Interlayering of OHAp
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OCP
(4) A. M. Thorndike, A. J. Wells, and E. B. Wilson, Jr., J . Chem. Phys., 15, 157 (1947); R. C. Golike, I.M. Mills, W. B. Person, and B. Crawford, Jr., ibid., 25, 1266 (1956). (5) F. Matossi and V. Hohler, Z. Naturforsch. A , 22, 1516, 1525 (1967). (6) C. Haas and D. F. tiornig, J . Chem. Phys., 26, 707 (1957). (7) K. H. Hellwege, W. Lesch, M. Plihal, and G. Schaack, Z. Phys., 232, 61 (1970). (8) J. C. DeCiuS, R. Frech, and P. Bruesch, J. Chem. Phys., 58, 4056 (1973). (9) J. C. Decius and R. M. Hexter, "Molecular Vibrations in Crystals",
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McGraw-Hill, New York, 1977. (10) H. Nichols and R. Frech, J . Chem. Phys., 68, 4083 (1978). (11) J. C. Decius, J. Chem. Phys., 49, 1387 (1968); R. Frech and J. C. Decius, ibid., 51, 1536 (1969). (12) R. Frech and J. C. Decius, J . Chem. Phys., 51, 5315 (1969). (13) R. Frech and J. C. Decius, J . Chem. Phys., 54, 2374 (1971). (14) I. W. Levin and S. Abramowitz, J . Chem. Phys., 43, 4213 (1965). (15) J. L. Duncan, J . Mol. Specfrosc., 22, 247 (1967). (16) L. Pauling, "The Nature of the Chemical Bond", 2nd ed, Cornell University Press, Ithaca, 1940.
Interlayering of Crystalline Octacalcium Phosphate and Hydroxylapatite Walter E. Brown,*+ American Dental Association Health Foundation Research Unit at the National Bureau of Standards, Washington, D.C. 20234
LeRoy W. Schroeder, Division of Chemistry and Physics, Food and Drug Administration, Washington,D.C. 20204
and James S. Ferris Wartburg Cdege, Waver&, Iowa 50677 (Received November 6, 1978) Publication costs assisted by the National Institute for Dental Research
Hydroxylapatite (OHAp), Calo(P04)6(OH)2,and octacalcium phosphate (OCP), Ca8H2(P04)6.5H20, form interlayered mixtures in which some of their hOO X-ray diffraction peaks interact to form combination peaks. Calculations reveal that the dzoo(OCP)-dlm(OHAp)peak shifts toward lower spacings from 9.34 A, the dzmof OCP, to 8.16 A, the dlm of OHAp, as the Ca/P ration increases from 1.33 to 1.67. The dSm(0CP)-dzm(OHAp) and the d7M)(OCP)-d3m(0HAp) combination peaks shift to higher spacings as the Ca/P ratio of the interlayered mixture increases. The calculated shifts in the d2~(OCP)-dlm(OHAp) peak are corroborated by experimental data in the literature. Although shifts in the X-ray diffraction peaks are common in clays, this is the first time this phenonmenon has been observed in calcium phosphates; it provides a new approach to the study of tooth and bone mineral in their formative stages when OCP is most likely to be present. TABLE I: Unit Cell Dimensions of Octacalcium The principal constituent of tooth and bone mineral and Phosphate and Hydroxylapatite many other physiological precipitates is impure, poorly crystallized hydroxylapatite (OHAp), Calo(P04)6(OH)2.Its OCP OHAp structure' projected on the ab plane is shown in Figure 1. 9.432 a, A 19.705 Another calcium phosphate, octacalcium phosphate (OCP), 9.432 b, A 9.529 CaEH2(P04)6.5H20, has a remarkable structural similarity2 6.855 6.881 c, A to OHAp, as seen in Figure 1,where the projection is also 90 CY, deg 90.13 92.19 90 4 , d% on the ab plane. Although first proposed3 as a compound 120 Y, deg 108.36 in 1845 on evidence that today could not be considered adequate, and rediscovered several times s u b ~ e q u e n t l y , ~ ~ ~ for incorporation of impurities and defects into dental the existence of OCP as a distinct compound was quesenamel,a ( 5 ) a mechanism of growth of biological apatite^,^ tionable until its optical properties, unit-cell dimensions, and (6) a rationale why teeth may vary in their caries and space group were determined.6 Subsequently its susceptibility. An accounting of all these phenomena is structure, as shown in Figure 1, revealed such a close made possible by the simple assumptions that OCP can similarity between their layers parallel to the bc plane (line be a precursor to OHAp and that the hydrolysis of OCP A-A in Figure 1)that, it was apparent that OHAp and OCP to produce OHAp takes place in situ. can form epitaxial interlayered mixtures with a minimum The existence of interlayered mixtures between these of interfacial energy. However, OCP is unstable relative two salts was first indicated6by the similarities in the unit to OHAp and tends l,o hydrolyze according to the reaction cell dimensions b, c, and a (Table I) and by variability in CaEH2(P04)6*5H20 + 2Ca2+ = Ca10(P04)6(OH)2+ 4Hf the optical properties of crystals which had indices of The chemistry of OHAp is greatly complicated by its refraction intermediate between those of the two end ability to form interlayered mixtures with OCP; on the members.8 These precipitates, which clearly were not other hand, this structural relationship provides a rational simple physical mixtures of the two salts, nevertheless basis for understanding physiologically important pheyielded powder diffraction patterns with lines from both nomena such as: (1)a mechanism for prevention of dental salts; qualitatively there was a correlation between the caries by fluoride i n drinking water,7 (2) the unusual intensities of these lines with the Ca/P ratio. Subsequently variability in the stoichiometry of apatitic precipitates,a it was shown in a "single-crystal" X-ray study using (3) the cause of ribbonlike and platy morphologies of Weisenberg geometry8 that a crystal with intermediate biological apatitic crystallites,6 (4) a possible mechanism properties contained both crystals with the expected 0022-3654/79/2083-1365$01 .OO/O
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A
W.
E. Brown, L. W.
Schroeder, and J.
TABLE 11: Comparison of t h e Calculated d-Spacings for t h e Possible
hOO
Peaks in OCP a n d O H A p
oc P
hOO 100
-n
200 300 400
500 600 700
Flgure 1. Structure of hydroxylapatite (OHAp) projected on the 001 (three unit cells are shown); structure of octacalcium phosphate (OCP) projected on the 001 (three half-unit cells are shown); plane where the structures of hydroxylapatite and octacalcium phosphate are joined epitaxially (A--A).
parallel orientations of their axes b and c, respectively. In this “crystal”, the two types of layers were so thick that they diffracted much the same as individual units in a physical mixture. It was postulated a t that time, however, that the individual layers could be much smaller. We present evidence here to show that some apatitic precipitates contain layers in random arrangement in which the individual layers are only a few unit cells thick. These produce diffraction effects in which the individual layers no longer scatter X-rays independently. This is a phenomenon that is frequently observed in interlayered clay~.~-l’ The evidence presented here for the existence of such interlayered structures is based on comparison of calculated one-dimensional, h00-type scattering curves with experimental X-ray diffraction data. The calculations were obtained through the use of a computer program provided by Hower;12the experimental observations were reported by Eanes and Meyer13 in which they showed that the apparent dloopeak of OHAp-containing precipitates appeared with spacings higher than expected for OHAp and which varied with the Ca/P ratio of the precipitate. This method provides a significant new approach to the study of calcium phosphate precipitates with compositions between those of OCP and OHAp, particularly hard-tissue minerals in their early stages of formation.
Methods The interlayered mixtures of interest here are in the form of small crystallites, with thicknesses varying from 50 to 500 A. The one-dimensional Fourier-transform method of MacEwanlO was used, because it is more convenient for calculation of diffraction patterns for crystallites of various sizes than the method developed by Hendricks and Tellereg The MacEwan method calculates the Fourier transform of the distribution of scattering atoms generated by a sequence, which may be partially ordered, of layers of differing composition. The transform is divided into a part due to interference within a layer, the layer structure factor, and a part due to interference between layers. Reynoldsll treated interference between layers by means of mixing functions that depend on the probability of occurrence for an assemblage of mixed layers, N in all, and on the phase factors generated by the corresponding layer spacings. The effect of particle size is simply treated by assigning probabilities of occurrence to the various values of N. For random interlayering the probability of occurrence of an assemblage of N layers is the product of a multinomial coefficient and n p , ” ~where p L is the probability of occurrence and m, the number of layers of type i. Nearest neighbor interactions among layers can be allowed
-
OHAp
OCP PHC
S.Ferris
dhoo,
a
18.68 9.34 6.23 4.67 3.74
hOO
dhon >
100
8.16
200
4.08
300
2.72
3.11 2.67
for by means of conditional probabilities; e.g., pv is the probability that given a layer of type i, then a layer of type j follows. Interactions between second and third neighboring layers can be allowed for by the conditional probabilities, PiJk and plJkldefined analogously to pLJ. The specific algorithm used in this study was written by Hower.12 Since the junction of layers for OHAp and OCP occurs parallel to the bc plane, the coordinates of OHApl and OCP were first projected onto the normal to the bc plane. Inasmuch as both structures are centrosymmetric, the phases in the layer structure factors are sums of cosines and are efficiently calculated. DebyeWaller factors taken from the published study of OHAp’ and an unpublished study of a refinement of the OCP14 structure were included together with appropriate scattering factors in the calculations. Compositions of the mixtures were varied by varying the POHAp/POCP ratio. Partially ordered and random arrangements of layers can be obtained by adjustments of the conditional probabilities, pi],etc. Generally, the maximum values of N were set to ten, since values of twenty produced very similar diffraction patterns. In the absence of further knowledge, the distribution of N (particle size distribution) values was assumed to be Gaussian. Patterns were calculated for Ca/P ratios from 1.33, that of OCP, to 1.67, that of OHAp. The Ca/P ratio is given in terms of the probabilities, p,, by Ca/P =
( ~ ~ P O C+P
~ O P O H A ~ ) / ( ~ ~ P O+ C PPOH HA^)
where POCP
+ POHAP = 1
Although the diffraction patterns are sensitive to partial ordering, random arrangements were considered to be most appropriate for this initial study.
Results Figure 2 shows calculated scattering curves for several representative compositions of interlayered crystals. It can be seen that the OCP curve has six peaks in this region, dzoothrough d7,, while OHAp has three peaks, dloothrough dSm.These are listed in Table I1 to permit comparison of their numerical values. The diffraction peaks for the pairs dzoo(OCP)and dloo(OHAp),d500(OCP)and dzoo(OHAp), and d,,(OCP) and dSoo(OHAp)are near each other. Thus, in interlayered mixtures they can produce combination peaks by phase interference between layers such that the peaks shift with the C a / P ratio. The d20o(OCP)-d100(OHAp) peak shifts toward lower d values as the C a / P ratio increases, while the d5m(OCP)-d2m(OHAp)and the d,,(OCP)-d,,(OHAp) peaks shift toward higher d values as the C a / P ratio increases (Figure 3), in accord with expectations in each instance. The calculated d spacings are seen to vary approximately linearly with Ca/P ratio. The low-angle regions of the calculated curves for materials
The Journal of Physical Chemistry, Vol. 83, No. 11, 1979
Interlayering of OHAp and OCP ' --.-
5
IO
15
25
20
30
1307
35 '0°1
Dilfraction Angle,
2t
Figure 2. Calculated X-ray scattering curves along the reciprocal axis a * for various Ca/P ratios (Cu Ka, radiation; abscissa in 28; ordinate in arbitrary intensity unit!jl: A, Ca/P = 1.67; B, Ca/P = 1.56; C, CaIP = 1.44; D, Ca/P = 1.37; E, CaIP = 1.33.
with low C a / P ratios (Figure 2) contain spurious peaks. These are attributable to the small number of unit cell thicknesses used in the calculations, and possibly due to the presence of ripples from the dlm(OCP) peak which is very intense. The errors in the d values are quite large for this composition range (Figure 3).
Discussion Eanes and Meyer13 prepared a series of calcium phosphate precipitates i n which the C a / P ratio had been allowed to increase by maintaining the precipitates in contact with mother liquor for varying lengths of time; the initial solutions were supersaturated with respect to both OHAp and OCP. During maturation of a precipitate, OHAp layers could have been formed by either of two processes: (1)direct deposition of OHAp on the 100 face of the OHAp-OCP crystal; the number of unit-cell thicknesses adjacent to an OCP layer could be either odd or even; or (2) by hydrolysis of OCP layers; only an even number of unit-cell thicknesses in an OHAp layer would be possible. As the simplest model for the present calculation, it was assumed that the numbers of unit-cell thicknesses in the two types of layers were completely random. Eanes and Meyer observed a peak in the X-ray diffraction pattern which had a substantially higher spacing than the dloo of OHAp, and noted that the spacing decreased as the C a / P ratio increased. The values were always less than 9.34 A, the d200of OCP, and more than 8.16 A, the dIooof OHAp. We identify this peak as the d2m(OCP)-dloo(OHAP)combination and have calculated its spacing as a function of the C a / P ratio as described above. Plots of the experiimental values in their Figure 7 and our calculated values of this spacing against the Ca/P ratio (Figure 3) show a reasonably good correlation with a linear dependence of spacing on Ca/P ratio. The calculated and experimental points deviate somewhat from the straight line, especially those of Figure 10; both entail errors due to selection of maxima in rather broad peaks. The spurious peaks in the materials with low Ca/P ratios contribute to the deviations from the straight line, and there appear to be some systematic effects due t o crystal size or t o un-
//
2 75,
1
lo
1 bo
160
160
C a / P Mole
1
70
Ratio
Figure 3. Shifts in the positions of the combination peaks as functions of Ca/P. The squares and triangles on the top line are experimental points from Figures 7 and 10, respectively, of Eanes and Meyer.I3
certainties in the Ca/P ratio arising from adsorption of calcium or phosphate ions crystallites because of their high specific surfaces. The d500(OCP)-d200(OHAp)and the d700(OCP)-d300(OHAp) combination peaks also vary almost linearly with C a / P ratio (Figure 3). Eanes and Meyer did not detect variations in the positions of the d,oo(OCP)-d200(OHAp) and the d7m(OCP)-d3m(OHAp)peaks; the former was too broad and too weak to measure, and the latter was partially overlapped by other peaks in the diffraction patterns. It may be possible to overcome both of these effects through the use of samples with highly preferred orientation for hOO diffraction. Powder diffraction data are frequently used to calculate the unit-cell dimensions of apatites through least-squares refinement procedures. It is notable that such calculations would be affected in opposite directions by shifts of the d2oo(OCP)-dloo (OHAp) peak compared to the dbO0(OCP)-d200(OHAp) and d700(OCP)-d300(OHAp)peaks. The presence of OCP layers would increase the apparent value of the a dimension of OHAp calculated from the d200(OCP-dloo (OHAp) peak and conversely for the dbO0(OCP)-d200(OHAp) and d700(OCP)-d300(OHAp) peaks. Thus, examination of the shifts in the dh,(OHAp) peaks may make it possible to distinguish between interlayering and actual changes in lattice dimensions due to incorporation of impurities in the form of a solid solution in the apatite structure. Another phenomenon could account for variability in dhoospacings. When OCP is dehydrated by heating or exposure t o a vacuum, its dloo spacing decreases.6
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The Journal of Physical Chemistry, Vo/. 83, No. 11, 1979
Newesely15has reported that the process is reversible and suggests that in this respect OCP behaves like a clathrate compound. However, it is unlikely that this phenomenon is related to the observations of Eanes and Meyer, because it is not affected by the Ca/P ratio, and their samples were not subjected to heat. Although their samples were dried in vacuo, the temperature was too low to permit loss of water of crystallization. The existence of finely dispersed interlayering in synthetic precipitates does not, of course, prove that enamel also contains such interlayers. There is, however, considerable evidence7”J6J7that suggests that OCP acts as a precursor in the formation of hard-tissue mineral. If OCP does act as a precursor, then interlayering would be a natural consequence and, as noted earlier, would affect the properties of enamel in many ways. For example, ingested fluoride could reduce dental caries by accelerating the hydrolysis of OCP to the less soluble apatite. The ribbonlike morphology of the initial enamel crystallites is more consistent with the morphology of OCP crystallites than it is with the hexagonal symmetry of OHAp. The hydrolysis of OCP to OHAp could be primarily a topotactic transition; it is in the hydrolysis process that impurities and defects might be incorporated into the apatite crystallites of enamel, thereby altering their susceptibility to caries attack. Thus, OCP appears to have several crucial effects on the properties of tooth enamel, even though its concentration in mature enamel is quite low. A more complete understanding of these effects could help lead to a solution of the caries problem. One way to achieve this goal is through additional studies of the type described here. Such studies should include (i) a closer fit of the
Sidney Golden
calculational model with the experimental crystallites (i.e., a model in which the OHAp layers comprise only even numbers of unit-cell thicknesses) and (ii) specific models of the crystallites deduced from electron microscopic examinations and chemical considerations.
Acknowledgment. This research was supported in part by the National Institutes of Health-National Institute of Dental Research Grant No. DE05030 to the American Dental Association Health Foundation and is part of the dental research program conducted by the National Bureau of Standards in cooperation with the American Dental Association Health Foundation. Substantial assistance in the computational aspects was provided by Dr. John Hower, National Science Foundation. References and Notes (1) M. I. Kay, R. A. Young, and A. S.Posner, Nature(London),204, 1050 (1964). (2) W. E. Brown, Nature (London), 196, 1048 (1962). (3) J. J. Berzelius, Am. Chem. Pharm., 53, 286 (1845). (4) R. Warington, J . Chem. Soc., 19, 296 (1886). (5) N. Bjerrum, Selected Papers, Einar Munksgaard, Copenhagen, 245 (1949). (6) W. E. Brown et al., J . Am. Chem. Soc., 79, 5318 (1957). (7) W. E. Brown, Ciin. Orthop., 44, 205 (1966). (8) W. E. Brown et al., Nature (London), 196, 1050 (1962). (9) S. B. Hendricks and E. Teller, J . Chem. Phys., 10, 147 (1942). (10) D. M. C. MacEwan, 11, Kolloid-Z., 156, 61 (1957). (11) R. C. Reynolds, Am. Mineral., 52, 661 (1967). (12) J. Hower, personal communication. (13) E. D. Eanes and J. L. Meyer, Caic. Tiss. Res., 23, 259 (1977). (14) Unpublished data. (15) H. Newesely, Monatsh Chem., 98 (2), 379 (1967). (16) Transactions of the First Conference on Biology of Hard Tissues, New York Academy of Science, held in Princeton, N.J., in 1965, p 310. (17) K. J. Mijnzenberg and M. Gebhardt, Clin. Orthop., 90, 271 (1973).
Generalized Hartree Theory. Upper Bounds to Hartree-Fock Energies Sidney Goldent Department of Chemistry, Brandeis University, Waitham, Massachusetts 02 154 (Received January 15, 1979)
A generalization of the Hartree theory is derived from a minimum principle applied to the ground-state energy of an electronic system of which a typical single-electrondistribution is constrained to fulfill Fermi-Diracstatistics. The resulting generalized Hartree energy is shown to provide a rigorous upper bound to the analogous Hartree-Fock energy.
1. Introduction Since it is the earliest procedure designed to yield reliable approximations for ground-state solutions of Schroedinger’s equation for polyelectronic systems, Hartree’s theory1 lacks the fundamental rigorous justification of the similarly intended theory of Fock2which soon followed it. The Hartree theory was constructed to satisfy the restrictions of Fermi-Dirac statistics expected for electrons, but fails to satisfy the fundamental exchange-antisymmetry required by Pauli’s exclusion principle; this is fulfilled by the Fock theory. As a consequence, the former theory has been superseded in virtually all atomic and molecular applications by the latter theory, which is now universally known as the HartreeFock approximation. Primarily because of the appeal of the physically intuitive “self-consistent field” it invokes and the relatively 0022-3654/79/2083-1388$01 .OO/O
simple mathematics it involves, however, Hartree’s method has continued to receive attention and interest over the year^.^,^ Despite its limitations, the accuracy of the results it has produced appears not to have been seriously aff e ~ t e d For . ~ these reasons, a suitable generalization and simplification of Hartree’s theory that provides rigorous upper bounds to Hartree-Fock energies can have some practical utility. It is to that end that the present paper is directed. The generalized Hartree theory is developed in the following section as a consequence of minimizing an appropriate energy expression which maintains Fermi-Dirac statistics for a single-electron distribution. The generalized Hartree ground-state energies are shown in the next section to provide a rigorous upper bound to ground-state Hartree-Fock energies. The final section gives a brief discussion of the essential differences between the original 0 1979 American
Chemical Society
