Fermi sea λ F F (non-interacting particles)
a Fermi sea λf
Fermi sea λf
a Fermi sea λf
a a Fermi sea λf
Question: What is the most favorable arrangement of these two spheres? a R=? a Answer: The energy of the system does not depend on R as long as R > a. NB Assuming that a liquid drop model for the fermions is accurate! This is a very strange answer! Isn t it? Something is amiss here. Fermi sea λ F
Let us try to think of this situation now in quantum mechanical terms. The dark blue region is really full of de Broglie s waves, which, in the absence of homogeneities, are simple plane waves. When inhomogeneities are present, there are a lot of scattered waves. Also, there are some almost stationary waves, which reflect back and forth from the two tips of the empty spheres. a R a As in the case of a musical instrument, in the absence of damping, the stable musical notes correspond to stationary modes. Problems: 1) There is a large number of such modes. ) The tip-to-tip modes cannot be absolutely stable, as the reflected wave disperses in the rest of the space.
Fermionic Casimir effect A force from nothing onto nothing
A few reasons why the crust of a neutron star is a fun and challenging object to study Aurel Bulgac (Seattle) Collaborators: P.H. Heenen (Brussels), P. Magierski (Warsaw), A. Wirzba (Bonn), Y. Yu (Seattle) Transparencies (both in ppt and pdf format) will be available at http://www.phys.washington.edu/~bulgac
What shall I talk about? I shall describe how this rather subtle quantum phenomenon, the Fermion counterpart of the Casimir effect, is affecting quite perhaps drastically the crystalline structure of the neutron star crust, leading likely to a more complex phase, with a richer structure. I shall also show that in low density neutron matter, when neutron matter becomes superfluid and vortices can form, the spatial profile of a vortex resembles more its Bose counterpart, and develops a strong density depletion along its axis.
Anderson and Itoh,Nature, 1975 Pulsar glitches and restlessness as a hard superfluidity phenomenon The crust of neutron stars is the only other place in the entire Universe where one can find solid matter, except planets. A neutron star will cover the map at the bottom The mass is about 1.5 solar masses Density 10 14 g/cm 3 Author: Dany Page
Es V E V E ~ ( n') E C tot = u σ d r = πn' = E = n' E e x e 0 r + ue ~ (n') + E 1 1 du u d + d K ' + s n 1 18 ns s + E c d + u surface energy Coulomb energy bulk energy of dense phase Lorenz, Ravenhall and Pethick Phys. Rev. Lett. 70, 379 (1993) Ravenhall, Pethick and Wilson Phys. Rev. Lett. 50, 066 (1983) Figure of merit to remember 0.005 MeV/fm 3 = 5 kev/fm 3
Let me now go back to my starting theme and consider spherical inhomogeneities in an otherwise featureless Fermi see
Quantum pinball machine
Casimir Interaction among Objects Immersed in a Fermionic Environment The ratio of the exact Casimir energy and the chemical potential for four equidistant spheres of radius a separated by r forming a tetrahedron and also the same ratio computed as a sum of interactions between pairs or triplets for two different separations. E E C C a µ j1[ kf ( r a) ] r>> a π r two spheres µ π ( r a) a r ( a) sphere next to a plane j 1 [ k ( r a) ] F kfa 8πm A. Bulgac and A. Wirzba, Phys. Rev. Lett. 87, 10404 (001). cos(kf ( r 3 r a))
What happens at the boundary of a normal and superfluid regions? outside inside Andreev reflection
After all this annoying theoretical detour let us look at neutron star crust now! Figure of merit to remember 0.005 MeV/fm 3 = 5 kev/fm 3
Quantum Corrections to the GS Energy of Inhomogeneous NM slabs tubes The Casimir energy for various phases. The lattice constants are: L = 3, 5 and 8 fm respectively. bubbles u anti-filling factor fraction of empty space average density ρ 0 A. Bulgac and P. Magierski Nucl. Phys. 683, 695 (001) Nucl. Phys, 703, 89 (00) (E)
The Casimir energy for the displacement of a single void in the lattice Rod phase Slab phase y x Bubble phase A. Bulgac and P. Magierski Nucl. Phys. 683, 695 (001) Nucl. Phys, 703, 89 (00) (E)
Deformation of the rod-like phase lattice α γ kev/fm 3 β αβ sin γ =1 volume conservation A. Bulgac and P. Magierski Nucl. Phys. 683, 695 (001) Nucl. Phys, 703, 89 (00) (E)
rods deformed nuclei Skyrme HF with SLy4, Magierski and Heenen, Phys. Rev. C 65, 045804 (00)
rippled slabs bcc scc bcc nuclei Skyrme HF with SLy4, Magierski and Heenen, Phys. Rev. C 65, 045804 (00)
bcc scc bcc E between spherical and rod-like phases E between spherical and slab-like phases Skyrme HF with SLy4, Magierski and Heenen, Phys. Rev. C 65, 045804 (00)
Spherical phase (scc) Rod-like phase Size of box d = 6 fm P. Magierski, A. Bulgac and P.-H. Heenen, nucl-ph/011003
Rod-like phase Spherical phase (bcc) Size of box d = 3.4 fm P. Magierski, A. Bulgac and P.-H. Heenen, nucl-ph/011003
Slab-like phase Bubble-like phase Size of box d = 0.8 fm P. Magierski, A. Bulgac and P.-H. Heenen, nucl-ph/011003
scc scc scc scc bcc Various contributions to the energy density as a function of the proton quadrupole moment. Magierski, Bulgac and Heenen, Nucl.Phys. A719, 17c (003)
E E E ~ (n') E E dense phase bulk energy of ' 1 18 E E ~ ( Coulomb energy 1 ' V E energy surface V E c s e tot 0 1 C s + + + = + = + + = = u n n K n' n') u d du d u r e x n r d u s s d π σ Ravenhall, Pethick and Wilson Phys. Rev. Lett. 50, 066 (1983) Lorenz, Ravenhall and Pethick Phys. Rev. Lett. 70, 379 (1993)
n 1, (n) n 1 n 1 Phase I Phase II [ ] [ ] 1 1 1 1 1 1 1 1 1 1 1 1 ) ( ) ( ) ( ) ( 0 ) 0 ) ) (1 0 ) ) ( ) (1 ) ( n n n n dn n d dn n d dx N d(e n n n n x dn N d(e nv V n x xn N dn N d(e V n x n x E = = = = = = = + = = + = ε ε ε ε µ µ µ µ ε ε Pure phase II Pure phase I Mixed phase I+II
Let us consider now two moving spheres in the superfluid medium at velocities below the critical velocity for the loss of superfluidity. r a 1 a u 1 u in out i in i ren i out ren ren kin i M m a M r u r u r u u r a a u M u M T ρ ρ γ γ γ γ γ ρ π γ γ πρ = = + = + = + + + = 1,, 1 ) (1 1 ) (1 3 4 ) )( ( 3 1 4 3 1 1 3 1 1 1 Kinetic energy has a similar 1/r 3 dependence as the Casimir energy!
Hamiltonian of a nucleus immersed in a neutron superfluid ( E < ): ˆ π H = lm + C ˆ α l lm lm, M l ( γ -1) M = mρ R 5 ; γ = l in γ ( l+ 1) + l N C = C surf + C coul l l l Spreading width of a quadrupole vibrational multiplet (l=): 1/ ( Ze) Rp 3 ω Γtot 0.169 RC RC C R p C proton radius R Wigner Seitz cell radius exc ρ out ρ in Neutrons Energy depends on the Vibrating orientation nucleus with respect to the lattice vectors
....... Spherical symmetry breaking due to the coupling between lattice and nuclear vibrations spherical nuclei..... deformed nuclei Nuclear quadrupole excitation energy in the inner crust
What have we established so far? Quantum corrections (Casimir( energy) to the ground state energy of inhomogeneous neutron matter are of the same magnitude or larger then the energy differences between various simple phases. Lattice defects and lattice distortions have characteristic energy ergy changes of the same order of magnitude. Only relatively large temperatures (of order of 10 MeV) ) lead to the disappearance of these quantum energy corrections. Fully self-consistent calculations confirm the fact that the pasta phase might have a rather complex structure, various shapes can coexist, at the same s time significant lattice distortions are likely and the neutron star crust could be on the verge of a disordered phase. Pethick and Potekhin,, Phys. Lett. B 47, 7 (1998) present argument in favor of a liquid crystal structure of the pasta phase. Jones, Phys. Rev. Lett. 83, 3589 (1999) claims that the thermal fluctuations are so large that the system likely cools down to an amorphous and heterogeneous ous phase. Dynamics of these structures is important
Now I shall switch gears and discuss some aspects of the physics of vortices in low density neutron matter. A vortex is just about the only phenomenon in which a true stable superflow is created in a neutral system I shall describe briefly the DFT-LDA extension to superfluid Fermi systems: SLDA (Superfluid LDA) I shall apply this theory to describe the basic properties of a vortex in low density neutron matter.
SLDA equations for superfluid Fermi systems: Energy Density (ED) describing the normal phase Additional contribution to ED due to superfluid correlations Typo: replace m by m(r) Y.Yu and A. Bulgac, PRL 90, 501 (003)
Screening effects are significant! s-wave pairing gap in infinite neutron matter with realistic NN-interactions BCS These are major effects beyond the naïve HFB from Lombardo and Schulze astro-ph/00109
Fayans s FaNDF 0 = e 7 /3 k m F exp π tanδ ( k An additional factor of 0.4 is due to induced interactions Naïve HFB/BCS not valid. F ) Y. Yu and A. Bulgac, PRL 90, 161101 (003) from Heiselberg et al Phys. Rev. Lett. 85, 418, (000)
Landau criterion for superflow stability (flow without dissipation) Consider a superfluid flowing in a pipe with velocity v s: E Nmvs Nmv E ε s + < 0 + p + vs p + s < 0 v ε p p no internal excitations One single quasi-particle excitation with momentum p In the case of a Fermi superfluid this condition becomes v v S F < ε F
Vortex in neutron matter u v α kn α kn ( r ) ( r ) = u v α α ( r)exp[ i( n ( r)exp[ i( n + 1/ 1/ ) φ ikz], ) ikz] φ n - half -integer ( r ) = ( r)exp( iφ), r = ( r, φ, z) [cyllindrical coordinates] Oz - vortex symmetry axis Ideal vortex, Onsager's quantization (one per Cooper pair) V v mr 1 π ( r ) = ( y, x,0) V ( r ) dr = v C m Y. Yu and A. Bulgac, PRL 90, 161101 (003)
Y. Yu and A. Bulgac, PRL 90, 161101 (003) Distances scale with λ F Distances scale with ξ F
Dramatic structural changes of the vortex state naturally lead to significant changes in the energy balance of a neutron star v v S F λf ξ ε F ε max 0.1, In low density region ε ( ρ F extremely fast vortical motion, ) ρ > ε ( ρ ) ρ which thus leads to a large anti - pinning energy out out in in E V pin > 0 : E V pin = [ ε( ρ ) ρ ε( ρ ) ρ ] out out in in V The energy per unit length is going to be changed dramatically when compared to previous estimates, by E L vortex [ ε( ρ ) ρ ε( ρ ) ρ ] out out in in πr Specific heat, transport properties are expected to significantly affected as well. Some similar conclusions have been reached recently also by Donati and Pizzochero, PRL 90, 11101 (003), NP A 74, 363 (004).
Main conclusions of this presentation: The crust of a neutron star has most likely a rather complex structure, among candidates: regular solid lattice, liquid crystal, significant number of defects and lattice distortions, disordered phase, amorphous and heterogeneous phase. The elastic properties of such structures vary, naturally, a lot from one structure to another. At very low neutron densities vortices are expected to have a very unusual spatial profile, with a prominent density depletion along the axis of the vortex. The energetics of a star is thus affected in a major way and the pinning mechanism of the vortex to impurities is changed as well.