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Structured mixed phase at charged kaon condensation
Nuclear Physics A 749 (2005) 186c–189c
Structured mixed phase at charged kaon condensation Toshiki Maruyamaa , Toshitaka Tatsumib , Dmitri N. Voskresenskyc , Tomonori Tanigawada , and Satoshi Chibaa a
Advanced Science Research Center, Japan Atomic Energy Research Institute, Tokai, Ibaraki 319-1195, Japan
b
Department of Physics, Kyoto University, Kyoto 606-8502, Japan
c
Moscow Institute for Physics and Engineering, Kashirskoe sh. 31, Moscow 115409, Russia d
Japan Society for the Promotion of Science, Tokyo 102-8471, Japan
Non-uniform structures of the mixed phase at the first-order phase transition to the charged kaon condensation are studied using the density functional theory within the relativistic mean-field model. Including the electric field effects and applying Gibbs conditions in a proper way, we numerically determine density profiles of nucleons, electrons and condensed kaons. The importance of the charge screening effects is elucidated and thereby we show that the Maxwell construction is effectively justified. 1. Introduction Considering phase transitions in nuclear matter, various scenarios have been discussed, like e.g. liquid-gas transition, various meson condensations, hadron-quark deconfinement transition, etc. In most cases these are first-order phase transitions (FOPT). In the systems with different charged species undergoing the FOPT the structured mixed phases might appear [1]. For example in case of the nuclear matter below the saturation density, where the liquid-gas FOPT is relevant, the “nuclear pasta” structures have been studied by many authors. The physical reason of the possibility of the mixed phases with charged structures is that the charge neutrality may hold only globally rather than locally, as in the Maxwell construction. Mechanically, the balance between the Coulomb force and the surface tension is responsible for the occurrence of the spatially non-uniform structures. There existed a naive view that not all Gibbs conditions can be satisfied in the description of the Maxwell construction [1]. However, it has been demonstrated in recent papers [2] that Gibbs conditions are actually fulfilled also in case of the Maxwell construction, if one properly includes the electric field effects. As an explicit example the hadron-quark deconfinement transition has been considered; the density region of the structured mixed phase might be largely limited as a result. Taking charged kaon condensation as a further example, we study in this paper whether the “pasta” structures appear in the high-density neutron-star matter. The appearance
0375-9474/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2004.12.031
187c
−3 1.0 3D, ρ B=0.45 [fm ]
0.5
p n e K
160 E/A−m B [MeV]
0.0 −3 1.0 2D, ρ B=0.55 [fm ] 0.5
0.0 −3 1.0 1D, ρ B=0.60 [fm ] 0.5
U K=−130 MeV
140
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droplet rod slab tube bubble uniform single phase
100
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0.0 −3 1.0 2D, ρ B=0.70 [fm ]
R, R cell [fm]
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T. Maruyama et al. / Nuclear Physics A 749 (2005) 186c–189c
0.5
0.0 −3 1.0 3D, ρ B=0.72 [fm ] 0.5
0.0
10
0 0
10 r [fm]
20
Figure 1. Density profiles of kaonic structures. A parameter Uk , the kaon optical potential at normal nuclear density, is set to be 130 MeV.
λ (e) D λ (p) D
20
0.4
0.5
0.6 0.7 ρ B [fm −3]
0.8
Figure 2. Top: binding energy per nucleon of nuclear matter in beta equilibrium. Bottom: structure size R (thick curves below) and the cell size Rcell (thick curves above).
of the “pasta” structures could have important consequences for the various neutron star phenomena. It causes a drastic change of the neutrino opacity at an early stage of the neutron star evolution and influences the subsequent neutron star cooling, it affects the matter resistance to the stress and consequently the glitch phenomena, etc. There are few works about this matter so far, and the finite-size effect and the Coulomb interaction effect are still not elucidated [3]. We demonstrate how the charge screening affects the structured mixed phase. 2. Kaon Condensation in High-density Matter We use the density functional theory (DFT) within the relativistic mean field (RMF) model [4]. We get the coupled equations of motion for meson fields (sigma, omega, rho and kaon), nucleon fields, the electron field and the Poisson equation for the electric field. Within this framework we can satisfy Gibbs conditions in a proper way. The numerical solution is obtained by employing the Wigner-Seitz cell approximation and a relaxation method. Our framework and its ability to reproduce properties of nuclear matter and finite nuclei have been reported by Maruyama et al. [5]. However, only uniform configurations were used as the starting ones for the relaxation procedure and some of the final density profiles proved to be trapped in meta-stable states. By trying different starting density profiles we have improved our framework to avoid missing the ground states.
188c
T. Maruyama et al. / Nuclear Physics A 749 (2005) 186c–189c
200
100
0
single phase Gibbs Maxwell
droplet rod slab tube bubble uniform 1100 1200 µ B [MeV]
1300
‘‘No Coulomb’’ µ e [MeV]
µ e [MeV]
Full calculation 200
100
0
1100
1200 µ B [MeV]
1300
Figure 3. Phase diagram in the µB -µe plane, where µB and µe denote baryon- and chargechemical potentials, respectively. Left: full calculation. Right: the electric potential is discarded in determining the matter structure. Curves calculated using Gibbs conditions disregarding finite size effects and the Coulomb interaction effect (“Gibbs”) and that for the Maxwell construction (“Maxwell”) are also presented for comparison.
We calculate the structure of the neutron star matter at zero temperature for a given average baryon density. For each baryon density, the cell radius and the geometrical dimension (indicated in Fig. 1 by “3D”, “2D”, “1D”) are chosen to satisfy the minimum of the energy. If Glendenning’s claim were correct, the structured mixed phase would develop in a broad density range from well below to well above the critical density determined from the Maxwell construction. In this density interval the matter should exhibit the structure change similar to the nuclear “pasta” phases [5]: the kaonic droplet, rod, slab, tube, bubble. Actually we observe such structures in our calculation (see Fig. 1). The horizontal axis is the distance from the cell center and the hatch shows the cell boundary. From the top of the figure, the matter structures correspond to droplet, rod, slab, tube, and bubble of kaon and proton. The neutron distribution proves to be rather flat. In the upper panel of Fig. 2 we plot the energy per nucleon of the matter. Pieces of solid curves indicate the energetically favored structures, while the dotted curve shows the case for the uniform matter. One can see the softening of matter by the appearance of “pasta” structures. In the lower panel plotted are the structure size (kaon-matter radius or kaonhole radius) and the cell size. The dashed lines show the Debye screening lengths of the (e) (p) (e) electron and the proton, λD and λD , respectively. In most cases λD is less than the cell size Rcell but it is larger than the size of the structure R. However the proton Debye (p) length λD is always shorter than Rcell and R. When the minimal value of the Debye length inside the structure is shorter or of the order of R, the charge screening effects should be pronounced [2]. To demonstrate the charge screening effect on the kaonic mixed phase, we show in Fig. 3 the phase diagrams in the µB -µe plane. Left panel exhibits the full calculation. In the right panel we show the case, when the electric potential is discarded in determining the density profile (“No Coulomb”) and the Coulomb energy, using the density profile thus determined, is then added to the total energy. We see that in the “No Coulomb” case
T. Maruyama et al. / Nuclear Physics A 749 (2005) 186c–189c
ρ[fm −3]
ρ[fm −3]
ρ[fm −3]
Full calculation
3D, ρ B=0.45 [fm −3]
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p n e K
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189c
0.0
1D, ρ B=0.60 [fm −3]
0
5
10 r [fm]
Figure 4. Comparison of density profiles of kaonic matter calculated with and without the Coulomb interaction. The cell size is not optimized in this comparison.
15
the pieces of solid curves are situated between the curves calculated for the case when the Gibbs conditions are applied disregarding finite size effects and the Coulomb interaction effect (indicated by “Gibbs”) and that one given by the Maxwell construction (indicated by “Maxwell”). The “Full calculation” case is more similar to the one given by the Maxwell construction. As follows from the density profiles shown in Fig. 4, the local charge neutrality is more pronounced in the case of the “Full calculation” (smaller difference of kaon and proton densities). These results suggest that the Maxwell construction is effectively meaningful owing to the charge screening effects. 3. Summary and Concluding Remarks We have discussed how the geometrical structure manifests in the kaon condensation FOPT. Using a self-consistent framework based on DFT and RMF, we have taken into account the Coulomb interaction in a proper way to demonstrate how the Coulomb interaction changes the phase diagram. We see that changes are more remarkable in the case of the kaon condensation than for the “nuclear pastas” at sub-nuclear densities [5]. The density range of the structured mixed phase is largely limited and thereby the phase diagram becomes similar to that given by the Maxwell construction. Although the importance of such a treatment has been demonstrated for the hadron-quark matter transition [2], one of our new findings here is that we can figure out the role of the charge screening effect without introducing an “artificial” input of the surface tension. REFERENCES 1. N. K. Glendenning, Phys. Rev. D46 (1992) 1274. 2. D. N. Voskresensky, M. Yasuhira and T. Tatsumi, Phys. Lett. B541 (2002) 93; Nucl. Phys. A723 (2003) 291. 3. T. Norsen and S. Reddy, Phys. Rev. C63 (2001) 065804. 4. Density Functional Theory, ed. E. K. U. Gross and R. M. Dreizler, Plenum Press (1995). 5. T. Maruyama et al., nucl-th/0311076.
Structured mixed phase at charged kaon condensation Toshiki Maruyamaa , Toshitaka Tatsumib , Dmitri N. Voskresenskyc , Tomonori Tanigawada , and Satoshi Chibaa a
Advanced Science Research Center, Japan Atomic Energy Research Institute, Tokai, Ibaraki 319-1195, Japan
b
Department of Physics, Kyoto University, Kyoto 606-8502, Japan
c
Moscow Institute for Physics and Engineering, Kashirskoe sh. 31, Moscow 115409, Russia d
Japan Society for the Promotion of Science, Tokyo 102-8471, Japan
Non-uniform structures of the mixed phase at the first-order phase transition to the charged kaon condensation are studied using the density functional theory within the relativistic mean-field model. Including the electric field effects and applying Gibbs conditions in a proper way, we numerically determine density profiles of nucleons, electrons and condensed kaons. The importance of the charge screening effects is elucidated and thereby we show that the Maxwell construction is effectively justified. 1. Introduction Considering phase transitions in nuclear matter, various scenarios have been discussed, like e.g. liquid-gas transition, various meson condensations, hadron-quark deconfinement transition, etc. In most cases these are first-order phase transitions (FOPT). In the systems with different charged species undergoing the FOPT the structured mixed phases might appear [1]. For example in case of the nuclear matter below the saturation density, where the liquid-gas FOPT is relevant, the “nuclear pasta” structures have been studied by many authors. The physical reason of the possibility of the mixed phases with charged structures is that the charge neutrality may hold only globally rather than locally, as in the Maxwell construction. Mechanically, the balance between the Coulomb force and the surface tension is responsible for the occurrence of the spatially non-uniform structures. There existed a naive view that not all Gibbs conditions can be satisfied in the description of the Maxwell construction [1]. However, it has been demonstrated in recent papers [2] that Gibbs conditions are actually fulfilled also in case of the Maxwell construction, if one properly includes the electric field effects. As an explicit example the hadron-quark deconfinement transition has been considered; the density region of the structured mixed phase might be largely limited as a result. Taking charged kaon condensation as a further example, we study in this paper whether the “pasta” structures appear in the high-density neutron-star matter. The appearance
0375-9474/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2004.12.031
187c
−3 1.0 3D, ρ B=0.45 [fm ]
0.5
p n e K
160 E/A−m B [MeV]
0.0 −3 1.0 2D, ρ B=0.55 [fm ] 0.5
0.0 −3 1.0 1D, ρ B=0.60 [fm ] 0.5
U K=−130 MeV
140
120
droplet rod slab tube bubble uniform single phase
100
80
0.0 −3 1.0 2D, ρ B=0.70 [fm ]
R, R cell [fm]
ρ[fm −3]
ρ[fm −3]
ρ[fm −3]
ρ[fm −3]
ρ[fm −3]
T. Maruyama et al. / Nuclear Physics A 749 (2005) 186c–189c
0.5
0.0 −3 1.0 3D, ρ B=0.72 [fm ] 0.5
0.0
10
0 0
10 r [fm]
20
Figure 1. Density profiles of kaonic structures. A parameter Uk , the kaon optical potential at normal nuclear density, is set to be 130 MeV.
λ (e) D λ (p) D
20
0.4
0.5
0.6 0.7 ρ B [fm −3]
0.8
Figure 2. Top: binding energy per nucleon of nuclear matter in beta equilibrium. Bottom: structure size R (thick curves below) and the cell size Rcell (thick curves above).
of the “pasta” structures could have important consequences for the various neutron star phenomena. It causes a drastic change of the neutrino opacity at an early stage of the neutron star evolution and influences the subsequent neutron star cooling, it affects the matter resistance to the stress and consequently the glitch phenomena, etc. There are few works about this matter so far, and the finite-size effect and the Coulomb interaction effect are still not elucidated [3]. We demonstrate how the charge screening affects the structured mixed phase. 2. Kaon Condensation in High-density Matter We use the density functional theory (DFT) within the relativistic mean field (RMF) model [4]. We get the coupled equations of motion for meson fields (sigma, omega, rho and kaon), nucleon fields, the electron field and the Poisson equation for the electric field. Within this framework we can satisfy Gibbs conditions in a proper way. The numerical solution is obtained by employing the Wigner-Seitz cell approximation and a relaxation method. Our framework and its ability to reproduce properties of nuclear matter and finite nuclei have been reported by Maruyama et al. [5]. However, only uniform configurations were used as the starting ones for the relaxation procedure and some of the final density profiles proved to be trapped in meta-stable states. By trying different starting density profiles we have improved our framework to avoid missing the ground states.
188c
T. Maruyama et al. / Nuclear Physics A 749 (2005) 186c–189c
200
100
0
single phase Gibbs Maxwell
droplet rod slab tube bubble uniform 1100 1200 µ B [MeV]
1300
‘‘No Coulomb’’ µ e [MeV]
µ e [MeV]
Full calculation 200
100
0
1100
1200 µ B [MeV]
1300
Figure 3. Phase diagram in the µB -µe plane, where µB and µe denote baryon- and chargechemical potentials, respectively. Left: full calculation. Right: the electric potential is discarded in determining the matter structure. Curves calculated using Gibbs conditions disregarding finite size effects and the Coulomb interaction effect (“Gibbs”) and that for the Maxwell construction (“Maxwell”) are also presented for comparison.
We calculate the structure of the neutron star matter at zero temperature for a given average baryon density. For each baryon density, the cell radius and the geometrical dimension (indicated in Fig. 1 by “3D”, “2D”, “1D”) are chosen to satisfy the minimum of the energy. If Glendenning’s claim were correct, the structured mixed phase would develop in a broad density range from well below to well above the critical density determined from the Maxwell construction. In this density interval the matter should exhibit the structure change similar to the nuclear “pasta” phases [5]: the kaonic droplet, rod, slab, tube, bubble. Actually we observe such structures in our calculation (see Fig. 1). The horizontal axis is the distance from the cell center and the hatch shows the cell boundary. From the top of the figure, the matter structures correspond to droplet, rod, slab, tube, and bubble of kaon and proton. The neutron distribution proves to be rather flat. In the upper panel of Fig. 2 we plot the energy per nucleon of the matter. Pieces of solid curves indicate the energetically favored structures, while the dotted curve shows the case for the uniform matter. One can see the softening of matter by the appearance of “pasta” structures. In the lower panel plotted are the structure size (kaon-matter radius or kaonhole radius) and the cell size. The dashed lines show the Debye screening lengths of the (e) (p) (e) electron and the proton, λD and λD , respectively. In most cases λD is less than the cell size Rcell but it is larger than the size of the structure R. However the proton Debye (p) length λD is always shorter than Rcell and R. When the minimal value of the Debye length inside the structure is shorter or of the order of R, the charge screening effects should be pronounced [2]. To demonstrate the charge screening effect on the kaonic mixed phase, we show in Fig. 3 the phase diagrams in the µB -µe plane. Left panel exhibits the full calculation. In the right panel we show the case, when the electric potential is discarded in determining the density profile (“No Coulomb”) and the Coulomb energy, using the density profile thus determined, is then added to the total energy. We see that in the “No Coulomb” case
T. Maruyama et al. / Nuclear Physics A 749 (2005) 186c–189c
ρ[fm −3]
ρ[fm −3]
ρ[fm −3]
Full calculation
3D, ρ B=0.45 [fm −3]
1.5
p n e K
1.0
0.5
0.0 1.5
0.5
2D, ρ B=0.55 [fm ]
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5
10 r [fm]
Figure 4. Comparison of density profiles of kaonic matter calculated with and without the Coulomb interaction. The cell size is not optimized in this comparison.
15
the pieces of solid curves are situated between the curves calculated for the case when the Gibbs conditions are applied disregarding finite size effects and the Coulomb interaction effect (indicated by “Gibbs”) and that one given by the Maxwell construction (indicated by “Maxwell”). The “Full calculation” case is more similar to the one given by the Maxwell construction. As follows from the density profiles shown in Fig. 4, the local charge neutrality is more pronounced in the case of the “Full calculation” (smaller difference of kaon and proton densities). These results suggest that the Maxwell construction is effectively meaningful owing to the charge screening effects. 3. Summary and Concluding Remarks We have discussed how the geometrical structure manifests in the kaon condensation FOPT. Using a self-consistent framework based on DFT and RMF, we have taken into account the Coulomb interaction in a proper way to demonstrate how the Coulomb interaction changes the phase diagram. We see that changes are more remarkable in the case of the kaon condensation than for the “nuclear pastas” at sub-nuclear densities [5]. The density range of the structured mixed phase is largely limited and thereby the phase diagram becomes similar to that given by the Maxwell construction. Although the importance of such a treatment has been demonstrated for the hadron-quark matter transition [2], one of our new findings here is that we can figure out the role of the charge screening effect without introducing an “artificial” input of the surface tension. REFERENCES 1. N. K. Glendenning, Phys. Rev. D46 (1992) 1274. 2. D. N. Voskresensky, M. Yasuhira and T. Tatsumi, Phys. Lett. B541 (2002) 93; Nucl. Phys. A723 (2003) 291. 3. T. Norsen and S. Reddy, Phys. Rev. C63 (2001) 065804. 4. Density Functional Theory, ed. E. K. U. Gross and R. M. Dreizler, Plenum Press (1995). 5. T. Maruyama et al., nucl-th/0311076.