the dominant scattering processes lead to the crossed po- larisation of in/out photons. Also, the selection rules in the presence of a magnetic field are found to be the same, with the n − → n + mode being the strongest. We estimate the intensity of one of the n − → n + scattering processes to be I$_{n}$ − → n + ∼ 10 − 12 for Ω ∼ 1eV photons in magnetic field B ∼ 10T. The most recent theoretical prediction for monolayer graphene of the intensity of the phonon-induced G peak, 36 a well known Raman feature in carbon materials, 37 estimates I$_{G}$ ∼ 10 − $^{11}$. This result is only one order of magnitude greater than the intensity of a single n − → n + peak. Therefore, spectral features of inter-Landau-level transitions in bilayer graphene pre- dicted in this paper may be observable experimentally. The electron Raman scattering in an external mag- netic field would complement infrared spectroscopy as it can give information about electronic excitations be- tween different pairs of Landau levels. The purpose of this paper was to identify and describe the dominant inter-Landau-level modes. However, additional correc- tions e.g., to the Raman-active magneto-exciton energies, due to the many-body effects neglected in the above con- siderations will be present in the spectra. Many-body corrections were, for example, observed in the infrared spectroscopy experiment performed in external magnetic fields. 31 Electronic Raman measurements could provide a test ground for some of the theoretical models proposed to account for these many-body effects. portional to the inverse square of the incoming photon energy Ω. Therefore, two peaks drawn with dot-dashed lines are roughly four times smaller on the right figure. The spectral density of the ( n ± 1) − → ( n ∓ 1) + transi- tions is smaller by a further factor ( γ$_{1}$ Ω ) 2 in comparison to the n − → n + transitions. Hence, this contribution, shown with dashed lines, is close to zero on the right fig- ure, while on the left, the first of the two smaller peaks corresponding to symmetric transitions 2 − → 0, 0 → 2 + and 3 − → 1, 1 → 3 + is still visible in the total spectral density. Because of the contrasting polarization factors in Eq. (11), contributions of different modes, n $^{−}$→ n + or ( n ± 1) $^{−}$→ ( n ∓ 1) $^{+}$, to the total spectral density could be separated using polarizers. If the polarizers were set as to collect only photons with circular polarization identical to that of the incoming photons, then the n − → n + con- tribution would be measured. However, if only the pho- tons with polarization opposite to the polarization of the incoming beam were detected, the ( n ∓ 1) − → ( n ± 1) + contribution would be determined. Increasing the filling factor leads first to the 2 − → 0 and 3 − → 1 transitions being blocked when LLs with n = 0 and n = 1 are completely filled. Therefore, the height of the two corresponding ( n ± 1) − → ( n ∓ 1) + peaks is halved (transitions 0 → 2 + and 1 → 3 + are still allowed). Next to disappear are the first n − → n + peak, that is 2 $^{−}$→ 2 $^{+}$, and the remains of the first ( n ± 1) $^{−}$→ ( n ∓ 1) + peak, (due to the 0 → 2 + transition) because of the filled LL 2 $^{+}$. Complete filling of each following Landau level results in the disappearance of the next n − → n + and ( n ± 1) $^{−}$→ ( n ∓ 1) + peaks. The contact interaction scattering amplitude δ R , il- lustrated using Feynman diagram in Fig. 1(d), results from the second term in the interaction Hamiltonian (3) and corresponds to the one-step Raman processes. It is characterised by operators ∂ 2 ˆ H/∂p$_{i}$∂p$_{j}$ and hence, the bilayer graphene Hamiltonian in (1) does not allow any contact interaction processes, as it includes only terms linear in the electron momentum p . The only contribu- tion to the contact interaction comes from the addition δ ˆ H , which contains terms quadratic in p . However, such contribution involves prefactor v 2 γ$_{0}$ Ω and therefore leads only to small corrections in the intensity of Raman scat- tering of photons with energy less than the band-width of graphene, ∼ 6 γ$_{0}$ . In fact, the contribution to the scat- tering amplitude due to the contact interaction obtained We presented a theory of inelastic scattering of pho- tons in bilayer graphene accompanied by the excitation of electron-hole pairs . Similar to monolayer graphene, This project has been funded by the EPSRC grants EP/G041954 and Science & Innovation Award EP/G035954. Acknowledgments Appendix: The contact interaction FIG. 3: Comparison of electronic contributions to the Raman spectra in neutral bilayer graphene for two different energies of incoming photons: (a) Ω = 1eV, and (b) Ω = 2eV. For each case, total spectral density g ( ω ) and contributions due to the n − → n + and ( n ± 1) − → ( n ∓ 1) + modes are shown in the solid, dot-dashed, and dashed line, respectively. Intensity scale is the same on (a) and (b); values of the parameters used: v = 10 $^{6}$m/s, γ 1 = 0 . 4eV, B = 10T, and Γ = 0 . 012eV. 6 GLYPH<G77> GLYPH<G77> GLYPH<G77> GLYPH<G77> IV. SUMMARY