Hartree-Fock-Bogoliubov atomic mass models and the description of the neutron-star crust

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1 Hartree-Fock-Bogoliubov atomic mass models and the description of the neutron-star crust Nicolas Chamel in collaboration with S. Goriely and J. M. Pearson Research Associate FNRS Institute of Astronomy and Astrophysics Université Libre de Bruxelles Belgium 3-7 August 2009, Trento

2 Punchline The self-consistent mean field theory is a method of choice for the global description of various nuclear systems from finite nuclei to supernova core and neutron star crust

3 Outline 1 Mean field theory with effective interactions HFB atomic mass models new effective interactions with improved pairing channel 2 Applications to the description of the neutron star crust ground-state composition neutron superfluidity

4 HFB atomic mass models using effective interactions The energy of a nucleus is expressed as (q = n, p for neutrons,protons) E = ] E HFB [ρ q (r), ρ q (r), τ q (r), J q (r), ρ q (r) d 3 r + E corr The various densities are calculated from 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 HFB + δe HFB i δe HFB σ δτ q δρ q δj q ) q δe HFB δ ρ q

5 Phenomenological corrections Wigner energy E W = V W exp { λ E corr = E W + E coll ( 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

6 Determination of the model parameters General fitting procedure The parameters are fitted to the 2149 measured atomic masses with Z, N 8 Additional constraints : isoscalar effective mass M s/m = 0.8 compressibility 230 K v 270 MeV charge radius of 208 Pb, R c = ± fm symmetry energy J = 30 MeV neutron-matter equation of state NEW : 1 S 0 pairing gap in infinite neutron matter

7 Density dependent contact pairing force v pair q (rr i r j r i,rr j ) = v π q [ρ n (r), ρ p (r)] δ(rr i r j ), r = (r i + r j )/2 r i r j standard ansatz ( ) v π q ρn + ρ α ) p [ρ n, ρ p ] = V πq (1 η ρ 0

8 Density dependent contact pairing force v pair q (rr i r j r i,rr j ) = v π q [ρ n (r), ρ p (r)] δ(rr i r j ), r = (r i + r j )/2 r i r j standard ansatz ( ) v π q ρn + ρ α ) p [ρ n, ρ p ] = V πq (1 η ρ 0 Drawbacks not enough flexibility to fit realistic pairing gaps in infinite nuclear matter and in finite nuclei ( isospin dependence) the fit is computationally expensive

9 Microscopically deduced pairing force 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 q [ρ q ] is the locally the same as in infinite nuclear matter with density ρ q

10 Microscopically deduced pairing force 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 q [ρ q ] is the locally the same as in infinite nuclear matter with density ρ q New procedure v π [ρ q ] = v π [ n (ρ q )] constructed so as to reproduce exactly a given pairing gap n (ρ n ) in neutron matter by solving directly the HFB equations in uniform neutron matter for each density ρ n Chamel, Goriely, Pearson, Nucl. Phys.A812,72 (2008).

11 Expression of the pairing force Cutoff prescription : s.p. cutoff above the Fermi level v π [ρ n ] = 8π 2 ( 2 ) 3/2 ( µn+ε Λ ) 1 ε dε (ε µn ) 2 + n (ρ n ) 2 2Mn 0 µ n = 2 2Mn (3π 2 ρ n ) 2/3

12 Expression of the pairing force Cutoff prescription : s.p. cutoff above the Fermi level v π [ρ n ] = 8π 2 ( 2 ) 3/2 ( µn+ε Λ ) 1 ε dε (ε µn ) 2 + n (ρ n ) 2 2Mn 0 µ n = 2 2Mn (3π 2 ρ n ) 2/3 exact fit of the given gap n (ρ n ) analytic expression without any free parameters! automatic renormalization of the pairing strength with any readjustments of the cutoff ε Λ

13 Choice of the pairing gap 4 n [MeV] BCS pairing gap obtained with realistic nucleon-nucleon potentials 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.

14 Neutron vs proton pairing Because of possible charge symmetry breaking effects, proton and neutron pairing strengths are not equal Neglect of polarization effects in odd nuclei (equal filling approximation) are 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 ]

15 f n fn 1.06 f p fp 1.05 Neutron vs proton pairing Note that fn /f n + fp /f p + neutron and proton pairing strengths are effectively equal the pairing strength is larger for odd nuclei in agreement with a recent analysis by Bertsch et al. Phys.Rev.C79(2009), (3) o (MeV) N=90 V T-odd =0 N=100

16 Nuclear matter properties predicted by the new Skyrme force BSk16 BSk16 BSk15 BSk14 a v [MeV] ρ 0 [fm 3 ] J [MeV] Ms/M Mv/M K v [MeV] L [MeV] G G G G ρ frmg /ρ Chamel, Goriely, Pearson, Nucl. Phys.A812,72 (2008).

17 Neutron-matter equation of state at subsaturation densities E/A [MeV] FP (1981) BSk16 Akmal et al. (1998) 0 0 0,02 0,04 0,06 0,08 0,1 0,12 0,14 0,16 ρ [fm -3 ] Chamel, Goriely, Pearson, Nucl. Phys.A812,72 (2008).

18 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 ] Chamel, Goriely, Pearson, Nucl. Phys.A812,72 (2008).

19 1 S 0 pairing gap in neutron matter The new model yields a much more realistic gap than previous 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 ]

20 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] Chamel, Goriely, Pearson, Nucl. Phys.A812,72 (2008).

21 Latest mass model HFB-17 Fit the 1 S 0 pairing gaps of both neutron matter and symmetric nuclear matter include medium polarization effects on the gaps Neutron matter Symmetric nuclear matter [MeV] ρ [fm -3 ] Pairing gaps from recent Brueckner calculations Cao et al.,phys.rev.c74,064301(2006). [MeV] ρ [fm -3 ]

22 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 M q = M to be consistent with the neglect of self-energy effects on the gap ( ) v π q [ρ n, ρ p ] = 8π 2 2 3/2 2M µq+ε Λ 0 ε dε (ε µ q ) q 1

23 Density dependence of the pairing force v πn η= η= ρ [fm -3 ] Goriely, Chamel, Pearson, EPJA (2009).

24 Isospin dependence of the pairing force 800 v πn η Goriely, Chamel, Pearson, EPJA (2009).

25 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) Goriely, Chamel, Pearson, PRL102, (2009).

26 HFB-17 mass predictions Differences between experimental and calculated masses as a function of the neutron number N for the HFB-17 mass model. Goriely, Chamel, Pearson, PRL102, (2009).

27 Predictions to newly measured atomic masses HFB mass model fitted to the 2003 Atomic Mass Evaluation. Compare its predictions to new 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) http ://research.jyu.fi/igisol/jyfltrap_masses/ gs_masses.txt

28 Nuclear matter properties predicted by the new Skyrme force BSk17 BSk16 BSk17 a v ρ J Ms/M Mv/M K v L G G G G ρ frmg /ρ Goriely, Chamel, Pearson, PRL102, (2009).

29 Comparison between HFB-16 and 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. Goriely, Chamel, Pearson, EPJA (2009).

30 Microscopic structure of neutron star crusts Chamel&Haensel, Living Reviews in Relativity 11 (2008), 10 http ://relativity.livingreviews.org/articles/lrr / LATEX

31 Ground-state structure of neutron star crust below neutron drip Baym, Pethick, Sutherland (BPS), Astr. J.170(1971)299. Perfect crystal with a single nuclear species (A,Z) at lattice sites

32 Ground-state structure of neutron star crust below neutron drip Baym, Pethick, Sutherland (BPS), Astr. J.170(1971)299. Perfect crystal with a single nuclear species (A,Z) at lattice sites minimise the energy per nucleon ε/n b ε = n N E{A, Z } + ε e + ε L ε e energy density of the electron gas (well-known) ε L lattice energy density (well-known) E{A, Z } energy of a nucleus taken from experiment otherwise has to be calculated

33 Ground-state composition of the neutron star outer crust Using experimental masses when available N HFB-16 HFB Z ρ [10 11 gcm -3 ] Goriely, Chamel, Pearson, EPJA (2009).

34 Description of neutron star crust beyond neutron drip Extended Thomas-Fermi+Strutinsky Integral method (fast approximation to Hartree-Fock method) Expand τ q (r) and J q (r) in terms of ρ q (r) and ρ q (r) Minimize the energy per nucleon [ ] E EETF ρ q (r), ρ q (r) d 3 r A = ρ(r) d 3 r Include proton shell effects via the Strutinsky integral Ep sc = [ ] n i ǫ i,p d 3 2 r i 2 M τ p + ρ p Ũ p + J p W p p Onsi et al., Phys.Rev.C77, (2008).

35 Ground-state composition of the inner crust of neutron stars Results of ETF+SI (and ETF) calculations with Skyrme BSk14 ρ (fm 3 ) Z A (38) 200 (146) (39) 460 (385) (39) 1140 (831) (38) 1215 (1115) (35) 1485 (1302) (33) 1590 (1303) (31) 1610 (1261) (30) 800 (1171) (29) 780(1105) Onsi et al., Phys.Rev.C77, (2008).

36 Crust composition vs effective interaction The composition of the inner crust is very sensitive to the effective interaction! Example : ETF calculations (no shell effects) Z BSk9 BSk ρ [fm -3 ]

37 Neutron star crust and neutron skin in 208 Pb Both models give comparably good mass fit for the most neutron rich nuclei but HFB-8 was not constrained to reproduce the neutron matter eos HFB-8 HFB-9 σ(2149 M) σ(m nr ) J L θ( 208 Pb) small (large) θ leads to neutron star crust with large (small) nuclear clusters

38 Neutron star crust and neutron skin in 208 Pb Both models give comparably good mass fit for the most neutron rich nuclei but HFB-8 was not constrained to reproduce the neutron matter eos HFB-8 HFB-9 σ(2149 M) σ(m nr ) J L θ( 208 Pb) small (large) θ leads to neutron star crust with large (small) nuclear clusters PREX A measurement of neutron skin thickness with an accuracy < 0.04 fm can put strong constraints on crust properties

39 Ground-state composition of the neutron star crust for HFB-17 Results of ETF+SI calculations with Skyrme BSk17 ρ (fm 3 ) Z A

40 Ground-state composition of the neutron star crust for HFB-17 Results of ETF+SI calculations with Skyrme BSk14 ρ (fm 3 ) Z A

41 Superfluidity in neutron star crust Theoretically, neutron superfluidity in neutron star crust is well established. It was suggested long before the discovery of pulsars by Migdal (1959) only two years after BCS and studied by Ginzburg and Kirzhnits (1964), Wolff (1966) and many others

42 Superfluidity in neutron stars and observations Observational evidence of superfluidity? pulsar glitches (long relaxation times, glitch mechanism) neutron star thermal X-ray emission (cooling) RXJ Vela pulsar

43 Effects of nuclear clusters on superfluidity? Most studies of superfluidity have been devoted to the case of pure neutron matter but...

44 Effects of nuclear clusters on superfluidity? Andersson s theorem Effects of the clusters are negligible if the superfluid coherence length is much larger than the lattice spacing. 100 ξ, R cell [fm] ξ (BCS) ξ (Brueckner) R cell ξ (RG) d n Pippard s definition 20 ξ = 2 k F πm n log ρ [g.cm -3 ] based on the results of Negele&Vautherin

45 Superfluidity in the crust within the Wigner-Seitz approximation The HFB equations have been already solved by several groups using the W-S approximation

46 Superfluidity in the crust within the Wigner-Seitz approximation The HFB equations 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 fm 3 ), in some cases the pairing gaps are almost completely suppressed Baldo et al., Eur.Phys.J. A 32, 97(2007).

47 Problem Limitations of the W-S approximation Results strongly depend on the choice of boundary conditions which are not unique 2.5 k F =1.1 fm -1 n, MeV Z=26 Z=20 Baldo et al., Eur.Phys.J. A 32, 97(2007) r, fm Solution Go beyond the W-S app. by using the band theory of solids Chamel et al., Phys.Rev.C75(2007)

48 Superfluidity in the inner crust beyond the W-S approximation Generalized BCS equations to account for the periodic lattice αk = 1 2 β k v pair βk k αkα kβk k β k k E βk k k tanh E βk 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 2 αk + 2 αk ε αk and ϕ αk (r) are obtained from band structure calculations Chamel, Nucl.Phys.A747(2005)109.

49 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 Usually only two bands are enough in superconductors but in neutron star crust, the number of bands can be very large up to thousand

50 Neutron pairing gaps vs single-particle energies Example at ρ = 0.06 fm 3 with BSk16 F [MeV] 2,2 2 1,8 1,6 1,4 1,2 1 0, ε [MeV]

51 Neutron pairing gaps vs single-particle energies Example at ρ = 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

52 Average neutron pairing gap vs temperature Example at ρ = 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

53 Average neutron pairing gap vs temperature Example at ρ = 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

54 Average neutron pairing gap vs temperature Example at ρ = 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

55 Neutron pairing gaps vs density ρ [fm 3 ] Z A ρ f n [fm 3 ] F [MeV] u [MeV] u [MeV] Nucleon density profiles for BSk16 at ρ = 0.06 fm r [fm] nuclear clusters lower F by 10 20% compared to uniform neutron matter ( u )

56 Impact on thermodynamic quantities : specific heat of superfluid neutrons Example at ρ = 0.06 fm 3 with BSk inhomogeneous superfluid homogeneous superfluid with T cu homogeneous superfluid with T c [fm -3 ] C V (n) T [10 9 K]

57 Impact on thermodynamic quantities : specific heat of superfluid neutrons Example at ρ = 0.06 fm 3 with BSk inhomogeneous superfluid homogeneous superfluid with T cu homogeneous superfluid with T c [fm -3 ] C V (n) T [10 9 K] multi-band effects are small and C (n) V uniform neutron matter is close to that of

58 Impact on thermodynamic quantities : specific heat of superfluid neutrons Example at ρ = 0.06 fm 3 with BSk inhomogeneous superfluid homogeneous superfluid with T cu homogeneous superfluid with T c [fm -3 ] C V (n) T [10 9 K] multi-band effects are small and C (n) V is close to that of uniform neutron matter provided T c is suitably renormalized

59 Dynamical effective mass and entrainment effects Due to the interactions with the nuclear clusters, the neutrons move in the crust as if they had an effective mass m n. For free electrons in solids m e 1 2m e

60 Dynamical effective mass and entrainment effects Due to the interactions with the nuclear clusters, the neutrons move in the crust as if they had an effective mass m n. For free electrons in solids m e 1 2m e In the crust rest frame p n = mnv v n therefore in another frame v n p n = mnv v n + (m n mn)v v c v c

61 Dynamical effective mass and entrainment effects Due to the interactions with the nuclear clusters, the neutrons move in the crust as if they had an effective mass m n. For free electrons in solids m e 1 2m e In the crust rest frame p n = mnv v n therefore in another frame v n p n = mnv v n + (m n mn)v v c entrainment effects are important for neutron star dynamics (oscillations, precession, glitches)

62 Dynamical effective neutron mass in the crust Using the band theory of solids, m n is given by m n = n n K K = 1 3 F d 3 k (2π) 3 Tr 1 m (k) n with the usual local effective mass tensor defined by ( ) 1 ij mn(k) = 1 2 E k 2. k i k j Note that this has been also applied in neutron diffraction Zeilinger et al., PRL57 (1986), 3089.

63 Dynamical effective neutron mass in the neutron star crust Chamel (2005) BSk16 transition to pastas crust-core interface crust-core interface ,01 0,02 0,03 0,04 0,05 0,06 0,07 0,08 0,09 ρ [fm -3 ] Chamel, Nucl.Phys.A749, 107 (2005) Preliminary results for BSk16

64 Quasi-periodic oscillations in Soft-Gamma Repeaters Detection of QPOs in giant flares of a few SGR crustquakes Example : SGR

65 Neutron star seismology The frequencies of the seismic modes depend on the internal structure of the neutron star. From L. Samuelsson Entrainment effects have a significant impact on the modes (among other things).

66 Summary of the first part We have developed effective nucleon forces that are very well-suited for astrophysics applications : they give an excellent fit to essentially all nuclear mass data (σ = 581 kev for HFB-17) they reproduce the neutron matter eos and 1 S 0 pairing gap obtained with realistic potentials. HFB atomic mass table http ://

67 Summary to the second part We have just started to calculate the properties of neutron star crust : the composition depends on the density dependence of the symmetry energy (neutron skin thickness in 208 Pb PREX) the neutron pairing gaps are reduced by 10 20% due to inhomogeneities but their temperature dependence is universal the nuclear clusters strongly affect the dynamics of the unbound neutrons pulsar glitches, QPOs in SGRs...