1. Corrections to the formula for Compton rotation in magnetic field A.I. Sery, Brest State A.S. Pushkin University (Brest, Belarus) XIII Gomel School «Actual Problems of Microworld Physics» 03.08.2015

2. I. Introduction

3. Model of the process: photon is moving in spin polarized electron gas in strong MF (at NS surface, for example). Rotation of the plane of polarization is possible (a kind of optical activity of plasma). What kind of rotation dominates – the Faraday or Compton one?

4. Compton rotation was predicted by V.G. Baryshevsky and V.L. Luboshitz in 1965; experimentally discovered in 1970s Барышевский, В. Г. Ядерная оптика поляризованных сред / В. Г. Барышевский. – М. : Энергоатомиздат, 1995. – 320 с. Лобашев, В. М. Экспериментальное наблюдение вращения плоскости линейной поляризации -квантов в намагниченных ферромагнетиках / В. М. Лобашев [и др.]. // Письма в ЖЭТФ. – 1971. – Т. 14. – С. 373–376. Lobashev, V. M. Rotation of the plane of polarization of quanta and leftright asymmetry of scattering by thick magnetized scatterers / V. M. Lobashev [et al.]. // Sov. Phys.·JETP. – 1975. – Vol. 41, No.4. – P. 606–609. Bock, P. Observation of the Faraday Effect with 230 keV and 330 keV Photons / P. Bock, P. Luksch // Lett. Nuovo cimento. – 1972. – Vol. 2, № 21. – P. 1081–1084.

5. The difference between Faraday and Compton rotation

6. Experiments for CR were carried out for the case when 2µ B B/ħ << . Astrophysical aspect is interesting because the effect is related to the study of the characteristics of circumstellar medium, the sources of X- and  -radiation, and the case 2µ B B/ħ ~  is possible (the idea of V.G. Baryshevsky and V.V. Tikhomirov)..

7. The algorithm from the following article has been used: Фомин, П.И. Резонансное комптоновское рассеяние во внешнем магнитном поле / П.И. Фомин, Р.И. Холодов // ЖЭТФ. – 2000. – Т.117, вып. 2. – С. 319–325. Phomin, P.I. Resonant Compton scattering in external magnetic field / P.I. Phomin, R.I. Kholodov // JETP. – 2000

8. Electron propagator contains the following expressions (multiplied by plane waves): The functions U n are expressed through Hermitian functions is substituted either for g (transferred 4-momentum for r- scattering) or for f ( transferred 4-momentum for s-scattering)  i are Dirac matrices,  i are expressed through 

9. The main task is to calculate the difference of the following constructions:

10. II. Results

11. If resonance widths on the intermediate Landau level are not considered then at total spin polarization of electrons we have (here  is the angle between k and B):

12. If ħ  < m e c 2 then only R-resonance is considered

13. The following changes are made for R-diagram (the pole is removed from the real axis on the complex energy plane)

14. Angle of rotation in p z =0 approximation at n e = 10 22 cm -3 when 2µ B B/ħ ~ 

15. If p z  0, then the averaging procedure gives (T=0)

16. for the integrals we have

20. Angle of Compton rotation

21. The difference between the previous and new results is very small (it is exactly zero at  = 0; see examples for  =10 -13, N = 200000 points for numerical integration):

22. III. Remarks and conclusions

23. If other conditions are equal then

24. Important remarks For high degree of ionization, measuring the position of polarization plane at different ,  and estimating B (due to Zeemann splitting, gyrolines, etc.), at low T (when kT/E F << 1), one can estimate n e (applying the formulae for Compton and Faraday rotation) The obtained formulae are useful in studying white dwarfs and neutron stars, namely, their magnetic fields and the structures of their atmospheres

25. Important remarks (continued) At finite T (it can be estimated by spectroscopic methods) another averaging procedure must be used, and it leads to another formula for the angle of rotation.

26. Important remarks (continued) The difference between the p z =0 approximation and the averaging on p z is small The contribution of  =1, =3 and  =3, =1 is negligibly small in comparison with the contribution of  =1, =2 and  =2, =1

27. Conclusions the value of Compton rotation changes its sign at resonance (  0 is the corresponding frequency) there are 2 peaks of the value of rotation - 1 at each side of  0  0 decreases with increase of  (the angle between k and B)  0 increases with increase of B the maximum value of Compton rotation at resonance decreases with increase of either  or B

28. Angle of Compton rotation

29. You can find the details of the calculations in the articles Серый, А.И. О комптоновском вращении при движении фотонов под произвольным углом к линиям индукции магнитного поля. / А.И. Серый // Веснiк Брэсцкага унiверсiтэта. Серыя 4 «Фiзiка. Матэматыка». – 2011. – № 2. – С. 43 – 48. Серый, А.И. О комптоновском вращении в магнитном поле с учетом ширины резонанса. / А.И. Серый // Веснiк Брэсцкага унiверсiтэта. Серыя 4 «Фiзiка. Матэматыка». – 2012. – № 2. – С. 30 – 36. Серый, А. И. О некоторых поляризационных эффектах в астрофизической плазме / А. И. Серый // Веснiк Брэсцкага унiверсiтэта. Серыя 4 «Фiзiка. Матэматыка». – 2014. – № 1. – С. 30–43. Sery, A. I. To the Problem of Compton Rotation of Photons in a Strong Magnetic Field: Limit of Total Spin Polarization of Electrons / A. I. Sery // Nonlinear Phenomena in Complex Systems. – 2014. – Vol. 17, № 4. – P. 420–422.

30. Thank you for your attention !