Age and Charge Effects in Cortical Bone: Raman Study of Phosphate Ions Stretching

1.1. General Remarks

Increasing attention has been given to biomineralization mechanisms [1-3] because of their important role in the biosphere, including the lithosphere, hydrosphere, and atmosphere [4-6], as well as in living organisms under physiological and pathological conditions and in the development of biomimetic analogues [7-9]. Current microscopic understanding of biomineralization is primarily based on mineralogical and biomedical studies and relies mainly on X-ray diffraction and electron microscopy investigations (see, e.g., [10,11]). They show that the hard biological tissues form unique “organism-material” complexes [12-14], in which the dominant role is played by hexagonal calcium carbonate hydroxyapatite (HAP, Ca_10-x(CO₃) _x(PO₄)_6-x(OH)_2-x, 0 ≤ x ≤ 2) with substantially variable stoichiometry that reflects the non-stationary conditions of their formation inside the voids of collagen fibers [15-19].

Variations in stoichiometry are governed by vacancies at Ca²⁺ and OH⁻ sites, as well as isomorphic substitutions at different crystallographic positions of HAP, with concentrations depending on the characteristics of the biogenic environment. Different authors have reported varying interpretations of these stoichiometric variations [13-15, 18-23]. In particular, a recent study on age-related changes in bone [17] showed that deviations in stoichiometry can act as a source of charge heterogeneity within the mineral matrix. The interplay of size, stoichiometry, and charge effects may lead to novel structure–function relationships with distinctive mechanical and electrophysical properties, thereby supporting further experimental and theoretical investigations into the atomic structure and dynamics of native bone.

1.2. The Bone-Battery Concept

A nanoscopic view of mineralized bone assembled from HAP nanocrystals (HAPNs) is presented in Figure 1. The HAPNs within a coplanar conglomerate are illustrated in the figure as shaded boxes separated by hydrated nanolayers. Because long-range order in HAPNs is disrupted, the electroneutrality condition that applies to individual unit cells in macro- and microcrystals may no longer hold for these nanostructures. Thus, owing to their small size and disrupted long-range order, these structures may behave as charged entities. To compensate for the charge distribution Q(r), the electroneutrality condition in bone can be reformulated in a more general form [17].

The integration region in Equation (1) is the supercell S, which is marked with white lines in Figure 1.

The supercell S includes both a HAPN and its adjacent hydrated nanolayers. Equation (1) suggests that the excess charge in a HAPN is compensated by opposite charges accumulated in the surrounding hydrated nanolayers. Charge distributions in mineralized bone are schematically illustrated in Figure 2.

Comparative analysis of temporal variations in electron binding energies, lattice constants, and calcium deficiency has predicted the accumulation of excess negative charge in HAPNs [17]. The built-in charge, Q, is reported to reach its maximum in newborn bone and decrease with age. The relationship

links the built-in charge Q with the spectral shift ${∆E}_{core}$ of core-electron BE and enables forecasting of charge alterations. The charge Q was estimated to be approximately 8 ± 3 e⁻ in young bone and 3 ± 2 e⁻ in mature bone [17]. In Equation (2) $<r^{\prime }>$ and $<r^{″}>$ represent the average distances to vacant Ca²⁺ positions within a nanocrystal and to Ca²⁺ ions in the adjacent nanolayers, respectively, whereas, ε′ and ε″ denote the dielectric constants of bioapatite and the intercrystallite environment.

The nanoscale heterogeneity of charge distribution supports the interpretation of bone mineral as a battery-like system composed of many supercells, each formed by a negatively charged HAPN embedded within a positively charged hydrated medium.

Using the extracted charges and HAPN dimensions [17], the density of electrostatic energy associated with charge inhomogeneity was estimated to be P > 10⁶ J/m³ for young bone and P < 10⁵ J/m³ for mature bone. This estimation indicates the substantial decrease (more than 10 times) of the built-in energy ℰ with age. These findings suggest that the effects of electrostatic fields in bone may have been substantially underestimated. Such fields may polarize atomic and molecular fragments, alter ionic dynamics, and potentially link aging processes with discharge-like phenomena in bone mineral.

1.3. Study Objectives

The “bone-battery” (BB) concept may open new perspectives for age- and patient-specific targeted drug delivery, the development of novel energy-related biomaterials, and advanced biomedical nanotechnologies aimed at promoting healthy aging.

The present study focuses on electrophysical phenomena associated with electrostatic fields induced by built-in charges and their impact on atomic dynamics in bone tissue. The primary objectives of this study are to characterize the strength and spatial distribution of electrostatic fields, quantitatively evaluate these fields, and clarify their role in aging-related phenomena.

Since the electrostatic field gradient is expected to be particularly strong at the interfaces between neighboring nanocrystals (see Figure 2), HAPN-associated atomic and molecular constituents can be approximately classified into two groups. The first group (A) comprises atomic and molecular constituents within the bulk, whereas the second group (B) includes atoms and molecular fragments at the interfaces, where electrostatic forces are expected to induce stronger distortions than in the bulk.

The nanocrystal-hydrated nanolayer interface is generally regarded as a more disrupted region [21]. However, electrostatic fields may promote dipole ordering of [PO₄]³⁻, [CO₃]²⁻, and [OH]⁻ molecular ions, with a particularly pronounced effect in young bone. To test this hypothesis, the present analysis focuses on Raman scattering associated with P–O stretching vibrations. This choice is motivated by the presence of a narrow, intense v₁[PO₄]³⁻ band at 960 cm⁻¹ in Raman spectra, which is widely accepted as a key indicator of bone mineral content (see, e.g., [24-26]).

Considering the partitioning of PO₄³⁻ ions into bulk and interfacial populations, the v₁(PO₄³⁻) Raman band in native bone may be expected to split into two components. One sub-band (A), associated with the bulk population, is expected to resemble the v₁(PO₄³⁻) band in HAP, whereas the second sub-band (B), associated with interfacial regions, is expected to exhibit a red shift and noticeable distortion due to electrostatic field effects.

Against this background, the main objectives are (i) to detect and characterize the splitting of the v₁[PO₄]³⁻ Raman band in young and mature bone and (ii) to compare the observed spectral behavior with that predicted by the bone-battery (BB) concept. Section 2 describes the methods used for sample preparation and Raman band analysis. Section 3 presents the evaluation of electrostatic fields in bone and reports the results of the experimental spectral analysis. Section 4 discusses Raman spectroscopic evidence supporting a relationship between built-in charges and age-related changes in bone tissue, consistent with predictions based on X-ray diffraction and X-ray photoelectron spectroscopy (XPS) [17,27].

2.1. Samples

Bone samples were prepared ex situ from the cortical layers of the mid-diaphysis of the femur, tibia, and humerus of white mongrel male rats, grouped as young (40–60 g; n = 8) and mature (250–280 g; n = 6). All animal-related procedures were conducted in accordance with applicable ethical requirements and reported following the ARRIVE guidelines. The study complied with applicable national regulations and international standards, including GOST R ISO 10993-2-2009 and relevant ISO 10993 standards related to animal welfare requirements. The study also followed GOST 33215-2014 and GOST 33216-2014, which define requirements for the maintenance and care of laboratory rodents and rabbits. The study protocol (10-10-2015) was approved by the Local Ethics Committee of the R.R. Vreden National Medical Research Centre of Traumatology and Orthopedics. All animals were maintained under standardized conditions in the vivarium of the National Medical Research Center of Traumatology and Orthopedics.

The cortical bone was thoroughly cleaned of soft tissue, rinsed with saline, blotted dry, and ground into a fine powder with particle sizes of approximately 1 mm. This sample preparation was not expected to affect the measured spectra, and no thermal treatment was applied to the bone samples. To compare bone mineral with reference hydroxyapatite, near-stoichiometric HAP crystals synthesized as described in [22,28] were used as reference compounds.

The structural parameters of the samples, determined by X-ray diffraction (XRD) [17,27], are summarized in Table 1. Table 1 presents the HAP lattice constants a = b and c, mean linear size <L> of HAPNs, crystallinity degree D, mean nanolayer thickness <d>, effective number of atoms N per HAPN, charge Q in young and mature bone, and the corresponding errors.

2.2. Raman Probing

Raman spectroscopy (see, e.g., [26]) enables phase identification in complex materials and detects subtle variations in light scattering, thereby providing insight into nanoscale effects. This method is a powerful tool for probing and mapping nanophases dispersed within matrices (see, e.g., [29,30]), surface nanophases [31], charge transport processes [32,33], film orientation [34], cluster size [35], configurational order [36], interfacial reactions [37], and mechanical loading effects [38]. This spectroscopic approach was used to investigate site-dependent coupling between P–O stretching vibrations and dipole polarization of [PO₄]³⁻ ions in young and mature bone.

Static disorder and dynamic dephasing represent two limiting mechanisms that give rise to Gaussian and Lorentzian broadening of Raman signals, respectively (see, e.g., [39]). The present study focuses on specific changes in Raman band shape. First, a two-component v₁(PO₄³⁻) band structure was expected, enabling assessment of the differential effects of electric fields on bulk and interfacial components. Second, the agreement between the observed spectral behavior and predictions from the bone-battery model was evaluated. It was hypothesized that [PO₄]³⁻ ions in the interfacial region undergo static disorder as well as polarization along the electric field lines. Accordingly, the Lorentzian character of sub-band B is expected to become more pronounced with increasing coherence of P–O stretching vibrations in the corresponding regions.

This expectation supported the use of Voigt functions to approximate the experimental bands. This approach allows direct extraction of the Gaussian and Lorentzian widths, W_G and W_L, which were considered markers of static disorder and vibrational coherence of the PO₄³⁻ units contributing to sub-bands A and B.

The fitting analysis showed that two Voigt functions provided an optimal model for the v₁(PO₄³⁻) band in the bone specimens. To characterize the band shapes in young and mature bone, two Voigt functions are employed, with peak positions and Gaussian and Lorentzian widths treated as free parameters determined through best-fit optimization of the experimental data. This approach enabled a quantitative assessment of the applicability of the BB concept by comparing the extracted spectral widths with model-based predictions. In addition to spectral widths, variations in relative intensities and Raman shifts of the sub-bands in young and mature bone were analyzed to evaluate the proposed concept.

3.1. Experimental

Raman measurements of bone samples and the reference hydroxyapatite crystal were performed over a wide spectral range of 300–4000 cm⁻¹ and, in greater detail, within the narrow interval of 920–990 cm⁻¹. Measurements were carried out using both a Horiba Jobin-Yvon LabRAM HR800 Raman spectrometer (Science Park of Saint Petersburg State University; within the frameworks of projects 101-22060, 112-22072, 113-22068, 119-13853) and an Olympus BX41 microscope (40× objective) with a 532 nm excitation laser [40] (Prof. E. Rühl’s laboratory, Freie University Berlin, within the framework of GRISC projects P-2017b-1 and P-2018b-12). Repeated Raman measurements of bone samples demonstrated good reproducibility. No sample charging or decomposition effects were observed.

The measured spectra of young and mature bone, together with the HAP reference, within the narrow spectral interval are shown in Figure 3. The smooth background, determined from wide-range Raman measurements, has been subtracted in the figure. The intense v₁(PO₄³⁻) band in bone was substantially broadened and asymmetric compared with that in the HAP reference crystal. This broadening is expected because the crystalline structure of bone mineral is more disordered than that of HAP. However, the observed band asymmetry and its underlying origin require further investigation.

3.2. Assessment of Electrostatic Effects

To evaluate electrostatic fields, the concept of double electrical layers (DELs) was employed. The DEL effect in bone was previously reported by Wilson et al. [41]. Based on Nuclear Magnetic Resonance (NMR) spectral analysis, this effect was attributed to [PO₄]³⁻ ions at the surfaces of HAPNs and their charge compensation by Ca²⁺ ions accumulated within hydrated nanolayers.

Within the BB concept, an alternative origin of DELs is proposed, where negatively charged layers arise mainly from the built-in charge Q rather than from broken phosphate bonds at the surface, Q. Considering the high mobility of hydroxide anions in HAP, it may be assumed that they are displaced from the bulk by intrinsic electrostatic repulsive forces and subsequently accumulate at the surfaces of HAPNs. Thus, the charge Q_surf attributed mainly to excess [OH]⁻ rather than to [PO₄]³⁻ groups. It is assumed that Q_surf ≥ Q, and the contribution of phosphate ions is considered negligible due to surface relaxation effects in HAPNs.

In the interfacial region between two HAPNs, the electrostatic field can be approximately described as a superposition of two mirror-symmetric DELs. Figure 4 illustrates this model, where the left and right DELs correspond to the opposite ends of two neighboring HAPNs. Black and white triangles, as well as white and green hexagons, refer respectively to hydroxides, hydroxide vacancies, Ca²⁺ vacancies, and Ca²⁺(H₂O)_n clusters.

Next, the DELs are described as flat capacitors, (−||+) and (+||−), with negative plates formed by [OH]⁻ anions at the surface and positive plates formed by Ca²⁺ cations compensating the total charge. This simple model makes it clear that hydrated nanolayers in bone represent potential wells $\phi (x)$ .

Assuming L >> ${\displaystyle \frac{1}{2}}$ d, uniform distribution of Q_surf, linear dependence of electric field strength F in the capacitors, and neglecting edge effects in them, we derive that $\phi (x)$ is a V-shaped potential well. In the interval − ${\displaystyle \frac{1}{2}}$ d ≤ x ≤ + ${\displaystyle \frac{1}{2}}$ d it is

where, $\Delta \phi$ is the potential difference:

between the negative and positive charged plates. Q_yz represents the part of Q_surf associated with the yz plate (see Figure 4), and the capacitance C_yz is

where S_yz is the surface area of the plate. The total capacity of a single HAPN is C = ${\sum }_{j,j\ne i}{C}_{ij}$ . The directions of F are marked in Figure 4.

Using the sizes and charges in Table 1, neglecting age variations of ε′ and ε″, considering that HAPNs in the bone samples are elongated along the c (||z) axis by about 5–6 times more than along the x and y axes and applying Equations (3) and (5), the potential difference and the electrostatic field strength are estimated as $\Delta {\phi }^{young}$ ≈ 140 ± 50 mV and F^young ≈ 230 ± 80 kV/mm for young bone, respectively. In mature bone, the characteristics are reduced by more than an order of magnitude, reaching values of Δφ^mature ≈ 10 mV and F^mature ≈ 25 kV/mm. Thus, it is inferred that, with aging, the potential well in the interfacial region becomes narrower and significantly shallower.

Figure 5 shows the potentials φ(r) for young and mature bone over a wider range. For comparison, it is assumed that φ(r) ≈ 0 in the inner region of the HAPN (see Figure 4 and Figure 5), because the Coulomb potentials created by Ca²⁺ and OH⁻ vacancies compensate each other due to the high mobility of hydroxide ions. Taking into account the charge distributions, the potential can be represented as:

Inside the hydrated nanolayers, φ(r) represents a V-shaped potential well of depth V_0. Potential barriers are formed at the surfaces of HAPNs, and inside the HAPNs the potential φ(r) decreases exponentially. Here, b (=d + L) is the mean size of the supercell S and α determines the rate of decrease of potential. The model potential in Figure 5 is shown on a meV scale. In reality, it is smoothed because of the finite spatial extent of the charge distribution and thermal atomic motion.

The assessment of electrostatic fields supports the predicted trends in atomic dynamics in bone, as described in Section 1.3. In addition to static disorder, the interfacial region is expected to exhibit dipole alignment along the direction of the electrostatic forces. As a result, sub-band B is expected to show a more pronounced Lorentzian character than sub-band A in the Raman spectra of bone.

3.3. Data Analysis

To test these predictions, the measured v₁(PO₄³⁻) bands in young and mature bone were analyzed in greater detail. The best-fit Voigt functions are shown as solid black lines in Figure 3. The extracted Raman shifts E_A and E_B, Lorentzian widths W_L, Gaussian widths W_G, full widths at half maximum of sub-bands A and B, and relative intensity ${\displaystyle \frac{B}{A+B}}$ are presented in Table 2. The spectroscopic parameters of the v₁[PO₄]³⁻-band in HAP are also shown. The χ² values are also presented in Table 2. Since the spectral positions and Gaussian and Lorentzian widths were not fixed a priori, the extracted parameters were treated as independent characteristics of the sub-bands under study.

The spectroscopic parameters in Table 2 make evident the resemblance of the position of the intense high-frequency sub-band A with the v₁[PO₄]³⁻-band in HAP crystal. The subband A exhibits a pronounced Gaussian profile ( $W_G^A$ >> $W_L^A$ ). As shown in Table 2, the Gaussian width $W_G^A$ decreases by more than a factor of two with age, which is qualitatively consistent with the increased degree of crystallinity in mature bone compared with young bone. As for the Lorentzian width $W_L^A$ varies weakly and is much smaller than $W_G^A$ . Thus, the similarity of the Raman shift of sub-band A to the v₁[PO₄³⁻] band in crystalline hydroxyapatite (HAP), together with its Gaussian profile, allows sub-band A to be assigned to [PO₄]³⁻ moieties in the bulk.

The sub-bands B in the spectra of both young and mature bone are shifted downward by 13.3 and 12.7 cm⁻¹ compared to the v₁[PO₄]³⁻-band in HAP. This pronounced shift indicates an association between sub-band B and interfacial PO₄³⁻ moieties.

Consistent with the proposed model, the extracted shapes of sub-bands A and B show clear differences. Sub-band B shows a predominantly Lorentzian character, with $W_G^B$ much smaller than $W_L^B$ , whereas sub-band A shows a predominantly Gaussian character in the Raman spectra of both young and mature bone. This difference may indicate strong static disorder in HAPNs, especially in young bone, as well as polarization of P–O dipoles in the contact region.

In addition to the distinctions in shape, Table 2 demonstrates (i) a minor but distinct downward displacement Δ_A of the sub-band A: E_A(mature) ≈ 960.7 cm⁻¹ < E_A(young) ≈ 961.3 cm⁻¹, and (ii) a decrease in the contribution of the sub-band B to the total signal with age:

4.1. Electrostatic Effects

The assessment of electrostatic fields generated by embedded charges within the BB concept indicates that ion polarization at interfaces and within hydrated nanolayers should not be neglected. The Raman analysis of symmetric P–O vibrations in young and mature bone is consistent with this assessment.

However, it should be noted that the estimates of Δφ and F were made using multiple approximations. Ca²⁺ vacancies in HAPNs were assumed to be uniformly distributed, and the dielectric constants ε′ and ε" were considered close to those of imperfect HAP crystals [42] and water [17], approximately 15 and 80, respectively. These values do not account for: (i) substantial distance-dependent variations in ε" near the HAPN surface [43], (ii) interface morphology, (iii) acceleration of molecular motion in external potentials [44,45], and (iv) the dependence of ε′ on specific imperfections in bioapatite. Besides, the condition L >> ${\displaystyle \raisebox{1ex}{$\mathrm{d}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$ assumed by Equations (4) and (5), requires further investigations as applied to nanoelements. The finite spatial extent of the charge distribution on the negative and positive plates was also ignored.

Since the contribution of PO₄³⁻ ions to Q_surf was not considered, both the potential difference and electrostatic field strength may be underestimated. Therefore, the actual electrostatic fields may be stronger than the values estimated in the present work.

The inferred polarization and ordering of PO₄³⁻ ions at the interface are notable because the prevailing view assumes significant structural disorder in this region (see, e.g., [21]). Although this does not imply restoration of crystalline order, it suggests that the interfacial region may not be fully comparable to amorphous calcium phosphate [46].

To clarify this feature, Raman signals from bone tissue can be compared with those from invertebrate exoskeletons. Raman spectra of the exoskeleton enable effective probing of CO₃²⁻ ions in different chemical environments [47] as CaCO₃ polymorphs form the mineral phase of invertebrates. The studies of the v₁[CO₃²⁻]-band shapes face similar difficulties in understanding their asymmetry compared to the band in the reference spectra of calcium carbonates. According to Wehrmeister et al. [47], the v₁[CO₃²⁻]-band is composed of two components: the high-frequency Lorentzian at 1088 cm⁻¹ and the low-frequency Gaussian at 1078 cm⁻¹. These components correspond, respectively, to nanocrystalline CaCO₃ and amorphous calcium carbonate phases. In particular, these compositions have been identified in the palm region of the claw of Callinectes sapidus and in the exoskeleton of Calappa granulata [48,49].

Examining the Raman v₁[CO₃²⁻] band in combination with Ca 2p NEXAFS spectroscopy and electron microscopy data, Katsikini and co-authors [48] confirmed the assignment of Lorentzian and Gaussian sub-bands to crystalline and amorphous CaCO₃ phases, respectively, and linked their intensities to site-dependent variations in exoskeleton crystallinity.

At first glance, the two-component structure of the v₁[PO₄³⁻] and v₁[CO₃²⁻] bands appear to indicate a similar origin. However, the different distributions of Gaussian and Lorentzian components suggest different underlying origins for these bands. To investigate this difference, it can be assumed that the v₁[PO₄³⁻] band may also have a mixed crystalline–amorphous origin. This assumption is supported by Raman studies of amorphous calcium phosphates [50,51], in which the v₁[PO₄³⁻] band is observed at 10–13 cm⁻¹ lower than in crystalline HAP, corresponding to the same Raman shift region as sub-band B.

To evaluate the proposed crystalline–amorphous origin of the v₁[PO₄³⁻] band asymmetry, Lorentzian and Gaussian profiles were assigned to the high- and low-frequency components, respectively. However, the resulting fit quality, with χ² ≈ 1817, was nearly eightfold poorer than that obtained for the fitting presented in Figure 3. This poorer fit does not necessarily exclude a contribution from mixed crystalline and amorphous phases to the v₁(PO₄³⁻) band in bone; however, it indicates that other dominant mechanisms may govern its spectral features. One possible interpretation is dipole polarization of PO₄³⁻ ions and their ordering within intrinsic electrostatic fields in bone. Accordingly, the results do not support the interpretation that the low-frequency shoulder arises mainly from amorphous mineral; rather, they are more consistent with ordered but distorted PO₄³⁻ groups at the interface under strong electric fields.

Despite the small magnitude of the displacement Δ_A reported in Section 3.3, its direction is important because it supports the proposed influence of electrostatic fields on atomic dynamics. This direction cannot be explained by adiabatic compression of PO₄³⁻ ions, because the HAP unit cell is less compressed in young bone than in mature bone [28]. In that case, an upward shift would be expected. Therefore, the observed behavior may instead be attributed to electrostatic interactions, particularly Coulomb repulsion between anions within HAPNs. As |Q(mature)| < |Q(young)|, the frequency ω of P-O stretching decreases with age, resulting in E_A(young) > E_A(mature). This behavior may serve as an indicator of age-related dissipation of charge Q.

The weakening of electrostatic fields and the growth of HAPNs with age are qualitatively consistent with inequality (7). However, the observed decrease in the relative intensity ${\displaystyle \frac{B}{A+B}}$ is smaller than expected based on the extracted changes in charge and size (see Table 2). This discrepancy may be associated with preservation of dipole orientation during the deposition of additional atomic layers as HAPNs grow, suggesting a possible memory effect in dipole orientation during bone aging.

Further confirmation of the memory effect can be obtained by analyzing the cumulative changes in the Lorentzian and Gaussian contributions to the v₁[PO₄³⁻]-band with age. The coefficient ${\displaystyle \frac{L}{G}}$ was introduced as follows:

This coefficient represents the relative Lorentzian-to-Gaussian contribution to the overall band shape. The ${\displaystyle \frac{L}{G}}$ coefficients computed for young bone, mature bone, and the HAP crystal using Equation (8) and experimental data are given in Table 2. The Lorentzian contribution is lowest in young bone; although static disorder decreases with age, the ${\displaystyle \frac{L}{G}}$ coefficient increases and, in mature bone, approaches the value characteristic of the HAP crystal. This result further confirms the applicability of the chosen method of v₁[PO₄]³⁻ bandshape analysis using Voigt functions.

4.2. Prospects

The assessment of electrostatic fields in mineralized bone suggests: (i) an important role of these fields at HAPN interfaces and (ii) a substantial reduction in their strength with age. It should be noted that the decrease is quite dramatic, because the expected potential difference Δφ for young bone noticeably exceeds the kinetic energy (k_BT) of the thermal motion of atoms, whereas in mature bone the difference Δ $\phi$ is virtually smoothed out by the thermal movements. It is also noted that Δ $\mathrm{\phi }$ in young bone exceeds the membrane potential (70–90 mV), but is substantially lower in mature bone.

The predicted variations in the potential φ(r) suggest the possibility of quantum effects such as electron trapping in potential wells and electron tunneling through potential barriers (see Figure 5) as charge carriers propagate through bone mineral. To quantify the valence band structure of bone, a three-dimensional superlattice model was previously proposed [52]. The model relates the valence band energy in bone to the electron-optical properties of the supercell S using the relation

This relation links the electron energy E to the wave vector k in the super periodic potential of a conglomerate, where T denotes the electron transmission amplitude through supercell S and bj represents the mean dimension of S in the j-direction. In solid-state physics, this relation is known as the Heine equation [53].

The three-dimensional superlattice model predicts the red shift Δ_n = E_n^HAP − E_n^bone ≈ 2E_n ${\displaystyle \frac{<\tilde{d}>}{<L>}}$ >0 of valence bands in bone mineral compared to those, E_n^HAP, in HAP crystal, and links the shift with super-long-order parameters of conglomerate and with peculiarities of electrons propagation through the conglomerate. $<\tilde{d}> \mathrm{and}$   $<L>$ are respectively the electron-optical length of hydrated nanolayers and the mean linear size of HAPN. As the potential φ(r) is found to vary substantially with age, an essential age dependence of the electron-optical length $<\tilde{d}>$ can be predicted. Therefore, the valence band energy E_n^bone in mineralized bone may be an age-dependent characteristic of its electronic structure.

X-ray dichroism observed in photoemission from cortical bone [54] also suggests that charge transfer and related quantum phenomena may depend on the orientation of HAPNs relative to the principal axes of bone. According to the Wolff paradigm [55], bone structural orientation is governed by locomotor function and mechanical loading against gravity. It may therefore be hypothesized that the quantization axes of bone nanostructures are related to the directions of gravitational forces within the skeleton. This suggests that studies of biomineralization mechanisms may need to consider quantum effects in relation to gravitational orientation. Accordingly, quantum phenomena may play a role in bone tissue, particularly in the propagation of molecular signals along its structure [56]. These phenomena warrant further detailed investigation and are therefore not discussed further here.

The built-in charge Q and the energy ℰ are determined not only by Ca²⁺ vacancies, calcium deficiency, and geometry of HAPN, but also by substitutions at all crystallographic positions, which affect the charge. Thus, for example, substitution of Ca²⁺ by Na⁺ and OH⁻ by O²⁻ increases Q, whereas carbonation decreases it. The effects of such substitutions, which may be influenced by environmental and nutritional factors, were not considered in the present study. Further research is required to evaluate their impact on the electrophysical properties of bone and to enable more rigorous quantitative validation of the BB concept.