Halos in heavy nuclei

A fascinating phenomenon of increased radial sizes of nuclei far from stability [41,42,43,44] was very intensely studied in recent years. In extreme situations, when nuclei consist of about thrice more neutrons than protons, the outer weakly bound neutrons may form halos of particle densities extending to rather large distances. For example, the flag-ship nucleus ¹¹Li exhibiting this kind of structure has the measured [45] root-mean-square (rms) radius of $R_{\rm rms}$=3.27(24)fm, while in ⁹Li one has $R_{\rm rms}$=2.43(2)fm. Supposing that in ¹¹Li the size of the ``core'' subsystem of 9 particles is the same as that of ⁹Li, one obtains the rms radius of the two-neutron subsystem equal to $\sqrt{(11*3.27^2-9*2.43^2)/2}$=5.67fm. Therefore, the outer two-neutrons occupy a volume that has the size similar to that of ²⁰⁸Pb, which is a nucleus with about 20 times more particles!

Sizes of several light neutron-rich nuclei were recently measured via the total interaction cross sections, see Refs. [46,47,48]. Although such measurements rely on a number of theoretical assumptions pertaining to the reaction mechanism, the increase of size of nuclei near the drip lines seems to be fairly well established. The question of increased sizes of medium heavy and heavy neutron-rich nuclei could be so far addressed only in theoretical calculations within mean-field models. The decisive role played by pairing correlations in establishing asymptotic density distributions has been recognized since a long time [49,50], and several calculations based on the Bogoliubov approach were already performed, both in the non-relativistic [51,52,53] and in the relativistic [54,55,56,57] framework.

The neutron rms radius constitutes only an indicative general measure of the neutron density distribution. In fact, three different physical properties of the density distribution can be singled out that strongly influence the rms radius, namely, (i) the global size of the nucleus, (ii) the surface diffuseness, and (iii) the rate of decrease of density in the asymptotic region. These three properties are almost directly related to (i) the number of nucleons, (ii) the surface tension or surface interaction terms, and (iii) the particle binding energy or the Fermi energy, respectively. The halo phenomena are obviously related to properties of particle distributions in the asymptotic region. One should keep in mind, however, that we cannot expect nuclear halos to correspond to what one might infer from the every-day-life meaning of the term. The nuclear density distributions always decrease when the distance from the center of nucleus increases, and by the halo we can only mean an unusually slow rate of such a decrease.

In order to disentangle the three effects mentioned above, one has to use a sufficiently rich model of the neutron distribution. For example, the density distributions obtained by filling with particles the harmonic-oscillator (HO) potential up to a given principal HO number cannot serve this purpose, because they are determined by a single parameter, the HO frequency $\hbar\omega_0$. Therefore, neither the surface diffuseness nor the asymptotic decrease rate can be properly described, even if $\hbar\omega_0$ is adjusted to reproduce the overall size of the nucleus. Similarly, the Fermi (or the Woods-Saxon) type of shape,

$$\rho(r)=\frac{\rho_0}{1+\exp[(r-R_0)/a]}\quad,$$ (1)

can reproduce the overall size and surface diffuseness by an adjustment of parameters R₀ and a, respectively. (See Ref. [58] for a recent application of this form to an analysis of experimental data.) However, here the asymptotic behavior, $\rho(r)$$\simeq$$\rho_0$ $\exp\{-r/a\}$, is dictated by the surface diffuseness a, and not by the Fermi energy. In order to describe this latter physical feature, that is crucially important in weakly bound systems, even more a complicated model distributions must be used. For example, the third parameter can be introduced in a very simple way by rising the denominator in Eq. (1) to some power $\gamma$. In this case, the surface diffuseness and asymptotic decrease rate can be independently adjusted by varying a and $\gamma$. Another parameterization achieving similar goal was recently analyzed in Ref. [59].

Density distributions obtained within microscopic self-consistent approaches are obviously able to independently model all three physical effects described above. Here, the quantitative reproduction of the nuclear size, surface diffuseness, and asymptotic decrease rate depends on properties of the underlying interactions, while the density distributions may in principle assume unrestricted forms. By employing the classic Helm model distributions [60,61,62], a method to analyze microscopic distributions was recently devised [53]. A comparison of Fourier transforms (form factors) of the Helm and microscopic densities allows to define the diffraction radius R₀ and the surface thickness $\sigma$ from the positions of the first zero and first maximum of the form factor, respectively. These characteristics of the density are entirely independent of the asymptotic decrease rate, and hence they define a suitable Helm rms reference radius,

$$R^{\rm (H)}_{\rm rms} = \sqrt{\frac{3}{5}\left(R_0^2 +5\sigma^2\right)}.$$ (2)

An increase of the rms radius with respect to the Helm rms reference radius can now be used as a signal for the appearance of the halo structure, and hence the size of the halo was in [53] defined as the difference $\delta{R}_{\rm halo}$= $\sqrt{\frac{5}{3}}(R_{\rm rms}-R^{\rm (H)}_{\rm rms})$. Such a halo parameter allows us to discuss and compare the results obtained within different models, and puts a definition of the halo phenomenon on quantitative grounds.

  

Figure 3: Left panel: neutron halo parameters (top), neutron Fermi energies (middle), and neutron pairing gaps (bottom) calculated in the HFB/SLy4 model for the two-neutron drip-line even-even nuclei (i.e., the heaviest even-even isotopes which are predicted to be two-neutron bound). From Ref. [53]. Right panel: matter density distributions calculated in sodium isotopes within the RMF+HB approach. Near the neutron drip line (⁴¹Na) the matter density at large distances is determined by the neutron density. (From Ref. [54]. Reprinted from Physics Letters B, Vol 419, 1998, Page 1, J. Meng et al, ``The proton and neutron distributions in...'', Copyright (1998), with permission from Elsevier Science.)

In Ref. [53] it was shown that in the mean-field description of medium heavy and heavy nuclei, the neutron halos appear gradually when the neutron numbers increase beyond the neutron magic numbers. The halos of different sizes can be obtained by changing the parameterizations of nuclear effective interactions, and in particular those of the pairing force [63]. Results of calculations presented in the left panel of Fig. 3 indicate that the sizes of halos at the two-neutron drip line are of the order of 1fm throughout the mass table. Fluctuations that are observed in function of the mass number are correlated with the values of the neutron Fermi energy $\lambda_n$. However, no correlation is observed with the positions of the Fermi energy with respect to the low-$\ell$ (s and p) single-particle orbitals. This latter fact reflects the so-called pairing anti-halo effect [64].

Indeed, the pairing correlations strongly modify the asymptotic properties of single-particle densities. In the extreme single-particle picture, the single-particle $\ell$=0 wave functions behave asymptotically as $\psi(r)$$\simeq$ $\exp(-\kappa{r})/r$, with the decrease rate of $\kappa$= $\sqrt{-2m\epsilon/\hbar^2}$ given by the single-particle energy $\epsilon$. Therefore, with $\epsilon$ $\rightarrow$0, these wave functions become infinitely flat, and the corresponding rms radii, and the halo sizes, become infinite. Nothing of the sort is observed in the Hartree-Fock-Bogoliubov (HFB) calculations, because here the lower components of the quasiparticle wave functions vanish exponentially with the decrease rate of $\kappa$= $\sqrt{2m(E-\lambda)/\hbar^2}$ given by the quasiparticle energy E [49,50]. Since the paring correlations do not vanish at drip lines, the nonzero values of E prevent $\kappa$from vanishing, even in the limit of zero binding given by $\lambda$=0. As the result, one of the conditions generally thought to be a prerequisite for the halo structures [65], i.e., the presence of the low-$\ell$ orbitals, does not apply to paired even-N systems. On the other hand, this condition does apply to odd-N systems, because the blocked HFB states correspond to the quasiparticle wave functions that vanish with the decrease rate of $\kappa$= $\sqrt{-2m(E+\lambda)/\hbar^2}$, which goes to zero at the one-neutron drip line given by $E+\lambda$=0.

The fact that the neutron densities of even-N systems decrease with a non-zero rate when approaching the drip line is well born out in all self-consistent Bogoliubov calculations. An example obtained within the relativistic mean field (RMF) approach [54] is shown in the right panel of Fig. 3. Results for all bound even-N sodium isotopes were calculated in space coordinates, and the resulting matter density distributions clearly show a common non-zero decrease rate when approaching the neutron drip line, which simply illustrates the pairing anti-halo effect. This is especially strongly accentuated in the RMF+HB method which gives a cluster of nearly degenerate states close to the Fermi surface [54], which results in strong pairing correlations near the neutron drip line. Exactly the same situation occurs in the zirconium isotopes studied in Ref. [55].