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PROTON IMPURITY in the NEUTRON MATTER: a NUCLEAR POLARQN PROBLEM Report INP No

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PROTON IMPURITY in the NEUTRON MATTER: a NUCLEAR POLARQN PROBLEM Report INP No PROTON IMPURITY IN THE NEUTRON MATTER: A NUCLEAR POLARQN PROBLEM Report INP No. 1603/PH Marek Kutschera H. Niewodniczański Institute of Nuclear Physics ui. Radzikowskiego 152 31-342 Kraków, Poland and Włodzimierz Wćjcik Institute of Physics. Technical University ul.Podchorażych 1 30-0S4 Kraków. Poland October 1992 Attract: We study interactions of a proton impurity with density oscillations of the neutron matter in a Debye approximation The protoii-phonou coupling l.s of tl.e defonnation- potential type at long wavelengths. It is weak at low density raid increases with tbr neutron matter density. We calculate the protons effective mass perturbatively for a weak coupling, and use a canonical transformation teclmique for stronger couplings The proton's effective mass grows significantly with density, and at higher densities the proton impurity can be localized. This behaviour is similar to that of the pnlarou in solids. We obtain properties of the i ocali zed proton in the strong coupling regime from variational calculations, treating the neutron matter in the Thomas-Fermi approximation. PACS number H65.rf 1. INTRODUCTION The number of protons present in the liquid core of neutron stars is of the order of a few percent [1] as compared with neutrons. The proton admixture, wliich is. required for the beta-stability of the system, is found to decrease with density |1]. It eventually vanishes at some density п.. Model calculations give nv in the range: 5n0 to 10n0 !lj. Hence protons can be regarded as impurities in the neutron matter at densities close to nc. Particularly important is the proton effective mass since it plays a crucial role fcr such phenomena in the neutron star matter as proton superfluidity [2j and/or possible spin instability [Z]. The contributions to the proton's effective mass are generally of two types The first one is due to two-body nuclear interactions while the otb,er one is due to the coupling to 1 excitations of the neutron matter. We shall denote corresponding values of the proton's effective mass m. and, mc//, respectively. At finite proton concentrations the latter con¬ tributions are expected to be small |4] and m, ss m«//. However, for a proton impurity, when the relevant energy scale is its vanishing kinetic energy, any phonon contribution can affect the effective mass profoundly, giving mf// much higher than m.. In this paper we study the coupling of proton impurities in dense neutron matter to small density oscillations. This coupling can be quite strong as a result of the behaviour of the proton chemical potential in the neutron matter. Various parametrizations show that it has н minimum close to the saturation density no. At higher densities the proton i'bemic&l potential increases and becomes positive at high densities (Fig. 1). This suggests that a uniform density neutron matter surrounding a proton might not be the lowest energy state. Let us consider a neutron matter of density higher than the one corresponding to the minimum ш Fig.l. and imagine a small density fluctuation, such that the density around the proton is slightly reduced. For the long-wavelength fluctuation the proton energy is thus also reduced, a? the proton moves now in an attractive potential well. This observation indicates that a single proton tends to disturb a uniform neutron matter i.e. there appears a <*oupliiK> o{ the proton to the (long-wavelength) neutron matter density oscillations The coupling has a deformation-potential form. The aim of this paper is to calculate the proton effective mass mr// due to the proion- phouon coupling. The "nuclear" effective, mass m. derived from nuclear matter calculation* will be used as a "Ыи-е'" mass in our calculations. We treat phonoim in the neutron matter ш a Debye approximation. We use a perturbation theory and a canonical transformation technique to calculate the proton effective mass for low and intermediate values of thr cou¬ pling, respectively. The proton effective mass is found to increase with the neutron deusity At higher densities, where the coupling becomes strong, one can expect a behaviour similar to that of the polaron. which becomes localized in the strong coupling limit. We perform i:i the case of stron? coupling variational Thomas-Fermi calculations which show iocali/.auou of a sinele proton in neutron rnatier above a critical density n- which, however, depends strongly on TU.. As we mentioned, the phenomenon we consider has a well known analogue in th«' *ohd state physics. In some crystals the electron-phonon interactions influence the effective masr, of electrons significantly [5.6]. The system consisting of the electron and the virtual phonon cloud is called polaron. Above a critical coupling the polaron becomes localized and the com-sponding effective mass is very large {5.6]. In Sect.2 we derive the interaction Hamiltonian which couples the proton to thr neu¬ tron matter density os-illations. In Sect.3 we calculate the proton effective mass in the first-order perturbation theory, for the weak coupling, and by a canonical transformation technique for stronger couplings. In Sect.4 a possibility of localization is the case of stiong coupling is considered. The results are discussed in Sect .5. 2. COUPLING OF A PROTON IMPURITY TO PHONONS IN NEUTRON MATTER Let us consider a single proton in a uniform neutron mutter of density >ijv. If the momentum к of the proton is low, the energy of toe proton is EP(k) S ^- + «.„, (1) where the effective potential is equal to the proton chemical potential in a neutron matter, We// = M"JV). (2) The chemical potential рр(пц) is the energy of a zero-momentum proton. Here m. is the effective mass of the proton due to nuclear interactions with neutrons and it does not include any phonon contribution. The effective potential can in principle depend on the proton momentum, however, this is higher order and can be neglected for email values of 4 We can thus write the Hamiltooian of the slow proton in the form Н(рв) m -^1 + ,,,,(„„), (3) Assume now that the neutron matter is slightly inhomogeneous, with the neutron density n(r,t) = п* + Л»(г,О, (4) where 6n is a small perturbation. If the density n(r,() varies sufficiently slowly, i.e. the wavelength of. the perturbation is sufficiently long and the frequency is low, the proton energy is still given by eq.(l). Hence the Hamiltonian of the single proton in the neutron mar ter with small long-wavelenght oscillations is The last term describes the coupling of the proton to the snail oscillations of the neutron matter density. For long-wavelength oscillations the neutron matter can be treated ae a continuous medium, and standard methods of describing phonon* in the medium [6] can be used. The density oscillations can be expteastd by tje displacement u(r,r) of the neutron matter from its equilibrium position (6) We can expand the displacement field a in the normal moda (phonons): where e(k) is the unit polarization vector, £ is tbe energy per neutron (including the rest maci) in neutron matter of density njv and u>(k) is the enc .-gy of the phonon of momentum k. For longitudinal acoustical phonons e<k) is parallel to к and w(k) = с, | Ч |, where c, is the velocity of acoustical waves in the neutron matter. Quantization of the phonon field can be carried out in a standard way replacing the Fourier component Ofc(t) ж aucxp(-iu>(k)t) by the pbooon annihilation operator. The interaction Hamiltonian becomes where B(ą) = -i^-(nN)nN*l-±2±-. (9) Expanding also tbe proton wave function in tbe plane wave basis, •Иг) - 4=УЧеЛг, (10) we can write the interaction Hamiltonian in terms of phooon and proton annihilation and creation operators СкК-в!,). (U) Tbe proton-phooon coupling (11) has the form of the defennation-potent.ial coupling in solids [6], with the deformation-potential constant (12) This quantity measures the strength of tbe proton-pbonon coupling. Since it is a product of tbe derivative of .he proton chemical potential, with respect to tbe neutron density, and the neutron density itself, it is small at low densities and near the minimum in Fig.l. For higher densities о increases linearly, Fig.2. In the following we shall consider only tbe high density range which extends to the right of tbe minimum in Fig.l. The proton-pbonon coupling is thus weak near tbe minimum and becomes strong at high density. 3. WEAK AND INTERMEDIATE PROTON-PHONON COUPLING The interaction Hamiltonian (11) can be treated as a small perturbation of the energy 0) of a proton in a uniform density neutron matter Hp , eq.(3), in case of low values of B(q). We now consider this case. Pertvrb*tivt ctleuUtions The lowest-order perturbation correction is obtained for single-phonon intermediate states [6]. The first order wave function, in the Fock representation, is -1M + (-DE 1 where the matrix element of the interaction Hamiltonian is <k,,lq|A««|k.0>= -^B(q)<5klik_q (M) and £>(k,q) = Ep(k-q) + w(q) - EP(k). (15) We use here a standard notation [6], in which the single-phonon states ( k, lq > are defined by the proton momentum к and the phonon momentum q. The wave function (13) is thus Using the wave function (16) we can calculate the energy of the proton impurity up to the second order in the interaction, which is F<2>^ kJ j. уЧ< к - q, !«, | H, | к, О E*(k) = sz + v"f - \ fcj nt Evaluating the last contribution in the expression (17) we can find the proton effective mass, as well as the change of the proton energy due to the coupling to phonons.
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