Examining barrier distributions and, in extension, energy derivative of probabilities for surrogate experiments

The energy derivatives of probabilities are functions suited to a best understanding of certain mechanisms. Applied to compound nuclear reactions, they can bring information on fusion barrier distributions as originally introduced [1,2], and also, as presented here, on fission barrier distributions and heights. Extendedly, they permit to access the compound nucleus spin-parity states preferentially populated according to an entrance channel, at a given energy.

Figure 1

Top panels: (a) cosinus shape potential: $V\left(x\right)=V_0(1-\cos \left(\alpha x\right))$, (b) Eckart potential: $V\left(x\right)=\frac{V_1\exp \left(\alpha (x-x_0)\right)}{1+\exp \left(\alpha (x-x_0)\right)}+\frac{V_2\exp \left(\alpha (x-x_0)\right)}{{(1+\exp \left(\alpha (x-x_0)\right))}^2}$, historically introduced to describe some chemical reactions, and (c) potential representing the application of a uniform static electric field perpendicular to a plane surface conductor in order to describe electron emission in intense electric fields: $V\left(x\right)=\frac{V_0}{1+\exp (-\alpha (x-x_0))}-V_0\frac{(x-x_0)}{x_1-x_0}$. [All the potentials (a), (b), and (c) were scaled in order to have electron-Volt (eV) magnitude; the dashed lines indicate the top barrier values.] Middle panels: (d), (e), and (f) tunneling probabilities as function of energy [respectively corresponding to the above potentials (a), (b), and (c)] calculated with the numerical Numerov method as in the Lambin-Vigneron approach [7, 8]. And finally for the bottom panels: (g), (h), and (i) barrier distributions directly obtained from the numerical energy derivative of the previous tunneling probabilities (d), (e), and (f). The dashed lines indicate the top barrier values as respectively obtained from (a), (b), and (c). The horizontal (bottom panels) and the vertical (top panels) arrows define the energy spreading of the barrier distributions.

Figure 2

Upper panel: Comparison of fission (viewed as a tunneling problem) probabilities, for the same ${}^{242}\mathrm{Am}$ CN: $P_{n,f}^{\mathrm{HF}}\left(E\right)$ calculated [11] (HF) for the system $n+{}^{241}\mathrm{Am}{}^{242}{\mathrm{Am}}^{*}$ fission (solid line), $P_{\gamma ,f}^{\mathrm{HF}}\left(E\right)$ calculated [11] (HF) for the system $\gamma +{}^{242}\mathrm{Am}{}^{242}{\mathrm{Am}}^{*}$ fission (dashed line) and measured $P_{SR,f}^{\exp }\left(E\right)$ via a surrogate reaction [12, 13]: ${}^{243}\mathrm{Am}{(}^3\mathrm{He},\alpha ){}^{242}{\mathrm{Am}}^{*}\mathrm{fission}$ (points). Lower panel: Comparison between fission barrier distributions respectively obtained from the energy derivative of their corresponding fission probability (same legend as in upper panel).

Figure 3

Schematic representation. Top panels: (Left) Heavy ion fusion barriers or more precisely, as usually called in these approaches, fusion eigenbarriers corresponding to coupling between the ground state and one (here) collective excited state of the target nucleus in the entance channel [2]. (Right) Corresponding fusion barrier distribution obtained from the energy derivative of the total fusion probability. Bottom panels: (Left) Fission barriers described as transition states, each associated to a fission path for a given spin-parity $J^{\pi }$ of the CN formed. (Right) By analogy to the fusion case, fission barrier distribution obtained from the energy derivative of the total fission probability. In this schematic representation fission barriers do not have any second well and, consequently resonant transmissions are avoided and fine peak structures too (we just look at the “gross” structure of the global fission barrier distribution, as identically done in Sec. 2 C). Contrary to the fusion case, individual contributions of each transition (barrier) state cannot be resolved for fission, due to quantum spreading effects (Sec. 1 B) and to the energy proximity of all these states. As previously mentioned, the wider the potential barrier, the narrower the barrier distribution is spread, and reciprocally. For this reason two short vertical lines on the top of fusion barrier distribution (right top panel) were reported, in order to indicate the spreading of the fission barrier distribution for comparisons with the fusion barrier distribution. In fact, these barrier distributions for fusion and fission processes could also respectively be called and viewed as energy distribution of excited collective states during the fusion process and, energy distribution of populated transition states during the fission process.

Figure 4

Comparison between energy distributions of populated states obtained from the derivative of a neutron emission probability, for the same ${}^{176}\mathrm{Lu}$ CN formed at the same excitation energies: $D_{n,n}^{\mathrm{HF}}\left(E\right)$ calculated [11] (HF) for the reaction ${}^{175}$Lu($n,n$)${}^{175}$Lu${}^{*}$ (solid line), $D_{\gamma ,n}^{\mathrm{HF}}\left(E\right)$ calculated [11] (HF) for the reaction ${}^{176}$Lu(${\gamma ,n)}^{175}$Lu${}^{*}$ (dashed line). $S_n$ arrow indicates the neutron separation energy value. Half-numbers indicate the corresponding spin-parity states populated in the residual nucleus ${}^{175}$Lu and their associated stick mark indicates their energy value taking into account the neutron separation energy ($S_n=6.288$ MeV) when transcribed in the CN system.

Figure 5

Energy zoom of the comparison between fission barrier distributions tails of the same ${}^{242}\mathrm{Am}$ CN formed through different entrance channels. Same legend as Fig. 2 was kept. The narrow peaks of both calculations correspond to resonant fission barrier transmissions as obtained from Ref. [14] treatment. In fact these narrow peaks can be viewed as the energy distributions of the populated bound states trapped in the multihumped barrier wells.

Figure 6

Dalitz representation of two evaluated probabilities sets $\{P_{\mathrm{EC},f}\left(E\right),P_{\mathrm{EC},n}\left(E\right),P_{\mathrm{EC},\gamma }\left(E\right)/S_n⩽E⩽S_{n,2n}\}$, for $n+{}^{241}\mathrm{Am}$ (solid curve $\mathrm{EC}=n$) and $\gamma +{}^{242}\mathrm{Am}$ (dashed curve $\mathrm{EC}=\gamma$) entrance channels. The very big points are there to indicate some energy values (corresponding also to the experimental values of Fig. 2), in order to appreciate the discrepancies on probabilities between these two systems. Closed symbols correspond to the $n+{}^{241}\mathrm{Am}$ system, and open symbols to the $\gamma +{}^{242}\mathrm{Am}$ system. Both circles correspond to the same excitation energy $E=5.581$ MeV $(E_n\sim 45$ keV), both triangles to $E=5.796$ MeV $(E_n\sim 0.26$ MeV), both squares to $E=6.225$ MeV $(E_n\sim 0.69$ MeV), and finally both stars to $E=6.870$ MeV $(E_n\sim 1.33$ MeV). $E_n$ means neutron incident energy. The meaning of horizontal lines in the equilateral triangle is described in the text.

Figure 7

Lack of information on the compound emission processes, $H_{\gamma }{(}^{242}\mathrm{Am},E)$ and $H_n{(}^{242}\mathrm{Am},E)$, related to a given entrance channel. These functions, $H_{\gamma }{(}^{242}\mathrm{Am},E)$ and $H_n{(}^{242}\mathrm{Am},E)$, are plotted versus the excitation energy above the last neutron separation energy $S_n$: $E-S_n$ ($S_n=5.538$ MeV). The very big point symbols (open and closed) are related to those of Fig. 6.

Figure 8

Lack of information on the compound emission processes, $H_{\gamma }{(}^{176}\mathrm{Lu},E)$ and $H_n{(}^{176}\mathrm{Lu},E)$, related to a given entrance channel. These functions, $H_{\gamma }{(}^{176}\mathrm{Lu},E)$ and $H_n{(}^{176}\mathrm{Lu},E)$, are plotted versus the excitation energy above the last neutron separation energy $S_n$: $E-S_n$ ($S_n=6.288$ MeV).

Figure 9

Total neutron compound emission probabilities $P_{{\mathrm{EC}}_i,n}(\mathrm{CN},E)$ ($i=1,2$), for a CN formed with two different hypothetical entrance channels ${\mathrm{EC}}_1$ (solid curve) and ${\mathrm{EC}}_2$ (dashed curve). The sticks, on the top of this figure, indicate the energy positions of the first seven hypothetical states (transcribed in the CN system: ${\epsilon }_k+S_n$) of the residual nucleus obtained after neutron emission, with their corresponding spin-parity values $J_k,{\pi }_k$.

Figure 10

Energy derivative of the total neutron compound emission probabilities $P_{{\mathrm{EC}}_i,n}(\mathrm{CN},E)$ ($i=1,2$), for a CN formed with two different hypothetical entrance channels ${\mathrm{EC}}_1$ (solid curve) and ${\mathrm{EC}}_2$ (dashed curve). The sticks, on the top of this figure, indicate the energy positions of the first seven hypothetical states (transcribed in the CN system: ${\epsilon }_k+S_n$) of the residual nucleus obtained after neutron emission, with their corresponding spin-parity values $J_k,{\pi }_k$.