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Relativistic correction of atomic scattering factors for high-energy electron diffraction
aErnst Ruska Centre, Forschungszentrum Jülich GmbH, 52425 Jülich, Germany
*Correspondence e-mail: m.lentzen@fz-juelich.de
Relativistic electron diffraction depends on linear and quadratic terms in the
the latter being neglected in the frequently used relativistically corrected Schrödinger equation. The quadratic term modifies atomic scattering amplitudes in particular for large-angle scattering and backscattering. The respective correction increases with increasing scattering angle, increasing and increasing Conventional tabulations for electron scattering and its large-angle extrapolations can be amended in closed form by a universal correction based on the screened Coulomb potential squared.1. Introduction
A frequently used framework for the calculation of high-energy electron diffraction by an atom or ion is the solution of the relativistically corrected Schrödinger equation (Molière, 1947; Fujiwara, 1961
) with a model for the atomic or ionic These model potentials are tabulated for a wide range of atomic numbers and frequently occurring ionic charges in the form of scattering factors (Doyle & Turner, 1968
; Doyle & Cowley, 1974
; Rez et al., 1994
, 1997
) or their parameterizations (Doyle & Turner, 1968
; Doyle & Cowley, 1974
; Fox et al., 1989
; Rez et al., 1994
, 1997
; Waasmaier & Kirfel, 1995
; Weickenmeier & Kohl, 1998
; Peng, 1998
; Lobato & Van Dyck, 2014
); see Kirkland (2010
) for a survey. Conventionally, tables of the scattering factors display the Born scattering amplitude, that is the Fourier transform of the times an interaction constant. A relativistic correction, dependent on the electron speed, is applied to the tabulated values, which can be directly used to determine scattering cross sections on the first Born approximation.
The normal form of the relativistically corrected Schrödinger equation (Molière, 1947; Fujiwara, 1961
) is linear in the yet the correct relativistic energy-momentum relation, which is the basis of the Klein–Gordon equation (Klein, 1926
; Gordon, 1926
; Kragh, 1984
), contains an additional quadratic term in the That term is neglected in the above conventional framework, and thus, to the best of our knowledge, no tabulations exist for fully corrected relativistic scattering factors.
The aim of this work is to explore the impact of the quadratic
term on atomic or ionic electron scattering amplitudes in particular at large angles, including backscattering. Furthermore, a method is proposed to amend the existing tables for the Born scattering factors. The work presents a brief survey of the required theory, calculations for a set of atoms of small, medium and large at small, medium and large electron energy, and concludes with a discussion of possible applications.2. Theory
2.1. Wave equations
The relativistic energy-momentum relation (Einstein, 1905)
with rest energy E0 = mc2, E in vacuum, , momentum p, speed of light c, rest mass m, e and
is divided by 2(E0 + E ) and rearranged:
Thus the quadratic energy relation (1) adopts a form akin to a linear energy relation with the parameters
,
, electron speed v,
and
the relativistically modified
in vacuum.The Klein–Gordon equation (Klein, 1926; Gordon, 1926
; Kragh, 1984
) for fixed is derived by substituting the momentum operator
for the momentum p, and the relativistically corrected Schrödinger equation (Molière, 1947
; Fujiwara, 1961
) by further neglecting the squared potential term. As usual,
denotes the Planck constant h divided by
.
The scattering amplitude is derived from the wave equations by an ansatz for the wavefunction,
which describes the scattering of a plane wave with wavevector into a spherical wave with an amplitude dependent on the scattering vector
. The wavevector of a plane partial wave after scattering is thus
, and
denotes a coordinate in real space. For
and
are equal in magnitude,
and
with the scattering angle between
and
, and
the Compton wavelength.
The amplitude of the spherical wave is determined in the far field, at large distance r from a scattering region bounded by a sphere of diameter d, with . If the bounded region contains a single atom, the scattering amplitude is called the atomic form factor. The far-field solution of the wave equation in the above sense is found on the first Born approximation (Born, 1926
) to the first order of an effective potential Veff, with the well-known result:
For the relativistically corrected Schrödinger equation Veff = V, and for the Klein–Gordon equation Veff = . The tabulations of atomic form factors according to equation (8)
are used in two ways. The first, and obvious, is the display of the atomic scattering amplitude, its modulus squared being the differential scattering (see the next section). The second is the indirect, through the Fourier transform in equation (8)
, but exact display of the atomic scattering potential.
2.2. Scattering amplitudes for a screened Coulomb potential
The integral (8) for the scattering amplitude of a screened atomic Coulomb potential (Wentzel, 1926
)
with Z, Eh, a0 and screening radius (Lenz, 1954)
can be solved in closed form, with the well-known result (Wentzel, 1926):
The scattering amplitude of the squared atomic Coulomb potential term
can be found in closed form as well:
with the .
Both and
have a maximum at
,
and
and for small scattering vectors is always much larger than
. A comparison of the asymptotes for large scattering vectors,
reveals that can become equal in magnitude to
for large Z. This is particularly clear for the asymptotic values of backscattering
and
when and g adopts the largest possible value 2k. The contribution f2 to the total scattering amplitude f = f1 + f2 becomes significant for large-angle scattering and backscattering.
The differential
isfor scattering into a solid angle and azimuthal symmetry.
3. Calculation of scattering amplitudes and cross sections
Born scattering amplitudes [equation (8)] were calculated for carbon (Z = 6), germanium (Z = 32) and gold (Z = 79) at kinetic energies of 20, 200 and 2000 keV over the full range of scattering angles,
. Two different models were used for the scattering potential: the screened Coulomb potential [equation (9)
] and the screened Coulomb potential extended by the squared Coulomb potential term [equation (12)
]. The scattering amplitudes for both models, f1 and f1 + f2, are displayed in Figs. 1
, 2
and 3
. The difference between the two scattering amplitudes increases with increasing scattering angle, increasing and increasing kinetic energy.
![]() | Figure 1 Born scattering amplitudes f1 (grey) and f1 + f2 (black) on a logarithmic scale versus scattering angle for carbon, germanium and gold at a of 20 keV. |
![]() | Figure 2 Born scattering amplitudes f1 (grey) and f1 + f2 (black) on a logarithmic scale versus scattering angle for carbon, germanium and gold at a of 200 keV. |
![]() | Figure 3 Born scattering amplitudes f1 (grey) and f1 + f2 (black) on a logarithmic scale versus scattering angle for carbon, germanium and gold at a of 2000 keV. |
The difference between the two models can be further expressed by calculating Born scattering cross sections for total scattering, , which denotes the total mismatch. The relative difference between the two models is pronounced for large scattering angles, and thus it is instructive to further calculate the for backscattering,
. A third important measure is the for scattering outside the typical acceptance angle of an electron microscope, θ = 250 mrad…π.
The respective cross sections on the two models, and the relative differences, are compiled in Tables 1, 2
and 3
, again for carbon (Z = 6), germanium (Z = 32) and gold (Z = 79) at kinetic energies of 20, 200 and 2000 keV. The relative differences of the total cross sections decrease for increasing energy; they increase for backscattering and scattering outside the microscope acceptance angle with increasing energy. With increasing the relative differences increase in any category.
|
|
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4. Discussion
Conventional tables of the scattering factors f1 (Doyle & Turner, 1968; Doyle & Cowley, 1974
; Rez et al., 1994
, 1997
; Kirkland, 2010
) are organized such that the Born scattering amplitude [equation (8)
] is only tabulated for a range of scattering vectors where Rutherford scattering is modified by the effects of screening, up to, e.g., s = g/2 = 60.0 nm−1. The amplitudes for larger scattering vectors are understood to be calculated with the Rutherford formula [equation (16)
]. In a last step the tabulated values have to be multiplied by γ as the interaction constant used in the tabulations conventionally contains m and not m*.
The above standard procedure can be amended to include the effects of the squared potential term [equation (12)], thus providing a proper relativistic correction. Although f2 was calculated for the screened Coulomb potential in equation (13)
, it can serve as a universal correction, because in the range of small scattering vectors, where the details of the screening would play a role, f2 is dominated by f1. For larger scattering vectors, in the regime of Rutherford scattering [equation (16)
], f2 has the proper asymptote [equation (17)
].
Thus the conventional tables can be used to derive the proper relativistic scattering amplitude:
(i) Multiply tabulated values by γ.
(ii) Extrapolate the tabulated range through the Rutherford formula [equation (16)].
(iii) Determine the screening parameter g0 using equation (10).
(iv) Add the squared potential term f2 using equation (13).
Once the scattering amplitude is determined, a Fourier transform to real space provides the effective potential to be used in diffraction calculations on the Klein–Gordon equation. The above treatment of the squared potential term allows, however, the use of simpler algorithms for the solution of the Schrödinger equation instead.
The implementation of the proper relativistic scattering amplitude is particularly suitable for the phase grating approximation of the multislice algorithm (Cowley & Moodie, 1957). The calculation of the phase grating requires a projection of the potential along the chief propagation direction, which is achieved by evaluating the of an atomic arrangement with the component of the scattering vector along the propagation direction set to zero. Including the squared potential term would now involve a Fourier transform to real space, calculation of the squared potential and line integrations along the propagation direction, or alternatively a numerically costly convolution in Compared with the latter, the prescription given in this work provides a numerically very efficient way to determine the respective additional based on the form factors f2.
The squared potential correction [equation (13)] is obviously most significant for backscattering, as can be deduced from the scattering cross sections displayed in the rightmost columns of Tables 1
, 2
and 3
. The error by neglecting the correction can be as large as 66.7% for the case of gold at a of 2 MeV. The modification of backscattering cross sections extends, however, into the region of medium electron energies and medium to small charge numbers. A striking example is the for knock-on damage in germanium, which involves scattering angles from 2.5 rad to π for a of 400 keV to transfer the required displacement energy of 15 eV to a germanium atom. The Born for this process is 0.00316 pm2, but only 0.00189 pm2 by neglecting correction (13)
, which is a difference of 40.2%. An example of knock-on damage of a light element is oxygen displacement in magnesium oxide at a displacement energy of 55 eV. The Born for this process at 400 keV electron energy is 0.000504 pm2, but only 0.000441 pm2 by neglecting correction (13)
, which is a difference of 12.5%.
The modification of the cross sections for scattering outside the acceptance angle of an electron microscope indicates that there is also an impact on an important parameter of f1 alone.
namely electron absorption. In the bore of the objective pole-piece limits the cone of scattered electrons to a semi-angle of around 250 mrad, and thus a certain fraction of scattered intensity is missing in the image plane underneath; it appears to be absorbed by the imaging system. For larger kinetic energies and larger atomic numbers the estimate of that apparent electron absorption would be in error on the linear model5. Conclusion
The conventional framework of electron scattering by an is considered. The respective modification of atomic scattering amplitudes increases with increasing scattering angle, increasing and increasing Conventional tabulations for electron scattering (Doyle & Turner, 1968
; Doyle & Cowley, 1974
; Rez et al., 1994
, 1997
; Kirkland, 2010
) and its large-angle extrapolations can be amended in closed form by a universal correction [equation (13)
] based on the screened Coulomb potential squared [equation (12)
].
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