Relativistic theory of low energy electron diffraction: Formalism and application to KKRZ‐pseudopotentials

Solid State Communications

Müller

1980

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In low-energy electron diffraction from Pt(1 11), the longitudinal component of the spin polarization vector and its transverse component normal to the scattering plane were measured by a Mott detector and found to agree very well with corresponding theoretical results. Rotation diagrams of the longitudinal and transverse components exhibit only a three-fold symmetry in contrast to the six-fold symmetry, which time reversal invariance dictates for intensities.IN ELASTIC electron-atom scattering with an high-precision two-axis manipulator for the Pt(l 11) unpolarized incident beam of momentum k 0, spin-orbit crystal allow to feed selected diffracted beams into the coupling leads to spin polarization such that for polarization analyzer based on Molt scattering at electrons of momentum k the polarization vector is 120 keV [I). On its path between the diffraction sysnormal to the scattering plane defined by k0 and k [11.tern and the Mott-detector, the diffracted beam under In low-energy electron diffraction from crystal surfaces consideration is deflected electrostatically by 90°about of large.Z materials effects of spin polarization normal the axis normal to the scattering plane (k0, k). Since to the scattering plane are well-established the deflection leaves the polarization vector P experimentally and theoretically 121. So far, however, unchanged, the transverse component P~normal to no experimental verification is known for in-plane the scatteringplane (P~= P . n; n = k0 x k/1k0 x ki) components, the existence of which was predicted by remains transverse with respect to the new beam theory [31as a consequence of multiple scattering. We direction, while the initial longitudinal component report here, for diffraction from Pt(l 11), the first P,~= P k/jkl becomes transverse (lying in the paper measurements of a spin polarization component parallel plane, still normal to n). The scattering chamber of the to the scattering plane and compare them to correspond-Mott detector can be rotated about the incoming beam. ing theoretical results. The present spin polarization Setting the scattering plane of the Molt detector versus azimuthal angle profiles (rotation diagrams) parallel to the original scattering plane (k0, k) one further clarify a lack of rotation symmetry indicated measures P~. The component I',, is analysed, if the two by earlier results on the normal component of polarscattering planes are perpendicular to each other. ization, which were measured for a slightly different Numerical calculations were performed according diffraction geometry [4], and provide experimental to a relativistic LEED theory [6) with model assumpverification of recent predictions based on symmetry lions as in a recent application to P1(1 11) [4). The arguments [5].effective ion-core potential contains an energyMost of the experimental set-up has been dependent local exchange contribution, the inner potendescribed previously [4].The principle of measuring tial has the real and imaginary parts 12 and 4eV, and the longitudinal spin polari...

Section: In Elastic Electron-atom Scattering With Anmentioning

confidence: 99%

Longitudinal spin polarization and symmetries in low-energy-electron diffraction: Experiment and theory for Pt(111)

Bauer

Solid State Communications

Müller

1980

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“…2 TmSb crystallizes in the cubic NaCl structure and the Tm 3+ J=6 CEF splitting gives a I\ singlet ground state followed by a T 4 triplet at 25 K and a T 5 triplet at 56 K, the other excited levels lying higher than 110 K. These higher levels are neglected in our considerations throughout this work. 4 In this Letter we would like to show the first experimental evidence of a CEF effect on the thermal expansion. We have chosen TmSb because of its properties mentioned above and because elec- We report a crystal-electric-field effect on the low-temperature thermal expansion of TmSb, manifesting itself as a Schottky-type anomaly.…”

Section: R Federmentioning

confidence: 91%

“…The Korringa-Kohn-Rostoker-Ziman (KKRZ) 5 type relativistic pseudopotential used in Ref. 4 depends on radii R t -associated with the spin-up and spin-down atomic phase shifts 5^-which have to be optimized individually in a numerical application of the method. Since calculations up to 200 eV require phase shifts up to about l-l (cf.…”

Section: R Federmentioning

confidence: 99%

“…4 In particular, the crystal is assumed to consist of a finite number of monatomic layers, for which the Dirac-equation boundary problem is solved to yield the fourspinor amplitudes and thence the intensities and the spin polarization vectors of the reflected beams as functions of the energy, the polar and azimuthal angles of incidence, and the polarization vector of the (normalized) primary beam. A modification was, however, made in the treatment of the monolayer part of the problem.…”

mentioning

confidence: 99%

See 1 more Smart Citation

Spin Polarization in Low-Energy Electron Diffraction from W(001)

1976

Phys. Rev. Lett.

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A relativistic low-energy electron-diffraction theory has been modified to allow computations for energies up to 200 eV. Its application to W(001) yields intensity profiles in reasonable correspondence with experiment, and pronounced polarization profiles which show encouraging agreement with novel and as yet limited experimental data. A high sensitivity of the polarization profiles to displacement of the topmost layer suggests that spin polarization measurements could be valuable for structure determination of surfaces involving heavy atoms.Theoretical studies of spin-polarization effects in low-energy electron diffraction (LEED) from the (001) surface of tungsten 1 * 2 were recently followed by an experiment, 3 in which significant degrees of spin polarization were found for electron energies from 45 to 190 eV at various small angles of incidence. Comparison of these novel data with the theoretical predictions 1 ' 2 is not possible, however, since the latter are confined to energies below about 40 eV. It is the aim of this Letter to report the first theoretical results at higher energies, to compare them with the recent experiment, 3 and to draw conclusions regarding the value of a polarization analysis in surface-structure determination.The theory underlying the present calculations follows to a large extent a relativistic LEED theory described earlier. 4 In particular, the crystal is assumed to consist of a finite number of monatomic layers, for which the Dirac-equation boundary problem is solved to yield the fourspinor amplitudes and thence the intensities and the spin polarization vectors of the reflected beams as functions of the energy, the polar and azimuthal angles of incidence, and the polarization vector of the (normalized) primary beam. A modification was, however, made in the treatment of the monolayer part of the problem. The Korringa-Kohn-Rostoker-Ziman (KKRZ) 5 type relativistic pseudopotential used in Ref. 4 depends on radii R t -associated with the spin-up and spin-down atomic phase shifts 5^-which have to be optimized individually in a numerical application of the method. Since calculations up to 200 eV require phase shifts up to about l-l (cf. also Van Hove and Tong), 6 I decided to avoid the resulting radii parameter problem by developing and applying the Dirac-equation-based analog 7 of a KKR-type Schrodinger-equation-based method, 8,9 which has been used with great success for LEED intensity calculations. 10 The ef-fect of thermal lattice vibrations is taken into account by replacing the actual (real) spin-up and spin-down atomic phase shifts 6j* by effective (complex) phase shifts fy*, which are obtained from the fy* by averaging the corresponding scattering amplitudes over a Debye spectrum. 11 In the computational application to W(001) the following specific model assumptions have been made. Effective phase shifts oV, obtained from Mattheiss's muffin-tin potential 12 and with use of the bulk Debye temperature of 380°K, are included up to 1 = 1. The number of surface reciprocal lattice ...

“…where M l is obtained as described in Section 4 of [2]. The problem of incorporating the above potentials is thus reduced to the determination of the propagation matrices Ma and M,, which proceeds as follows.…”

Section: Theory : Incorporation Of Surface Potential Contributionsmentioning

confidence: 99%

Relativistic Theory of Low Energy Electron Diffraction: Application to the (001) and (110) Surfaces of Tungsten

Physica Status Solidi (b)

1974

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A relativistic theory for calculating low energy electron diffraction intensities and spin polarization properties is applied to the (001) and (110) surfaces of tungsten for several angles of incidence. Intensity-energy profiles are obtained in generally good agreement with experimental data. Relativistic effects on intensities are found t o be noticeable, and appreciable spin polarization features are predicted. I n particular, large degrees of spin polarization in conjunction with sizeable intensities are found in the specular beam for large angles of incidence on W(OO1).