Hydroxyl ordering in igneous apatite

American Mineralogist, Volume 89, pages 1411–1421, 2004 Hydroxyl ordering in igneous apatite R. CHRIS TACKER* North Carolina State Museum of Natural Sciences, 11 West Jones Street, Raleigh, North Carolina 27601-1029, U.S.A. ABSTRACT Apatites from several pegmatites and two alkaline igneous environments were analyzed by electron microprobe and by micro-FTIR spectroscopy in the region 3000–4000 cm–1. Hydrogen bonding between OH and adjacent halogens shifts the OH-stretching modes and proves sensitive to the ordering of anions and cations within the apatite structure. With few exceptions, the spectrum of the stretching vibration can be modeled as a combination of peaks observed in synthetic apatites. Relative concentrations of OH populations can be determined from unpolarized spectra of oriented specimens. Analysis of populations of OH-Cl and OH-F pairs shows that OH-Cl pairs occur at a frequency greater than expected from a random sequence. This finding confirms earlier results that Cl incorporation into the hexagonal fluorapatite structure requires a hydroxyl nearest neighbor. Additional component bands suggest cationic ordering into the Ca2 site. Comparison of random mixing models with spectroscopic data suggests that Mn preferentially orders with OH. Limited data for the Kola apatite suggest the same for Sr and REE, in accord with previous results on the influence of OH on REE uptake. The small change in bond valence sum produced by substitution of an OH may explain the pairing of OH and Sr, but not with Mn. Activity of HAp in apatite is proportional to the area of the peak at 3575 cm–1. Activity calculated from mole fraction of HAp in the normalized microprobe analysis may overestimate the activity of water in associated exchange equilibria. Ordering of Cl-OH-F sequences produces potential complications for thermodynamic models. INTRODUCTION The apatite mineral group is a valuable petrogenetic indicator of volatile behavior in igneous and metamorphic rocks. The geochemistry of the solid-solution series [(Ca5(PO4)3(F, OH, Cl)] can be linked to the thermodynamic fugacities or activities of F-, OH- and Cl-complexes through a variety of exchange equilibria (e.g., Tacker and Stormer 1991). However, thermodynamic models of apatite are limited by the understanding of atomic-scale mixing of the end-members. Order-disorder phenomena will affect the thermodynamic activities of end-member apatite phase components. Practical application of the models to petrogenetic problems is likewise limited by the difficulties with electron microprobe analysis (Stormer et al. 1993), and the associated lack of measurement of the OH concentrations. End-member fluorapatite (FAp) is hexagonal with the space group P63/m. Along the sixfold c-axis, F atoms lie at the center of triangles of Ca2 atoms. The F atoms and the Ca2 triangles lie on the mirror plane at z = 1/4 and z = 3/4, although Fleet et al. (2000a, 2000b) noted that the F atoms can be displaced slightly from the special position. Pure end-member chlorapatite (ClAp) and hydroxylapatite (HAp) are monoclinic (space group P21/c, first setting). Chlorine and OH are ordered above or below the Ca2 triangle. In both space groups, the halogens and OH may be considered as a one-dimensional sequence along c (Hughes et al. 1989). * E-mail: [email protected] 0003-004X/04/0010–1411$05.00 X-ray structural refinements of ternary apatites show two possible ordering sequences for OH and Cl in mixed apatites (Hughes et al. 1990). When Cl and OH are ordered into crystallographically equivalent sites above or below the plane of Ca2 triangles, the monoclinic structure is preserved. In hexagonal structural sequences, the OH and Cl lie in close proximity between two adjacent Ca2 triangles. It is this hexagonal sequence that allows accommodation of a large Cl atom in a fluorapatite (Hughes et al. 1990). This proximity allows hydrogen bonding between Cl and OH, which will be apparent in the infrared signal of these apatites. The stretching mode of OH in hydroxylapatite is found at 3575 cm–1 (Fowler 1974). This peak, or mean wavenumber, will be referred to as OHOH. Hydrogen bonding with an adjacent Cl shifts this band to 3494 cm–1 (Dykes and Elliott 1971; Maiti and Freund 1981), referred to as OHCl. Hydrogen bonding with a nearest-neighbor F shifts the absorbance to 3535–3540 cm–1, referred to as OHF (R.A. Young et al. 1969; Levitt and Condrate 1970; Freund and Knobel 1977; Baumer et al. 1985). Hence there are three possible populations of hydroxyls in apatites that can be discerned with FTIR: OHOH, OHCl, and OHF. The position of the stretching mode also varies with cation substitutions on Ca2 (Engel and Klee 1972; Fowler 1974). The objective of the present work was to analyze the OHstretching region in well-crystallized apatites to identify anionic ordering. The apatite mineral group also shows extensive cationic substitution. A second objective was to characterize effects of these substitutions on the OH-stretching signal. 1411 1412 TACKER: HYDROXYL ORDERING IN IGNEOUS APATITE METHODS Sample preparation and analysis FTIR of igneous apatites from a variety of environments was undertaken to determine the nature of the hydroxyl signal. Apatites from a variety of pegmatites were procured from the Harvard Mineralogical Museum and from David London, University of Oklahoma. The pegmatites were the Bennett pegmatite mine, Oxford County, Maine (Wise et al. 1994; Wise and Rose 2000); the Harding pegmatite, Taos County, New Mexico (Chakoumakos and Lumpkin 1990); Strickland-Cramer pegmatite, Portland, Connecticut; Midnight Owl pegmatite, White Picacho District, Arizona (London and Burt 1982); the Black Mountain pegmatite, Rumford, Maine (Brown and Wise 2001); and the Pulsifer quarry, Androscoggin County, Maine (King and Foord 1994; Harvard Mineralogical Museum number 122454). Two apatites from alkaline sources were examined, a Sr-rich apatite from the Kola Peninsula (USNM no. 136827; Wolf and London 1995; Rakovan and Hughes 2000), and one from the Dwyer Mine, Wilberforce, Ontario. Durango fluorapatite (E.J. Young et al. 1969) came from the personal collection of the author. Single crystals were mounted in Crystalbond, cut parallel to (100), and doubly polished. An exception was the apatite from the Midnight Owl mine, a microcrystalline mass that could not be oriented crystallographically. It was included because preliminary data indicated higher Mn concentrations. Crystalbond was dissolved in acetone, and a slight residue removed manually with a Kimwipe and propanol. Thickness of the finished sections was between 0.07 and 0.2 mm, measured on a Mitutoyo digital micrometer to ±0.001 mm. Finished samples were examined under the petrographic microscope, found to be within a few degrees of [001], and then mapped. Most samples were single phases (some of gem quality), with no intergrowths of other minerals. Fluid inclusions were present in some samples, but avoided during analysis. As a check for homogeneity, apatites were given a rough polish and then examined under cathodoluminescence (CL), then ground and repolished to remove any area affected by the electron beam (Stormer et al. 1993). No zonation of Mn, which imparts a bright orange color in CL, was observed. Electron microprobe analyses (EMPA) of several of the apatites were performed at the University of Oklahoma (Table 1) by George B. Morgan III, on a Cameca SX-50 instrument. Sections were oriented with the c-axis perpendicular to the beam to avoid diffusion of F atoms documented by Stormer et al. (1993), and confirmed by Ottolini et al. (2000). Standard conditions were 20 keV acceleration and a 20 μm spot. Major and minor elements (Mn, Fe, F, Cl, Na, P, and S) were analyzed with a 10 nA beam to further avoid problems with light elements. Trace elements (La, Ce, Nd, Sm, Si, Y, Sr, and Gd) were then analyzed with a 20 nA beam to maximize count rates. Standards and sources used were Mn-rhodonite (USNM), Fe-augite (Penn State), F-topaz (Utah), Cl-tugtupite (CCNM), Ca and Si-bytownite (Stillwater), Na-albite (Amelia, VA), P-Durango fluorapatite, S-barite (C. Taylor), Y-synthetic Y2SiO5, and Sr-strontiantite (USNM). The standard for La, Ce, Nd, Sm, and Gd was a synthetic REE glass from the University of Oregon (Drake and Weill 1972). Published analytical data for the Pulsifer apatite (Sudarsanan et al. 1972; Dunn 1977) and Durango apatite (E.J. Young et al. 1969) were used in calculations presented below (Table 1). Analyses were normalized to a total charge of 25, assuming a full complement of 12.5 (O2– + Cl– + F–). This procedure yielded lower F and higher OH than other methods of normalizing the analysis, but did not produce excess F. OH was determined by difference XOH = 1– XF – XCl, where X is mole fraction. Samples were analyzed on an Analect Micro-FTIR at the University of Tulsa, with doubly polished (100) sections placed on top of manufactured KBr disks. Typical analyses involved 514 scans. The resolution of the FTIR was 4 cm–1. If a peak was present at a mean wavenumber more than a 2 σ error of 8 cm–1 away from a recognized peak position, it was assumed to be new or unassigned. Each mineral was analyzed at least 5 times with unpolarized radiation, although some data were lost in transfer. Four oriented spectra of Durango FAp and Wilberforce apatite were obtained with polarized radiation at the Department of Chemistry, University of Tulsa, using a Nicolet 510P FT-IR. Acquisition of polarized spectra with this instrument was not practical for most specimens due to the small size and thinness of the sections. The same areas of specimens were subjected to both electron microprobe and FTIR analysis. A small uncertainty results in that the two techniques sample different analytical volumes of the material: electron microprobe analysis is predominantly a shallow surface analysis while FTIR samples the entire thickness of the specimen. The effects of differing analytical volumes are observed in apatites from the Kola Peninsula and from the Black Mountain pegmatite, where small OHCl peaks are observed, but no Cl is measured by EMPA. This feature may reflect sample heterogeneity perpendicular to the c axis. It is also possible that the sample is homogeneous, and Cl is below detection limits of EMPA (<0.006 wt%), but present in amounts large enough to be detected as OHCl in the larger analytical volume of FTIR. Preliminary data from Secondary Ion Mass Spectrometry (SIMS) of these samples suggests that this may be the case (Tacker, unpublished data, 2003) for the Kola, Pulsifer, and Black Mountain apatites. Correction of spectra for polarization effects Quantification of OH in a mineral requires use of polarized infrared radiation (Libowitzky and Rossman 1996). Total absorbance (Atotal) is equal to the peak area Ai (cm–1) of an absorber measured in orthogonal directions (Libowitzky and Rossman 1996) Atotal = Aa + Ab + Ac, where the subscripts refer to crystallographic axes. For hexagonal minerals, measurement of total absorbance requires two measurements. If parallel to crystallographic axes: Atotal = Aa + Ac. For apatites, absorbance of OH-stretching bands is zero when polarization is parallel to the a-axis (Elliott 1965, 1994; Levitt and Condrate 1970). Maximum absorbance is found parallel to the c-axis. This phenomenon was confirmed for Durango and Wilberforce apatites in this study, and is also observed for laser Raman spectroscopy of oriented apatite sections (Tacker unpublished data). Hence, with polarization parallel to the c-axis: Atotal = Ac. Data presented herein were obtained with unpolarized radiation. If crystallographically oriented sections are used, the absorbance of OH in apatites represents a special case where a “polarized” spectrum may be calculated from the unpolarized data: Ac = –log[(2 × 10–Aunpolarized) – 1] (1) The derivation of this equation is given in Appendix 1. The effects of this recalculation are shown in Figure 1 for the apatite from the Bennett pegmatite. An interesting result is that this calculation places a limit on the unpolarized spectrum of an absorber polarized parallel to c. There cannot be a base ten log of zero or a negative number, so [(2 × 10–Aunpolarized) – 1] > 0 and Aunpolarized < 0.3010 This result is another aspect of the “plateau effect” observed by Libowitzky and Rossman (1996) whereby unpolarized absorbance cannot be scaled linearly with thickness. Absorbance will be approximately linear only at small thicknesses. As thickness of the specimen increases, unpolarized absorbance shows a “plateau” that is insensitive to the concentration of the absorber. For the apatites, unpolarized spectra with absorbances in excess of 0.3010 cannot be used to determine Ac. This limit also has immediate implications for interpreting the spectra. Absorbances above this limit indicate the presence of multiple absorbers polarized parallel to the c-axis, or the presence of absorbers that have an additional component not polarized parallel to c. The first instance was observed as multiple OH moieties, with individual A < 0.3010, that combine to produce a single signal with Aunpolarized > 0.3010. This was the case for 2 spectra from the Harding pegmatite. The second instance was observed as broad absorbances contributing to background, interpreted as probable interstitial water. In all other cases, TACKER: HYDROXYL ORDERING IN IGNEOUS APATITE 1413 FIGURE 1. Unpolarized spectrum from (100) section of Bennett pegmatite apatite, recalculated according to Equation 1, to determine Ac of the polarized spectrum. Aunpolarized was below this limit, so the calculated Ac is linearly scaled to with thickness. Assignment of component bands in the OH-stretching region Peaks were modeled with the Voight function with a linear background correction, using the PeakFit software (SPSS, Inc.). The shape of the Voight function differs from the Gaussian primarily in the shape of the base, or tail, of the peak. Peakfitting with the Gaussian shape resulted, in some cases, in small peaks present at the basal edge of the larger peaks. For example, Gaussian peak forms might yield small OHCl absorbances at the base of the OHF, which were eliminated by the use of a Voight function. In the end, identical results were obtained using a Gaussian peak shape, primarily because the results rely on the major observable peaks. Peakfitting of spectra followed a specific routine, proceeding from simple model to more complex. Peaks were placed at 3535 cm–1 (OHF), 3575 cm–1 (OHOH), and 3494 cm–1 (OHCl), if needed, and the model was then refined. Additional peaks were observed in positions identical to those of the synthetic endmembers Sr5(PO4)3OH (Sr-OHOH), Sr5(PO4)3(OH,F) (Sr-OHF), and Mn5(PO4)3OH (Mn-OHOH). Each spectrum was subjected to peakfitting at least three times in different sessions to assess the errors inherent in this procedure. Peak positions were reproducible to within ±2 cm–1 with this procedure. The standard deviation of the averaged peak area is given as the error in Tables 2, 3, and 4. Weak or poorly resolved peaks were not reproducible in multiple peakfitting sessions, leading to larger standard errors. In all specimens, the spectra could be resolved into different components. The relative concentrations of the various hydrogenbonded OH groups was variable within each crystal, indicating a heterogeneous distribution. Representative fitted spectra are shown in Figures 2, 3, and 4. Figure 2 shows the simplest spectra. Good results were obtained using Voight function peak forms in combination with published peak positions, giving confidence in use of this method for more complex spectra (Figs. 3–5). The 3535–3540 cm–1 absorbance of OHF is the principal peak seen in most spectra. It is not surprising that the majority of OH groups in fluorapatite are hydrogen bonded to F atoms. However, this single peak is insuf- F IGURE 2. FTIR spectra of the OH-stretching region of (a) Wilberforce, (b) Black Mountain, (c) Harding 23E, and (d) Durango apatites. In addition to the prominent OHF component band, OHCl and OHOH components can be resolved. Note that the Black Mountain spectrum shows no Mn-OH components, even though the apatite contains highest manganese measured by electron microprobe. All spectra are “polarized,” recalculated according to Equation 1. 1414 TACKER: HYDROXYL ORDERING IN IGNEOUS APATITE TABLE 1. Electron microprobe analyses of apatite specimens P2O5 SiO2 Y2O3 La2O3 Ce2O3 Sm2O3 Nd2O3 FeO MnO CaO SrO Na2O SO3 F Cl Sum O=F+Cl Total P Si Y La Ce Sm Nd Fe Mn Ca Sr Na S F Cl OH Black Mountain wt% 1σ 41.94 0.39 0.01 0 0 0 0.03 0.01 0.03 0.02 0 0 0.01 0.01 0.5 0.03 5.18 0.14 49.5 0.44 0.01 0.01 0.01 0.01 0.01 0.01 3.55 0.1 0 0 100.78 1.49 99.29 2.914 0.001 0 0.001 0.001 0 0 0.034 0.36 4.352 0 0.002 0.001 0.921 0 0.079 0.0525 0 0 0 0 0 0 0.0002 0.0025 0.0304 0 0 0 0.007 0 0.007 Bennett wt% 42.15 0.01 0 0.06 0.13 0 0.06 0.65 2.24 51.5 0.02 0.06 0.01 3.71 0.01 100.61 1.56 99.05 2.916 0.001 0 0.002 0.004 0 0.002 0.044 0.155 4.509 0.001 0.01 0.001 0.959 0.001 0.04 1σ 0.44 0 0 0.01 0.03 0.01 0.03 0.14 0.12 0.51 0.01 0.01 0.01 0.12 0.01 0.0604 0 0 0 0 0 0 0.0005 0.0012 0.036 0 0 0 0.008 0 0.008 Harding 2 wt% 1σ 41.87 0.32 0 0 0 0 0.02 0.02 0.01 0.02 0 0 0.01 0.01 0 0 3.51 0.15 51.85 0.79 0.02 0.01 0.02 0.01 0.02 0.01 3.57 0.04 0.01 0.01 100.91 1.51 99.4 Harding 23 wt% 1σ 41.89 0.39 0 0 0 0 0.03 0.02 0.03 0.02 0 0 0.01 0.01 0 0 3.56 0.36 51.55 0.82 0.03 0.01 0.02 0.01 0.01 0.01 3.6 0.06 0.01 0.01 100.74 1.52 99.22 Kola wt% 39.01 0.17 0 1.75 2.43 0.13 0.46 0.01 0.03 43.12 8.86 0.78 0.1 3.36 0 100.21 1.41 98.8 Formula proportions based on 12.5 (O, C, F) 2.898 0.055 2.903 0.064 2.883 0 0 0 0 0.015 0 0 0 0 0 0.001 0 0.001 0 0.056 0 0 0.001 0 0.078 0 0 0 0 0.004 0 0 0 0 0.014 0 0 0 0 0.001 0.243 0.002 0.247 0.0029 0.002 4.542 0.037 4.521 0.0423 4.033 0.001 0 0.001 0 0.448 0.003 0 0.003 0 0.132 0.001 0 0.001 0 0.007 0.923 0.007 0.932 0.008 0.928 0.001 0 0.001 0 0 0.076 0.007 0.067 0.008 0.072 1σ 0.18 0.01 0 0.05 0.07 0 0.02 0.01 0.03 0.48 0.08 0.03 0.01 0.04 0 0.0347 0.0001 0 0.0004 0.0006 0 0.0001 0 0 0.0209 0.0021 0.0003 0 0.004 0 0.004 Midnight Owl wt% 1σ 42.30 0.39 0.04 0.04 0 0 0.01 0.01 0.01 0.02 0 0 0.02 0.02 0.02 0.02 0.17 0.17 55.17 0.49 0.59 0.06 0.01 0.01 0.02 0.01 3.73 0.08 0 0.01 102.09 1.57 100.52 2.884 0.003 0 0 0 0 0.001 0.001 0.012 4.761 0.028 0.002 0.001 0.95 0 0.05 0.0514 0.0001 0 0 0 0 0 0 0.0004 0.0332 0.0002 0 0 0.007 0 0.007 Strickland-Cramer wt% 1σ 41.89 0.36 0.01 0 0 0 0.06 0.02 0.01 0.02 0.01 0.01 0.03 0.02 0.25 0.05 3.28 0.1 51.28 0.5 0.01 0.01 0.07 0.01 0.03 0.02 3.55 0.08 0.01 0.01 100.49 1.5 98.99 2.908 0.001 0 0.002 0 0 0.001 0.017 0.228 4.506 0 0.011 0.002 0.921 0.001 0.078 0.0499 0 0 0 0 0 0 0.0001 0.0015 0.0305 0 0 0 0.006 0 0.006 Notes: Durango and Pulsifer from published data. FIGURE 3. FTIR spectra with prominent bands at about 3550 cm–1. (a) Tamil Nadu B, (b) Tamil Nadu C, and (c) Harding 2B. All spectra are “polarized,” recalculated according to Equation 1. ficient to describe the signal in the OH-stretching region. Although Cl concentrations are low in all specimens, most spectra (Fig. 2) show OHCl. The Wilberforce, Black Mountain, and Harding 23 spectra show this peak as a shoulder. In the Durango apatite spectra, the peak is shifted from 3495 to 3482–3488 cm–1. It is not clear if the shift is due to substitution of a single cation in the Ca2 triangle, or if two or more cations have been replaced. The statistically more likely interpretation is the first, which is preferred here. Because of the uncertainties in the assignment of this peak, the Durango results were not used in later calculations. TACKER: HYDROXYL ORDERING IN IGNEOUS APATITE 1415 TABLE 1. —Continued Tamilnadu wt% 1σ 41.94 0.3 0.2 0.01 0.18 0.18 0.02 0.02 0.05 0.02 0 0 0.06 0.02 0.05 0.05 0.08 0.08 54.82 0.48 0.05 0.01 0.05 0.01 0.13 0.02 2.33 0.05 0.29 0.02 100.25 1.05 99.2 2.91 0.016 0.012 0.001 0.002 0 0.002 0.003 0.006 4.814 0.002 0.008 0.008 0.604 0.04 0.356 0.0431 0.0002 0.0005 0 0 0 0 0 0.0001 0.0285 0 0 0 0.004 0 0.004 Wilberforce wt% 1σ 41.34 0.36 0.13 0 0 0 0.23 0.02 0.5 0.02 0.02 0.01 0.21 0.02 0.01 0.02 0.03 0.1 53.93 0.06 0.53 0.01 0.25 0.01 0.36 0.02 3.49 0.08 0.01 0.01 101.04 1.47 99.57 2.86 0.011 0 0.007 0.015 0.001 0.006 0.001 0.002 4.721 0.025 0.04 0.022 0.902 0.001 0.097 0.0414 0.0001 0 0.0001 0.0001 0 0.0001 0 0.0002 0.0261 0.0001 0.0001 0 0.005 0 0.005 Durango Pulsifer wt% 1σ wt% 1σ 40.78 2.04 42.22 2.11 0.34 0.02 0 0 0.076 0 0 0.42 0.02 0 0 0.47 0.02 0 0 0.03 0 0 0 0.2 0.01 0 0 0 0 0 0 0.01 0 0.62 0.031 54.02 2.7 55.29 2.76 0.07 0 0.23 0.0115 0 0.37 0.02 0 3.53 0.18 3.55 0.18 0.41 0.02 0 0 101.05 101.68 1.58 1.49 99.47 100.19 2.826 0.028 0.005 0.013 0.014 0.001 0.006 0 0.001 4.737 0.003 0.036 0.023 0.914 0.057 0.029 0.482 0.002 0 0.001 0.001 0 0 0 0 0.25 0 0.001 0.002 0.033 0.002 0.033 2.887 0 0 0 0 0 0 0 0.042 4.786 0 0 0 0.907 0 0.093 0.488 0 0 0 0 0 0 0 0.002 0.251 0 0 0 0.033 0 0.033 Peakfitting revealed the presence of OHOH (3568–3577 cm–1). In Figures 3 and 4, all specimens show the presence of OHOH except for the apatite from Bennett. Although these are predominantly fluorapatites, there are still regions that resemble end-member HAp. In Figure 5, the Kola apatite shows the presence of Sr-OHOH identical to the synthetic end-member (Engel and Klee 1972; Fowler 1974). The OH in the apatite structure is coordinated with three Ca2. Replacement of a single Ca2 with another cation could be expected to shift or broaden the position of the OH-stretching band. If two Ca2 are replaced, then the peak should shift further; and if all three are replaced, the stretching band should resemble that of the synthetic end-member. In these spectra, Srand Mn-apatites show absorbances akin to those of the synthetic end-members. Durango (Fig. 2), Tamil Nadu, and Harding 2E specimens (Fig. 3) have a peak at 3550 cm–1 with three possible interpretations. First is that it represents Mn-OHOH (Mahapatra et al. 1990). Second, Engel and Klee (1972) found the 3550 cm–1 band to be present in synthetic Sr F-HAp, and assigned it to SrOHF. Strontium orders into the Ca2 site in apatite (Hughes et al. 1991a; Rakovan and Hughes 2000). Microprobe analysis (Table 1) shows that the Tamil Nadu apatites contain approximately equivalent amounts of Sr and Mn, whereas Mn exceeds Sr in the Harding 2E analysis. The peak at 3550 cm–1 cannot be assigned conclusively in Durango or Tamil Nadu apatite, but is assigned to Mn-OHOH in the Harding 2E. FIGURE 4. FTIR spectra with prominent bands at about 3520 cm–1, interpreted to be Mn-OHF. (a) Midnight Owl, (b) Pulsifer, (c) Bennett, and (d) Strickland-Cramer. All spectra are “polarized,” recalculated according to Equation 1, except for the Midnight Owl spectrum of an unoriented microcrystalline mass of apatite. 1416 TABLE 2. TACKER: HYDROXYL ORDERING IN IGNEOUS APATITE Comparison of concentration data (Eq. 3) derived from FTIR results for OHCl and OHF, with probabilities derived from electron microprobe data (Equation 4) Bennet A Bennet B Bennet C Bennet D Black Mountain A Black Mountain B Harding 2A Harding 2B Harding 23A Harding 23B Harding 23C Harding 23E Kola A Kola B Pulsifer A Pulsifer B Pulsifer C Pulsifer D Strickland-Cramer A Strickland-Cramer B Strickland-Cramer C Strickland-Cramer D Strickland-Cramer E Tamil Nadu A Tamil Nadu B Tamil Nadu C Wilberforce A Wilberforce B Wilberforce C Wilberforce D Wilberforce polarized TABLE 3. OHCl/OHF 0.1141 0.1406 0.3444 0.0998 0.2697 0.2696 0.2096 0.3735 0.1200 0.3932 0.1520 0.5022 0.2069 0.2093 0.0511 0.0511 0.1097 0.0158 0.1611 0.1068 0.3343 0.0381 0.1573 0.1442 1.1068 no OHF 0.1158 0.1644 no OHCl 0.1203 0.3723 1σ 0.0326 0.0598 0.0394 0.0217 0.0461 0.1661 0.0918 0.2795 0.0332 0.3316 0.0649 0.4333 0.0118 0.2031 0.0069 0.0129 0.0168 0.0057 0.0702 0.0770 0.1584 0.0660 0.0575 0.1442 0.3435 0.0288 0.0394 0.0237 0.2516 XCl/XF 0.0014 0.0014 0.0014 0.0014 0.0000 0.0000 0.0015 0.0015 0.0015 0.0015 0.0015 0.0015 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0015 0.0015 0.0015 0.0015 0.0015 0.0667 0.0667 0.0667 0.0015 0.0015 0.0015 0.0015 0.0015 1σ 0.00002 Ordered 0.00002 Ordered 0.00002 Ordered 0.00002 Ordered 0.00000 Ordered 0.00000 Ordered 0.00003 Ordered 0.00003 Ordered 0.00002 Ordered 0.00002 Ordered 0.00002 Ordered 0.00002 Ordered 0.00000 Ordered 0.00000 Ordered 0.00000 Ordered 0.00000 Ordered 0.00000 Ordered 0.00000 Ordered 0.00002 Ordered 0.00002 Ordered 0.00002 Ordered 0.00002 Disordered 0.00002 Ordered 0.00053 Disordered 0.00053 Ordered 0.00053 0.00001 Ordered 0.00001 Ordered 0.00001 0.00001 Ordered 0.00001 Ordered Comparison of concentration data (Eq. 3), derived from FTIR results for Mn-OHF and OHF, with probabilities derived from electron microprobe data (Eq. 5) Bennet A Bennet B Bennet C Bennet D Black Mountain A Black Mountain B Harding 2A Harding 2B Harding 23A Harding 23B Harding 23C Harding 23E Pulsifer A Pulsifer B Pulsifer C Pulsifer D Strickland-Cramer A Strickland-Cramer B Strickland-Cramer C Strickland-Cramer D Strickland-Cramer E Mn-OHF/OHF 0.4723 0.5446 0.5805 0.5249 0.8689 0.4366 0.6227 0.5924 0.7071 2.4796 0.6413 0.8870 0.5926 0.5884 0.5926 0.3049 0.8778 0.7416 1.7905 0.7009 0.4243 1σ 0.2221 0.1355 0.1029 0.0969 0.6087 0.2285 0.2553 0.1361 0.1070 2.0017 0.2532 0.3917 0.0022 0.0460 0.0022 0.0315 0.2655 0.3207 0.9818 0.1249 0.1421 X3Mn /X3Ca 0.0012 0.0012 0.0012 0.0012 0.0178 0.0178 0.0050 0.0050 0.0053 0.0053 0.0053 0.0053 0.000023 0.000023 0.000023 0.000023 0.0041 0.0041 0.0041 0.0041 0.0041 1σ 0.00019 0.00019 0.00019 0.00019 0.00246 0.00246 0.00041 0.00041 0.00102 0.00102 0.00102 0.00102 0.00002 0.00002 0.00002 0.00002 0.00055 0.00055 0.00055 0.00055 0.00055 Ordered Ordered Ordered Ordered Ordered Ordered Ordered Ordered Ordered Ordered Ordered Ordered Ordered Ordered Ordered Ordered Ordered Ordered Ordered Ordered Ordered The third possibility is that the 3550 cm–1 peak represents an OH-F-HO configuration (Baumer et al. 1985), which is observed in synthetic FAp-HAp mixtures at 3547 cm–1. This peak is observed in low-F specimens only, where the samples presented here are high-F types. Baumer et al. (1985) did not use peakfitting methods to resolve the relative contributions of OHOH and OHF. Their assignment is reinterpreted here as a combination of OHF with the edges of the predominant OHOH peak, which acts to shift the apparent maximum. Figure 4 shows fitted spectra of Pulsifer and Midnight Owl apatites, which have a small peak clearly present with a mean wavenumber at 3520 cm–1 (3514–3520 cm–1). In the fitted spectra of apatites from the Black Mountain and Strickland-Cramer pegmatites, a shoulder on the absorbance is present, resolved as a prominent band from 3515–3520 cm–1. Microprobe analysis of the pegmatite apatites indicates that they are high in Mn. Hughes et al. (1991a) demonstrated that Mn atoms in apatites from the Harding pegmatite enter both Ca1 and Ca2 sites. The substitution of the smaller Mn cation (Shannon 1976) in Ca2 produces a shift of the OH-stretching mode to lower wavenumbers, from 3495 cm–1 in OHCl to 3380 cm–1 in Mn-OHCl (Engel and Klee 1972), and from 3575 cm–1 in OHOH to 3550 cm–1 in Mn-OHOH (Mahapatra et al. 1990). Hence, this peak is attributed to a MnOHF configuration as compared to the Ca-OHF configuration at 3535 cm–1. In the Kola Sr-apatite (Fig. 5), there are three peaks observed in addition to those assigned to OHOH, OHF, and OHCl. The band with a mean at 3591 cm–1 has been observed in Sr-HAp as the stretching mode of OH groups (Sr-OHOH) (Engel and Klee 1972; Fowler 1974). Two unassigned peaks are observed at 3468 and 3432 cm–1. In britholite, which is isostructural with apatite, there is a broad band centered at 3434 cm–1 attributed to REE-OHF (Oberti et al. 2001). The 3468 cm–1 peak observed here is tentatively assigned to REE-OHOH, given the preference of REE for the Ca2 site in FAp and HAp (Hughes et al. 1991b; Fleet et al. 2000a, 2000b), and the influence of anion chemistry on the site occupancies of REE (Fleet et al. 2000b). In several spectra, broad absorptions were present that were interpreted to be interstitial water. These may be water in fluid inclusions too small to be observed by petrographic microscope, because fluid inclusions were mapped and avoided during analysis. Alternatively, these absorbances may represent clusters of water molecules similar to those observed in quartz (Kronenberg 1994). The resolution of OHCl absorbances in apatites that are low in both Cl and OH supports the conclusions that an OH group might be required as an intermediate between F and a larger Cl atom (Hughes et al. 1990). The presence of OH-stretching bands identical to those observed in synthetic Mn- and Sr-apatites TABLE 4. Results for Kola Sr-apatite REE-OHF ± 1σ X3REE ± 1σ Sr-OHOH ± 1σ X3Sr ± 1σ /X3Ca /OHOH /X3Ca /OHF Kola A 0.0974 ±6.86x10-9 0.0014±0.00007 Ordered no CaOHOH Kola B 0.0738 ±0.008 0.0014±0.00007 Ordered 0.3807±0.0554 0.037±0.0018 Ordered Kola C 0.0478 ±0.0065 0.0014±0.00007 Ordered 0.6553±0.2084 0.037±0.0018 Ordered Kola D 0.142 ±0.1839 0.0014±0.00007 Disordered 0.287±0.3477 0.037±0.0018 Disordered Kola E 0.1392 ±0.0141 0.0014±0.00007 Ordered no CaOHOH Note: Comparison of concentration data (Eq. 3), derived from FTIR results for Sr-OHOH and OH OH and for REE-OHF and OHF, with probabilities derived from electron microprobe data (Eq. 5). TACKER: HYDROXYL ORDERING IN IGNEOUS APATITE 1417 beam by the spectrometer optics can affect the integrated absorbance (Libowitzky and Rossman 1996) and data are lacking here to quantify those effects. Comparing the integrated area of the OHCl peak to that of the OHF yields: A3494 cm –1 cOHCl cOHF = A3535 cm –1 ε i( 3494 cm –1 ) (3) ε i( 3535cm –1 ) Random ordering of anions and cations F IGURE 5. Spectrum of Kola strontium-apatite, recalculated according to Equation 1. implies the existence of some degree of ordering between the cations and anions. Relative concentrations of the OH populations and probabilities of configurations can be quantified to test the hypotheses of ordering. Relative quantification of OH populations Determination of the relative concentrations of populations of OHOH and OHCl and OHF pairs from the FTIR data is essential for analysis of ordering. Libowitzky and Rossman (1996, 1997) offer a methodology for quantifying OH (as H2O) in minerals, which requires measurement of the total intensity of the integrated absorbance of the OH band at orthogonal directions in a crystal, using polarized radiation. The calculations that follow utilize their derivations and conventions. Concentration is determined from: 1.8 c = Ai · t·ρ·ε l (2) where c is the concentration of H2O (in wt%), Ai is integrated absorbance of a peak (cm–1), t is thickness in cm, ρ is density (gm/cm3), and εi is integrated molar absorption coefficient (L/ mol·cm2). In the simplest case, the area of the various absorbance bands can be compared directly. This comparison is not possible for bands with differing mean wavenumbers, as it requires knowledge of the integrated molar absorption coefficients (εi), the values of which have not yet been determined explicitly for the apatites. However, the relationship between mean wavenumber and ει has been determined for OH-O hydrogen bonding (Libowitzky and Rossman 1997) as ει = 246.6(3753 – ν), where ν is mean wavenumber of the OH-stretching band in cm–1. Values of εi were calculated for the mean wavenumbers 3433, 3494, 3520, 3535, 3550, 3575, and 3592 cm–1. The applicability of this equation to OH-F and OH-Cl pairs awaits further research, but represents the best approximation currently available. The assumption that εi values for all mean wavenumbers were approximately equal was tested, yielding relative concentrations of the lower-wavenumber moieties that were slightly and systematically higher. Use of this assumption did not change the conclusions. At this point it is essential to emphasize that the quantities determined are the relative concentrations of OHOH, OHCl, and OHF within each spectrum. Partial polarization of the infrared Hughes et al. (1990) concluded that in the ternary apatites, an OH might be required between Cl and F in order for the apatite structure to accommodate the large Cl anion, which suggests non-random anionic ordering. FTIR of the apatites give insight into the ordering of hydroxyl anions in this sequence. For comparison, random ordering must be calculated, which is similar to a configurational entropy calculation of random distribution of atoms on sites within a crystal structure (Price 1985). In a random, one-dimensional sequence along the sixfold axis there are three anions, but two separate configurations of hydroxyl groups: the hydrogen in the OH group is either pointed toward or away from the neighbor under consideration. For example, Cl-OH sequencing, in crystallographically similar sites (Hughes et al. 1990) leads to monoclinic structure, but no hydrogen bonding between the two. A Cl-HO sequence, on the other hand, leads to hexagonal structure and hydrogen bonding. For convenience, these two configurations of OH will be named OHhb and OHnhb for hydrogen-bonded and non-hydrogen-bonded, respectively. In analogous configurations, an F-OH sequence shows no hydrogen bonding, whereas an F-HO sequence does. When the two configurations of the hydroxyl ion in the random population are considered XOHtotal = XOHhb + XOHnhb where X is mole fraction, and the subscript i is the anion. If XF + XCl + XOHtotal = 1, then XF + XCl + XOHhb + XOHnhb = 1 Xi is also the probability, P, of finding an anion i on the sixfold axis site (Price 1985), for an apatite formula written as Ca12(Ca23(PO4)6(OH,F,Cl): P(X1 = i) = Xi The probability of finding Ca on the Ca2 sites is equal to one for OHOH, OHCl, and OHF, and the probability of finding Ca atoms on Ca1 and P atoms on the tetrahedral site are assumed to be unity. Along the hexad, there is one way for an OH group to point out of two possible orientations, analogous to the problem of flipping a coin. From the classical definition of probability (Spiegel 1961), P = 1/2 = 0.5 1418 TACKER: HYDROXYL ORDERING IN IGNEOUS APATITE Hence the probability P(X1 = OHhb) = 0.5XOHtotal The initial hypothesis is that each site is independent of the others. The probability, P, of finding a hydrogen-bonded OH ion on site X1, followed by an adjacent Cl on the site X2 is P(X1 = OHhb, X2 = Cl) = P(X1 = OHhb)·P(X2 = Cl) = XOHhbXCl and for F: P(X1 = OHhb, X2 = F) = P(X1 = OHhb)· P(X2 = F) = XOHhbXF As determined above, the relative concentrations of OHF and OHCl are determined from FTIR data. These relative concentrations can be compared to the probabilities, which are POHCl P ( X1 = OH hb , X2 = Cl) XOHhb XCl = = = POHF P ( X1 = OH hb , X2 = F ) XOHhb XF (4) 0.5 XOHtotal XCl XCl = 0.5XOHtotal XF XF Comparison of these probabilities determined with Equation 4 with the FTIR data derived from Equation 3 is given in Table 2. The equations of Price (1985) may also be used to consider the probability of cation-anion ordering. The probability of finding three Mn atoms on the Ca2 site is given by RESULTS where Σni = sum of the component atoms i on the Ca2 site, from the normalized analysis. This sum will be equal to three by the formula chosen Ca2 [Ca12Ca23(PO4)3(F,OH,Cl)], but the value of XCa will vary depending on the way atoms are assigned to the Ca2 site. Previous studies (Hughes et al. 1991a; Rakovan and Hughes 2000) found that Sr partitions almost entirely into the Ca2 site, so all Sr will be assigned to that site. Manganese atoms partition between the Ca1 and Ca2 sites (Hughes et al. 1991a), but probability was calculated by assigning all Mn to Ca2 to give a maximum value. Calcium fills the remainder of the Ca2 site so that 3 – nMn – nSr 3 The ratio of the probability of finding Mn-OHF to that of finding OHF is then Ca2 3 P(Mn-OH)·P(Ca-F) (XMn ) ·X ·(X Ca2 )3 ·XF = Ca2 3 OH Ca Ca2 3 P(Ca-OH)·P(Ca-F) (XCa ) ·XOH ·(XCa ) ·XF = j where ε is the error in the measurement and F is a function of Xi (Shoemaker et al. 1974). P(MnCa2) = (XMnCa2)3 where n n Ca2 X Mn = Mn = Mn and Σni 3 Ca2 XCa = of probabilities for Mn-OHOH to that for OHOH. Results are given in Tables 3 and 4. An implicit assumption in this calculation is that all three Mn atoms occupy a single Ca2 triangle. The validity of this assumption is supported by the similarities of the spectra to end-member synthetic apatites. Rare earth elements partition between Ca1 and Ca2, depending on a variety of factors including anion chemistry (Fleet et al. 2000b), but all were assigned to Ca2 in order to give the maximum probability of finding REE on that site. A similar calculation was carried out comparing the ratio of probabilities of Sr-OHOH to OHOH, and REE-OHOH to OHF vs. those observed with FTIR. A similar calculation could be performed using the peak at 3550 cm–1, but given that the peak could represent contributions from Mn-OHOH and Sr-OHF, this calculation was not attempted. The probability ratios were compared with FTIR data at the one sigma level. There should be 1:1 agreement for the two different numbers. A specimen was considered to be ordered if the concentration ratio measured from FTIR, minus the one sigma error was greater than the ratio of probabilities plus the one sigma error. The initial assumption in calculating the probabilities is that the sites are independent of each other. If the concentration ratio and the probabilities are different, it indicates that the sites are not independent, but are associated or ordered. Error in the normalization and in all calculations was determined by: 2  ∂F  2 ε 2 (F) = Σ   ε (Xi )  ∂Xi X i (5) Ca2 3 ) (XMn Ca2 3 (XCa ) Note that Equation 5 would yield the same result for the ratio Comparison of the FTIR and electron microprobe data indicates that pairing of OH and Cl nearest neighbors takes place at a frequency greater than expected from independent random ordering of columnar anions (Table 2). In the cases where the apatites are found to be disordered, it is usually due to relative error in the FTIR measurement produced by a poorly resolved OHCl peak. Similar results are found in the comparison of FTIR and microprobe data for Mn, Sr, and REE. FTIR results show that the concentrations of these cation-hydroxyl pairs are much larger than expected from random ordering. Again, apatites that do not show ordering usually have larger standard errors in the measurements than those that show ordering. In the case of Mn-apatites, there appears to be a clear association between Mn in the Ca2 site and OH in the anion position. A similar association is observed in the Kola apatite for Sr and REE. Although interesting, the Sr and REE results are based on a single specimen and could reflect other influences such as fluctuations in conditions of formation or temperature. DISCUSSION Incorporation of a large Cl anion into the FAp structure in synthetic apatites is accomplished by the movement of the F away from mirror plane (Mackie and Young 1974). Fleet et al. (2000a, 2000b) noted that the thermal parameter of F parallel to the c-axis is larger than expected for thermal motion alone and TACKER: HYDROXYL ORDERING IN IGNEOUS APATITE attributed this to movement out of the plane of the Ca2 triangle. In contrast, Hughes et al. (1990) found that accommodation in a natural apatite was produced by the presence of an OH between the Cl and the F on the hexad. The difference between the calculated probabilities and the concentrations determined from FTIR demonstrates that an OH intermediate is necessary for incorporation of Cl into a fluorapatite. In every case except that of the Durango fluorapatite, the concentration of Cl is subordinate to that of OH. The effects of the disordered, hexagonal OH-Cl sequence have implications for thermodynamic models of the apatite solid-solution series. In the ordered monoclinic form of the hydroxyl-chlorapatite, local crystal structure is similar to that seen in the end-members as described by Hughes et al. (1989). This similarity may be interpreted as a case of thermodynamic ideal mixing: the local structure of the mixed form is not significantly different from a mixture of the two end-members. In the disordered hexagonal sequence, the Cl atom is pushed nearer to the Ca2 triangle, which expands to accommodate the larger anion, leading to tilting of adjacent polyhedra (Hughes et al. 1990). Volume mismatch and lattice distortion gives rise to non-ideal thermodynamic mixing terms (Davies and Navrotsky 1983; Ghiorso and Sack 1991). Hence, non-ideal mixing of the ClAp and HAp end-members may be expected in hexagonal sequences, proportional to the concentration of OHCl. The hydrogen bonding that results in OHCl can be expected to persist to at least 800 °C (Hitmi et al. 1988). Furthermore, if hydroxyl intermediaries are required for the incorporation of Cl atoms into fluorapatite, then the mixing of FAp and ClAp end-members is dependent on HAp. This dependence indicates that thermodynamic models of the FApClAp-HAp solid solution may not be described adequately as a combination of three binary solid solutions, but may require ternary interaction parameters. Thermodynamic models should account for the fact that the activity of HAp in apatite, related to activity or fugacity of water or hydroxyl in many exchange equilibria, is proportional only to the mole fraction of OHOH. Thermodynamic models that do not account for this may considerably overestimate the activity of water in associated exchange equilibria. Tacker and Stormer (1989) concluded that the available data would permit that FAp-HAp solid solutions be treated as ideal above 773 K. However, the available data were scant, and no data are available on the relative populations of OHOH and OHF in the run products. At 298 K, HAp-FAp solid solutions show asymmetric and negative deviations from ideal mixing behavior (Duff 1971) in Gibbs free energy, although not in molar volume (Schaeken et al. 1975). Hydrogen bonding contributes to the negative free energy of mixing, hence the persistence of the hydrogen bonds at higher temperatures may be a key factor in understanding the energetics of mixing for these end-members. The ordering of Mn and Sr between Ca1 and Ca2 in natural apatites has been explained by bond-valence calculations (Hughes et al. 1991a): In the Ca1 site, Mn is less underbonded than in Ca2, and Sr is less overbonded in Ca2 than in Ca1. Results presented here indicate that the presence of OH influences the occupancy of the Ca2 site by Mn. Manganese is found on the Ca2 site in excess of the random site occupation, even when all Mn 1419 is assigned to Ca2 to produce a maximum probability. In crystal-structure refinements, Hughes et al. (1991a) showed a slight increase in Mn on the Ca2 site of the apatite with higher OH. To test the effects of anion chemistry on Mn site occupancy at Ca2, an O was substituted for a F and the bond valence sum calculated using the methods of Brown (1981), in conjunction with the interatomic distance data of Hughes et al. (1991a). This substitution produced little change. Moving the OH farther from the Ca2 produces a slight decrease in the bond valence sum, when an increase is necessary to reduce the underbonding of the Mn on that site. The absence of Mn-OH bands from the Black Mountain apatite spectra is at odds with the spectra of other pegmatitic apatites, even though the Black Mountain apatite contains the highest concentration of Mn. This raises the possibility of external controls on the ordering of Mn and OH, such as temperature, magma structure, or fluid chemistry. The exchange of F and OH between apatite and various sources is highly temperature dependent (Tacker and Stormer 1991; Tacker 1992). Data from the Kola apatite suggest ordering between Sr and OH and between REE and OH. Earlier studies found that REE partitioning between Ca1 and Ca2 is influenced by bond valence (Hughes et al. 1991b; Fleet and Pan 1995) as well as by anion chemistry (Fleet et al. 2000a, 2000b). Using the data of Hughes et al. (1991a), and of Sudarsanan and Young (1972), there again seems to be little difference between the bond valence sums for an M2 cation in Sr-FAp vs. Sr-HAp. The slight reduction might reduce overbonding for the Sr in the Ca2 site. Although the results for Mn ordering into Ca2 are based on several specimens, results for Sr and REE are based on a single specimen. It is possible that the ordering observed for Sr and REE in the Kola apatite reflects changes in fluid chemistry during the growth of the apatite. In either case, this paper provides a methodology for further investigation, and FTIR will provide insight into the influence of anions on REE site occupancies. ACKNOWLEDGMENTS The spectroscopy presented here was originally supported by an EPSCoR program grant EHR-91089771 to the University of Tulsa and the University of Oklahoma. The loan of an apatite from the Harvard Mineralogical Museum, and gifts from D. London are gratefully acknowledged. A warm “thank you” is due to P. Michael, University of Tulsa, for access to the FTIR microspectrometer, for training in its use, and for discussion of the results. Thanks are also due to D. Teeters, Department of Chemistry, University of Tulsa, for access to the FTIR in their laboratory. The work, discussion, and assistance of G. 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Libowitzky and Rossman (1996) demonstrate that the orthogonal directions need not be coincident with crystallographic axes (or major axes of the indicatrix). For convenience herein, the subscripts refer to crystallographic axes. For hexagonal minerals, determination of total absorbance would require two measurements. If parallel to crystallographic axes: Atotal = Aa + Ac. As discussed above, polarized absorbance TACKER: HYDROXYL ORDERING IN IGNEOUS APATITE of OH-stretching bands in apatite is zero parallel to the a-axis and at a maximum parallel to the c-axis (Elliott 1965, 1994; Levitt and Condrate 1970). Hence, with polarization parallel to the c-axis: Atotal = Ac. If (100) sections of apatite crystals are analyzed, then Ac may be calculated from unpolarized spectra. First, consider the transmission of unpolarized radiation. Absorbance, A, is related to transmittance by A = –log10T, where transmittance (T) is the intensity (I) of radiation that comes through the specimen, divided by the intensity of incident radiation (Io). The unpolarized transmittance (Libowitzky and Rossman 1996, their Eq. 8) is given by: Tunpolarized = (Tmax + Tmin ) . 2 2π Tunpolarized = 2π ∫ T (φ)dφ = ∫ (T min 0 2∀ Taverage = ) 1 ∫ Tmin cos2 φ + Tmax sin2 φdφTunpol = 2π − 0 0 (Tmin + Tmax ) 2 Figure 6 of Libowitzky and Rossman (1996) demonstrates that the averaged value fits observed data. Unpolarized absorbance can then be calculated from polarized transmittance: Aunpolarized = − log [(Tmax + Tmin ) 2] Aunpolarized = − log [ 10 − Amin + 10 − Amax ( 10 Libowitzky and Rossman (1996) presented this equation (their Eq. 8) as the result of the integration of: 1421 − Aunpolarized ( = [ 10 − Amin + 10 − Amax ) 2] ) 2] For the apatite OH, Amax is parallel to the c-axis, and parallel to a, Amin = 0. Therefore, 10–Aunpolarized = [(1 + 10–Amax)/2] ·cos 2 φ + Tmax ·sin 2 φ dφ 0 and However, the integration of this equation yields: Amax = Ac = –log[(2 × 10–Aunpolarized) – 1] Tunpolarized = (Tmin + Tmax)π. Equation 8 of Libowitzky and Rossman (1996) is actually the average of the function: Unpolarized spectra were collected from (100) sections of apatite crystals, sections that contained both Amax and Amin.