Deuteron Structure and Form Factors: Using Inverse Potentials for S-waves

The radial time independent Schr$\ddot{\text{o}}$dinger equation (TISE), for $\ell$ = 0 (S-wave), is given by

where $\mu$ is reduced mass of neutron and proton. The analytical solution of TISE is derived by Morse [28] and ground state energy expression is given by

where

is called well-depth parameter and is dependent only on $V_{0}$ and $a_{m}$.

Utilizing experimental binding energy (BE) for Deuteron as, E₀= $-2.224589(22)$ MeV [29], $V_{0}$ can be expressed in terms of $a_{m}$ as

To fix the other two parameters $a_{m}$ and $r_{m}$, we utilise experimental SPS. Out of an infinite set of values for $V_{0}$ and $a_{m}$ that could give rise to experimental BE, only one set in consonance with a particular $r_{m}$ should give rise to observed experimental SPS. To determine SPS, Morse [28] suggested phase function method.

The Schr$\ddot{\text{o}}$dinger wave equation for a spinless particle with energy E and orbital angular momentum $\ell$ undergoing scattering is given by

The second order differential equation Eq. 6 has been transformed to the first order non-homogeneous differential equation of Riccati type [30, 31] given by

Prime denotes differentiation of phase shift with respect to distance and the Riccati Hankel function of first kind is related to $\hat{j_{\ell}}(kr)$ and $\hat{\eta_{\ell}}(kr)$ by $\hat{h}_{\ell}(r)=-\hat{\eta}_{\ell}(r)+\textit{i}~{}\hat{j}_{\ell}(r)$ . In integral form the above equation can be written as

for $\ell=0$,the Riccati-Bessel and Riccati-Neumann functions $\hat{j_{0}}$ and $\hat{\eta_{0}}$ get simplified as $sin(kr)$ and $-cos(kr)$, respectively and the above equation is written simply as

The function $\delta_{0}(k,r)$ is called phase function. Here, $k=\sqrt{E/(\hbar^{2}/2\mu)}$ and $\hbar^{2}/2\mu$ = 41.47 MeVfm². Centre of mass energy $E_{c.m.}$ is related to laboratory energy by $E_{c.m.}=0.5E_{\ell ab.}$. SPS have been obtained by numerically integrating above equation starting from origin upto asymptotic region using Runge-Kutta (RK) 5th order method [32] with initial condition $\delta_{0}(k,0)=0$. Advantage of PFM is, SPS are directly obtained from potential without recourse to wave-function. So, the Morse function is incorporated into the phase equation and its model parameters are optimised by calling fifth order RK-method in an iterative fashion within an optimisation procedure.

The procedure utilised for optimisation is broadly as follows:

1. 1.

   Model parameters are given certain bounds. For example, both $a_{m}$ and $r_{m}$ are chosen to be having values within an interval (0,1).

2. 2.

   Define a cost function that needs to be minimised. We have chosen mean squared error (MSE), between the two data sets, given by

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

   (10)

   where $\delta_{i}^{sim}$ are SPS obtained using PFM solved via RK-5 method and ($\delta_{i}^{expt}$) are experimental SPS from mean energy partial wave analysis data MEPWAD of Granada [33].

3. 3.

   Call the optimisation routine to determine the best parameters that fit the experimental data with minimum MSE.

The detailed procedure for obtaining final optimised parameters using TDA is discussed in results and Appendix section. Once the model parameters are obtained, one can determine the DWF.

To determine deuteron charge and magnetic FFs measured from electron scattering experiments, the knowledge of ground state wavefunction is a basic requirement. The analytical solution for ground state wave function due to Morse potential [26] is given by

where

and $N_{0}$ is to be determined from normalisation of Deuteron wave-function (DWF) $\psi_{D}(r)$. Considering ${}^{3}D_{1}$ wave-function $w_{2}(r)$ to be proportional to $u_{0}(r)$ [19, 34], $N_{0}$ has been determined, such that

It should be noted that this equation has two unknowns and hence one more condition needs to be utilised to fix them. This is done by choosing one of the static properties of deuteron from experimental data, typically electric quadrupole moment [2]. Considering relativistic effects and deuteron finite size to be negligible the quadrupole moment is given by following expression:

Once the DWF is determined one can determine both static properties and form factors.

In case of singlet ${}^{1}S_{0}$ the exact relation which we have used for ${}^{1}S_{0}$ SPS calculation is given by Matsumoto and Guerin [26, 45] as

Here, $\gamma$ is Euler constant ($\gamma=0.57721$) and $\epsilon_{0}$ is given by Eq. 12. It is interesting to note that SPS are linearly dependent on $r_{m}$, but have a non-linear dependence on parameters $V_{0}$ and $a_{m}$, for different values of lab energy parameter $k$. SPS can be directly obtained using this analytical expression, as an iterative equation, to determine best model parameters that give minimum mean squared error (MSE) with respect to experimental mean energy partial wave analysis data (MEPWAD) [33]. Alternatively, one can obtain optimised parameters using PFM technique inside the iterative loop. Thus, analytical expression Eq. 25 provides a good cross-check to validate efficacy of PFM.
It is important to remember that the main contribution to total scattering cross-section is from singlet ${}^{1}S_{0}$ state. The results for SPS are shown in 1(a)

Partial cross section $\sigma_{\ell}$ can be calculated using

and total cross-section [46] is calculated as

where $\sigma_{t}$ and $\sigma_{s}$ are partial cross-sections for ${}^{3}S_{1}$ and ${}^{1}S_{0}$ respectively.

Low energy scattering parameters i.e. scattering length (a) and effective range ($r_{e}$) are determined from slope and intercept of $kcot(\delta)~{}vs~{}0.5~{}k^{2}$ relation while the low energy SPS can be obtained using using following relation [47] for any potential

A linear regression plot of $k$ vs $kcot(\delta)$ allows for calculation of required scattering parameters $a$ and $r_{e}$ from intercept and slope respectively.

Initially, model parameters are optimised by choosing to minimize MSE for entire data set consisting of 12 points. First, analytical expression for SPS of singlet (${}^{1}S_{0}$) scattering state, given by Eq. 25 has been numerically implemented and model parameters are optimised by minimizing MSE between obtained and experimental values. Then, optimisation is performed using 5th order RK method within an iterative loop to minimize MSE. Both procedures have resulted in exactly same values for model parameters, and are shown in Table 1. This cross-verifies the correctness of PFM.
In case of triplet ground state (${}^{3}S_{1}$), only two parameters $a_{m}$ and $r_{m}$ are varied and $V_{0}$ is calculated via energy constraint Eq. 5. The optimised values obtained are shown in Table 1. The MSE values obtained are $<0.1$ and to quantify the performance, we have chosen mean absolute error (MAE) as a measure. The triplet and singlet SPS have been obtained with MAE of 0.35 and 0.70 respectively.

The uncertainties, $\Delta\delta(E)$, in SPS data at different energies specified in Granada MEPWAD [33] have been utilised to create two extreme data sets. One by adding $\Delta\delta(E)$ to $\delta(E)$ and the other by subtracting $\Delta\delta(E)$ from $\delta(E)$. The model parameters obtained for these two respective sets are:
${}^{3}S_{1}$: [116.040, 0.832, 0.347] $\&$ [112.306, 0.850, 0.354]
${}^{1}S_{0}$: [72.463, 0.891, 0.367] $\&$ [68.521, 0.911, 0.377]
These model parameter sets are used to obtain uncertainties in SPS for triplet and singlet states. While the obtained SPS are utilised to determine low energy scattering parameters and total cross-section, the model parameters give rise to deuteron wave function(DWF) from which various static properties are determined. The DWF also helps in calculation of its various em form factors.

This kind of analysis is akin to data fitting as in MLA, wherein best parameters are obtained by including all available experimental values, at validation stage, to obtain model interaction. One should be aware that there is a good possibility that MLA might lead to over-fitting [25]. Also, optimised parameters could be sensitive to data set. This aspect is being studied.

In principle, for modeling in physics context, one would expect that number of data points to be chosen for optimisation should be equal to number of model parameters. The obtained parameters must be able to explain rest of the data reasonably well. Then, a question arises as to, what would be right way to choose two(three) data points from among the twelve available for triplet(singlet) states. This would result in a total of ${}^{12}C_{2}(=66)$ and ${}^{12}C_{3}(=220)$ combinations, for triplet and singlet states, respectively. Initially, we have obtained optimised parameters for each of these available combinations and then carefully analyzed the results. Here, are some important observations:

1. 1.

   Depth of potential is energy dependent. That is, data points from low energy region [0.1, 25] have resulted in lower $V_{0}$ values as compared to those from higher energy regions, [200, 350].

2. 2.

   The model has good predictive power for interpolated data points but errors increase due to extrapolation, especially at far away points.

For instance, considering data points from low energy region [0.1, 10] have resulted in better prediction of SPS for immediate data points in range [25, 150] as compared to those in high energy region [200, 350] where the SPS obtained had larger errors. Similarly, considering 3 data points in high energy region [200, 350] for optimisationhave resulted in poor accuracy in scattering parameter values reflecting that SPS of low energy region, important for calculating scattering length and effective range, are not determined to good accuracy.
Based on these observations, we have deduced that data points consisting of end points 0.1 and 350 along with an intermediate data point preferably chosen from [25,150] range would give best results. To accommodate more choices, mean absolute error (MAE) was utilised as a quantitative measure. After carefully analysing the results, at various stages of our calculations for the possible combinations, we have applied the following criteria:
In the first step, for fixing $r_{m}$ value, we have considered those combinations for which MAE $\leq$ 2.
Then, in second step, scattering and static properties as well as scattering phase shifts were obtained for each of the combinations, with MAE $\leq$ 1, and their averages and standard deviations are determined and tabulated.
This is traditional data analysis (TDA), typically expected, in validation stage of a model.

There are a total of 12 experimental data points. As discussed in the main paper, to obtain two model parameters of ${}^{3}S_{1}$ state, there are a total of ${}^{12}C_{2}=66$ possible combinations. The model parameters are determined by choosing two lab energies at a time and minimising the mean squared error. Then, the SPS were obtained at remaining 10 energies from the data set and overall mean absolute error (MAE) is determined. The data has been sorted with ascending values of MAE and is presented in Table LABEL:3S1combinations.

Discussion:

• •

  From the above 66 combinations, 64 of them have $MAE<2$. The average value for $r_{m}$ from these 64 combinations is determined to be 0.8427 fm.

• •

  Keeping $r_{m}$ fixed, one needs to vary only one parameter $a_{m}$ (because $V_{0}$ is dependent on $a_{m}$). So, only ${}^{12}C_{1}$, i.e. only one of the energies from the data set needs to be considered for optimising the parameter $a_{m}$.

• •

  Utilizing the optimised model parameters, the ${}^{3}S_{1}$ wave function has been determined using equation 11.

• •

  Then, the proprotionality factor for ${}^{3}D_{1}$ wavefunction has been determined such that quadrupole moment $Q_{D}=0.2589fm^{2}$ is obtained using Eq. 14 and deuteron wave function (DWF) from Eq. 13 is normalised.

• •

  Utilizing the obtained DWF, other static properties of deuteron have been determined and tabulated for all 12 energies in Table 4.

• •

  One can observe that the depth of the potential keeps increasing with increasing energy except for the first two values. This might be because the first data point is added from Arndt data and is not part of mean energy analysis data of Granada.

• •

  The values of depth $V_{0}$ and width $a_{m}$, given in bold, are utilised for obtaining possible range of parameters in our calculations.

• •

  Since, we have three parameters to be determined for ${}^{1}S_{0}$ state, a total of 220 combinations need to considered. All these are shown in Table LABEL:220combi, where the data have been once again presented in ascending order of overall MAE.

• •

  Out of these 220, only 130 of them have $MAE<2$. The average value for $r_{m}$ from these 130 combinations is determined to be 0.897 fm.

• •

  Keeping $r_{m}$ fixed, one needs to vary only two parameter $V_{0}$ and $a_{m}$. So, only ${}^{12}C_{2}$, that is 66 combinations need to be worked out. These are given in Table 6.

• •

  A total of 14 combinations are having $MAE<1$. These have been considered for determining energy scattering parameters ($a_{s}$ and $r_{s}$) and are shown in Table LABEL:1S0final.

• •

  The values of depth $V_{0}$ and width $a_{m}$, given in bold, are utilised for obtaining possible range of values for scattering parameters in our calculations.