Pöschl-Teller-Ginocchio potential

The PTG potential depends on four parameters: $\Lambda$ which determines its shape and diffuseness, $\nu$ which defines its depth, s (in fm^-1) which is a scaling factor that can be adjusted to obtain a given mean radius of the potential, and a, which allows to account for an effective mass of the particle. In the present study, the effective mass is not discussed, and therefore we take a=0, for which the radial PTG potential is defined as:

$$V_{PTG}(R) = \frac{\hbar^2 s^2}{2m}\left( v(R) + c(R)\right) ,$$ (1)

where m is the mass of free particle, v(R) is the central part,

 

$$\Lambda^2\nu_{Lj}(\nu_{Lj}+1)(1-y^2) +\left({1-\Lambda^2\over 4}\right)$$ v(R) = - (2)

$$\times (1-y^2) \left(2-(7-\Lambda^2)y^2+5(1-\Lambda^2) y^4\right)$$

and c(R) is a (nonstandard) centrifugal barrier,

$$c(R) = L(L+1)\biggl({1-y^2\over y^2}\biggr) \bigl(1+(\Lambda^2-1)y^2\bigr).$$ (3)

Dimensionless variable $\eta$ is proportional to the radial coordinate R,

$$\eta=sR ,$$ (4)

and y is implicitly defined by the expression:

$$\eta={1\over \Lambda^2}\biggl[\hbox{arctanh}(y) +\sqrt{\Lambda^2-1} \arctan\bigl(y\sqrt{\Lambda^2-1}\bigr)\biggr] ,$$ (5)

where $0\le \eta \le +\infty$ and, consequently, $0 \le y \le 1$.

Parameter $\Lambda$ can take any positive value. (Potentials with $\Lambda$<1, for which in Eq. (5) function $\arctan$changes into $\hbox{arctanh}$, will not be considered in the present study.) In principle, parameters $\Lambda$, $\nu$, and s can take different values for every value of L and j. However, below we use (Lj)-independent values of shape parameter $\Lambda$ and scaling factor s. On the other hand, as indicated in Eq. (2), values of depth parameter $\nu$ are independently chosen for each L and j, so as to obtain 'reasonable' single-particle spectra, and in particular, to simulate the spin-orbit splitting of single-particle levels which is absent in the PTG potential.

Problem of the nonstandard centrifugal barrier (3) deserves a few words of discussion. At the origin ( $\eta \rightarrow 0$), y decreases as $\eta$, $y \rightarrow \eta$, and therefore,

$$s^2c(R) \longrightarrow {L(L+1)\over R^2}$$ (6)

becomes the standard centrifugal barrier. On the other hand, at large distances ( $\eta \rightarrow \infty$), 1-y decreases exponentially, i.e.,

$$c(R) {\longrightarrow} 4\Lambda^2e^{-2\Lambda^2(\eta-\eta_0)} ,$$ (7)

where

$$\eta_0={\sqrt{\Lambda^2-1}\over\Lambda^2}\arctan{\sqrt{\Lambda^2-1}} ,$$ (8)

and the nonstandard PTG centrifugal barrier exponentially disappears. Consequently, in the PTG potential all partial waves behave asymptotically as the s waves. Therefore, in the following analysis we concentrate on the L=0 bound states and resonances for which the centrifugal barrier is absent (see discussion in Secs. 4 and 5). In some numerical calculations we replace the PTG centrifugal barrier s²c(R) (3) by the physical barrier L(L+1)/R², i.e.,

$$V_{PTG'}(R) = \frac{\hbar^2}{2m}\left( s^2 v(R) + \frac{L(L+1)}{R^2}\right) .$$ (9)

Unfortunately, in such a case the analytical solutions do not exist.

For small values of $\Lambda$, the PTG potential is wide and diffuse. In Fig. 1 (lower panel) we present the PTG potentials for L=0 and $\Lambda$=1, 3, and 7, with parameters s and $\nu_{Lj}$ (Table 8) chosen in such a way as to keep the depth and radius of the potential fixed. For $\Lambda=1$ and $\ell$=0 (see Fig. 1), one obtains the Pöschl-Teller potential [45,46] which has been widely studied, e.g., in the molecular physics. For larger values of $\Lambda$, the PTG potential gets steeper and resembles the Woods-Saxon potential. For relatively large values of $\Lambda$ and relatively small values of the depth parameter, one may find a small barrier at the edge of the potential well. Finally, for still larger $\Lambda$, the PTG potential resembles the finite-depth square well potential. The PTG potentials presented in Fig. 1 for $\Lambda$=3 and 7 correspond to the profiles which are interesting from the point of view of simulating the nuclear mean field within a physical range of the diffuseness.

In the middle panel of Fig. 1 we present the L=4 radial PTG potentials, again for $\Lambda$=1, 3, and 7, and parameters s and $\nu_{Lj}$ given in Table 8. Similarly, the top panel (with the corresponding parameters s and $\nu'_{Lj}$) shows the L=4 potentials for the PTG centrifugal barrier s²c(R) replaced by the physical centrifugal barrier. One can see that up to a little beyond the minimum, the PTG potential reproduces fairly well the potential with the physical barrier. However, at larger distances, the PTG centrifugal term disappears too quickly.

Nature of the energy eigenstates inside the potential well, as well as the nature of scattering solutions, depend strongly on the shape parameter $\Lambda$. The smaller is the value of $\Lambda$, the broader are the single-particle resonances. Below the critical value of ${\Lambda}_{\mbox{\scriptsize {crit}}}=\sqrt{2}$, the resonances entirely disappear, i.e., there are no single-particle resonant states in any partial wave anymore.