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Rigid strange lattices in Proto-Neutron Stars
Juergen J. Zach
Ohio State/UCSD
13 May, 2002
INT Nucleosynthesis workshop
Physical Environment: Core Collapse Supernovae
• Protoneutron star: core at late stages of Kelvin-Helmholtz cooling phase
• Times: t3s after bounce mostly deleptonized
• Densities: C~2-5nuclear
• Temperature: T~10MeV conditions for which many
studies (e.g. Heiselberg et al. 2000) indicate a 1st-order phase transition to matter with macroscopic strangeness
Deconfined quark matter (Pons et al. 2001), formation of strange quarks through weak reactions, e.g.:
Forms of macroscopically strange matter
(-, --, … ) Hyperons (Balberg et al. 1999)
Bose-Einstein condensate of mesons (K-) (Pons et al. 2000)
duus
dlu
slu
l
l
mesonep me
*n :condition threshold
e
e
en
ep
/ vs. Kpn
lpn l
nnn density thresholddetermines
...2
... :equ. n
pn
sm*d :threshold
Formation of a Phase Transition Lattice
Gibbs conditions in mixed phase determine strange phase fraction :
B,s=B,N ; e,s=e,N;
Ps = PN ; Ts = TN ; global charge neutrality (Glendenning 1992). Geometrical structure determined
by minimum of E=Ecoulomb+Esurface+Ecurvature+…
(Heiselberg et al. 1993, Glendenning 2001)
Spherical droplets of minority phase at =0/1.
Rods, platelets at ~0.5.
Properties of the Phase Transition Lattice at Finite Temperature
Solution of the whole lattice: equivalent to general problem of solid state physics from first principles
intractable OCP-model for the limit of small droplet sizes: - structure-less point charges - uniform charge distribution (no screening)
Solving the OCP Model
Displacement:
1993)(Chabrier ))(4
1(2
31122
2
D
Mdd
u
plasmadroplet
1~/ TkBplasma
222 / du
442
23
10092.205.01
1031.4096.0249.0)(
(Debye))/6( ; )( :Dispersion
))1/((/1)(
)/1/( :with
3/1cellunit
2plasma
0
1
2/1 0
Vkk
kk
etdtxxD
VMeZ
DD
xt
cellunitdropletdropletplasma
Degeneracy parameter:
Lindemann parameter : Monte Carlo simulations (Stringfellow 1990, Ceperley 1980):
intermediate range between classical and quantum limits
Solid-liquid coexistence curve:
Melting Curves – Charge Dependence
M=0.4fm-3; R=3fm;
protoneutronstar cools through melting temperature during Kelvin-Helmholtz cooling phase
3/8.0 fmeC
3/7.0 fmeC
3/1.0 fmeC
3/6.0 fmeC
Melting Curves – Size Dependence
C=0.4fm-3; R=3fm;
no crystallization below Rdroplet ~ 1fm
lattice crystallizes first for deeper layers
Limits of OCP Model
• Deformation (“wobble”) modes:
;1052
9MeV
M dropletS
;13
;10
;5
fm
fm
fm
e
N
quarkK
freeze-out around lattice crystallization for small droplets
• Screening effects; Debye lengths (Heiselberg et al. 1993, Norsen et al. 2001):
;]5
[2
1dM(r)v
2
1
]2
9[
2
1
222
2
dRM
dRSE
Sdroplet
S
Mechanical Stability of the Crystallized Lattice
• Shear constant of bcc - Coulomb lattice:
)energy lattice : Wangle, distortion :( lxy2
2
44 xy
l
d
Wdc
);/0.1)((477.4 344 afmRMeVc C
;10~)(1
energy);on (deformati )(2
33
344
max
244
fm
MeV
a
c
dx
dU
A
a
xcU
crit
• Obtain Wl with Ewald’s method (Ewald 1921).
• Critical shear stress:
Lattice Crystallization and Hydrodynamics
• Lepton number gradient dominant driving force of convection (Epstein 1979) at late stages of PNS evolution:
criterion)(Ledoux 0)()( ,, dr
dY
Ydr
dS
SC e
SPe
YPL e
critrot
CC V
E
s
m
km
r ~10)(1
~v 6
convection and differential rotation can prevent crystallization convection can break up lattice formed during transient quiet period
• Differential rotation (Goussard et al. 1998); min. period ~1ms
critconv
fm
MeV
V
E
s
m ~10~ 1996) al.et (Keil 10~v3
36C
Possible effects on neutrino transport ~3-20sec. post-bounce?
• Reddy et al. 2000: coherent scattering off strange droplets increase in -opacity of mixed phase by 1-2 orders of magnitude
Knee in -luminosity after 1st-order phase transition?
Rearrangements of solid lattice during PNS evolution irregularities in -emission?
Localized fractures of lattice by convection asymmetric -transport?
Work in progress - other observational signatures?
• Gravity wave signature of anisotropic neutrino transport pattern detectable for Galactic SN.
• “Settling” of lattice defects might cause some pulsar glitches.
• Interaction with magnetic field in PNS?• Phase transition lattice might be responsible
for non-spherical features in core collapse supernovae?
Conclusions:
• Crystallization of the lattice formed during a first order phase transition in protoneutronstars possible for temperatures T~1-10MeV.
• Deformation modes of the lattice droplets freeze out around the same temperature.
• Critical shear stress ~10-3MeV/fm3 complex interaction between lattice crystallization and hydrodynamics (convection and differential rotation).
• Solid lattice could lead to spatial anisotropies and temporal irregularities in -transport.
References:
• (Heiselberg et al. 2000): H. Heiselberg, M. Hjorth-Jensen, Phys. Rep. 328 (2000) 237-327.
• (Glendenning 1992): N.K. Glendenning, Phys. Rev. D 46 (1992) 1274.
• (Heiselberg et al. 1993): H. Heiselberg, C.J. Pethick, E.F. Staubo, Phys. Rev. Lett. 70 (1993) 1355.
• (Glendenning 2001): N.K. Glendenning, Phys. Rep. 342 (2001) 393-447.
• (Pons et al. 2001): J.A. Pons, A.W. Steiner, M. Prakash, J.M. Lattimer, Phys. Rev. Lett. 86 (2001) 5223-5226.
• (Pons et al. 2000): J.A. Pons, S. Reddy, P.J. Ellis, M. Prakash, J.M. Lattimer, Phys. Rev. C 62 (2000) 035803.
• (Balberg et al. 1999): S. Balberg, I. Lichtenstadt, G.B. Cook, ApJS. 121 (1999) 515-531.
• (Chabrier 1993): G. Chabrier, ApJ. 414 (1993) 695-700.
• (Stringfellow 1990): G.S. Stringfellow, H.E. DeWitt, Phys. Rev. A 41 (1990) 1105.
• (Ceperley 1980): D.M. Ceperley, B.J. Alder, Phys. Rev. Lett. 45 (1980) 566.
• (Norsen et al. 2001): T. Norsen, S. Reddy, Phys. Rev. C 63 (2001) 065804.
• (Ewald 1921): P.P. Ewald, Ann. Phys. 64 (1921) 253-287.
• (Epstein 1979): R.I. Epstein, Mon. Not. R. Astr. Soc. 188 (1979) 305-325.
• (Goussard et al. 1998): J.O. Goussard, P. Haensel, J.L. Zdunik, Astron. Astrophys. 330 (1997) 1005-1016.
• (Reddy et al. 2000): S. Reddy, G. Bertsch, M. Prakash, Phys. Lett. B 475 (2000) 1-8.