PAIRING AND EXCITATIONS IN NEUTRON-RICH NUCLEAR SYSTEMS Neutron-Rich Systems

Neutron-Rich Systems

bound nuclei weakly bound nuclei nuclei+ neutron gas neutron matter Pairing in Neutron-Rich Systems

nuclei+ neutron gas uniform matter (weakly) bound nuclei

1 1 • Crust: - neutron S0 superfluidity • neutron S0 superfluidity in non-uniform nuclear matter • Consequences • Core : - neutron 3PF superfluidity 1 - binding (e.g., neutron skins and halos) - proton S0 superconductivity

- quasiparticle excitations • Consequences : - giant glitches - cooling Pairing and excitations in neutron-rich systems

Content

• General properties of superfluid Fermi systems: pair condensation

• Treatment of pairing correlations: BCS, Bogoliubov de Gennes, linear response (QRPA)

• Applications: - quasiparticle excitations in neutron-rich nuclei - superfluid and thermal properties of neutron stars Pair Condensation in Nuclear Systems

Outline

I) One- Cooper- pair problem

II) Condensate of pairs: a few properties III) BCS condensate: resume IV) Non-uniform condensate: Bogoliubov equations Moment of inertia : Migdal

One of the first hints on the nuclear superfluidity in neutron stars

One- Cooper-pair problem (I)

Physical system: two fermions subjected to an atractive interaction and situated on top (k >kF) of a free gas of fermions

1 ikir • Free states ( box of length L) (r,k ) 3 / 2 e . . ϕ i =L

• The two-electrons with CM at rest, i.e., k1 = −k2 ≡ k φ 2 2 ikr1 −ikr2 = = φε(r1,r2 ) = g(k)e e ↑↓ ε ∑ [− ∆1 − ∆2 +V (r1,r2 )] = (E + 2 F )φ k >k F 2m 2m ε ε ε G ε ' V ' ≈ − if < , ' < + ε 2 + g(k )V ' = (E + 2ε )g(k) kk 3 F k k F cut k ∑ kk F L ' k >kF 0 otherwise ε (E + 2 − 2ε )g (k ) = C F k G 1 G 1 = ε C = − g (k ) 3 ∑ 3 ∑ L k >kF 2( k −ε F ) − E L k > k F One- Cooper-pair problem (2)

ε G 1 F +ε cut Nε( )d 1 = 1 = G ε 3 ∑ ε ∫ ε L k>kF 2( k −ε F ) − E 2( −ε ) − E F ε F ε ε +ε F cut dε − 2ε cut 1 ≅ GN( ) E = 2 F ∫ ε 2( −ε F ) − E GN (ε ) F ε e F − 1 −2

GN (ε F ) if GN(ε ) <<1 E ≡ −∆ ≈ −2ε cute F

E < 0 exist for an arbitrarly small interaction strength, in variance with the two - body problem. This fact is due to the condition k > kF

and due to the degeneracy of N(ε F ). One- Cooper-pair problem (3)

The eigenvalue is in fact W = 2ε F + E ≡ 2ε F − ∆. Thus ∆ is the binding energy of the pair (relative to the non -interacting particles).

If the pair is moving with the momentum =q , the binding energy is [1]: v =q ∆φ = ∆ − F . Thus the binding energy is maximum for the pair with (k,-k) . q 2 The wave function of the pair is : k cut N (k ) (r = r − r ) = (const ) ε e ikr dk 1 2 ∫ 2 k − ( E + 2 F ) k F ε -r ∆ e 2 At large r it has the form [3] : φ ∝ . r

The average extension (' coherence length' ) is [1] : 2 =v ξ ≡< r 2 > 1 / 2 = F 3 ∆ One-Cooper-pair in finite nuclei

Physical system: two nucleons extra a closed shell

The most exotic system: 11Li n n

9Li Condensate of Cooper pairs

The type of the condensate depends on the ratio between the size of the Cooper pairs ,ξ, and the average distance between pairs, d .

I) Condensate of Bose-Einstein type: if the pairs are well-separated, i.e., ξ <

II) BCS-type condensate: if the pairs are strongly overlapping, i.e., ξ >>d. For a typical superconductors, in the region of a given pair (of size ξ) one can find located the center of mass of about 106 other pairs.

• The condensate cannot be considered anymore as formed by bound Cooper pairs. However, as in the one-Cooper-pair problem, in the condensate there is a strong corelation between the states (k,-k).

• The strong correlations among pairs are generated almost entirey by the Pauli principle restrictions and much less by the true dynamical interactions between pairs. This fact allows one to treat the system in the lowest order as if dynamical interactions existed only between the mates of the pair. The structure of a condensate

Physical system: N fermions in the presence of an attractive force Cooperφ pair instability >>> system of identical pairs

ΨN (r1,...,rN ) = A (r1,r2 )....φ(rN −1,rN )(1↑,2 ↓)...(N −1↑, N 7)

• each pair is described by the same wave function φ

ikr1 −ikr2 φ(r1,r2 ) = g(k)e e ∑ k is not restricted at k>kF k

ik1r1 −ik1r2 ΨN (r1 ,...rN ) = ∑ g(k1 )...g(k N / 2 )A e (1 ↑)e (2 ↓)...... k1, ..kN / 2 Slater determinant

| Ψ >= g (k )... g (k )c + c + ....c + c + | − > N ∑ 1 N / 2 k1 ↑ − k1 ↓ k N / 2 ↑ − k N / 2 ↓ k1... k N / 2 + N / 2 + + + | ΨN > ∝ (S ) | − > ; with S ≡ ∑ g(k)ck↑c−k↓ k

Note: S+ is not a boson operator, so the w.f. is not a Bose-Einstein condensate Pairs condensate in finite nuclei

The case of Sn isotopes : neutrons in the major shell N=50-82

+ N / 2 + + + | ΨN > ∝ (S ) | − > ; S ≡ ∑ g j (c j c j ) J =0 j 2 Z =|< SM (J = 0) | ΨN >| | SM (J = 0) > ⇔ shell model w.f. (exact)

N. Sandulescu et al, PRC55 (1997)2708 Superfluid Flow

ikr1 −ikr2 • Center of mass at rest: φ(r1,r2 ) = ∑ g(k)e e φ 0 ΨN (r1,...,rN ) = A (r1,r2 )....φ(rN −1,rN )(1↑,2 ↓)...(N −1↑, N 7)

• Translation motion: if the system is Galilean invariant, one can simply shift the CM φof each pair with a given amount in momentum space i N i i qr j q(r1 +r2 ) q(rN −1 +rN ) 2 ∑ q 2 2 j=1 0 ΨN (r1,...,rN ) = A (r1,r2 )e ....φ(rN −1,rN )e ... ≡ e ΨN

valid in general if q does not change much over a coherence length (London)

• Superfluid flow: arbitrary motion (Feynman) G = N v = grad(s); i s ∑ s( j) m s 2 j=1 0 G ΨN (r1,...,rN ) ≡ e ΨN curl(vs ) = 0

Note: a stationary condensate corresponds to a metastable equilibrium since a change of it would involve a simultaneous transition of many pairs Pair condensate: BCS ansatz

• Condensate with a given number of pairs | Ψ >= g (k )... g (k )c + c + ....c + c + | − > N ∑ 1 N k1 ↑ − k1 ↓ k N / 2 ↑ − k N / 2 ↓ k1... k N / 2 • BCS ansatz : distribution of pairs v g ≡ k k u | BCS>=C (1+ g c+ c+ )| − > k | BCS (u v c+ c+ )| ∏ k k↑ −k↓ >=∏ k + k k↑ −k↓ −> ν k k

=∞ 1 + | BCS > ∝ ∑ν (S + )ν | − >= eS (coherent distribution) =0 ν! condition : N =< BCS | Nˆ | BCS >

• Essential properties: - the width of the distribution is Ν1/2 - the relative fluctuations is 1/Ν1/2 BCS equationsδ

| BCS >= (u + v c+c+ ) | − > ˆ ˆ ∏ k k k k u ,v < BCS | H − µN | BCS >= 0 k k k

2 1 ε k − µ ∆l v = (1− ) ∆k = − Vkl k ε 2 2 ∑ ε 2 2 2 l ( k − µ) + ∆k 2 ( k − µ) + ∆l δε 2 ∆ is the energy regionδ where the pairing is important, i.e, v goes from 1 to 0 2 m∆ = k F k ≈ ∆ ⇒ k ≈ 2 ⇒ δr ≈ coherence length (Pippard) = k F m∆ ∆ • Condensation amplitude: k κk ≡< BCS | ck ck | BCS >= uk vk = 2Ek 1 • Gap for constant interaction: ∆ ≈ 2 ε cut exp[ − ] N F G

1 2 2 2 • Condensation energy: Wn −Ws = N F ∆ ≈ 2ε cut N F exp[− ] 2 N F G (compare to the binding energy of Cooper’s pair )