Localizations and phase shifts

Since the pairing potentials depend self-consistently on the pairing densities, they are concentrated inside the nucleus, i.e., they are nonzero inside, and go to zero outside the nuclear volume. Therefore, the more the continuum wave functions are concentrated inside the nucleus, the more they are sensitive to the pairing coupling. In order to quantitatively characterize the wave functions from this point of view, we introduce their 'localization' as the norm inside the sphere of radius $R_{\mbox{\scriptsize {Loc}}}$,

$$L\left[\psi\right] = \int_0^{R_{\mbox{\scriptsize {Loc}}}} \left\vert\psi(r)\right\vert^2 dr,$$ (38)

where the radius $R_{\mbox{\scriptsize {Loc}}}$ is, for the purpose of the present study, arbitrarily fixed at $R_{\mbox{\scriptsize {Loc}}}=1.5 \times R_{1/2} = 7.578$fm, while R_1/2 is the radius where the potential drops to its half value. [Note that the volume element $4\pi r^2$ is included in the definition of wave function $\psi(r)$.] For the bound states, localization is just the probability to find the particle inside the sphere R< $R_{\mbox{\scriptsize {Loc}}}$. For the continuum wave functions which are not normalizable, localization depends on the chosen normalization condition.

Below we present results for the continuum wave functions normalized in two ways. Firstly, we may normalize them to unity inside the box of a large radius. (The value of $R_{\mbox{\scriptsize {box}}}$=30fm is chosen for all results obtained in the present study.) Outside the box, the wave functions oscillate and have an infinite norm. In this case, the localization is a fraction of the probability to find the particle inside the nucleus relative to the probability to find it inside the box. Absolute values of such a localization depend, of course, on the radius of the box, however, we are only interested in comparing relative localizations of the wave functions with different energies. Secondly, we may normalize the continuum wave functions in such a way that they all have a common arbitrary amplitude in the asymptotic region. For wave functions in which the volume element $4\pi r^2$ is included, as is the case here, their amplitudes do not asymptotically depend on r. Again, the absolute values of this localization depend on the value of the common amplitude, but the relative values tell us at which energy the given wave function is better concentrated inside the nuclear volume. One may chose other normalization conditions (to a delta function in energy or in momentum, for example), but we did not find any advantage in looking at the corresponding localizations, and we discuss here only the two ways described above.

Fig. 4a shows values of localizations of the s_1/2PTG continuum states, calculated analytically, for the normalization in the box (solid line) and for the normalization to a constant amplitude (dashed line). Calculations are performed for the PTG potential for which there is a weakly bound 3s_1/2 state (cf. Table 2), and therefore, the lowest resonance in the s_1/2 channel appears high in energy. The wide bump in the localization, which is seen near 40MeV reflects the known fact that the S-matrix pole at (39-18i)MeV generates continuum states which are more localized than the continuum states far from the resonant energies. However, values of localizations near the maximum are only a factor of 2-3 larger than those far from the maximum, i.e., one cannot à priori expect that the only continuum states which can couple to the pairing field are those close to resonances.

One can see that the localization obtained from normalizing the continuum states in the box (solid line) presents numerous wiggles, which appear when the consecutive half-waves enter the box. On the other hand, the localization obtained from the amplitude normalization is given by a smooth curve. Both localizations are fairly similar, and therefore, different normalization prescriptions do not affect our conclusion about the relative localizations of the continuum states.

There is some difference between these two different normalization prescriptions when $\epsilon$ $\rightarrow$0. At the origin, $\phi(r)$=$A\sin(kR)$. In case of the normalization in the box, we can assume for k $\rightarrow$0 that this expression is valid in the whole box. Then : $\int_0^{R_{\mbox{\scriptsize {Loc}}}} {\phi}^2(r) dr/ \int_0^{R} {\phi}^2(r) dr... ...size {Loc}}}} r^2 dr/\int_{0}^{R} r^2 dr = (R_{\mbox{\scriptsize {Loc}}}/R)^3 =$ const. We have verified this assertion also numerically. In the case of normalizing to an amplitude A, for k $\rightarrow$0 one gets: $\int_{0}^{R_{\mbox{\scriptsize {Loc}}}} {\phi}^2(r) dr \simeq \vert A\vert^2k^2r^3/3 \longrightarrow 0$. Therefore, the dashed curve turns sharply down when approaching $\epsilon$=0.

In Fig. 4a we also present localizations of the continuum states calculated numerically (circles) in the same box of $R_{\mbox{\scriptsize {box}}}$=30fm, and using the discretization of wave functions on a mesh of nodes equally spaced by H=0.25fm. One can see that the numerical results perfectly reproduce the analytical calculations for the same normalization. In the numerical calculations, the box plays merely a role of selecting from the infinite continuous set of positive-energy solutions a discrete subset of wave functions which vanish at the box boundary. Apart from that, the numerically calculated wave functions are very precise representations of the exact wave functions for some specific discretized values of the energy.

Fig. 4b shows the phase shifts of the same continuum states calculated analytically (solid line) and numerically (circles). (In this case, normalization of the continuum wave functions does not play any role.) Again, one can see that the numerical results very precisely reproduce the analytical ones in the whole range of studied energies.

Usually, one identifies the resonance when the phase shift passes $\pi /2$ (or better $n \pi /2$, as the phase shift is defined modulo $2\pi$). This definition works well for narrow, well separated resonances. In the case of PTG potential, the resonances are broad and this definition is inadequate. It is then better to identify the resonance with the inflection point in the derivative $d\delta (\epsilon)/d\epsilon$. This also demonstrates real difficulty in identifying resonances in potentials with diffuse surfaces and proves again the advantage of the soluble models where the S-matrix is analytically known and the poles can be analytically studied.