Mathematical Techniques in Crystallography and Materials Science

Prince, E.

doi:10.1007/978-1-4684-0136-3

Cited by 196 publications

(272 citation statements)

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“…Three cycles of full-matrix least-squares refinement (on F) with unit weights, isotropic temperature factors son, 1970) were included in subsequent cycles. In the final stages of robust/resistant refinement (Prince, 1982), each reflection was given a weight computed from a Chebyshev series with three parameters 45.3, -50.7, 25.9 (Carruthers & Watkin, 1979). This weighting scheme resulted in the down-weighting of one reflection (888); in the final listing 10Fo(888)= 3470, 10Fc(888) = 3877.…”

Section: Methodsmentioning

confidence: 99%

Absolute configuration and optical activity of laevorotatory Bi₁₂TiO₂₀

Swindells¹,

González²

1988

Acta Crystallogr Sect B

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Section: Methodsmentioning

confidence: 99%

Absolute configuration and optical activity of laevorotatory Bi₁₂TiO₂₀

Swindells¹,

González²

1988

Acta Crystallogr Sect B

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“…Inclusion of any weaker reflections failed to decrease errors or help correlation problems and resulted only in an artificially high R factor. The function minimized was r~o(IFol -IFcl) 2, where the weights were calculated using a robust/resistant weighting scheme (Prince, 1982). A separate refinement was also carried out against Fo 2, but this refinement showed no significant differences and on this occasion a better convergence was achieved when the refinement was carried out against Fo.…”

Section: Data Analysis and Refinement Proceduresmentioning

confidence: 99%

“…A separate refinement was also carried out against Fo 2, but this refinement showed no significant differences and on this occasion a better convergence was achieved when the refinement was carried out against Fo. (Prince, 1982), three parameters: 19.9, -20.1, 15.0 0.0001 3.4 -4.1 Larson (1970) 8.7 (6) Source of atomic scattering factors International Tables for X-ray Crystallography (1974, Vol. IV) physically unreasonable or did not converge back to their original positions when shifted away.)…”

Section: Data Analysis and Refinement Proceduresmentioning

confidence: 99%

Investigation into the Crystal Structure of the Perovskite Lead Hafnate, PbHfO₃

Corker¹,

Glazer²,

Kaminsky³

et al. 1998

Acta Crystallogr B Struct Sci

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The room‐temperature crystal structure of the perovskite lead hafnate PbHfO3 is investigated using both low‐temperature single crystal X‐ray diffraction (Mo Kα radiation, λ = 0.71069 Å) and polycrystalline neutron diffraction (D1A instrument, ILL, λ = 1.90788 Å). Single crystal X‐ray data at 100 K: space group Pbam, a = 5.856 (1), b = 11.729 (3), c = 8.212 (2) Å, V = 564.04 Å3 with Z = 8, μ = 97.2 mm−1, F(000) = 1424, final R = 0.038, wR = 0.045 over 439 reflections with F >1.4σ(F). Polycrystalline neutron data at 383 K: a = 5.8582 (3), b = 11.7224 (5), c = 8.2246 (3) Å, V = 564.80 Å3 with χ2 = 1.62. Although lead hafnate has been thought to be isostructural with lead zirconate, no complete structure determination has been reported, as crystal structure analysis in both these materials is not straightforward. One of the main difficulties encountered is the determination of the oxygen positions, as necessary information lies in extremely weak l = 2n + 1 X‐ray reflections. To maximize the intensity of these reflections the X‐ray data are collected at 100 K with unusually long scans, a procedure which had previously been found successful with lead zirconate. In order to establish that no phase transitions exist between room temperature and 100 K, and hence that the collected X‐ray data are relevant to the room‐temperature structure, birefringence measurements for both PbZrO3 and PbHfO3 are also reported.

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“…Because Vy is positive definite, its Cholesky factor, a lower triangular matrix, L, such that Vy = LL r, exists (Prince, 1994), and the difference matrix V x -V~ can be written in the form…”

Section: Least-squares Fittingmentioning

confidence: 99%