Characterizing the fission events

We start with the analysis of the orientation of the fission plane with respect to the beam direction. The fission plane is reconstructed by the velocity of two FFs, using ${\overrightarrow{n}}_{\text{FF}}=\left({\overrightarrow{v}}_{{\text{F}}_1}\times {\overrightarrow{v}}_{{\text{F}}_2}\right)/\left|{\overrightarrow{v}}_{{\text{F}}_1}\right|\left|{\overrightarrow{v}}_{{\text{F}}_2}\right|$ to denote the normal vector of the fission plane, as shown in Fig. 3(a). Defining α₁ as the angle between ${\overrightarrow{n}}_{\text{FF}}$ and the beam direction ${\overrightarrow{v}}_{\text{beam}}$, one can characterize how much the fission plane deviates from the beam. The distribution of $\left|\text{cos}\left({\alpha }_1\right)\right|$ is peaked at 0 with a rather small width ${\sigma }_{{\alpha }_1}\approx 6^{\circ }$, as shown in Fig. 3(b), inferring that the fission plane keeps approximately the memory of the initial angular momentum of the rotating system. With $Z\ge 10$ as the condition to identify FFs for theoretical calculations, the transport model prediction about the distribution of $\left|\text{cos}\left({\alpha }_1\right)\right|$ is in rather agreement with the experiment. The scattering plots of folding angle vs. $\left|\text{cos}\left({\alpha }_1\right)\right|$ provide the information of fission and detection geometry. With the detector filter of PPACs on both θ_lab and ${\varphi }_{\text{lab}}$ according to the setup, the experimental folding angle in Fig. 3(c) can be approximately described by the model simulation in Fig. 3(d).

Fig. 3Full-screen View

(Color online) (a) Geometric diagram of fission plane of FFs and LCP emission. (b) Angular distribution between the normal vector ${\overrightarrow{n}}_{\text{FF}}$ of the fission plane and the beam direction ${\overrightarrow{v}}_{\text{beam}}$. The experimental (c) and simulation (d) results of the folding angle vs. $\left|\text{cos}\left({\alpha }_1\right)\right|$ are shown in the bottom panels

The characteristics of this rotating fissioning system was obtained using the experiment data and theoretic simulations. First, to estimate its charge and mass, the linear momentum transfer (LMT) should be estimated experimentally. Assuming a symmetric fission processes, the velocity of the fissioning system (FS) can be simply calculated by ${\overrightarrow{v}}_{\text{FS}}=\frac{1}{2}\left({\overrightarrow{v}}_{\text{F}1}+{\overrightarrow{v}}_{\text{F}2}\right)\mathrm{,}$ (2) where ${\overrightarrow{v}}_{\text{F}1}$ and ${\overrightarrow{v}}_{\text{F}2}$ are the velocities of the two FFs, respectively. The LMT is defined as $\text{LMT}=\frac{A_{\text{tar}}\cdot v_{\text{FS}}^Z}{A_{\text{pro}}\cdot \left(v_{\text{pro}}-v_{\text{FS}}^Z\right)}\mathrm{,}$ (3) Here the subscripts pro and tar denotes the projectile and the target, respectively. $v_{\text{FS}}^Z$ is the projection of ${\overrightarrow{v}}_{\text{FS}}$ on the beam direction. As shown in Fig. 4, the distribution of the LMT derived from the experimental data is peaked in the vicinity of 0.4. The small peak below LMT<0.2 is contributed by the fission events triggered by PPAC 1 and PPAC 3. Accordingly, the typical charge and mass of the rotating fission system are $Z_{\text{FS}}\approx Z_{\text{tar}}+Z_{\text{pro}}\cdot \text{LMT}=96$ and $A_{\text{FS}}\approx A_{\text{tar}}+A_{\text{pro}}\cdot \text{LMT}=242$.

Fig. 4Full-screen View

Experimental distribution of LMT

Second, to estimate the angular momentum of the rotating fission system, one needs the most probable impact parameter, which can determined by the event weigh obtained from transport model simulations filtered by experimental conditions. Defining the fission event weight by $W_{\text{F}}=b\frac{n_{\text{F}}}{N_{\text{tot}}\left(b\right)}\mathrm{,}$ (4) where $n_{\text{F}}$ is the number of fission events among $N_{\text{tot}}$ events simulated at a given impact parameter b.

Figure 5 shows the distribution of $W_{\text{F}}$, where the most probable impact parameter $b_{\text{m}}$ is located in the vicinity of 7 fm.

Fig. 5Full-screen View

The weight of the fission events as a function of impact parameter b in ImQMD05 simulations

The distance between the transferred part of the projectile and the mass center of the fissioning system is defined as $D=b_{\text{m}}\frac{A_{\text{tar}}}{A_{\text{tar}}+A_{\text{pro}}\cdot \text{LMT}}\mathrm{,}$ (5) where $b_{\text{m}}=7$ fm, $A_{\text{tar}}=208$, $A_{\text{pro}}=86$ and LMT=0.4, respectively.

The angular momentum is written as $J=P_{\text{pro}}\cdot \text{LMT}\cdot D\mathrm{,}$ (6) where $P_{\text{pro}}=18700$ MeV/c and $D\approx 6$ fm was derived with LMT=0.4. Then, the angular momentum of the rotating system is approximately $J\approx 200\hslash$.

Third, to estimate the excitation energy of the rotating fission system, the moment of inertia I of a spherical nucleus with the mass $M_{\text{FS}}$ is $I=\frac{2}{5}{M}_{\text{FS}}{r}_{\text{FS}}^2\mathrm{,}$ (7) where $r_{\text{FS}}=1.4A_{\text{FS}}^{1/3}$ fm is the radius of the fissioning system. The rotating energy $E_{\text{rot}}=J^2/2I\approx 100$ MeV is approximately obtained. Ignoring the reaction Q value, the excitation energy could be extracted by $E^{*}=E_{\text{kin}}^{\text{i}}-E_{\text{kin}}^{\text{f}}-E_{\text{rot}}\mathrm{,}$ (8) where $E_{\text{kin}}^{\text{i}}$ and $E_{\text{kin}}^{\text{f}}$ are the initial state kinetic energy and the final state kinetic energy, respectively. Approximately, one has $E^{*}\approx 600$ MeV. The excitation energy is close to the one of the fission system formed in 25 MeV/u Ar+Au at $\text{LMT}\approx 80\%$, where the E^* was calculated by the pre-scission α multiplicity [100]. For additional properties of fission systems, such as Viola systematics and angular distribution of the fission axis, please refer to our previously published paper [87].

Analysis of the energy spectra of t and ³He

We now present the analysis of the emission of triton and ³He in the (fast)fission events. The energy spectra of LCPs in coincidence with FFs contain thermal and dynamical information of the particles emitted from the fission events. Fig. 6 presents the energy spectra of triton (open circles) and ³He (open triangles) emitted from fission events in different angular ranges corresponding to SSDTs 2 to 4. To reduce the contamination of quasi-projectiles, the data of SSDT1 covering 10-20° in the laboratory is not counted here. It is shown that the spectrum of ³He is generally harder than that of triton, leading to a larger average kinetic energy of the former. The difference between triton and ³He is more pronounced at forward angles than at large angles. This observation of “³He-puzzle” is in accordance with the previous inclusive measurements at high beam energies [73, 75-77, 81, 101-104].

Fig. 6Full-screen View

(Color online) The experimental energy spectra of triton (circle) and ³He (triangle) in ${20}^{\circ }\le {\theta }_{\text{lab}}\le {60}^{\circ }$ covered by SSDT2 to SSDT4 in coincidence with two FFs. The arrows represent the peak position of each experimental energy spectrum

The “³He-puzzle” has been interpreted by two possible scenarios: sequential decay [74] and coalescence model [78]. In the sequential decay scenario, the difference between ³He and triton is influenced by the Coulomb barrier, for which ³He is emitted at an earlier stage with high temperature to overcome the Coulomb barrier higher than that of triton [74]. In coalescence scenario, which was applied to interpret the difference between ³He and α particles [78], the former is dominantly produced by the coalescence of preequilibrium nucleons, delivering larger mean kinetic energy. These two explanations are qualitatively in agreement, supporting that ³He is predominantly emitted at earlier stage. Our experimental results show that the “³He-puzzle” exists in the events tagged by fission. It suggests that the puzzle exists in both inclusive and fission events.

Out-of-plane emission and the effect of $E_{\text{sym}}\left(\rho \right)$

Benefiting from the wide angular coverage of the SSDTs and PPACs in laboratory reference frame, the angular behavior of the particle emission can be analyzed. To compare the yields of particles with different energy spectrum behaviors and avoid the influence of the possible experimental distortion caused by the energy threshold in each SSDTs, a data adaptive energy spectrum peak cut scenario is applying. We focus on the descending part on the high energy side of the energy peak. The energy peak positions (E_p) are listed in Table 2. Meanwhile, using the energy condition $E\ge E_{\text{p}}$ as the low limit cut, one can suppress the interference of the evaporation process and emphasize the feature of the dynamic emission.

The angular distribution of R(t/³He) as a function of the polar angle in laboratory θ_lab is generated with events of one LCP in coincidence with two FFs, as shown in Fig. 7. The same energy threshold, geometry and folding angle cuts are applied to both experimental and simulation results. It is shown that for the wide angular range, the distribution exhibits a rising trend. This feature is consistent with the moving source picture, where the neutron richness of particle emission increases from the projectile-like source to the medium velocity source corresponding to the neck, as predicted by various transport model simulations [40, 41, 46, 48, 49, 51, 105-109], and experimentally observed in a specific angular window [42, 45, 50, 54, 79, 110-112] or a parallel velocity window [45, 79, 80, 113-118].

Fig. 7Full-screen View

(Color online) The ratio R(t/³He) as a function of θ_lab. The black solid squares and black line represent the experiment data and fitting result with $E\ge E_{\text{p}}$ cuts in coincidence with fission events. The red and blue cross markers represent the ImQMD05 calculations data of γ = 0.5 and 1.0 in the inset. The red dot and blue dash lines are the fitting results of γ = 0.5 and 1.0 which is normalized with experimental fitting result of p₀

In order to see the symmetry energy effect, a soft (γ = 0.5) and a stiff (γ = 1.0) symmetry energy are adopted in the ImQMD05 simulations. These two γ values correspond to slope parameter of $E_{\text{sym}}\left(\rho \right)$ with L=51 and 77 MeV at ρ₀, respectively. Although the predicted value of R(t/³He) is far off to the experiment, the rising trend of R(t/³He) as a function of θ_lab was reproduced by model simulations in both γ cases. In order to quantify the increasing rate, the function of $f(x)=e^{\left(p_0+p_{1}x\right)}$ is applied to fit the data and the model predictions, respectively. The parameter p₁ describes the increasing rate of R(t/³He) to θ_lab. As shown in Fig. 7, the rising trend depends on γ. Visibly, a softer $E_{\text{sym}}\left(\rho \right)$ causes a relative larger increasing rate. When comparing the fitting results between experiment and model in Table 3, the value of experimental p₁ is marginally located between γ = 0.5 and 1.0. Nevertheless, the large uncertainty here reduces the sensitivity and hinder to make a convincing constraint of $E_{\text{sym}}\left(\rho \right)$.

It is then motivated to go a further step to find a novel probe, of which the fission event topology is better controlled and the sensitivity on $E_{\text{sym}}\left(\rho \right)$ can be enhanced. Fig. 8 presents the angular distribution of R(t/³He) with respect to the fission plane. The α₂ on the abscissa is the relative angle between ${\overrightarrow{n}}_{\text{FF}}$ and the velocity of the coincident triton or ³He ${\overrightarrow{v}}_{\text{LCP}}$ as shown in Fig. 3(a), with $\left|\text{cos}\left({\alpha }_2\right)\right|=0$ (1) corresponding to in-plane (out-of-plane) emission. Again, the same cuts are applied for both experimental and theoretical results. The increasing trend of R(t/³He) with $\left|\text{cos}\left({\alpha }_2\right)\right|$ indicates that the neutron rich particles emitted from out-of-fission-plane is enhanced. This phenomenon is the consequence of the competition between the isospin migration and the centrifugal motion of the particles in the rotating fission system. When the reaction system is viewed as a rotating emission source, particles emitted near the fission plane are subjected to stronger centrifugal potential during the emission process, weakening the difference between neutrons and protons under the isovector potential. From the in-plane to out-of-plane, more neutron rich particles are emitted due to the effect of isospin fractionation [67], indicating that the effect of the isovector potential becomes more significant compared to centrifugal potential. This observation gives us the chance to explore the properties of isospin transport and the density dependence of $E_{\text{sym}}\left(\rho \right)$ in (fast)fission reactions.

Fig. 8Full-screen View

(Color online) The ratio R(t/³He) as a function of $\left|\text{cos}\left({\alpha }_2\right)\right|$. The black solid squares and black line represent the experiment data and fitting result. The red and blue cross markers represent the theoretical calculations data of γ = 0.5 and 1.0 in the inset. The red dot and blue dash lines are the fitting results of γ = 0.5 and 1.0 which is normalized with experimental fitting result of p₀

Similarly, to describe the increasing trend of the angular distribution of $\left|\text{cos}\left({\alpha }_2\right)\right|$, the function of $f(x)=p_0+p_1{x}^4$ is used to fit data and the simulations. Again, p₀ is far off to the experiment due to the clustering difficulty of transport model, but the parameter p₁ can be used to describe the increasing rate of the ratio with out-of-plane angle. In Fig. 8, the fitting curves exhibit a different increasing behavior between γ = 0.5 and 1.0, indicating that the enhancement of neutron rich particle emission out of fission plane is sensitive to the form of $E_{\text{sym}}\left(\rho \right)$. Inspecting the increasing curves and the values of p₁ as listed in Table 4, one finds that the experimental increasing rate situates between the theoretical prediction with γ = 0.5 and 1.0, in accordance with the conclusion of our previous work [36], where a totally different probe was used. The comparison seems to exclude very soft (γ <0.5) and very stiff (γ >1.0) candidates of symmetry energy. The results indicate that the ratio R(t/³He) as a function of $\left|\text{cos}\left({\alpha }_2\right)\right|$ seems to be a sensitive probe for density dependent symmetry energy, especially in the larger $\left|\text{cos}\left({\alpha }_2\right)\right|$ range, which is very close to the boundary of the current detector coverage. Hence, more events in the larger $\left|\text{cos}\left({\alpha }_2\right)\right|$ range are preferentially requested in the further experiments. Data analysis of a new measurement of ⁸⁶Kr+¹²⁴Sn at 25 MeV/u is ongoing [119].

Figure 9 shows in addition the relationship between $\left|\text{cos}\left({\alpha }_2\right)\right|$ and θ_lab with the experiment events of triton in coincidence with two fission fragments. Visibly, there is a weak positive correlation between $\left|\text{cos}\left({\alpha }_2\right)\right|$ and θ_lab. The origin of the correlation is partly due to the fact that the azimuth coverage of the PPAC is quite limited. With such weak correlation, one infers that the two distributions shown in Fig. 7 and Fig. 8 have their own implications. Namely, the distribution of R(t/³He) as a function of θ_lab indicates that the low density and neutron rich medium velocity emission source (neck) is formed, while the distribution of R(t/³He) as a function of $\left|\text{cos}\left({\alpha }_2\right)\right|$ characterizes the fine out-of-plane properties of the isospin transport in a fissioning process. Upon comparing the results presented in Table 3 and Table 4, it becomes evident that the enhancement of R(t/³He) vs. $|\cos ({\alpha }_2)|$, particularly at larger out-of-plane angles, appears to be a more sensitive probe for studying nuclear symmetry energy than the polar angular distribution of R(t/³He). In another word, in the properly characterized fission events, the effect of $E_{\text{sym}}\left(\rho \right)$ can be magnified, supporting the previous predictions by transport model simulations [40].

Fig. 9Full-screen View

(Color online) The scattering plot between $|\cos ({\alpha }_2)|$ and laboratory angle with the experiment events of triton in coincidence with two fission fragments

Currently we do not attempt to make a fine tuning and constraint of γ parameter in the simulations, since the absolute value of R(t/³He) is not yet well reproduced, as indicated by Fig. 7 and 8. Further studies are required in transport model in order to elucidate the origin and the formation mechanism of light clusters including triton and ³He. Recently, the yield of light clusters is better reproduced by introducing Mott effect in transport model [120]. Meanwhile, the cooling process of the rotating fissioning system with similar E^* and J is of high interest. We are going to make further calculations on particle emission from a rotating system with inclusion of deuteron, triton and ³He apart of neutron, proton and α particles, as done in [121]. The emission of other particles than A=3 isobars may bring significant effect to the featured distribution of the latter in the cooling process of the fissioning system.