Resonance State Wave Functions of $^{15}$Be using Supersymmetric Quantum Mechanics

TL;DR

This paper employs supersymmetric quantum mechanics with a microscopic potential to calculate resonance wave functions of unbound $^{15}$Be, providing theoretical insights that complement experimental efforts.

Contribution

It introduces a novel application of supersymmetric quantum mechanics to unbound nuclei, specifically calculating resonance states of $^{15}$Be with a microscopic potential.

Findings

Identified the 1.8 MeV resonance state with specific energy and width.

Found no other nearby resonances for $^{15}$Be.

Demonstrated the framework's effectiveness in studying unbound nuclei.

Abstract

The theoretical procedure of supersymmetric quantum mechanics is adopted to generate the resonance state wave functions of the unbound nucleus $^{15}$Be. In this framework, we used a density dependent M3Y microscopic potential and arrived at the energy and width of the 1.8 MeV (5/2$^+$) resonance state. We did not find any other nearby resonances for $^{15}$Be. It becomes apparent that the present framework is a powerful tool to theoretically complement the increasingly important accelerator based experiments with unbound nuclei.

Equations

$$\left(-\frac{{\hslash }^2}{2\mu }\frac{d^2}{d{r}^2}+v(r)-E\right){\psi }_E(r)=0$$

$$\hat{v}(r;\lambda )=v(r)-\frac{{\hslash }^2}{2\mu }\left[\frac{4{\psi }_E(r){\psi }_E^{\prime }(r)}{I_E(r)+\lambda }+\frac{2({\psi }_E(r){)}^4}{{(I_E(r)+\lambda )}^2}\right],$$

$$C(E)={\int }_{r_a}^{r_b}{[{\hat{\psi }}_E(r^{\prime })]}^{2}d{r}^{\prime },$$

$$T=exp[-2{\int }_b^c\sqrt{\frac{2\mu }{{\hslash }^2}(\hat{v}(\lambda ;r)-E_R})dr]$$

$$\Gamma =2\sqrt{\frac{{\hslash }^2}{2\mu }}\frac{exp(-2{\int }_b^c\sqrt{\frac{2\mu }{{\hslash }^2}(\hat{v}(\lambda ;r)-E_R})dr)}{{\int }_a^b\frac{dr}{\sqrt{(E_R-\hat{v}(\lambda ;r))}}}.$$

Full text

I Introduction

The study of neutron-unbound nuclei are extremely important to probe the neutron-drip line in connection with astrophysical problems and other important issues BO10 ; BA12 . With the increasing availability of exotic rare isotope beams at state of the art accelerators, researchers are encouraged to carry out studies on structure of such unbound nuclei SN13 . To complement such experimental findings, robust theoretical frameworks are indispensable.

The beryllium isotopic chain is a very attractive arena to make a systematic study extending from stable nuclei to its neutron-unbound rare isotopes KU15 . The first attempt to populate the unbound nucleus 15Be was carried out in 2011 by Spyrou et al SP11 using a two proton knockout reaction from a 17C secondary beam. The search for a resonance was unsuccessful and it was concluded that 15Be is unbound by more than 1.54 MeV. Attempts were made to predict states of 15Be in the shell-model framework WA92 ; BR07 ; PO85 . The shell-model calculations in Ref. WA92 ; BR07 predicted a $\frac{3}{2}^{+}$ ground state for 15Be together with a low lying $\frac{5}{2}^{+}$ excited state at approximately 300 keV and additional states above 1.2 MeV whereas Ref. PO85 predicted just the opposite, namely a $\frac{5}{2}^{+}$ ground state with an excited $\frac{3}{2}^{+}$ state at 70 keV. Clearly, the existing theoretical calculations of 15Be level structure, in the shell-model framework, is very uncertain. To observe the 15Be nucleus, Snyder et al SN13 carried out a 14Be(d,p)15Be reaction in 2013 leading to the first observation of 15Be. It is pertinent that the selected channel is expected to populate both the $\frac{3}{2}^{+}$ and $\frac{5}{2}^{+}$ states in 15Be. The 15Be, produced in this neutron transfer reaction, being unbound decayed immediately into 14Be and a neutron. The reconstructed decay energy spectrum exhibits a resonance at 1.8(1) MeV and the spin parity of this state is tentatively assigned to be $\frac{5}{2}^{+}$. However, the experiment does not resolve the question of which of the two states among $\frac{3}{2}^{+}$ and $\frac{5}{2}^{+}$ corresponds to the ground state. More recently in 2015, Kuchera et al KU15 performed a two proton knockout reaction using 17C secondary beam on thick 9Be target to find the predicted 15Be $\frac{3}{2}^{+}$ state. However, the results indicate that a 15Be component was not needed to describe the data. Thus, we see that experiments on 15Be are very few and there are substantial ambiguities in the results of theoretical calculations relevant to experimental findings.

II Theory

In the present work, we resort to a very effective technique of supersymmetric quantum mechanics (SQM) for the study of unbound nuclear systems. Earlier, we successfully applied SQM to detect low-lying broad resonances of the weakly bound nucleus 11Be DU14 . The effectiveness of our theoretical procedure is due to its ability to circumvent the numerical challenges posed by the shallow potential of such nuclear systems. The study of resonance states of 15Be is a challenging problem as it has no bound state while its two-body potential for a resonance state is a shallow well, followed by a very low and wide barrier. The low barrier does temporarily trap the system leading to a broad resonance. This inadvertently hinders accurate calculation of resonance energy masked by the broad resonance width. Under the circumstances, we decided to adopt SQM to study the resonance states of the unbound nucleus 15Be. We treat it in the framework of a two-body model consisting of an inert core of 14Be and a single valence neutron. The density distribution used for 14Be consists of separate core and halo components. The parameterization used is named Gaussian-Halo, reflecting an rms radius of 3.1 fm for 14Be IL09 . The two-body potential $v(r)$ is generated microscopically in a single folding model using the density dependent M3Y (DDM3Y) effective interaction DU14 . The DDM3Y interaction was used also in our earlier work DU14 , to study the weakly bound nucleus 11Be. In this context, it may be noted that the DDM3Y effective interaction was found earlier also to give a satisfactory description of radioactivity, nuclear matter and scattering BA04 ; BA05 ; GU06 .

From the constructed microscopic potential, SQM generates a family of isospectral potentials (IP), which have a normalizable positive energy solution at a selected energy. This is a lesser known result of SQM, namely a bound state in the continuum (BIC) DA82 ; NI84 ; KH89 ; PA93 . This IP has desirable properties which can be utilized to extract information about unbound resonance states. The microscopic potential constructed from single folding calculation is in general a shallow well followed by a low and wide barrier. For a finite barrier height, in principle, a system may be temporarily trapped inside the shallow well when its energy is close to the resonance energy. In reality, there is a very high probability for tunnelling through the barrier which gives rise to broad resonance widths. Our technique bypasses this problem and obtains accurate resonance energies. This is achieved by construction of isospectral potentials that are deep enough to supress the tunnelling probability of the system and yet provide an accurate location of resonance states. This becomes even clearer by the plots of the resulting resonance state wave function.

Going further, an isospectral partner potential could be constructed by following the ideas extended by Pappademos et al PA93 to scattering states with positive energy in the continuum. Wave functions in the continuum are non-normalizable but following PA93 , one can construct normalizable wave functions at a selected energy, which represents a BIC. The BIC represents a solution of the equation with an isospectral potential $\hat{v}(r;\lambda)$, where $\lambda$ is a parameter which affects the strength of IP DU14 . It follows from theory as well as in practice that resonance energy does not depend on the choice of $\lambda$. So a suitable choice of $\lambda$ ensures the stability of the resonance state. It preserves the spectrum of the original potential while adding a discrete BIC at a selected energy.

As already mentioned, we have considered quasi-bound 15Be to be a two-body system (14Be + $n$), and for microscopically constructed $v(r)$, inclusive of the centrifugal barrier, it is possible to construct a family of strictly isospectral potentials $\hat{v}(r;\lambda)$ for arbitrarily chosen parameter $\lambda$. This is done by the BIC formalism as follows. The equation for positive energy resonance state wave function $\psi_{E}(r)$ is

[TABLE]

where $E$ is the energy of the resonance state. Here we follow the procedure described in PA93 and solve the two-body Schrödinger equation for a positive energy $E$ subject to the boundary condition $\psi_{E}(0)=0$ and normalized to a constant (fixed) amplitude of oscillation in the asymptotic region. The solution $\psi_{E}(r)$ is not square integrable and it oscillates as r increases. It can be verified by direct substitution that $\hat{\psi}_{E}(r;\lambda)$ satisfies Eq. (1), where

$\hat{\psi}_{E}(r;\lambda)=\frac{\psi_{E}(r)}{I_{E}(r)+\lambda}\>$, and $I_{E}(r)={\displaystyle\int}_{0}^{r}{[\psi_{E}(r^{\prime})]}^{2}dr^{\prime}$ with $v(r)$ replaced by

[TABLE]

Now the potential $\hat{v}(r;\lambda)$ given by Eq. (2) depends on the arbitrary parameter $\lambda$, $\hat{\psi}_{E}(r)$ is the solution to the energy $E$. As $\psi_{E}(r)$ oscillates with a constant amplitude in the asymptotic region, it is observed $I_{E}(r)$ increases approximately linearly with $r$ for large $r$ and from above equations we find $\hat{\psi}_{E}(r;\lambda)$ to be normalizable. Thus $\hat{\psi}_{E}(r;\lambda)$ represents a BIC of $\hat{v}(r;\lambda)$, which is isospectral with $v(r)$. Hence $\hat{v}(r;\lambda)$ develops a deep and narrow well followed by a high barrier which advances towards the origin for $\lambda\rightarrow 0+$ and approaches $v(r)$ for $\lambda\rightarrow+\infty$.

The deep well and high barrier combination effectively traps the system giving rise to a quasi-bound state. We calculate the probability of the system to be trapped within this enlarged well-barrier combination as

[TABLE]

where $r_{a}$ and $r_{b}$ are radial distances at the classical turning points $a$, $b$ within the potential well. Our method is advantageous when it comes to highly accurate calculation of resonance energy along with numerical ease. In our procedure, the probability $C(E)$ of the system to be trapped in well-barrier combination of $\hat{v}(r;\lambda)$ shows a sharp peak at the resonance energy for appropriate choice of $\lambda$. Resonance state in the original potential $v(r)$ is apparent but not prominent within the well. It gets enhanced by a very large amount in $\hat{v}(r;\lambda)$ giving rise to a sharp peak in probability $C(E)$. Although the probability plots $C(E)$ of the system exhibits independence of resonance energy on the choice of $\lambda$ values, eventually a judicious choice of $\lambda$ is necessary to eliminate numerical errors in the wave function.

Width ($\Gamma$) of the resonance is calculated from its mean life ($\tau$), using the time energy uncertainty relation. The mean life is given by the reciprocal of the decay constant, which is expressed as a product of the number ($n_{c}$) of impacts on the barrier and the transmission probability ($T$). Semiclassical estimation of $n_{c}$ is obtained as the reciprocal of time of flight within the well of $\hat{v}$ between the classical turning points $a$ and $b$. Here, $T$ is given by the WKB approximation for the transmission through the barrier of $\hat{v}$ with resonance energy $E_{R}$,

[TABLE]

where $b$ and $c$ are the classical turning points of the barrier. The final expression for $\Gamma$ is

[TABLE]

We have verified by direct calculation that $\Gamma$ is independent of $\lambda$ within numerical errors.

III Results

A plot of $C(E)$ as a function of $E$ for various $\lambda$ values shows how the trapping effect of $\hat{v}(r;\lambda)$ increases as $\lambda$ decreases. For appropriate choice of $\lambda$, suitable isospectral potentials are constructed. Probability $C(E)$ of the system for $\frac{5}{2}^{+}$ state is plotted in Fig. 1. We present $C(E)$ for $\lambda=5\times 10^{-6},1\times 10^{-6},7\times 10^{-7}{\rm~{}and~{}}3\times 10^{-7}$ in the same figure, normalizing each curve to a peak value of 100. Sharpness of the peak increases rapidly as $\lambda$ decreases towards smaller values. It is evident from the plot that there is a resonant state at energy $E$ = 1.8 MeV for 15Be. The Fig. 2 shows the family of isospectral potentials $\hat{v}(r;\lambda)$ at resonance energy $E_{R}$ for the same $\lambda$ values as in Fig. 1 for the $\frac{5}{2}^{+}$ state of 15Be along with the original single folded potential $v(r)$. Enhancement of trapping probability of the system for the $\frac{5}{2}^{+}$ state of 15Be as $\lambda$ decreases could be seen from Fig. 3. Wave functions $\hat{\psi}_{E}(r;\lambda)$ at the resonance energy 1.8 MeV for the same $\lambda$ values as in Fig. 1 are presented in Fig. 3. The asymptotic region represents a free particle while a peak appears within the well-barrier combination, representing the enhanced probability of the particle being trapped inside the well. This peak increases rapidly as $\lambda$ decreases towards lower values. The inset of Fig. 3 shows the wave function plot for $\lambda=1\times 10^{-6}$ in an expanded scale up to 100 fm. These wave function plots are signatures of resonance states. They display appreciable amplitude within the well and behave as free particles exhibiting sinusoidal wave function in asymptotic region once it leaks out of the well-barrier trapping effect. We tried the same procedure to locate the presence of $\frac{3}{2}^{+}$ state. There was no confirmation regarding any contribution from $\frac{3}{2}^{+}$ state as evident from Fig. 4. Within the well, the low amplitude of the $\frac{3}{2}^{+}$ wave function in comparison to the $\frac{5}{2}^{+}$ state and in asymptotic region nearly similar oscillatory behavior of the two states justifies our point. Wave functions in Fig. 4 have small amplitude in the well and sinusoidal nature in asymptotic region, representing an unbound state. Plot of wave function for $\frac{3}{2}^{+}$ state of 15Be clearly rules out any presence of resonance state. A similar search for $\frac{7}{2}^{+}$ resonance state was carried out and its wave functions represented unbound system with no resonance effect.

The accuracy in locating the resonance energy $E_{R}$ could be increased by the choice of an optimum value of $\lambda$, although in general $E_{R}$ is independent of $\lambda$. This optimized $\lambda$ value could be used in all further calculations cutting down numerical computational time. The procedure adopted to study resonances in binary systems can also be used in calculating differential cross sections as a function of energy. Wave functions for resonance states are readily available in our procedure which could be advantageous for further extended calculations. The resonance width obtained is $\Gamma=470$ keV as compared to the experimental finding SN13 of $\Gamma_{exp}$ = 575 (200) keV. We also found that $\Gamma$ is independent of $\lambda$.

The novelty of our method lies in the extraction of resonant state wave functions with the help of SQM procedure. We would like to reiterate that the isospectral potentials are generated from an effective potential $v(r)$ which represent the same energy quasi-bound state. The wave function shown in the inset of Fig. 3 establishes firmly the quasi-bound nature of the state. All the wave functions have the same nature namely a high probability inside the potential well-barrier and free particle nature once it tunnels out from the trap. The depth and width of the isospectral potential well, and the height and width of the barrier adjust itself in such a way that they are able to reproduce the resonance energy as well as the width. Since isospectral potentials could represent quasi-bound states and reproduce the quasi-bound state energies, it is not surprising to expect that they will represent another property of such states viz. width of the states. Our calculation also points out that different shaped wave functions modified by $\lambda$ not only represent the position of resonance but also the width of the state.

IV Conclusion

In conclusion, we have been able to generate the wave functions in the SQM framework with a DDM3Y microscopic potential and arrive at the unbound state energy and width, of the 15Be nucleus. Our procedure confirmed the existence of $\frac{5}{2}^{+}$ state and its experimentally observed unbound resonance energy KU15 . In the present formalism, though $\lambda$ appears as a parameter to enhance resonance effect but it has no role in locating the exact resonance energy. Excellent agreement with the experimental results for 15Be can be ascribed to the realistic two-body (14Be + $n$) folded potential. The SQM is the only procedure by which resonant state wave functions are extracted and utilized to effectively reproduce an experimental observable ($\Gamma$). Wave functions of same $J^{\pi}$ but different $\lambda$ values are in a sense equivalent as they reproduce the same resonance energy and width of the state.

Physics of exotic unbound nuclei would pervade the field of nuclear physics in the coming years and a robust theoretical framework exclusively to study such nuclei is necessary. Earlier, we successfully applied this procedure to an unstable nucleus 11Be. The present work shows that the same procedure works also on an unbound nucleus with excellent results.