# An Investigation of the Nuclear Reaction Near the Three-Body Break-up Threshold: As a Ultra Low Energy Nuclear Synthesis

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## Abstract

We investigate the possibility of lanthanum (La)-nucleus creation via the reaction Cs(2d, \(\gamma \))La on the three-ion quasi-molecule \(\hbox {CsD}_2\) in the \(\hbox {CsD}_2\hbox {Pd}_{{12}}\)-cluster. In order to calculate d–Cs–d (or D–Cs–D) three-body bound states and wave functions, we adopt a very accurate three-body variational method with more than 80 figures. The \(\hbox {CsD}_2\) and La binding energies and wave functions are obtained with an S-wave trial functions. The *wave function overlap* (WFO) value between the La nuclear excited state and the several \(\hbox {CsD}_2\) quasi molecular modified states is calculated. It should be stressed that the WFO value is of critical importance for the existence of the electro-magnetic transition in the Cs(2d, \(\gamma \))La reaction. We found that the WFO value is very sensitive to the nuclear potential tail with a two- or three-body \(1/r^2\)-type long range hadron potential, and gives a promising result for the reaction. The result is also followed by the three-body Faddeev calculation.

## 1 Introduction

In the traditional nuclear fusion approach, the most important problem is whether the incident energy can penetrate the Coulomb barrier or not, because the reaction is usually started in the free field. Recently we pointed out that the Cs–d penetration in the quasi-molecule which is developed in the Pd-crystal could easily occur [1], since the energy levels of such a quasi-molecule could close to the top of Cs–d Coulomb barrier or over the barrier in the “Pd-cage” which is a repulsive Coulomb barrier between the ions (Cs/D) and outer Pd. Therefore, the penetration problem is resolved in the quasi-molecule: \(\hbox {CsD}_2\) in the \(\hbox {CsD}_2\hbox {Pd}_{{12}}\) system, and \(\hbox {CsD}_4\) in \(\hbox {CsD}_4\hbox {Pd}_{{12}}\), and also \(\hbox {CsD}_6\) in the system \(\hbox {CsD}_6\hbox {Pd}_{{12}}\) which are given by the hydrogen storage in the \(\hbox {Pd}_{{13}}\) cub octahedron/\(\hbox {D}_6\), or Pd-cluster where the central Pd is substituted by Cs which was precisely investigated by Watari et al. [2]. In their paper, the numerical calculation was done with the molecular ADF program, which is based on the linear combination of atomic orbital’s method using density functional theory with all-electron Slater-type basis sets [2]. They found that the external electron configuration of \(\hbox {CsD}_2\hbox {Pd}_{{12}}\) is very similar to that of La [2]. Therefore one could confirm that such a quasi-molecule exists independently from the Pd-cluster. By this reason, we call \(\hbox {CsD}_2\) in the \(\hbox {CsD}_2\hbox {Pd}_{{12}}\) crystal the *quasi molecule* (see Fig. 1). However, their calculation was done in the region of \(10^{4}~\mathrm{fm}\le r<\infty \), therefore, it is very hard to investigate the nuclear fusion problem which is performed in the region of \(0\le r\le \infty \).

On the other hand, in the nuclear cluster physics, it is known that the highest excited state of the nucleus is the linear chain of the clusters [3, 4, 5, 6, 7, 8, 9], which is very similar to the molecular system in the present case (see Fig. 1). However, such a nuclear energy level is still far from the molecular state. Therefore, one could imagine that the nuclear states with the long range potential, which have predicted in the hadron three-body system [10, 11, 12, 13, 14] could interpolate such a region between the nuclear state and the molecular state.

In order to investigate a very shallow nuclear state, we would like to study nuclear d–Cs–d three-body problem by a very accurate three-body variational (HDV) method with 80 to 100 figures based on a usual variational approach [15]. For the nuclear potentials, the WS potential is adopted with the repulsive Coulomb potentials for Cs–D, D–D, and the repulsive Pd-cage, and also we have to take into account the electron effects for the attractive *Ze*-Cs, *Ze*-D Coulomb potentials.

The total Hamiltonian of the quasi-molecule D–Cs–D with the Pd-cage and the nuclear d–Cs–d (or D–Cs–D) system is discussed in Sect. 2. In Sect. 3, the quasi molecular low-lying wave function \(\psi _m\) and the nuclear wave function of the highest excited state of La \(\psi _n\) are obtained for the reaction: Cs(2d, \(\gamma \))La. The WFO value between both wave functions: \(\psi _n\), and \(\psi _m\) will be compared for the cases where the two- and three-body long range potentials are included or not. The conclusion and discussion will be given in Sect. 4.

## 2 Effective Three-Body Hamiltonian In Molecular D–Cs–D and Nuclear d–Cs–d Systems in Pd-Cluster

*Z*-electrons (Zel) and ion: \(V^{M}_{\mathrm{N-Zel}}\) where N stands for {Cs, D} ions, and the electron-electron Coulomb repulsive potential: \(V_{\mathrm{el-el}}^M\) which is the sum of the

*i*th-electron and the

*j*th-electron: \(V_{c}^{\mathrm{e}_i-\mathrm{e}_j}\). For the three-nuclear (hadron) system: d–Cs–d, the two-body nuclear potentials are given by \(V_{\mathrm{3-N}}^{had}\) especially for the Cs–d potential: \(V_N^{\mathrm{Csd}}\), and for the d–d potential: \(V_N^{\mathrm{dd}}\). The two-body Coulomb interactions among three ions: \(V_{\mathrm{3-N}}^C\), especially for Cs–H interaction: \(V_c^{\mathrm{CsH}}\), and also for H–H interaction: \(V_c^{\mathrm{HH}}\), respectively.

*Z*-electron kinetic energy, respectively.

*i*th-N and

*j*th-N nuclear potential: \(V_{N}^{\mathrm{N}_i\mathrm{N}_j}(\mathbf{r}_{ij})\),

*Ze*-charged ion and the \(-Z'e\)-charged electron cloud constructs the \(\hbox {CsD}_2\hbox {Pd}_{{12}}\) cluster with the characteristic lattice constants. Therefore, we can also omit the potential Eq. (6) when we take into account our three-cluster system. Although, the potential is formally given by using the electron density function \(\rho (r)\) with the fine structure constant \(\alpha \),

We obtain a very good fit to the experimental ground state energy \({{\mathcal {E}}}_0=-\,32.3\) MeV, the root mean square (rms) radius \(R_0=6.25\) fm, the highest excited energy \({{{\mathcal {E}}}_{max}=-\,3.5134\times 10^2}\) keV for the negative value, and the rms radius \(R_{max}=165.5\) fm, for the usual nuclear potential with the Coulomb plus three-cluster force by the HDV calculation. Therefore, the highest energy level is far from that of the usual molecular state, although the higher partial waves and the non-central part of the WS potential have been omitted for the first time.

On the *k*th-particle transfer diagram in the three-body AGS Born term [19], \(V_e\) becomes (A) \(V_e=V_{e0}a_e^2/[r_{ij}^2+a_e^2]\) for (\(r_{ki}=0\) and \(r_{jk}=0\)), \(V_e=V_{e0}a_e^2/[2r_{ij}^2+a_e^2]\) for (\(r_{ki}=0\) or \(r_{jk}=0\)) which are the three-body long range potential between a pair and the spectator particle; and (B) \(V_e=V_{e0}a_e^2/[2r_{kj}^2+a_e^2] =V_{e0}a_e^2/[2r_{ki}^2+a_e^2]\) (for \(r_{ij}=0\)) which indicates the two-body long range potential. (C) While, putting \(\mathbf{r}_k=(\mathbf{r}_i+\mathbf{r}_j)/2\), it gives \(V_e=V_{e0}a_e^2/[3r_{ij}^2/2+a_e^2]\) which is due to the three-body long range potential as well as (A) [10].

Some related ions and the shallow spin-parities: \(J^\pi \), and masses by (u) and (MeV) units, where \(1\hbox {u}=931.5019\) MeV is the universal constant

Ions | \(J^\pi \) | Mass (u) | Mass (MeV) |
---|---|---|---|

\(^2\)H (deuteron) | 1\(^+\) | 2.0141 | 1876.1 |

\(^4\)He | 0\(^+\) | 4.0026 | 3728.4 |

\(^{135}\)Cs | \(7/2^+\) | 134.91 | \(1.2567\times 10^5\) |

\(^{137}\)Ba | \(3/2^+\) | 136.91 | \(1.2753\times 10^5\) |

\(^{139}\)La | \(7/2^+\) | 138.91 | \(1.2939\times 10^5\) |

## 3 Numerical Results

The two-cluster potential between Cs and d could be constructed by the three-body reaction: \(\mathrm{Cs}+\mathrm{d}\equiv \mathrm{Cs}+(\hbox {N},\hbox {N}) \rightleftharpoons (\mathrm{Cs},\mathrm{N})+\mathrm{N}\) which gives the general particle transfer (GPT) potential [14] or the AGS Born term [19]. However, in such a way, it is very hard to make the two-cluster form-factors with the core excitation for the three-cluster problem: d–Cs–d. Therefore, we adopt the WS-type Cs–d potential of Eq. (9). By using the WS and the Coulomb potentials, the two-body energy levels are shown in Table 2, where the ground states are well fitted. It is found that the number of the calculated energy levels considering only the central part of the WS potential are rather small. Therefore, additional WS components should be incorporated containing for example spin-orbit and tensor terms for the two-cluster interactions.

Calculated two-body binding energies by using the WS potential with the above mentioned parameters and the Coulomb potential, where \(E_1\) values are compared with the 2-body type in Table 3

reaction type | \(E_1\) (MeV) | \(E_2\) (MeV) | \(E_3\)(MeV) | \(E_4\) (MeV) |
---|---|---|---|---|

d–d | \(-\) 23.772716 | |||

Cs–d | \(-\) 16.31894 | \(-\) 13.43531 | \(-\) 9.003575 | \(-\) 2.975595 |

Some reaction types of binding energies by the mass relation using Table 1 with (MeV) unit

Reaction type | Binding energy | (MeV) |
---|---|---|

2-Body | 2\(\times m_{^2\mathrm{H}}-m_{^4\mathrm{He}}\) | 23.8 |

2-Body | \(m_{^2\mathrm{H}}+m_{^{135}\mathrm{Cs}}-m_{^{137}\mathrm{Ba}}\) | 16.1 |

3-Body | 2\(\times m_{^2\mathrm{H}}+m_{^{135}\mathrm{Cs}}-m_{^{139}\mathrm{La}}\) | 32.3 |

3-Body | \( m_{^4\mathrm{He}}+m_{^{135}\mathrm{Cs}}-m_{^{139}\mathrm{La}}\) | 8.4 |

The traditional nuclear interaction appears in the region of \(0\le r< 10^4\) fm, however, the molecular levels are calculated in the region of 10\(^4 {~\mathrm fm}\le r <10^{7}\) fm. Therefore, the nuclear wave functions and the molecular wave functions do not overlap to give rise to the electro-magnetic (EM) transition. This means that the molecular states are stable and never go down to the nuclear states.

However, the binding energy of \(\hbox {CsD}_2\) in the \(\hbox {Pd}_{{12}}\)-cage is completely different from the free field. We solve the three-ion system in the \(\hbox {Pd}_{{12}}\)-cage for the full region: \(0\le r<\infty \) by the HDV method. In other words, for the nuclear and the molecular systems we can solve the eigen-equation on the basis of a common field in the full region with the 80-100 figures accuracy, however it seems to be very hard for the usual FORTRAN program in this stage. The second promising method is shown in Ref. [1] where the entire wave function could be made by the nuclear state and the molecular state: \(|\varPsi >=|\psi \otimes \psi _M>\). Both energy states could be corrected by the \( {boost ~up ~effect}\) which is shown in Ref. [1], however we do not take this method in this paper.

- (1)
The first one is the three-body calculation for the full potential

with the traditional short range WS potential,$$\begin{aligned} V_{\mathrm{3-N}}^{had}+V_{\mathrm{3-N}}^C+V_c^{\mathrm{Pd}}+V_t. \end{aligned}$$(19) - (2)
The second one is the three-body calculation for the full potential

with the two- and three-body long range potential \(V_e\):$$\begin{aligned} V_{\mathrm{3-N}}^{had}+V_{\mathrm{3-N}}^C+V_c^{\mathrm{Pd}}+V_t+V_e. \end{aligned}$$(20)

Negative highest excited state of La binding energy (\({{\mathcal {E}}}_{max}\)), and the positive energy (\({{\mathcal {E}}}_{+}\)) which give max-\(W_{nm}\), and the value of max-\(W_{nm}\) (maximum value of WFO)

potentials | \({{\mathcal {E}}}_{max}\) (eV) | \({{\mathcal {E}}}_{+}\) for max-\(W_{nm}\) (eV) | max-\(W_{nm}\) | \(W_n\) |
---|---|---|---|---|

Equation (20) (L) | \(-\,58.975(n=20)\) | \(1.4633\times 10^{4} ~(m=52)\) | 0.10 | 0.070 |

Equation (19) (S) | \(-\,3.5134\times 10^5 (n= 5)\) | \(2.2726\times 10^5 ~(m=53)\) | 0.03 | 0.0057 |

(L)/(S) | \(1.6786\times 10^{-4}\) | \(6.4389\times 10^{-2}\) | 3.33 | 12.3 |

The calculated results by the HDV method show a series of negative binding energy levels which starts from \(-\,32.3\) MeV to the negative highest nuclear level: \({{\mathcal {E}}}_{max}=-\,3.5134\times 10^5\) eV for the short range Eq. (19), and to \({{\mathcal {E}}}_{max}=-\,58.975\) eV for the long range Eq. (20). On the other hand, there is a series of the positive energy states over the Coulomb barrier through to the Pd-cage which is located at \(1.57\times 10^6 \) fm and 2.73 MeV height. Let us call such positive energy states the “quasi molecular states” conventionally.

Therefore, for Eq. (19) the energy gap between the negative nuclear state and the quasi molecular state is very large. However, the calculations with the long range nuclear potential Eq. (20) give a series of discrete negative energy band from \(-\,32.3\) MeV to \(-\,58.975\) eV (HDV-method: see Fig. 2), or \(-\,44\) eV (Faddeev-method: see Fig. 3, where \(-\,2.9\) eV is unstable). Therefore, we can conclude that the three-body Faddeev calculation with the *r*-space method gives qualitatively similar results with those by the HDV method in the negative energy region, although the Faddeev method [21] is rather hard to reach the Pd-cage. These results show that the energy gap between the negative nuclear state and the quasi molecular state is rather small for the long range nuclear potential.

Finally, in Table 4, the WFO results of max-\(W_{nm}\) with the long range potential are 3.33 times, and of \(W_{n}\) with 12.3 times larger values than those with the usual short range nuclear potential. Therefore, the long range nuclear potential could be favorable to bring about the low energy Cs(2d, \(\gamma \))La reaction.

## 4 Conclusion and Discussion

This work is based on the assumption that a quasi-molecular system exists in a Pd-cluster. Such a quasi-molecule is almost isolated in the Pd-cage. Therefore, the excited state of the quasi-molecule: \(\hbox {CsD}_2\) does not break-up, because it is not in the free space but restricted in the Pd-cage. Therefore, some excited states could easily penetrate the Coulomb barrier of the Cs–D interaction or even over the top of it. Therefore, they could go into the nuclear *quasi-resonance states* by means of “stable” states in the Pd-cage. Hence, a quasi-resonance wave packet of \(\hbox {CsD}_2\) could overlap with that of the highest nuclear excited state or the most shallow bound state of La nucleus. In such a case, the WFO has a non zero value, allowing the EM transitions to occur. The possible transition will be started from the E2 or M1 transition, because the E1 transition is forbidden by the same \(J^\pi \) values for Cs and La. Once, the transfer to the nuclear state occurs, then the nuclear transition to the next states is automatically continued. In our present calculations for the ordinary short range nuclear potential, and for the nuclear potential with a long range, we found that the former case gives a small WFO, but the latter case could offer a larger value by an order of magnitude. We mentioned that the WFO value is critical for the occurrence of the Cs(2d, \(\gamma \))La reaction.

Recently, one of the authors (S.O) proposed a new long range nuclear potential of the form: \(1/r^n\) [10, 11, 12, 13, 14]. We pointed out why such a long range part in the one pion transfer interaction or the Yukawa potential comes from [14]. Our conclusion was that any particle transfer in the three-body Faddeev theory [19] generates not only the short range, but also the long range interactions which was called the *general particle transfer* (GPT) potential. Therefore, any two-body nuclear potential in the three-body constituents is constructed by the GPT potential [10, 11, 12, 13, 14]. Furthermore, the three-body problem with the GPT potential generates also a GPT potential between the two-body center of mass and the spectator. This idea is concretely represented by the potential form of Eq. (18). If the long range part contains a \(1/r^2\)-type potential, the scattering length will become infinity [16, 17, 18]. The GPT-potential with the \(1/r^2\)-type gives the three-body bound states near at the three-body threshold which would be very close to the molecular states.

There are some reports of the experimental results about the low temperature nuclear transition (LTNT) [22, 23, 24, 25, 26, 27, 28]. Unfortunately, it seems that a reasonable theoretical description for such reactions has not been given as of yet, since almost all of the traditional theories are based on the short range nuclear forces which could reach as far as 100 fm. Therefore, the nuclear reaction rate is too small for the LTNT, where the Coulomb repulsive force dominates the attractive nuclear force. Our theory may give a hint to the experimental physicists to investigate a mysterious phenomena such as inexplicable thermal generation measured by a special arrangement in the materials, without any usual nuclear reaction phenomena or radioactive substances. One could imagine that such a high energy \(\gamma \)-ray emission does not come from the *slow reaction* caused by the condensed states, or the \(\gamma \)-ray radiation would be diminished due to the multiple Compton scattering off the electrons in the Pd-cluster, although many nuclear physicists still wonder why some radioactive particles or high energy \(\gamma \)-rays are not observed in the LTNT.

Finally, it should be noticed that the present calculation adopted only the S-wave trial function, and the central nuclear potential. Therefore, we can not give the final conclusion in this paper. However, our results seem to suggest that if the LTNT experiment could be realized, then the *long range* potential in the nuclear system could be observed.

## Notes

### Acknowledgements

The authors would like to acknowledge to Drs. N. Watari, N. Hamada, and A. Kodama for valuable discussions regarding the theoretical aspects of the molecular system, as well as I. Toyoda, S. Tsuruga and H. Kakigami for sharing with us the experimental insight. One of the authors (SO) would like to express his thanks to Profs. I. Lagaris and K. Kato for their helpful suggestions. We are indebted to Mitsubishi Heavy Industry (MHI) Co. Ltd. for significant financial support.

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