Pairing time-reversal states J = 0 + (a) +m -m k F k F K k - - PowerPoint PPT Presentation
Nuclear pairing from microscopic forces (with applications) Paolo Finelli Dept. of Physics and Astronomy, University of Bologna paolo. fj nelli@bo.infn.it Based on Phys. Rev. C 90, 044003 Published 21 October 2014 Pairing time-reversal
University of Bologna paolo.fjnelli@bo.infn.it
Based on Phys. Rev. C 90, 044003 – Published 21 October 2014
kF + δ kF + δ kF − δ kF − δ
kx ky kz
−~ k ↓ ~ k ↑
Filled Fermi sea (a) (b) ω
K
L.N. Cooper considered the scattering of two particles which have an attractive interaction in the presence of a Fermi sea (restricting the possible momenta where the particles can scatter). He showed that even for arbitrarily small interactions, pairs will be formed in the system. Instability of the Fermi sea: pair formation happens for any non-zero, attractive interaction.
time-reversal states
+m
Jπ= 0+
∆(k) = X
k0
hk|V |k0i ∆(k0) 2E(k0)
E(k)2 = ✏(k)2 + |∆(k)|2
hk|V |k0i = 4π X
L
(2L + 1)PL(ˆ k · ˆ k0)VL(k, k0)
∆L(k) = − 1 ⇡ Z 1 k02dk0 VL(k, k0) p ✏(k0)2 + [P
L0 ∆L0(k0)2]
∆L(k0)
∆(k) = X
L,M
r 4π 2l + 1YLM(ˆ k)∆LM(k)
|∆(k)|2 → D(k)2 ≡ 1 4π Z dˆ k |∆(k)|2 = X
L,M
1 2L + 1|∆LM(k)|2
the different M-components become uncoupled and all equal. No dependence on the quantum number M, but L components are coupled by tensor terms
we used the method suggested by Khodel where the original gap equation is replaced by a coupled set of equations for the dimensionless gap function χ(p) defjned by Δ(p) = ΔF χ(p) and a non-linear algebraic equation for the gap magnitude ΔF = Δ(pF) at the Fermi surface.
V.V. Khodel, V.A. Khodel, J.W. Clark, NPA 679 (2000) 827
Vll0(k, k0) = vll0φll0(k)φll0(k0) + Wll0(k, k0) ,
φll0(k) = Vll0(k, kF )/Vll0(kF , kF ) vll0 = Vll0(kF , kF )
The potential is written in the following way
∆l(k) − X
l0
(−1)Λ Z dτ 0 Wll0(k, k0)∆l0(k0) E(k0) = X
l0
Dll0φll0(k)
The gap equation becomes
Dll0 = (−1)Λvll0 Z dτ φll0(k)∆l0(k) E(k)
dτ = k2dk/π
vanishes at the Fermi surface
V.V. Khodel, V.A. Khodel, J.W. Clark, NPA 679 (2000) 827
Dll0 = (−1)Λvll0 Z dτ ∆l0 E φll0
∆l = X
l1l2
Dl1l2χl1l2
l χl1l2
l
− X
l0
(−1)Λ Z dτ 0 Wll0 χl1l2
l0
E = δll1φl1l2
∆l(k) = X
l1l2
Dl1l2χl1l2
l
(k)
χl1l2
l
(k) − X
l0
(−1)Λ Z dτ 0 Wll0(k, k0)χl1l2
l0
(k0) E(k0) = δll1φl1l2(k)
scale factor (calculations must be δ-independent)
χl1l2
l
(k) − X
l0
(−1)Λ Z dτ 0 Wll0(k, k0) χl1l2
l0
(k0) p ξ2(k0) + δ2 = δll1φl1l2(k)
Self-consistent scheme
LO (Q/Λχ)0 NLO (Q/Λχ)2 … Medium-range Long-range
r P
e n t i a l
The nuclear force at large distances is governed by the exchange of one or multiple pions.
Short-range
LO (Q/Λχ)0 NLO (Q/Λχ)2 …
LO (Q/Λχ)0 NLO (Q/Λχ)2
… Medium-range Long-range
r P
e n t i a l
The nuclear force at large distances is governed by the exchange of one or multiple pions.
The short-range part
driven by physics not resolved explicitly in reactions with typical nucleon momenta of the order of Mπc. It can be mimicked by zero- range contact interactions with an increasing number of derivatives.
parametrized
Short-range
… … Short-range
contact interactions and 1π-exchange
including 2π-exchange
+ ... + ... + ...
including 3π-exchange
+ ...
LO NLO N2LO N3LO Phase shifts of np scattering as calculated from NN potentials at different orders of ChPT (black dots are experimental data)
see also Epelbaum, Hammer and Meissner, RMP 81 (2009) 1773
V = V2B + V3B ' V2B + V eff
2B (ρ)
3 body ➜ 2 body density dependent Holt et al., [PRC 81 (2010) 024002] in-medium nucleon propagator
2 3 4 5 6
In-medium NN interaction generated
exchange component (c1,c3,c4)
exchange (cD) and short- range component (cE)
Lippmann-Schwinger cutoff
f Λ = exp
with n = 2, 3
cutoff the short-range part of the 2PE contribution
V (k, k0) ! V (k, k0)f Λ(k, k0)
Machleidt (N3LO) Epelbaum (N3LO)
dimensional regularization spectral function
Λ = 450, 500, 600
{Λ, ˜ Λ} = {450, 500}, {450, 700}, {550, 600}, {600, 600}, {600, 700}
k k k′ k′
One can use RG transformations to evolve to lower Λ while preserving the truncation error of the original Hamiltonian ➩ eliminate coupling between high- and low-momenta components (kF ≤ 2 fm−1)
Renormalization Group could help for
Features
Vlowk Vsrg
d dΛTΛ(p, q; q2) = 0
Bogner, Kuo and Schwenk, PR 386 (2003) 1
The SRG is based on a continuous sequence of unitary transformations that suppress off- diagonal matrix elements, driving the Hamiltonian towards a band-diagonal form.
k k k′ k′
Bogner, et al., PPNP 65 (2010) 94-147
∆(k) = − 1 ⇡ Z 1 dk0k02 V (k, k0)∆(k0) p (✏(k) − ✏(kF ))2 + ∆(k)2
✏(k) − ✏(kF ) = k2 − k2
F
2MN ✏(k) − ✏(kF ) = k2 − k2
F
2M ∗
N
0.05 0.1 0.15 0.2
ρ [fm
0.6 0.7 0.8 0.9 1
M
*(ρ) / M
effective nucleon mass
Holt, Kaiser and Weise, EPJA 47 (2011) 128 Substantial agreement with non relativistic Skyrme phenomenology at saturation density (0.7 < M*/M < 1). Effective nucleon mass M*/M
1S0 1S0
nuclear matter neutron matter
FINITE TEMPERATURE
still in progress
and neutron stars
3SD1 3PF2
SRG evolutions
nuclear matter neutron matter
5 10 15 20
1D2
5 10 15 20
1D2
10 20 30
10 20 30
10 20 30
3P2
10 20 30
3P2
100 200 300
2
100 200 300
2
5 10 100 200 300 Elab [MeV]
3D3
5 10 100 200 300 Elab [MeV]
3D3
20 40 60 [deg]
1S0
20 40 60 [deg]
1S0
10 20
3P0
10 20
3P0
10 [deg]
1P1
10 [deg]
1P1
3P1
3P1
60 120 180 [deg]
3S1
60 120 180 [deg]
3S1
5 10
1
5 10
1
100 200 300
100 200 300
100 200 300 Elab [MeV]
100 200 300 Elab [MeV]
R = 0.9 fm NLO N2LO N3LO N4LO Epelbaum, HK, Meißner, arXiv: 1412.4623
New renormalisation technique in the coordinate space with the cutoff R being chosen in the range of R = 0.8 . . . 1.2 fm. For contact interactions, they use a non- local Gaussian regulator in momentum space with the cutoff Λ = 2R-1
f ⇣ r R ⌘ = 1 − exp ✓ − r2 R2 ◆6
1S0
neutron matter
3SD1
nuclear matter
3PF2
neutron matter
Paolo Finelli, unpublished, Ramanan et al. 1606.09053v2, Drischler et al. 1610.05213
momentum cutoff (450-550 MeV) momentum cutoff (450-550 MeV) momentum cutoff (450-550 MeV) coordinate cutoff (0.8-1.2 fm)
3PF2 channel is relevant for cooling
inner core?
core
inner crust
ν c
i n g
ν ν N N
cooper pair
http://www.astroscu.unam.mx/neutrones/NSCool/
Neutrino emission from the formation (and breaking) of Cooper pairs
∆F /✏F
ξRMS/d
P(d)
The probability density to fjnd two nucleons at a distance r in the 1S0 neutron matter channel evaluated using VNN only
P(r) = Z r dr0 ρ(r0)
For nuclear systems the following parameters have been suggested
BCS-BEC 1 kF as 1 1 kF as ⌧ 1
kF |as| < 1
∆F /✏F
ξRMS/d
P(d)
The probability density to fjnd two nucleons at a distance r in the 1S0 neutron matter channel evaluated using VNN only
P(r) = Z r dr0 ρ(r0)
Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú ÚÚÚÚÚÚÚ Ú Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê ‡ ‡ ‡‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ 0.2 0.4 0.6 0.8 1.0 1.2 0.4 0.5 0.6 0.7 0.8 0.9 1.0
kF @fm-1D PHdL
BEC bound BCS bound
‡
CONT.
Ê
VNN+3N+M*
Ú
VNN
1 S0 nn
Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ë Ë Ë Ë Ë Ë Ë Ë Ë Ë Ë Ë Ë Ë Ë Ë Ë Ë Ë Ë Ë Ë Ë Ë Ë Ë Ë Ë Ë Ë Ë Ë Ë Ë Ë Ë ËËËËËËËËËËËËËËËËËËËË ü ü ü ü ü ü ü ü ü 0.2 0.4 0.6 0.8 1.0 1.2 0.02 0.05 0.10 0.20 0.50 1.00 2.00
kF @fm-1D xRMSêd
BEC bound BCS bound
ü
CONT.
Ë
VNN+3N+M*
Ú
VNN
1 S0 nn
Crossover area
For nuclear systems the following parameters have been suggested
(500 MeV cutofg)
(b) (a) (c) dineutron deuteron
p h
c
p l i n g
✓ h − µ ∆ ∆ −h + µ ◆ ✓ U V ◆
k
= Ek ✓ U V ◆
Δ: pairing fjeld In the hamiltonian h we have Vph: particle-hole potential μ: chemical potential
¯ ∆ =
P
k ∆kv2 k
P
k v2 k
∆(5)(N0) = −1 8 [E(N0 + 2) − 4E(N0 + 1)+ 6E(N0) − 4E(N0 − 1) + E(N0 − 2)]
State-dependent single-particle states Theory Occupation factors Empirical Estimates
Δ Δ kF
N,Z
Paolo Finelli et al, PRC 86 (2012) 034327
N3LO N3LO and 3-body
Z = 82
with exp. data.
gaps are reduced by 30/40%[See also PRC 80 (2009) 044321].
82 126
N
1 2
ω