The electron distribution in silicon - II. Theoretical interpretation

Abstract

The absolute structure factors of silicon measured in Part I are examined in detail using Dawson’s structure factor formalism with recent theoretical estimates of free atom Hartree-Fock wavefunctions (Clementi 1965). It is not possible to describe the structure of silicon with only Kubic Harmonic expansions having physically reasonable coefficients. If the valence-shell form factor is radially compressed by 6.8% then we can obtain an excellent fit using only the third- and fourth-order Kubic Harmonics with amplitudes F 3 ═ + 1.012 and G 4 ═ — 0.206. There is some evidence that the anomalous dispersion corrections for silicon which are available in the literature are in error by 0.02 electron units. From the high-order reflexions we have measured the X-ray Debye temperatures of silicon as ⊝ M ═ 536.3 ± 5.0 K at 92.2 K and ⊝ M ═ 532.5 ± 2.0 K at 293.2K . The increase in ⊝ M at low temperature is smaller than had been expected theoretically. Our measurements of the 555 structure factor permit us to confirm Nunes’s (1971) upper-bound estimate of the anharmonic force constant for silicon. Finally, we have compared our results with the self-consistent orthogonalized-plane-wave band structure calculations of Stukel & Euwema (1970) for silicon atoms in a crystal. We find that the Kohn-Sham-Gaspar model of exchange results in structure factors which are in much better agreement with those measured than are structure factors calculated using Slater’s exchange term. As in the interpretation based on Dawson’s formalism there is evidence here that the theoretical anomalous dispersion corrections may be inadequate.

Thermal vibrations in convergent-beam electron diffraction

“…This new estimate agrees with the extremely accurate X-ray measurement of 0.0764(2)A by Aldred & Hart (1973).…”

A new theory for X-ray diffraction

“…This B factor is difficult to reconcile with the expected value of $0.02 nm 2 (Reid & Pirie, 1980). The experimentally measured parameter on bulk Si of B $0.046 nm 2 , by Aldred & Hart (1973), represents the maximum value, because their experimental conditions were highly biased towards the core value. Reid & Pirie (1980) have critically reviewed the literature as well as having calculated this value for Si by numerous approaches, and concluded that B $0.02 nm 2 is the most likely value.…”

“…In strongly bonded Si the shell will be damped compared with the vibrations of the core, because it is linked via the bonding to other atoms (Reid & Pirie, 1980). The core Debye-Waller factor has been calculated and measured accurately, see for example the compilation and arguments in Reid & Pirie, and the measurements of Aldred & Hart (1973), who give a value close to the calculations at room temperature of B = 0.04613 nm 2 using high-order reflections; this value is therefore more closely related to the core. This puts an upper limit on the value of B under these conditions.…”

Picometre-precision analysis of scanning transmission electron microscopy images of platinum nanocatalysts

“…Frozen phonon multislice simulations as implemented by Kirkland 31 were used for the standardless atom counting of the Pt nanocatalyst, the source size determination using GaN data and the Si dislocation distorted image series analysis. At 300 K, the root-mean-square displacements of Si, Ga and Pt atoms in the structures we studied are 0.076, 0.060 and 0.067 Å, respectively [32][33][34][35] .…”