In the following, crystallographic axes are referred to as a, b, and c. For orthorhombic and monoclinic minerals, the refractive indices are labelled as nα, nβ, and nγ, and the axes of the optical indicatrix are labelled as X, Y, and Z. For orthopyroxene α=β=γ=90°, and the crystallographic axes overlap with the principal optical axes (Table 1 and Fig. 1a). Therefore, in crystal sections parallel to (100), (010), and (001), two of the three optical axes are in the plane (Table 1 and Fig. 1b).

In the monoclinic system of clinopyroxene, the angles α=γ=90° and β≠90° and, thus, the a axis are not perpendicular to the (b–c) plane. Therefore, the vector a* was defined as the normal vector in relation to the (b–c) plane, resulting in three orthogonal vectors together with the b and c axes (Table 1 and Fig. 1c). To avoid confusion with previous notations, a new nomenclature is introduced for the directions of the polarised measurements of specific crystal sections, following a practical approach based on features which are common in a wide range of natural samples and that can be easily identified in polarised spectra of different samples (e.g. idiomorphic large single crystals, grain separates or thick sections). In addition to optical features that can be used to find the principal optical axes, morphological features can be used to obtain a reference orientation and two additional perpendicular planes. For example, clinopyroxene displays two cleavage planes, orthogonal to each other, and both contain the c axis; thus, the direction of the c axis can be easily identified (Table 1 and Fig. 1d).

Previous nomenclature was often constrained to the axis of the optical indicatrix in various measurement directions, which, for clinopyroxenes, has the limitation that samples with different compositions are not easily comparable. The reason for this is that the angle between the crystallographic c axis and the axes of the optical indicatrix X and Z, which are all parallel to (010), depends on the major element composition, as reflected by the extinction angle, which can vary from 4 to 94° (Tröger, 1956).

The new orientation nomenclature combines two parameters and is stated as [Vector]_(Plane). The first parameter [Vector] describes the polarisation direction of the incident infrared beam, which is set by rotating the polariser on the instrument. The second parameter (Plane) is the crystal section orthogonal to the [Vector], making it easier to quickly identify how a crystal needs to be prepared in order to measure a specific direction. To keep the definition compact, (Plane) is formatted as a subscript. This nomenclature is highly flexible as it can be used for any crystal system, combining Miller indices; the axes of the optical indicatrix X, Y and Z; the crystallographic axes a, b, and c; or any other defined vectors, for example, a*, being the orthogonal vector in relation to the (c–b) plane.

The important benefit of the new nomenclature is that different samples become directly comparable in terms of absorption behaviour and anisotropy. When referring to the polarisation direction, the electric field vector of the incident wave E of the polarised infrared beam is alternately oriented parallel in the corresponding planes [a*]_(a^*–b), [b](a*–b), [Y](b–c), [c](b–c), [X](a–c), or [Z](a–c), with the propagation direction as defined in Table 1.

To quantify the H₂O content of NAMs with FTIR, a modified version of the Beer–Lambert law is commonly used and can be expressed as follows: 1Abs=c×t×E,2A*=AxtAx+AytAy+AztAz,3CH2O=MH2O×A*ρ×Ei×106. CH2O is the H₂O content (ppm H₂O by weight; µg g⁻¹), MH2O is the molecular weight of H₂O (18.02 g mol⁻¹), A* is the total linear or integrated absorbance of three orthogonal polarised measurements normalised to 1 cm (cm⁻¹ for linear values or cm⁻² for integrated values; for isotropic materials Aunpolarised×3=A*), ρ is the density of the mineral (in g L⁻¹), E is the linear or Ei integrated molar absorption coefficient (in L mol(H₂O)⁻¹ cm⁻¹ or L mol(H₂O)⁻¹ for absolute values, cm⁻² for integrated values), t is the path length that the infrared beam travels through the sample, defined by the thickness of the sample, to the IR light beam (in cm), and 10⁶ is the conversion factor to obtain the results in µg g⁻¹. When working with samples that have an inclined or rough surface (e.g. loose grains) relative to the direction of the infrared beam, it is important to consider the fact that the actual path length t can differ from the sample's thickness due to variations in the beam trajectory caused by the mineral's refractive index or scattering effects.

In homogeneous samples, it is possible to obtain the total absorption of a crystal calculated as the sum of three polarised spectra along the three axes of the optical indicatrix (Bell et al., 2003; Asimow et al., 2006) or by measuring any three mutually orthogonal polarised spectra in two perpendicular sections (Libowitzky and Rossman, 1996). These can be obtained either by using the morphologic features of a crystal, such as, for example, crystal habitus and cleavage of a clinopyroxene (Eq. 4), or the optical features (Eq. 5) or by using spectral features such as Si–O overtones to identify sets of known spectra acquired perpendicular to three orthogonal vectors. 4A*=[a*](a*–b)+[b](a*–b)+[c](c–b)5A*=[Y](c–b)+[X](a–c)+[Z](a–c) Libowitzky and Rossman (1996) demonstrated that the unpolarised spectrum can be calculated from any orthogonal measurements (acquired with the polariser at x and x+90°) in a single cut. By using the approximation Aunpol≈(Amax+Amin)/2, the following equation can be used to approximate the total integrated absorbance by ideally using orthogonal sections that contain minimum and maximum absorbance of the main defects. This will then average and minimise the effect of different indicatrices, especially if the sections and spectra used are close to the maximum and minimum, and, consequently, the following formulation is used for estimating the approximated total integrated absorbance: 6A*=[Y](c–b)+[c](c–b)2+[X](a–c)+[Z](a–c)2+[b](a*–b)+[a*](a*–b)2, where [Y](c–b)+[c](c–b)2≈⟂(100)unpol, [X](a–c)+[Z](a–c)2≈⟂(010)unpol, and [b](a*–b)+[a*](a*–b)2≈⟂(a*–b)unpol.

It is important to keep in mind that, in measurements of natural samples, this is often only an approximation due to errors in the orientation of the samples and should be applied using spectra acquired within the validity of the Beer–Lambert law (i.e. under nearly linear, non-saturating conditions) with absorbance below 1 before thickness normalisation, and, in samples with strong anisotropy, this approximation cannot be applied (see Withers, 2013).

For clinopyroxene in particular the determination of total absorbance comes with additional complications. Some of the OH-related defects show a polarisation behaviour that is aligned with the axes of the optical indicatrix (X, Y, and Z), while others show an orientation close to the crystallographic b and c axes and the vector a* (Dowty, 1978; Strens et al., 1982). Therefore, a framework must be established relating to how minimum, maximum, and intermediate absorbances of the main OH defects vary as a function of clinopyroxene composition. The usefulness of Si–O overtones in FTIR spectra to obtain information on mineral orientation has been recognised for several years (e.g. Jamtveit et al., 2001; Johnson et al., 2002; Lemaire et al., 2004; Asimow et al., 2006; Férot and Bolfan-Casanova, 2012; Karimova and Stalder, 2013; Mosenfelder and Rossman, 2013; Aulbach et al., 2023). However, Si–O overtones have not been systematically investigated for minerals with lower symmetries. To date, there is no general agreement in the community regarding the publication of the full spectra used in FTIR studies. Therefore, many studies do not report polarised spectra, publish only few representative spectra of the O–H region, and neglect the Si–O overtones, which could be useful in distinguishing different orientations and polarisation directions.

The approximated theoretical transmission and absorption behaviours for small absorbances as a function of polarisation angle, following Malus's Law, have been formulated and applied previously (Eq. 8, e.g. Libowitzky and Rossman, 1996; Kovács et al., 2008; Asimow et al., 2006; Sambridge et al., 2008). Theoretical equations that are important for this empirical work are summarised in Eqs. (7)–(9). Note that the true physical theoretical background and mathematical formulations, which take dispersion, refraction, scattering, and other effects into account and which are also valid for high absorbances, are far more complex than presented here. The empirical validation of the relationship between sample thickness, linear absorbance, and integration was carefully tested and is valid for absorption bands and thicknesses with low absorbances and low anisotropy (see Kovács et al., 2008; Asimow et al., 2006; Sambridge et al., 2008; Withers, 2013; Shuai and Yang, 2017).

From the absorbance of the polarised endmember spectra, the theoretical absorbance values can be calculated. A first conclusion of this work is that an orientation mismatch of ±10° close to a maximum or minimum intensity is insignificant up to a maximum absorbance of 0.5, whereas the same mismatch midway between the maximum and minimum intensity has a significant effect on the estimated absorption value and deviates even more strongly for higher absorbance (Fig. 2). An additional fundamental observation is that spectra with low absorbance, measured close to the absorption maximum and minimum, show the least deviation from the theoretical approximation (Eqs. 8–9), which is particularly useful for practical applications of anisotropic materials. Conversely, spectra recorded at intermediate angles can deviate significantly and should therefore be avoided (if possible) for quantitative analysis. This shows the importance of identifying the actual maximum and minimum OH absorbance in low-symmetry crystals rather than assuming a priori their orientations. This implies that, for natural samples, errors due to misalignment of the crystal are negligible close to a maximum or minimum but can result in a large error in intermediate domains. Additionally, the relationship of Eq. (8) only holds within the validity of the Beer–Lambert law (i.e. absorption linear to thickness and low anisotropy), which is empirically valid below an absorption of 1 for non-thickness-corrected spectra. In the case of higher absorption, Eq. (9) must be used. For most NAMs investigated, when using double-sided polished thick sections of <500 µm, the absorbance values of OH-related defects are low enough that the approximations of Eq. (8) and, thus, also [Y](c–b)+[c](c–b)2≈⟂(100)unpol, [X](a–c)+[Z](a–c)2≈⟂(010)unpol, and [b](a*–b)+[a*](a*–b)2≈⟂(a*–b)unpol are applicable. In larger single crystals, NAMs with very high H₂O content and hydrous phases, absorbance can easily exceed 1, and the approximations should not be applied anymore. Figure C2 in the Supplement shows how the strong absorption of Si–O overtone bands leads to a deviation from Eq. (8) for the angular polarisation, while the weaker absorption of the OH bands can be very well approximated by Eq. (8). As a result, the calculated unpolarised spectrum of the OH bands matches the measured one very well, whereas a significant derivation can be observed between the calculated and measured spectra of the Si–O overtone bands (Fig. C3). 7A(θ)=A×cos⁡2θ8Aϕ=Amax×cos⁡2ϕ+Amin×sin⁡2ϕ9Aϕ=-log⁡1010-Amaxcos⁡2ϕ+10-Aminsin⁡2ϕ The equations above are as presented in Libowitzky and Rossman (1996); θ is the angle between the electric field vector E of the incident light and the transition dipole moment of the absorber. ϕ is the angle between the electric field vector of E of the incident light and the maximum absorbance value (Amax) of the absorbance figure.

Seven different types of natural pyroxene samples were selected as being representative of commonly investigated sample types (e.g. single crystals, thick sections, homogeneous, zoned) out of n>100 investigated samples. These selected samples were used to demonstrate workflows for investigating absorption behaviour and the relationship of Si–O overtones and OH-related defects with respect to different crystal chemistries, orientations, and polarised angles in transmission FTIR measurements. Further data from additional samples are used in an application example to demonstrate the variability of the presented approach and are presented in Table B1. Selected representative optical images are compiled in Fig. 4 (additional images in the Supplement). The average major element composition measured with electron probe micro analyser (EPMA) is presented in Table B1. EPMA methods are documented in the Supplement. Mineral formula recalculation was performed with MinPlotX (Walters and Gies, 2025), and results are displayed in Figs. B1 and B2. For the clinopyroxene samples, prior to sample selection, a large number of samples available in our collection, as well as spectra of previously investigated samples, were screened. Out of these pre-screened samples, more than 50 different samples were investigated in detail, and, based on grain size, H₂O content, number of inclusions, and contamination by inclusion or alteration of minerals, the following selection of samples was made (Table B1 and the Supplement), representing an increasing degree of complexity: (1) an oriented, metamorphic gem-quality orthopyroxene from Tanzania; (2) an oriented. gem-quality diopside from Binntal, Switzerland; (3) an idiomorphic, magmatic augite single crystal from Mount Guntur, western Java, Indonesia; (4) well-equilibrated omphacite fragments from two lightly crushed cratonic eclogite xenoliths from Koidu, West African craton; and (5) a double-sided, polished, thick section of omphacite from an orogenic eclogite from the western Gneiss region, Norway. Additionally, the data set has been supplemented with omphacitic and diopsidic clinopyroxene samples, which were prepared as double-sided, polished, thick sections, to apply the developed method and to show the results for a larger set of samples. Samples for which major element composition was determined are included in Table B1.

Transmission FTIR analyses were conducted at the Institute of Geological Sciences at the University of Bern using a Bruker Tensor II spectrometer with a globar infrared source and a KBr beam splitter, coupled to a Bruker Hyperion 3000 microscope with a dry-air-purged sample chamber, XY-stage, 15× objective, and a variable aperture. The instrument is equipped with a liquid-nitrogen-cooled mercury cadmium telluride detector and a KRS-5 (thallium bromide iodide) infrared polariser.

For each sample, polarised measurements were acquired in a wavenumber range from 6000 to 600 cm⁻¹, with a resolution of 4 cm⁻¹ and a variable number of scans and aperture sizes depending on the sample. The clinopyroxene in the eclogite sample CdG_19_40 (Table 2) showing representative absorbance bands in the region between 3000 and 3800 cm⁻¹ was chosen to compare between spectra acquired with 4, 8, 16, 32, and 64 scans (Fig. C1). A total of 8 or 16 scans were chosen as the ideal time in relation to signal noise, depending on the sample thickness and spectra quality.

Multiple analytical positions were programmed and then sequentially measured with incrementally changing orientation of the polariser between 0 and 180°, followed by a background measurement using the automatic polariser described above. Hereafter, such a set of equal polariser orientations will be referred to as a “PolGroup”. The automatic polariser was incremented in equal steps between 2 and 5°, depending on the sample, until 180° was reached, resulting in a total number of 36 to 90 spectra per measurement point.

The thickness of each measurement point was measured depending on the sample size and shape using the calibrated z position of the microscopy stage, with an uncertainty of ±3 µm for small samples, and a Mitutoyo micrometre, with an accuracy of ± 2–5 µm for larger samples.

For the data processing of the FTIR, spectra were cut to the range of 4000–1250 cm⁻¹, and the atmospheric correction and concave rubber band baseline correction with 64 points and four iterations were performed in OPUS^® version 8.5. Note that, for samples with spectra showing broad peaks, for example, glasses, carbonates, and feldspar, this method is too conservative and will lead to an underestimation of the resulting integrated absorbance. Spectra were then exported and processed further with the software SpecXY (Gies et al., 2024). All spectra used in this paper are normalised to 1 cm thickness. The corresponding polarised background spectrum was subtracted from the sample spectrum, and the absorbance of characteristic bands and the integrated absorbance for the O–H region were extracted for all spectra at each measurement position in the SpecDB module of SpecXY following the workflow of Gies et al. (2024). Scatterplot matrices were generated with MinPlotX (Walters and Gies, 2025).

The polarisation effect on the baseline was tested by comparing measurements without and with background measurements in between the rotation of the measurements, and only minor differences in the Si–O overtone region between the two methods could be identified. However, these could be removed by the concave rubber band baseline correction performed in OPUS. To be able to completely rule out any effect of the corrections on the polarisation effect, it is advised to acquire the background for each corresponding polarisation angle and to correct all spectra of the corresponding PolGroup accordingly.

From the Tanzania orthopyroxene cuboid described in Turner et al. (2015), several smaller cuboids with variable thicknesses oriented parallel to (100), (010), and (001) were prepared (Fig. 4a), and angle-resolved polarised spectra were acquired. This sample is used to investigate the polarisation behaviour of a well-described orthopyroxene, oriented parallel to the crystallographic axes, with sharp and non-overlapping O-H absorption bands, representing the simplest possible case of pyroxene polarisation behaviour. The orthorhombic crystal symmetry allows for the orientation of the crystal using either cleavage or optical extinction. This sample can be used to identify the orthopyroxene endmember patterns in polarised spectra acquired parallel to the crystallographic axes.

Orthopyroxene spectra show a strong polarisation behaviour of the O–H region and Si–O region, with clear endmember spectra acquired with the polariser set parallel to the crystallographic axes (Fig. 5). In the Si–O overtone region, [a](a–c) has a clear maximum between 2100–1800 cm⁻¹ and a minimum between 1750–1300 cm⁻¹, whereas [b](b–c) shows the highest absorbance in the higher-wavenumber Si–O overtone bands > 2100 cm⁻¹ (for the definition of the endmembers [a](a–c), [b](b–c), and [c](a–c), see Fig. 1 and Table 1). [c](a–c) is defined by the lowest absorbance in the higher-wavenumber Si–O overtones between 2300–1800 cm⁻¹ and the highest absorbance in the lower-wavenumber Si–O overtones between 1800–1400 cm⁻¹. The polarisation behaviour shows an excellent agreement between measurements and the theoretical approximation estimated using Eq. (8) (Fig. 6). Bands in the Si–O overtone region that are influenced by multiple absorbance bands with different polarisation behaviour or thick samples that exceed the thickness for which the Beer–Lambert law can be applied deviate from the theoretical approximation (Fig. 2).

The O–H-related bands at 3513 and 3410 cm⁻¹ and the Si–O overtones with lower wavenumbers between 1800–1400 cm⁻¹ show the same polarisation behaviour, with the strongest absorption parallel to [c](a–c) (Fig. 7). In contrast, the absorption of the OH-related defects parallel to [a](a–c) and [b](b–c) is weaker and similar.

This example illustrates the polarisation behaviour of the spectral endmembers in the orthorhombic system. Furthermore, it demonstrates how the endmember of the Si–O overtone region can be used to find the principal axes of absorbance in orthopyroxene, to investigate the orientation of an orthopyroxene crystal or preferred orientations in a sample, and to obtain three orthogonal sections for H₂O determinations.

A gem-quality diopside crystal of sample NMBE_8755 was selected from the collection of the Natural History Museum in Bern. Oriented angular polarised-transmission FTIR measurements consisting of 38 spectra were measured at 11 different positions from the top to the bottom of the crystal in two orthogonal directions ((100) and (010)) to investigate the spatial variation of O–H absorbance across the crystal. Except for the crystal base, which showed minor traces of amphibole, the concentration was found to be homogeneous. A section of the crystal was then cut out with a wire saw, resulting in an 825×830×1025 ± 10 µm cuboid with all faces orthogonal to the (010) plane. This sample was chosen to investigate the increased complexity of the monoclinic system but with the simple case of a homogeneous H₂O concentration and known crystal orientation.

Spectra of the oriented diopside cuboid acquired parallel to (100), (010), and the (a*–b) plane show similar Si–O overtone endmembers for [a](a–c), [b](b–c), and [c](a–c) as the previously described orthopyroxene. However, with more complex monoclinic symmetry, two more endmembers ([X](a–c) and [Z](a–c)) can be identified in the (010) plane (Fig. 8). The orthogonal endmembers [Y](b–c), [X](a–c), and [Z](a–c) correspond to the axes of the optical indicatrix X, Y, and Z with similar extinction angles. The spectral endmember patterns of [Y](b–c) and [b](a*–b) are very similar in terms of Si–O overtones and contribution to the total integrated O–H absorption. They are thus treated as one for the H₂O quantification but need to be considered separately when investigating OH incorporation mechanisms and defect assignment. The presented example also demonstrates the effect of sample thickness, where, for thick samples (>400 µm), the relation of Si–O overtone absorption no longer behaves purely linearly, and polarised spectra with high absorption in the Si–O overtone region can no longer be used quantitatively (see Fig. 7, [a*]_(a^*–b), [b](a*–b), and [Z](a–c)). In the O–H region, there is still a good agreement between the unpolarised measured spectrum and the theoretical unpolarised spectrum approximated by the mean of the maximum polarised spectra acquired at the polarisation angles x and x+90° (see Fig. C3). However, it is not always possible to prepare an oriented micro-cuboid, especially for minerals with a well-developed cleavage, such as clinopyroxenes. Ideally, the sample should be 80–350 µm thick to investigate the Si–O overtones and to study absorption in the O–H region. The chosen sample thickness is always a trade-off between absorption in the O–H and Si–O region and the feasibility of the sample preparation. It should, therefore, be verified whether the Beer–Lambert law still applies to a particular sample in the wavenumber region of interest by comparing the measured unpolarised spectrum with the mean of the maximum polarised spectra. Nevertheless, in a section parallel to (100) or the (a*–b) plane, it is possible to distinguish between [Y](b–c) and [b](a*–b) if the +90° spectrum is either [a*]_(a^*–b) or [c](b–c).

Hydrogen incorporation mechanisms of diopsides are complex, and at least seven different overlapping absorption bands are identifiable in the diopside from Binntal, Switzerland. From the polarisation behaviour it is possible to distinguish two groups of defect types by comparing the polarisation behaviour of OH absorption bands with that of Si–O overtones (Fig. 9). These two families of bands correspond to different incorporation mechanisms in clinopyroxenes, i.e. incorporation on the tetrahedral T site (3660–3600 cm⁻¹) and octahedral M site (3590–3300 cm⁻¹), as reported in Balan et al. (2020). Most importantly, the absorption maxima and minima of OH-related defects correspond to spectral endmembers in the Si–O overtone region. This makes the combination of Si–O overtones and OH-related defects a valuable tool in finding sets of spectra acquired perpendicular to three orthogonal vectors to estimate the total integrated absorbance and further investigate OH incorporation mechanisms. For example, in the (100) and (a*–b) sections, it is evident that the band 3645 cm⁻¹ maximum corresponds to the maximum of the Si–O overtone band 2105 cm⁻¹, which is indicative of [Y](b–c) (Figs. C3 and C4). The minimum of the Si–O overtone band 1973 cm⁻¹, which is indicative of [c](b–c), correlates with the maximum of the OH-related defects between 3050 and 3580 cm⁻¹ and is only detectable in a (100) section. Additionally, in the (010) section, the maxima and minima of both OH-related defects and Si–O overtone bands correlate with the optical extinction behaviour of diopside and, thus, the axes of the optical indicatrix (X, Y, and Z).

OH-related defects do not necessarily correspond to the maximum and minimum absorbance of the different endmembers in the Si–O overtone region. This is because the maximum absorbance of OH-related defects does not necessarily have to be parallel to any of the endmembers, the principal optical directions, or the crystallographic axes and can have a different orientation. Additionally, the relationship between Si–O overtone bands and OH bands can be further investigated; however, for simplicity, only the total integrated absorbance is used here. This allows further investigation of the two sets of spectra acquired perpendicular to three orthogonal vectors, [a*]_(a^*–b), [b](a*–b), and [c](b–c) and [Y](b–c), [X](a–c), and [Z](a–c). For example, the diopside from Binntal shows the total integrated absorbance behaviour in the O–H region of [c](b–c) > [Z](a–c) > [Y](b–c) ≈ [X](a–c) > [b](a*–b) > [a*]_(a^*–b). Finally, this example provides the important observation that the M-site (3590–3300 cm⁻¹) maxima and minima [c](b–c) and [a*]_(a^*–b) are present in the (100) and (a*–b) sections, whereas the maximum and minimum of the T-site (3660–3600 cm⁻¹) defects are present in the (a*–b) and (010) sections.

An idiomorphic single crystal from Mount Guntur (western Java, Indonesia) from the mineral collection of the Institute of Geological Sciences at the University of Bern was prepared as a wafer parallel to (010) and the (a*–b) plane, with a thickness of 135±10 and 175±10 µm, respectively. The augite crystal shows sector zoning and oscillatory growth zoning, as well as a twinning plane parallel to the (a*–b) plane and a second smaller crystal fragment attached on one side (Fig. 10). The preparation of an inclusion-free 100-section crystal was not possible due to the complexity of the oscillatory and sector zonation in combination with the small size of the crystal. Nevertheless, this sample is valuable in investigating the polarisation behaviour in the two prepared sections and demonstrates relative changes in absorption within a single grain. The augite illustrates how complex crystals, grown in a complex and evolving environment, can be studied using orientation and polarisation effects of OH-related defect groups and Si–O overtones to (i) understand the systematics of OH-related defect groups, (ii) study relative concentration changes within a single section, and (iii) evaluate the contribution of specific polarised measurements to the calculation of the total O–H absorption value. The purpose of this example is to highlight the potential for future research, and it should not be interpreted as being applicable to all magmatic clinopyroxenes.

Three different zones, free from visible inclusions or cracks, which were identifiable in all prepared sections, and round polarised spectra were acquired and investigated. Spectra show the typical absorbance behaviour of magmatic clinopyroxenes, with a strong peak around 3634 cm⁻¹, polarising [b](a*–b)>[X](a–c)>[a*](a*–b)≫[Z](a–c) and at least two peaks between 3600–3400 cm⁻¹ with a polarisation behaviour of [Z](a–c)≫[X](a–c)>[b](a*–b)≈[a*](a*–b) (Fig. 11). It is noteworthy that zonation of H₂O is preserved in this sample at magmatic temperatures and is strongly coupled to the chemical zonation (Figs. 10, C5). When comparing the integrated absorbance maps representative of the two different families of bands with the chemical maps, it is evident that the higher-wavenumber bands (Fig. 11a) related to the exchange of M³⁺ + H⁺ for Si⁴⁺ (Balan et al., 2020) anticorrelate with Fe and correlate with Cr and Ca, showing no sector zoning, whereas the M-site-related defects (Fig. 11b) show similarities with Al, Ti, Si, and Mg, showing strong sector zoning (Fig. 10).

In the section prepared parallel to the (a*–b) plane, polarisation behaviour of the Si–O overtones correlates perfectly with the visible OH bands (see Fig. 11a; band 3634 is strong, and band 3514 is weak). In the section parallel to (a–c) the polarisation behaviour of the OH-related defects does not exhibit the same pattern as the Si–O bands. This is particularly evident when comparing the OH absorption band at 3634 cm⁻¹ with the polarisation of the overtone band at 1525 cm⁻¹. This relationship occurs through both sides of the twinned crystal and is consistent with polarised mapping of the transition (Fig. C6). Thus, this observation suggests that the infrared indicatrices of the 1525 cm⁻¹ Si–O overtone bonds have a different orientation compared to the infrared indicatrices of the O–H-related defects, which coincide with the axes of the optical indicatrix (Fig. 11). This agrees with the notion of Strens et al. (1982), where the indicatrices of different bands can have different orientations, which do not necessarily have to coincide with the axes of the optical indicatrix or crystallographic axes. Clearly, more detailed high-resolution angular polarisation measurements of magmatic clinopyroxene are needed to investigate this fundamental phenomenon.

Handpicked clinopyroxene fragments from two lightly crushed cratonic eclogite xenoliths (Koidu, West African craton), K5-5 and K5-14, described in Aulbach et al. (2019) and Aulbach et al. (2024), were mounted in epoxy and double-sided polished to thicknesses of 260 ± 15 and 240 ± 15 µm, respectively. These samples were used to investigate the polarisation behaviour of multiple randomly oriented grains (Fig. 4c and d). H₂O contents were shown to be homogeneous, previously demonstrated with high-resolution FTIR mappings using the focal plane array (FPA) detector, showing no gradients in several mapped grains. These crystals are usually difficult to orientate due to the small grain size and irregular grain shape. Homogeneous clinopyroxene fragments of multiple eclogite xenolith samples were investigated with the acquisition of angle-resolved polarised measurements of 40–60 grains depending on the sample. In the following, the endmember spectra for two representative samples (K5-5 and K5-14) with different OH absorption patterns are presented, along with a detailed analysis of variation in H₂O quantification results from different approaches. Finally, selected classified spectral endmembers of the two samples are compared, showing different OH absorption patterns, polarisation behaviour, and families of OH absorption bands.

To obtain the five spectral endmembers [a*]_(a^*–b), [Y](b–c), [c](b–c), [X](a–c), and [Z](a–c), the previously described endmember finding approach of SpecXY was used for both samples in a range of 2200–1450 cm⁻¹. Spectra with >90 % of endmember component were labelled as corresponding endmembers. The resulting spectra show clear spectral endmember features for [a*]_(a^*–b), [Y](b–c), and [c](b–c), whereas [X](a–c) and [Z](a–c) are in between [a*]_(a^*–b) and [c](b–c). Nevertheless, in these specific samples, [X](a–c) and [Z](a–c) can be identified by unique spectral patterns in the lower wavenumbers of the Si–O overtone region between 1800–1450 cm⁻¹ (Fig. 12). It is now possible to determine different families of bands in the O–H region with similar polarisation behaviour and also to identify minor absorption bands, which only appear in specific endmember spectra. Surprisingly, the higher wavenumbers consist of at least two separate defects with different polarisation behaviour: (1) the higher-wavenumber peak between 3635–3655 cm⁻¹, behaving similarly to the observed peak in diopside-rich clinopyroxenes (e.g. Fig. 8, Sample 2, Binntal) identifiable in [Y](b–c) and [a*]_(a^*–b), and (2) an additional band at 3622–3635 cm⁻¹, with a maximum [X](a–c) and [c](b–c) similar to that seen in the previous magmatic clinopyroxene example. The higher-wavenumber defects are more dominant in sample K5-14, whereas the lower-wavenumber defects are more dominant in sample K5-5. The lower-wavenumber defects in the O–H region show similar polarisation behaviour, and only minor differences can be observed between the two samples. For example, in the [a*]_(a^*–b) spectra of K-5-14, two additional bands around 3480 and 3432 cm⁻¹ are identifiable, which remain hidden under the more dominating bands around 3450 cm⁻¹ in sample K5-5. By extracting the absorbance of the characteristic Si–O overtone bands and plotting them against each other and the total integrated absorbance, the orientation effect of polarised FTIR spectra becomes evident (Figs. 13, 14, C7, and C8). With this method it is possible to identify spectral endmembers, evaluate the relationship between orientation and OH absorbance, and check for homogeneity of different grains by comparing OH absorbance on grains that display identical overtones. A key observation from this example is that these two clinopyroxenes need to be oriented differently if the maximum and minimum absorbance should be present in the prepared sections. While the ideal sections for K5-14 are (a–c) and (a*–b), with the maximum in [Z](a–c), the ideal sections for K5-5 are (b–c) and (a*–b), with the maximum and minimum in [c](c–b) and [a*]_(a^*–b), respectively. To investigate the anisotropy of the different defects, theoretically, four spectra should be enough. However, in the complex case of clinopyroxenes with multiple overlapping absorption bands with differently oriented indicatrices, more sections are needed to see clear patterns for different families of bands corresponding to different defects.

A double-sided polished thick section of an orogenic eclogite from the western Gneiss region in Norway (MKS_A83) was prepared, with a thickness of 365±10 µm. It is frequently observed that natural samples exhibit inhomogeneous H₂O contents of different origins (e.g. chemical zonation, diffusion gain or loss of H, alteration to and/or exsolution of hydrous phases). This example demonstrates the potential of using spectral endmembers to investigate H₂O variations within a sample. The inhomogeneity of this sample can also be seen in the multi-grain high-resolution FPA mapping (Fig. C9).

For sample MKS_A83, we selected 49 optically clear spots in the core and rim domains of different, randomly selected grains and made angular polarised measurements as described previously. The endmembers [a*]_(a^*–b), [Y](b–c), [c](b–c), [X](a–c), and [Z](a–c) were classified using the endmember finding approach of SpecXY. The results were verified by comparing the characteristic bands of the Si–O overtones from the previously presented examples (K5-5 and K5-14), and spectra with less than 10 % deviation of the characteristic bands were selected to represent the endmembers (Fig. 15a). The endmembers [X](a–c) and [Z](a–c) are more difficult to identify because the characteristic maxima in the absorption patterns of the lower-wavenumber Si–O overtones between 1400–1800 cm⁻¹ were not present in the sample MKS_A83 (Fig. C10). By comparing spectra with nearly identical measurement and polarisation direction for [a*]_(a^*–b), [Y](b–c), and [c](b–c), the integrated absorbance of the OH-related bands reveals inhomogeneities in H₂O content across the sample (Fig. 15). Furthermore, it is now also possible to identify and distinguish overlapping minor absorption bands, such as the one only visible in [a*]_(a^*–b) and [Y](b–c) at 3432 cm⁻¹, which would be invisible using unpolarised spectra. A comparison of characteristic bands and the integrated absorbance of the O–H region for all PolGroups of 49 measurement positions, with a total number of 2205 polarised spectra of sample MKS_A83, reveals a distribution of nearly random orientations, as can be seen by the ratio of the absorbance at the bands 2105 and 1973 cm⁻¹ (Fig. 15b). The variable absorbance observed within the same endmember group in the O–H region further confirms the presence of inhomogeneities in the H₂O content of the measured clinopyroxene grains, which can also be observed in the endmember spectra (Figs. 15 and C10).

This example illustrates how the selection of spectra for comparison based on the endmembers of the Si–O overtones can be used to analyse variations across different grains with different orientations. It allows quantification of core and rim H₂O contents, a feature that cannot be obtained when using the mean of unpolarised spectra, where it is not clear if variation in absorbance is related to orientation or difference in H₂O contents. Furthermore, this approach can be combined with textural and compositional information (e.g. different grain sizes, chemical zonation) to statistically investigate the within-sample or within-zone differences of H₂O content.

Technical developments of the automatic polariser enable the acquisition of all polarisation angles in programmable spots, thus allowing the acquisition of a large number of polarised spectra of anisotropic samples. The open-source hardware design is easy to adapt to work with various instruments and can be further customised to improve data quality. There are two critical benefits of the new workflow.

First, for the quantification of H₂O in anisotropic nominally anhydrous minerals, it is important to recognise families of bands with a systematic polarisation behaviour. This can be verified by investigation of the spectral endmembers, which will also lead to the identification of the appropriate endmembers to be used for the quantification approximation.

Second, because measurements can be automated, a larger number of spots can be acquired with significantly less manual effort. Human bias is eliminated when setting the polarisation angle. The larger number of spots can be investigated in greater detail, allowing for improved statistics and data verification. To improve the quantification results and avoid overestimations caused by contamination, altered and contaminated domains, as well as hydrous phases, should also be included when performing automatic angular polarised measurements. Consequently, their polarisation behaviour and contribution to the spectrum of the phase of interest can be constrained even if, in some polarisation angles, the O–H absorbance is saturated. This makes it easier to identify obvious contamination of samples by rapidly plotting endmember spectra together, for example, to identify antigorite in olivine, chlorite in garnet, epidote in feldspar, amphibole and sheet silicates in pyroxenes, and melt or fluid inclusions in all nominally anhydrous minerals. Additionally, in low-symmetry minerals like clinopyroxene, it is possible to determine the directions of maximum and minimum absorption with respect to the axes of the optical indicatrix and the crystallographic axes for each family of defects, which is a prerequisite for the accurate determination of H₂O contents.

For orthopyroxenes and clinopyroxenes from various geological settings and with different chemical compositions, it is possible to observe similar patterns in Si–O overtones for the specific orthogonal endmembers [a*]_(a^*–b), [Y](b–c), and [c](b–c), equivalent to the previously described Si–O endmember spectra in olivine (see Lemaire et al., 2004; Asimow et al., 2006; Giuliani et al., 2025). The characteristic spectral patterns are best defined by the sharp Si–O overtone bands in diopsides, whereas the magmatic/augitic and omphacitic samples show less well-defined, smoothed, overlapping peaks. For all investigated samples, [a*]_(a^*–b) can be identified by a clear maximum between 2100–1800 cm⁻¹ and a minimum between 1750–1300 cm⁻¹. [Y](b–c) shows the highest absorbance in the Si–O overtone bands with wavenumbers > 2100 cm⁻¹. [c](b–c) can be identified by the lowest absorbance in the higher-wavenumber Si–O overtones between 2300–1800 cm⁻¹, and the highest absorbance in the lower-wavenumber Si–O overtones is between 1800–1400 cm⁻¹ (e.g. Figs. 8 and 12).

[X](a–c) and [Z](a–c) are not identifiable purely by the FTIR spectra in some investigated samples; however, since [X](a–c) and [Z](a–c) are aligned with axes of the optical indicatrix, it is possible to orient crystals optically or to identify suitable grains, for example, in a thick section, using optical microscopy before FTIR analysis. Additionally, for large crystals, morphological features such as the habitus in idiomorphic crystals or the cleavage allow orientation of the crystals parallel to the (100), (010), and (a*–b) plane and the acquisition of spectra of [Y](b–c) and [c](b–c), [X](a–c) and [Z](a–c), and [a*]_(a^*–b) and [b](a*–b), respectively. The endmember spectra in the Si–O overtones, which show maximum and minimum absorption in different wavenumbers, coincide with a*, b, and c. In orthorhombic crystals, X, Y, and Z are aligned with a, b, and c, and, therefore, each endmember can be acquired in two different directions (Fig. 1 and Table 1). In monoclinic crystals only Y and b are aligned, and most observed indicatrices show either a maximum or a minimum in [b](a*–b) or [Y](c–b) spectra. Consequently, any measurements with the b axes in the plane and the polariser parallel have very similar absorbance. Therefore [b](a*–b) is equivalent to [Y](c–b) and can be acquired in all sections with the polarisation parallel to b. Once endmember spectra in the overtone region are known, synthetic spectra in any direction can be obtained using linear combination following Eq. (9). However, the inverse approach to estimate endmember spectra from a small number of randomly oriented, intermediate spectra as proposed by Asimow et al. (2006) and Jackson et al. (2018) is complicated because the angular relationship of absorption for the relatively strong absorption of the overtone bands is a function of (the unknown) maximum and minimum absorption (Figs. 2 and C2).

The presented examples clearly show the complexity of the numerous different overlapping OH-related defects in clinopyroxenes. These compilations, controversies, and limitations in the understanding of H incorporation mechanisms in clinopyroxenes have been presented in several publications (e.g. Skogby and Rossman, 1989; Skogby et al., 1990; Ingrin et al., 1995; Bromiley and Keppler, 2004; Skogby, 2006; Stalder and Ludwig, 2007; Koch-Müller et al., 2007; Ferriss et al., 2016; Balan et al., 2020; Gose and Schmädicke 2022, Aulbach et al., 2024).

A detailed investigation and discussion of all samples and their complex, anisotropic OH absorption behaviour, as well as the published calibration coefficients, would go beyond the scope of this paper. The focus here lies in using selected samples to highlight the potential of the newly presented workflows for a much more detailed investigation of anisotropic defects. This also provides an opportunity to account for the fact that measuring only three directions is insufficient for a full description of monoclinic samples (Dowty, 1978). Furthermore, it is possible to identify families of bands with similar polarisation behaviour or to find bands that overlap with different polarisation behaviour. This provides additional information for meaningful deconvolution of spectra such as peak positions and peak width (Gies et al., 2024). Even though the two samples K5-5 and K5-14 show nearly the same Si–O overtone signatures, they differ significantly in terms of OH absorption (see Aulbach et al., 2024). Whilst K5-14 shows both, absorption between 3700–3600 cm⁻¹ associated with T-site exchanges and absorption between 3600–3400 cm⁻¹ associated with exchanges on the M sites, sample K5-5 shows dominant absorption between 3600–3400 cm⁻¹, and only a minor peak is present between 3600–3700 cm⁻¹. Nevertheless, both spectra also show similarities in the polarisation behaviour of different bands, for example, a local maximum at wavenumber 3540 cm⁻¹, which can best be seen in [c](b–c) and [Z](a–c). Another interesting observation is the different position of the maximum in the range 3700–3600 cm⁻¹, where the maximum in [Y](b–c) is located at 3630 cm⁻¹, and the maximum in [c](b–c) is located at 3610 cm⁻¹. It is still questionable whether, in these two samples, the band at 3610–3613 cm⁻¹ is an intrinsic OH band or reflects exsolution of amesite and clinochlore, and both cases have been discussed previously (Bromiley and Keppler, 2004; Koch-Müller et al., 2004; Koch-Müller et al., 2007; Andrut et al., 2007; Jiang et al., 2022; Gose and Schmädicke, 2022; Skogby et al., 2016; Aulbach et al., 2023; Aulbach et al., 2024; Schmädicke et al., 2025). However, the strong polarisation effect documented here points towards an intrinsic OH band in this specific case as it is not observed in an eclogite xenolith showing optically visible cloudy exsolution features or in sample O-175 reported in Koch-Müller et al. (2004).

An advantage of using polarised measurements is that it allows comparison with calculated polarisation of different substitutions from the Density Functional Theory (DFT) modelling results of Balan et al. (2020). The coupled substitutions on the M sites between 3600–3400 cm⁻¹ involving trivalent cations and H show a polarisation behaviour [c](b–c)≫[a*](a*–b)≈[Y](b–c) in both calculations and natural samples, which further indicates an orientation of the OH indicatrix close to the crystallographic b and c axes and the vector a*. The observed bands in diopside and augite around wavenumbers 3650–3620 cm⁻¹ (exchange of Si for a trivalent cation and H⁺ on the T site) show a polarisation behaviour [Y](b–c)>[a*](a*–b)≫[c](b–c), in agreement with the DFT modelling. This is also the reason for the clear correlation in the scatterplot matrices, between the maximum of the M-site-related defects and endmember spectra [c](b–c), as well as for the observation that the maximum of the T-site defects is strongest in [Y](b–c) and [a*]_(a^*–b) endmember spectra (e.g. Figs. 9, 13, and 15). In sample MKS_A83, a minor absorption band at 3432 cm⁻¹ can be revealed by the endmember spectra. This peak and the polarisation behaviour are in good agreement with the synthetic Fe- and Na-doped samples of Purwin et al. (2009) and the Density Functional Theory (DFT) modelling results of Balan et al. (2020) for an Na-related defect in diopside.

OH-related absorption defects in clinopyroxenes are associated not with one trivalent cation but rather with a distribution of all available trivalent cations that can be incorporated. The relative partitioning of which M³⁺ cations fit into these defects depends on many parameters such as crystal chemistry, temperature, pressure, and H₂O activity. In general, the relationship of certain defects with respect to major and trace element contents often shows a clear correlation with Al, which simply reflects that Al is the dominant element involved in coupled substitutions in clinopyroxenes, and not necessarily that Al is directly involved in the defect; therefore, it is difficult to attribute defects in natural samples (Gies et al., 2024). Additionally, The DFT results in Table 1 of Balan et al. (2020) for diopside show that similar defects involving different M³⁺ cations (e.g. Al³⁺, Fe³⁺, Cr³⁺, Ti³⁺, Sc³⁺) have a different pleochroism because the infrared absorption indicatrices are oriented differently depending on the cation involved. The position of the OH-stretching mode for each M³⁺ also shifts slightly as a function of the local atomic environment (e.g. changes in O–H bond length and nearby cations), causing small but systematic differences in wavenumber positions between different trivalent cations (Table 1, Balan et al., 2020). The measured absorption is therefore the sum of all of these infrared indicatrices, each contributing with different intensities and orientations. Since the wavenumber position of all these defects are still very close together, the resulting observed absorption bands are often broad Gaussian-like peaks or are asymmetric, as can be seen in clinopyroxenes with T-site OH defects involving Al³⁺, Fe³⁺, and potentially Cr³⁺ (Figs. 8 and 11). The symmetry and distribution of all the present defects produce a single absorption figure, which can be recognised in the angular polarised measurements. As a result, the correlation of quantified H₂O contents with major and trace elements contents, such as the frequently observed clear correlation with Al, is not diagnostic and simply reflects that Al is the dominant element involved in coupled substitutions in clinopyroxenes (Gies et al., 2024).

The quantification of H₂O content of homogeneous, anisotropic samples can, principally, be done in bulk with unpolarised spectra (see Fig. C3, Sambridge et al., 2008; Kovács et al., 2008) if certain criteria are satisfied. However, with zoned samples showing crystal-preferred orientation (CPO) and contaminated samples resulting in H₂O broadband contribution to the spectra, this approach should not be applied. A key point is also the choice of the baseline correction and the used range of integration, which need to be consistent across a set of investigated samples to ensure comparability (Ferriss et al., 2016). The use of unoriented and unpolarised measurements and the uncertainty of the calibration coefficients furthermore lead to a dispersion of the calculated H₂O content, potentially masking a clear correlation between H₂O content and other compositional data.

The ideal sections to measure the spectra containing maximum and minimum absorbance, required to obtain the total absorbance cannot, in principle, be determined before the measurement of a specific sample. If the goal is to obtain the best possible measurement for the total absorbance, two cuboids containing [a*]_(a^*–b), [b](a*–b), [c](c–b) and [Y](c–b), [X](a–c). [Z](a–c) need to be prepared. More practical is the approach presented here, where maximum and minimum absorption directions are determined using morphological, optical, and overtone characteristics of clinopyroxene. With maximum polarised measurements and the identified spectral endmembers, it is possible to statistically test for homogeneity and obtain information about the polarisation behaviour of different defects or defect families. This approach also accounts for effects in monoclinic and triclinic crystals where the direction of the principal direction of absorption of specific defects and their corresponding absorption bands does not always coincide with either the crystallographic or the principal refractive index directions (see Dowty, 1978; Strens et al., 1982; Libowitzky and Rossman, 1996).

Theoretically, polarised measurements along any three mutually perpendicular sections can be used to obtain the total absorbance. Thus, the sums of polarised absorption along [a*]_(a^*–b)[b](a*–b), [c](c–b) (A + B + C) and along [Y](c–b), [X](a–c). [Z](a–c) (B + D + E) are theoretically equivalent. Practically, it is advantageous to choose the orientation where maximum and minimum absorption occurs. The reason for this is that, when the measurement orientation deviates significantly from the maximum O–H absorption, small misorientations will result in a larger uncertainty of the absorption (Fig. 2) and, hence, total absorbance. Additionally, maximum and minimum absorption are helpful in the determination of the defect type, information that is lost when intermediate sections are studied.

To help identify suitable grains and endmember spectra during the measurements of different types of pyroxenes, the spectral endmembers in the O–H and Si–O overtone region and their OH polarisation behaviour were compiled in Fig. 16. Furthermore, the use of available textural (e.g. core vs. rim) and other information (e.g. colour, chemical data) in combination with the presented endmember approach allows us to subdivide the data set and to find the spectral endmembers for each group. This enables us to quantify the H₂O content of inhomogeneous samples (with distinguishable zones) for all groups individually and generates much more detailed information on the variation in the H₂O content of a given sample. For samples with a CPO, spectra can be measured in multiple thick sections of different orientations to obtain all needed spectral endmembers. The combination of all the spectra can then be used to classify the endmembers (also in different zones), which are then used for H₂O quantification. In addition, it is possible to compare the differences not only in H₂O content but also in the absorption spectra of the different OH-related defects (Fig. 15). Thus, it is possible to study incorporation mechanisms in the context of their spatial distribution and absorption anisotropy, which provides significantly more information than previous measurement methods.

For pyroxenes, the combination of [a*]_(a^*–b), [b](a*–b) or [Y](c–b), [c](c–b) is feasible with samples that allow the use of morphologic information for orientation (e.g. cleavage or habitus) or where it is possible to use a big-data-polarised FTIR approach to identify the orthogonal Si–O spectral endmembers [a*]_(a^*–b), [b](a*–b), [c](c–b). Large inclusion-free grains; thick sections containing randomly oriented grains; or multiple prepared sections of a sample with CPO, containing all endmembers, are ideal for the combination of [Y](c–b), [X](a–c). [Z](a–c) if either [X](a–c) or [Z](a–c) is distinguishable in the spectral Si–O endmember or if the axes of the optical indicatrix are identifiable with optical microscopy.

It is not always possible to discriminate between [b](a*–b) and [Y](b–c) spectra with an Si–O overtone region endmember-finding approach because there is very little difference between spectra in any orientation taken to be polarised with the electric vector parallel Y. However, it is possible to distinguish [b](a*–b) and [Y](b–c) when the +90° polarised spectrum is a pure endmember spectrum of either [a*]_(a^*–b) or [c](b–c). Fortunately, for all clinopyroxenes, the deviation between the total integrated absorbance is very similar for [Y](b–c) and [b](a*–b), and, therefore, any mixture of [Y](b–c) and [b](a*–b) can be used as 2×[Y](b–c)measured≈ [Y](b–c) + [b](a*–b) with little effect on the estimate of the calculated O–H absorbance. Therefore, for clinopyroxenes, the discrimination between [Y](b–c) and [b](a*–b) can be neglected for H₂O quantification but is important when investigating OH incorporation mechanisms.

To illustrate the effect of using different sets of three perpendicular sections, the total integrated absorbance was calculated employing Eq. (6) using the baseline-corrected endmember spectra to account for pyroxenes with M- and T-site defects (see previous section). Where [b](a*–b) could not be identified, [Y](b–c) was used instead of [b](a*–b). The average deviation for all investigated samples between the total integrated absorbance from the combinations of the orthogonal endmembers [a*](a*–b)+[b](a*–b)+[c](b–c) or [Y](b–c)+[X](a–c)+[Z](a–c) is smaller than 7 %. The maximum observed deviation of the total integrated absorbance and the combination of the orthogonal endmembers [a*](a*–b)+[b](a*–b)+[c](b–c) or [Y](b–c)+[X](a–c)+[Z](a–c) of all investigated samples is <16 % (Fig. 17). Ideally, all identified endmembers should be used to determine the total integrated absorbance because the OH absorption indicatrix is not always aligned with the axis of the optical indicatrix, and, hence, this method gives the smallest measurement errors.

If the endmembers [a*](a*–b)+[Y](b–c)+[c](b–c)+[X](a–c)+[Z](a–c) can be identified from a large data set of polarised spectra, 3 times the average of all identified endmembers can be used to determine the total absorbance and further reduce the influence of a misidentified endmember or of the variation across different endmember combinations with regard to the total integrated absorbance. In the investigated samples, 3×([a*](a*–b)+[Y](b–c)+[c](b–c)+[X](a–c)+[Z](a–c))/5 and the total integrated absorbance agree within < 12 % (Figs. 17, C11, and C12). This works remarkably well for clinopyroxenes because the dominating absorbance proportions of the endmembers follow the behaviour of either [c](b–c)≈[Z](a–c)≫[Y](b–c)≈[b](a*–b)≈[X](a–c)≈[a*](a*–b) (or [c](b–c)≈[X](a–c)≫[Y](b–c)≈[b](a*–b)≈[Z](a–c)≈[a*](a*–b)) in samples with dominating M-site defects in the lower wavenumber peaks between 3600–3400 cm⁻¹ or [Y](b–c)≈[b](a*–b)≈[a*](a*–b)≈[Z](a–c)≫[X](a–c)≈[c](b–c) in samples with dominating absorbance in the tetrahedral site in the higher-wavenumber range between 3660–3550 cm⁻¹ (Fig. 18). This is especially important when orienting a clinopyroxene crystal for the estimation of the total integrated absorbance or when investigating defects (Dowty, 1978; Strens et al., 1982). While the orientation [a*]_(a^*–b), [b](a*–b), [c](c–b) or [Y](c–b), [X](a–c). [Z](a–c) does not make a big difference for samples with the highest absorbance parallel to the b axis (as, for example, in mantle pyroxenes), it will have a larger effect for samples with the highest absorbance in [c](b–c), [X](a–c), or [Z](a–c) (Figs. C13 and C14). Therefore, in studies investigating the hydrogen incorporation mechanism or comparing FTIR absorption with other methods of quantifying the water content of a sample to obtain calibration coefficients, it is crucial to follow the recommendations of Dowty (1978) in order to obtain a full description of the sample's absorption anisotropy and thus the determination of total integrated absorbance with the smallest measurement error.

In addition to the use of the previously presented combinations of spectral endmembers, it is possible for some samples to obtain single-spot H₂O contents if the section is in a special orientation. For that, the polarisation behaviour of the OH-related defects and the grain orientation must be known. Once the polarisation behaviour of a sample is well understood and the maximum and minimum absorbances are known, additional combinations of endmembers can be used together with an orientation factor to approximate the total integrated absorbance (e.g. Hu et al., 2026).

If the integrated absorbance is significantly higher in one direction and nearly equal in the corresponding two orthogonal directions, as shown in Giuliani et al. (2025) for olivine (2×[Y](b–c)+[a*](a*–b) or 2×[c](b–c)+[a*](a*–b)) or presented here for magmatic and metamorphic pyroxenes, two polarised measurements can be combined to estimate the total integrated absorbance (e.g. 2×[Y](b–c)+[c](b–c)). This can, however, only be applied for mineral grains in an orientation with the spectral endmember needed for the quantification in the section or orientations with known orientation factors. Interestingly, in the data set investigated here, using the endmember with the lowest integrated absorbance twice plus the endmember with the highest integrated absorbance works well for omphacitic and diopside-rich clinopyroxene samples (Fig. 16). Conversely, using the endmember with the lowest integrated absorbance twice plus the endmember with the lowest integrated absorbance works best for augitic/magmatic clinopyroxenes.