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Rigid strange lattices in Proto-Neutron Stars Juergen J. Zach Ohio State/UCSD 13 May, 2002 INT Nucleosynthesis workshop

Rigid strange lattices in Proto-Neutron Stars Juergen J. Zach Ohio State/UCSD 13 May, 2002 INT Nucleosynthesis workshop

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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.

Liquid Lattice – Upper Layers

Crystallized Lattice – Upper Layers

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.