Bremsstrahlung of fast electron on graphene Astapenko V.A. Moscow Institute of Physics and Technology, Russia, astval@mail.ru Krotov Yu.A. “Polyus” R&D Institute named after M.F. Stelmakh, Russia, kryuri@yandex.ru

Fig.1. Static and Polarization Channels of Bremsstrahlung. M,Q Pf M,Q Pf P,M,Q-momentum, k mass and charge of a projectile; Pi Ra Pi Ra m,e k – momentum of 2 k a Bs photon, dW= ASBs APBs Ra – atomic radius, 2 -projectile particle, k m,e -target particle. Pi M,Q Pf Pi Pf k The are two components in Bremsstrahlung of charged particle on a atom. One component is the Bremsstrahlung of a projectile on Coulomb force center (Bethe and Heitler). The second component is the radiation caused by the fact that the incident particle polarizes electronic shells of atom and this polarization is also the sours of the radiation. This second component is called “polarization Bs” (Buimistrov, Amusia, Tsytovich, Zon). These two channels of Bs simultaneously exist and it is importance not only to calculate and estimate each of them but also to take into account the interference between them.

For relativistic projectile particles the essential distinction in the bremsstrahlung directional diagrams for both channels was shown on Fig. 2. A relativistic projectile emits in a narrow cone of angles Δθ~Mc2/E~1/ γ along its momentum, while the polarization of target electrons emits practically isotropically, and this allows the interference contribution of both bremsstrahlung channels in the ultra-relativistic case to be neglected

3-D crystal The cross-section of a photo process on an ensemble of target atoms looks like (in case of a monatomic target) (1), where the sum is over all target atoms being in the volume of interaction, is the differential cross-section of the process on one atom. The structure factor of a medium in a three-dimensional case (a three-dimensional single crystal, the angle brackets mean averaging over atom positions): (2), where N is the full number of atoms in the volume of interaction, g is the wave vector of a reciprocal lattice, S(q) is the normalized structure factor of a unit cell of a crystal on the wave vector q ,S(q=0)=1, u2 is the mean square of thermal displacement of an atom. Following M.L. Ter-Mikaelyan who has solved this problem for the static bremsstrahlung of the relativistic electron in a single crystal, we will refer to the bremsstrahlung channel which is described by the first term in (2) as the incoherent channel, and to the second channel as the coherent channel. We see that each of these channels is subdivided into static and polarization channels. In case of the coherent channel, not only the law of conservation of energy takes place, but also the law of conservation of quasi-momentum takes place.

The coherent contribution of the static bremsstrahlung amplitude (CSBs) corresponds to the well known coherent bremsstrahlung in a crystal (CBs) [Ter-Mikaelyan, Uberal, Feretty ], and the coherent contribution of the polarization bremsstrahlung amplitude (CPBs) in higher frequency limit corresponds to parametric X-ray radiation (PXR) [ Finberg, Thignyk, Garibyan, Yan Shi, Baryshevsky]. These two channels of photon emission had been long studied theoretically and experimentally as independent channels. The reasons why the CPBs and CSBs channels had been so long considered independently are probably that in crystals the photon emission of fast particles is usually considered, and for them the directional diagrams of these channels differ essentially. The characteristic frequency ranges, in which CBs and PXR are usually observed, differ considerably also. Now the situation changes. The experiments with nonrelativistic (50-100 keV) and low-relativistic (1-5 MeV) electrons in crystals began to be calculated with consideration for the interference of both bremsstrahlung channels (Nasonov’s group -Belgorod, Feranchuk’s group –Minsk……. In fact, the consistent theory of bremsstrahlung on an individual atom with consideration of the polarization contribution now has became the microscopic basis for the majority of mechanisms of photon emission from a charged particle in different media.

2D crystal – graphene Fig. 4. The crystal structure of graphene. A unit cell (CDEF) and elementary translation vectors (e1, e2) are shown   nm is the lattice constant for graphene, nm is the distance between the nearest atoms (the distance between the atoms in a unit cell, graphene has two atoms in a unit cell). Graphene is a two-dimensional allotropic modification of carbon formed by a layer of carbon atoms with a thickness of one atom, these atoms are connected to a hexagonal two-dimensional crystal lattice. Graphene can be presented as one graphite plane separated from a bulk crystal. Why graphene? First of all, it is a perfect two-dimensional crystal where a transmitted momentum is discrete in the plane and is arbitrary across it. What can it give in coherent bremsstrahlung processes? What is a the structure factor of a unit cell of graphene? The question of the temperature dependence of the squared deviation of an atom from the equilibrium position in a graphene film is also important. How will it be manifested in coherent and incoherent channels? Finally, active work is now underway on obtaining large-size loose-hanging (self-supporting) graphene films, which allows looking forward to the performance of an experiment.

Structure factor of unit cell of graphene We assume that an atom A is at the origin of the coordinates, then It follows from the figure that Then for the vectors of the reciprocal lattice of graphene we have , and the structure factor of a unit cell of graphene is Here from we find (3) .

Generalized dynamic polarizability of a carbon atom The sum over the virtual states of atomic electrons can be expressed in terms of the generalized dynamic atom polarizability that shows a resonance nature at atomic frequencies. In the multiplicative approximation the generalized dynamic polarizability of an atom is expressed in terms of dipole polarizability and an atomic form factor: (4), the atomic form factor is calculated on the Slater wave functions of atomic orbitals for carbon. , (5)   where Nj is the number of equivalent electrons in the j th atomic shell, β and µ are the Slater parameters. As it was shown in the work [Shevelko V.P., Tolstikhina I.Yu., Stolker Th. Stripping of fast heavy low-charged ions in gaseous targets // NIM B. - 2001. V. 184. P. 295-308.], the form factor calculated in such a manner differs from its Hartree-Fock analog by no more than units of percents.

To obtain the frequency dependence of the imaginary part of the dipole polarizability, we proceed from its relation with the emission absorption cross-section given by the optical theorem: The data on the emission absorption coefficient given at the site of the Berkeley National Laboratory are used. The real part of the atom polarizability can be restored with the use of the Kramers-Kronig relation. The results of calculation of the dynamic polarizability of a carbon atom are presented here.

Bremsstrahlung of electron scattering by the two-dimensional plane of graphene. The process geometry is shown in Fig. 5 is the angle of photon emission with respect to the normal to the plane of graphene, is the polar angle of electron incoming with respect to the normal to the plane of graphene, is the azimuth angle of electron incoming. v k z ( 00 1 ) x 0) y (0 It was found that the weak bonding of graphene planes in graphite makes it possible to a good accuracy (about 5-10%) to consider the phonon spectrum of graphene coinciding with that of graphite, neglecting its logarithmic divergence for large values. Thus it is possible to consider temperature deviations of atoms in graphene as those in graphite (Yu.E. Lozovik, UPS 2008)

Incoherent PB on graphene The cross-section of incoherent PB on a target (in terms of one atom) is (6) where , , μ is the reduced mass of an electron and an atom of the target, (7) here results is presented

Coherent PB on graphene The cross-section of coherent PB on a two-dimensional target (in terms of one atom) is (8) where (9) , (10) Let us note that in contrast to coherent PBs in a three-dimensional single crystal, when the emitted frequency is fixed, the frequency of coherent PBs in a two-dimensional single crystal is not a fixed value. Nevertheless, in a two-dimensional case with fulfillment of certain conditions the PBs spectrum has sharp maxima. The frequencies of these maxima are determined by the zeros of the denominator in the expression (9). leaves on a mass envelope

Frequency of a spectral maximum in coherent PB on graphene The resonance condition in the cross-section of coherent PB in the general case gives: (11) In case of the zero angle of electron incoming (ψ=0 ) into a two-dimensional single crystal, we have for the function determining the dependence of the resonance frequency of emission on the electron velocity and the angle of photon emission: (12)

Zero line on this figure is a boundary between resonance and nonresonance areas. Above zero line resonance conditions are executed Under zero line the resonance does not exist Fig. 6 The dependence of the determinant in the expression (12) on the electron velocity (β) for different angles of emission (α).

Fig. 7 In low frequency limit we see the manifestation of generalized dynamic polarizability of a carbon atom/ The comparison of the cross-sections of coherent and incoherent PB on graphene and on a carbon atom for an electron energy of 58 keV (v = 60 atomic units, the normal incidence of an electron on the graphene plane (=0), and an angle of emission of 30).

Fig. 9. The dependence of the cross-section of coherent PB on graphene on the angle of electron incoming, =30, for the electron velocity v = 60 atomic units (energy = 58 keV) and different photon energies.

Coherent static (ordinary) bremsstrahlung on graphene The expression for the differential cross-section of coherent SB on graphene is: (13) where , . Summation over the reciprocal lattice vectors g in the formula (13) reduces to summation over the set of the integers (n1 , n2 ) .

Incoherent static (ordinary) bremsstrahlung on graphene For the incoherent part of the cross-section of SB on graphene we have (14) The difference of incoherent channels of PB and SB on graphene (Fig. 15) is about two and a half orders of magnitude at the maximum of the spectral dependence of PB. The latter fact is connected with the fact that the polarization channel is formed at great distances from a target, the contribution of these distances to incoherent emission is suppressed by the Debye-Waller factor that is small for q<1/u (high impact parameters).

Fig. 10. The comparison of the spectral cross-sections of incoherent PB (continuous red curve) and SB (dotted blue curve) of an electron scattered on graphene, v = 100 atomic units, (energy= 240 кэВ) the angle of emission is α=30. The difference of incoherent channels of PB and SB on graphene is about two and a half orders of magnitude at the maximum of the spectral dependence of PB. The latter fact is connected with the fact that the polarization channel is formed at great distances from a target, the contribution of these distances to incoherent emission is suppressed by the Debye-Waller factor .

Results Polarization Bremsstrahlung (PB) and Static (ordinary) Bremsstrahlung (SB) of fast electron scattered on graphene are investigated theoretically. The coherent and incoherent interactions between electron and two-dimensional graphene lattice so as dynamical polarizability and form-factor of carbon atom core are taken into account. In specified region of the problems parameters sharp maxima in PB spectrum is predicted . For normal electron incidence on graphene plane the analytical description of resonance frequencies on velocity and radiation angles is obtained. Our analysis show that dominated Bremsstrahlung channels of fast electron on graphene are coherent PB and incoherent SB.

Fig The comparison of the cross-sections of coherent and incoherent PB on graphene and on a carbon atom for an electron energy of 4 MeV (v = 136 atomic units), the normal incidence of an electron on the graphene plane (=0), and an angle of emission of 30.