On the Problem of Normal Nucleon Fermi-liquid with Fermi pseudopotential and One-Pion-

Exchange Potential

A. I. Sery,

Brest State Pushkin University, Belarus

Xth Gomel School July 2009

The origin of neutron stars’ magnetic fields can be explained both by the presence of proton

superconducting currents, and by spontaneous nucleon spin polarization in the liquid core

• V. G. Baryshevsky and M. I. Podgoretzky predicted nuclear pseudomagnetism in 1964

• After that V. G. Baryshevsky proposed the idea that spin-polarized state of nucleon system can turn out energetically preferrable in comparison with non-polarized one

• I. e. it’s an example of spontaneous symmetry violation

The approach to the problem of spin polarization of nucleons and its link with ordinary magnetic field

before and after the idea of V. G. BaryshevskyBefore 1960s The essence of the idea of V. G. Baryshevsky

external magnetic field

must be present certainly (i. e. it is

primary)

can be absent

spin polarization of nucleons

owing to external magnetic field (i. e. it is secondary)

can turn out energetically profitable owing to nuclear interaction between nucleons (i. e.

it is primary)

«polarizational» magnetic field

as a correction for the external

one

in the absence of the external one can be the only one

nuclear pseudomagnetic

field (NPMF)

it hadn’t been taken into

consideration at all

owing to spin polarization of nucleons; its effect on an individual nucleon energy is

much stronger than the effect of ordinary magnetic field

The difference between MF and NPMF can be considered by the questions

• 1. What fundamental interaction is the field caused by? 2. Can it influence an electron, proton, neutron? 3. Can it be created by moving charged particles? 4. Can it exist in nucleon medium without spin polarization? 5. The relationship between field induction B, degree of particles’ polarization р0 and their concentration N (μi – particles’ intrins. magn. moments, μn – intrins. magn. mom. of a moving neutron). 6. Is particle energy quantization (after Landau) in such a field possible? 7. Can it exist in vacuum (i. e. far from sources)? 8. Is it correct to speak about such a field, generated by a single particle (either a one at rest, or a moving one)?

The main differences between ordinary magnetic field and nuclear pseudomagnetic

field

Magnetic field NPMF

1 ElectromagneticNuclear (namely spin

dependence of nuclear forces)

2Yes, because they have

intrinsic magnetic momentsNo (doesn’t come in nuclear

interaction); yes; yes

3Yes (according to Biot-Savart-

Laplace law)No, electric charge has

absolutely nothing to do here

4Yes, if there is an ordered

motion of protons

No, because the presence of spin polarization is the only factor

causing NPMF formation

The main differences between ordinary magnetic field and NPMF (continued)

Magnetic field NPMF

5В ~ Nр0, namely:

В = 4πμnNр0

В ~ Nр0, namely:

В = πħ2Nβр0/(mμn) (in case of NPMF of nucleons,

which have spin ½); m – nucleons’ mass, β depends on nuclear scattering amplitudes

6 Yes No

7

Yes, because electromagnetic

interaction radius is infinite

No, because nuclear interaction radius is finite

8

Yes; even a single neutron at rest,

owing to its intrinsic magnetic moment, creates ordinary

magnetic field around itself

No; an individual nucleon creates a short-range field of nuclear forces around itself, while NPMF is a

collective phenomenon: it arises as a result of spatial averaging of nuclear forces’ fields, created by individual nucleons, the averaged component

of potentials, which depends on directions of spins, turns out to be nonzero

The first papers on the problem of the possibility of spontaneous spin polarization of nucleons appeared in late 1960s, i. e. already after the

discovery by V. G. Baryshevsky and M. I. Podgoretzky. The results are very different. Let’s observe the main methods and results for neutrons

methods examples of models and potentials spontaneous polarization is possible

Fermi-gas a) hard spheres; b) NPMF in the framework of Fermi

pseudopotential or OPE potential

a) at nn = 0.41 fm– 3 (M. J. Rice,

1969); b) most likely, no (*2006, 2008)

Fermi-liquid different Skyrme models at n ~ 0.1 1 fm– 3 (A. A. Isayev et al., 2004)

magnetic susceptibility calculation

a) Skyrme effective forces; b) Argonne v18 (two-body), Urbana

IX (three-body); c) Stoner criterion for Fermi

pseudopotential

a) at n = 0.18 0.26 fm– 3 (A. Viduarre et al., 1984); b) most

likely, no (S. Fantoni et al., 2001); c) most likely, no (*2006)

variants of relativistic

Hartree-Fock (HF)

Dirac-HF with effective nucleon-meson Lagrangian

at n ~ 1038 1039 cm– 3 (S. Marcos et al., 1991)

Brueckner-HF with Nijmegen II, Nijmegen NSC97e and Reid93

most likely, no (I. Vidana et al., 2002)

And now let’s observe the main results for neutron-proton matter

method examples of models and potentials

spontaneous polarization is possible

Fermi-gas NPMF in the framework of Fermi pseudopotential or OPE

potential

at nn ~ np ~ 1035 cm– 3 (*2006); in

the 2d case it can even be -equlibrium (np ~ 1037 cm– 3, np ~

1034 cm– 3) (**2008, 2009)

Fermi-liquid different Skyrme models at n ~ 1038 1039 cm– 3 (A. A. Isayev et al., 2004)

magnetic susceptibility calculation

a) Skyrme effective forces; b) Stoner criterion for Fermi

pseudopotential

a) at n ~ 1038 1039 cm– 3 (A. Viduarre et al., 1984); b) at nn

~ np ~ 1035 cm– 3 (*2006)

relativistic Hartree-Fock

(HF)

a) Dirac-HF to strongly asymmetric matter; b)

Brueckner-HF with Nijmegen II, Nijmegen NSC97e and

Reid93

a) for protons if neutron-proton spin interaction exceeds some threshold value; b) most

likely, no (I. Vidana et al., 2002)

For example, let’s see the results for beta-equlibrium polarization with oppositely directed spins of protons and neutrons in the framework of

Fermi-gas approach with NPMF in case of OPE potential

Fermi-liquid can be either normal or superfluid. We’ll consider a normal one.

• Similarity of description for ideal Fermi-gas and Fermi-liquid:

• а) distribution functions formally have the same appearance:

• n(p) = ((E(p) - µ)(kT)-1 + 1) -1 ,• but for Fermi-liquid the expression for E(p) is not

known beforehand;• б) Fermi momentum is determined by the same

formula (according to Landau theory, 1956)

• pF = ħ(32)1/3(N/V)1/3

The main differences for ideal Fermi-gas and Fermi-liquid

• Fermi-gas• Particle• Interaction function

is zero• Particle energy

doesn’t depend on momentum distribution of other particles

• Fermi-liquid• Quasiparticle• Interaction function

is nonzero• Quasiparticle

energy depends on momentum distribution of other quasiparticles

A considerable contribution to working out Fermi-liquid theory was made by Landau,

Akhiezer, Peletminsky and others

А. А. Isayev elaborated the algorithms of calculation of polarization conditions for normal neutron-proton (and simply neutron) Fermi-liquid ignoring beta-equilibrium in the framework of Skyrme model, where a quadratic dependence on relative momenta of 2 nucleons takes place for the Fourier transforms of nucleon-nucleon

potentials

However, V.G.Baryshevsky and V.V. Tikhomirov proposed to apply a superposition of Fermi

pseudopotential and one-pion-exchange potential

Why just so? • Fermi pseudopotential is very simple (delta-function,

multiplied by a constant containing scattering length), and it was specially constructed to explain cross-sections of low-energy nucleon-nucleon scattering

• OPE potential at distances of 2-4 fm gives practically the same results, as does Hamada-Johnson potential, which regards one the most detailed and precise for nucleon-nucleon scattering description

• But it contains not quadratic, but more complex dependence on momenta, as a result – integrating becomes more difficult

• Besides, the approach assumes the absence of tensor terms in potentials, and the absence of the dependence on orbital momentum also

A positive result for Fermi-gas method with the same potentials allows to hope, that it will be similar for Fermi-liquid method

The main points of the method (see the details in posters)

• Neglecting the tensor part of OPE potential, because averaging over angles gives zero

• Expanding the Fourier transform of the potential over Pauli matrices

• Constructing normal Fermi-liquid amplitudes• Writing expressions for 2 or 4 components of

normal distribution and for normalization condition

• Obtaining of self-consistent system of equations• Substitution of integration for summation

There are 4 equations and 6 unknown quantities in self-consistent system for neutron liquid (the

dimensions of matrices and so-called «vectors» are 2 times larger for the neutron-proton one)

Temperature T can be taken as free parameter

There are 8 equations and 11 unknown quantities in

self-consistent system for neutron-proton liquid

• Temperature and isospin asymmetry parameter can be taken as free parameters Then total concentration, 2 degrees of polarization, 2 chem. potentials, mathem. expectations of the squares of the transferred momenta over 4 components of Fermi-Dirac distribution function remain for neutron-proton system

• I’m sorry, but I had no enough time to make precise calculations; though preliminary estimates show, that if at nn ~ 102 np polarization really exists, than, most likely, the spins of protons and neutrons are oppositely directed (the result is qualitatively similar to the one obtained in Fermi-gas method), besides neutron polarization is negligibly small, and protons are almost totally polarized; if it is really so, then ordinary magnetic field, generated by protons, can reach the order of 1014 Gs in pulsars or magnetars

The main conclusions• if we apply Fermi pseudopotential or OPE potential then the

threshold density for spontaneous neutron-proton system polarization (ignoring beta-equilibrium) is about 3 orders lower (0.0001-0.001 fm-3) in case of Fermi-gas method with NPMF or in case of magnetic susceptibility calculation in comparison with any other method predicting nuclear ferromagnetism (0.1-1 fm-3)

• there is only one type of polarization predicted for these potentials – when proton spins are directed oppositely to neutron spins (though co-directed spins are predicted for some Skyrme models in the Fermi-liquid approach)

• these potentials and methods do not predict spontaneous polarization for pure neutron matter (though some Fermi-liquid methods with Skyrme models do)

• there is a hope that the application of these potentials to Fermi-liquid method is going to give similar results (the corresponding self-consistent systems of equations have been already obtained)

Thank you for your attention !