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1010 LINUS PAULING Vol 51 Reaction 12K 25 17K 375 Heat evolved H3130 gt 11quot H2PO439 210 216 2000 H2PO4quot gt H HPO4quot 7127 7058 2300 The calculated heats of ionization which are in calories per mole refer of course to the ionization at in nite dilution The values of pKa calculated are insui ciently accurate for the heat effect of the third ionization step to be calculated Heat however is evolved in this ionization Summary With the quinhydrone electrode measurements of the PH values of phosphate mixtures have been made which lead to the following values for the ionization constants of phosphoric acid pK 25 210 pK1375 pK225 713 pK375 216 706 These gures are compared with those derived from other data From the data of Britton pKa at 20 is calculated to be about 121 LIVERPOOL ENGLAND CONTRIBUTION FROM GATES CHEMICAL LABORATORY CALIFORNIA INSTITUTE OF TECHNOLOGY No 192 THE PRINCIPLES DETERMINING THE STRUCTURE OF COMPLEX IONIC CRYSTALS BY LINUS PAULING Recevan SEPTEMBER 5 1928 PUBLISHED APRIL 5 1929 1 The Relative Stability of Alternative Structures of Ionic Crystals The elucidation of the factors determining the relative stability of alter native crystalline structures of a substance would be of the greatest signi cance in the development of the theory of the solid state Why for ex ample do some of the alkali halides crystallize with the sodium chloride structure and some with the cesium chloride structure Why does ti tanium dioxide under different conditions assume the different structures of rutile brookite and anatase Why does aluminum uosilicate AlgSiOt F2 crystallize with the structure of topaz and not with some other struc ture These questions are answered formally by the statement that in each case the structure with the minimum free energy is stable This answer however is not satisfying what is desired in our atomistic and quantum theoretical era is the explanation of this minimum free energy in terms of atoms or ions and their properties Efforts to provide such a treatment for simple alternative structures such as the sodium chloride and cesium chloride structures and the uoride and rutile structures have been made with the aid of the Born potential expression and modi cations of it Assuming that all ions repel each April 1929 STRUCTURE OF COMPLEX IONIC CRYSTALS 1011 other according to a high inverse power of the distance between them the repulsive potential being proportional to Fquot the equilibrium energy of a crystal is given by q zzezA 1 i 1 R n in which R is the equilibrium distance between two adjacent ions in the crystal and A is the Madelung constant characteristic of the structure A knowledge of how R changes from structure to structure for a given sub stance would then allow the prediction of which structure is stable if Equa tion 1 were accurate Methods of calculating R have been suggested1 It is found however that Equation 1 is in error by at least 2 in some cases1b and this error of around 5000 cal mole suf ces to invalidate the theory in applications of this kind An explanation of the sodium chloride cesium chloride transition which accounts for the observed properties of the alkali halides has been reported2 but the considerations involved including deformation phenomena have not yet been given quantitative formulation The application of these methods to more complex crystals would in volve the highly laborious calculation of the Madelung constant for a number of complicated ionic arrangements Furthermore the methods provide no way of determining the possible structures for which calcu lations should be made An in nite number of possible atomic arrange ments for a complicated substance such as a silicate are provided by the theory of space groups There is desired a set of simple rules which need not be rigorous in their derivation nor universal in their application with the aid of which the few relatively stable structures can be identi ed among the multitude possible for a given substance These rules could be used in the prediction of atomic arrangements for comparison with x ray data They would also provide a criterion for the probable correctness of struc tures suggested by but not rigorously deduced from experimental measure ments Finally they would permit the intuitive understanding of the stability of crystals in terms of visualizable interionic interactions 2 The Application of the Coordination Theory in the Determination of the Structures of Complex Crystals As a result of the recent increase in knowledge of the effective radii of various ions in crystals3 Professor W L Bragg has suggested and applied a simple and useful theory leading to the selection of possible structures for certain complex crystals His fundamental hypothesis is this if a crystal is composed of large ions and small ions its structure will be determined essentially by the large ions and may approximate a closepacked arrangement of the large ions alone 1 a F Hund Z Physik 34 833 1925 b Linus Pauling THIS JOURNAL 50 1036 1928 Z Krish 67 377 1928 Linus Pauling ibid 69 35 1928 3 Wasastjerna Soc Sci Farm Comm Phys Math 38 1 1923 1012 LINUS PAULING Vol 51 with the small ions tucked away in the interstices in such a way that each one is equidistant from four or six large ions In some cases not all of the closepacked positions are occupied by ions and an open structure results To apply this theory one determines the unit of structure in the usual way and nds by trial some closepacked arrangement of the large ions of known crystal radius usually oxygen ions with a radius of 135 140 A compatible with this unit The other ions are then introduced into the possible positions in such a way as to give agreement with the ob served intensities of re ection of x rays the large ions being also shifted somewhat from the closepacked positions if necessary With the aid of this method Bragg and his co workers have made a very signi cant attack on the important problem of the structure of silicates involving the de termination of structures for beryl BEaAlgsl60184 chrysoberyl BeA12045 olivine Mg Fe2SiO46 chondrodite HzMgaslzolo humite HzMg7Sl3014 Cll O humite HgMg9i40137 phenacite Be2 SiO48 etc During the investigation of the struc ture of brookitef the orthorhombic form of titanium dioxide a somewhat di er ent method for predicting possible struc nlme39 Large antics present tlta39 tures for ionic crystals was developed mum ions small circles oxygen ions An octahedron with a titanium ion at based upon the assumption 0f the 3039 its center and oxygen ions at its corners ordination Of the anions in the crystal is shown The two edges marked about the cations in such a way that With arrows are Shared With adjOining each cation designates the center of a Getahedm39 polyhedron the corners of which are occupied by anions This method leads for a given substance to a small number of possible simple structures for each of which the size of the unit of structure the spacegroup symmetry and the positions of all ions are xed In some cases but not all these structures correspond to close packing of the large ions when they do the theory further indicates the amount and nature of the distortion from the close packed arrangement The structures of rutile and anatase the two tetragonal forms of ti tanium dioxide have been determined by rigorous methods Figs 1 and 2 They seem at rst sight to have little in common beyond the fact 4 W L Bragg and J West Proc Roy Soc London 111A 691 1926 5 W L Bragg and G B Brown ibid 110A 34 1926 a W L Bragg and G B Brown Z Krish 63 538 1926 7 a W L Bragg and J West Proc Roy Soc London 114A 450 1927 b W H Taylor and I West nd 117A 517 1928 3 W L Bragg ibid 113A 642 1927 9 Linus Pauling and I H Sturdivant Z Krist 68 239 1928 Fig l The unit of structure for April 1929 STRUCTURE OF COMPLEX IONIC CRYSTALS 1013 that each is a coordination structure with six oxygen atoms about each titanium atom at octahedron corners From a certain point of View however they are closely similar They are both made up of octahedra sharing edges and corners with each other in rutile two edges of each octahedron are shared and in anatase four In both crystals the titanium oxygen distance is a constant with the value 195196 A The basic octahedra are only approximately reg ular in each crystal they are deformed in such a way as to cause each shared edge to be shortened from 276 A the value for regular octahedra to 250 A the other edges being correspondingly lengthened As a result of these considerations the following assumptions were made 1 Brookite is composed of octahedra each with a titanium ion at its center and oxygen ions at its corners 2 The octahedra share edges and corners with each other to such an extent as to give the crystals the correct stoichiometric composition 3 The titaniumoxygen distances throughout are 195 196 A Shared edges of octahedra are shortened to 250 A Two structures satisfying these re quirements were built out of octahedra The rst was not the structure of brook ite The second however had the same spacegroup symmetry as brookite V1115 and the predicted dimensions of the unit of structure agreed within 05 with Fig 2 The unit of structure of those observed Structure factors cal anatase39 The titanium OCtahEdfon culated for over fty forms with the use Shares tile fair gdges marked wnh arrows w1th adjoining octahedra of the predicted values of the nine para meters determining the atomic arrangement accounted satisfactorily for the observed intensities of re ections on rotation photographs This ex tensive agreement is so striking as to permit the structure proposed for brookite shown in Fig 3 to be accepted with con dence The method was then applied in predicting the structure of the ortho rhombic crystal topaz A12SiO4F210 It was assumed that each aluminum ion is surrounded by four oxygen ions and two uorine ions at the corners 1 Linus Pauling Proc Nat Acad Sci 14 603 1928 1014 LINUS PAULING Vol 51 of a regular octahedron and each silicon ion by four oxygen ions at the corners of a regular tetrahedron The length of edge of octahedron and tetrahedron was taken as 272 3 correspondng to crystal radii of 136 A for both oxygen and uorine ions Unc structure was built up of these polyhedra On studying its distribution of microscopic symmetry ele ments it was found to have the spacegroup symmetry of Vt which is that of topaz Its unit of structure approximates that found experimen tally and the predicted values of the fteen parameters determining the atomic arrangement account for the observed intensities of re ection from the pinacoids This concordance is sntl icient to make it highly probable that the correct structure of topaz has been found Fig 4 3 The Principles Determining the Structure of Complex Ionic CrystalsriThe success of the coordination method in predicting struc tures for brookite and topaz has led to the proposal of a set of principles govern ing the structure of a rather extensive class of complex ionic crystals The crystals considered are to contain only small cations with relatively large electric charges that is usually trivalent and tetravalent cations with crystal radii not over about 08 A All anions are large over 13quot A and univalent or divalent Furthermore they should not be too highly deformable The most important anions satisfying this restriction are the oxygen ion and the uorine ion with crystal radii 135 140 A This physical differentiation of the anions and cations under discussion in regard to size and charge nds expression throughout this paper Mark cdly different roles are attributed anions and cations in the construction of a crystal as a result a pronounced distinction between them has been made in the formulation of the structural principles Throughout our discussion the crystals will be referred to as composed of ions This does not signify that the chemical bonds in the crystal are necessarily ionic in the sense of the quantum mechanics they should not however be of the extreme nonpolar or shared electron pair type l hus compounds of copper and many other eighteenshell atoms cannot be Fig 3 The structure of brookite Professor W L Bragg has written the author that the same ideal structure has been found by I W39est paper to be published in the Proceedings of the Royal Society 1 The crystal radii used in this paper are those of Pauling THIS JOURNAL 49 765 1927 1 Ft London Z Physik 46 455 1928 b L Pauling Proc Nat Acad Sci 14 359 1928 Such as K2Cuclquot2H20 whose structure has been determined by S B Hendricks and R G Dickinson THIS JOURNAL 49 2149 1927 April 1929 STRUCTURE on COMPLEX IONIC CRYSTALS 1015 treated in this way Shared electron pair bonds are also present in com plexes containing large atoms with a coordination number of four such as the molybdate ion Mood 1 the arsenate ion AsOd Z etc The principles described in the following six sections have been deduced in part from the empirical study of known crystal structures and in part from considerations of stability involving the crystal energy Fig 4 The structure of topaz The layers are to be superposed in the order abcd with d uppermost The crosses are the traces of the corners of the unit of structure in the plane of the paper Large circles represent oxygen large double circles uorine small open circles aluminum and small solid circles silicon ions 4 The Nature of the Coordinated Polyhedra I A coordinated polyhedron of anions is formed about each cation the cation anion distance being determined by the radius surn and the codrdination number of the cation by the radius ratio In the case of crystals containing highly charged cations the most im portant terms in the expression for the crystal energy are those representing 1016 LINUS PAULING Vol 51 the interaction of each cation and the adjacent anions The next terms in importance are those representing the mutual interaction of the anions The negative Coulomb energy causes each cation to attract to itself a num ber of anions which approach to the distance at which the Coulomb at traction is balanced by the characteristic cation anion repulsive forces This distance is given with some accuracy by the sum of the crystal radii of cation and anion12 If too many anions are grouped about one cation the anion anion re pulsion becomes strong enough to prevent the anions from approaching this closely to the cation The resultant increase in Coulomb energy causes such a structure to be unstable when the anion cation distance is increased to a value only slightly greater than the radius sum Approxi mate lower limits of the radius ratio the ratio of cation radius to anion radius leading to a stable structure with given coordination number can accordingly be calculated purely geometrically s 12 The minimum radius ratios for tetrahedra octahedra and cubes are given in Table I TABLE I RADIUS RATIOS AND CooRDINATION NUMBERS Polyhedron Coordination number Minimum radius ratio Tetrahedron 4 x x 1 0225 Octahedron 6 1 0414 Cube 8 1 0732 Since the repulsive forces are determined by the true sizes of ions and not their crystal radii the radius ratios to be used in this connection are the ratios of the univalent cation radii to univalent anion radii12 Values of this ratio for small ions are given in Table II together with predicted and observed coordination numbers the agreement between which is excellent TABLE II CooRDINATION NUMBERS FOR IONS IN OXIDES Predicted Observed Strength of coordination coordination electrostatic Ion Radius ratio number number bonds B 020 301 4 30r4 101 34 Be 25 4 4 12 Li 34 4 4 U Si l 37 4 4 l A1 41 4or6 6 34 or z Mg 47 6 G 13 Ti t 55 6 6 2a Sc quot3 60 6 6 V2 Mo 53 6 6 1 Nb 57 6 6 53 Zr 62 6 6 or8 23 01 12 5 Such calculations were rst made and substantiated by comparison with observed structures in some cases by V M Goldschmidt Geochemische Verteilungsgesetze der Elemente 0510 1927 April 1929 STRUCTURE OF COMPLEX IONIC CRYSTALS 1017 The radius ratio for B 3 is only a little less than the lower limit for tetra hedra The usual coordination number for boron with oxygen is 3 in the borate ion Bod 3 It is four however in the 12tungstoborate ion16 in which a stabilizing in uence is exerted by the tungsten octahedra So far as I know Alt3 has the coordination number 6 in all of its com pounds with oxygen the structures of which have been determined The cobrdination number 4 would also be expected for it however it is probable that it forms tetrahedra in some of its compounds as for example 7 alumina the cubic form of A1203 and the feldspars in which there occurs replacement of Na and Si4 by Ca and Al This possibility is fur ther discussed in Section 11 Zr has the coordination number 8 in zircon The polyhedron of oxy gen ions about it is however not a cube It is on account of the ease with which these polyhedra are distorted that large cations with coordi nation numbers greater than six are not included in the eld of applica tion of the suggested principles Octahedra and tetrahedra retain their approximate shapes even under the action of strong distorting forces and moreover rules have been formulated governing the distortion that they do undergo Section 9 5 The Number of Polyhedra with a Common Corner The Electro static Valence Principle The number of polyhedra with a common corner can be determined by the use of an extended conception of electro static valence Let ze be the electric charge of a cation and v its coordi nation number Then the strength of the electrostatic valence bond going to each corner of the polyhedron of anions about it is de ned as S EIN 2 Let e be the charge of the anion located at a corner shared among several polyhedra We now postulate the following electrostatic valence principle II In a stable coordination structure the electric charge of each anion tends to compensate the strength of the electrostatic valence bonds reach ing to it from the cations at the centers of the polyhedra of which it forms a corner that is for each anion Zr E2s 3 In justi cation of this principle it may be pointed out that it places the anions with large negative charges in positions of large positive potentials for the bond strength of a cation gives approximately its contribution to the total positive potential at the polyhedron corner the factor 1 v ac counting for the larger cation anion distance and the greater number of adjacent anions in the case of cations with larger coordination number 1 Pauling unpublished material 1018 LINUS PAULING Vol 51 and the application of the principle requires that the sum of these poten tials be large in case the valence of the anion is large It is not to be anticipated that Equation 3 will be rigorously satis ed by all crystals It should however be always satis ed approximately As a matter of fact almost all crystals which have been investigated conform to the principle Equation 3 is necessarily true for all crystals the anions of which are crystallographically equivalent such as corundum A1203 rutile anatase spinel MgA1204 garnet CagAlzsi301218 cryolithionite N asLiaAlan19 etc It is also satis ed by topaz each oxygen ion common to one silicon and two aluminum ions has Es 2 see Table II for a list of values of 5 while each uorine ion attached to two aluminum ions only has Es 1 Similarly in beryl some oxygen ions are shared be tween two silicon ions and some between one silicon one beryllium and one aluminum ion in each case Es 2 In chondrodite HzMgssnom humite HgMg7Si3014 and clinohumite H2Mg9i4018 there are oxygen ions common to one silicon tetrahedron and three magnesium octahedra Es 2 and OH groups common to three magnesium octahedra Es 1 This list of examples could be largely extended 6 The Sharing of Edges and Faces The electrostatic valence principle indicates the number of polyhedra with a common corner but makes no prediction as to the number of corners common to two polyhedra that is whether they share one corner only two corners de ning an edge or three or more corners de ning a face In rutile brookite and anatase for example each oxygen ion is common to three titanium octahedra and hence has Es 2 satisfying Equation 3 but the number of edges shared by one octahedron with adjoining octahedra is two in rutile three in brook ite and four in anatase In corundum on the other hand each aluminum octahedron shares one face and three edges with other octahedra The reason for this di erence is contained in the following rule III The presence of shared edges and particularly of shared faces in a coordinated structure decreases its stability this e 39ect is large for cations with large val ence and small coordination number and is especially large in case the radius ratio approaches the lower limit of stability of the polyhedron This decrease in stability arises from the cation cation Coulomb terms The sharing of an edge between two regular tetrahedra brings the cations at their centers to a distance from each other only 058 times that obtain ing in case the tetrahedra share a corner only and the sharing of a face decreases this distance to 033 times its original value Fig 5 The corre sponding positive Coulomb terms cause a large increase in the crystal energy and decrease in the stability of the structure especially for highly 1 Linus Pauling and S B Hendricks THIS JOURNAL 47 781 1925 1 G Menzer Z Krist 63 157 1926 w G Menzer ibtd 66457 1927 April 1929 STRUCTURE OF COMPLEX IONIC CRYSTALS 1019 charged cations The effect is not so large for regular octahedra amount ing to a decrease in the cation cation distance to the fractional value 071 for a shared edge and 058 for a shared face These calculated decreases are valid only in case the change in structure is not compensated by deformation of the polyhedra Some compensating deformation will always occur the rules governing deformation Section 9 show that it will be small in case the radius ratio approaches the lower limit of stability for the polyhedron and will increase with the radius ratio Fig 5 a b and 6 show two tetrahedra of oxygen ions with a corner an edge and a face in common d e and f show two octahedra of oxygen ions with a corner an edge and a face in common In agreement with expectation silicon tetrahedra tend to share only corners with other polyhedra when this is possible as in topaz AleiO4F2 etc titanium octahedra share only corners and edges while aluminum octahedra when constrained by the stoichiometrical formula of the sub stance will share faces in some cases as in corundum A1203 The effect of large radius ratio in diminishing the instability due to an increase in the number of shared edges is shown by the approximate equality in free energy of rutile brookite and anatase with two three and four shared edges respectively As a matter of fact the order of stability is just that of the number of shared edges rutile being the most stable20 in agreement with expectation Many other dioxides also crystallize with the rutile structure but no other is known with the brookite or anatase structure The effect of small valence and large coordination number is further shown by the observation that silicon tetrahedra which share corners only with aluminum octahedra share edges with magnesium octahedra in olivine chondrodite humite clinohumite and with zirconium poly hedra With coordination number eight in zircon 7 The Nature of Contiguous Polyhedra I V In a crystal can C Doelter quotHandbuch der Mineralchemie Theodor Steinkop Dresden 1918 Vol III Part 1 p 15 1020 LINUS PAULING Vol 51 taining diferent cations those with large valence and small coordination num ber tend not to share polyhedron elements with each other This rule follows directly from the fact that cations with high electric charges tend to be as far apart from each other as possible in order to reduce their contribution to the Coulomb energy of the crystal The rule requires that in silicates the silicon tetrahedra share no ele ments with each other if the oxygen silicon ratio is equal to or greater than four topaz zircon olivine orthosilicates in general If stoichiomet rically necessary corners will be shared between silicon tetrahedra but not edges or faces In the various forms of silicon dioxide all four corners of each tetrahedron are shared with adjoining tetrahedra In diorthosili cates the Si207 group is formed of two tetrahedra sharing a corner The metasilicates should not contain groups of two tetrahedra with a common edge but rather chains or rings each tetrahedron sharing two corners as in beryl with a ring of six tetrahedra stable because of the approximation of the tetrahedral angle to 120 Other silicates are no doubt similar It is of interest that the electrostatic valence principle requires that cor ners shared between two silicon tetrahedra be not shared also with other polyhedra this is true for beryl 8 The Rule of Parsimony V The number of essentially di erent kinds of constituents in a crystal tends to be small First the electrostatic bonds satis ed by all chemically similar anions should be the same if pos sible topaz all oxygen ions common to two aluminum octahedra and one silicon tetrahedron all uorine ions common to two aluminum octahedra This does not require the anions to be crystallographically equivalent in topaz the oxygen ions are crystallographically of three kinds in brook ite of two kinds crystallographic non equivalence does not imply essen tial difference from the standpoint of the coordination theory Often the preceding rules do not permit all anions to be alike as for example in the case of silicates with an oxygen silicon ratio greater than four in which the four orthosilicate oxygens are necessarily different from the others In these cases the number of different kinds of anions will however be small Second the polyhedra circumscribed about all chemically identical cations should if possible be chemically similar and similar in their con tiguous environment that is in the nature of the sharing of corners edges and faces with other polyhedra For example each aluminum octahedron in topaz has as corners four oxygen and two uorine ions and each shares two edges with other octahedra and four corners with silicon tetrahedra The titanium octahedron in rutile shares two edges in brook ite three and in anatase four but no structure is known in which these different octahedra occur together The polyhedra which are similar in these respects may or may not be crystallographically equivalent for they April 1929 STRUCTURE or COMPLEX IONIC CRYSTALS 1021 may differ in their remote environment Thus the contiguously similar tetrahedra of silicon atoms about carbon atoms in carborundum are crystal lographically of several kinds ve in carborundum I 9 Distortion of the Polyhedra The above rules su ice to indicate the nature of the structure of a given crystal so that a structure can be composed of regular polyhedra in accordance with them and its space group symmetry and approximate dimensions compared with those found by xray analysis In this way a structure can be identi ed as giving ap proximately the correct atomic arrangement as was done for brookite and for topaz but the actual atomic arrangement may differ consider ably from this ideal arrangement corresponding to regular polyhedra as a resultant of distortion of the polyhedra The investigation of the agreement between observed intensities of xray re ections and structure factors calculated for all atomic arrangements involving small displace ments from the ideal arrangement would be extremely laborious It is accordingly desirable to be able to predict with some accuracy the nature and the amount of the distortion to be expected for a given structure In not too complicated cases this can be done theoretically by nding the minimum in the crystal energy with respect to variations in the para meters determining the structure with the use of a theoretical expression for the interionic repulsion potential Such calculations have been carried out for rutile and anatase21 leading to the result that in each case the shared edges of the titanium octahedra are shortened to the length 250 A other edges being compensatorily lengthened This distortion is ac tually found experimentally for these crystals It was accordingly as sumed to hold for brookite also and the atomic arrangement derived in this way was shown to be in complete agreement with the observed in tensities of xray re ection In general it is not possible to make such calculations on account of the excessive labor involved It can be seen however that the cation cation repulsion will shorten shared edges and the edges of shared faces and reasonably con dent application may be made of the following rule Poly hedra of oxygen ions about trivalent and tetravalent cations are distorted in such a way as to shorten shared edges and the edges bounding shared faces to a length of about 250 A Edges bounding shared faces have been observed to be shortened to 250 A in corundum A1203 and to 255 A in hematite Fezog in agreement with the foregoing rule It is furthermore to be anticipated that the cation cation repulsion will operate in some cases to displace the cations from the centers of their coordinated polyhedra This action will be large only in case the radius ratio approaches the lower limit for stability so that the size of the polyhe dron is partially determined by the characteristic anion anion repulsive 1 Linus Pauling Z Kris 67 377 1928 1022 LINUS PAULING Vol 51 forces the distribution of closely neighboring cations must of course be onesided in addition Hematite and corundum provide an example of this effect In these crystals each octahedron shares a face with another octahedron Now in an iron octahedron with radius ratio about 048 the repulsive forces principally effective in determining the interionic dis tances are those between iron and oxygen ions The Coulomb repulsion of the two iron ions accordingly can produce only a small displacement of these ions from the octahedron centers the iron ions in hematite are observed to be 206 A from the oxygen ions de ning the shared face and 199 A from the other oxygen ions In an aluminum octahedron on the other hand with radius ratio 041 the characteristic repulsive forces be tween oxygen ions as well as those between oxygen and aluminum ions are operative as a result of this double repulsionH12 the distance from the center of the octahedron to a corner is somewhat greater than the sum of the crystal radii of aluminum and oxygen The aluminum ions are correspondingly mobile and the aluminum aluminum Coulomb re pulsion is to be expected to cause a large displacement in their positions This is observed the two aluminum oxygen distances in corundum are 199 A and 185 A The nature and approximate amount of the distortion to be expected in other cases can be similarly estimated an example will be given in a later paper16 10 The ClosePacking of Large Ions In piling together polyhedra in the attempt to predict a possible structure for a crystal with the aid of the principles described above the recognition from the observed di mensions of the unit of structure that the atomic arrangement is probably based on a closepacked arrangement of the anions is often of very con siderable assistance for it indicates the probable orientation of the polyhe dra which can then be grouped together to form the completed structure This was done in the determination of the structure of topaz and contrib uted considerably to the ease of solution of the problem The approxima tion of the dimensions of the unit of structure of topaz to those of a Close packed structure does not however suffice to determine the positions of the anions for there are two types of closepacking simple hexagonal and double hexagonal2 which have these dimensions This ambiguity 22 An in nity of equally closepacked arrangements of spheres can be made from the closepacked layers A with a sphere at X 0 Y 0 X Y and Z being hexagonal coordinates B with a sphere at X 13 Y 23 and C with a sphere at X ala Y 13 For simple hexagonal closepacking these layers are superposed in the order ABABAB for cubic closepacking in the order ABCABC for double hexagonal closepacking in the order ABACABAC and so on with everincreasing complexity In only the rst two are the spheres crystallographically equivalent and only these two have been generally recognized in the past this restriction is however undesirable from the standpoint of the coordination theory April 1929 STRUCTURE OF COMPLEX IONIC CRYSTALS 1023 was no serious obstacle in the prediction of the structure by the coordi nation method the octahedra and tetrahedra were suitably piled together and the resultant arrangement of oxygen and uorine ions was found to be double hexagonal close packing Brookite is also based upon a double hexagonal closepacked arrange ment of the oxygen ions The dimensions of the unit of structure di er so much on account of distortion from those for the ideal arrangement however that the existence of close packing was recognized only after the structure had been determined It may be pointed out that in some structures easily derivable with the coordination theory such as the rutile structure the anion arrangement approximates no type of closepacking whatever 11 Applications of the Theory As an illustration of the application of the foregoing principles some predictions may be made regarding the structure of cyanite andalusite and sillimanite the three forms of Assn From the rule of parsimony we expect all aluminum octahedra to be similar and all silicon tetrahedra to be similar Let the number of octahedra one corner of which is formed by the ith oxygen ion be 1 then the stoi chiometrical oxygen aluminum ratio 52 requires that 1 5 i3 E 4 in which the sum is taken over the six oxygen ions forming one octahedron Four out of ve oxygen ions in accordance with Rule IV will be distin guished through being attached to silicon ions this fact is expressed by the equation 1 I i 2 5 in which the prime signi es that the sum is to be taken over these oxygen ions only Let us now assume that the oxygen ions are of only two kinds with respect to their values of 1 those attached to silicon ions 72 in num ber forming one class and those not attached to silicon ions 2 in number forming the other class Equations 4 and 5 then become 52 0 aquot with mm 6 6 112 on The only solution of these equations involving integers is m 4 a1 2 712 2 a2 4 Thus about each aluminum ion there will be four oxygen ions common to two aluminum octahedra and one silicon tetrahedron and two oxygen ions common to four octahedra23 For both kinds of oxy 3 It is possible that the aluminum octahedra may be of more than one kind In this case average values of the sums would have to be used in Equations 4 and 5 and the equations would no longer possess a single solution 1024 LINUS PAULING Vol 51 gen ions Es 2 so that the principle of electrostatic valence which was not used in the derivation of Equation 6 is satis ed This result while limiting considerably the number of possible structures for these crystals by no means determines their structures Further in formation is provided by Rule III from which it is to be expected that the silicon tetrahedra share only corners with aluminum octahedra and the octahedra share only corners and edges and possibly one face with each other These predictions are not incompatible with Professor Bragg s assign ment of a cubic closepacked arrangement of oxygen ions to cyanite They are however in pronounced disagreement with the complete atomic arrangement proposed by Taylor and Jackson whose suggested structure con icts with most of our principles Their structure is far from parsi monious with four essentially different kinds of octahedra and two of tetrahedra Each silicon tetrahedron shares a face with an octahedron contrary to Rule III The electrostatic valence principle is not even approximately satis ed one oxygen ion common to four aluminum octahe dra and one silicon tetrahedron has Es 3 while another common to two octahedra only has Es 1 For these reasons the atomic arrange ment seems highly improbable The coordination theory and the principles governing coordinated struc tures provide the foundation for an interpretation of the structure of the complex silicates and other complex ionic crystals which may ultimately lead to the understanding of the nature and the explanation of the proper ties of these interesting substances This will be achieved completely only after the investigation of the structures of many crystals with xrays To illustrate the clari cation introduced by the new conception the follow ing by no means exhaustive examples are discussed Let us consider rst the silicates of divalent cations with coordination Number 6 and hence with electrostatic bond strength 5 13 An oxygen ion forming one corner of a silicon tetrahedron would have 25 2 if it also formed a corner of three R octahedra if it were not attached to a silicon ion it would have to form a corner of six Rquot39 octahedra to satisfy the electrostatic valence principle But six octahedra can share a corner only by combining in the way given by the sodium chloride structure and this arrangement involving the sharing of many edges is expected not to be stable as a part of the structure of a complex silicate25 Ac cordingly we conclude that no oxygen ions not attached to silicon occur in these silicates that is the oxygen silicon ratio cannot be greater than 24 W H Taylor and W W Jackson Proc Roy Soc London 119A 132 1928 5 The instability of the sodium chloride structure for oxides is shown by the heat of the reactions Ca0 H20 CaOH2 15500 cal and MgO H20 MgOH2 9000 cal April 1929 STRUCTURE OF COMPLEX IONIC CRYSTALS 1025 41 The silicates which will occur are orthosilicates R2SiO4 metasili cates RSi03 etc This is veri ed by observation no basic silicates of such cations are known although many normal silicates such as forsterite Mg28i04 etc exist A univalent anion F OH may be shared among three R octahe dra alone so that compounds may occur in which these anions are present in addition to the SiO groups Such compounds are known MgasiOi F OH2 prolectite Mg5Si042F OH2 chondrodite etc Compounds of simple structure according to the coordination theory are those in which the number of essentially different kinds of anions is small In a simple orthosilicate containing aluminum and a divalent cation each oxygen ion would form the corner of a silicon tetrahedron s 1 an aluminum octahedron s 12 and one or two R polyhedra one tetrahedron or two polyhedra with u 8 The composition of the substance would then be given by the formula R3A128i3012 Similarly in simple metasilicates there would occur such oxygen ions in addition to those common to two silicon tetrahedra the corresponding formula is R3A128i6013 This result is in striking agreement with observation The most important double orthosilicates of divalent and trivalent metals are the garnets Ca3Al2Si3012 grossular Ca3Cr2Si3012 uvarovite CagFezsiSOm topazolite etc and the only double metasilicate is beryl BegAlgSieOn The radius ratio of potassium ion and oxygen ion is 076 so that the coordination number to be expected for potassium ion in silicates is 8 the corresponding electrostatic bond strength being 13 In potassium aluminum silicates containing aluminum octahedra the electrostatic valence principle would require at least four potassium polyhedra to have a com mon corner together with a silicon tetrahedron and an aluminum octahe dron this is not spatially possible It is accordingly highly probable that in these compounds the structure of none of which has yet been satis factorily investigated with x rays the aluminum ions have a coordination number of 4 There could then occur oxygen ions with 25 2 com mon to a silicon tetrahedron an aluminum tetrahedron and two potassium polyhedra The potassium aluminum ratio would then be 11 In a large number of silicates in particular the important feldspars this ratio is observed KAlSiO4 phakelite KAlSizos leucite KAlSlgOg potassium feldspar microcline orthoclase K Na3A13Sl9034 nepheline etc In other silicates containing more than this amount of aluminum such as muscovite HgKAlgsiaom etc it is probable that the excess aluminum ions usually have the coordination Number 6 Other alkali ions except lithium also probably have the coordination Number 8 as a rule and should similarly have a tendency to a 11 ratio with aluminum this is shown in NaAlSiaog albite H2N32A12Sl3012 natro lite H2CS4A14Sl9027 pollucite etc 1026 H L CUPPLES Vol 51 In spodumene LiAlSi205 and petalite LiAlShOm it is possible that oxygen ions with Es 2 are common to a silicon tetrahedron an alumiv num tetrahedron and a lithium tetrahedron the radius ratio for lithium ion is 033 No aluminum silicates of alkali metals are known in which the Al L3zR1 ratio is less than 11 Summary A set of principles governing the structure of complex ionic crystals based upon the assumption of a coordinated arrangement of anions about each cation at the corners of an approximately regular polyhedron is formulated with the aid of considerations based upon the crystal energy Included in the set is a new electrostatic principle which is of wide appli cation and considerable power It is shown that the known structures of many complex crystals in par ticular the complex silicates satisfy the requirements of these principles As an illustration of the application of the principles in the prediction of structures with the coordination theory some properties of the structures of the three forms of A12Si05 cyanite andalusite and sillimanite are predicted It is further shown that the theory requires that no stable basic silicates of divalent metals exist and that in aluminum silicates of alkali metals there should be at least one aluminum ion for every alkali ion The structures of aluminum silicates of divalent metals which are sim plest from the coordination standpoint are shown to correspond to the formulas R3A12i3012 and R3A128i6018 which include the most im portant minerals of this class the garnets and beryl PASADENA CALIFORNIA CONTRIBUTION FROM FERTILIZER AND FIXED NITROGEN INVESTIGATIONS BUREAU or CHEMISTRY AND SOILS U S DEPARTMENT or AGRICULTURE SOLUBILITY IN THE GASEOUS PHASE ESPECIALLY IN THE SYSTEM NH3lNHag H2g Mg BY H L CUPPLES RECEIVED SEPTEMBER 7 1928 PUBLISHED APRIL 5 1929 A number of investigators have found that a solubility effect may be shown within a gaseous phase especially under circumstances in which the density of the gaseous phase is relatively high Pollitzer and Strebel1 were perhaps the rst to mention this phenomenon The twophase system NH31NH3g H2g N2g in which the H2 and N were maintained at the constant mole ratio of 3H2N2 has been experimentally investigated by Larson and Black2 Concentrations 1 Pollitzer and Strebel Z physik Chem 110 785 1924 2 Larson and Black THIS JOURNAL 47 1015 1925
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