An empirical relation between hardness and bond-ionicity in a crystal

Julg, André

doi:10.1007/bf00357446

Cited by 13 publications

(11 citation statements)

References 13 publications

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“…For instance, the hardness, the elastic constants and the cohesive energy increase, the hygroscopy and the polarizability and hence the relative permittivity O 1987 International Union of Crystallography decrease with decreasing cation atomic number (Huggins & Sakamoto, 1957;Julg, 1978;Miura, Murata & Shiro, 1978;Chitra, Vempati & Jacobs, 1983). For instance, the hardness, the elastic constants and the cohesive energy increase, the hygroscopy and the polarizability and hence the relative permittivity O 1987 International Union of Crystallography decrease with decreasing cation atomic number (Huggins & Sakamoto, 1957;Julg, 1978;Miura, Murata & Shiro, 1978;Chitra, Vempati & Jacobs, 1983).…”

Section: Introductionmentioning

confidence: 99%

“…The anharmonicity of atomic vibrations in wurtzitestructure compounds has been studied with X-ray diffraction by Mair & Barnea (1975), Moss & Barnea (1976) and Whiteley, Moss & Barnea (1977, 1978. The anharmonicity of atomic vibrations in wurtzitestructure compounds has been studied with X-ray diffraction by Mair & Barnea (1975), Moss & Barnea (1976) and Whiteley, Moss & Barnea (1977, 1978.…”

Section: Introductionmentioning

confidence: 99%

“…The anharmonicity of atomic vibrations in wurtzitestructure compounds has been studied with X-ray diffraction by Mair & Barnea (1975), Moss & Barnea (1976) and Whiteley, Moss & Barnea (1977, 1978. Allowing for electronic expansion of ions in the model improved the agreement with observed macroscopic quantities but still revealed some incompatibilities (Miura, Murata & Shiro, 1977, 1978Miura, Sat6, Murata & Shiro, 1980). Allowing for electronic expansion of ions in the model improved the agreement with observed macroscopic quantities but still revealed some incompatibilities (Miura, Murata & Shiro, 1977, 1978Miura, Sat6, Murata & Shiro, 1980).…”

Section: Introductionmentioning

confidence: 99%

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Multipole analysis of X-ray diffraction data on BeO

Vidal-Valat¹,

Vidal²,

Kurki-Suonio³

et al. 1987

Acta Crystallogr A Found Crystallogr

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X-ray diffraction intensities were measured from a BeO single crystal with wurtzite structure specially treated to reduce extinction effects. Preliminary leastsquares refinement yielded experimental structure factors on an absolute scale, corrected for extinction and anomalous dispersion, and a reference model Be2+O 2-with position parameter z = 0.3778 (4) and anisotropic Debye-Waller factors corresponding to atomic mean-square displacements with spherical average (U 2) "--0"0045 (3) and 0.00353 (1) A 2 and prolateness (u 2)-(u 2) = 0.0008 (4) and 0.00005 (9) A2 for Be 2+ and O z-, respectively. The residual differences Fo-Fc (of these experimental structure factors and those of the reference model) were analysed in terms of site-symmetrized multipole expansions. The phases of Fo were refined iteratively to take into account the effect of the significant multipole components found. The contribution of the Be 2+ ion was subtracted to make possible a study of the oxygen lattice. Significant multipole components of third, fourth and sixth order indicate that the oxygen in BeO forms bonded hexagonal 0 2-planes. They build up diffuse O-O bridges strongly bent towards the stabilizing Be z+ ions above the plane. The Be 2+ ions exhibit a stretching quadrupole-like deformation suggesting some covalency of the bonding they build between the O 2-planes. There is some evidence of anharmonicity of atomic vibrations, particularly in the direction of the hexagonal axis, AbstractA simple relationship has been developed for determining values for the moduli of normal order deviates for sample size n from the values of the normal order deviates for sample size 2n + 1. This relationship can be expressed as a quadratic 'correction' to the normal order values. This approximation is in error by <0.001 for values of n > 8.

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Section: Introductionmentioning

confidence: 99%

“…The anharmonicity of atomic vibrations in wurtzitestructure compounds has been studied with X-ray diffraction by Mair & Barnea (1975), Moss & Barnea (1976) and Whiteley, Moss & Barnea (1977, 1978. The anharmonicity of atomic vibrations in wurtzitestructure compounds has been studied with X-ray diffraction by Mair & Barnea (1975), Moss & Barnea (1976) and Whiteley, Moss & Barnea (1977, 1978.…”

Section: Introductionmentioning

confidence: 99%

“…The anharmonicity of atomic vibrations in wurtzitestructure compounds has been studied with X-ray diffraction by Mair & Barnea (1975), Moss & Barnea (1976) and Whiteley, Moss & Barnea (1977, 1978. Allowing for electronic expansion of ions in the model improved the agreement with observed macroscopic quantities but still revealed some incompatibilities (Miura, Murata & Shiro, 1977, 1978Miura, Sat6, Murata & Shiro, 1980). Allowing for electronic expansion of ions in the model improved the agreement with observed macroscopic quantities but still revealed some incompatibilities (Miura, Murata & Shiro, 1977, 1978Miura, Sat6, Murata & Shiro, 1980).…”

Section: Introductionmentioning

confidence: 99%

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Multipole analysis of X-ray diffraction data on BeO

Vidal-Valat¹,

Vidal²,

Kurki-Suonio³

et al. 1987

Acta Crystallogr A Found Crystallogr

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“…This is a crucial feature for alloys that will be machined or subjected to friction. Although numerous papers have been published attempting to correlate hardness with fundamental properties, like bulk modulus [1], electronic structure [2], interatomic distances [3] and bonds [4], there is not yet a priori method to obtain a desired hardness when making new alloys. For tetrahedrally bonded covalent solids a rule of thumb indicates that the maximum hardness is attained when the valence electron concentration per atom (VEC) is about 4.…”

Section: Introductionmentioning

confidence: 99%

The role of valence electron concentration in the cohesive properties of YBxN1−x, YCxN1−x and YNxO1−x compounds

Soto

Moreno-Armenta

Reyes-Serrato

2008

Journal of Alloys and Compounds

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“…a=lp~-p.. 1 (1) pmm + p.. where p,.,. and p.. respectively are the Mulliken electron population carried by the atomic orbitals X,.…”

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confidence: 99%

Determination of the Charge Distribution in Nonmetallic Crystals II. Compounds with d‐orbitals. Corundum

Julg

Pellégatti

Marinelli

1980

Israel Journal of Chemistry

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Abstract. A S.C.F. method using localized molecular orbitals, previously proposed to determine net charges in zincblende-and wurtzite-type crystals, has been adapted to the corundum case (AI 203-a )~. The presence of d -orbitals in hybrid atomic orbitals and the difference between coordination numbers give rise to some problems. Then, the choice of Slater-type orbital exponents, the introduction of corrective factors in the calculation of charges, the molecular orbitals orthogonalization and the introduction of Madelung-type constants are discussed. The net charge found for Al atoms (Q~0.57) is, given the hypotheses used in the calculation, in good agreement with the one (0.33) determined from an empirical relation connecting hardness and ionicity of a crystal.Although corundum is not as precious a stone as diamond, it is a much appreciated gem, because of its colours when metallic ion impurities are inserted: Fe." or Ti 4 + (blue sapphire), Cr" (ruby), Mn " (oriental amethyst), ... on the one hand, and for its high hardness (9 in Mohs' scale), on the other. In a recent paper, one of us' established a connection between hardness of a crystal and bond ionicity of the corresponding lattice, a high hardness corresponding to a strongly covalent framework. In order to define the degree of ionicity of a crystal, we generalized thewhere p,.,. and p.. respectively are the Mulliken electron population carried by the atomic orbitals X,. and X. used to build the bond molecular orbital. In a crystal with a stoichiometric composition MfJ.,N~", where M is the metal atom the coordination number of which in the lattice is II,., we define the ionicity as (2) where 0,. is the actual net point charge of the atom M and O~O) is the net point charge it would have if the lattice was perfectly ionic.On the other hand, we improved by smoothing the hardness Mohs' scale by a comparison with results obtained by various physical methods. Starting from some experimental and theoretical data we proposed an empirical relation between the improved hardness Hand ionicity as defined by relation (2):where p and q are the principal quantum numbers of the valence orbitals of the atoms M and N. In the corundum case (Ah03-a)~, H = 8.9 and by relation (3) we obtain OAI = 0.33, 00 = -0.22. These values look a priori small. But in quartz, for example, the hardness of which is 7.0. the charge on Silicon also obtained by relation (3) and by various theoretical calculations':' is about 3.0. For corundum we do not have any experimental data about charges. Therefore a theoretical determination of these charges presents a double interest: to obtain a correct estimation of these quantities on the one hand and to test relation (3) in the case of a mineral of strong hardness on the other. Indeed, we can find, in many high school books and even in University lectures, ions A1 3 + and 0 2 -to describe the structure of corundum. Therefore, even qualitatively, it is interesting to know whether we are closer to the ionic structure or to the covalent one. Moreover, now ...

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An empirical relation between hardness and bond-ionicity in a crystal

Cited by 13 publications

References 13 publications

Multipole analysis of X-ray diffraction data on BeO

Multipole analysis of X-ray diffraction data on BeO

The role of valence electron concentration in the cohesive properties of YBxN1−x, YCxN1−x and YNxO1−x compounds

Determination of the Charge Distribution in Nonmetallic Crystals II. Compounds with d‐orbitals. Corundum

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