2. Cyclotron Scattering Resonance Features

An electron moving in a uniform magnetic field will be acted upon by the Lorentz force

$$\bf F {{\bf v}\times{\bf B}\over c}$$ (25)

and will describe a helix about the field lines with radius $$R = {v_\perp m_e\over e B}$$
where v_$\perp$ is the component of the electron's velocity normal to the magnetic field. A quantum treatment is required if the de Broglie wavelength of the electrons is comparable to R. In this case the transverse energy of the electrons is quantized into Landau levels. An electron in a Landau level is in a bound state so it does not emit synchrotron radiation. Because the lifetimes of the excited Landau levels are short compared to the time for collisional de-excitation, radiation is scattered out of the beam rather than being absorbed. The transition energy is

$${e\hbar\over m_e}$$ (26)

where B₁₂ is the magnetic field in units of 10¹² G. In the accretion column of an accreting X-ray pulsar we expect that the magnetic field is strong enough that E_Cy > kT. In this case almost all of the electrons will be in the ground Landau level.

The scattering cross-sections of the magnetized plasma depend on the polarizations of the incident and scattered photons. The polarization of an electromagnetic wave in a magnetized plasma can be represented as a linear combination of two circularly polarized normal modes. The right-hand circularly polarized mode is known as the extraordinary wave. Its E-field vector rotates in the same sense as the electron gyration. Thus the cross-section for scattering the extraordinary wave will have a resonance as $\omega$$\to$$\omega_{\rm Cy}^{}$ (Mészáros (1992)). The left-hand circularly polarized mode is the ordinary wave. In a cold, magnetized plasma its cross-section is not resonant. Vacuum polarization effects will modify the cross-sections. Virtual pairs of electrons and positrons will participate in the radiation scattering. The result is that both the ordinary and extraordinary modes will be resonant (Ventura et al. (1979)). Ventura et al. (1979) also found that scattering with conversion between the ordinary and extraordinary modes is important. This process enhances the depth of the cyclotron resonance feature in the emergent spectrum. The cross-section for cyclotron scattering of an extraordinary-mode photon to an extraordinary-mode photon is $$A_{\rm Cy} = {\tau_{\rm Cy}\left({{W_{\rm Cy}\over E_{\rm Cy}}E}\right)^2\over (E-E_{\rm Cy})^2+W_{\rm Cy}^2}.$$
Natural broadening due to the short lifetime of the excited landau levels will make W_Cy at least 100 eV. If the plasma is hot, Doppler broadening will make

$$\sqrt{kT\over m_ec^2}$$ (27)

Other factors that will broaden the resonance are a non-uniform magnetic field or contributions from scattering regions at different heights above the neutron star's surface. If the width of the resonance was due to Doppler broadening alone, the electron temperature of the plasma would be

$$\left(\vphantom{{W_{\rm Cy}\over E_{\rm Cy}}}\right. {W_{\rm Cy}\over E_{\rm Cy}} \left.\vphantom{{W_{\rm Cy}\over E_{\rm Cy}}}\right)^{2}_{}$$ (28)

This is strictly an upper limit because of the other broadening mechanisms. Figure 4.2 shows a calculated spectrum for an AXP from Alexander and Mészáros (1991). The fundamental resonance is at $\omega_{c}^{}$ and the harmonics at 2$\omega_{c}^{}$ and 3$\omega_{c}^{}$ are due to transitions into higher Landau levels. The calculation includes the effects of vacuum polarization, two-photon processes, and stimulated emission.

Figure 7: Calculated spectrum of an AXP with a CSRF (Alexander and Mészáros (1991)).