Topological complexity of crystal structures: quantitative approach

Krivovichev, Sergey V.

doi:10.1107/s0108767312012044

Cited by 176 publications

(133 citation statements)

References 42 publications

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“…According to this approach developed by Krivovichev (2012Krivovichev ( , 2013, the complexity of a crystal structure can be quantitatively characterized by the amount of Shannon information it contains measured in bits (binary digits) per atom (bits/atom) and Eby and Hawthorne (1990); coordinates of H atoms are optimized using DFT method per unit cell (bits/cell), respectively. The concept of Shannon information, also known as Shannon entropy, used here originates from information theory and its application to various problems in graph theory, chemistry, biology, etc.…”

Section: Structural Complexity Calculationsmentioning

confidence: 99%

“…Recently, we have developed an approach to quantitative evaluation of structural complexity of minerals by the Shannon information theory that allows determination of this important parameter in terms of information amount per atom and per unit cell (Krivovichev 2012(Krivovichev , 2013. Using statistical arguments, Krivovichev (2016) demonstrated that structural information per atom provides a negative contribution to configurational entropy of crystals and therefore is a physically important parameter.…”

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confidence: 99%

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Hydrogen bonding and structural complexity of the Cu3(AsO4)(OH)3 polymorphs (clinoclase, gilmarite): a theoretical study

Krivovichev¹

2017

J. Geosci.

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Density functional theory (DFT) is used to determine positions of H atoms and to investigate hydrogen bonding in the crystal structures of two polymorphs of Cu 3 (AsO 4 )(OH) 3 : clinoclase and gilmarite. Hydrogen bonds in clinoclase involve interactions between hydroxyl groups and O atoms of arsenate tetrahedra, whereas the crystal structure of gilmarite features OH … OH bonding, which is rather uncommon in copper hydroxy-oxysalts. Information-based parameters of structural complexity for clinoclase and gilmarite show that the former is more complex (I G,total = 213.212 bits/cell) than the latter (I G,total = 53.303 bits/cell), which indirectly points out that gilmarite is metastable. This suggestion is supported by the lower density of gilmarite (4.264 g/cm3 ) compared to that of clinoclase (4.397 g/cm 3). The hypothesis of metastable character of gilmarite is in agreement with the Goldsmith's simplexity principle and the Ostwald-Volmer rule.Keywords: hydrogen bonding, clinoclase, gilmarite, density functional theory, structural complexity, metastability Received: 4 January, 2017; accepted: 3 March, 2017; handling editor: J. Plášil clinoclase and gilmarite that would explain the difference in their abundance in nature.Recently, we have developed an approach to quantitative evaluation of structural complexity of minerals by the Shannon information theory that allows determination of this important parameter in terms of information amount per atom and per unit cell (Krivovichev 2012(Krivovichev , 2013. Using statistical arguments, Krivovichev (2016) demonstrated that structural information per atom provides a negative contribution to configurational entropy of crystals and therefore is a physically important parameter.One of the interesting implications of quantitative measures of structural complexity is that it provides the possibility to evaluate Goldsmith's (1953) simplexity principle in numerical terms. This principle states that, under appropriate kinetically favored crystallization regime (such as crystallization from supersaturated solutions or supercooled melts), the crystalline phase that may form in the system is not the most stable one, but very frequently metastable with the structure simpler than that of the stable phase. This principle found many examples in geological systems (Goldsmith 1953;Morse and Casey 1988) and its general validity was confirmed by means of the information-based measures (Krivovichev 2013(Krivovichev , 2015.The note is in order that structural complexity (or simplexity in Goldsmith's terms) is not the only parameter governing formation of metastable phases, but other parameters such as the structure of precursors and prenucleation clusters and influence of other phases during heterogeneous nucleation should be taken into account as

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Section: Structural Complexity Calculationsmentioning

confidence: 99%

mentioning

confidence: 99%

Hydrogen bonding and structural complexity of the Cu3(AsO4)(OH)3 polymorphs (clinoclase, gilmarite): a theoretical study

Krivovichev¹

2017

J. Geosci.

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“…Here, we provide a brief overview for the purpose of identifying the utility of such measures as part of a system measure of complexity for systems that can be modeled as a graph. Broadly, graph complexity measures can be classified into entropy-based [42][43][44], and non-entropic classes [42,[45][46][47][48].…”

Section: Complexity Measures For Graphsmentioning

confidence: 99%

A general framework for measuring system complexity

Efatmaneshnik

Ryan

2016

Complexity

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In this work, we are motivated by the observation that previous considerations of appropriate complexity measures have not directly addressed the fundamental issue that the complexity of any particular matter or thing has a significant subjective component in which the degree of complexity depends on available frames of reference. Any attempt to remove subjectivity from a suitable measure therefore fails to address a very significant aspect of complexity. Conversely, there has been justifiable apprehension toward purely subjective complexity measures, simply because they are not verifiable if the frame of reference being applied is in itself both complex and subjective. We address this issue by introducing the concept of subjective simplicity—although a justifiable and verifiable value of subjective complexity may be difficult to assign directly, it is possible to identify in a given context what is “simple” and, from that reference, determine subjective complexity as distance from simple. We then propose a generalized complexity measure that is applicable to any domain, and provide some examples of how the framework can be applied to engineered systems. © 2016 Wiley Periodicals, Inc. Complexity 21: 533–546, 2016

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“…Such structural diversity is caused by different P-T conditions of crystallization, amount of incorporated Fe 3+ or different mechanisms of vacancies distribution in the structure. According to the structural complexity classification [79,80], 4C polytype is "simple", while 5C and 6C modifications are "intermediate" (Figure 16). Relatively low-temperature formation of pyrrhotite-5C in comparison with pyrrhotite-4C is in good agreement with a common tendency of structural complexity growth with crystallization temperature decrease.…”

Section: Mo(sm) + Fes(mms) = Ms(mss) + Feo(sm)mentioning

confidence: 99%

Three-D Mineralogical Mapping of the Kovdor Phoscorite-Carbonatite Complex, NW Russia: II. Sulfides

et al. 2018

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Abstract:The world largest phoscorite-carbonatite complexes of the Kovdor (Russia) and Palabora (South Africa) alkaline-ultrabasic massifs have comparable composition, structure and metallogenic specialization, and can be considered close relatives. Distribution of rock-forming sulfides within the Kovdor phoscorite-carbonatite complex reflects gradual concentric zonation of the pipe: pyrrhotite with exsolution inclusions of pentlandite in marginal (apatite)-forsterite phoscorite, pyrrhotite with exsolution inclusions of cobaltpentlandite in intermediate low-carbonate magnetite-rich phoscorite and chalcopyrite (±pyrrhotite with exsolution inclusions of cobaltpentlandite) in axial carbonate-rich phoscorite and phoscorite-related carbonatite. Chalcopyrite (with relicts of earlier bornite and exsolution inclusions of cubanite and mackinawite) predominates in the axial carbonate-bearing phoscorite and carbonatite, where it crystallizes around grains of pyrrhotite (with inclusions of pentlandite-cobaltpentlandite and pyrite), and both of these minerals contain exsolution inclusions of sphalerite. In natural sequence of the Kovdor rocks, iron content in pyrrhotite gradually increases from Fe 7 S 8 (pyrrhotite-4C, Imm2) to Fe 9 S 10 (pyrrhotite-5C, C2 and P2 1 ) and Fe 11 S 12 (pyrrhotite-6C) due to gradual decrease of crystallization temperature and oxygen fugacity. Low-temperature pyrrhotite 2C (troilite) occurs as lens-like exsolition inclusions in grains of pyrrhotite-4C (in marginal phoscorite) and pyrrhotite-5C (in axial phoscorite-related carbonatite). Within the phoscorite-carbonatite complex, Co content in pyrrhotite gradually increases from host silicate rocks and marginal forsterite-dominant phoscorite to axial carbonate-rich phoscorite and carbonatite at the expense of Ni and Fe. Probably, this dependence reflects a gradually decreasing temperature of the primary monosulfide solid solutions crystallization from the pipe margin toward its axis. The Kovdor and Loolekop phoscorite-carbonatite pipes in the Palabora massif have similar sequences of sulfide formation, and the copper specialization of the Palabora massif can be caused by higher water content in its initial melt allowing it to dissolve much larger amounts of sulfur and, correspondingly, chalcophile metals.

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Topological complexity of crystal structures: quantitative approach

Cited by 176 publications

References 42 publications

Hydrogen bonding and structural complexity of the Cu3(AsO4)(OH)3 polymorphs (clinoclase, gilmarite): a theoretical study

Hydrogen bonding and structural complexity of the Cu3(AsO4)(OH)3 polymorphs (clinoclase, gilmarite): a theoretical study

A general framework for measuring system complexity

Three-D Mineralogical Mapping of the Kovdor Phoscorite-Carbonatite Complex, NW Russia: II. Sulfides

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