I. N. STRANSKI 21 SURFACE STRUCTURES OF IONIC CRYSTALS BY K. MOLI~RE, W. RATHJE AND I. N. STRANSKI Received 16th February, 1949 There is no doubt that the structural peculiarities of crystal surfaces play a role in many physical and chemical processes in which the surfaces of solid bodies participate. Up to the present no experimental method exists which is dependable enough for the determination of surface structures. The only way to decide how the positions of equilibrium of the atoms in the surfaces can differ from those of the infinitely extended space lattice is to employ theoretical considerations with the use of simple models.22 SURFACE STRUCTURES OF IONIC CRYSTALS In the following report is an account of calculations on the model of the rocksalt lattice, the aim of which was to determine the surface structures caused by the effect of the ionic polarizability.Our work* follows on an investigation by Lennard- Jones and Dent 1 published twenty years ago, but goes further in that not only the displacement of the ions in the direction of the surface normals are taken into consideration, but also the tangential distortion of the surface lattice. As in the case of Lennard-Jones, besides the electrostatic forces, only forces of repulsion of small range according to Born’s power function are taken into account; here also a simplified surface structure is assumed where only the ions of the uppermost lattice face undergo deformation and displacement, while the rest of the lattice below it remains undeformed. Only average ionic properties of the two components are taken into account (except for the signs of the charges), i.e., the potentials of repulsion and the polarizabilities for both kinds of ions are regarded as equal.13 c FIG. I. The idea directing our calculation was the consideration that the polariza- bility of the ions can have the effect of lowering the symmetry. This idea was applied by Born and Heisenberg2 and by Hund3 to ionic molecules, and by the latter to three-dimensional ionic lattices. If one imagines the polarizability of these to be continuously increased, the co-ordination lattices, which at first are stable, change at a certain polarizability into less sym- metrical structures, namely, to layer- or molecule-lattices. These polariza- tion sub-structures” are distinguished by the fact that the ions prefer those positions where the electrical field has the highest possible intensity, in other words, where the gain in polarization energy is as great as possible.From the fact that an isolated (001)-lattice face in equilibrium must have a lower lattice constant compared with a compact space lattice (2.68 A for NaC1, calculated using the function of repulsion which is obtained from the lattice constant 2-81 A and the compressibility of the space lattice) Lennard- Jones and Dent l came to the conclusion in their above-mentioned investi- gation that a tendency to contract exists in the surface, which they treated as analogous to the surface tension of a liquid. The question of how such a contraction could be brought about was not followed up by Lennard- Jones and Dent ; above all, the part played by the ionic polarizability in a tangential deformation of the surface was not considered.We will show that no tendency towards tangential contraction exists in the case of low polariza- bility in the surface. Only a decrease in the distance between the two uppermost lattice faces is to be expected, similar to that already calculated by Lennard-Jones and Dent. With more highly polarizable ions, on the 1 Lennard-Jones and Dent, Proc. Roy. Soc. A , 1928, 121, 247 ; see also Madelung, Physik. Z., 1919, 20, 494 ; Zwicky, Helv. physic. Acta, 1930, 3, 269 ; Stranski, J b . Univ. Sofia, 1927-28, 24, 297 ; Z . physik. Chem., 1928, 136, 259. Hund, 2. Physik, 1925, 34, 833. 2 Born and Heisenberg, Z.Physik, 1924, 23, 388. * An account of our calculations which have been supplemented in the meantime by further results has already been published in 2. Physik, 1948, 124, 421 and 429.K. MOLIERE, w. RATHJE AND I. N. STRANSKI 23 other hand, tangentially deformed surface structures are favoured from the standpoint of energy. These are also adapted to the periodicity of the space lattice situated below, but in such a way that adjacent ions collect together to form small isolated molecular complexes.* That is to be seen best from a consideration of one- and two-dimensional structures. One- and Two-dimensional Lattices.-The investigations on stability carried out by Hund3 in three dimensions, but only approximately, can easily be calculated exactly for one- or two-dimensional lattices.How will a chain consisting of alternate positive and negative ions and an isolated lattice face of the rocksalt (001)-type behave respectively by a continual rise in the ionic polarizability ? Certain limiting assumptions must first be made concerning the form of the sub-structures of low symmetry which result from the co-ordination structures, and which one can imagine as being produced from the first by a homogeneous deformation. The chain or lattice face will break up into single insular complexes, and it seems plausible to expect that similar crystal units in the structure thus formed will assume equivalent positions as regards energy and structure and that the ionic complexes formed are electrically neutral, Thus, for the alternating chain only a division into double ion molecules comes into question, whilst for the lattice face four-ion insular complexes are con- ceivable too.Furthermore, " chain lattices " can also be formed here (analogous to the layer lattices in three dimensions), i.e., the complexes formed can extend over the whole surface in simple co-ordinative relationship. If one introduces the additional assumption that the complexes formed by a division of the original co-ordination lattice in the manner suggested are deformed by the effect of the polarization forces with no loss in their own symmetry, one is bound to arrive, in the case of the (001)-lattice face, a t the types shown in Fig. 2 . These structures may be characterized, as indicated in the figures, by means of a relative parameter y (the relative approach between immediate neighbours, referred to the distance between the ions 6 in the undeformed structure).The lattice energy of the chain or lattice face must now be formulated as a function of y. I n addition to the coulomb ionic effect and the energy of repulsion we introduce the polarization energy for one ion in the form ++ + - p . + + 2 / 2 C C -+ + where is the dipole moment produced in the single ion ; E is the " self-field intensity " arising from all other charges and dipoles ; a is the mean polar- izability. The components of p can be eliminated by putting the partial differential quotients equal to zero as was done by Born and Heisenberg.2 The following equation t is then obtained for the lattice energy for an ion Dair : + e is here the elementary charge, A .r-fi the repulsion potential of two ions (we use n = 9) ; VB) is Born's repulsion potential, V P ) , F(p), Fjd) self- potentials or self-field intensities for corresponding forms with the lattice constant 6 = I, originating from the poles ($) and dipoles (d). These * A similar hypothesis was already put forward by one of us in 1928. t The easily proved rule must be taken into account that for all configurations of the type under consideration the mutual effect between one dipole and all other ionic charges is equal in value to that between one ionic charge and all other dipole moments.24 SURFACE STRUCTURES OF IONIC CRYSTALS FIG. 2 b. FIG. 2 c. FIG. 2 d . "1K. MOLIERE, W. RATHJE AND I. N. STRANSKI 25 values are functions of y, and can be calculated according to the well-known methods of Madelung and Ewald.5 From the condition s u / a Y 2 = 0, one can find the " critical polarizability " above which the tangentially deformed structures must be more stable than the co-ordination structures.If, for the lattice constants of the latter, one inserts the equilibrium values of 6 = 2-49 A for the chain and 6 = 2.68 A for the lattice face, one obtains : for the chain . . .. CG$\ = 1-82 A3 ; for the lattice face . . Fag:: = 1-95 As. But if one stretches the chain or lattice face under force, so that their lattice constant adjusts itself to the value 6 = 2-81 A of the compact space lattice, acdt. becomes much smaller, namely : for the chain . . . . c ~ $ \ = 0.58 A 3 ; The mean value of the polarizabilities of Na+ and C1- amounts to CC = 1-61 A.The function U(y) for this value of a and for 6 = 2-81 A was plotted graphi- cally. for the lattice face . . Fazi8'; = 1-50 A'. It shows an energy minimum : for the chain at . . .. y = z o y o ; for the lattice face at .. ~ " 6 % . For the (001)-lattice face the structure type (a), the chain lattice parallel to the edge, proves to be the most stable. The figures quoted refer to this type. If one puts the lattice face or edge as the surface in a space lattice, the deformation of the surface is diminished through the effect of the undeformed remaining part of the lattice, as is shown below. The (001) -Surface.-The expression for the lattice energy of the lattice face must now be completed by the terms which express the potential energy of the ions of the uppermost lattice face in the field of the half-lattice lying below, which is undistorted and infinitely extended.* The field intensity where the surface ions are situated now has components perpendicular to the surface (z-direction).The lattice energy depends therefore upon a further parameter 5, for which we choose the relative distance (referred to 8) between the two uppermost lattice faces (see Fig. 3). The lattice energy per pair of ions - U(,,, is now The values FV, FF represent the self-potentials and self-field intensities of the surface already introduced above, V , HF the potentials and field inten- sities induced by the ions of the half-lattice at the points where the surface ions are situated; these are defined, as previously, for lattices with a distance between the ions of I.They can be calculated best according to Madelung's method. 4 Madelung, Physik. Z., 1918, 19, 524. Ewald, Ann. Physik, 1921, 64, 253- * These terms, of course, must be substituted in the expression for the lattice energy of the uppermost lattice face in their full amounts (or if one refers to a pair of ions, with the factor 2), and must not be halved like the self-mutual effect of the surface ions. This fact was overlooked by Lennard-Jones and Dent.'26 SURFACE STRUCTURES OF IONIC CRYSTALS U 0'1 I b 0.08 C 0.05 d 0'01 As the direct application of the conditions of equilibrium 3U/3y = o and 3U/3< = o is too complicated, we determine the minimum potential energy (maximum lattice energy) by a graphical method.If (- U ) for certain values of the tangential parameter y is plotted against the vertical distortion co-ordinate <, curves are obtained of the type in Fig. 4 a. The figure refers to the type (a) (Fig. 2 ) , the chain lattice parallel to the edge, and the polarizability (a = 1-61 A3), all the maxima lie at OSg7 0.97 y = 0, i.e., the lattice face retains its 0.96 complete symmetry. At the most, the 0.96 f23 f#-f3 1 jZ6 1 1 i f2.5 f25 'IK. MOLIERE, W. RATHJE AND I. N. STRANSKI 27 predominating polarization sub-structure found for the (001)-face. Then follow the types (b) and (c) and with a greater difference the diagonal chain lattice (a). From the result one can conclude that for NaCl probably no decrease in the symmetry of the face is to be expected on the cube faces.This should only set in with higher polarizability. The [OOlI-Edge.-It is in line with the approximate method used up t o now to deform independently only the ion chain which forms the edge of the crystal, whilst all inner atoms of the crystal quadrant remain unpola- rized in their normal positions. Those structures in the surface lattice faces adjacent to the edge are to be fixed which, according to the calculation above, represent the configurations of minimum energy for the infinitely extended surface. In this case also we allow the whole row on the edge t o move its position relative to the rest of the crystal, confining ourselves to a displacement in the plan which divides the angle between the adjacent cube faces.0.00 0.05 0.10 055 aa0 - r - r FIG. 5 a . FIG. 5 b. According to the previous section, when the polarizability is low the faces remain undistorted tangentially, their ions are polarized perpendicularly to the face and the distance between the two uppermost lattice faces is some- what decreased. In this case the critical polarizability for a conversion of the edge to a linear molecular lattice amounts to = 1-4 pi3. If one substitutes the polarizability 1-61 A3 (corresponding to NaC1) the energy minimum then lies at y = 0.04, which indicates a 4 yo approach between neighbouring ions. The edge is displaced about 5 yo in the direction of the remaining crystal quadrant. If we now use the higher polarizability of 2-1 A3, we must assume the chain lattice structure type (a) (see Fig. 2) in the cube surfaces.As the chain structure can take up different positions relative to the edge, different types of combinations must be considered, from which we pick out the three types in Fig. 6 as those most favoured from the standpoint of energy. According to the calculation carried out, type (a), in which the chains run vertically to the edge on both sides, is found to be the most stable. A molecular structure is obtained in the edge with an 18 yo approach between the nearest neighbours (compared with 11 yo in the adjacent28 SURFACE STRUCTURES OF IONIC CRYSTALS faces). In equilibrium, the whole edge alters its position relative to the rest of the crystal by 4 yo (compared with 2.5 yo €or the face). The Surface Structures for Crystals of the Rocksalt Type.-The formulz for the lattice energy of faces and edges contain the polarizability in the combination a/S3 only.If one makes the assumption (somewhat rough, of course) that the repulsion potentials for all lattices of the rocksalt type can be represented by means of a power function with the same repulsion exponent (n = g), Alp-' is a common constant for all lattices. (A is calculated from the equilibrium lattice constant of the space lattice which is obtained experimentally.) Thus the value 6.U is a pure function of the quotient afS3. It is thus now possible to specify critical values for this quotient, which are determining factors in a sub-structure formation in the faces and edges. These are (a/S3)crit. = 0.064 for the [OOI]-edge ; (a/83)crit-= 0.073 for the (001)-face. No decrease in symmetry in the faces and edges is to be expected in the alkaline halides for : Salt KF NaF RbF CsF KC1 LiF a/S3 0.047 0.047 0.056 0.061 0.062 0.063 The cube faces retain their full symmetry, whilst the edges show molecular structure for the salts : ---I- RbCl KBr RbBr NaCl 3 l L I a/ 63 0.065 0.069 0.070 0.073 The edges have strongly defined molecular structures, the faces exhibit chain lattice structures in the following : Salt RbI KI NaBr LiCl NaI LiBr LiI a/63 0.079 0.080 0.082 0.091 0.096 0.101 0.117 The way in which the lattice energy behaves at the polarizability a = 2.1 Hi3 shown above in Fig.4 and 5 might fit the case NaI. The anion is the chief determining factor for the values of the mean polarizability.On the other hand, the denominator of the quotient a/S3 is the lattice constant 6, which has especially low values for salts with small (strongly polarizing) cations. Therefore, the most strongly defined polarization sub-structures in the surfaces are to be expected in salts with large anions and small cations. ---I Decrease in Symmetry in Individual Faces.-It must be assumed that the decrease in symmetry in individual faces will show itself externally in some way in the physical properties of the faces. No direct method for the determination of surface structures exists, however, as yet; it is known that when electron diffraction is brought about, space lattice regions of considerable thickness are always involved. Clues to a decrease in sym- metry in the surface are given by numerous experiments using the etching method,6 in which a lower symmetry actually was found than that corre- sponding to the space lattice.But up to the present, no clear connection Brauns, N. Jahrb. Miner., 1886, I, 224 ; 1889, I, IZI. Rosicky, N . Jahrb. Miner., 1916, z, 15.K. MOLIERE, w. RATHJE AND I. N. STRANSKI 29 with our theory of polarization sub-structures could be established, as in the etching method apparently there are too many unpredictable conditions playing a part. FIG. 6 b . FIG. 6 4 b . Reorientation Processes ; Decrease of Symmetry of the Habitus.- It can be seen from Fig. 5 that the different sub-structure types of the (001)-face are in fairly keen competition with one another from the stand- point of energy (ordinate scale 10-l~ erg/ion pair), One must further note that each type of structure possesses four possible forms (e.g., orientation of the chain lattice in the x- and y-direction).I t must be assumed that spontaneous reorientations take place between the different structures.30 SURFACE STRUCTURES OF IONIC CRYSTALS It is possible that, a s co-operative processes, such changes would require considerable energy of nucleus formation or activation. As a result the frequency of such reorientation processes would be dependent upon temperature. As the molecular structures are especially well defined on the edges, as was shown above, there should be the fewest reorientation processes taking place there. Thus the edges function as nucleus-forming centres, i.e., they determine the structures of the adjacent parts of the faces, and the most favoured are chain lattice structures of the type (a) (Fig.6 a) with the chain running perpendicular to the edge. i coofl FIG. 7. rhombic dodecahedron. In the long needles was observed. If one considers the complete cubic. crystal-body, it is conceivable that parallel sub-structures will be formed on the cube faces of a zone through mutual influence across the common edge. Such a case is shown diagrammatically for the zone (001) in Fig. 7. According to the tem- perature, a more or less frequent change in structure will set in on the two faces which do not belong to the zone, and this can also lead to occasional reorientation of the complete zone structure beyond the edge, but on the whole the parallel surface structures of the faces of the zone in question would be especially stable.The difference in the degree of orientation of the two faces could perhaps show itself in their growing and adsorption properties, by which a lower symmetry of the whole crystal body (crystal habitus), in this case a tetragonal form, could be simulated. Nothing like this is known up to the present for the rocksalt lattice, but an analogous case might be found in the cubic space-centred lattice of crystallized ammonia. The growth form here is the presence of pectic acid, crystallization in Influence of External Forces ; Tensile Strength.*-There is no doubt that surface structures are sensitive to the effect of external forces, in which, incidentally, one may include the forces which proceed from an adsorption layer or a neighbouring phase, mentioned in the previous example.Tangential electric fields, for example, would probably favour the structure types (c) and (d) of Fig. 2 which possess a tangential electric moment, from the standpoint of energy. This might express itself in dielectric or optical anomalies, unless the effects are too small to be observed. If one submits the crystal to a tensile force in the direction of a cube edge, the tendency to form molecular lattice structures in the edges and faces running parallel to the direction of the force will increase greatly with only a small stretch. Molecular cracks, perpendicular to the direction of the force, are thus produced in the surface. It is conceivable that at a certain tensile force, which lies far below the theoretical tensile strength 7 Ehrlich, 2.anorg. Chem., 1932, 203, 26. 8 Stranski, Ber., 1942, 75, 1667.K. MOLIERE, W. RATHJE AND I. N. STRANSKI 3= of the infinitely extended crystal, these surface cracks will extend further into the inside of the crystal. The cracks produced would coincide with the breaking-faces of the crystal. Thus the presence of atomic sharp edges would be a decisive factor for the lowering of the tensile strength, compared with the theoretical value, as is actually confirmed experimentally. The experimental data of the Joff6 effect are in agreement with this, namely, FIG. 8 a. FIG. 8 b. if the edges are removed by superficial solution of the crystal, the tensile strength (that is, its upper limit in a great number of experiments) increases until almost at the theoretical value. With a renewed growing process (new formation of the atomic sharp edges) the tensile strength again greatly decreases.9 Joffg, Kirpitschewa and Lewitzky, 2. Physik, 1924, 22, 286.32 SURFACE STRUCTURES OF IONIC CRYSTALS By carrying out the necessary calculations, we have convinced ourselves how the surface structure of a superficially dissolved crystal might be expected to behave. This is known to consist of numerous atomic cube face steps. It was found that the ion chain which forms the steps does not exercise a directing influence upon the chain structures of the two adjacent parts of the face, as does the edge. The crack structures produced in the face are isolated to a certain extent by the steps.This result could be supported by means of calculations on the equilibrium structure of the (011)-face. This might be taken as representative for the structure of the superficially dissolved surface, as it is made up exclusively of atomic cube steps. The only sub-structure types of the (oII)-face which come into question are shown in Fig. 8 a, b. The calculation of the lattice energy shows that the type A is z x 10-l~ erg/ion pair more stable than type B. The distortion parameter (defined according to Fig. 8) amounts in equilibrium to 8 yo for NaCI, 20 yo for NaI. Thus for the (oII)-face it would seem that surface structures will be chiefly formed in which the molecular surface cracks do not lie in the track of possible breaking surfaces, as is the case in type B.In the more stable type A, on the other hand, the surface cracks follow [ I ~ I ] , i.e., in the track of rhombic dodecahedron faces; these are possible sliding-faces of the crystal. This could explain the increase in plasticity of superficially dissolved crystals. From our considerations, however, we are not able to produce a mathe- matical theory for the cracking. Among other things, the influence of statistically distributed lattice disturbances would have to be included. But it seems certain to us that structural irregularities in the surface will have to be taken into consideration in any exact theory of the future. Considerations Concerning the Justification of the Assumptions Made.-One objection which could be made to our way of calculating refers to the use of the linear polarization expression, + + $ = a E . This is known to be valid for homogeneous fields and small field intensities only. I t is certain that neither of these conditions are fulfilled in crystal surfaces. Estimates as regards energy, which refer to alkaline halide mo1eculesJ10 give cause for the assumption that, in reality, the share of the polarization energy in the total bond energy is considerably greater than the share calculated from the polarizability in the homogeneous field. If one subtracts the energy of repulsion, calculated from crystal lattice data, from the energy of dissociation known from spectroscopic data, and makes a correction for the effect of van der Waals’s forces, the remainder is more than twice as great as the classic polarization energy. The term remaining contains the quantum-mechanical mutual effect of the electron-clouds, which is difficult to estimate, but it can scarcely be assumed that this is very great. It can therefore be assumed that in the mutual effect of the ions in a crystal surface also the polarization share is still greater than that calculated by us according to the classical method. The data which we give for the surface distortions probably represent, therefore, a lower limit of the structural deviations realized in nature. Institut f u r Physikalische Claemie und Elektrochemie, Berlin-Charlottenburg 2, Hardenbergstrasse 34, Germany. Hellmann and Pschejetzkij, Acta Physicochiwz., 1937, 7, 621.