about the existence of the third barrier were later obtained with different models.In Ref. [45] the MDC-RMF model was used to investigate the triple-humped barriers in light even-even actinides 232,234,236,238 U and 226,228,230,232 Th.The relativistic functionals PC-PK1 [84] and DD-ME2 [87] were adopted.Next we will mainly discuss the results from DD-ME2.
A third minimum was found on the PES's of 232,234,238 U: The pocket is rather shallow for 232 U and 234 U and very pronounced for 238 U.These results are quite similar to those obtained in recent calculations based on the macroscopic-microscopic model [88][89][90][91] and the Skyrme Hartree-Fock-Bogoliubov model [92].
In the case of Th isotopes, our calculation predicts a pronounced third minimum for 226 Th with a pocket depth of 2-3 MeV.As the neutron number N increases, both the energy of the third minimum and the height of the third barrier decrease, and the depth of the third well is also reduced.This trend has also been predicted in macroscopicmicroscopic calculations [91].For 230 Th only a shallow pocket of depth less than 1 MeV occurs around the third minimum.The pocket around the third minimum is very shallow for 232 Th.

Summary
In this contribution we present the formalism and some applications of the MDC-CDFTs in which all shape degrees of freedom β λµ with even µ are allowed.The potential energy (curves) of actinide nuclei and the effect of the triaxiality on the first and second fission barriers were investigated.It is found that besides the octupole deformation, the triaxiality also plays an important role second fission barriers.The third of actinide nuwere also studied.A pronounced third minimum were predicted for 226 with a pocket depth of 2-3 MeV.

Figure 1 .
Figure 1.(Color online) Potential energy curves of240 Pu with various self-consistent symmetries imposed.The solid black curve represents the calculated fission path with V 4 symmetry imposed: the red dashed curve that with axial symmetry (AS) imposed, the green dotted curve that with reflection symmetry (RS) imposed, the violet dot-dashed line that with both symmetries (AS & RS) imposed.The empirical inner (outer) barrier height[83] is denoted by the grey square (circle).The energy is normalized with respect to the binding energy of the ground state.The parameter set used is PC-PK1[84].Taken from Ref.[40].

Figure 2 .
Figure 2. (Color online) Potential energy curves of 240 U.The solid black curve represents the calculated fission path with V 4 symmetry imposed, the blue dotted curve represents that with axial symmetry (AS) imposed.The empirical inner barrier height[85] is denoted by the dots.The energy is normalised with respect to the binding energy of the ground state.The parameter set used is PC-PK1[84].

Figure 3 .
Figure 3. (Color online) Potential energy curves of even-even actinides nuclei in the outer barrier regions from the MDC-RMF calculations.The reflection asymmetric shapes are allowed.The triaxial results are displayed by solid lines, while the results from the axial symmetric calculations are shown by dashed lines.The binding energies are normalized with respect to the binding energies of the ground states of each nucleus.The empirical values of fission barrier heights are shown as red dots[83,85].Taken from Ref.[42].