Neutron-Proton Interaction
Modeled using Morse Function:
Constructing Inverse Potentials Using Variational Monte-Carlo and Phase Function Method

The first order non-homogeneous differential equation for SPS [16, 17, 18] is given by

$$\delta_{\ell}^{\prime}(k,r)=-\frac{V(r)}{k}\bigg{[}cos(\delta_{\ell}(r))\hat{j}_{\ell}(kr)-sin(\delta_{\ell}(r))\hat{\eta}_{\ell}(kr)\bigg{]}^{2}$$

(2)

This is numerically solved using Runge-Kutta 5th order method. For $\ell=0$, the Ricatti-Bessel and Riccati-Neumann functions $\hat{j_{0}}$ and $\hat{\eta_{0}}$ get simplified as $sin(kr)$ and $-cos(kr)$, respectively.

The greatest advantage of this method is that, the phase-shifts are directly expressed in terms of the potential and have no relation to the wavefunction. This has been utilised in this paper to obtain inverse potentials in an innovative way by implementing a modified VMC in tandem with PFM. The technique optimizes the model parameters of the potentials to obtain the best match with respect to the experimental values.

Typically, VMC is utilised for obtaining the ground state energy for a given potential. The method starts with a trial wavefunction, which is varied at a random location by a random amount in the Monte-Carlo sense. Then, the energy is determined using the newly obtained wavefunction and variational priniciple is applied. This process is done iteratively till one converges to the ground state. Here, we consider to vary the potential instead of wavefunction and minimise variance with respect to experimental data, as follows:

Initialisation step:

To begin the optimisation procedure, Morse parameters $V_{0}$, $r_{m}$ and $a_{m}$ are given some initial values. The phase equation is integrated using RK-5 method for different values of k, a function of lab energies E, to obtain the simulated SPS, say $\delta^{sim}_{k}$. The mean percentage error (MPE) has been determined with respect to the experimental mean energy analysis data of Arndt[19], say $\delta^{exp}_{k}$, as

$$MPE=\frac{1}{N}\sum_{i=1}^{N}\frac{|\delta^{exp}_{k}-\delta^{sim}_{k}|}{|\delta^{exp}_{k}|}*100$$

(3)

This is named as $MPE_{old}$ and is also assigned to $MPE_{min}$.

Monte-Carlo step:

A random number $r$, generated in an interval [-I, I], is added to one of the parameters, say $V_{0new}=V_{0}+r$.

PFM step:

Again, the phase equation is integrated with new set of parameters $V_{0new}$, $r_{m}$ and $a_{m}$ to obtain new set of simulated SPS, say $\delta^{sim-new}_{k}$, using which $MPE_{new}$ is determined.

Variational step:

If $MPE_{new}<MPE_{old}$, then $V_{0}=V_{0new}$, $MPE_{min}=MPE_{new}$, else old values are retained.

The final three steps are repeated for each of the parameters to complete one iteration. The size of interval is reduced after a certain number of iterations, if there is no significant reduction in $MPE_{min}$. The process is completed when $MPE_{min}$ does not change any further, that is, convergence is reached.

The analytical solution of time independent Schrodinger equation for Morse potential[20], in Eq. 1, is given by

$$E_{n}=-\frac{\hbar^{2}}{2\mu a_{m}^{2}}(p-n-1/2)^{2}$$

(4)

where

$$p^{2}=\frac{2\mu V_{0}a_{m}^{2}}{\hbar^{2}}$$

(5)

The energy expression is only consisting of $V_{0}$ and $a_{m}$. Using the binding energy of deuteron $E_{D}=-2.224589$ MeV for the ground state, one can impose a constraint on $V_{0}$ for a given value of $a_{m}$. So, we need to vary only two parameters, $a_{m}$ and $r_{m}$ in VMC while obtaining the SPS using PFM. The MPE has been reduced to less than $4\%$ for both MEAD of 350 and 1050 MeV for parameters shown in Table 1. This procedure is akin to obtaining the inversion potential that results in best SPS, while retaining the binding energy of deuteron.
A comparative analysis parameter that represents relative mean square error, is defined as

$$c^{2}=\frac{1}{N}\sum_{i=1}^{N}\frac{\left(\delta_{i}^{expt}-\delta_{i}^{sim}\right)^{2}}{\delta_{i}^{expt}}$$

(6)

This parameter has been computed by comparing simulated SPS with MEAD of Arndt [19] and found to be 0.24(0.16) for data upto 350(1050) MeV. Typically, a value less than 1 is considered to be a good match between the two sets of data.

The singlet S-state, for anti-parallel spins given by ${}^{1}S_{0}$, is not a bound state. Starting with ${}^{3}S_{1}$ potential parameters, SPS of ${}^{1}S_{0}$ are determined using PFM. Then, MPE is calculated w.r.t MEAD of ${}^{1}S_{0}$. Now, VMC and PFM are iterated till MPE converges to a constant value. The resultant parameters of inverse potential, see Fig 1(a), are given in Table 1. The obtained SPS for all energies upto 1050 MeV are shown in Fig. 1(b). The MPE and c² for data upto 350 (1050) MeV are found to be 2.8$\%$ (3.6$\%$) and 0.03 (0.05) respectively.

To describe pp interaction, an extra Coulomb term[21] is added to np interaction potential, as

$$V_{Coulomb}=\frac{e^{2}}{r}erf(\beta r)$$

(7)

Where $\beta$ is a parameter to be adjusted for obtaining correct experimental SPS. Starting with parameters obtained for ${}^{1}S_{0}$ SPS, $\beta$ is also varied in Monte-Carlo sense to minimize MPE w.r.t. experimental SPS due to pp interaction. For $\beta=0.75fm^{-1}$, minimum MPE value of 2.8$\%$(3.8$\%$) and $c^{2}$ value of 0.03(0.07) for data up to 350(1050) MeV are obtained. The potential and SPS plot are shown in Fig. 1.

In order to obtain SPS shifts for higher $\ell$ values, one needs to bring in a non-local potential form that contains $\ell$ dependence. This is proposed to take care of all possible spin-spin, isospin-isospin, and spin-isospin dependencies. Here, in this paper, we assume non-local potential to have a similar form as that of local potential and utilized Aldrich-Green approximation [22] as

$$V_{2}(r)=\frac{\hbar^{2}}{2\mu}\ell(\ell+1)\bigg{[}\frac{e^{-r/a_{m}}}{a_{m}^{2}}\frac{1}{(1-e^{-r/a_{m}})^{2}}\bigg{]}$$

(8)

There are no new parameters incorporated into it. Since we are dealing with partial wave analysis of individual channels, the tensor part, and its components, resulting from various spin and isospin dependencies, have been assumed to be internal to central part of the potential. While this suffices for singlet P and D channels, one has to include spin-orbit interaction for triplet states, given by

$$V_{LS}(r)={\left(\frac{r_{0}}{\hbar}\right)}^{2}\frac{1}{r}\frac{2V_{0}}{a_{m}}\bigg{[}e^{-(r-r_{m})/a_{m}}-e^{-2(r-r_{m})/a_{m}}\bigg{]}(\vec{L}.\vec{S})$$

(9)

Here, $\vec{L}.\vec{S}=\frac{\hbar^{2}}{2}[J(J+1)-L(L+1)-S(S+1)]$. The proportionality constant $r_{0}^{2}$ takes care of dimensional analysis and also adds an extra parameter for optimization.
The phase function equation for $\ell=1$, i.e. P-partial wave, is of the form

$$\delta_{1}^{\prime}(k,r)=-\frac{V(r)}{k}\bigg{[}cos(\delta_{1}(r))\big{(}\frac{sinkr}{kr}-coskr\big{)}+sin(\delta_{1}(r))\big{(}\frac{coskr}{kr}+sinkr\big{)}\bigg{]}^{2}$$

(10)

and that for $\ell=2$, i.e. D-partial wave, is of the form

$$\delta_{2}^{\prime}(k,r)=-\frac{V(r)}{k}[cos(\delta_{2}(r))f(kr)-sin(\delta_{2}(r))g(kr)]^{2}$$

(11)

where f(kr) is

$$f(kr)=\left(\frac{3}{(kr)^{2}}+1\right)sinkr-\frac{3}{kr}$$

(12)

and g(kr) is

$$g(kr)=\left(\frac{-3}{(kr)^{2}}+1\right)coskr-\frac{3}{kr}sinkr$$

(13)

The procedure of VMC in tandem with PFM equations as given above has been applied to all states of $\ell$ = 1 $\&$ 2 respectively. One has to include the non-local potential for singlet states and add the spin-orbit potential as well for the triplet states, while determining the SPS. The obtained inversion potential parameters, along with MPE and $c^{2}$ values, are shown in Table 1 .

It is observed that triplet P-states are obtained to an accuracy of less than $8\%$ for data up to 350 MeV and singlet is having an MPE of about 16$\%$.

Similarly, results obtained for triplet D-states are slightly better than that of singlet D-state. This clearly shows that having an extra parameter due to the spin-orbit term has given more options for VMC to converge better as compared to singlet terms which did not have any extra parameter to adjust. The obtained SPS for various P and D states have been shown in Fig. 3(b) and Fig. 3(b), respectively. The matching trends w.r.t MEAD of Arndt[19] indicate the merit of our procedure in obtaining the inverse potentials. Especially, the trends in various curves reveal a good match for data up to 350 MeV. The potentials do show resemblance to what has been obtained by Nijemen group[4].

1. 1.

   We have undertaken a comparative study[9] by considering phenomenological two term Malfiet-Tjon [6] and Manning-Rosen [8], potentials and found that ${}^{3}S_{1}$ SPS obtained with these saturate to a constant value above a certain lab energy.

2. 2.

   An attempt has been made to include tensor interaction due to ${}^{3}D_{1}$ state, modeled as proportional to ($1/r^{2}$)[23], so that the actual central part of potential could be obtained. But, it turned out that ${}^{3}S_{1}$ SPS are better represented with Morse function. Hence, tensor interaction has been assumed to be an inherent part of it. We also observe that best convergence for SPS has been obtained for ${}^{3}D_{1}$ state as compared to all other partial D-wave channels.

1. 1.

   We have shown that ${}^{1}S_{0}$ np SPS can be obtained using same local interaction potential with albeit slightly different model parameters.

2. 2.

   Also, SPS for ${}^{1}S_{0}$ pp interaction have been successfully matched with MEAD of Arndt, by modelling the Coulomb interaction as proportional to an $erf()$ function with only one extra parameter.

3. 3.

   It is simplicity of Morse form that has been able to achieve this convergence. One of the important characteristics of a good model is parsimony.

The non-local interaction to model spin and iso-spin dependencies based on Aldrich-Green approximation [22] of centrifugal term, does provide impetus to the inversion technique, as shown in our results. But, there are other considerations that might further improve convergence. For example, the methodology of super-symmetric quantum physics(SUSY) to construct partner super-potentials[7] for $\ell\neq 0$ scattering states from its hypergeometric type ground state wave-functions might be a better choice and is being attempted.

While performing VMC, initial values play an important role in the convergence process. Of course, same starting points could lead to a slightly different set of parameters on convergence, for a chosen number of decimal places, as is expected in a VMC procedure. Also, completely different starting points might also end up giving similar convergences as well, in which case, it is important to look at the corresponding potential plots to get a better understanding. Here are a few observations based on our multiple runs with regard to the construction of inversion potentials for P and D channels.:

1. 1.

   In case of ${}^{3}D_{3}$, it was found that by starting with a completely different set of starting values, VMC ended up with $V_{0}$ = 269.743, $r_{m}$ = 0.671, $a_{m}$ = 0.286 and $r_{0}$ = 0.1995 yielding a better convergence for MPE as 8$\%$ as compared to current values quoted in Table 1 which have resulted in MPE of 11.2 $\%$. On plotting the potentials with both sets of parameters, it was observed that potential due to former parameters has resulted in an extremely large attractive part with a magnitude close to $10^{5}$ for r values between 0.03-0.04 fm, followed by another low attractive potential between 1-1.3 fm and finally the decaying tail part. This is certainly an aberration, as such strong attraction, is not anticipated in any physical interaction. A very similar occurrence has been observed in the case of ${}^{3}P_{2}$ as well.

2. 2.

   Multiple runs for ${}^{3}D_{2}$ have shown that, as initial parameters values are slightly varied, the convergence to final parameters lie within a small range. For instance, values of [$V_{0},r_{m},a_{m}$, $r_{0}$] have been obtained as [272.172, -0.070, 0.657, 1.148] and [252.431, -0.234, 0.719, 1.344] with MPE values of 17.5$\%$ and 17.3$\%$ respectively. Even though $V_{0}$ in Table 1 appears to be the mean of these two quoted sets, all other parameters are higher. So there are no particular correlations to guide the process of convergence.

3. 3.

   In case of singlet channels, ${}^{1}P_{1}$ and ${}^{1}D_{2}$, we have included a proportionality constant $r_{1}$ to act as a parameter to check if the convergence improves. This resulted in very small gains. For example, the value of MPE reduced from 16.1$\%$ to only 15.8$\%$ in case of ${}^{1}P_{1}$ SPS, for data up to 350 MeV. Here, we have noted that three sets of completely different parameters [95.507, 0.642, 0.422, 2.238], [194.245, 0.366, 0.451, 2.464] and [240.561, 0.354, 0.417, 2.755] have all resulted in MPE values which are close to 15.9$\%$. Even though the parameter sets are extremely different, their potential plots are all very closely overlapping and are hardly distinguishable. We were expecting that there would be a possibility of obtaining isospectral potentials, just as in case of Marchenko technique [9], but such a scenario does not appear to be occurring in our procedure. One might have to perform many more such trials before one can rule out completely such a possibility.

Our group has applied this technique to obtain best model parameters for Woods-Saxon potential[24] for determining single-particle energies of neutron and proton states in Shell model of the nucleus. Similarly, we have also applied this technique to obtain best model parameters for Morse potential[14] to determine ro-vibrational frequencies of diatomic molecule HCl. It has been observed that results obtained are better than those from regression analysis performed on analytical expressions based on theoretical considerations. In fact, one can conclude that VMC is an equivalent methodology for least-squares minimization-based optimization. Even though VMC procedure looks very promising, one must keep in mind that it strongly depends on initial conditions. There is every possibility that optimization process could get stuck in a local minima and not lead to convergence. This is one of the reasons why a good theoretical framework could be conisdered to act as a guide to VMC procedure.

Finally, to conclude Morse function seems to be an ideal phenomenological potential that is able to explain np interactions in the weakly bound deuteron system. Taken along with Coulomb term as proportional to an $erf()$ it is able to represent pp interaction as well. The inversion methodology based on VMC needs to be performed for higher $\ell>2$ channels. This could lead to reconstruction of various internal interaction potentials that in turn could give complete quantitative description of experimental data. Further, our VMC least square error minimization procedure might prove very useful in chemical and biological systems, economics, management, and mathematics and where ever optimization of model parameters is involved.