Calcif.Tiss. Res. 24, 105-111 (1977)
Calcified Tissue Research 9 by Springer-Verlag 1977
The Effects of Crystal Size Distributions on the Crystallinity Analysis of Bone Mineral A.G. Miller and J.M. Burnell Department of Mining,Metallurgicaland CeramicEngineeringand the Departmentof Medicine,Universityof Washington,Seattle, Washington98195, USA Summary. The mechanisms by which crystal size distributions affect the usual method of quantitative Xray diffraction analysis of bone mineral have been determined on synthetic crystals. It was observed that each component of a crystal-size distribution diffracts independently. This independence causes systematic nonlinear behavior in the plot of integrated intensity vs. broadening parameter curves. The nonlinearity resulted in an overestimation of the amount of nondiffracting material present in bone mineral. Because crystal size distributions may vary for different crystallographic directions, it is strongly suggested that the usual practice of adding the c-axis and a-axis integrated intensities to estimate the crystallinity of the sample be discontinued. Methods of understanding the crystal size distribution function in bone mineral are discussed and evaluated.
Key words: Bone mineral - - X-ray diffraction - Crystallinity.
Introduction Quantitative x-ray diffraction (XRD) phase analysis has been applied to the study of bone mineral during the last several years. The integrated intensity of bone mineral was always less that that of size-matched synthetic hydroxyapatite (HA) preparations [11]. To account for this observation it was proposed that bone mineral contained a second, nondiffracting phase [41. This phase was thought to be an amorphous calcium phosphate (ACP) [4, 13, 17], The clear existence of such a phase had been seen in vitro in the reaction sequence of HA formation when such processes Send offprint requests to
A.G. Millerat the above address.
occurred in near physiological environments [2, 3]. Studies of bone mineral utilizing infra-red spectroscopy [17], X-ray diffraction, and electron spin resonance [13, 14] yielded data that were initially interpreted to be supportive of the view that ACP was present in bone mineral in significant amounts. However, transmission electron microscopy (TEM) investigations of bone mineral did not find the high amounts of ACP predicted by the above techniques [14]. Subsequent investigations suggested that the apparent ACP pool could also consist in part of nondiffracting or poorly diffracting crystals as well as ACP [16]. A recent study utilizing radial distribution function analysis has estimated that there may be no more than 10% max. ACP in mature bone mineral [ 11 ]. This paper investigates the effects of varying the crystal size distribution on percent crystallinity analysis. Conventional methods to date, in order to determine the crystallinity of a bone mineral sample, utilize a plot made of integrated intensities vs. linebroadening parameters for synthetic HA crystals of varying size/distortion characteristics [12, 13, 14, 16, 17]. With decreasing size and/or increasing crystalline distortion one observes broader diffraction profiles and decreased integrated intensities. (For simplicity we will refer hereafter to this size/distortion effect as only a size effect). Past studies have found that for synthetic HA preparations a plot between the integrated intensities and the broadening parameters yields a linear relationship. The crystallinity of a bone mineral Sample is obtained by dividing the summed integrated intensity of the sample's (002), or c-axis, and (310), or a-axis reflections by the summed (002) and (310) integrated intensities of a synthetic HA preparation that displays the same line-broadening parameters as the sample [12, 13, 16]. To investigate the applicability of this method when different crystal size distributions were present, we chose two sets of HA standards, and by mechanically
106
mixing them together in varying amounts obtained samples having known crystal size distributions and yet possessing no ACP [11]. The effect of such distributions was to create the false impression of the presence of ACP in the sample when it was analysed by the above method. The data obtained show that the usual assumption of linear behavior of the integrated intensity vs. broadening parameter curve did not hold for mixtures of crystals of varying sizes. This behavior is in accord with the fundamental XRD behavior of crystal size distributions. The results suggest that the previous overestimated values for the ACP content of bone mineral may be due in part to the effects of crystal-size distributions. They also suggest that past data analysis techniques have skewed systematically the results toward an overestimation of the amorphous pool.
A.G. Miller and J.M. Burnell" Crystal Size Distribution Effects to within 0.01 ~ 20 for the (002) reflection and to within 0.015 ~ 20 for the (310) reflection. A comparison of the results obtained by the above computer program and one used previously by Drs. J.D. Termine and E.D. Eanes [ 16] has shown that the two programs are equivalent in determining the I, fit, and fl} values.
Crystal Size Distribution Mixes. In this paper results are given for bimodal crystal size distributions. The original study investigated trimodal distributions, and those experiments have the same exact results as the ones observed for bimodal crystal size distributions. For clarity of presentation, we have restricted ourselves to the bimodal case, although the results are directly applicable to the case of n distinct constituents. Samples of A and B crystals (see above) were weighed and combined so that the percent of A crystals present increased by 20% intervals. Random mixing of the samples was accomplished by mechanically stirring the mixture and then passing the mixture through a 200 mesh screen two times.
Results Materials and Methods Synthetic Crystals. The salts of CaCI2, K2HPO 4, and Na2CO 3 were combined in aqueous solution in concentrations that resulted in formation of a hydroxyapatite with chemical composition very close to that of bone mineral and with a surrounding fluid of the same electrolyte concentration as that reported for bone extracellular fluid 110]. In that way, the surface ion content as well as the crystal composition would resemble that of bone. Part of the solution was aged for 6 days at 23 o resulting in "A crystals", and part of the solution was aged for 6 days at 45 ~ resulting in "B crystals". The aging at 45 ~ resulted in larger crystals, B crystals, than did the aging at 23 ~ A crystals. The samples were then washed with alcohol and spread on a Petri dish as a thin slurry, and the slurry quickly dried in a hood. The samples were ground to less than 200 mesh and stored under vacuum in a dessicator until used. Sizing the particles to below two hundred mesh size, approx. 74 #m, results in particles sufficiently small for satisfactory powder mounting. The actual crystal sizes involved are several orders of magnitude smaller.
When the mixes of the crystal populations were analyzed, the observed integrated intensities indicated that the crystals diffracted independently of each other. The curves in Figure 1 show that the observed integrated intensities were simply the sum of the integrated intensities of each constituent. This result is not surprising in light of the basic equations concerning XRD behavior of powder mixtures. When the observed integral breadth values, ill, and the width at one-half max. intensity values, fl89 were plotted against composition as in Figures 2 and 3, the resultant curves showed pronounced nonlinear behavior. Because fl_~is defined as the width at 1/2 peak
50-
/- e
X-Ray Diffraction Analysis., X-ray diffraction analyses were done on a Phillips vertical diffractometer (Phillips Electronic Instruments, Mount Vernon, New York) utilizing a scintillation counter and a graphite monochromator. A copper target X-ray tube was operated at 40 KV and 20 mA to produce the Cu-Ktz radiations used. The samples were step-scanned at 0.05 ~ 20/step, 200 s per step. The hydroxyapatite c axis (002) reflections were scanned from 28.00 ~ to 23.50 ~ 20, and the a-axis (310) reflections were scanned from 43.50 ~ to 36.00 ~ 20. The (310) reflections contained weak (212) and (221) reflections. The output was captured on paper tape by a Teletype ASR-33 (Teletype Corp., Skokie, Illinois) and then converted to a magnetic tape on a PDP-9 (Digital Equipment Corp,, Cambridge, Massachusetts) or read through an ASR-33 directly to a disk storage site. The data were processed by a computer program on a CDC 6400/Cyber 73 computer (Control Data Corp., Minneapolis, Minnesota). This program calculated for both (002) and (310) reflections the integrated intensity (peak area), the integral breadth (peak area/peak max. intensity), and the width at one-half max. intensity. The integrated intensity (I) was determined with a coefficient of variation of 1.5%. Integral breadths (ill) and widths at one-half maximum intensity (fl~) were determined
/ /
//I /
45. ,,...
I (002~o.~//%" ~ j,1"
9~ 4 0 -
1/
"•
/
/
/,-" ~'~I(3101
35/
30-
25
0
20
410 Weight
610 %B
810
1OO
Crystals
Fig. 1. Plot of integrated intensity of the (002) plane, I(002), or the (310) plane, I(310), vs. composition
A.G. Miller and J.M. Burnell: Crystal Size Distribution Effects
107
o8I
2.0- " ' ,
9
\ "'~\
0.7
pl/2 (
-...
)
9
oo
~IA-
Pl/2(310)---~ ~
,r
o 04
0
1.2-
o
2b
4b
eb
8b
, 40 Weight
o
1.1
, 20
~bo
%B
,
,
60
80
100
CrystaJs
Fig. 3. Plot of~1(002 ) or ill(002) vs. composition
Weight %B Crystals
Fig. 2. Plot of fix(310) or f189
vs. composition
intensity, one might infer that fl} is more heavily weighted by larger crystals than is Pr This apparently heavier weight occurs because the fl-I quantity is measured in the central region of the profile which the larger crystals dominate. The //1 quantity, while still strongly dependent upon the peak max. intensity, accounts for the presence of the smaller crystals and also by using the total area which would include the tails where the smaller crystals have more influence on the profile. This trend is seen in Table 1, which shows the deviation from linear behavior for both the//1 and fl89 parameters. When the integrated intensity of a crystal size mixture was plotted against the broadening parameters
(Figs. 4 and 5), as others have done in the past [12, 13, 16], one observes pronounced nonlinear behavior. In this case, we used the sum of ]/1(310) and ill(002), ~fll, as the definition of the broadening parameter of the system as has been used by previous investigators [12,
151. These results show that differing crystal size distributions can produce nonlinear curves of intensity vs. the broadening parameter, even though the pure constituents display a linear relationship. In terms of the mathematics of the diffraction processes, such behavior is not difficult to reconcile. In Figure 1 the data of intensity vs. composition shows that the intensities of crystal size distribution mixes were algebraically additive. This behavior extended to
Table 1. Systematic errors in broadening parameters due to crystal size distributions
%"A"
p,(310)o 2O
p§
o 2O
/~,(002)~2O
~}(oo2)o 2o
crystalsa
100 80 60 40 20 0
Obs b
Linc
Error
Obs
Lin
Error
Obs
Lin
Error
Obs
Lin
Error
2.03 1.86 1.74 1.65 1.58 1.53
2.03 1.93 1.83 1.73 1.63 1.53
0.00 0.07 0.09 0.08 0.05 0.00
1.77 1.52 1.38 1.28 1.25 1.17
1.77 1.65 1.53 1.41 1.29 1.17
0.00 0.13 0.15 0.13 0.04 0.00
0.729 0.675 0.630 0.592 0.561 0.534
0.729 0.690 0.651 0.612 0.573 0.534
0.000 0.025 0.021 0.020 0.012 0.000
0.642 0.582 0.541 0.522 0.498 0.489
0.642 0.611 0.581 0.550 0.520 0.489
0.000 0.029 0.040 0.028 0.022 0.000
Weight percent "A" (small) crystals in A p/us B crystal mix used in experiment b//~b~or pob~ are the observed integral breadth or width at one-half maximum intensity c fitIIn o r p~tin are the computed values that would occur if the broadening parameter varied linearly
108
A.G. Millerand J.M. Burnell:Crystal Size DistributionEffects
55-
function of composition when the latter is expressed on a weight percent basis. Recalling that the integral breadth, sit, is defined as the observed peak area divided by the observed peak maximum, equation (1) shows that the governing form of the equation for the//i quantity is:
50~
,~. 45-
I,...I 4 0 -
Y Z x:i(O)
35-
,8~bs _ , o i=, 30
2.0
211
2'.2 213
2~
i.s
~.6
='.7
Z X'r[i(O=
2~s
50-
45-
.......
40-
3~ 2.0
i
i
i
i
i
t
i
i
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
0ma x)
i=1
Fig. 4. Plot of integratedintensityof the (310) plane, I(310), vs. the sum of tl(002) and it(310), Sil t
..... 9
Fig. 5. Plot of integratedintensityof the (002) plane, [(002), vs. the sum of fl1(002) and it(310), )-~1
where fl}~bs is the observed integral breadth; Omax corresponds to the angle of the maximum intensity of the sample; li(O= Omax) is the intensity of the ith constituent at the angular position corresponding to the observed peak maximum; the other terms are as previously defined. Here It(B) is defined as the net intensity above background. It should be noted that the observed peak maximum of the sample may not correspond exactly with the observed peak maximum of any given one of the constituents. Small peak shifts in very small growing crystals in the hydroxyapatite system have been seen in vitro by Eanes (private communication, E.D. Eanes), and we have seen a similar shift in vivo in very young rats. Unfortunately, the equation governing the behavior of the fl89 values is not easily derived because the definition of fl~ is an empirical one lacking a convenient mathematical formulation [ 18]. The equation indicating linear behavior, as used previously for conventional crystallinity analysis (Figs. 2 and 3), is given by
Z xAo)
R,,.- r x : , , = ~'t
the profile on a point-by-point basis such that for any angle:
l (o) O.S= ~ x/i(O)
(2)
(1)
i=1
where l~ is the observed intensity at an angle 0; X i is the weight fraction of the /th component of the crystal size distribution; I/O) is the intensity at an angle 0 for the ith component; and n is the limiting number of constituent crystal size components. One might expect equation (1) to be on a volume percent basis rather than a weight percent basis from consideration of the general case of diffraction from a group of components [5]. However, since the samples here have approx, the same mass absorption coefficients and presumably similar densities, the weight percent value is identical to the volume percent value. Thus, Figure 1 shows that the intensity is a linear
-
i=1
x i=1
o_
(3)
' 1 i ( 0 = 0re,x) i
where p~" is the linearly varying/~t parameter for the ith constituent crystal size, and 0max is the angular location of the peak maximum intensity of the ith component. Comparing equations (2) and (3), it is noted that in general they are not equal. It is this inequality that causes the systematic nonlinear behavior shown in Figures 2-4, and 5. It must be remembered that the definition of fl~i,, equation (3), is an empirical one without theoretical basis. For the system under consideration, fl)~"has been observed to be valid empirically for the case of synthetic HA crystal preparations of various sizes as long as the size variations were not large [1 1, 12, 13, 15]. However, when mixtures of these preparations are formed to create large, known crystal size variations, the assumption of linearity in the fit or fl_~ values no longer holds. For the HA system under consideration the value for the peak maximum location
A.G. Miller and J.M. Burnell: Crystal Size Distribution Effects
109
i of the constituents, 0max, are relatively constant with respect to the line broadening, so that the observed peak maximum location of the mixtures is approx. equal to that of the constituents. For this situation the inequality in equation (4) is such that fl},bs is always less than or equal to fl~in. This arises because the larger crystals dominate the peak maximum intensity, resulting in a fl}~bsvalue always less than that predicted by equation (3). In the general case, if there were to exist very large peak displacements such that the peak maxima for some of the constituents were located far from the peak maxima of other constituents, then it is theoretically possible for fl}~bs to be larger than fl~in. Displacements large enough to produce this latter result have not been reported in this system, so that apparently, for the case of crystal size distributions related to bone mineral, fl}'bSwillalways be less than fl~in, resulting in nonlinear curves of the general form shown in Figures 4 and 5.
Discussion
As described above, in the standard method of quantitative phase analysis as applied to the study of bone mineral, the following effective formulation has been used [4, 12, 14, 161: % C = ~I~
sta
x 100
(4)
where Z/~ is the sum of the integrated intensity of the (310) plane, I(310), and the (002)plane, It0o2); where ~istd is the sum of/(o02 ) and Im0~ of an appropriate standard synthetic hydroxyapatite mixture; and, % C is defined as the "crystallinity" of the sample [4, 14]. The appropriate synthetic standard for the ~/std value has been chosen on the basis of a single synthetic HA crystal system that displayed the same ~_flt value (fll(002) plus fl,m0)) as observed in the bone mineral sample [12, 151. Based upon the results herein reported this methodology can now be seen to have two systematic errors. First, the method does not account for crystal size distribution effects. While the intensity values from a crystal size distribution are the weighted algebraic sum of the constituent intensities (Fig. 1), this relationship does not hold for the broadening parameters (Figs. 2 and 3). Differing crystal size distributions result in broadening parameters of lower value determined experimentally than would be predicted if the broadening parameters were algebraically additive (Figs. 2 and 3). Thus, when a synthetic crystalline standard is selected for use in equation (4) by the criteria of matched broadening characteristics, one is biased to
choose a standard with a higher intensity than is appropriate for the sample. This results in the prediction of nondiffracting material in the sample when in fact there may be none. The existence of a relatively widespread distribution of crystal sizes in bone mineral has support from observation and logic. Transmission electron microscopy by Termine et al. [15], has directly demonstrated crystals varying through a wide-size range in the same sample. One might also argue that since younger bone crystals are smaller than older ones, at least up to some limiting point [9], then the normal process of crystal formation and dissolution will necessarily create crystals of different ages and, therefore, sizes in any given bulk sample. The second objection to the use of equation (4) in analysing bone mineral is the addition of the integrated intensity of the (002) and (310) reflections to give a single value. We have shown in Figures 4 and 5 that crystal size distributions affect the curve of the integrated intensity vs. broadening parameter differently for different reflections. This is simply a function of the starting crystals used. There is no a priori reason why the crystal-size distribution function for an arbitrary sample of bone mineral should be the same for both the c-axis direction or the a-axis direction. This independence of the crystal size distribution function, for at least the perpendicular directions in this case, is obscured by the addition of the I(002) or 1(310) values, as is done in equation (4). The fundamental equations of quantitative phase analysis are explicit in that the choice of reflection is irrelevant for use in determining the quantity of a crystalline phase present in a sample [5]. The existence of an amorphous phase would cause both reflections to be decreased in intensity to the same degree. However, we have shown in Figures 4 and 5 that different reflections are influenced to different extents. These ideas suggest strongly that it might be possible to identify, at least partially, the changes in the crystal-size distributions in bone mineral samples if the (002) and (310) reflections were treated separately. Based upon the above, one need not necessarily invoke the concept of nondiffracting crystals or of very poorly diffracting crystals that escape detection in order to explain why crystal size distributions can simulate the presence of ACP in the analysis. Figure 1 shows that--aft of the crystals in the mixtures were diffracting as efficiently as they were in the starting unmixed state. Figures 2 and 3 show that the mixtures displayed broadening parameters which, when plotted against integrated intensity (Figs. 4 and 5) yielded results that would lead an unsuspecting observer to predict the presence of ACP when actually there was none. Thus, crystal size distribution effects may be at least partially responsible for the apparently
110 overestimated amounts of ACP previously reported to be present in bone mineral. There has been a disparity between the amounts of ACP previously predicted by XRD methods and the amounts observed by transmission electron microscopy [141. Much of the earlier work in transmission electron microscopy (TEM) has been criticized for use of aqueous preparations which were feared to have converted ACP to HA. Similarly, there had been a question about the resolution capabilities of TEM in identifying amorphous deposits. Nonetheless, it was difficult to accept the idea that 40% of bone mineral exists in an amorphous state and yet not have observation by TEM methods of such a large pool of material. Crystal size distribution effects may offer a partial reconciliation of these different analytical results. The final result will depend upon our ability to develop XRD techniques that can handle more precisely the phenomenon of anisotropic crystal size distributions. One approach that has been used for crystal size distribution analysis for systems such as ruthenium [7], nickel [61, and other materials [191 has been the use of the second derivative of the Fourier size-broadening coefficients [1, 181. This method is not well suited for the bone mineral system because the multiple order reflections needed to determine the Fourier size coefficients are not easily and reliably available [8]. Also, the second derivative of such data after the necessary number of manipulations is a somewhat suspect figure unless enormous efforts have been made to minimize the effect of error propagation in the analysis. The origin and propagation of such errors have been studied by Young et al. [191. They have shown that extremely small experimental errors result in significantly altered final results. An alternate method that is currently being investigated in our laboratories to determine the crystal size distribution function makes use of equation (1) as the basis for an empirical decomposition of the observed diffraction profile into its constituent parts. The use of entire diffraction profiles of synthetic crystals of different sizes in conjunction with equation (1) can be formulated into a least square fitting procedure. One can include the observed diffraction profile of amorphous calcium phosphate as well as synthetic crystals of varying sizes to determine simultaneously the amount of amorphous material and the crystal size distribution function. Preliminary studies have shown that the starting equations may be illconditioned in many cases; thus, one must be cautious in the selection of the constituent components. Based upon the above observations, it would appear that future XRD analysis of bone mineral must not ignore the effects of crystal size distributions. Care
A,G. Miller and J.M. Burnell:Crystal SizeDistribution Effects should be exercised in the selection of the bone sample to be analysed. A whole bone contains different types of bone tissue, metaphyseal and diaphyseal. In the growing rat the metaphysis has a smaller average size crystal than the diaphysis [13]. If one mixes the two types of bone tissue, the resultant answer may overestimate the amount of "nondiffracting" material because of the bias of the larger crystals. Therefore, it would seem advisable that comparative studies be performed on only one type of bone, metaphyseal or diaphyseal, rather than the whole bone. In summary, the mechanism has been presented by which the presence of larger crystals can dominate the XRD-phase analysis of bone mineral so that there is a systematic error creating the impression of the existence of an amorphous phase. This effect may be different in the (002) and (310) reflections. It is suggested that no useful purpose is served either theoretically or empirically by the addition of the (002) and (310) reflection for the crystallinity analysis of bone mineral since this summation only obfuscates potentially useful information, These data should not be construed to indicate that XRD results are not valuable. Our conclusions simply indicate some caution in interpretation of results, particularly with reference to quantification of percent crystallinity. The large pool of existing data clearly demonstrates significant alterations in the crystallographic composition of bone mineral as a function of growth and disease states. The studies reported herein do argue for separate analyses of peaks and further refinement of data reduction techniques in order to advance our knowledge of the physiologic mechanisms involved in the very complex system of bone.
Acknowledgement. The authors gratefully acknowledge the technical assistance and advice of M.R. Chapman, Dr. L. Larsen, and E. Teubner.This work was supportedby contract NO 1 AM-32208 from the National Institutes of Health. References
1. Bienenstock, A.: Calculation of crystalline size distribution from x-ray line broadening.J. Appl. Phys. 34, 1391 (1963) 2. Eanes, E.D., Gillessen, I.H., Posner, A.S.: Intermediate states in the precipitation of hydroxyapatite. Nature 208, 365-367 (1965) 3. Eanes, E.D., Posner, A.S.: Kinetics and mechanism of conversion of non-crystalline calcium phosphate to crystalline hydroxyapatite. Trans. N.Y. Acad. Sci. Ser. II, 28, 233-241 (1965) 4. Harper, R.A., Posner, A.A.: Measurement of non-crystalline calcium phosphate in bone mineral. Proc. Soc. Exp. Biol.Med. 122, 137-142 (!965) 5. Klug, H.P., Alexander, L.E.: X-ray diffractionprocedures for polycrystalline and amorphous materials, 2nd ed. New York: Wiley 1974
A.G. Miller and J.M. Burnell: Crystal Size Distribution Effects 6. Langford, J.l.: The variance and other measures of line broadening in powder diffractometry. II. The determination of particle size. J. Appl. Cryst. 1, 131-138 (1968) 7. Lele, S., Anantharam, T.R.: Domain size distribution in deformed ruthenium. Acta Cryst. A24, 654-656 (1968) 8. Lundy, P.R., Eanes, E.D." An x-ray line broadening study of turkey leg tendon. Arch. Oral Biol. 18, 813-826 (1973) 9. Menczel, J., Posner, A.S., and Harper, R.A.: Age changes in the crystallinity of rat bone apatite. Israel J. Med. Sci. 1, 251252 (1965) 10. Neuman, W.F.: The Milieu Interieur of bone: Claude Bernard revisited. Fed. Proc. 25, 1846-1850 (1969) 11. Posner, A.S., Betts, F.: Synthetic amorphous calcium phosphate and its relation to bone mineral structure. Accounts Chem. Res. 8, 273-281 (1975) 12. Russell, J.E., Termine, J.D., Avioli, L.V.: Abnormal bone mineral maturation in the chronic uremic state. J. Clin. Invest. 52, 2848-2852 (1973) 13. Termine, J.D.: Amorphous Calcium Phosphate: The Second Mineral of Bone. Cornel University, Ph.D. Thesis (1966)
111 14. Termine, J.D.: Mineral chemistry and skeletal biology. Clin. Orthop. 85, 207-241, 1972 15. Termine, J.D., Eanes, E.D.: Comparative chemistry of amorphous and apatite calcium phosphate preparations. Calcif. Tiss. Res. 10, 171-197 (1972) 16. Termine, J.D., Eanes, E.D., Greenfield, D.J., Nylen, M.U.: Hydrazine-deproteinated bone mineral. Calcif. Tiss. Res. 12, 73-90 (1973) 17. Termine, J.D., Posner, A.S.: Infrared analysis of rat bone: age dependency of amorphous and crystalline fractions. Science 153, 1523-1525 (1966) 18. Warren, B.E.: X-ray diffraction. Reading, Mass.: AddisonWesley 1969 19. Young, R.A., Gerdes, R.J., Wilson, A.J.C.: Propagation of some systematic errors in x-ray line profile analysis. Acta Cryst. 22, 155-162 (1967)
Received December 30, 1976 / Accepted May 16, 1977
Calcified Tissue Research 9 by Springer-Verlag 1977
The Effects of Crystal Size Distributions on the Crystallinity Analysis of Bone Mineral A.G. Miller and J.M. Burnell Department of Mining,Metallurgicaland CeramicEngineeringand the Departmentof Medicine,Universityof Washington,Seattle, Washington98195, USA Summary. The mechanisms by which crystal size distributions affect the usual method of quantitative Xray diffraction analysis of bone mineral have been determined on synthetic crystals. It was observed that each component of a crystal-size distribution diffracts independently. This independence causes systematic nonlinear behavior in the plot of integrated intensity vs. broadening parameter curves. The nonlinearity resulted in an overestimation of the amount of nondiffracting material present in bone mineral. Because crystal size distributions may vary for different crystallographic directions, it is strongly suggested that the usual practice of adding the c-axis and a-axis integrated intensities to estimate the crystallinity of the sample be discontinued. Methods of understanding the crystal size distribution function in bone mineral are discussed and evaluated.
Key words: Bone mineral - - X-ray diffraction - Crystallinity.
Introduction Quantitative x-ray diffraction (XRD) phase analysis has been applied to the study of bone mineral during the last several years. The integrated intensity of bone mineral was always less that that of size-matched synthetic hydroxyapatite (HA) preparations [11]. To account for this observation it was proposed that bone mineral contained a second, nondiffracting phase [41. This phase was thought to be an amorphous calcium phosphate (ACP) [4, 13, 17], The clear existence of such a phase had been seen in vitro in the reaction sequence of HA formation when such processes Send offprint requests to
A.G. Millerat the above address.
occurred in near physiological environments [2, 3]. Studies of bone mineral utilizing infra-red spectroscopy [17], X-ray diffraction, and electron spin resonance [13, 14] yielded data that were initially interpreted to be supportive of the view that ACP was present in bone mineral in significant amounts. However, transmission electron microscopy (TEM) investigations of bone mineral did not find the high amounts of ACP predicted by the above techniques [14]. Subsequent investigations suggested that the apparent ACP pool could also consist in part of nondiffracting or poorly diffracting crystals as well as ACP [16]. A recent study utilizing radial distribution function analysis has estimated that there may be no more than 10% max. ACP in mature bone mineral [ 11 ]. This paper investigates the effects of varying the crystal size distribution on percent crystallinity analysis. Conventional methods to date, in order to determine the crystallinity of a bone mineral sample, utilize a plot made of integrated intensities vs. linebroadening parameters for synthetic HA crystals of varying size/distortion characteristics [12, 13, 14, 16, 17]. With decreasing size and/or increasing crystalline distortion one observes broader diffraction profiles and decreased integrated intensities. (For simplicity we will refer hereafter to this size/distortion effect as only a size effect). Past studies have found that for synthetic HA preparations a plot between the integrated intensities and the broadening parameters yields a linear relationship. The crystallinity of a bone mineral Sample is obtained by dividing the summed integrated intensity of the sample's (002), or c-axis, and (310), or a-axis reflections by the summed (002) and (310) integrated intensities of a synthetic HA preparation that displays the same line-broadening parameters as the sample [12, 13, 16]. To investigate the applicability of this method when different crystal size distributions were present, we chose two sets of HA standards, and by mechanically
106
mixing them together in varying amounts obtained samples having known crystal size distributions and yet possessing no ACP [11]. The effect of such distributions was to create the false impression of the presence of ACP in the sample when it was analysed by the above method. The data obtained show that the usual assumption of linear behavior of the integrated intensity vs. broadening parameter curve did not hold for mixtures of crystals of varying sizes. This behavior is in accord with the fundamental XRD behavior of crystal size distributions. The results suggest that the previous overestimated values for the ACP content of bone mineral may be due in part to the effects of crystal-size distributions. They also suggest that past data analysis techniques have skewed systematically the results toward an overestimation of the amorphous pool.
A.G. Miller and J.M. Burnell" Crystal Size Distribution Effects to within 0.01 ~ 20 for the (002) reflection and to within 0.015 ~ 20 for the (310) reflection. A comparison of the results obtained by the above computer program and one used previously by Drs. J.D. Termine and E.D. Eanes [ 16] has shown that the two programs are equivalent in determining the I, fit, and fl} values.
Crystal Size Distribution Mixes. In this paper results are given for bimodal crystal size distributions. The original study investigated trimodal distributions, and those experiments have the same exact results as the ones observed for bimodal crystal size distributions. For clarity of presentation, we have restricted ourselves to the bimodal case, although the results are directly applicable to the case of n distinct constituents. Samples of A and B crystals (see above) were weighed and combined so that the percent of A crystals present increased by 20% intervals. Random mixing of the samples was accomplished by mechanically stirring the mixture and then passing the mixture through a 200 mesh screen two times.
Results Materials and Methods Synthetic Crystals. The salts of CaCI2, K2HPO 4, and Na2CO 3 were combined in aqueous solution in concentrations that resulted in formation of a hydroxyapatite with chemical composition very close to that of bone mineral and with a surrounding fluid of the same electrolyte concentration as that reported for bone extracellular fluid 110]. In that way, the surface ion content as well as the crystal composition would resemble that of bone. Part of the solution was aged for 6 days at 23 o resulting in "A crystals", and part of the solution was aged for 6 days at 45 ~ resulting in "B crystals". The aging at 45 ~ resulted in larger crystals, B crystals, than did the aging at 23 ~ A crystals. The samples were then washed with alcohol and spread on a Petri dish as a thin slurry, and the slurry quickly dried in a hood. The samples were ground to less than 200 mesh and stored under vacuum in a dessicator until used. Sizing the particles to below two hundred mesh size, approx. 74 #m, results in particles sufficiently small for satisfactory powder mounting. The actual crystal sizes involved are several orders of magnitude smaller.
When the mixes of the crystal populations were analyzed, the observed integrated intensities indicated that the crystals diffracted independently of each other. The curves in Figure 1 show that the observed integrated intensities were simply the sum of the integrated intensities of each constituent. This result is not surprising in light of the basic equations concerning XRD behavior of powder mixtures. When the observed integral breadth values, ill, and the width at one-half max. intensity values, fl89 were plotted against composition as in Figures 2 and 3, the resultant curves showed pronounced nonlinear behavior. Because fl_~is defined as the width at 1/2 peak
50-
/- e
X-Ray Diffraction Analysis., X-ray diffraction analyses were done on a Phillips vertical diffractometer (Phillips Electronic Instruments, Mount Vernon, New York) utilizing a scintillation counter and a graphite monochromator. A copper target X-ray tube was operated at 40 KV and 20 mA to produce the Cu-Ktz radiations used. The samples were step-scanned at 0.05 ~ 20/step, 200 s per step. The hydroxyapatite c axis (002) reflections were scanned from 28.00 ~ to 23.50 ~ 20, and the a-axis (310) reflections were scanned from 43.50 ~ to 36.00 ~ 20. The (310) reflections contained weak (212) and (221) reflections. The output was captured on paper tape by a Teletype ASR-33 (Teletype Corp., Skokie, Illinois) and then converted to a magnetic tape on a PDP-9 (Digital Equipment Corp,, Cambridge, Massachusetts) or read through an ASR-33 directly to a disk storage site. The data were processed by a computer program on a CDC 6400/Cyber 73 computer (Control Data Corp., Minneapolis, Minnesota). This program calculated for both (002) and (310) reflections the integrated intensity (peak area), the integral breadth (peak area/peak max. intensity), and the width at one-half max. intensity. The integrated intensity (I) was determined with a coefficient of variation of 1.5%. Integral breadths (ill) and widths at one-half maximum intensity (fl~) were determined
/ /
//I /
45. ,,...
I (002~o.~//%" ~ j,1"
9~ 4 0 -
1/
"•
/
/
/,-" ~'~I(3101
35/
30-
25
0
20
410 Weight
610 %B
810
1OO
Crystals
Fig. 1. Plot of integrated intensity of the (002) plane, I(002), or the (310) plane, I(310), vs. composition
A.G. Miller and J.M. Burnell: Crystal Size Distribution Effects
107
o8I
2.0- " ' ,
9
\ "'~\
0.7
pl/2 (
-...
)
9
oo
~IA-
Pl/2(310)---~ ~
,r
o 04
0
1.2-
o
2b
4b
eb
8b
, 40 Weight
o
1.1
, 20
~bo
%B
,
,
60
80
100
CrystaJs
Fig. 3. Plot of~1(002 ) or ill(002) vs. composition
Weight %B Crystals
Fig. 2. Plot of fix(310) or f189
vs. composition
intensity, one might infer that fl} is more heavily weighted by larger crystals than is Pr This apparently heavier weight occurs because the fl-I quantity is measured in the central region of the profile which the larger crystals dominate. The //1 quantity, while still strongly dependent upon the peak max. intensity, accounts for the presence of the smaller crystals and also by using the total area which would include the tails where the smaller crystals have more influence on the profile. This trend is seen in Table 1, which shows the deviation from linear behavior for both the//1 and fl89 parameters. When the integrated intensity of a crystal size mixture was plotted against the broadening parameters
(Figs. 4 and 5), as others have done in the past [12, 13, 16], one observes pronounced nonlinear behavior. In this case, we used the sum of ]/1(310) and ill(002), ~fll, as the definition of the broadening parameter of the system as has been used by previous investigators [12,
151. These results show that differing crystal size distributions can produce nonlinear curves of intensity vs. the broadening parameter, even though the pure constituents display a linear relationship. In terms of the mathematics of the diffraction processes, such behavior is not difficult to reconcile. In Figure 1 the data of intensity vs. composition shows that the intensities of crystal size distribution mixes were algebraically additive. This behavior extended to
Table 1. Systematic errors in broadening parameters due to crystal size distributions
%"A"
p,(310)o 2O
p§
o 2O
/~,(002)~2O
~}(oo2)o 2o
crystalsa
100 80 60 40 20 0
Obs b
Linc
Error
Obs
Lin
Error
Obs
Lin
Error
Obs
Lin
Error
2.03 1.86 1.74 1.65 1.58 1.53
2.03 1.93 1.83 1.73 1.63 1.53
0.00 0.07 0.09 0.08 0.05 0.00
1.77 1.52 1.38 1.28 1.25 1.17
1.77 1.65 1.53 1.41 1.29 1.17
0.00 0.13 0.15 0.13 0.04 0.00
0.729 0.675 0.630 0.592 0.561 0.534
0.729 0.690 0.651 0.612 0.573 0.534
0.000 0.025 0.021 0.020 0.012 0.000
0.642 0.582 0.541 0.522 0.498 0.489
0.642 0.611 0.581 0.550 0.520 0.489
0.000 0.029 0.040 0.028 0.022 0.000
Weight percent "A" (small) crystals in A p/us B crystal mix used in experiment b//~b~or pob~ are the observed integral breadth or width at one-half maximum intensity c fitIIn o r p~tin are the computed values that would occur if the broadening parameter varied linearly
108
A.G. Millerand J.M. Burnell:Crystal Size DistributionEffects
55-
function of composition when the latter is expressed on a weight percent basis. Recalling that the integral breadth, sit, is defined as the observed peak area divided by the observed peak maximum, equation (1) shows that the governing form of the equation for the//i quantity is:
50~
,~. 45-
I,...I 4 0 -
Y Z x:i(O)
35-
,8~bs _ , o i=, 30
2.0
211
2'.2 213
2~
i.s
~.6
='.7
Z X'r[i(O=
2~s
50-
45-
.......
40-
3~ 2.0
i
i
i
i
i
t
i
i
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
0ma x)
i=1
Fig. 4. Plot of integratedintensityof the (310) plane, I(310), vs. the sum of tl(002) and it(310), Sil t
..... 9
Fig. 5. Plot of integratedintensityof the (002) plane, [(002), vs. the sum of fl1(002) and it(310), )-~1
where fl}~bs is the observed integral breadth; Omax corresponds to the angle of the maximum intensity of the sample; li(O= Omax) is the intensity of the ith constituent at the angular position corresponding to the observed peak maximum; the other terms are as previously defined. Here It(B) is defined as the net intensity above background. It should be noted that the observed peak maximum of the sample may not correspond exactly with the observed peak maximum of any given one of the constituents. Small peak shifts in very small growing crystals in the hydroxyapatite system have been seen in vitro by Eanes (private communication, E.D. Eanes), and we have seen a similar shift in vivo in very young rats. Unfortunately, the equation governing the behavior of the fl89 values is not easily derived because the definition of fl~ is an empirical one lacking a convenient mathematical formulation [ 18]. The equation indicating linear behavior, as used previously for conventional crystallinity analysis (Figs. 2 and 3), is given by
Z xAo)
R,,.- r x : , , = ~'t
the profile on a point-by-point basis such that for any angle:
l (o) O.S= ~ x/i(O)
(2)
(1)
i=1
where l~ is the observed intensity at an angle 0; X i is the weight fraction of the /th component of the crystal size distribution; I/O) is the intensity at an angle 0 for the ith component; and n is the limiting number of constituent crystal size components. One might expect equation (1) to be on a volume percent basis rather than a weight percent basis from consideration of the general case of diffraction from a group of components [5]. However, since the samples here have approx, the same mass absorption coefficients and presumably similar densities, the weight percent value is identical to the volume percent value. Thus, Figure 1 shows that the intensity is a linear
-
i=1
x i=1
o_
(3)
' 1 i ( 0 = 0re,x) i
where p~" is the linearly varying/~t parameter for the ith constituent crystal size, and 0max is the angular location of the peak maximum intensity of the ith component. Comparing equations (2) and (3), it is noted that in general they are not equal. It is this inequality that causes the systematic nonlinear behavior shown in Figures 2-4, and 5. It must be remembered that the definition of fl~i,, equation (3), is an empirical one without theoretical basis. For the system under consideration, fl)~"has been observed to be valid empirically for the case of synthetic HA crystal preparations of various sizes as long as the size variations were not large [1 1, 12, 13, 15]. However, when mixtures of these preparations are formed to create large, known crystal size variations, the assumption of linearity in the fit or fl_~ values no longer holds. For the HA system under consideration the value for the peak maximum location
A.G. Miller and J.M. Burnell: Crystal Size Distribution Effects
109
i of the constituents, 0max, are relatively constant with respect to the line broadening, so that the observed peak maximum location of the mixtures is approx. equal to that of the constituents. For this situation the inequality in equation (4) is such that fl},bs is always less than or equal to fl~in. This arises because the larger crystals dominate the peak maximum intensity, resulting in a fl}~bsvalue always less than that predicted by equation (3). In the general case, if there were to exist very large peak displacements such that the peak maxima for some of the constituents were located far from the peak maxima of other constituents, then it is theoretically possible for fl}~bs to be larger than fl~in. Displacements large enough to produce this latter result have not been reported in this system, so that apparently, for the case of crystal size distributions related to bone mineral, fl}'bSwillalways be less than fl~in, resulting in nonlinear curves of the general form shown in Figures 4 and 5.
Discussion
As described above, in the standard method of quantitative phase analysis as applied to the study of bone mineral, the following effective formulation has been used [4, 12, 14, 161: % C = ~I~
sta
x 100
(4)
where Z/~ is the sum of the integrated intensity of the (310) plane, I(310), and the (002)plane, It0o2); where ~istd is the sum of/(o02 ) and Im0~ of an appropriate standard synthetic hydroxyapatite mixture; and, % C is defined as the "crystallinity" of the sample [4, 14]. The appropriate synthetic standard for the ~/std value has been chosen on the basis of a single synthetic HA crystal system that displayed the same ~_flt value (fll(002) plus fl,m0)) as observed in the bone mineral sample [12, 151. Based upon the results herein reported this methodology can now be seen to have two systematic errors. First, the method does not account for crystal size distribution effects. While the intensity values from a crystal size distribution are the weighted algebraic sum of the constituent intensities (Fig. 1), this relationship does not hold for the broadening parameters (Figs. 2 and 3). Differing crystal size distributions result in broadening parameters of lower value determined experimentally than would be predicted if the broadening parameters were algebraically additive (Figs. 2 and 3). Thus, when a synthetic crystalline standard is selected for use in equation (4) by the criteria of matched broadening characteristics, one is biased to
choose a standard with a higher intensity than is appropriate for the sample. This results in the prediction of nondiffracting material in the sample when in fact there may be none. The existence of a relatively widespread distribution of crystal sizes in bone mineral has support from observation and logic. Transmission electron microscopy by Termine et al. [15], has directly demonstrated crystals varying through a wide-size range in the same sample. One might also argue that since younger bone crystals are smaller than older ones, at least up to some limiting point [9], then the normal process of crystal formation and dissolution will necessarily create crystals of different ages and, therefore, sizes in any given bulk sample. The second objection to the use of equation (4) in analysing bone mineral is the addition of the integrated intensity of the (002) and (310) reflections to give a single value. We have shown in Figures 4 and 5 that crystal size distributions affect the curve of the integrated intensity vs. broadening parameter differently for different reflections. This is simply a function of the starting crystals used. There is no a priori reason why the crystal-size distribution function for an arbitrary sample of bone mineral should be the same for both the c-axis direction or the a-axis direction. This independence of the crystal size distribution function, for at least the perpendicular directions in this case, is obscured by the addition of the I(002) or 1(310) values, as is done in equation (4). The fundamental equations of quantitative phase analysis are explicit in that the choice of reflection is irrelevant for use in determining the quantity of a crystalline phase present in a sample [5]. The existence of an amorphous phase would cause both reflections to be decreased in intensity to the same degree. However, we have shown in Figures 4 and 5 that different reflections are influenced to different extents. These ideas suggest strongly that it might be possible to identify, at least partially, the changes in the crystal-size distributions in bone mineral samples if the (002) and (310) reflections were treated separately. Based upon the above, one need not necessarily invoke the concept of nondiffracting crystals or of very poorly diffracting crystals that escape detection in order to explain why crystal size distributions can simulate the presence of ACP in the analysis. Figure 1 shows that--aft of the crystals in the mixtures were diffracting as efficiently as they were in the starting unmixed state. Figures 2 and 3 show that the mixtures displayed broadening parameters which, when plotted against integrated intensity (Figs. 4 and 5) yielded results that would lead an unsuspecting observer to predict the presence of ACP when actually there was none. Thus, crystal size distribution effects may be at least partially responsible for the apparently
110 overestimated amounts of ACP previously reported to be present in bone mineral. There has been a disparity between the amounts of ACP previously predicted by XRD methods and the amounts observed by transmission electron microscopy [141. Much of the earlier work in transmission electron microscopy (TEM) has been criticized for use of aqueous preparations which were feared to have converted ACP to HA. Similarly, there had been a question about the resolution capabilities of TEM in identifying amorphous deposits. Nonetheless, it was difficult to accept the idea that 40% of bone mineral exists in an amorphous state and yet not have observation by TEM methods of such a large pool of material. Crystal size distribution effects may offer a partial reconciliation of these different analytical results. The final result will depend upon our ability to develop XRD techniques that can handle more precisely the phenomenon of anisotropic crystal size distributions. One approach that has been used for crystal size distribution analysis for systems such as ruthenium [7], nickel [61, and other materials [191 has been the use of the second derivative of the Fourier size-broadening coefficients [1, 181. This method is not well suited for the bone mineral system because the multiple order reflections needed to determine the Fourier size coefficients are not easily and reliably available [8]. Also, the second derivative of such data after the necessary number of manipulations is a somewhat suspect figure unless enormous efforts have been made to minimize the effect of error propagation in the analysis. The origin and propagation of such errors have been studied by Young et al. [191. They have shown that extremely small experimental errors result in significantly altered final results. An alternate method that is currently being investigated in our laboratories to determine the crystal size distribution function makes use of equation (1) as the basis for an empirical decomposition of the observed diffraction profile into its constituent parts. The use of entire diffraction profiles of synthetic crystals of different sizes in conjunction with equation (1) can be formulated into a least square fitting procedure. One can include the observed diffraction profile of amorphous calcium phosphate as well as synthetic crystals of varying sizes to determine simultaneously the amount of amorphous material and the crystal size distribution function. Preliminary studies have shown that the starting equations may be illconditioned in many cases; thus, one must be cautious in the selection of the constituent components. Based upon the above observations, it would appear that future XRD analysis of bone mineral must not ignore the effects of crystal size distributions. Care
A,G. Miller and J.M. Burnell:Crystal SizeDistribution Effects should be exercised in the selection of the bone sample to be analysed. A whole bone contains different types of bone tissue, metaphyseal and diaphyseal. In the growing rat the metaphysis has a smaller average size crystal than the diaphysis [13]. If one mixes the two types of bone tissue, the resultant answer may overestimate the amount of "nondiffracting" material because of the bias of the larger crystals. Therefore, it would seem advisable that comparative studies be performed on only one type of bone, metaphyseal or diaphyseal, rather than the whole bone. In summary, the mechanism has been presented by which the presence of larger crystals can dominate the XRD-phase analysis of bone mineral so that there is a systematic error creating the impression of the existence of an amorphous phase. This effect may be different in the (002) and (310) reflections. It is suggested that no useful purpose is served either theoretically or empirically by the addition of the (002) and (310) reflection for the crystallinity analysis of bone mineral since this summation only obfuscates potentially useful information, These data should not be construed to indicate that XRD results are not valuable. Our conclusions simply indicate some caution in interpretation of results, particularly with reference to quantification of percent crystallinity. The large pool of existing data clearly demonstrates significant alterations in the crystallographic composition of bone mineral as a function of growth and disease states. The studies reported herein do argue for separate analyses of peaks and further refinement of data reduction techniques in order to advance our knowledge of the physiologic mechanisms involved in the very complex system of bone.
Acknowledgement. The authors gratefully acknowledge the technical assistance and advice of M.R. Chapman, Dr. L. Larsen, and E. Teubner.This work was supportedby contract NO 1 AM-32208 from the National Institutes of Health. References
1. Bienenstock, A.: Calculation of crystalline size distribution from x-ray line broadening.J. Appl. Phys. 34, 1391 (1963) 2. Eanes, E.D., Gillessen, I.H., Posner, A.S.: Intermediate states in the precipitation of hydroxyapatite. Nature 208, 365-367 (1965) 3. Eanes, E.D., Posner, A.S.: Kinetics and mechanism of conversion of non-crystalline calcium phosphate to crystalline hydroxyapatite. Trans. N.Y. Acad. Sci. Ser. II, 28, 233-241 (1965) 4. Harper, R.A., Posner, A.A.: Measurement of non-crystalline calcium phosphate in bone mineral. Proc. Soc. Exp. Biol.Med. 122, 137-142 (!965) 5. Klug, H.P., Alexander, L.E.: X-ray diffractionprocedures for polycrystalline and amorphous materials, 2nd ed. New York: Wiley 1974
A.G. Miller and J.M. Burnell: Crystal Size Distribution Effects 6. Langford, J.l.: The variance and other measures of line broadening in powder diffractometry. II. The determination of particle size. J. Appl. Cryst. 1, 131-138 (1968) 7. Lele, S., Anantharam, T.R.: Domain size distribution in deformed ruthenium. Acta Cryst. A24, 654-656 (1968) 8. Lundy, P.R., Eanes, E.D." An x-ray line broadening study of turkey leg tendon. Arch. Oral Biol. 18, 813-826 (1973) 9. Menczel, J., Posner, A.S., and Harper, R.A.: Age changes in the crystallinity of rat bone apatite. Israel J. Med. Sci. 1, 251252 (1965) 10. Neuman, W.F.: The Milieu Interieur of bone: Claude Bernard revisited. Fed. Proc. 25, 1846-1850 (1969) 11. Posner, A.S., Betts, F.: Synthetic amorphous calcium phosphate and its relation to bone mineral structure. Accounts Chem. Res. 8, 273-281 (1975) 12. Russell, J.E., Termine, J.D., Avioli, L.V.: Abnormal bone mineral maturation in the chronic uremic state. J. Clin. Invest. 52, 2848-2852 (1973) 13. Termine, J.D.: Amorphous Calcium Phosphate: The Second Mineral of Bone. Cornel University, Ph.D. Thesis (1966)
111 14. Termine, J.D.: Mineral chemistry and skeletal biology. Clin. Orthop. 85, 207-241, 1972 15. Termine, J.D., Eanes, E.D.: Comparative chemistry of amorphous and apatite calcium phosphate preparations. Calcif. Tiss. Res. 10, 171-197 (1972) 16. Termine, J.D., Eanes, E.D., Greenfield, D.J., Nylen, M.U.: Hydrazine-deproteinated bone mineral. Calcif. Tiss. Res. 12, 73-90 (1973) 17. Termine, J.D., Posner, A.S.: Infrared analysis of rat bone: age dependency of amorphous and crystalline fractions. Science 153, 1523-1525 (1966) 18. Warren, B.E.: X-ray diffraction. Reading, Mass.: AddisonWesley 1969 19. Young, R.A., Gerdes, R.J., Wilson, A.J.C.: Propagation of some systematic errors in x-ray line profile analysis. Acta Cryst. 22, 155-162 (1967)
Received December 30, 1976 / Accepted May 16, 1977
