Figure 7.1. Schematic spectrum of lowest energy charged-pion excitations in the npep matter at densities lower than the n- condensation threshold. Elementary excitation energy w is plotted versus k2, square of excitation wavenumber. Left: Density is lower than the condensation threshold. Right: Density is higher than the condensation threshold, and at k ~ k0 matter becomes unstable with respect to the formation of zero-sound-like excitations (see text for details). After Migdal et al. (1990).
Let us first reconsider the stability of a spatially uniform state of the npei matter in beta equilibrium. As shown in § 3.5, such a state is unstable with respect to the formation of periodic "nuclear structures" (density waves) at P ^ 0.5po. This instability signals a phase transition to a spatially-ordered non-uniform state which is the neutron star crust. Now consider the stability of the uniform npei matter at p > p0. This can be done by studying excitations of the npei matter associated with perturbations of the ground state of the system. In the context of pion condensation, there are two types of relevant excitations. Both are boson-type excitations which differ in their low-momentum dispersion relations. Excitation quanta have well defined spin and charge quantum numbers S and q. Pion condensation is related to excitations with the pion quantum numbers S = 0 and q = ±1 or q = 0.
Excitations of the first type correspond to collective modes (analogous to the well known zero-sound in liquid 3He). They have an acoustic large-wavelength dispersion relation ws — csk at k 0, where cs is the zero-sound speed. These three zero-sound-like excitations will be denoted as ^St'0.
Excitations of the second type correspond to "real pions" and have a characteristic "resonance" ("optical") long-wavelength behavior. In the limit of a "dilute medium" (p —> 0) their energy spectrum — \Jm2 c4 + k2c2 is similar to the energy spectrum of real pions in vacuum. These three excitations will be denoted as n±'0.
Using the Fermi-liquid theory one can calculate the dispersion relations (k) for all six excitation modes in the npei matter, provided the gas of excitations is diluted, so that the their number density nn ^ nb. A schematic representation of these dispersion relations is given in Figs. 7.1 and 7.2.
Let us start with excitations of zero-sound type. The npei matter turns out to be most susceptible ("the softest") to excitations. As seen from the right panel of Fig. 7.1, for this mode the function (k) acquires a minimum, wmin < o, at some k = k0(p). As soon as
n s the mode is spontaneously excited with some (or even all) protons converted into neutrons. This instability with respect to p —► + n, (7.2)
if it occurs at all, starts at a well defined critical density pn+. In the particle-hole language, the system becomes unstable with respect to the formation of pairs of protons and neutron-holes (which are just excitations). Simultaneously, the npei matter looses translational invariance.
Let us turn to excitations of which n- is relevant for us. For sufficiently high densities the function w = (k) has a minimum at a finite k = kn-(Fig. 7.2). If wm-n < |w™n|, then there exists such a value of k at which pairs of excitations n+n- with opposite momenta of |fc| = k can be spontaneously created,
In this way the instability initiates n- condensation.
As in a symmetric nuclear matter, the spontaneous creation of n0 quasi-particles becomes possible when wmin < 0 . (7.4)
This instability, if it happens at all, starts at a well defined density pn° .
In thermodynamic equilibrium, elementary excitations (boson quasipar-ticles) n+, n-, and n0 form Bose-Einstein condensates, and their chemical potentials are equal to the energy of the occupied momentum state with minimum quasiparticle energy (wn+, , and wno = 0, respectively).
The mechanism of spontaneous transition to a pion-condensed state deserves a comment. As pointed out by Migdal many times, pion condensation in a pure neutron matter does not imply an instability of the form n —> p + n- . (7.5)
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