Possible Bose-Einstein condensation of $\alpha$ particles in the ground state of nuclear matter

The phase diagram of isospin-symmetric chemically equilibrated mixture of $\alpha$ particles and nucleons ($N$) is studied in the mean-field approximation. Skyrme-like parametrization is used for the mean-field potentials as functions of partial densities $n_{\alpha }$ and $n_N$. We find that there is a threshold value $a_{*}$ of the parameter $a_{N\alpha }$ which describes the attractive interaction between $\alpha$ particles and nucleons. At $a_{N\alpha }<a_{*}$ the ground state of nuclear matter at zero temperature consists only of interacting nucleons, whereas at $a_{N\alpha }>a_{*}$ the nuclear ground state includes also an admixture of $\alpha$'s. We demonstrate that the equation of state of such $\alpha \text{−}N$ system contains both the first-order liquid-gas phase transition and the Bose-Einstein condensation of $\alpha$ particles.

Figure 1

The binding energy per baryon $W$ for cold $\alpha \text{−}N$ matter on the $(n_B,\chi )$ plane. White contours correspond to constant $W$ values (given in MeV inside white boxes). Panel (a) shows the results for $a_{N\alpha }=1.9\mathrm{GeV}{\mathrm{fm}}^3<a_{*}$. The GS corresponds to a pure nucleon matter ($n_{\alpha }=0$) with parameters given in Eq. (1). The metastable state of a pure $\alpha$ matter is shown by the white diamond at the $\chi =1$ axis. The results in (b) are obtained for $a_{N\alpha }=2.13\mathrm{GeV}{\mathrm{fm}}^3>a_{*}$. In this case the system has only one minimum of $W$ and the GS contains both nucleons and $\alpha$'s.

Figure 2

(a) Relative change of baryon density (the solid line) and binding energy (the dashed curve) in the GS of $\alpha \text{−}N$ matter as a function of $a_{N\alpha }$. (b) Fraction of $\alpha$'s in the GS of $\alpha \text{−}N$ matter at different $a_{N\alpha }$. The solid (dashed) line is obtained with (without) readjusting $a_N,b_N$.

Figure 3

(a) Phase diagram of $\alpha \text{−}N$ matter for $a_{N\alpha }=2.13$ GeV ${\mathrm{fm}}^3$ on the $(\mu ,T)$ plane. The full square shows the GS position, whereas the full and open circles corresponds to the critical and triple points, respectively. Shading marks states with BEC of $\alpha$ particles. (b) Fractions of $\alpha$ particles in $\alpha \text{−}N$ matter as functions of $T$ for $a_{N\alpha }=2.13\mathrm{GeV}{\mathrm{fm}}^3$. The solid and dashed lines correspond to $\mu =225$ and 230 MeV, respectively. The thick lines show fractions of all $\alpha$'s, while the thin ones give the contribution of the condensate.

Figure 4

Same as Fig. 3 but for the $(n_B,T)$ plane. MP denotes the mixed-phase region.

Figure 5

(a) The temperatures at the CP (the solid line) and TP (the dashed curve) for $\alpha \text{−}N$ matter as functions of $a_{N\alpha }$. The horizontal upper and lower dash-dotted lines correspond to the critical temperatures of the pure nucleonic- and $\alpha$ matter, respectively. (b) Same as (a) but for relative fractions of $\alpha$'s.