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A.I. Sery Brest State A.S. Pushkin University
XXVII International Seminar βNonlinear
Phenomena in Complex Systems" 19-22 May 2020 Minsk-Sosny, Belarus
Baryshevsky-Luboshits effect is the rotation of photon polarization plane in spin-polarized electron gas explained by the difference of Compton scattering amplitudes for the same and opposite spin directions of photon and electron. Baryshevsky-Luboshits effect was predicted in 1965 and experimentally discovered in 1970s.
V.G. Baryshevsky. Nuclear Optics of Polarized Media [in Russian]. Moscow, Energoatomizdat, 1995. β 320 p. β P. 88-99.
Spin polarization in astrophysics (especially in the outer layers of DA white dwarfs) is predicted not only for electron gas but for proton gas also.
Sery, A. I. Spin Polarization of Nucleons: Limits of Low and High Temperatures / A. I. Sery // Bulletin of the Russian Academy of Sciences. Physics. β 2015. β Vol. 79, No 4. β P. 506β512.
As far as Compton effect for protons is also possible (like for electrons), itβs interesting to estimate the contribution of spin-polarized protons to Baryshevsky-Luboshits effect.
If a photon passes the distance ππ₯ and its plane of linear polarization turns on the angle ππ , the following denotations will
be used for ππ
ππ₯ calculation:
- electromagnetic coupling constant; B β Bohr magneton;
= B 2 β anomalous magnetic moment of electron;
ne β electron number density; np β proton number density;
π e β electron spin polarization vector; π p β proton spin polarization vector;
me β electron mass; mp β proton mass; - photon frequency;
π β unit vector in the direction of photon propagation;
Denotations
Here ππ π is the spin-dependent part of Compton electron scattering amplitude, πββ π
β² and πββ(Οβ²) are total Compton cross-sections, respectively, for the same and opposite spin directions of photon and electron.
ππ
ππ₯=
2ππππ
ππ eπ π πππ π (1a)
Imππ πβ² =πβ²
4ππ πββ πβ² βπββ πβ²
2 (1c)
The formulae for ππ
ππ₯ calculation in electron gas are the following:
π πππ π =Δ§π2
()2+2π
Imππ πβ²
πβ²2βπ2
β
0dΟβ² (1b)
is used instead of (1b) and (1c).
V.G. Baryshevsky. Nuclear Optics of Polarized Media [in Russian]. Moscow, Energoatomizdat, 1995. β 320 p. β P. 93-94.
For the π limit the formula (1a) doesnβt change but the approximate expression
π πππ π 2
8
Δ§
πππ (1d)
The approximate expressions for Reππ π at the limits of low
and high photon frequencies are the following
ππ
ππ₯=
2ππππ
ππ ππ π πππ π (2a)
Similarly to (1a), the formulae for ππ
ππ₯ calculation in proton gas can
be obtained:
π πππ π 73
723
32
Δ§
πππ
Δ§π
ππ2
2 , Δ§ βͺ2πππ2 (2b)
π πππ π 73
18
Δ§
πππΔ§π
ππ2
, Δ§ β«2πππ2 (2c)
A.I. Akhiezer, V.B. Berestetsky. Quantum Electrodynamics [in Russian]. Moscow, Nauka, 1969. β 624 p. β P. 578.
π΅ = 4ππππππN, π =2.7928 (3a)
Proton spin polarization degree is estimated as ππ=0.9 for temperatures T=7104 K and number densities np=ne=1026 cm-3 .
Sery, A. I. Spin Polarization of Nucleons: Limits of Low and High Temperatures / A. I. Sery // Bulletin of the Russian Academy of Sciences. Physics. β 2015. β Vol. 79, No 4. β P. 506β512.
The expression magnetic field strength is (where N is nuclear magneton)
For the corresponding values of ππ and np one obtains B=2.9104 Gs . If we consider electrons to be paramagnetic and non-relativistic, then their spin polarization degree is estimated as
ππ = π‘β
π΅π΅
ππ (3b)
For the corresponding values of π΅ and T one obtains pe~310-5 .
V.S. Sekerzhitsky. Equlibrium Systems of Fermions and Bosons in Magnetic Fields [in Russian]. Brest, BrSU, 2008. β 198 p. β P. 53.
Comparing (1a) to (2a) at np=ne, one needs to compare ppπ πππ π to pππ πππ π .
If we assume the value of photon energy as 100 keV (such energies are preferable for Baryshevsky-Luboshits effect), then , according to (1b),(1c) and (2b), for absolute values we obtain
|ppπ πππ π | ~ 310-23 cm,|pππ πππ π | ~ 2210-18 cm
|pππ πππ π | >> |ppπ πππ π |
At very high photon energies, according to (1d) and (2c), one obtains that
|pππ πππ π | = |ppπ πππ π | at Δ§8.5πππ27.9 GeV.
Such energies are not realistic for photons emitted by the outer layers of DA white dwarfs.
1. The contribution of polarized protons to Baryshevsky-Luboshits effect in comparison with that of polarized electrons is small in keV and MeV range even if proton spin polarization degree is several orders higher than that of electrons.
2. Both contributions can be equal only in GeV range (if proton spin polarization degree is 5 orders higher than that of electrons) where their magnitudes are very small in comparison with that of elecrons in keV and MeV range. GeV range is not realistic for DA white dwarfs.
Conclusions
Thank you for your attention!