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Calculation of inelastic helium atom scattering from H2/NaCl(001)
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10.1063/1.3589259
/content/aip/journal/jcp/134/19/10.1063/1.3589259
http://aip.metastore.ingenta.com/content/aip/journal/jcp/134/19/10.1063/1.3589259

Figures

Image of FIG. 1.
FIG. 1.

Partial dispersion relation of phonons of p(1 × 1)para-H2/NaCl(001) in an extended zone scheme. The data points, with ±0.3 meV uncertainties, are from the Traeger and Toennies HAS experiments4 with scattering plane aligned (nearly) along the ΓM axis of the Brillouin zone. Also shown are the position of the Rayleigh wave (RW) of the NaCl(001), the position of the first Brillouin zone edge (BZ), and the estimated dispersion relations of the LA (solid line marked ωLA) and SH modes (solid line marked ωSH) of the square H2 lattice on a static substrate, Eqs. (6) and (7). Note that the energies and wavenumbers in the HAS experiments usually are the changes for the He atom and then are the negative of these values. Energies in meV and wavenumbers in Å−1.

Image of FIG. 2.
FIG. 2.

Schematic of the NaCl(001) surface showing the positions of the Na+ and Cl ions. The hydrogen molecules are atop the sodium ions and form a square lattice with one molecule per unit cell and lattice vectors a 1 and a 2 of length ℓ = 3.96 Å. The primitive reciprocal lattice vectors b 1 and b 2, of magnitude g 0 = 2π/ℓ = 1.59 Å−1 of the square substrate surface lattice are directed along the [110] and [10] azimuths, i.e., along the azimuths between nearest neighbor Na+-Na+ pairs. The dashed square shows the Brillouin zone. The ΓM axis is directed from Brillouin zone center to the midpoint of a primitive reciprocal lattice vector b 1 or b 2. The -axis of the Cartesian coordinates is parallel to a 2 and b 2.

Image of FIG. 3.
FIG. 3.

Time of flight spectra for a monolayer of p-H2 recorded at incident energies E i and angles θ i of 15.1 meV and 43.1° (top) and 26.5 meV and 46° (bottom).

Image of FIG. 4.
FIG. 4.

Experimental strengths of the longitudinal (LA) and shear horizontal (SH) one-phonon excitations along the ΓM direction in the reciprocal lattice as function of the wave vector q.

Image of FIG. 5.
FIG. 5.

A 2D contour plot of the potential energy surface (PES) based on the analytical expressions for the Tao He-H2 and Vargas He-NaCl potentials (TSM model) Eq. (2) at z = 0.4 Å. The x and y coordinates are given in units of the lattice constant ℓ = 3.96 Å and span therefore the range between zero and one; the focus is on the central region of the unit cell. A static hydrogen molecule is at each corner. Potential at the center is 0.52 meV. (a) The “exact potential energy surface.” (b) Potential energy surface approximated by a 25 Fourier coefficient expansion with Lagrange multiplier constraints, fitted to the cell center, atop site and bridge site.

Image of FIG. 6.
FIG. 6.

Potential energy surface (PES) based on the analytical expressions for the Tao He-H2 and Vargas He-NaCl potentials with zero-point averaging of the hydrogen molecules positions (TSMz model) at z = 0.8 Å. Identifications as in Fig. 5. Potential at the center is 12.76 meV.

Image of FIG. 7.
FIG. 7.

The calculated strengths, including the Debye Waller reduction, of the LA and SH specular one-phonon inelastic excitations along the ΓM direction for the LA calculations and with a misalignment of 1° for the SH calculations as a function of the wave vector transfer q with both a monochromatic (NWI = 600) and polychromatic (NWI = 100) He-atom beam. The calculations are based on the TSM potential model.

Image of FIG. 8.
FIG. 8.

Comparison between experimental intensities (left axis) and calculated strengths (right axis) with the TSM model, including the Debye-Waller reduction, of the LA and SH specular inelastic excitations along the ΓM direction for the LA calculations and with a misalignment of 1° for the SH calculations as a function of the wave vector transfer q for a monochromatic (NWI = 600) He-atom beam.

Image of FIG. 9.
FIG. 9.

Comparison between experimental intensities (left axis) and calculated strengths (right axis) with the TSMz model, including the Debye-Waller reduction, of the LA and SH specular inelastic excitations along the ΓM direction for the LA calculations and with a misalignment of 1° for the SH calculations as a function of the wave vector transfer q for a monochromatic (NWI = 600) He-atom beam.

Tables

Generic image for table
Table I.

Corrugation measures of the potential energy surface for He to monolayer plus substrate. z C height at center of unit cell with potential energy V C (z C ) equal to E ; V A  (z C ) is the atop site energy there; z A is height where atop site potential V A (z A ) equals E . NVG is g-shell number where . The corresponding scaled reciprocal lattice vector squared is also given, to facilitate comparing triangular and square lattices. Energies in meV and lengths in Å.

Generic image for table
Table II.

SH excitations at E i = 26.5 meV. Raw specular SH inelastic channel norms N q (x, y, ϕ) for specified model x = TSM or TSMz, packet width y = NWI = 600 or 100, and alignment ϕ of the scattering plane relative to the ΓM axis as a function of q with the angle of incidence θ i for the scan curve condition. The notation for the channel norms in Sec. IV has been simplified by dropping the polarization index λ and reciprocal wave vector (g = 0). Also shown are the raw total SH inelastic norm ) and the ratio I q /N q of the final and raw calculated SH one-phonon inelastic specular strengths,3 including the estimate of the DW factor F DW, and the same for the total SH inelastic strengths . Only results for the TSM model with NWI = 600 and ϕ = 1° have been included to limit the amount of data since the conversion factor from raw to final strengths is similar for the other cases.

Generic image for table
Table III.

LA excitations at E i = 26.5 meV. Raw specular (g = 0) LA inelastic channel norms N q (x, y, ϕ) for specified model x = TSM or TSMz, packet width y = NWI = 600 or 100, and alignment ϕ of the scattering plane relative to the ΓM axis as function of q with the angle of incidence θ i for the scan curve condition. Notation as in Table II. Also shown are the raw total LA inelastic norm N q, tot(TSM,600,0°) and the ratio of the final calculated LA inelastic specular strength,3 including the estimate of the DW factor F DW, for the one-phonon excitations, to the raw LA inelastic specular strength and the same for the total LA inelastic strengths . Only results for the TSM model with NWI = 600 and ϕ = 0° have been included to limit the amount of data since the conversion factor from raw to final strengths is similar for the other cases.

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/content/aip/journal/jcp/134/19/10.1063/1.3589259
2011-05-19
2014-04-16
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Calculation of inelastic helium atom scattering from H2/NaCl(001)
http://aip.metastore.ingenta.com/content/aip/journal/jcp/134/19/10.1063/1.3589259
10.1063/1.3589259
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