Superfluidity in neutron-star crusts

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1 Superfluidity in neutron-star crusts Nicolas Chamel Institut d Astronomie et d Astrophysique Université Libre de Bruxelles, Belgique INT, Seattle, May 2011

2 Why studying superfluidity in neutron stars? Neutron stars are by nature quantum systems: they contain highly degenerate matter which can therefore exhibit various phenomena observed in condensed matter physics like superfluidity. Superfluidity affects the evolution of neutron stars: pulsar glitches, pulsations, precession, cooling, magnetic field... Cassiopeia A (NASA)

3 Are electrons in neutron stars superconducting? The surface layers of non-accreting neutron stars are mostly composed of ordinary iron which is not superconducting. It was found in 2001 (Shimizu et al., Nature 412, 316) that iron under pressure can become superconducting at densities ρ 8.2 g.cm 3 with T ce 2 K. But T ce is much smaller than the temperature in neutron stars. Yakovlev et al, proceedings (2007), arxiv:

4 Are electrons in neutron stars superconducting? In the deeper layers of neutron stars at densities ρ 10 4 gcm 3, atoms are fully ionised by the pressure. The critical temperature of a uniform non-relativistic electron gas (jelium) is given by (T pi is the plasma temperature) ( T ce = T pi exp 8 v Fe /πe 2) T ce exp( ζ(ρ/ρ ord ) 1/3 ) with ρ ord = m u /(4πa 3 0 /3). At densities above 106 g.cm 3, electrons become relativistic v Fe c so that (α = e 2 / c 1/137) T ce = T pi exp( 8/πα) 0 Ginzburg, J. Stat. Phys. 1(1969),3. Electrons in neutron stars are not superconducting.

5 Nuclear superfluidity in neutron stars The BCS theory was applied to nuclei by Bohr, Mottelson, Pines and Belyaev Phys. Rev. 110, 936 (1958). Mat.-Fys. Medd. K. Dan. Vid. Selsk. 31, 1 (1959). N.N. Bogoliubov, who developed a microscopic theory of superfluidity and superconductivity, was the first to explore its application to nuclear matter. Dokl. Ak. nauk SSSR 119, 52 (1958). Superfluidity in neutron stars was suggested long ago (before the actual discovery of neutron stars) by Migdal in It was first studied by Ginzburg and Kirzhnits in Ginzburg and Kirzhnits, Zh. Eksp. Teor. Fiz. 47, 2006, (1964).

6 t Superfluidity and superconductivity in neutron stars In spite of their names, neutron stars are not only made of neutrons! As a consequence, they could contain various kinds of superfluids and superconductors. hyperon star strange star quark hybrid star s u p e r c o color superconducting strange quark matter (u,d,s quarks) 2SC 2SC+s Σ,Λ,Ξ, n,p,e, µ n d u,d,s quarks 2SC CFL H CFL CFL K + CFL K 0 CFL π 0 N+e N+e+n u c t K i n,p,e, µ n π g p r o n o s traditional neutron star n superfluid crust nucleon star neutron star with pion condensate Fe g/cm 3 Hydrogen/He atmosphere g/cm 3 g/cm 3 picture from F. Weber R ~ 10 km Neutron stars are expected to contain at least a neutron superfluid in their crust.

7 (MeV) Superfluidity in neutron-star crusts Most microscopic calculations have been performed in uniform neutron matter. k F [fm -1 ] BCS 2.5 Chen 93 Wambach 93 Schulze 96 2 Schwenk 03 Fabrocini 05 Cao dqmc 09 AFDMC 08 QMC AV k F a Microscopic calculations using different methods predict different density dependence of the 1 S 0 pairing gaps. Gezerlis & Carlson, Phys. Rev. C 81, (2010). Is the neutron superfluid in the crust really uniform? What is the effect of the nuclei?

8 Nuclear energy density functional theory in a nut shell The nuclear energy density functional theory allows for a tractable and consistent treatment of nuclear matter, atomic nuclei and neutron-star crusts. The energy of a lump of matter is expressed as (q = n, p) [ ] E = E ρ q (r), ρ q (r),τ q (r), J q (r), ρ q (r) d 3 r where ρ q (r), τ q (r)... are functionals of ϕ (q) 1k (r) and ϕ(q) 2k (r) ( ) ( ) hq (r) λ q q (r) ϕ (q) 1k (r) q (r) h q (r)+λ q ϕ (q) 2k (r) = E (q) k ( ϕ (q) 1k (r) ϕ (q) 2k (r) ) h q δe δτ q + δe δρ q i δe δj q σ, q δe δ ρ q

9 Effective nuclear energy density functional In principle, one can construct the nuclear functional from realistic nucleon-nucleon forces (i.e. fitted to experimental nucleon-nucleon phase shifts) using many-body methods E = 2 2M (τ n +τ p )+A(ρ n,ρ p )+B(ρ n,ρ p )τ n + B(ρ p,ρ n )τ p +C(ρ n,ρ p )( ρ n ) 2 +C(ρ p,ρ n )( ρ p ) 2 +D(ρ n,ρ p )( ρ n ) ( ρ p ) + Coulomb, spin-orbit and pairing Drut et al.,prog.part.nucl.phys.64(2010)120. But difficult task so in practice, we use phenomenological (Skyrme) functionals Bender et al.,rev.mod.phys.75, 121 (2003).

10 Phenomenological corrections for atomic nuclei For atomic nuclei, we add the following corrections: E corr = E W + E coll Wigner energy E W = V W exp { λ ( N Z A ) 2 { ( ) 2 } }+V W N Z exp A rotational and vibrational spurious collective energy { } E coll = Erot crank b tanh(c β 2 )+d β 2 exp{ l( β 2 β2 0 )2 } A 0

11 Experimental data: Construction of the functional 2149 measured nuclear masses with Z, N 8 compressibility 230 K v 250 MeV charge radius of 208 Pb, R c = 5.501±0.001 fm N-body calculations with realistic forces: isoscalar effective mass M s/m = 0.8 equation of state of pure neutron matter 1 S 0 pairing gaps in symmetric and neutron matter Landau parameters (stability against spurious instabilities) Chamel, Goriely, Pearson, Phys.Rev.C80, (2009). Chamel&Goriely, Phys.Rev.C82, (2010) With these constraints, the functional is well suited for describing neutron-star crusts.

12 Empirical pairing energy density functionals E pair = 1 4 v π q [ρ n,ρ p ] = V Λ πq q=n,p v πq [ρ n,ρ p ] ρ 2 q ( ( ) ρn +ρ αq ) p 1 η q ρ 0 Drawbacks not enough flexibility to fit realistic pairing gaps in infinite nuclear matter and in finite nuclei ( isospin dependence) the global fit to nuclear masses would be computationally very expensive

13 Microscopically deduced pairing functional Assumptions: v π q [ρ n,ρ p ] = v π q [ρ q ] depends only on ρ q Duguet, Phys. Rev. C 69 (2004) isospin charge symmetry v π n = v π p = v π v π [ρ q ] is the locally the same as in infinite nuclear matter with density ρ q v π [ρ q ] = v π [ q (ρ q )] constructed so as to reproduce exactly a given pairing gap q (ρ q ) in infinite homogeneous matter by solving directly the HFB equations Chamel, Goriely, Pearson, Nucl. Phys.A812,72 (2008).

14 Pairing in nuclei and in nuclear matter Inverting the HFB equations yields v π [ρ q ] = 8π 2 ( 2 2M q ) 3/2 µq+ε Λ 0 ε dε (ε µ q ) 2 + q (ρ q ) 2 1 µ q = 2 2Mq (3π 2 ρ q ) 2/3 Cutoff prescription: s.p. energy cutoff ε Λ above the Fermi level This procedure provides a one-to-one correspondence between the pairing strength in finite nuclei and the 1 S 0 pairing gap in infinite nuclear matter.

15 Analytical expression of the pairing strength In the weak-coupling approximation q µ q and q ε Λ ( v π [ρ q ] = 8π2 2 I q (ρ q ) 2Mq(ρ q ) I q = ( ) 2µq µ q [2 log +Λ q Λ(x) = log(16x)+2 1+x 2 log Chamel, Phys. Rev. C 82, (2010) ) 3/2 ( ελ µ q )] ( 1+ ) 1+x 4 exact fit of the given gap function q (ρ q ) no free parameters automatic renormalization of the pairing strength with ε Λ

16 Pairing gaps from contact interactions The weak-coupling approximation can also be used to determine the pairing gap of a Fermi gas interacting with a contact force ( ) 2 = 2µ exp g(µ)vreg π µ is the chemical potential, g(µ) is the density of states and v π Λ is a regularized interaction 1 v π reg = 1 v π + 1 v π Λ v π Λ = 4 g(µ)λ(ε Λ /µ)

17 Accuracy of the weak-coupling approximation This approximation remains very accurate at low densities because the s.p. density of states is not replaced by a constant as usually done. symmetric nuclear matter 0 0 neutron matter v πn [MeV fm 3 ] numerical integration weak coupling approx. (one term) weak coupling approx. (all terms) ρ [fm -3 ] v πn [MeV fm 3 ] numerical integration weak coupling approx. (one term) weak coupling approx. (all terms) ρ [fm -3 ] Chamel, Phys. Rev. C 82, (2010)

18 Pairing in dilute neutron matter At very low densities, the pairing gap is given by n = ( ) 2 7/3 ( ) π µ n exp e 2k F a nn Gorkov&Melik-Barkhudarov, Sov. Phys. JETP, 13, 1018, (1961). /E FG GFMC BCS Gorkov /(ak F ) -E mol /2 :cosh -E mol /2 :δ Chang et al. Phys.Rev.A70, (2004). ) 3/2 ( v π [ρ n ] = 8π2 2 I n (ρ n ) 2Mn(ρ n ) I n = [ 14 µ n 3 8 ( ) π log 2 +Λ 3 k F a nn ( ελ µ n )]

19 Pairing cutoff and experimental phase shifts In the limit of vanishing density, the pairing strength ( ) v π [ρ q 0] = 4π2 2 3/2 ελ 2M q should coincide with the bare force in the 1 S 0 channel. A fit to the experimental 1 S 0 NN phase shifts yields ε Λ 7 8 MeV. Esbensen et al., Phys. Rev. C 56, 3054 (1997). rms [kev] s.p. cutoff [MeV] On the other hand, a better mass fit can be obtained with ε Λ 16 MeV while convergence is achieved for ε Λ 40 MeV. Goriely et al., Nucl.Phys.A773(2006),279.

20 Choice of the pairing gap Fit the 1 S 0 pairing gap obtained with realistic NN potentials at the BCS level (no medium effects) 4 3 n [MeV] 2 1 Pairing gap obtained with Argonne V 14 potential ρ n [fm -3 ] n (ρ n ) essentially independent of the NN potential n (ρ n ) completely determined by experimental 1 S 0 nn phase shifts Dean&Hjorth-Jensen,Rev.Mod.Phys.75(2003)607.

21 Neutron vs proton pairing Because of possible charge symmetry breaking effects, proton and neutron pairing strengths may not be equal v π n [ρ] v π p [ρ] The neglect of polarization effects in odd nuclei (equal filling approximation) is corrected by staggered pairing we introduce renormalization factors f ± q (f + n 1 by definition) v π n [ρ n ] = f ± n v π [ρ n ] v π p [ρ p ] = f ± p v π [ρ p ]

22 Neutron vs proton pairing What comes out of the global mass fit? f n fn 1.06 f p fp 1.05 neutron and proton pairing strengths are effectively equal fn /f n + fp /f p + the pairing strength is larger for odd nuclei fq > f q + N=90 This is in agreement with a recent analysis by Bertsch et al. Phys.Rev.C79(2009), (3) o (MeV) V T-odd =0 N=100

23 1 S 0 pairing gap in neutron matter This new mass model yields a much more realistic gap than our previous mass models! 5 4 n [MeV] HFB-16 HFB ,02 0,04 0,06 0,08 0,1 0,12 0,14 0,16 0,18 0,2 ρ n [fm -3 ]

24 Neutron-matter equation of state at subsaturation densities FP (1981) BSk16 Akmal et al. (1998) E/A [MeV] ,02 0,04 0,06 0,08 0,1 0,12 0,14 0,16 ρ [fm -3 ]

25 Dilute neutron-matter equation of state E / E FG k F [fm -1 ] Lee-Yang Friedman-Pandharipande APR GFMC v6, v8 Baldo-Maieron Schwenk-Pethick χeft NLO DBHF Hebeler-Schwenk AFDMC QMC AV k F a Gezerlis & Carlson, Phys. Rev. C 81, (2010).

26 Neutron-matter equation of state at high densities E/A [MeV] FP (1981) BSk16 Akmal et al. (1998) ,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 ρ [fm -3 ]

27 Constraints from heavy-ion collisions Our functional BSk16 is consistent with the pressure of symmetric nuclear matter inferred from Au+Au collisions SLy4 BSk16 P [MeV fm -3 ] ρ/ρ 0 Danielewicz et al., Science 298, 1592 (2002).

28 HFB-16 mass table Results of the fit on the 2149 measured masses with Z, N 8 HFB-16 HFB-15 HFB-14 HFB-8 σ(m) [MeV] ǫ(m) [MeV] σ(m nr ) [MeV] ǫ(m nr ) [MeV] σ(s n ) [MeV] ǫ(s n ) [MeV] σ(q β ) [MeV] ǫ(q β ) [MeV] σ(r c ) [fm] ǫ(r c ) [fm] θ( 208 Pb) [fm]

29 HFB-17 mass model: microscopic pairing gaps including medium polarization effects Fit the 1 S 0 pairing gaps of both neutron matter and symmetric nuclear matter obtained from Brueckner calculations taking into account medium polarization effects Neutron matter Symmetric nuclear matter [MeV] ρ [fm -3 ] [MeV] Cao et al.,phys.rev.c74,064301(2006) ρ [fm -3 ]

30 New expression of the pairing strength the pairing strength is allowed to depend on both ρ n and ρ p v π q [ρ n,ρ p ] = v π q [ q (ρ n,ρ p )] q (ρ n,ρ p ) is interpolated between that of symmetric matter (SM) and pure neutron matter (NM) q (ρ n,ρ p ) = SM (ρ)(1 η )± NM (ρ q )η ρ q ρ η = ρ n ρ p ρ n +ρ p M q = M to be consistent with the neglect of self-energy effects on the gap Goriely, Chamel, Pearson, PRL102, (2009). Goriely, Chamel, Pearson, Eur.Phys.J.A42(2009),547.

31 Density dependence of the pairing strength v πn η= η= ρ [fm -3 ]

32 Isospin dependence of the pairing strength 800 v πn η

33 HFB-17 mass table Results of the fit on the 2149 measured masses with Z, N 8 HFB-16 HFB-17 σ(2149 M) ǫ(2149 M) σ(m nr ) ǫ(m nr ) σ(s n ) ǫ(s n ) σ(q β ) ǫ(q β ) σ(r c ) ǫ(r c ) θ( 208 Pb)

34 HFB-17 mass predictions Differences between experimental and calculated masses as a function of the neutron number N for the HFB-17 mass model.

35 Predictions to newly measured atomic masses HFB mass models were fitted to the 2003 Atomic Mass Evaluation. The predictions of these models are in good agreement with new mass measurements HFB-16 HFB-17 σ(434 M) ǫ(434 M) σ(142 M) ǫ(142 M) Litvinov et al., Nucl.Phys.A756, 3(2005) gs_masses.txt

36 Nuclear masses: HFB-16 vs HFB-17 Differences between the HFB-16 and HFB-17 mass predictions as a function N for all 8 Z 110 nuclei lying between the proton and neutron drip lines.

37 Superfluidity in neutron-star crusts with the Wigner-Seitz approximation The HFB equations in neutron-star crusts have been already solved by several groups using the W-S approximation The effects of the clusters are found to be dramatic at high densities ( 0.03 nucleons per fm 3 ), in some cases the pairing gaps are almost completely suppressed Baldo et al., Eur.Phys.J. A 32, 97(2007).

38 Problems Limitations of the W-S approximation the results of HFB calculations depend on the boundary conditions which are not unique the nucleon densities and pairing fields exhibit spurious fluctuations due to box-size effects 2.5 k F =1.1 fm -1 n, MeV 2.0 Z= Z= r, fm Baldo et al., Eur.Phys.J. A 32, 97(2007). Spurious shell effects 1/R 2 are very large in the bottom layers of the crust and are enhanced by the self-consistency of the calculations.

39 Solution Nuclear band theory Go beyond the W-S app. by using the band theory of solids Chamel et al., Phys.Rev.C75(2007) The band theory takes consistently into account both nuclear clusters and free neutrons ϕ αk (r) = e i k r u αk (r) u αk (r + T) = u αk (r) α rotational symmetry around the lattice sites k translational symmetry of the crystal

40 Anisotropic multi-band neutron superfluidity In the decoupling approximation, the Hartree-Fock-Bogoliubov equations reduce to the BCS equations αk = 1 2 β k v pair βk αkα kβk k β k k E k βk k k tanh Eβk k 2T v pair = αkα kβk k β k k d 3 r v π [ρ n (r),ρ p (r)] ϕ αk (r) 2 ϕ βk k (r) 2 E αk = (ε αk µ) αk ε αk, µ and ϕ αk (r) are obtained from band structure calculations Chamel et al., Phys.Rev.C81, (2010).

41 Validity of the decoupling approximation The decoupling approximation means that d 3 r ϕ αk (r) (r)ϕ βk (r) δ αβ d 3 r ϕ αk (r) 2 (r) This approximation is justified whenever (r) varies slowly as compared to ϕ αk (r) for those states in the vicinity the Fermi level. bad for weakly bound nuclei (delocalized continuum states involved while q (r) drop to zero outside nuclei) good for strongly bound nuclei exact for uniform matter reasonable for dense layers of neutron-star crusts

42 Analogy with terrestrial multi-band superconductors Multi-band superconductors were first studied by Suhl et al. in 1959 but clear evidence were found only in 2001 with the discovery of MgB 2 (two-band superconductor) In neutron-star crusts, the number of bands can be huge up to a thousand! both intra- and inter-band couplings must be taken into account

43 Description of neutron star crust beyond neutron drip The equilibrium structure of the inner crust is determined with the Extended Thomas-Fermi (up to 4th order)+strutinsky Integral method (ETFSI). Pairing is expected to have a small impact on the composition and is therefore neglected. Nuclei are assumed to be spherical. Onsi et al., Phys.Rev.C77, (2008). Advantages of ETFSI method very fast approximation to the full Hartree-Fock method avoids the difficulties related to boundary conditions but include proton shell effects (neutron shell effects are generally much smaller and are therefore omitted) Chamel et al.,phys.rev.c75(2007),

44 Ground-state composition of the inner crust Results for BSk14 n b (fm 3 ) Z A n n (r), n p (r) [fm -3 ] 0.1 n b = n b = n b = n 0.05 b = n b = r [fm] Onsi, Dutta, Chatri, Goriely, Chamel and Pearson, Phys.Rev.C77, (2008).

45 Ground-state composition of the inner crust ETFSI calculations for two different functionals with HFB-14 n b (fm 3 ) Z A with HFB-17 n b (fm 3 ) Z A

46 Neutron pairing gaps vs single-particle energies Example at n b = 0.06 fm 3 with BSk16 F [MeV] 2,2 2 1,8 1,6 1,4 1,2 1 0, ε [MeV] The presence of clusters reduces αk but much less than predicted by previous calculations

47 Average neutron pairing gap vs temperature Example at n b = 0.06 fm 3 with BSk16 (T)/ (0) 1 0,9 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0, ,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 T/T c αk (T)/ αk (0) is a universal function of T The critical temperature is approximately given by the usual BCS relation T c F

48 Neutron pairing gaps vs density n f n is the density of unbound neutrons u is the gap in neutron matter at density n f n u is the gap in neutron matter at density n n n b [fm 3 ] Z A nn f [fm 3 ] F [MeV] u [MeV] u [MeV] the nuclear clusters lower the gap by 10 20% both bound and unbound neutrons contribute to the gap

49 Pairing field and local density approximation The effects of inhomogeneities on neutron superfluidity can be directly seen in the pairing field n (r) = 1 2 vπn [ρ n (r),ρ p (r)] ρ n (r), ρ n (r) = Λ α,k ϕ αk (r) 2 αk E αk Neutron pairing field for n b = 0.06 fm 3 at T = 0 ξ (r) [fm] ρ = 0.06 fm r [fm] Chamel et al., Phys.Rev.C81(2010) (c) (r) [MeV] (c) 2 ρ = 0.06 fm LDA r [fm]

50 Pairing field at finite temperature At T > 0, the neutron pairing field is given by n (r) = 1 2 vπn [ρ n (r),ρ p (r)] ρ n (r), ρ n (r) = Λ α,k ϕ αk (r) 2 αk E αk tanh E αk 2T Neutron pairing field for n b = 0.06 fm 3 The superfluid becomes more and more homogeneous as T approaches T c (r) [MeV] Chamel et al., Phys.Rev.C81(2010) ρ = 0.06 fm (c) r [fm]

51 (r) [MeV] (a) ρ = 0.07 fm MeV 16 MeV 8 MeV 4 MeV 1 MeV Impact of the pairing cutoff r [fm] (r) [MeV] (e) ρ = 0.05 fm MeV 16 MeV 8 MeV 4 MeV 1 MeV r [fm] n b [fm 3 ] F0 (16) [MeV] F0 (8) F0 (4) F0 (2) F0 (1) Pairing gaps (hence also critical temperatures) are very weakly dependent on the pairing cutoff.

52 Impact on thermodynamic quantities : specific heat inhomogeneous superfluid homogeneous superfuid renormalized homogeneous superfluid [fm -3 ] C V (sn) ρ = 0.06 fm (c) T [10 9 K] Band structure effects are small. This remains true for non-superfluid neutrons. Chamel et al, Phys. Rev. C 79, (R) (2009) The renormalization of T c comes from the density dependence of the pairing strength.

53 How free are neutrons in neutron-star crusts? Due to the interactions with the periodic lattice, neutrons move in the inner crust as if they had an effective mass m n. This is a well-known effect in solid-state physics (typically m e 1 2m e ). m e is related to the current-current correlation function. This entrainment effect is very important for the hydrodynamics of the neutron superfluid. Carter, Chamel, Haensel, Int.J.Mod.Phys.D15(2006)777. Pethick, Chamel, Reddy, Prog.Theor.Phys.Sup.186(2010)9.

54 Unbound neutrons vs conduction neutrons n f n is the density of unbound neutrons n c n = n f nm n /m n is the density of conduction neutrons n b (fm 3 ) Z A nn/n f n (%) nn/n c n f (%)

55 Entrainment effects in cold atoms Simiar entrainment effects are also predicted in unitary Fermi gases and could thus be potentially measured in laboratory. Example: unitary Fermi gas in a 1D optical lattice Watanabe et al., Phys. Rev. A78(2008),063619

56 Summary 1 The nuclear energy density functional (EDF) theory allows for a consistent treatment of superfluid neutrons in neutron-star crusts. 2 We have developed semi-local EDF constrained by experiments and N-body calculations: they give an excellent fit to essentially all nuclear mass data (σ = 581 kev for HFB-17) they reproduce various properties of infinite nuclear matter (EoS, pairing gaps, etc) 3 Using the band theory of solids, we have shown that the nuclear lattice affects both the static and the dynamic properties of the neutron superfluid in the dense layers of neutron-star crusts.