For acceleration voltages routinely used in the Transmission Electron Microscope (TEM), the kinetic energies of the beam electrons encompass a significant portion of their rest energies. Consequently relativistic corrections can become important for image calculation, as seen for example in relativistic calculations of electron energy loss images [1,2]. Therefore one has to describe the scattering process between a beam electron and an electron in the specimen using a completely relativistic formalism.

However, since the electrons in the specimen are usually non-relativistic, the question arises which relativistic corrections are important for a non-relativistic specimen electron excited by a relativistic beam electron. To examine this problem, one can describe the excitation of a specimen electron using a transition potential, as is often done in multislice simulations of inelastic scattering [3,4]. We derive a relativistic generalization of the transition potential to elucidate which relativistic corrections are important for the specimen electron.

In first order perturbation theory, an excitation event can be described by eq. (1), where H_{e} is the Hamiltonian of the beam electron and H_{i} is the Hamiltonian of the interaction between beam electron Ψ and specimen atom χ. The last term in eq. (1) can be used to define the transition potential, which in this case describes the transition of the specimen atom from the ground state 0 to the excited state m≠0.

Inserting non-relativistic quantities for H_{e}, H_{i}, χ and Ψ yields the well known Yoshioka equations [5]. Using relativistic quantities in eq. (1), we obtain the Dirac equation (2). Here, A_{μ} is the electromagnetic four-potential of the specimen, while {A_{m0}}_{μ} is the relativistic transition potential and γ^{μ}=(γ_{0}, γ_{1}, γ_{2}, γ_{3}) are the Dirac matrices. Since the relativistic transition potential has to satisfy an inhomogeneous wave equation, one way to obtain it is to solve eq. (3), for example with the help of the Green’s formalism. This leads to eq. (4) (see also [6]), where φ is the four-spinor of an electron in the specimen and ΔE_{m0 }is the energy difference between the states 0 and m divided by hc.

To calculate the matrix element I^{μ}(q) (see eq. (6)), the four-spinors of the excited specimen electron before and after the excitation by the beam electron are required. We describe the specimen electrons using Darwin wavefunctions [7]. These wavefunctions, shown in eq. (5) for the spin up and the spin down case, are an approximation of the four-component spinor in the non-relativistic limit. They relate the (relativistic) four-spinor φ to the (non-relativistic) Schrödinger wavefunction Φ.

One can now insert, for example, the spin up Darwin wavefunction into I^{μ}(q), obtaining eq. (6). Here, I_{0} is a matrix element also occurring in the non-relativistic calculation, while the other terms are relativistic corrections. To assess which correction terms are important, we will evaluate the integrals I_{0} ,I_{x }, I_{y} ,I_{Z} and I_{Δ}. The resulting transition potentials can be used for relativistic multislice simulations.

[1] R. Knippelmeyer et al., J. Microsc. 194 (1999) 30

[2] C. Dwyer, Phys. Rev. B 72 (2005) 144102

[3] C. Dwyer, Ultramicroscopy 104 (2005) 141

[4] L. J. Allen et al., Ultramicroscopy 151 (2015) 11

[5] H. Yoshioka, J. Phys. Soc. Jpn. 12 (1957) 618

[6] A. Lubk, PhD thesis, TU Dresden (2010)

[7] C. G. Darwin, Proc. Roy. Soc. A 118 (1928) 654

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