Introduction

Infrared (IR) and Raman spectroscopy rely on the interaction of nuclei with electromagnetic radiation (vibrations of atoms/bonds). IR spectroscopy entails the absorption of specific frequencies, whereas Raman spectroscopy involves inelastic scattering. The resulting infrared and Raman spectra capture lattice vibrations, reflecting the combined influence of chemical compositions and structures. These unique spectra facilitate the identification and characterization of materials, making them invaluable in Earth and planetary sciences. With advancing techniques, IR and Raman spectroscopy have not only contributed to the studies of earth and extraterrestrial samples but have also expanded their applications to remote planetary exploration (Karr 1975; Wang et al. 1995; Cooney et al. 1999; Ling et al. 2011; Kayama et al. 2018). Various space missions, such as the Mars 2020 Perseverance rover, ExoMars rover, and the Martian Moons eXploration mission, equip Raman spectrometers for mineral identification, composition analysis on planetary surfaces, and the search for evidence of life (Korablev et al. 2017; Rull et al. 2017; Cho et al. 2021; Lalla et al. 2022).

However, despite their versatile applications, several constraints limit the widespread use of infrared and Raman spectra. One notable limitation is the insufficient availability of databases for various minerals. Vibrational spectra are highly sensitive to even minor perturbations within chemical bonds, influenced by factors such as temperature, pressure, chemical composition, isotopic substitution, cation distribution, and spin states. To effectively utilize these spectra for mineral identification and differentiation, a thorough examination of the impact of each factor on vibrational modes is essential.

This study focuses on olivine, a common rock-forming mineral found in many planetary bodies (Earth, Moon, Mars, Mercury, etc.). The structural and vibrational properties of olivine have been extensively studied through both experimental and physical–mechanical computations (Iishi 1978; Akaogi et al. 1984; Hofmeister 1987; Rao et al. 1988; Chopelas 1990, 1991; Gillet et al. 1991; Ghose et al. 1992; Wang et al. 1993; Gillet 1996; Noel et al. 2006; Li et al. 2007; McKeown et al. 2010; Stangarone et al. 2017). However, most studies of infrared absorption and Raman spectroscopy for olivine have been conducted either under ambient conditions (Paques-Ledent and Tarte 1973; Iishi 1978; Hofmeister 1987, 1997; Kuebler et al. 2006; Dyar et al. 2009), high pressure at room temperature using diamond-anvil cells (Besson et al. 1982; Chopelas 1990; Durben et al. 1993), or high temperature using a heating stage at ambient pressure (Kolesov and Geiger 2004). To our knowledge, there are no published studies on simultaneous changes in temperature and pressure for olivine due to difficulties in experimental setup and computational modeling.

The experimental setup for simultaneous changes in temperature and pressure poses significant challenges. Computational methods provide a complementary approach for studying high-temperature and high-pressure mineral properties, allowing a deeper understanding of macro-properties at the atomic level and visualization of each normal mode. While simulating high pressures is relatively straightforward, simulating high temperatures requires careful consideration of anharmonicity. Before the temperature reaches levels where anharmonicity becomes significant, the QHA is accurate enough to model the temperature effect. Additionally, changes in temperature and pressure mainly affect the vibrational frequencies of a crystal by altering cell parameters and internal coordinates when anharmonicity is insignificant. Therefore, three steps are taken to compute the vibrational modes: using QHA to obtain cell parameters under temperature and pressure, performing geometry optimizations under fixed cell parameters to optimize the internal coordinates of atoms, and lastly, computing the vibrational frequencies of infrared and Raman. Additionally, we briefly examine the isotope and chemical compositional effect and cation ordering on the vibrational frequencies.

This study is based on Chapter 3 of S.C.'s thesis (Chen 2022).

Vibrational modes and crystal symmetry

Olivine belongs to the orthorhombic crystal system with the point group mmm in Hermann-Mauguin notation or D2h in the Schönflies notation. The space group is Pbnm, a non-conventional setting of Pnma with space group number 62. The chemical formula for olivine can be expressed as M1M2SiO4, featuring two non-equivalent types of octahedral sites, M1O6 and M2O6. There are four formula units in the unit cell, resulting in multiplicities of 4 for both octahedral sites. The M1 octahedron is distorted and smaller than M2. The most common metal cations occupying the octahedral sites are Fe and Mg. Due to the similar ionic radii of Fe and Mg, they form a complete solid solution, with the two end members being forsterite Mg2SiO4 and fayalite Fe2SiO4. A unit cell comprises 28 atoms, leading to 84 vibrational modes corresponding to the total degree of freedom. The symmetry decomposition of the 84 normal modes at the center of the Brillouin zone Γ, derived theoretically, is

$${\Gamma }_{\text{total}}=11{\text{A}}_{\text{g}}+11{\text{B}}_{1\text{g}}+7{\text{B}}_{2\text{g}}+7{\text{B}}_{3\text{g}}+10{\text{A}}_{\text{u}}+10{\text{B}}_{1\text{u}}+14{\text{B}}_{2\text{u}}+14{\text{B}}_{3\text{u}}$$
(1)

A Brillouin zone represents the primitive cell of a crystal in its reciprocal space. Mulliken symbols, such as Ag, B1g, etc., denote different symmetry types or irreducible representations. The character table of a point group provides information on which irreducible representations are infrared-active or Raman-active. Irreducible representations corresponding to linear functions, namely B1u, B2u, and B3u, are infrared active, while those corresponding to quadratic functions, Ag, B1g, B2g, and B3g, are Raman active. Symmetry Au modes remain silent. In total, there are 35 IR-active and 36 Raman-active vibrational modes. Three of these (acoustic) modes involve translations of the entire system (\({\text{B}}_{1\text{u}}+{\text{B}}_{2\text{u}}+{\text{B}}_{3\text{u}}\)).

$${\Gamma }_{\text{infrared}}=9{\text{B}}_{1\text{u}}+13{\text{B}}_{2\text{u}}+13{\text{B}}_{3\text{u}}$$
(2)
$${\Gamma }_{\text{Raman}}=11{\text{A}}_{\text{g}}+11{\text{B}}_{1\text{g}}+7{\text{B}}_{2\text{g}}+7{\text{B}}_{3\text{g}}$$
(3)

The olivine structure consists of isolated SiO4 tetrahedra and two different MgO6 octahedra. The tetrahedra exhibit three different Si–O distances, and their skewing is induced by the Coulomb force exerted by Mg on Si and O atoms (Lam et al. 1990). Each of the two Mg-octahedra has three distinct Mg1–O distances and four distinct Mg2–O distances.

Aside from the three acoustic modes, the remaining vibrational modes can be attributed to the translations of M1, M2, and SiO4, as well as the rotation and internal movement of SiO4. The internal movements of SiO4, involving bending and stretching, can be categorized into four types based on their symmetries: ν1, ν2, ν3, and ν4 (Lam et al. 1990). The ν1 and ν2 modes represent symmetric stretching and bending of the Si–O bond, respectively, with the Si atom remaining stationary. The ν3 and ν4 modes are antisymmetric stretching and bending modes, respectively, with the Si atom not stationary. The type of vibration is correlated with the symmetry derived from the correlation method and is summarized in Table 1 (Paques-Ledent and Tarte 1973; Hofmeister 1987).

Table 1 Classification of normal modes (81) of olivine for Pbnm, excluding the three acoustic modes
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Methods

Temperature and pressure effects on vibrational frequencies and QHA

The dependence of a given frequency on temperature and pressure can be caused by two aspects: (1) volume change due to compressibility (P) and thermal expansion (T); (2) a volume-independent contribution from lattice anharmonicity (T) (Gillet et al. 1989; Dai et al. 2013; Deshpande et al. 2014). At lower temperatures when anharmonicity is negligible, vibrational frequencies, with fixed composition, are a function of volume (Liu 1993; Gillet 1996). However, at very high temperatures, the interaction between phonons becomes significant, necessitating non-static methods, such as molecular dynamics and Monte Carlo. While these methods can be useful, they are strictly only valid for solids at elevated temperatures as they neglect the effect of vibrational quantum effects such as the zero-point energy. Therefore, anharmonicity must be checked in order to determine if QHA is a valid approximation. Gillet et al. (1989) stated that the anharmonic effect for mantle minerals becomes noticeable above 1000 K, leading to a 2% difference in heat capacity, Cv, at 1000 K and a 7% difference at 2000 K. Their experiments were conducted at room pressure. The valid domain of QHA in temperature increases with higher pressure (Carrier et al. 2007). Several studies (Baroni et al. 2010; Dovesi et al. 2018; Allen 2020) show that simulation with QHA is remarkably accurate in a wide temperature range up to close to the melting line. The temperature and pressure range of olivine in the upper mantle is approximately 1800 °C and 14 GPa. Since our simulations never exceed this range, the temperature and pressure conditions simulated in this study should be accurate enough using QHA. Erba et al. (2015a) also concluded that the QHA is reliably accurate in describing the thermodynamic properties of forsterite within its entire stability domain, as evidenced by comparing their computational results with experimental data from the literature.

Another approach to evaluate anharmonicity is by using the parameter mode anharmonicity, \({a}_{i}\), defined as \({a}_{i}={\left(\frac{\partial ln{(\nu }_{i})}{\partial T}\right)}_{V}\) (Gillet et al. 1989). If \({a}_{i}\) is not equal to 0, it indicates anharmonicity. Gillet (1996) concluded that the parameter \({a}_{i}\) typically ranges from − 1·10–5 to − 1·10–4 K−1 in various silicates and under different high-temperature and high-pressure conditions. The absolute value tends to be larger for low wavenumber modes, typically lattice modes. To estimate the effect of anharmonicity on vibrational frequencies, consider the following calculation. Suppose \({a}_{i}\) is near the maximum of measured experimental data, i.e., − 10–4 K−1. When the volume is constant (i.e., the pressure and temperature effect cancel each other out, keeping the volume unchanged), the frequency change increases with temperature change. For instance, if the temperature change is 1000 K, the relative change of \({\nu }_{i}\) would be approximately 10%. This offset caused by anharmonicity is acceptable when the temperature is below 1000 K, as it may be smaller than the computational error.

The QHA serves as a reliable approximation at low temperatures, before temperatures become high enough to exhibit anharmonic behavior. QHA finds widespread use in studying thermal properties and the temperature effects on vibrational frequencies (Reynard et al. 1992; Li et al. 2007; Wu and Wentzcovitch 2007; Deringer et al. 2014; Wen et al. 2017). QHA is expected to be a valid approximation for most of the mantle and only becomes problematic near the core-mantle boundary based on the measured melting temperature and expected mantle temperature (Wentzcovitch et al. 2010). QHA assumes that all bonds within a given crystal volume behave harmonically, with the equilibrium length capable of changing. Additionally, it posits that temperature and pressure impact vibrational frequencies solely through volume changes. However, it's crucial to note that this assumption is only strictly accurate for cubic structures. Anisotropy in compressibility and thermal expansion has been reported for olivine in the literature (Andrault et al. 1995; Trots et al. 2012). Beyond cell parameters, accurate fractional coordinates of atoms in the unit cell are equally crucial. Maintaining static fractional coordinates while adjusting cell parameters leads to incorrect proportional changes in all bond lengths, as observed in experimental data. SiO4 tetrahedra exhibit more rigidity than MgO6 octahedra. Hazen (1976) concluded that the thermal expansion and compressibility of Mg octahedra are comparable to those of bulk forsterite, while the stronger Si–O bonds undergo minimal expansion or compression. Therefore, this study considers the optimization of both cell parameters and fractional coordinates at the equilibrium cell volume.

Calculation of vibrational frequencies under changing T and P conditions

In this study, we conducted three series of calculations to examine the temperature and pressure effect: (1) pressure up to 15 GPa at static temperature (0 K); (2) temperature up to 900 K at static pressure (no hydrostatic pressure) to avoid anharmonicity; (3) simultaneous temperature and pressure. The computation is relatively straightforward for the sole pressure effect at 0 K. The optimized structure under external hydrostatic pressure can be easily obtained, and then the vibrational frequencies can be computed. When the temperature changes, the QHA must be used to obtain the volume (only cell parameters a, b, and c, no internal coordinates) under T and P first. This is followed by geometry optimization of the internal atomic positions while fixing the cell parameters, and finally, infrared and Raman frequencies are computed for the optimized structure. The pressure at a given volume V and temperature T has two parts: \(P(V,T)=P{({V}_{0}\to V)}_{T={T}_{0}}+{P}_{th}{({T}_{0}\to T)}_{V}\). The first term is the pressure needed to change the volume at a reference temperature, and the second term is thermal pressure, defined as the pressure change at constant volume V due to the temperature change.

Our ultimate goal is to compute infrared and Raman spectra under simultaneous T and P changes, where relevant data from both experiments and computations is lacking. Olivine, a dominant mineral phase in Earth's upper mantle down to the 410-km seismic discontinuity, undergoes a phase transition at this depth from α-(Mg, Fe)2SiO4 (olivine) to β-(Mg, Fe)2SiO4 (wadsleyite). The pressure at 410 km is approximately 14 GPa, while determining the temperature is more challenging. Various methods provide different temperature estimates, but it is approximately 1800 K (Akimoto et al. 1976; Ito and Katsura 1989; Trubitsyn and Trubitsyn 2020). For forsterite, the phase transition occurs at 1600 °C (1873 K) at 15 GPa and at 1200 °C (1473 K) at 14 GPa (Katsura and Ito 1989). Considering the stability range of forsterite, we selected the following temperature/pressure pairs: (1000 K, 5 GPa), (1500 K, 5 GPa), (1500 K, 10 GPa), and (2000 K, 10 GPa).

The Crystal17 program is employed for the DFT + QHA computations (Erba 2014; Erba et al. 2015a, b, c). As discussed earlier, vibrational frequencies under changing temperature and pressure are essentially a function of equilibrium geometry, under the temperature and pressure ranges we consider. Therefore, geometry optimizations under specific pressures or temperatures precede respective frequency computations. The initial forsterite structure is obtained from Bostroem (1987) with cell parameters of 4.749 Å, 10.1985 Å, and 5.9792 Å. The functional B3LYP is employed, which is a hybrid between a Hartree–Fock approach for the exchange term, and a BLYP functional for correlation. The basis sets used are 8-511d1G_valenzano_2006 for Mg (Valenzano et al. 2007), 86-311G for Si (Pascale et al. 2005; Noel et al. 2006), and 8-411d11G for O (Valenzano et al. 2006). The default isotopes used in the frequency calculations are 23.9850 for Mg, 27.9769 for Si, and 15.9949 for O.

Vibrational frequencies are computed at the Γ point of the reciprocal unit cell within the harmonic approximation at static temperature and pressure. The Hessian matrix, Hij, represents the second derivative matrix of energy due to the displacement of any atom from its equilibrium position. The mass-weighted Hessian matrix is calculated from the dynamic matrix divided by the square root of the mass of any given atom pair. The eigenvalues of the mass-weighted dynamic matrix are the frequencies, and the eigenvectors are the normal modes. Transverse optical (TO) modes are computed by default, and both infrared and Raman frequencies and intensities are calculated. Harmonic vibrational frequencies are computed and fitted for different volumes, with four volumes used by default. Once the equations of frequencies with volume are determined, all other thermodynamic properties can be analytically derived by minimizing the free energy. For example, the Helmholtz free energy F of the system in QHA can be expressed as a function of crystal structure X and temperature T:

$$F(X, T) = U(X)+\sum_{i}\frac{1}{2}h{\nu }_{i}(X)+{k}_{B}T\sum_{i}ln\left(1-{e}^{- \frac{h{\nu }_{i}\left(X\right)}{{k}_{B}T}}\right)$$
(4)

where the three terms are the internal energy of the system (\(U(X)\)), the zero-point vibrational energy \(\left( {\mathop \sum \nolimits_{i} \frac{1}{2}h\nu_{i} \left( X \right)} \right)\), and the temperature effect of the vibration part \(\left( {k_{B} T\mathop \sum \nolimits_{i} ln\left( {1 - e^{{ - \frac{{h\nu_{i} \left( X \right)}}{{k_{B} T}}}} } \right)} \right)\).

Results

Crystal structure changes with temperature and pressure

The crystal structure of forsterite was examined under changing temperature and pressure to explore its thermal expansion and compressibility, respectively. Table 2 presents the computed cell parameters and volume data for various temperature and pressure combinations. Cell parameters and volume with a single parameter change (either pressure or temperature) are plotted and compared to the literature data in Fig. 1. To emphasize unit cell changes resulting from variations in pressure and temperature conditions only, we have normalized the data by plotting the ratios of the cell parameters at a given temperature and pressure divided by those under ambient conditions (300 K, 0 GPa). This normalization eliminates subtle differences in initial cell parameters due to the computational parameters chosen, focusing on the impact of changes in pressure and temperature. Our calculated data demonstrate good agreement with the literature data.

Table 2 Unit-cell parameters of forsterite calculated under varying temperature (in K) and pressure (in GPa)
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Fig. 1
figure 1

Lattice parameter changes with pressure (left panel) and temperature (right) in this study are compared with the literature. Cell parameters (a, b, c) and volume (V) are normalized by dividing them by their respective values at ambient conditions (a0, b0, c0, V0) to eliminiate subtle differences in initial cell parameters. A second-order polynomial equation is used to fit the logarithm of cell parameters or volume with pressure or temperature. Only data from this study is used for the fitting

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To assess changes in volume and cell parameters in response to temperature and pressure, two crucial parameters come into play: bulk modulus and thermal expansion coefficient. Bulk modulus serves as a metric for gauging the volume change resistance to pressure, reflecting the material's compressibility. The thermal expansion coefficient delineates the volume variation induced by temperature fluctuations. Specifically, the isothermal bulk modulus, denoted as KT, is defined as

$${K}_{T}=-V{\left(\frac{\partial P}{\partial V}\right)}_{T}=-{\left(\frac{\partial P}{\partial lnV}\right)}_{T}$$
(5)

The adiabatic bulk modulus, KS, is defined analogously but under adiabatic conditions, signifying a process without heat exchange. Compressibility is the reciprocal of bulk modulus. The bulk thermal expansivity, denoted as α, is defined as

$$\alpha =\frac{1}{V}{\left(\frac{\partial V}{\partial T}\right)}_{P}={\left(\frac{\partial lnV}{\partial T}\right)}_{P}$$
(6)

and the linear thermal expansivity αi is defined as

$${\alpha }_{i}=\frac{1}{{x}_{i}}{\left(\frac{\partial {x}_{i}}{\partial T}\right)}_{P}={\left(\frac{\partial ln{x}_{i}}{\partial T}\right)}_{P}$$
(7)

where xi is the cell parameter.

The logarithm of volume exhibits a correlation with cell parameters, prompting the utilization of logarithmic fitting for the data presented in this study, as illustrated in Fig. 1. To capture the intricate relationship, a second-order polynomial is employed instead of a linear function for the expansion range of pressure and temperature. Notably, the considerably smaller coefficients in the T2 and P2 terms, two to four orders of magnitude lower than the linear terms, allow for a meaningful comparison of compressibility and linear thermal expansivity along different axes of olivine.

Compression and thermal expansion exert opposing influences on cell parameters; an increase in pressure diminishes cell volume, while elevated temperature produces the opposite effect. The rates of change with respect to pressure (dln/dP) for the a, b, and c axes are − 0.0018, − 0.0034, and − 0.0026 GPa−1 at 0 GPa, respectively. This indicates that compressibility is highest along the b-axis, followed by c and a, aligning with findings in the literature (Andrault et al. 1995; Finkelstein et al. 2014). Similarly, the rates of change with respect to temperature (dln/dT) for the a, b, and c axes are 1.04 × 10–5, 2.02 × 10–5, 1.54 × 10–5 K−1 at 0 K, representing linear thermal expansion at 0 K. Axis b is the most temperature sensitive, followed by c and a, mirroring the order of compressibility (Erba et al. 2015a).

The system's two bulk moduli, KS and KT, are computed in this study at various pressures and temperatures by minimizing the second derivative of energy with respect to volume. The results are presented in Table 3 and Fig. 2. Notably, the values of KS and KT coincide only at zero temperature, with KS consistently exceeding KT at any finite temperature. The bulk modulus and its derivative to pressure, computed by fitting the third-order Birch-Murnaghan equation at 0 K and 0 GPa are 123.5 GPa and 4.35, respectively, demonstrating close agreement with literature values (Table 3). The bulk modulus at 0 K is reported at different pressures, and a linear fit reveals the relationship K = 123.3 + 4.30P (in GPa). This value falls within the reported range of literature values, which exhibits significant variability. The KT and KS data at different temperatures are also provided. To characterize the bulk modulus as a function of temperature, a second-order polynomial is employed to fit KS and KT with T, resulting in Eqs. 8 and 9:

$$K_{T} = { 121}.{1} - 0.0{15}\,T - {1}.0{3} \times {1}0^{{ - {5}}} \,T^{{2}}$$
(8)
$$K_{{\text{S}}} = {121}.{5} - 0.0{12}\,T - {8}.{36} \times {1}0^{{ - {6}}} \,T^{{2}}$$
(9)
Table 3 Comparison of bulk modulus calculated in this study with literature data
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Fig. 2
figure 2

a Bulk modulus at various pressures at 0 K with displayed linear fitting equation. b Isothermal and adiabatic bulk modulus at different temperatures with shown second-order polynomial equation

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In this study, both volumetric and linear thermal expansion coefficients are computed up to 1000 K. The thermal expansivity increases with temperature while the rate of change \(\frac{\partial \alpha }{\partial T}\) decreases with temperature, as illustrated in Fig. 3. Forsterite exhibits pronounced anisotropy, with αb > αc > αa, indicating that axis a is the most rigid among the three axes. Several equations describing the thermal expansion coefficient with temperature have been proposed. Our data is fitted using the equation α = a0 + a1 × Τ + a2/T2, and the results are compared with literature values (Table 4).

Fig. 3
figure 3

Thermal expansion (α) and linear thermal expansion coefficients (αa, αb, and αc). The connection between the bulk and linear thermal expansion coefficients is \({\alpha }_{V}={\alpha }_{a}+{\alpha }_{b}+{\alpha }_{c}\)

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Table 4 Comparison of thermal expansion coefficient equations from this study and literature
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The bond lengths in the crystal structures for various temperature and pressure conditions are plotted in Fig. 4. The ratio is used in reference to the 0 K or 0 GPa conditions. Generally, Mg2–O bonds exhibit higher sensitivity to changes in pressure and temperature than Mg1–O bonds, while Si–O bonds display the smallest sensitivity.

Fig. 4
figure 4

a Bond lengths ratios (bond length at a given T/bond length at 0 K) change with temperature for calculated structures in this study for forsterite; b bond lengths ratios (bond length at given P/bond length at 0 GPa) with pressure. Blue, green, and red symbols represent Si–O, Mg2–O, and Mg1–O, respectively

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For thermal expansion, different Mg–O expansions with temperature vary from (1.7 ± 0.2) × 10–5 to (5.1 ± 0.5) × 10–5 Å/K with the average of 2.9 × 10–5 Å/K, which agrees well with the value range of 2–5 × 10–5 Å/K in Hazen (1976). The temperature effect on Si–O, however, is less pronounced. The Si–O bond lengths initially increase with temperature up to 700 K, followed by a decrease at a temperature of 900 K, though the variations are modest. Linear fitting yields the expansion rate of Si–O from (2.3 ± 4.8) × 10–6 to (6.1 ± 2.5) × 10–6 Å/K, with an average of 4.1 × 10–6 Å/K. The expansion of Mg–O is approximately seven times higher than that of Si–O. Hazen (1976) concludes that Si–O neither expands nor compresses significantly with temperature, while the change in Mg–O is comparable to the volume change. The limited change observed in Si–O bonds with temperature is attributed to the presence of isolated tetrahedra in olivine. In silicates, the bond lengths and volumes for tetrahedra with four bridging O atoms seem to increase with temperature more rapidly than those with four non-bridging O atoms.

The Mg–O compression with pressure varies from (− 7.7 ± 0.4) × 10–3 to (− 3.5 ± 0.06) × 10–3 Å/GPa with an average of − 5.4 × 10–3 Å/GPa. In comparison, the compression rate for Si–O varies from (− 1.5 ± 0.02) × 10–3 to (− 1.0 ± 0.02) × 10–3 Å/GPa with an average of − 1.29 × 10–3 Å/GPa. On average, the compression of Mg–O is approximately four times higher than Si–O.

Under conditions resembling the Earth's interior, compression due to pressure has a more pronounced effect on olivine cell parameters and bond lengths than thermal expansion. Taking the pressure at the 410 km discontinuity to be 14 GPa and the temperature around 1800 K, the average Mg–O bond changes by − 0.0756 Å due to pressure and lower bond change due to temperature (0.0522 Å). The same is true for Si–O bonds and volume, resulting in generally smaller volumes and bond lengths at greater depths.

Considering the change in bond length relative to volume, from 0 to 900 K, the volume change is 3.2%. Si–O experiences a 0.21% change, while Mg1–O and Mg2–O change by 0.9% and 1.54%, respectively. With pressure ranging from 0 to 15 GPa, the volume change is − 9.4%. Si–O exhibits a − 1.2% change, while Mg1–O and Mg2–O bonds change by − 3.1% and − 4.2%, respectively.

Comparison of vibrational spectra with the literature and mode assignments

Infrared and Raman spectra have been intensely investigated by FTIR and Raman spectroscopy, along with computational methods. However, most studies have focused on ambient temperature and pressure conditions through experimental measurements and static calculations (0 K, 0 GPa). This study compares infrared and Raman frequencies calculated at 300 K and 0 GPa with experimental data from various sources. Frequencies calculated at static conditions are compared with computational data found in the literature (Tables 5 and 6). The IR and Raman spectra are consistent with the literature in general. However, even the reported data exhibit some inconsistencies, which could arise from polarization mixing or errors in both experimental and computational approaches.

Table 5 Comparison of infrared vibrational frequencies with literature and mode assignments
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Table 6 Comparison of Raman vibrational frequencies with literature and mode assignments
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For Raman spectra, there is a typical doublet consisting of two high-intensity modes in the Raman spectra between 800 and 880 cm–1. The peak positions of this doublet, known to be a function of composition, have been widely used to determine olivine composition (Wang et al. 2004; Kuebler et al. 2006; Breitburg et al. 2018). For forsterite under static conditions (0 K, 0 GPa), the highest-intensity peak (DB1) is calculated to be 837.8 cm−1, and the second peak (DB2) is 855.8 cm−1 (Fig. 5). A visualization of the DB1 is provided in Fig. 6. Both modes exhibit Ag symmetry, consistent with the literature.

Fig. 5
figure 5

Infrared (IR) and Raman spectra (relative intensity versus wavenumber) under static conditions

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Fig. 6
figure 6

The first peak of the Raman doublet at a frequency of 837.8 cm−1. Symmetry Ag. Mg atoms are represented by green balls, O by red, and Si by apricot. Arrows depict the movement of each atom, if any. This mode is associated with the symmetric stretching of SiO4, with the Si atom being approximately stationary (ν1)

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For the mode assignment of vibrational spectra, energy considerations are crucial. The energy required to excite a phonon is the most for stretching, followed by bending, rotation, and translation. Bending movement changes the bond angle between two neighboring O atoms and a central cation while stretching changes the bond length. Typically, stretching modes require higher energy (i.e., higher wave number) than bending modes. The frequency of stretching vibrations is influenced by both atom mass and bond stiffness, with heavier atoms vibrating at lower frequencies. Stretching modes can be symmetric or asymmetric, with the latter usually having higher energy. Kuebler et al. (2006) have divided them into ranges of wave numbers < 400, 400–700, and 700–900 cm−1 for Raman. Others, like Breitburg et al. (2018), use 500 and 800 cm−1 as separating wavenumbers. In general, Raman-active phonons below 500 cm−1 are lattice modes, where SiO4 tetrahedra act as rigid units and Mg2 translations mix with SiO4 rotations and translations. Raman modes above 500 cm−1 primarily involve atomic displacements within the SiO4 tetrahedra. Paques-Ledent and Tarte (1973) summarized that the infrared region of 800–1050 cm−1 corresponds to stretching vibrations of the SiO4 tetrahedron, 650–475 cm−1 to the bending of SiO4, and below 475 cm−1 are external modes mainly due to the vibrations of the MgO6 octahedra.

For the static calculations in this work, mode assignments for infrared and Raman are summarized in Tables 5 and 6. Predicted vibration types from symmetry (Table 1) are used as a constraint, although the visualization of each mode shows extensive mode mixing, especially between translation and rotation. Lower modes are attributed to the translation of Mg cations and the rotation of SiO4, with a boundary between 400 and 500 cm−1. Many modes involve a combination of movements, making clear distinctions challenging. This is understandable due to the shared O atoms between Mg–O bonds and SiO4 tetrahedra, leading to the perturbation of SiO4 tetrahedra movements caused by Mg atoms.

Pressure dependence of the vibrational frequencies for forsterite

The pressure dependence of vibrational frequencies is computed up to 15 GPa at 0 K, and the peak positions for infrared and Raman active modes are summarized in Tables 5 and 6. The infrared spectra exhibit a consistent pattern across different pressures, shifting to higher frequencies with increasing pressure (Figs. 7 and 8). Notably, they follow a generally linear trend, except for one B1u mode at 209.7 cm−1 for 0 GPa (Fig. 8). This mode barely changes its peak position with pressure. The linear fitting gives a slightly negative slope (− 0.18 ± 0.29). For Raman, a similar trend is observed, albeit with some deviations. For instance, frequencies at 5 GPa are higher than those at 10 GPa, and four modes do not show a good linear trend (Fig. 8). All the four Raman modes deviating from linear trends share the symmetry of Ag. This rough linear trend agrees well with findings from literature studies. Chopelas (1990) and Wang et al. (1993) demonstrated that all Raman bands move to higher frequencies with increasing pressure, displaying a linear relationship for pressures up to ~ 7–9 GPa. Then, there is a decrease in the pressure derivative of some Raman modes. They attributed this change in slope to the compression mechanism change in forsterite under pressure, which may offer insights into the deviations observed in our study.

Fig. 7
figure 7

Infrared (left) and Raman (right) spectra under various pressures up to 15 GPa, all at 0 K

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Fig. 8
figure 8

Linear frequency fitting with pressure. Four Raman modes (#4, 23, 24, and 35) are removed from the figure due to high scattering with pressure. All the four modes share the symmetry of Ag

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Except for one B1u mode being invariant with pressure, the pressure dependence dνi/dP for infrared varies from 1.5 to 4.7 with an average of 3.0 ± 1.0 (σ) cm−1/GPa. In the case of Raman, excluding the mode with a frequency of 211 cm−1 at 0 GPa, the dνi/dP ranges from 0.6 to 4.8 with an average of 2.6 cm−1/GPa. While there is no significant difference observed in dνi/dP for low-frequency and high-frequency modes (Fig. 9), the relative frequency change (dνi/νi/dP) shows a notably higher value for low-frequency modes compared to high-frequency modes. The isothermal mode Grüneisen parameter can be used to quantify the relative frequency change with pressure.

Fig. 9
figure 9

a Infrared and Raman frequencies shift (dνi/dP) with pressures, all at 0 K. The lowest frequency data for Raman is excluded. The frequency data in the plot is from 0 GPa. b The isothermal mode Grüneisen parameter (γiT) at 0 K. The bulk modulus of 123.3 GPa is used in this study. The frequency data in the figure is from 0 GPa. The two negative modes (one infrared, one Raman) are not shown

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The Grüneisen parameter connects the volume and vibrational frequencies directly. The ith vibrational mode of a crystal has a mode Grüneisen parameter \({\gamma }_{i}\), defined as

$${\gamma }_{i}=-\frac{\text{d ln}{\upnu }_{\text{i}}}{\text{d lnV}}$$
(10)

If we combine the definition of the bulk modulus and Eq. 9, the Grüneisen parameter of ith mode at a specific temperature (isothermal mode Grüneisen parameter) \({\gamma }_{i{\rm T}}\) can be written as

$${\upgamma }_{\text{i{\rm T}}}={\text{\rm K}}_{\text{\rm T}}{\left(\frac{\partial \text{ln}{\upnu }_{\text{i}}}{\partial \text{P}}\right)}_{\text{T}}$$
(11)

Similarly, if we combine the definition of thermal expansion coefficient with Eq. 9, the Grüneisen parameter of the ith mode at a given pressure P (isobaric mode Grüneisen parameter) can be written as

$$\gamma_{{{\text{iP}}}} = \frac{1}{\alpha }\left( {\frac{{\partial \ln \nu_{i} }}{\partial T}} \right)_{P}$$
(12)

In the context of the QHA, vibrational frequencies are influenced by pressure and temperature solely through volume changes, therefore \({\gamma }_{i{\rm T}}={\gamma }_{iP}\) (Gillet 1996).

The isothermal mode Grüneisen parameter, \({\gamma }_{i{\rm T}}\), is significantly smaller for modes above 600 cm−1, corresponding to internal stretching and bending within the SiO4 tetrahedra. Therefore, we can conclude that the internal movement of SiO4 is less affected than the external lattice modes. This observation aligns with findings by Chopelas (1990), who attributed it to a more substantial volume change in the MgO6 octahedra than the SiO4 tetrahedra (Hazen 1976).

Temperature dependence of the vibrational frequencies for forsterite

The infrared and Raman vibrational modes for different temperatures are grouped based on their irreducible representation (symmetry). As the temperature increases, the frequency of each mode shifts to a lower wavenumber, displaying a generally linear relationship (Fig. 10). This trend aligns with findings in Raman studies from the literature (Gillet et al. 1991; Stangarone et al. 2017; Liu et al. 2021). No infrared spectra with increasing temperature have been reported, to our knowledge. For each mode, linear fitting was applied to assess the frequency shift with temperature Δνi/ΔT. The Δνi/ΔT values for infrared range from − 0.033 to − 0.007 cm−1/K, with an average of − 0.020 ± 0.005 cm−1/K (Fig. 11a). A few Raman modes deviate from a good linear trend. These outlier modes, characterized by Ag symmetry, are associated with the internal movement of SiO4. The remaining Raman modes have Δνi/ΔT values ranging from − 0.029 to − 0.009 cm−1/K, averaging − 0.019 ± 0.005 cm−1/K (Fig. 11). These findings are consistent with reported ranges of − 0.029 to − 0.008 cm−1/K (average − 0.017 cm−1/K) in experimental studies by Gillet et al. (1991) and − 0.024 to − 0.005 cm−1/K (average − 0.016 cm−1/K) in computational studies by Stangarone et al. (2017).

Fig. 10
figure 10

Linear frequency fitting with temperature. Four Raman modes (#21, 23, 24, and 31) are removed from the figure due to their non-linear trend with temperature. All four modes share the symmetry of Ag

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Fig. 11
figure 11

a Infrared and Raman frequency shifts (dνi/dT) with temperature. All pressures are 0 GPa. The lowest frequency data for Raman is excluded. b Relative frequency shifts with temperature (dlnνi/dT)

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Similar to the pressure effect, the absolute frequency shift with temperature (Δνi/ΔT) does not show a dependence on frequencies (Fig. 11). However, the relative change, Δνi/ΔT/νi, reveals that temperature has a more pronounced effect on lower frequencies associated with transition and rotation than on the higher-frequency modes linked to the internal stretching and bending of SiO4 (Fig. 11).

Vibrational frequencies of forsterite under simultaneous high T and P conditions

Simultaneous effects of temperature and pressure were simulated on vibrational modes for five pairs of T and P: (1000 K, 5 GPa), (1500 K, 10 GPa), (1500 K, 5 GPa), (2000 K, 10 GPa), and (2000 K, 20 GPa). The infrared and Raman spectra are displayed in Fig. 12. Infrared spectra are similar for various temperature and pressure pairs in terms of peaks and relative intensities, with only the peak frequencies change with T and P. However, Raman spectra show more variability.

Fig. 12
figure 12

Infrared and Raman spectra under simultaneous high temperatures and pressures

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Compositional and isotope effect on vibrational frequencies

In order to examine the compositional effect, we computed IR and Raman vibrational frequencies for the forsterite-fayalite solid solution. To maintain the symmetry of olivine, we specifically simulated the spectra for MgFeSiO4, where all four Fe ions are placed either in M1 or M2 sites within a unit cell. An important consideration is the spin states for Fe. We examined both low-spin and high-spin states for Fe in fayalite, and for the high-spin state, we considered both ferro- and antiferromagnetic configurations. Our energy computations reveal that fayalite with high-spin Fe is more than 180 kJ more favorable than low-spin fayalite per mol formula unit, and the antiferromagnetic spin configuration is 2.5 kJ more favorable than its ferromagnetic counterpart (Table 7). Furthermore, fayalite has been reported to be a noncollinear antiferromagnetic compound below a Neel temperature of approximately 65 K with Mossbauer spectroscopy and neutron diffraction studies (Santoro et al. 1966; Hayashi et al. 1987). Above the Neel temperature, it is paramagnetic. Therefore, high-spin Fe should be used in olivine calculations. Due to the close energy of antiferromagnetic and ferromagnetic structures and the simplicity of the ferromagnetic setting, we employed ferromagnetic configurations in our computations for fayalite and MgFeSiO4 for the IR and Raman spectra.

Table 7 Cell parameters and energetics calculation of olivine. All calculations started from the same structure
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All vibrational modes in fayalite exhibit lower frequencies than their respective counterparts in forsterite (see Fig. 13 for infrared spectra), potentially caused by the different masses of Fe and Mg, changes in the cell parameters of fayalite and forsterite, or chemical/electronic differences of the two cations or their respective bonds to O atoms. The differences range from − 8 to − 126 cm−1, with a slightly more pronounced distinction for higher frequency modes. In the calculations for the MgFeSiO4 solid solutions, intermediate frequencies are observed compared to the two end members. However, for the same composition, the two configurations (Fe in M1 or M2 sites) exhibit different frequency modes, with Δνi from 28 to − 41.5 cm−1.

Fig. 13
figure 13

IR vibrational frequencies for forsterite (fo)-fayalite (fa) solid solutions

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To isolate the mass effect on vibrational frequencies, we assigned different isotopes for metal cations. The default isotope for Mg used in the frequency calculations of Crystal17 is 24Mg. We replaced all eight cations in the unit cell with a mass of 24 with isotopes 26 or 56. It's important to note that there is no mass of 56 for Mg. The calculations used here allow us to separate out the mass effect by artificially increasing the mass of Mg to 56 while maintaining Mg chemistry. Both calculations maintained the same crystal structure as forsterite with the default 24Mg, differing only in isotope composition. The results indicate that 26Mg shifted wavenumbers up to 12 cm−1 lower (Fig. 14). High vibrational frequencies (> 800 cm−1), attributed to Si–O vibrations, seem to be less impacted (within − 1 cm−1). The “56Mg” substitution causes a more significant shift (up to 80 cm−1) than 26Mg due to a more considerable mass difference. For low-frequency modes, the Δνi could be as much as approximately − 62.5 cm−1 due to the heavy mass.

Fig. 14
figure 14

Vibrational frequencies change due to the isotope effect. All three calculations have the same cell parameters and coordinates but different masses in all eight cations in the unit cell. The intensities of 56Fe and 26Mg shift up for better visualization

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The mass effect on vibrational frequencies is not a constant value across different modes, and the shift is not directly proportional to the mass difference. To quantify the isotope effect, we consider a simple diatomic molecule harmonic oscillator. The frequency (ν) can be calculated as \(\nu =\frac{1}{2\pi }\sqrt{k/\mu }\) with \(\mu =\frac{{m}_{1}{m}_{2}}{{m}_{1}+{m}_{2}}\) representing the reduced mass and k as the force constant. In the case of metal cations bonded to O atoms (with a default mass of 16) to form octahedral sites, assuming the force constant remains constant, ν is approximated as \(\sqrt{1/\mu }=\sqrt{\frac{{m}_{1}+{m}_{2}}{{m}_{1}{m}_{2}}}=\sqrt{\frac{{m}_{1}+16}{{16m}_{1}}}\). For each mode, the frequency with 24Mg is 1.016 times that of 26Mg and 1.139 times that of 56Fe (Fig. 15).

Fig. 15
figure 15

Vibration frequencies change due to the isotope effect. All three calculations have the same cell parameters and coordinates but different masses. The isotope substitution applies to all eight cations in the unit cell

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The lighter frequencies in fayalite compared to forsterite result from a combined effect of mass and cell parameter differences. By isolating the mass effect, it becomes possible to assess the percentage contribution of each factor. Specifically, the mass effect accounts for approximately 50% for the low-frequency modes and significantly lower percentages for high-frequency modes (Fig. 16). The cation substitution in octahedral sites primarily impacts the vibration of MO6 and has a lesser effect on SiO4. Although metal ions affect SiO4 vibrations due to shared oxygen atoms, the influence is comparatively minor.

Fig. 16
figure 16

The contribution of mass difference to the mode shift between forsterite and fayalite. The observed difference in vibrational modes between the two is influenced by a combined effect of mass difference and cell parameters

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Discussions and implications

The vibrational properties of infrared and Raman are influenced by factors such as composition, temperature, pressure, isotope, spin, etc. In this study, we evaluate the effects of temperature and pressure in detail and briefly examine the compositional and cation distribution effects. The findings reveal significant impacts on normal modes, particularly the peak positions of certain vibrational modes like the doublet Raman modes. These modes have been used to predict olivine compositions. For instance, forsteritic (high molar Mg/(Mg + Fe)) olivine is indicative of a primitive magma source, while more fayalitic olivine suggests a more evolved source. However, caution should be taken in cases of substantial temperature and pressure variations. The two endmembers of (Fe, Mg)2SiO4 show an approximate difference of 10 cm−1 for the doublet modes. Assuming an average of Δνi/ΔT of – 0.02 cm−1 K−1and Δνi/ΔP of 3 cm−1 GPa−1, a 500 K temperature difference or 3 GPa pressure difference can completely override the composition indication if failing to consider the temperature and pressure effects. The surface temperature of Mars may vary by 200 K, leading to a potential 40% error in composition identification when using Raman to analyze mineral compositions or structures on the Martian surface. An alternative approach, yet to be tested widely, could involve using lower frequencies controlled by substitution. However, these are challenging to measure due to the relatively low intensities of these modes. Advances in experimental techniques may open avenues for broader applications in the future.

This study helps develop a more comprehensive data set of infrared and Raman spectra in temperature, pressure, and compositional space. The findings underscore the significant influence of factors such as temperature, pressure, composition, and cation ordering on peak positions. Frequency changes for each mode can be effectively parameterized using a simplified equation, enhancing our understanding of vibrational behavior in diverse temperature, pressure, and composition ranges as

$$d{\nu }_{i}= {a}_{i}dT+{b}_{i}dP+{c}_{i}X$$
(13)

where X is the fractional contribution of fayalite in the olivine. This is a combined effect of temperature–pressure-composition (P–T−X). Or, if the specific occupation of iron on the two different cation sites is taken into account, this equation could be expanded to

$$d{\nu }_{i}= {a}_{i}dT+{b}_{i}dP+{c}_{i}{X}_{Fe}^{M1}+{d}_{i}{X}_{Fe}^{M2}$$
(14)

The methodology employed in this study provides insights into the conditions under which specific phonons or vibrational modes are responsible for pronounced changes in thermal expansion and compression under pressure. This enhances our understanding of the intricate dynamics governing these phenomena.

Wide applications can also be found in the reported bulk modulus and thermal expansion data. One important application is in seismic data interpretation. Seismic wave velocities are influenced by the density and elastic properties of materials. The compressional-wave velocity \(\left( {V_{p} = \sqrt {\frac{{K + \frac{4\mu }{3}}}{\rho }} } \right)\) and shear-wave velocity \(\left( {V_{s} = \sqrt {\frac{\mu }{\rho }} } \right)\) are functions of bulk modulus K, shear modulus μ, and density ρ. While bulk modulus data has been reported in the literature, it is primarily at ambient conditions. However, olivine's bulk modulus varies significantly with pressure and temperature. Our study not only provides data under different temperature and pressure conditions but also offers linear/second-order polynomial fitting. This analytical form of the equation of state is valuable compared to discrete data points. Additionally, the bulk modulus data at high temperatures and pressures find applications in equations of state, such as the third-order Birch-Murnaghan equation. The Birch–Murnaghan isothermal equation of state is usually used at room temperature. With the bulk modulus K and its derivative K′ data at different temperatures, it can then be used for deep Earth conditions.

The methodology introduced in this study provides a systematic approach to compute infrared absorption and Raman spectra for minerals under simultaneous high temperature and high pressure, provided anharmonicity is not significant. Its applicability extends beyond olivine and can be effectively employed for studying other minerals as well.

Conclusions

We employed DFT in conjunction with the QHA to simulate the crystal structure change and vibrational properties of forsterite under high temperature and pressure conditions. Assuming the temperature is within the quasi-harmonic validity range, vibrational frequencies become solely dependent on the crystal structure, including both cell parameters and fractional positions. For a given temperature–pressure pair, we used QHA to obtain cell parameters, optimized the crystal structure, and computed infrared and Raman vibrational frequencies. This marks the first report, to our knowledge, of vibrational frequencies under simultaneous high-temperature and high-pressure conditions. A detailed examination of vibrational frequencies under varying temperature and pressure conditions reveals significant shifts in peak positions. The temperature- and pressure-induced frequency shifts vary for each mode, with an average of 3 cm−1/GPa and − 0.017 cm−1/K. Caution is advised when interpreting olivine compositions based on peak positions, especially in the presence of significant changes in environmental conditions (temperature, pressure). In addition to the temperature and pressure effect, we also evaluated the compositional effect and isotopic effect on vibrational frequencies. This study contributes to the development of a comprehensive dataset for infrared and Raman spectra in temperature, pressure, and compositional space, enhancing our ability to predict and interpret olivine data across diverse planetary conditions.