1. Introduction
Neutron–proton (n–p) scattering plays a crucial role in unraveling the fundamental processes in nuclear physics, contributing to the study of nuclear forces, the properties of nucleons and the underlying principles of nuclear structure. In previous research [1–5], theoretical models for n–p interactions at various energy scales have successfully explained experimental phenomena. There are two spin channels including the angular momentum l = 0 state and l ≠ 0 states since nucleons have a spin of 1/2. The n–p scattering cross-section data in past experiments [3] indicate that the interaction is almost entirely from the spin singlet state 1S0 below 5 MeV. When the incident energy is higher, the n–p interaction becomes spin-dependent and its form also depends on the energy scale. The Yukawa potential [6] is widely used to describe the n–p interaction with 1S0 state. In the Yukawa picture, the interaction between two nucleons is mediated by the exchange of pions. The force range is one of parameters of the Yukawa potential and its value is related to the mass of pions. The accurate value of the force range can be only determined by the scattering cross-section. It may be possible to find an additional method such as the electron dynamics to measure accurate parameters.
The momentum spectrum of ionized electrons can be precisely measured by the highly efficient cold target recoil ion momentum spectroscopy (COLTRIMS) technique [7–9]. To obtain the dynamics of ionized electrons during the scattering process, the gas target has to be used instead of a solid target. In past experiments [10, 11], the gas target has been used to study the 1S0 n–p scattering by detecting the dynamics of neutrons and protons. Based on existing experiments, a gas target composed of hydrogen atoms can be utilized in our envisaged experiment to determine the n–p interaction parameters with 1S0 state. When neutrons collide with hydrogen atoms, this results in recoiling protons and ionized electrons. Due to the significantly smaller mass of electrons compared to nucleons, changes in electron dynamics have a negligible effect on dynamics of neutrons and protons. However, electron dynamics are sensitive to the specific form of the n–p interaction during the scattering process. Therefore, electron dynamics can offer additional information beyond the scattering cross-section to obtain accurate parameters of the n–p interaction with 1S0 state.
In low-energy scattering, electrons move much faster than nucleons so that the nucleon motion can be solved classically based on the Bor–Oppenheimer (B–O) approximation [12, 13]. The electronic state can be obtained through numerically solving the time-dependent Schrödinger equation (TDSE) [14, 15]. The position of the nucleon is treated as a parameter in the TDSE for the electron. In this work, we will explore whether electron dynamics are sensitive to nuclear force parameters during the scattering process. If the electron dynamics are different in various n–p interaction parameters, the accuracy value of parameters can be determined through the momentum spectrum of the electron. It is clearer for qualitative and quantitative analysis to employ the Yukawa potential as the n–p interaction with 1S0 state. We present the momentum spectrum of ionized electrons at various force ranges of the Yukawa potential. The accurate force range can be determined by comparing the numerical results with the experimental data. In the head on n–p scattering process, the ionization probability reaches its maximum when the distance between the two nuclei is nearly at its minimum. By calculating the probability of the ionization and the excitation at various force ranges, we find that the electron is in the ground state or continuum state.
The rest of this paper is organized as follows. In section 2 , we introduce the calculation process of the three-body scattering in detail. In section 3 , we present the numerical results and discussions. A summary is given in section 4 .
2. Model and method
In our theoretical model, the three-body system consists of a neutron, a proton and an electron. There are only two types of interactions in this system: the n–p interaction and the Coulomb interaction between the proton and the electron. Masses of the neutron, the proton and the electron are denoted as mn, mp and me, respectively. Coordinates of the neutrons, the protons and the electrons are represented by rn, rp and re, respectively. The Hamiltonian of this system in the atomic unit is written as
$\begin{eqnarray}\begin{array}{rcl}H & = & -\displaystyle \frac{1}{2{m}_{{\rm{n}}}}{{\rm{\Delta }}}_{{{\boldsymbol{r}}}_{{\rm{n}}}}-\displaystyle \frac{1}{2{m}_{{\rm{p}}}}{{\rm{\Delta }}}_{{{\boldsymbol{r}}}_{{\rm{p}}}}-\displaystyle \frac{1}{2{m}_{{\rm{e}}}}{{\rm{\Delta }}}_{{{\boldsymbol{r}}}_{{\rm{e}}}}\\ & & +{V}_{N}\left({{\boldsymbol{r}}}_{{\rm{n}}}-{{\boldsymbol{r}}}_{{\rm{p}}}\right)-\displaystyle \frac{1}{\left|{{\boldsymbol{r}}}_{{\rm{e}}}-{{\boldsymbol{r}}}_{{\rm{p}}}\right|},\end{array}\end{eqnarray}$
where Δ represents the Laplace operator and VN denotes the n–p interaction. The last term is the Coulomb potential.The laboratory coordinates $\left({{\boldsymbol{r}}}_{{\rm{n}}},{{\boldsymbol{r}}}_{{\rm{p}}},{{\boldsymbol{r}}}_{{\rm{e}}}\right)$ can be transformed into the Jacobi coordinates [16] $\left({\boldsymbol{r}},{\boldsymbol{R}},{{\boldsymbol{R}}}_{{\rm{c}}}\right)$. Here, r, R, and Rc represent vectors denoting the relative distances between the electron and proton, the neutron and the center of mass of the hydrogen atom, and the center of mass of the three-body system, respectively. Then the transformation can be shown as
$\begin{eqnarray}{\boldsymbol{r}}={{\boldsymbol{r}}}_{{\rm{e}}}-{{\boldsymbol{r}}}_{{\rm{p}}},\end{eqnarray}$
$\begin{eqnarray}{\boldsymbol{R}}={{\boldsymbol{r}}}_{{\rm{n}}}-{{\boldsymbol{r}}}_{{\rm{p}}}-\alpha {\boldsymbol{r}},\end{eqnarray}$
$\begin{eqnarray}{{\boldsymbol{R}}}_{{\rm{c}}}=\displaystyle \frac{{m}_{{\rm{n}}}{{\boldsymbol{r}}}_{{\rm{n}}}+{m}_{{\rm{p}}}{{\boldsymbol{r}}}_{{\rm{p}}}+{m}_{{\rm{e}}}{{\boldsymbol{r}}}_{{\rm{e}}}}{M},\end{eqnarray}$
where M is the total mass of the three-body system and expressed as M = mn + mp + me. The symbol α is the ratio of the electron's mass to that of the hydrogen atom, given by $\alpha ={m}_{{\rm{e}}}/\left({m}_{{\rm{p}}}+{m}_{{\rm{e}}}\right)=1/1837$. In figure 1, the schematic diagram shows coordinates in the three-body system. The blue (green, red) circle represents the neutron (the proton, the electron).
Figure 1. Schematic illustration of the envisaged experiment: an atomic hydrogen gas target is exposed to a neutron beam. The blue (green, red) sphere represents the neutron (the proton, the electron). r denotes the relative position vector from the proton to the electron. R denotes the relative position vector from the center of mass of the hydrogen atom to the neutron. α represents the ratio of the electron's mass to that of the hydrogen atom, which is equal to 1/1837. |
In the Jacobi coordinates $\left({\boldsymbol{r}},{\boldsymbol{R}},{{\boldsymbol{R}}}_{{\rm{C}}}\right)$, the Hamiltonian is rewritten as
$\begin{eqnarray}H=-\displaystyle \frac{1}{2M}{{\rm{\Delta }}}_{{{\boldsymbol{R}}}_{{\rm{c}}}}-\displaystyle \frac{1}{2{\mu }_{N}}{{\rm{\Delta }}}_{{\boldsymbol{R}}}-\displaystyle \frac{1}{2{\mu }_{{\rm{e}}}}{{\rm{\Delta }}}_{{\boldsymbol{r}}}+{V}_{N}\left({\boldsymbol{R}}+\alpha {\boldsymbol{r}}\right)-\displaystyle \frac{1}{r},\end{eqnarray}$
where μN is the reduced mass of the neutron and hydrogen atom, μe is the reduced mass of the proton and the electron. μN and μe are represented as follows: $\begin{eqnarray}{\mu }_{N}=\displaystyle \frac{{m}_{{\rm{n}}}\left({m}_{{\rm{p}}}+{m}_{{\rm{e}}}\right)}{{m}_{{\rm{n}}}+{m}_{{\rm{p}}}+{m}_{{\rm{e}}}},\end{eqnarray}$
$\begin{eqnarray}{\mu }_{{\rm{e}}}=\displaystyle \frac{{m}_{{\rm{p}}}{m}_{{\rm{e}}}}{{m}_{{\rm{p}}}+{m}_{{\rm{e}}}}.\end{eqnarray}$
Here, we select the center-of-mass (CM) coordinate system, where Rc = 0 and ${{\rm{\Delta }}}_{{{\boldsymbol{R}}}_{{\rm{c}}}}=0$. Then, the Hamiltonian H is written as
$\begin{eqnarray}H=-\displaystyle \frac{1}{2{\mu }_{N}}{{\rm{\Delta }}}_{{\boldsymbol{R}}}-\displaystyle \frac{1}{2{\mu }_{{\rm{e}}}}{{\rm{\Delta }}}_{{\boldsymbol{r}}}+{V}_{N}\left({\boldsymbol{R}}+\alpha {\boldsymbol{r}}\right)-\displaystyle \frac{1}{r}.\end{eqnarray}$
As R becomes larger, VN approaches zero. Hence, the initial state of the three-body system ${\rm{\Psi }}\left({\boldsymbol{r}},{\boldsymbol{R}}\right)$ is approximated as follows: $\begin{eqnarray}{\rm{\Psi }}\left({\boldsymbol{r}},{\boldsymbol{R}}\right)=\displaystyle \frac{1}{{\left(2\pi \right)}^{3/2}}{\varphi }_{0}\left({\boldsymbol{r}}\right)\exp \left({\rm{i}}{{\boldsymbol{P}}}_{N}\cdot {\boldsymbol{R}}\right),\end{eqnarray}$
where φ0 is the ground state of the electron in the hydrogen atom and PN is the momentum of the reduced nucleon.In the case of low-energy scattering, relativistic effects are neglected. The electron moves much faster than the nucleon. It is difficult to solve the three-body scattering process quantum mechanically. To simplify this problem, we solve the motion of the nucleon classically based on the B–O approximation. The position of the nucleon is treated as a parameter in the electronic Schrödinger equation. In the CM coordinate system, the potential VN depends on both R and r. However, the reduced mass of the electron is significantly smaller than that of the nucleon $\left({\mu }_{e}/{\mu }_{N}\approx 1/918\right)$, electron's motion has a negligible influence on the nucleon's movement. In terms of the nucleon, VN can be considered solely as a function of R and denoted as ${V}_{N}\left({\boldsymbol{R}}\right)$. The classical movement of the nucleus can be expressed as15 ).
$\begin{eqnarray}{\mu }_{N}\ddot{{\boldsymbol{R}}}=-\displaystyle \frac{\partial }{\partial {\boldsymbol{R}}}{V}_{N}\left({\boldsymbol{R}}\right).\end{eqnarray}$
Then the value of R at time t can be determined. Concerning the electron, the electronic state is dependent on the vector R. Denoting $\psi \left({\boldsymbol{r}},t\right)$ as the electron's wave function, the TDSE for the electron is expressed as follows: $\begin{eqnarray}{\rm{i}}{\hslash }\displaystyle \frac{\partial }{\partial t}\psi \left({\boldsymbol{r}},t\right)=\left[-\displaystyle \frac{1}{2{\mu }_{{\rm{e}}}}{{\rm{\Delta }}}_{{\boldsymbol{r}}}-\displaystyle \frac{1}{r}+{V}_{N}\left({\boldsymbol{R}}+\alpha {\boldsymbol{r}}\right)\right]\psi \left({\boldsymbol{r}},t\right).\end{eqnarray}$
The ionized wave function of the electron is: $\begin{eqnarray}{\psi }_{\mathrm{ion}}\left({\boldsymbol{r}},t\right)=\psi \left({\boldsymbol{r}},t\right)-\displaystyle \sum _{i}\left\langle {\varphi }_{i}| \psi \right\rangle {\varphi }_{i}\left({\boldsymbol{r}}\right),\end{eqnarray}$
where φi represents the ith (i = 0, 1, 2...) bound state of the electron in the hydrogen atom. Through the Fourier transform, the ionized wave function of the electron in the momentum representation is given by: $\begin{eqnarray}{\phi }_{\mathrm{ion}}\left({\boldsymbol{p}},t\right)=\displaystyle \frac{1}{{\left(2\pi \right)}^{3}}\int {\psi }_{\mathrm{ion}}\left({\boldsymbol{r}},t\right){{\rm{e}}}^{-{\rm{i}}{\boldsymbol{p}}\cdot {\boldsymbol{r}}/{\hslash }}{\rm{d}}{\boldsymbol{r}}.\end{eqnarray}$
It is well known that the n–p interaction is spin-dependent. There are two spin channels including the angular momentum l = 0 state and l ≠ 0 states since nucleons have a spin of 1/2. The n–p scattering cross-section experimental data below 20 MeV is in [3]. It indicates that the interaction is almost entirely from a spin singlet below 5 MeV. To simplify the problem, we only study the 1S0 n–p scattering in this paper. The Yukawa potential [6] is typically employed to describe the n–p interaction with 1S0 state. Therefore, the potential VN is expressed as $\begin{eqnarray}{V}_{N}\left({\boldsymbol{R}}+\alpha {\boldsymbol{r}}\right)={V}_{0}\displaystyle \frac{{{\rm{e}}}^{-\left|{\boldsymbol{R}}+\alpha {\boldsymbol{r}}\right|/\lambda }}{\left|{\boldsymbol{R}}+\alpha {\boldsymbol{r}}\right|},\end{eqnarray}$
where λ represents the force range, and V0 serves as an integral constant that characterizes the field strength. When V0 > 0, VN acts as a repulsive potential. When V0 < 0, VN becomes an attractive potential. Under the Born approximation [17], the scattering cross-section σ is derived as $\begin{eqnarray}\sigma =\displaystyle \frac{16\pi {\mu }^{2}{V}_{0}^{2}{\lambda }^{4}}{{{\hslash }}^{4}\left(1+4{k}^{2}{\lambda }^{2}\right)},\end{eqnarray}$
where μ is the reduced mass of the proton and the neutron, i.e., $\mu ={m}_{{\rm{n}}}{m}_{{\rm{p}}}/\left({m}_{{\rm{n}}}+{m}_{{\rm{p}}}\right)$. k represents the relative incident momentum between the neutron and the proton. When fixing the incident energy, σ is obtained from the experimental data. The constant V0 can be derived as a function of λ according to equation (3. Results and discussions
3.1. Ionized electron momentum spectrum
First, calculate the value of R at time t using equation (10 ). Due to the rotational symmetry along the neutron incident direction, our numerical computations are conducted within a two-dimensional framework. We designate the positive x-axis direction as the orientation of the incident momentum. The incident energy of neutrons is much greater than the thermal kinetic energy of hydrogen atoms. Hence, it is assumed that the relative velocity of the nucleons is equal to the incident velocity of the neutron. The incident momentum is denoted as ${{\boldsymbol{P}}}_{N}=\left({P}_{N},0\right)$. The impact parameter is represented as b. We regard the time when the neutron is just affected by the Yukawa potential as the initial time. At this time, the position of the neutron is ${\boldsymbol{R}}=\left(-\sqrt{{R}_{0}^{2}-{b}^{2}},b\right)$, where R0 corresponds to the spatial boundary of the potential VN. The range of the impact parameter is −R0 ≤ b ≤ R0. The initial state of the electron is its ground state. We obtain the electron's wave function at time t through the TDSE. The symbol Δb represents the interval between the impact parameter bi and bi+1. The ionized electron momentum spectrum at a given impact parameter bi is denoted as ${f}_{{\boldsymbol{p}}}\left({b}_{{i}}\right)$. We then aggregate the momentum spectra corresponding to various impact parameters, and the total ionized electron momentum spectrum can be expressed as follows:
$\begin{eqnarray}{f}_{\mathrm{total}}=\displaystyle \frac{2{\rm{\Delta }}b}{{R}_{0}^{2}}\displaystyle \sum _{{i}}{b}_{{i}}{f}_{{\boldsymbol{p}}}\left({b}_{{i}}\right).\end{eqnarray}$
Figures 2(a)–(d) illustrate the momentum distribution of ionized electrons in the final scattering state for various force ranges λ = 0.8 fm, 1.0 fm, 1.2 fm and 1.4 fm through our semi-classical method. Since the momentum of ionized electrons is dependent on the incident neutron's effective kinetic energy and the force range λ, we have fixed the incident effective kinetic energy at 1.078 MeV [3]. The x-axis and the y-axis in figures 2(a)–(d) correspond to the momentum along the incident direction and perpendicular to it, respectively. The color bar represents the value of ftotal. The lower limit of the force range (λ = 0.8 fm) is roughly estimated from the nucleon's radius. The upper limit of the force range is estimated based on the mass of the pion: ${\lambda }_{\max }={\hslash }/\left({m}_{\pi }c\right)\approx 1.4$ fm, where c represents the speed of light and mπ represents the mass of the pion. A notable distinction emerges in the momentum distribution with different λ: for smaller λ values, the momentum of ionized electrons tends to cluster around p = 0, while higher λ values cause the momentum distribution of ionized electrons to broaden towards both positive and negative directions along the x-axis. To elucidate the differences between the momentum distribution with various force ranges, figure 2(e) displays the momentum intensity distribution along the x direction with py = 0. As the force range decreases, the field strength increases. The momentum spectrum of the ionized electron becomes more concentrated around p = 0. As the force range increases, the field strength and scattering time increase, causing the electron to gain more momentum. Consequently, the peak of the electron momentum spectrum will shift away from p = 0.
Figure 2. The momentum spectrum of ionized electrons. The effective incident energy fixed at 1.078 MeV. (a)–(d) Correspond to the cases λ = 0.8 fm, 1.0 fm, 1.2 fm and 1.4 fm, respectively. The x-axis and the y-axis correspond to the momentum along the incident direction and perpendicular to the incident direction. The color bar represents the value of ftotal. (e) shows the momentum intensity distribution along the x direction with Py = 0. The red, yellow, blue and green lines correspond to λ = 0.8 fm, 1.0 fm, 1.2 fm and 1.4 fm, respectively. (f) depicts the variation of the contrast value, denoted as Ic, as a function of the force range λ. Ic illustrates the contrast between the maximum and minimum values of momentum distribution intensity near px = 0, as shown in (e). |
The contrast value Ic in figure 2(f) is defined as
$\begin{eqnarray}{I}_{{c}}=\displaystyle \frac{{I}_{\max }-{I}_{\min }}{{I}_{\max }+{I}_{\min }},\end{eqnarray}$
where ${I}_{\max }$ and ${I}_{\min }$ represent the maximum and minimum values of momentum distribution intensity near px = 0 in figure 2(e), respectively. The difference in contrast values can be used to determine the force range of the Yukawa potential by comparing numerical calculations with experimental results. Additionally, all results are in the center of mass of the three-body system reference frame. In the laboratory frame of reference, the momentum of the electron shifts along the x-axis with the velocity of the center of mass.To illustrate the influence of the force range λ on the electronic state, figure 3 depicts the total ionization probability (the red dotted curve) and the total excitation probability (the blue solid curve) of scattering electrons as a function of the force range λ, respectively. These probabilities are the same in both center of mass frame and laboratory frame. In this figure, as the force range λ decreases, the excitation probability decreases, while the ionization probability increases first and then decreases. In figures 2(a)–(d), the momentum spectrum distribution first broadens and then becomes narrower, with the extremum occurring between 1.0 fm and 1.2 fm. This is consistent with the result shown in figure 3. However, the change in the excitation probability is approximately two orders of magnitude smaller than that in the ionization probability. The impact of the force range on the excitation probability is negligible compared to that on the ionization probability. The scattering process primarily involves ground states and continuum states of the hydrogen atom.
Figure 3. The red dotted line shows the ionization probability of electrons as a function of the force range λ. The blue solid line depicts the excitation probability of electrons varying with the force ranges λ. |
3.2. Temporal evolution of the electronic state
In previous studies of scattering problems, the scattering time is typically neglected. The S-matrix element only involves the initial state, the final state and the interaction. In our approach, the time taken by the reduced system, with the reduced mass μN, to traverse the potential range VN is defined as the scattering time. We can determine the electronic state of the system at each time t during the scattering process. This allows us to establish a relationship between the ionization probability and the scattering time, providing insights into how the ionization probability evolves throughout the scattering event.
The red lines in figure 4 represent the ionization probability as a function of the the scattering time. The blue lines show the temporal evolution of the ratio of relative distance between two nuclei to the maximum distance of potential energy, denoted as $\gamma \left(t\right)=R\left(t\right)/{R}_{0}$. We set the impact parameter b = 0 for simplicity. The dotted lines and the dashed lines correspond to cases of λ = 0.8 fm and λ = 1.4 fm, respectively. The situation of $\gamma \left(t\right)=1$ means that there is no interaction between the two nuclei. The ionization rate is near its maximum when the distance between the two nuclei is at its minimum. Then the ionization probability stays almost unchanged. Until now, attosecond pulses [18–21] have been used to probe the dynamics of electrons within atoms and molecules [22, 23]. The time duration of the scattering process is less than one zeptosecond according to the results from figure 4, which means observing the nuclear scattering process would require lasers even faster than zeptosecond timescales. Through this theoretical approach, it might be possible to indirectly access the ultrafast nuclear scattering process by probing dynamics of electrons.
Figure 4. The red lines show the ionization probability as a function of the the scattering time. The blue lines depict the temporal evolution of the ratio of relative distance between two nuclei to the maximum distance of potential energy, $\gamma \left(t\right)=R\left(t\right)/{R}_{0}$. the temporal evolution of the ratio of nuclear relative distance to the maximum potential energy range, denoted as $\gamma \left(t\right)=R\left(t\right)/{R}_{0}$. The minimum value of γ represents the smallest nuclear relative distance. The red lines and blue lines represent the variation of ionization probability and γ(t) with time, respectively. The dotted lines and dashed lines represent λ value of 0.8 fm and 1.4 fm, respectively. The horizontal coordinates corresponding to the endpoints of each line indicate the total scattering time. |
4. Conclusion
In this article we focus on the three-body scattering process involving a neutron, a proton and an electron. Neutrons collide with protons, resulting in recoiling protons and ionized electrons. Electron dynamics are sensitive to the specific form of the neutron–proton interaction during the scattering process, offering additional information beyond scattering cross-section to determine parameters in nuclear interaction models. We report calculations of this process through numerically solving the ordinary differential equation and the time-dependent Schrödinger equation based on the Born–Oppenheimer approximation. We find momentum spectra of ionized electrons exhibit significant differences at various Yukawa potential parameters. Therefore, these parameters can be accurately determined by comparing numerical calculations with experimental results.
Numerical results also show the temporal evolution of electronic states during the scattering process. Detailed analyses are made on factors that affect the ionization probability, including the scattering time and the relative distance between two nucleons. Additional discussions have been made on the ionization probability and excitation probability at various Yukawa potential parameters. Our theoretical approach should be useful for further experiments or theoretical analyses.