$\alpha$-cluster matter reexamined

We examine in detail two alternative descriptions of a system of $\alpha$ particles interacting via local interactions of different character, highlighting the fact that a faithful microscopic description of such systems demands a consistent treatment of both short- and long-range correlations. In preparation, we examine four different versions of modern microscopic many-body theory and conclude by emphasizing that these approaches, although a priori very different, actually lead to the same equations for their efficient application. The only quantity that depends on the formulation of many-body theory chosen is an irreducible interaction correction. In the language of Green's functions and Feynman diagrams, it is the set of both particle-particle and particle-hole irreducible diagrams, and in variational Jastrow-Feenberg theory it is determined by multipartite correlations and elementary diagrams. We apply these theoretical methods to the calculation of the energetics, structure, thermodynamics, and dynamics of $\alpha$ matter, as well as its condensate fraction. In dimensionless units, $\alpha$ matter appears to be remarkably similar to the much-studied ${}^4\mathrm{He}$ quantum fluid, its low-temperature properties now basically solved in the Jastrow-Feenberg framework. Accordingly, one can have confidence in the results of application of the same procedure to $\alpha$ matter. Even so, closer examination reveals significant differences between the physics of the two systems. Within an infinite nuclear medium, $\alpha$ matter is subject to a spinoidal instability. Extended mixtures of nucleons and $\alpha$ particles are yet to be given rigorous consideration in a corresponding theoretical framework.

Figure 1

Aziz interaction [28] for ${}^4\mathrm{He}$ (dash-dotted line) and Ali-Bodmer interaction [23] (solid line) for $\alpha$ matter, in normalized units. Also shown are curves for versions of the $\alpha \text{−}\alpha$ interaction derived from versions D1 and D1N of the Gogny variety of two-nucleon interactions.

Figure 2

Comparison between results for the equation of state of $\alpha$ matter for the Ali-Bodmer interaction within the HNC framework in HNC-EL//0 approximation as well as when triplets and elementary diagrams are included (left scale). Also shown is the long-wavelength limit, with $m{c}_s^2={\tilde{V}}_{\mathrm{p}-h}(0+)$ (right scale).

Figure 3

Same as Fig. 2 but for the D1N interaction.

Figure 4

Comparison of Ali-Bodmer (solid line) and D1N (dashed line) equations of state for $\alpha$ matter with that of liquid ${}^4\mathrm{He}$ for the Aziz interaction (dash-dotted line), in dimensionless units, energy in units of potential depth $\epsilon$, and lengths in units of core size $\sigma$.

Figure 5

Pair distribution function $g\left(r\right)$ as a function of density $\rho$ for the Ali-Bodmer interaction.

Figure 6

Static structure function $S\left(k\right)$ as a function of density $\rho$ for the Ali-Bodmer interaction.

Figure 7

Pair distribution function $g\left(r\right)$ at the density $\rho =0.07{\mathrm{fm}}^{-3}$, for the Ali-Bodmer interaction. The solid line shows the result including three-body and elementary-diagram corrections; the dashed line, the result including only three-body correlations; and the dotted line, the HNC-EL//0 approximation.

Figure 8

Same as in Fig. 7, but for the particle-hole interaction $V_{\mathrm{p}\text{−}\mathrm{h}}\left(r\right)$. The two solid lines show $V_{\mathrm{p}\text{−}\mathrm{h}}\left(r\right)$ when the semiphenomenological modifications discussed in Sec. 2f are, respectively, included or omitted. Also shown is the bare Ali-Bodmer potential (dash-dotted line).

Figure 9

Free energy $F/N$ per particle for the Ali-Bodmer interaction in the temperature range $k_{\mathrm{B}}T=0,1,...,10\mathrm{MeV}$.

Figure 10

Same as Fig. 9, but for the D1N interaction. Note that the fact that the higher-temperature curves of $F/N$ are concave does not contradict the stability condition $m{c}_s^2>0$, since $\rho F/N$ must be convex [see Eq. (2.31)].

Figure 11

Condensate fraction of $\alpha$ matter as a function of density. The reference to “3-body and elems” refers to the input function $g\left(r\right)$, but no three-body correlations or elementary diagrams were retained in the explicit expression, i.e., Eqs. (2.42) and (2.43) have been used for all calculations. The labeled solid lines depict the most complete calculations for the two interactions considered in this work; simpler versions for the D1N interaction are not shown.

Figure 12

Zero-sound dispersion relation $e_0\left(k\right)$ for the Ali-Bodmer interaction in Feynman approximation (long-dashed lines), and DMBT [107] (solid lines), for the densities $\rho =0.06,0.07,0.08,0.09{\mathrm{fm}}^{-3}$. The highest density corresponds to the highest roton minimum.

Figure 13

Same as Fig. 12, but for the D1N interaction and densities $\rho =0.03,0.04,0.05,0.06{\mathrm{fm}}^{-3}$. The highest density corresponds to the highest roton minimum.

Figure 14

Maps of $S(k,\omega )$ for the Ali-Bodmer potential for a sequence of densities $\rho =0.06{\mathrm{fm}}^{-3}$,...,$\rho =0.09{\mathrm{fm}}^{-3}$. The solid red line is the phonon dispersion relation, also shown in Fig. 12.

Figure 15

Same as Fig. 14 for the D1N interaction and a sequence of densities $\rho =0.03{\mathrm{fm}}^{-3}$,..., $\rho =0.06{\mathrm{fm}}^{-3}$.