Determination of $\alpha$–$\alpha$ interaction from experiment

Abstract

Fixed-energy quantum inversion method is applied to determine model independent $\alpha$-particle scattering potentials within the energy range between 0 and 50 MeV. Only experimentally allowed even phase shifts are used in the calculation. Detailed results exhibited at low and higher energies show up a large interaction domain, a thin strong Pauli-blocking part and a smooth merging region between the colliding particles. Over-all agreement between input and re-calculated phase shifts justifies the procedure.

1 Introduction

Quantum inverse scattering methods offer a possibility to determine interaction potentials from experiment. The procedure is preceded by a phase shift analysis of scattering data. The inverse scattering calculation can be accomplished either at a fixed angular momentum l or at a specified energy $E=\hbar ^2k^2/2\mu$.

In the first case one starts with a selected partial wave of $S-$matrix elements $S_l(k)$ which should be known at all $E>0$ values as well as at the pole positions (bound state energies if any) together with the corresponding normalisation factor of the bound state wave function. Both types of information can be inferred form experiments.

In the fixed energy case one works with an infinite set of phase shifts $\delta _l(k=fix)$ or, equivalently, $S_l=\exp (2i\delta _l)$ elements given in the range $l=0-\infty$. In principle, these elements can be obtained from scattering measurements by analyzing differential cross section data.

However, the limited experimental information poses severe problems concerning stability of the quantum inversion methods, i.e., the precise reproduction of input data from the resulted (inverse) interaction potential. Therefore, powerful methods have been developed to complete the experimental information. In the fixed-l case, extrapolating the data to high energies have been used [1]. In the fixed-E case the original theory of Newton and Sabatier (NS) [2, 3] is modified by Münchow and Scheid [4] in such a way that the experimentally irrelevant (not measurable) phase shifts at high l-s do not play a sensitive role.

In this letter we shall apply the modified fixed energy inversion scheme [4] to the case of $\alpha$-particle scattering in order to derive energy-dependent (but l-independent) effective $\alpha$–$\alpha$ cluster potential from analyzed experimental data [5,6,7]. Some l-dependent inverse potentials have already been derived [8] and a large number of theoretical works have been done to determine microscopically the ’true’ (E- and l- independent) $\alpha$–$\alpha$ interaction. (For an earlier review see [6], for microscopical models see [9] and references therein.)

In the case of identical $\alpha$-particles collision, the scattering amplitude results in a partial wave sum where only even ($l=0,2,...)$ waves contribute. Neglecting Coulomb effect which is weak in comparison to the nuclear force, the scattering amplitude can be expanded in terms of partial waves as

$$\begin{aligned} f(\theta ) = {1\over 2ik} \sum _{l=0(2) } (2l+1) (S_l-1) P_l(\cos \theta ), \end{aligned}$$

(1)

where $P_l$ denotes the Legendre polynomial. The wave number k is expressed by $k=(2\mu E_{})^{1/2}/\hbar$ where the reduced mass is $\mu ={m_P m_T}/{(m_P+m_T)}={{m_\alpha ^2}/{2 m_\alpha }}=m_\alpha /2=2m_n$ with obvious notations ($m_n$ being the nucleon mass). In the case of absorption, the S-matrix elements $S_l$ can be expressed either by the complex phase shifts ${\tilde{\delta }}_l$ or by the nuclear absorption coefficients $\eta _l=\exp (-2$Im$({\tilde{\delta }}_l))$ and the real nuclear phase shifts $\delta _l=$Re$({\tilde{\delta }}_l)$ as

$$\begin{aligned} S_l=e^{2i {\tilde{\delta }}_l}=\eta _le^{2i \delta _l}. \end{aligned}$$

(2)

In identical particle collisions it has become customary to ’complete’ the experimental information encoded within the even angular momenta by interpolating phase shifts also in the nonphysical partial waves. On some physical grounds (e.g., by requiring smoothness of the potential), also those phase shifts are included, by assumption, which belong to the odd partial waves [10, 11]. However, this procedure introduces some artifact in the inversion potentials, especially around the resonance energies.

Therefore attempts of inverting only the physically accessible even phase shifts have been suggested [12, 13] in the case of carbon nuclei. Interesting physical effects of a strong Pauli-repulsion, and a scaling law (interpretable with the partial fulfillment of the virial theorem) have been observed in the potentials obtained. In these studies the modified NS inverse method has been applied by using only even phase shifts as experimental input. The relevance of the inversion potentials obtained in such a way should be tested by independent methods. An obvious test is to calculate back the phase shifts from the inverse potential obtained and compare them with those of the input data. The physical picture suggested by the $^{12}$C+$^{12}$C inversion potentials has been a very attractive one showing up a relatively large interaction domain, a thin but strong Pauli blocking region [12] where the nuclei are touching each other. Then a smooth behavior in the interior (merging) domain of the nuclei follows which might be explained by a partial fulfillment of virial theorem valid for particles forming a stationary state and being held together by (screened) Coulomb, i.e. Yukawa forces [13].

We want now to extend a similar investigation to the $\alpha +\alpha$ case, providing an additional information on the fixed energy inverse $\alpha +\alpha$ potential by calculating also the respective wave functions in question.

2 Theoretical background

2.1 Newton–Sabatier (NS) theory

In fixed energy quantum inversion one solves an inhomogeneous linear integral equation of Gel’fand-Levitan type

$$\begin{aligned} K(\rho ,\rho ')=g(\rho ,\rho ')+\int \limits _0^\rho d\rho ''\rho ''^{-2} g(\rho ',\rho '')K(\rho '',\rho ') \end{aligned}$$

(3)

first derived by R.G.Newton [2, 3]. Here the input kernel $g(\rho ,\rho ')$ is the known quantity

$$\begin{aligned} g(\rho ,\rho ')=\sum _{l=0,2,...}^\infty c_l\varphi _l^{U_0}(\rho )\varphi _l^{U_0}(\rho ') \end{aligned}$$

(4)

composed from the regular wave functions belonging to the known reference potential $U_0(\rho )$. The solution kernel

$$\begin{aligned} K(\rho ,\rho ')=\sum _{\ell =0,2,..}^\infty c_l\varphi _l^U(\rho )\varphi _l^{U_0}(\rho ') \end{aligned}$$

(5)

contains the experimental data via the asymptotic form of the regular wave function

$$\begin{aligned} \varphi _l^U(\rho \ge \rho _0)=A_l\left[ \cos \left( {\tilde{\delta }}_l^{exp}\right) \sin \left( \rho -l{\pi \over 2}\right) -\sin \left( {\tilde{\delta }}_l^{exp}\right) \cos \left( \rho -l{\pi \over 2}\right) \right] . \end{aligned}$$

(6)

In the above equations the expansion coefficients $c_l(k)$ and the normalizing factors $A_l(k)$ depend only on the energy and these quantities can be determined by solving the system of Eqs. (3)–(6) at two outer radial points $\rho _1 ,\rho _2>\rho _0$.

The solution of Eq. (3) is unique if the following relation holds

$$\begin{aligned} D_l^U(\rho )K(\rho ,\rho ')=D_l^{U_0}(\rho ')K(\rho ,\rho ') \end{aligned}$$

(7)

where $(V=U,U_0)$

$$\begin{aligned} D_l^V(\rho )=\rho ^2\left[ {\text {d}^2\over \text {d}\rho ^2}+1-V(\rho )\right] . \end{aligned}$$

(8)

Finally, the inversion potential is obtained as

$$\begin{aligned} U(\rho )=U_0(\rho )-{2\over \rho }{\text {d}\over \text {d}\rho } {1\over \rho }K(\rho ,\rho ). \end{aligned}$$

(9)

The expansion coefficients $c_l$ appearing in Eqs. (4) and (5) are determined by the (experimental) phase shifts via the Regge-Newton (RN) equations

$$\begin{aligned} \varphi _l^U(\rho )=\varphi _l^{U_0}(\rho )-\sum _{l'=0,2,..}^{\infty } c_{l'}L_{ll'}(\rho )\varphi _{l'}^U(\rho ), \end{aligned}$$

(10)

where $L_{ll'}(\rho )=\int _0^\rho \text {d}\rho '\varphi _l^{U_0}(\rho ') \varphi _{l'}^{U_0}(\rho ')/\rho '^2$.

2.2 Modified Newton–Sabatier (mNS) theory

The NS procedure suffers from convergence problem when applied to practical cases where only a finite amount of terms can be retained in the expansions as a consequence of the limitation introduced by the experiment.

Münchow and Scheid [4] have modified the original NS theory by using the fact that in most cases of physics the form of the potentials is known from a certain finite distance $r_0$.

The technical parameter $r_0$ is designated to make a reasonable compromise between the insufficient information provided by the experiment and the wealth of information supplied by the known tail of the potential. In other words, the distance parameter $r_0$ balances between the experimental information (phase shifts) and the theoretical knowledge (potential tail). Of course the final result, the inversion potential must not depend (much) on the choice of $r_0$, and the practical requirement is always to find a region of this parameter where the results are insensitive to its values.

It has been shown by Münchow and Scheid [4] that in case of an infinite set of partial waves (or phase shifts) the inversion potential resulted by the modified theory does not depend on the parameter $r_0$ and the potential obtained is unique.

In practical cases the range of $r_0$ giving stable result is around the values given by the rule $\rho _0\equiv kr_0\sim l_{max}$ with k being the wave number and $l_{max}$ denotes the maximal (presently even) angular momentum quantum number beyond which the nuclear interaction of the colliding $\alpha$–particles is negligible (and the Coulomb interaction is also weak).

According to this practical restriction, the modified RN equation to be used in the inversion calculation reads as follows

$$\begin{aligned} \varphi _l^U(\rho )=\varphi _l^{U_0}(\rho )-\sum _{l'=0,2,..}^{l_{max}} c_{l'}L_{ll'}(\rho )\varphi _{l'}^U(\rho ), \end{aligned}$$

(11)

with $L_{ll'}(\rho )=\int _0^\rho \text {d}\rho '\varphi _l^{U_0}(\rho ') \varphi _{l'}^{U_0}(\rho ')/\rho '^2$.

Because of the asymptotic normalizing constant $A_l$, one needs at least two outer points $r_1,r_2>r_0$ (or $\rho _1,\rho _2>\rho _0$) to determine the unknown expansion coefficients $c_{l'}$-s appearing in Eq. (11). Sometimes also a least squares technique is applied to arrive at a stable solution for the $c_{l'}$-s.

3 Results

3.1 Inversion of experimental $\alpha$–$\alpha$ phase shifts

Figure 1 shows the fixed energy potentials V(E, r) obtained by inverting the experimental $\alpha$–$\alpha$ scattering phase shift data given in the systematic survey of Afzal et al [6]. Some of these phase shifts data are listed in Table 1, and can also be found in figure form in the Brown–Tang article [7]. From the potentials exhibited at many energies in Fig. 1, we shall analyze here only two, those corresponding to data $\{\delta _l^{re},\delta _l^{im}\}$ appearing in the first and third data columns given in Table 1, namely belonging to $E= 5$ and 38.71 MeV being characteristic for scattering at low and higher energies, respectively. The calculated inverse potentials for the four sets of data at energies $E= 5, 29, 38.71,$ and 49.9 MeV given in Table 1 appear also in Fig. 1, as the four thick lines $V(E=fix,r)$ vertically crossing the many other parallel lines showing the inverse potential lines $V(E,r=fix)$.

The selected two inverse potentials at $E=5$ and 38.71 MeV can be seen in Figs. 2a and 3a, b where also the usual technical parameters $(r_0, \rho _1, \rho _2, l_{max})$ for the inversion are indicated. The input phase shift data can be found, for completeness, once more in Tables 2 and 3, together with the resulted output phase shifts that the two inverse potentials yield, respectively.

Fig. 1

Inversion potentials generated by experimental $\alpha$–$\alpha$ phase shifts listed in Refs. [5,6,7]. a Real part. b Imaginary part

The potential at low energy (Fig. 2a) is real and very shallow while those at higher energy (Fig. 3a, b) are complex and oscillatory showing the plateau behavior. Moreover both potentials exhibit also the Pauli repulsion effect in their real parts.

Table 1 Experimental phase shifts (in degrees) deduced from $\alpha$–$\alpha$ scattering by Darriulat et al. [5] at several c.m. energies E indicated

Fig. 2

a Inversion potential generated by Brown experimental $\alpha$–$\alpha$ phase shifts at 5 MeV using technical parameters r0=11.7 fm, rho1=7.5, drho=0.5, nn=2, lmax2=6; b Wave functions in the partial waves $l=0, 2, 4, 6, 8.$

Fig. 3

a Real part of inversion potential generated by experimental $\alpha$–$\alpha$ phase shifts at 38.71 MeV; b Imaginary part of inversion potential generated by experimental $\alpha$–$\alpha$ phase shifts at 38.71 MeV. The technical parameters are r0=7.3 fm, drho=0.5, nn=2, lmax2=12

Table 2 Input real phase shifts $\delta _l^{re}$ in degrees as given by Brown and Tang [7] as a close reproduction of the measured differential cross section data of elastic scattering of $\alpha$-particles at c.m. energy of $E=5$ MeV ($k=0.69$/fm)

Table 3 Experimental (complex) input phase shifts ${\tilde{\delta }}_l^{exp}$ in degrees given by Darriulat et al [5] as a result of a phase shift analysis performed on the measured differential cross section data of elastic scattering of $\alpha$-particles at c.m. energy of $E=38.71$ MeV ($k=1.92/$fm)

As we see these potentials are similar to those obtained for the $^{12}$C+$^{12}$C inversion [12, 13], showing the Pauli-blocking and the plateau behavior that scales with energy as $V\sim 2E$ at smaller distances. Since the wave functions have not been investigated in the case of the carbon collisions, it seems interesting to see, what are the wave functions which these $\alpha +\alpha$ scattering (inverse) potentials generate.

3.2 Wave functions given by the inverse potentials

The corresponding wave functions are shown in Figs. 2b and 4a, b. We can observe that the wave functions are in harmony with the physical picture suggested by the potentials. In that region where the collective (c.m.) coordinate r looses its meaning, the wave functions are almost zero. Only at the very surface region begin the wave functions to emerge out of this ’nothing’ (collective) state by exhibiting the characteristics encoded in the various partial waves corresponding to the individual phase shifts.

For example, at 5 MeV the wave function with $l=0$ dominates over those with $l=2,4,$ and 6 whereas the wave function corresponding to the $l=8$th partial wave is negligible. Contrary to this, at 38.71 MeV the $l=6$th wave function overwhelms the others suggesting that here we might be facing with a special effect.

It is interesting to note that in order to obtain a partially satisfactory fit at $E=38.71$ MeV to the imaginary part of the $l=6$ phase shift value, Brown and Tang [7] should have used a very complicated combination for the imaginary potential of a volume and a surface absorption form possessing approximately of equal strengths.

In our case we observe in Table 3 that the inverse potential provides an over-all satisfactory reproduction of the input phase shifts. However, the calculated wave functions exhibited in Fig. 4 show that the $l=6$th partial wave is indeed behaving in a peculiar manner. This may explain also the difficulties encountered before to reproduce this imaginary part of the $l=6$ phase shift at the scattering energy $E=38.71$ MeV by using a modified resonating-group formalism [7].

Fig. 4

a Real part of wave function of the $\alpha$–$\alpha$ inversion potential at 38.71 MeV; b Imaginary part of wave function of the $\alpha$–$\alpha$ inversion potential at 38.71 MeV. The wave function corresponding to the $l=6$th partial wave overhelms the others

4 Summary

A modified inverse scattering method [4] has been applied to calculate the $\alpha$–particle interaction potentials by using the scattering phase shift data measured within the c.m. energy range of E=0–50 MeV [5,6,7].

The results obtained using two data sets at $E=5$ MeV and 38.71 MeV have been investigated in detail. The nuclear potentials obtained show up similar characteristics as in the case of earlier findings for carbon nucleus scattering [12, 13], i.e., a large interaction domain between r=4–8 fm, a thin repulsive region of $\sim 0.5$ fm depth around $r\sim 3.8$ fm, and a smooth merging region between r=0–3.5 fm.

As before, these three regions can be described as the territory of nuclear reaction events, the region of the Pauli blocking, and the merging domain where the stationary beryllium nuclear matter is (virtually) forming and also the virial theory is dominating.

The inverse potentials are tested by re-calculating the (input) phase shifts and an over-all agreement is obtained. A finer reproduction would be expected when taking account also of the weak Coulomb interaction between the colliding $\alpha$–particles in the inversion procedure.

This task is postponed to a future work.