...r"
`
`Volume 37, Number 9
`September 2001
`
`ISSN: 0020-1685
`CODEN: INOMAF
`
`A Journal of Original and Review Papers Devoted to Chemistry, Physics,
`and Applications of Various Inorganic Materials
`
`http://wvvw.maik.ru
`
`
`
`Translated and Published by
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`1
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`
`Inorganic Materials
`
`(Neorganicheskie Materialy)
`ISSN:0020-1685 _—7
`
`Editor-in-Chief
`Prof. Grigorii G. Devyatykh
`Dr. Sci. (Chem.), Academician of the Russian Academy of Sciences (RAS),
`Institute of High-Purity Substances, Russian Academy of Sciences, Nizhni Novgorod, Russia_
`Address for correspondence:
`Neorganicheskie Materialy, Leninskii
`pr. 31, Rm. 71, Moscow, 119991 Russia; Phone: 7 (095) 954-3483
`Executive Editor-in-Chief
`Deputy Editors-in-Chief
`Prof. Yurii D. Tret’yakov
`Prof. Vitalii M. Skorikov,
`Dr. Galina F. Gubskaya
`Dr. Sci. (Chem), Academician’, RAS,“
`Dr. Sci. (Chem.). Kumakov Institute of General
`Cand. Sci. (Chem), Kumakov Institute of General
`and Inorganic Chemistry, RAS, Moscow,_Russia
`and Inorganic Chemistry, RAS, Moscow, Russia
`Moscow State University, Moscow, Russia
`
`Prof. Vadim I. Nefedov
`Dr. Sci. (Chem.), Corresponding Member, RAS,
`Kumakov Institute of General and Inorganic
`Chemistry, RAS, Moscow, Russia
`
`. Prof. Mikhail F. Churbanovw
`Dr. Sci. (Chem.). Corresponding Member, RAS,
`Institute of High—Purity Substances, RAS,
`Nizhni Novgorod, Russia
`
`-
`
`Vladimir V. Boldyrev
`Dr. Sci. (Chem.), Academician, RAS, Institute of
`Mechano— and Solid-State Chemistry, RAS,
`Novosibirsk. Russia
`
`Gennadii S. Burkhanov
`Dr. Sci. (Eng), Pmf., Corresponding Member. RAS,
`Baikov Institute of Metallurgy and Materials Research,
`RAS, Moscow, Russia
`Valentin A. Fedorov
`Dr. Sci. (Chem.), Prof., Kumakov Institute of General
`and Inorganic Chemistry. RAS, Moscow, Russia
`
`Iosif N. Fridlyander
`Dr. Sci. (Eng), Academician, RAS, Division of
`Physical Chemistry and Technology of Inorganic
`Materials, RAS, Moscow, Russia
`
`*
`
`Anatolii N. Georgobiani
`Dr. Sci. (Phys.-Math), Lebedev Physical Institute,
`RAS, Moscow, Russia
`
`EDITORIAL BOARD
`Sergei P. Gubin
`~
`.
`-
`Dr. Sci. (Chem.), Kumakov Institute of General and
`Inorganic Chemistry. RAS, Moscow, Russia
`Anatolii V. Gusev
`.
`>
`Dr. Sci. (Chem.), Institute of High-Purity Substances,
`RAS, Nizhni Novgorod, Russia
`Aleksandr D. Izotov _
`Dr. Sci. (Chem.)_. Corresponding Member, RAS,
`Kumakov Institute of General and Inorganic Chemistiy,
`RAS, Moscow, Russia
`’
`Yurii A. Karpov
`Dr. Sci. (Chem), Prof., Corresponding Member: RAS,
`GIREDMET, Moscow, Russia
`Fedor A. Kuznetsov
`Dr. Sci. (Chem.), Academician, RAS, Institute of
`Inorganic Chemistry, RAS, Novosibirsk, Russia
`Georgii A. Maximov
`Dr. Sci. (Chem), Lobachevskii State University. Nizhni
`Novgorod, R_ussia
`Staff Editor Svetlana S. Baikova
`
`A
`Lev A. Nisel’son
`Dr. Sci. (Eng), Prof., GIREDMET, Moscow, Russia
`Vladimir P. Orlovskii
`Dr. Sci. (Chem), Kumakov Institute of General and
`Inorganic Chemistry, RAS, Moscow. Russia
`Vladimir Ya. Shevchenko
`Dr. Sci. (Chem.), Academician, RAS,
`Grebenshchikov Institute of Silicate Chemistry, RAS,
`St. Petersburg, Russia
`'
`Gennadii P. Shveikin
`Dr. Sci. (Chem), Academician. RAS, Lristitute of
`Solid—State Chemistry, RAS, Yekaterinburg. Russia
`Yakov A. Ugai
`'
`Dr. Sci. (Chem), Prof., Voronezh State University,
`Voronezh, Russia
`Anatolii S. Vlasov
`Dr. Sci. (Eng.), Prof., Mendeleev University of
`Chemical Technology, Moscow, Russia
`Vladimir P. Zlomanov
`Dr. Sci. (Chem), Prof., Department of Chemistry,
`Moscow State University, Moscow, Russia
`
`Editor of the English Translation Oleg M. Tsarev
`
`SCOPE
`Inorganic Materials (Neorganicheskie Materialy) was established in 1965; the journal contains reviews, original papers, and news about chemistry, physics,
`and applications of various inorganic materials. The journal discusses phase equilibria, including P—T—X diagrams, and the fundamentals of inorganic materials
`science, which determines preparatory conditions for compounds of various compositions with specified deviations from stoichiometry. Inorganic Materials
`is a multidisciplinary journal covering all classes of inorganic materials. It is indispensable to researchers in a variety of fields.
`Inorganic Materials is abstracted and/or indexed in Chemical Abstracts, Chemical Titles, Current Contents, Energy Research Abstracts, Solid State Abstracts
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`The journal was founded in 1965.
`Original Russian Edition Copyright © 2001 by the Russian Academy of Sciences, Division of Physical Chemistry
`and Technology of Inorganic Materials. and the Kumakov Institute of General and Inorganic Chemistry.
`English Translation Copyright © 2001 by MAIR “Nauka/Interperiodica” (Russia).
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`

`
`Contents
`
`Vol. 37, No. 9, 2001
`Simultaneous English language translation of the journal is available fro
`worldwide by Kluwer Academic/Plenum Publishers. Inorganic Materials ISSN 0020-1685.
`
`Van der Waals Radii of Elements
`
`S. S. Batsanov
`
`Crystal Energetics of Inorganic Compounds with a Close Packing of Anions
`Ya. A. Kesler and D. S. Filimonov
`
`Thermally Stimulated Currents in Si(P,Au): Analysis of Rate Equations
`
`V. M. Skorikov, V. I. Chmyrev, E. VLarina, and V. V.Zuev
`Thermally Stimulated Currents in Si(P,Au): Exponential Heating Profile
`V. M. Skorikov, V I. Chmyrev, E. V. Larina, V. V Zuev,
`V. V. _Grigor’ev, and A. D. Kiryukhin
`Effect of Oxidation Conditions on the Phase Composition, Structure,
`and Properties of Photosensitive Lead Sulfide Layers
`M. I. Kamchatka, Yu. M. Chashchinov, and D. B. Chesnokova
`
`— 1
`009.,
`A I
`
`9 NU’
`,
`
`§\‘32Q‘
`
`/
`K
`
`CrS—Yb3S4 Phase Diagram and Magnetic Properties of Yb3Cr2S6, Yb3CrS5, and Yb9CrS13
`F. I. Rustamova, A. V. Einullaev, M. R. Allazov,
`R. Z. Sadykhov, K. G. Ragimov, and M. B. Babanly
`Structural and Thermal Properties of Cu1_xAgxInS2 Chalcopyrite Solid Solutions
`N. S. Orlova and I. V. Bodnar’
`
`Formation of Graphite Structure in Carbon Crystallites
`E. A. Belenkov
`
`Morphology and Structure of BN and B4C Nanocrystals
`V Ya. Shevchenko and G. S. Yur’ev
`
`Formation of Nickel Silicide Films from Nickel Acetylacetonate and Organosilicon Compounds
`0. N. Mittovl, N. I. Ponomareva, 1. Ya. Mittova, and M. N. Bezryadin
`
`Plasmochemical Preparation of NiO—Al2O3 Catalysts
`I. Sh. Normatov, N. Shermatov, and U. Mirsaidov
`Coexistence of Cubic and Tetragonal Structures in Yttria—Stabilized Zirconia Nanoparticles
`V Ya. Shevchenko, 0. L. Khasanov, G. S. Yur’ev, and Yu. F. Ivanov
`
`High—Pressure Phase Transitions of M205 (M = V, Nb, Ta)
`and Thermal Stability of New Polymorphs
`
`V. P. Filonenko and I. R Zibrov
`
`Plasmochemical Preparation of Ni—Containing Zeolites
`N. Shermatov, I. Sh. Normatov, U. Mirsaidov, and U. Z. Rasulov
`
`Luminescence Spectra of Eu3*—Activated Potassium Lanthanum
`and Potassium Gadolinium Phosphate Vanadates
`
`V. F. Klzarsika, L. N. Komissarova, A. N. Kirichenko,
`E. N. Murav’ev, V. P. Orlovskii, and A. P. Chernyaev
`
`'
`
`Growth and Properties of K2TiNb2P2O13 Crystals
`T. Yu. Losevskaya, V. I. Voronkova, V. K. Yanovskii, and N. I. Sorokina
`
`_
`
`.
`
`963.
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`968
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`Physicochemical Properties of R, __,Ba_VMnO3i 5 (R = Rare Earth) Solid Solutions
`D. A. Lundin, E. A. Erenzina, N. N. Oleinikov, and V. A. Kersko
`
`Molecular Dynamics Simulations of Ba1_xGd‘,F2 H Solid Solutions
`over a Wide Temperature Range: 1. Thermodynamic and Transport Properties
`
`1. Yu. Gotlib, I.
`
`l/. Murin, E. M. Piotrovskaya, and E. N. Brodskaya
`
`Synthesis and Phase Composition of Na_\.M.,Ti8 _ X016 (0.67 S x S 2.0; M = Al, Ga, In)
`L. N. Fomina, A. D. Neuimin, S. F. Pal ’guev,
`S. V. Vakarin, and S.
`l/. Plaksin.
`
`On the Occasion of the Seventieth Birthday
`of Academician N. T. Kuznetsov
`
`Chronicle
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`

`
`Inorganic Materials. Vol. 37, No. 9, 2001, pp. 871-885. Translatedfrom Neorgnnicheskie Materialy, Vol. 37, No. 9, 2001, pp. 1031-1046.
`Original Russian Text Copyright © 2001 by Batsanov.
`
`N This material may be proleuerl by Copyright law (Title 17 us. Code)
`
`Van der Waals Radii of Elements
`
`S. S. Batsanov
`
`Centerfor High Dynamic Pressures, Mendeleevo, Solnechnogorskii raion, Moscow oblast, 141570 Russia
`Received February 14, 2001
`
`Abstract—The available data on the van der Waals radii of atoms in molecules and crystals are summarized.
`The nature of the continuous variation in interatomic distances from van der Waals to covalent values and the
`mechanisms of transformations between these types of chemical bonding are discussed.
`
`INTRODUCTION
`
`The notion that an interatomic distance can be
`thought of as the sum of atomic radii was among the
`most important generalizations in structural chemistry,
`treating crystals and molecules as systems of interact-
`ing atoms (Bragg, 1920). The next step forward in this
`area was taken by Mack [1] and Magat [2], who intro-
`duced the concept of nonvalent radius (R) for an atom
`situated at the periphery of a molecule and called it
`the atomic domain radius [1] or Wirkungsradius [2],
`implying that this radius determines intermolecular dis-
`tances. Later, Pauling [3] proposed to call it the van der '
`Waals radius, because it characterizes van der Waals
`interactions between atoms. He also showed that the
`van der Waals radii of nonmetals coincide with their
`ionic radii and exceed their covalent radii (r), typically
`by 0.8 A.
`
`Initially, only x-ray diffraction (XRD) data, molar
`volume measurements, and crystal—chemical consider-
`ations were used to determine R. Later studies extended
`the range of experimental approaches and culminated
`in a complete system of the van der Waals radii of free
`and bound atoms. Comparison of the results obtained
`by various physical methods made it possible to assess
`the accuracy and locate the applicability limits of the
`van der Waals radii and to reconcile the concept of van
`
`der Waals radius with the quantum-mechanical require-
`ment that the electron density vary continuously at the
`periphery of atoms.
`
`In this review, the van der Waals radii of atoms eval-
`uated from XRD data, molar volumes, physical proper-
`ties, and crystal-chemical considerations are used to
`develop a universal system of van der Waals radii.
`
`ISOTROPIC CRYSTALLOGRAPHIC
`VAN DER WAALS RADII
`
`Kitaigorodskii [4, 5] was the first to formulate the
`principle of close packing of molecules in crystalline
`phases. He assumed that the van der Waals areas of
`peripheral atoms in neighboring molecules are in con-
`tact but do not overlap (rigid—atom model), because the
`repulsive forces between closed electron shells rise
`sharply with decreasing intermolecular distance. He
`made up a system of van der Waals radii as consistent
`as possible with the intermolecular distances in organic
`compounds. His radii differed little from Pauling’s
`(Table 1).
`
`The system of van der Waals radii was further
`refined by Bondi [6, 7]. His detailed tables were very
`popular among chemists, even though the values of R
`were criticized in a number of works [8]. Bondi not
`
`Table 1. Crystallographic van der Waals radii of nonmetals
`
`Author, year
`
`Pauling, 1939
`
`Bondi, 1964
`
`Zefirov, 1974
`
`Gavezzotti, 1983-1999
`
`Batsanov, 1995
`
`Wieberg, 1995
`
`Rowland, 1996
`
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`
`
`
`872
`
`BATSANOV
`
`
`
`Fig. 1.. Schematic showing the formation of an'A2 molecule.
`
`Fig. 2. Schematic showing the formation of an M—X bond.
`
`only determined R from structural data but also calcu-
`lated R by adding 0.76 A to covalent radii and evaluated
`R from thermodynamic and physical properties. The R
`values recommended by Bondi are also listed in
`Table 1.
`
`Zefirov and Zorkii [9, 10] corrected some of the van
`der Waals radii and made a number of new suggestions,
`in particular, that R should be calculated from the short-
`est
`intermolecular distances (D), ensuring a three-
`dimensional system of contacts, since other peripheral
`atoms of the molecule may be not in contact, and the
`determination of R by averaging all intermolecular dis-
`tances may be unjustified. Although they highlight the
`statistical nature of van der Waals radii and their vari-
`ability within a few tenths of an angstrom, depending
`on various structural factors, the radii in their system
`are given with an accuracy of a few thousandths of an
`angstrom (Table 1).
`
`A classical approach to the determination of the van
`der Waals radii of organogens was proposed in [1 1-14]
`(Table 1). The system of van der Waals radii of nonmet-
`als elaborated by Rowland and Taylor [15] for struc-
`tural organic chemistry was based on a wealth of statis-
`tical data.
`
`The van der Waals radii of the halogens and carbon
`in inorganic compounds were first determined by Paul-
`ing [3], who used data on layered compounds such as
`CdC12 and graphite. Later, the van der Waals radii of
`halogens and chalcogens were evaluated from a large
`body of experimental data [16]: OH 1.51, Cl 1.80,
`_Br 1.90, I 2.10, S 1.80, Se 1.85, and Te 2.02 A.
`The van der Waalsrradii of metals are difficult to
`determine directly, because there is only a small num-
`' ber of structures in which metal atoms can be in contact
`with another molecule; with the development of struc-
`tural
`chemistry,
`the number of
`such structures
`increases. Table 2 lists the crystallographic van der
`Waals radii determined to date [l6—l9].
`
`Recently, the van derWaals radii of metals were cal-
`culated from structural data for metals [17] and their
`
`molecular compounds [20, 21]. Let us outline these
`approaches and the obtained results.
`Since the valence
`state of a metal
`remains
`unchanged upon a polymorphic transformation, while
`the bond number increases, the electron density redis-
`tributes between ligands,
`Av, = 2Av, = 3Av‘_,.'..,
`
`(1)
`
`_
`
`where AV is the volume common to the two atomic
`spheres (Fig.
`l), and the subscripts specify the bond
`number. It is easily seen that
`
`AV = §1c(R — r)2(2R +1‘).
`
`(2)
`
`Given that, in polymorphic transformations, a decrease
`in the intermolecular contact area is accompanied by an
`increase in covalent bond length [22], to each coordina-
`tion number N,, there correspond particular R and r.
`Therefore, Eq. (1) takes the form
`
`<R.—r.>2(2R. +r.> = 2<R2—r.)2<R2+r.) = ....<3)
`
`valid for univalent atoms. In the case of multivalent
`metals, the left—hand side of Eq. (3) should be multi-
`plied by valence vif single—bond radii are used as r [3].
`As a result, we obtain
`
`(R. -r.>2<R. +r.> = zz<R,.—r,.)2<R,.+r,.>.
`
`(4)
`
`where n = NC/v. Taking into account that the maximal
`NC in metals is 12, one can use Eq. (4) to calculate R
`corresponding to the largest change in NC upon, the
`polymorphic transformation. Such calculations were
`carried out [17] under the assumption that, upon an
`increase in NC, the decrease in R is proportional to the
`increase in r (Table 2).
`
`In another approach to determining R, the M—X
`bond is also thought of as the region common to the
`atomic spheres M and X, with an internuclear distance
`dMx (Fig. 2). It is easily seen that
`
`RM = (R§(+dl2VlX"2dMXrM)”2-
`
`(5)
`
`INORGANIC MATERlALS
`
`Vol. 37 No.9
`
`2001
`
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`
`VAN DER WAALS RADII OF ELEMENTS
`
`Table 2. Crystallographic van der Waals radii of metals
`
`R,
`
`.—
`
`[21]
`
`2.24
`
`2.1
`
`<9)
`
`2.25-2.41
`2.33-2.46
`2.30-2.44
`1.91-2.00
`2.29-2.38
`1.95-2.02
`2.09-2.15
`2.01-2.07
`
`2.11-2.19
`2.20-2.29
`2.04-2.12
`2.10-2.22
`1.91-2.02
`2.02-2.10
`2.00-2.08
`
`2.05-2.19
`1.96-2.10
`1.93-2.05
`1.92
`2.06-2.10
`
`2.04-2.08
`2.06-2.20
`1.92-2.00
`1.85-1.93
`
`2.00-2.14
`1.93-2.02
`1.78-1.86
`2.05-2.25
`1.94-2.14
`1.88-2.11
`1.89-1.97
`1.83-1.90
`1.87-1.96
`1.76-1.82
`1.76-1.82
`1.77-1.85
`2.25
`1.94
`
`2.17
`2.36
`2.52
`2.1
`
`1.75 [18]
`
`1.97
`2.06
`2.07
`
`1.96
`2.04
`2.05
`
`2.27
`2.29
`2.23
`
`2.15
`
`2.21
`2.19
`2.16
`
`2.23 [24]
`
`_
`
`2.05
`2.16
`2.14
`
`2.00
`2.11
`2.11
`
`2.0
`2.2
`2.2
`
`« 2.1
`
`2.40 [19]
`
`.
`
`.
`2.0
`2.1
`2.1
`
`2.0
`2.1
`2.1
`.
`.
`
`* Minimal radius from M»-H van derWaa1s distances for R(H) = 1.2 1
`
`[32].
`
`INORGANIC MATERIALS
`
`Vol. 37
`
`No. 9
`
`2001
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`
`
`
`874
`
`BATSANOV
`
`Using an orbital radius for rx and knowing RX, one can
`calculate RM.
`
`Since the metals in organic compounds are coordi-
`nated rnost frequently by C, S, N, Cl, and Br, with elec-
`tronegativities xc = X5 and XN = xcl, the RM listed in
`Table ‘2 are averaged over the structures of M(CH3),,,
`MCl,,, and MBr,, [20]. Table 2 also gives the RA" calcu-
`lated from the recent data on the bond lengths in the
`AuCl and AuBr molecules [23].
`
`The intermolecular contact radii R,C were calculated
`in [21]. It was shown that the values of R,C in tetrahedral
`crystals coincide with the van der Waals radii of the ele-
`ments of the fifth period and their compounds (Table 2,
`left column under Ref. [21]). The contact radii deter-
`
`mined from the M-~~C(CH3) distances in M(C5Me5),,
`molecules coincide with or are close to the van der
`
`radii determined by independent methods
`Waals
`(Table 2, right column under Ref. [21]). The likely rea-
`son is that the CH3 group deviates from the plane of the
`C5 ring without significant energy changes, and, hence,
`the repulsion between M and C(CH3) is similar to the
`intermolecular interaction.
`
`If the bond length d(M—X) is close to rM + rx and
`rM 2 rx, we obtain from Eq. (5) for tetrahedral struc-
`tures
`
`R = [d2+(O.8166d)2—2a'r]'/2 = 1.633r.
`
`(6)
`
`For r = 1.2 A (average radius of organogens), we obtain
`R — r = 0.633 x 1.2 = 0.76 A (the rule proposed by Paul-
`ing and confirmed by Bondi). The values of R calcu-
`lated in [25] as r+ 0.8 A are also given in Table 2.
`
`It can be seen in Fig. 1 that the overlap region, situ-
`ated between symmetrically arranged, positively
`charged atomic cores, will extend perpendicular to the
`bond direction with decreasing bond length. This is
`supported by experimental data [26, 27]. From electro-
`static considerations, it follows that the thickness of the
`high—electron-density region must
`increase with the
`distance from the bond line. Clearly, this effect will be
`more pronounced in the N2, 02, and F2 molecules [28].
`
`To which extent
`
`can the electron cloud be
`
`deformed? Clearly, electrons can be promoted no far-
`ther than to the next shell. Since the atomic size
`
`depends on the principal quantum number /1,
`
`r = (10112/Z*,
`
`(7)
`
`where (10 is the Bohr radius for hydrogen, and Z:-”‘ is the
`effective charge on the nucleus [3], the largest value of
`the atomic radius is
`
`Hence,
`
`R = now + 1)?/2*.
`
`R/r = [(n+ 1)/n]2.
`
`(8)
`
`(9)
`
`Equation (9) can be used to calculate R from the
`known covalent radii of atoms in pa bonds. Table 2
`summarizes the van der Waals radii calculated by
`Eq.(9) from the normal
`(minimal) and crystalline
`(maximal) covalent radii [29].
`The scatter in the R of metals in Table 2 is due not
`
`only to experimental errors and inaccuracy in calcula-
`tions but also to the effect of bond polarity, a positive
`charge which reduces the size of thevatom. It was found
`empirically [20, 21] that the van der Waals radius of a
`metal can be written as
`
`RM(X) = Rg1‘bAXiiixa
`.
`.
`0
`where RM IS the van der Waals radius of an uncharged
`
`(10)
`
`atom, AXMX is the difference in electronegativity, and b
`and m. are constants.
`
`Whereas the radii of metals depend mainly on bond
`polarity, those of nonmetals are determined for the most
`part by the structural features of molecules, because
`they are close to the anion radii. Besides, since there is
`a relationship between R and r, and r depends on the
`valence of the atom, R also must depend on the valence,
`as shown by Pyykko [18].
`The variations in R across homologous series of
`molecules were examined using AX4 tetrahalides as
`examples. The details of the calculations based on the
`principle of close packing are described in [30]; the
`
`Table 3. Intermolecular contact (RIC) and van der Waals (R) radii (A) of halogens in AX4 crystals
`
`
`
`CC14(II)
`SiCl4
`
`CCl4(I)
`
`Note:
`
`I and II are the cubic and monoclinic forms. respectively.
`
`INORGANIC MATERIALS
`
`Vol. 37 No.9
`
`2001
`
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`

`
`VAN DER WAALS RADII OF ELEMENTS
`
`, 875
`
`results are summarized in Table 3. The average van der
`Waals radii of halogens in tetrahalides agree with the
`values given in Table l.
`The data in Table 3 indicates the tendency for Rx to
`decrease in going from CX4 to SnX4 because of the
`stronger van der Waals interaction between molecules
`at larger electronic polarizabilities. A similar situation
`is observed in liquid tetrahalides:
`
`MCI4
`RC1, A [31]
`
`CCI4
`1.75
`
`SiC14
`1.63
`
`GCCI4 _
`1.53
`
`Accordingly, the distance between the central atom
`of a molecule and the nearest atom X of the neighbor-
`ing molecule also decreases in going from SiC14 to
`GeCl4 and_ to SnCl4: 4.90, 4.60, and 4.57 A, respec-
`tively [32].
`The van der Waals radii of halogens deduced from
`the structure of X2 molecules in the liquid state are
`notably larger, since the polarizabilities of X2 are higher
`than those of the corresponding AX4 molecules:
`
`X
`RX,A[33]
`
`F
`1.54
`
`Cl
`1.89
`
`Br
`2.03
`
`V
`
`I
`2.23
`
`DETERMINATION OF VAN DER WAALS RADII
`FROM MOLAR VOLUMES OF SOLIDS
`
`Since the van der Waals equation incorporates the
`molecular volume, imposing a lower limit to the inter-
`molecular distance, this volume can be used to calcu-
`late the latter. Indeed, the molar volumes calculated
`from gas-kinetics data, molecular refractions, and van
`der Waals radii are in reasonable agreement [34] and
`can, therefore, be used in dealing with structural prob-
`lems [4, 5, 35]. The determination of isotropic van der
`Waals radii from molar volumes in structures where the
`intermolecular distances are direction-dependent is the
`most appropriate procedure for averaging experimental
`data.
`
`Treating atoms as rigid spheres, one can calculate
`the packing factor of molecules in crystals, which
`ranges, according to Kitaigorodskii [4, 5], from 0.65 to
`0.77. As shown later, the packing factor in organic com-
`pounds can be much larger, up to p = 0.9 [10]. In com-
`plex molecules whose rigid structure prevents close
`packing, p is below that of close packing (0.74); the p
`values above 0.74 suggest that the concept of close
`packing should be revised. Indeed, the packing factor of
`homodesmic (diamond, [3-Sn, bcc, or fee) structures
`can be calculated as the ratio of the covalent atomic vol-
`
`ume to the unit-cell volume per atom. The packing fac-
`tor of heterodesmic (molecular) structures is equal to
`the ratio of the van der Waals molecular volume (sum
`of the van der Waals volumes of the constituent atoms)
`to the unit-cell volume per molecule. In view of this,
`strictly speaking, the packing factors of organic and
`inorganic structures cannot be compared.
`
`INORGANIC MATERIALS
`
`Vol. 37
`
`No. 9
`
`2001
`
`However, one can combine the “organic” and “inor-
`ganic” concepts by taking into account the covalent and
`van der Waals contribution to p. As a first approxima-
`tion, it can be taken that, in the plane defined by the
`intersection of van der Waals spheres, p is unity (bar
`packing), whereas in the other directions p = 0.7405
`(close packing of atomic spheres). The area of the base
`of the spherical segment cut from the van der Waals
`sphere distance r from its center is 1t(R2 — r3), and the
`surface area of the van der Waals sphere is 41tR2. The
`ratio of these values is G = (R2 — I2)/4R2; for NC sections,
`
`we have S = CNC . The total packing factor is then given
`by
`
`p* = S><l.00+(1—S‘)><0.7405.
`
`(11)
`
`This model can be checked as follows: With increas-
`
`ing NC, 3' approaches unity, attaining it at NC = 12: S =
`12(R2 — r2)/4R2. Therefore,
`
`R = (3/2)"2r.
`
`(12)
`
`Table 4 presents the results of calculations using
`Eq. (12) and the metallic radii (r) for NC = 12 [3, 36],
`except for Cu, Zn, Cd, Hg (metallic radii corrected by
`the Pauling method for the divalent state), Group V and
`V VII transition metals, Ga, In, Tl, Cr, Fe, Co, Ni, Rh, Ir
`(trivalent state), Mo, and W (tetravalent state).
`It can be seen from Tables 2 and 4 that the values
`
`calculated by Eq. (12) agree with independent determi-
`nations to within 5—l0%. This is the accuracy to which
`the packing factor in inorganic substances can be deter-
`mined using Eq. (11).
`The accuracy of relation (11) can also be assessed
`from the scatter in the p of isostructural crystals. In the
`diamond structure, the atomic volume is
`
`V, =
`
`7.‘
`3
`
`[4R3 —4(R — r)2(2R + r)],
`
`(13)
`
`which is equal to the van der Waals atomic volume
`minus four segments cut distance r from the center.
`Since we use here R = RIC = 1.633r, and r = 0.772,
`1.176, 1.225, and 1.405 A in the tetrahedral structures
`of C, Si, Ge, and Sn, respectively [37], V, is 5.10, 18.02,
`20.36, and 30.73 A3, respectively, or 0.90 of the unit-
`cell volume per atom in all cases. In the graphite struc-
`ture, V, = 8.797 A3, R = 1.677 A, and r = 0.760 A; V0
`can be calculated by an equation analogous to (13) in
`which three, rather than four, segments are subtracted
`from the van der Waals volume, because of the three-
`fold coordination of carbon. In this way, we find V0 =
`7.769 A3 and p* = 0.88. This small difference in p*
`between diamond and graphite is better correlated with
`the low energy of the phase transition (0.3% of the
`atomization energy) than the large (by a factor of 2) dif-
`ference in the classical (covalent) packing factors: 0.34 V
`and 0.17.
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`
`
`
`876
`
`BATSANOV
`
`Table 4. Van der Waals radii (A) of metals calculated by Eq. (12)
`
`
`
`1.42-1.62
`
`1.66-1.92
`
`
`
`l.66—l .99
`
`1.78-2.16
`
`
`S(ortho)
`
`1.63-1.71
`1.63-1.85
`
`
`
`S(monocl)
`
`Se(hex)
`
`Se(monocl)
`
`1 .72
`
`1.74-2.00
`
`1.75
`
`1.64-1.72
`
`
`1.796
`
`1.56
`
`1.68
`
`1.76
`
`
`
`P
`
`OI-As
`
`Ot-Sb
`
`'
`
`from XRD data. It can be seen that the mean packing
`Using the known covalent bond lengths and molar
`factor is 0.788 (il %) for isostructura1A2 molecules and
`volumes of Group V—VII nontransition elements in the
`0.826 (i1.7%) for the ring and framework structures of
`crystalline state and solving coupled Eqs. (1 1) and (13),
`the Group V and VI nontransition elements, whereas
`one can determine R and the corresponding (‘5 and p*.
`the p calculated for the A2 molecules by the classical
`Table 5 lists the values of V0 and r taken from [18, 38],
`R and p* calculated as described above, and R extracted method from covalent radii ranges from 0.043 to 0.256,
`
`INORGANIC MATERIALS
`
`Vol. 37
`
`No. 9
`
`2001
`
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`

`
`VAN DER WAALS RADII OF ELEMENTS
`
`Table 6. Molar volumes, bond lengths, and atomic and van der Waals radii in AX4 crystals
`
`"(A-M
`
`CF40)
`
`CF4(II)
`
`SiF4
`
`GeF4
`
`CCl4(I)
`
`CCl4(II)
`
`SiCl4
`
`SnCl4
`
`CBr4(I)
`
`CBr4(II)
`
`GeBr4
`
`SnBr4
`
`with an average of0.135 (-1.56%), and the p for the other
`structures ranges from 0.172 to 0.429, with an average
`of 0.303 (i32%). Thus,
`the scatter in p* is much
`smaller than that in p.
`The molar volume method can also be used to deter-
`
`mine the van der Waals radii of the halogens in AX4
`molecules by calculating the volumes of these mole-
`cules by the equation
`
`Vm = “;"<tR3.—<R,.—rA)2(2R,.+r,.)1
`
`<14)
`
`+ MRS: — (Rx — "x)2(2Rx + "x)l>
`
`and equating them to the experimentally determined
`volume V0 multiplied by 0.7405. Taking the average of
`the van der Waals radii of the Group IV nontransition
`elements [39] as RA and the reduced orbital radii [27] as
`r, we calculated the Rx values listed in Table 6. These
`values coincide with the crystallographic van der Waals
`radii of halogens (Table 3).
`
`EQUILIBRIUM VAN DER WAALS RADII
`OF ISOLATED ATOMS
`
`The minimum in the potential of van der Waals
`interaction between two isolated atoms corresponds to
`an equilibrium van der Waals radius Re. Since different
`interatomic potentials were used in calculations of the
`van der Waals energy [40], there is a significant scatter
`in the reported distance corresponding to E = 0
`
`INORGANIC MATERIALS
`
`Vol. 37
`
`No. 9
`
`2001
`
`(Table 7). In view of this, Re is sometimes regarded as
`merely an adjustable parameter [4, 5, 40].
`Allinger er al. [41], using experimental R data for
`inert gases (G) and the effective charges of atomic
`cores, calculated, by an interpolation method, R, for all
`chemical elements, which proved close to the values of
`R determined in [42] from the structural data for GM,
`GX, Znz, Cdz, and Hgz molecules, in which the bonds
`are weak and the atoms are in a nearly isolated state.
`The more detailed results obtained later in [43, 44]
`were also in close agreement with the equilibrium radii.
`
`An important point is that the distances in heteronu—
`clear van der Waals molecules turned out to be larger
`than the sum of the van der Waals radii because of the
`
`polarization effects:
`
`DAB = RA+RB+ARAB,
`
`where
`
`ARAB = am. — a'B)/0l'A]2/3-
`
`(16)
`
`Here, a = 0.045, on is the electronic polarizability, and
`OLA < OLB. Since the interatomic distances in AB van der
`Waals molecules (DAB) are larger than %(DAA + DBB),
`the dissociation energy of heteronuclear molecules is
`less than half the sum of the energies of the correspond-
`ing homonuclear bonds [45, 46]. Recall that the length
`of normal chemical bonds is smaller than the sum of
`covalent radii, and the energy is higher than the additive
`
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`
`878
`
`BATSANOV
`
`Table 7. Equilibrium van der Waals radii of nonmetals
`
`Author, year
`
`
`
`
`Allinger, 1976
`
`Mundt et al., 1983
`
`Himan et al., 1987
`
`Gavezzotti, 1993
`
`Parkani et al., 1994
`
`Venturelli, 1994
`
`Cornell et al., 1995
`
`Schmidt, 1996
`
`Gavezzotti, 1999
`
`2.10
`
`1.90
`
`2.11
`
`Table 8. Van der Waals radii (A) of isolated atoms
`
`A jjl Re
`C
`H
`1.56
`1.56
`1.85
`1.97
`
`Li
`
`Na
`K
`
`Rb
`
`Cs
`
`Cu
`Ag
`Be
`
`Mg
`Ca
`
`Sr
`
`Ba
`
`Zn
`
`A
`
`.
`
`2.7
`
`2.8
`2.9
`
`3.0
`
`3.1
`
`2.0
`
`2.4
`
`2.2
`
`2.46
`
`2.68
`3.07
`
`3.23
`
`3.42
`
`2.24
`2.41
`2.14
`
`2.41
`2.79
`
`2.98
`
`3.05
`