**3.1.13. Optical properties**

The analysis of optical properties is essential for each mineral examined, and through the use of microscopy, the optical properties of individual minerals may be interpreted in great detail. Optical mineralogy investigates the interaction of light (usually is limited to visible light) with minerals and rocks. Optical mineralogy concerns mainly the use of polarizing (petrographic) microscope which has two Nicol prisms, polarizer and analyzer (polarizer below the stage and the analyzer above the objective) [133]. Human eye is the most sensitive for viewing a solid in the wavelength symmetrically spread in intensity around 550 nm (**Fig. 11**(**a**)). Optical microscopy in visible light (from 700 nm (red) to 420 nm (violet), **Fig. 11**(**b**,**c**)) helps study the objects of smaller sizes up to lower limit of ~ 1 μm (c). **Fig. 11**(**d**) shows the scope of spectro‐ scopic techniques associated with electromagnetic spectrum [134].

**Property Technique and abbreviation Notes**

128 Apatites and their Synthetic Analogues - Synthesis, Structure, Properties and Applications

thermoacoustimetry

Thermospectrometry

The measurement should be performed as follows:

**Table 1** Methods of thermal analysis according to measured property or physical quantity.

time. This can include thermal and nonthermal analytical techniques.

transformation is termed as controlled-rate thermal analysis (CRTA) [130].

scopic techniques associated with electromagnetic spectrum [134].

Optical properties Thermoptometry or thermos optical analysis

Acoustic properties Thermosonimetry or

at the same time.

**3.1.13. Optical properties**

Thermobarometry – Pressure exerted by dense sample on the walls of a

Thermoluminescence TL Light emitted by the sample is measured

Structure Thermodiffractometry – Techniques where the compositional or chemical

**1. Combined**: the application of two or more techniques to different samples at the same

**2. Simultaneous**: indicates the measurement of two or more properties of a single sample

The sample-controlled method where the feedback used to control the heating is the rate of

Simultaneous thermogravimetry (thermogravimetric analysis) and differential thermal analysis (TG-DTA) are mostly used to investigate the course of synthesis and the characteri‐ zation of prepared apatites or to investigate the process of thermal decomposition of apatites, i.e. the processes such as dehydroxylation (e.g. **Section 1.5.2**), defluorination (**Section 3.2.4** and

The analysis of optical properties is essential for each mineral examined, and through the use of microscopy, the optical properties of individual minerals may be interpreted in great detail. Optical mineralogy investigates the interaction of light (usually is limited to visible light) with minerals and rocks. Optical mineralogy concerns mainly the use of polarizing (petrographic) microscope which has two Nicol prisms, polarizer and analyzer (polarizer below the stage and the analyzer above the objective) [133]. Human eye is the most sensitive for viewing a solid in the wavelength symmetrically spread in intensity around 550 nm (**Fig. 11**(**a**)). Optical microscopy in visible light (from 700 nm (red) to 420 nm (violet), **Fig. 11**(**b**,**c**)) helps study the objects of smaller sizes up to lower limit of ~ 1 μm (c). **Fig. 11**(**d**) shows the scope of spectro‐

**8.6**), decarbonation (thermal decomposition of carbonate-apatites, **Section 4.6.1**), etc.

constant volume cell is studied

TS Emitted (sonimetry) or absorbed (acoustimetry)

nature of the sample is studied.

characteristics or property of the sample is studied

TOA A family of techniques in which optical

sound is measured.

**Fig. 11.** Electromagnetic spectrum of solar radiation and black body (a), the resolution of various techniques (b) and the regions associated with spectroscopic techniques (c) [134].

Refractive index (*n*) is related to the angle of incidence (*i*) and the angle of refraction (*r*) according to the Snell´s law:

$$n = \sin i / \sin r \tag{4}$$

The refractive index increases as the wavelength of light decreases [134]. The absorption coefficient is related to the imaginary part of the refractive index. It was found that since the refractive index of a medium depends on the density of electrons in that medium, the index increases with the density of matter [135].

The luster of mineral (R) depends of the way in which the light is reflected from the surface of a mineral. The reflection is again dependent on the refractive index. Normally, the greater is the index of refraction, the brighter is the luster. The luster is classified into two broad classes [134]:


Optical properties of apatites can be determined using the complex dielectric function [136], [137],[138]

$$
\varepsilon\left(\left.\alpha\right> = \varepsilon\_1\left(\left.\alpha\right>\right) + i\varepsilon\_2\left(\left.\alpha\right>\right) \tag{5}
$$

in the range of linear response. By calculating the wave function matrix and using Kramers– Krönig relations, the imaginary and real part of the dielectric function *ε*1(*ω*) and *ε*2(*ω*) can be derived respectively as follows:

$$\begin{split} \mathcal{L}\_{2}\left(\hbar o\right) &= \frac{e^{2}}{\pi m o \sigma^{2}} \int\_{\overline{\kappa}\mathbb{Z}} dk^{3} \sum\_{\kappa,l} \left| \left< \Psi\_{\kappa}\left(\overline{k}, \overline{r}\right) \right> - i\hbar \left| \left< \Psi\_{1}\left(\overline{k}, \overline{r}\right) \right> \right|^{2} f\_{1}\left(\overline{k}\right) \\ & \left[ 1 - f\_{\kappa}\left(\overline{k}\right) \right] \delta \left[ E\_{\kappa}\left(\overline{k}\right) - E\_{1}\left(\overline{k}\right) - \hbar o \right] \end{split} \tag{6}$$

where *f*(*k̅*) is the Fermi distribution function, *l* and *n* mark occupied state and unoccupied state, respectively. *Ψ*n(*k̅*,*r̅*) is the Bloch wave function for the *n*th band with the energy *E*n(*k̅*) at the Brillouin zone point *k*. The matrix element momentum transition corresponds to the term | *Ψn*(*k* ¯, *<sup>r</sup>*¯) <sup>|</sup>−*i*ℏ<sup>|</sup> *<sup>Ψ</sup>*<sup>1</sup> (*k* ¯, *r*¯) |2.

For each apatite, the real part *ε*<sup>1</sup> and imaginary part *ε*<sup>2</sup> of dielectric function have similar features with some subtle differences. The real part *ε*1 has two main peaks:


The *ε*2 curve in the energy range from 5 to 15 eV characterizes three main peaks:


The refractive index *n* can be obtained by *n* = √*ε*1(0). The *ε*1(0) for FAP, ClAP and BrAP are 1.38, 1.41 and 1.46, respectively. Therefore, the refractive indexes for FAP, ClAP and BrAP are 1.17, 1.19 and 1.21. Moreover, analogous dielectric function curves and similar refractive index values show that the optical property of each apatite has some independence from the *c*-axis ion [136].

#### **3.1.14. Measuring of surface area, porosity and pore size distribution**

There is a conventional mathematical idealization that asserts that a cube of edge length *a* possesses a surface area of 6*a*<sup>2</sup> and that a sphere of radius *r* exhibits 4**πr**<sup>2</sup> . In reality, however, mathematical perfect or ideal geometric forms are unattainable (**Fig. 12**) since all real surfa‐ ces exhibit flaws under microscopic examination. Real surface irregularities (voids, pores, steps, etc.) make the real surface area greater than corresponding theoretical area. When the

**Fig. 12.** Description of particle shapes [140].

ew

130 Apatites and their Synthetic Analogues - Synthesis, Structure, Properties and Applications

( ) ( ) ( )

é ùé - -- <sup>ù</sup> ë ûë <sup>û</sup>

*f k E k Ek*

(*k* ¯, *r*¯) |2.

*n n*

¯, *<sup>r</sup>*¯) <sup>|</sup>−*i*ℏ<sup>|</sup> *<sup>Ψ</sup>*<sup>1</sup>

tions from O–*2p* and P–*3p* levels.

d

p w

*BZ n l*

h h ò å

features with some subtle differences. The real part *ε*1 has two main peaks:

derived respectively as follows:

e w

1

the term | *Ψn*(*k*

tively.

ion [136].

possesses a surface area of 6*a*<sup>2</sup>

 e w e w

in the range of linear response. By calculating the wave function matrix and using Kramers– Krönig relations, the imaginary and real part of the dielectric function *ε*1(*ω*) and *ε*2(*ω*) can be

( ) ( ) ( ) ( )

= áY ñ - áY ñ

2 2 1 1 , 1

*n*

<sup>2</sup> <sup>2</sup> <sup>3</sup>

 w

where *f*(*k̅*) is the Fermi distribution function, *l* and *n* mark occupied state and unoccupied state, respectively. *Ψ*n(*k̅*,*r̅*) is the Bloch wave function for the *n*th band with the energy *E*n(*k̅*) at the Brillouin zone point *k*. The matrix element momentum transition corresponds to

For each apatite, the real part *ε*<sup>1</sup> and imaginary part *ε*<sup>2</sup> of dielectric function have similar

**1.** First peak lies near the energy of 7 eV. Obviously, the first peak is caused by the transi‐

**2.** The second peak is located at 8.51, 8.62 and 8.76 eV for FAP, ClAP and BrAP, respectively.

**3.** The third peak is situated at 10.99, 11.08 and 11.45 eV for FAP, ClAP and BrAP, respec‐

The refractive index *n* can be obtained by *n* = √*ε*1(0). The *ε*1(0) for FAP, ClAP and BrAP are 1.38, 1.41 and 1.46, respectively. Therefore, the refractive indexes for FAP, ClAP and BrAP are 1.17, 1.19 and 1.21. Moreover, analogous dielectric function curves and similar refractive index values show that the optical property of each apatite has some independence from the *c*-axis

There is a conventional mathematical idealization that asserts that a cube of edge length *a*

mathematical perfect or ideal geometric forms are unattainable (**Fig. 12**) since all real surfa‐ ces exhibit flaws under microscopic examination. Real surface irregularities (voids, pores, steps, etc.) make the real surface area greater than corresponding theoretical area. When the

and that a sphere of radius *r* exhibits 4**πr**<sup>2</sup>

**2.** The second peak lying at ~25 eV results from the transitions from Ca–*3p* levels.

**1.** The first peak lies in 7.10, 7.16 and 7.43 eV for FAP, ClAP and BrAP, respectively.

The *ε*2 curve in the energy range from 5 to 15 eV characterizes three main peaks:

**3.1.14. Measuring of surface area, porosity and pore size distribution**

h

*<sup>e</sup> dk k r i k r f k <sup>m</sup>*

, ,

( ) = + 1 2 ( ) *i* ( ) (5)

(6)

. In reality, however,

cube, real or imaginary, of one meter edge length is subdivided into smaller cubes each one micrometer (10−6 m) in length, there will be 1018 particles formed, each exposing an area of 6 × 10−12 m<sup>2</sup> . Thus, the total area of all particles is 6 × 106 m<sup>2</sup> . This operation increases by million‐ fold the exposed area of fine powder compared to undivided material. Whenever the matter is divided into smaller particles, new surfaces must be produced with corresponding in‐ crease in surface area [139],[140].

The particle size distribution (PSD) was usually determined by sieve analysis, sedimentation methods (gravitational or centrifugal), microscopic techniques, light scattering, multiangle laser light scattering (MALLS), etc. [141].

The range of specific surface area, i.e. area per gram of matter, can vary widely depending on the particles' size, shape and porosity. The influence of pores can often overwhelm the size and external shape factors. The powder consisting of spherical particles exhibits total surface (*S*t ) and volume (*V = M/ρ*) [139]:

$$S\_i = 4\pi \left( r\_1^2 N\_1 + r\_2^2 N\_2 + \dots + r\_n^2 N\_n \right) = 4\pi \sum\_{i=1} r\_i^2 N\_i \tag{7}$$

$$V = \frac{4}{3}\pi \left(r\_1^3 N\_1 + r\_2^3 N\_2 + \dots + r\_n^3 N\_n\right) = \frac{4}{3}\pi \sum\_{i=1}^3 r\_i^3 N\_i \tag{8}$$

$$S = \frac{S\_r}{M} = \frac{3\sum\_{i=1}^{N} N\_i r\_i^2}{\rho \sum\_{i=1}^{N} N\_i r\_i^3} \tag{9}$$

where *r*<sup>i</sup> and *N*<sup>i</sup> are the average radii and numbers of particles in the fraction *i*. For spheres of uniform size, **Eq. 9** becomes the law:

$$S = \frac{\mathfrak{J}}{\rho r} \tag{10}$$

Besides the calculation of specific surface from its geometry and PSD curve, the adsorption isotherm (predominantly Langmuir and BET13), air permeability methods,14 and monolayer sorption methods were used to determine the specific surface area [142],[143].

The porosity is defined as the ration of pore volume to total volume. Porous material is defined as solids containing pores **Fig. 13**(**a**), which are classified into two major types: open and closed pores. Penetrating open pores (interconnected pores) are permeable for fluid and therefore are important in applications such as filters. Pores accessible from only one end are referred to as dead-end pores. Noninterconnected (closed) pores are not accessible at all. The classification of pores according to their size is shown in **Fig. 13**(**b**). Pores can be also classified as the pores among agglomerates and pores among primary particles (**Fig. 13**(**c**)) [144],[145].

**Fig. 13.** Schematic illustration of different morphology of pores (a) and classification of porous materials based on pore size (b) and schematic illustration of pores among agglomerates and primary particles (c) [144].

<sup>13</sup>**BRUNAUER**–**EMMETT**–**TELLER** (BET) theory of multilayered physical adsorption of gas molecules on a solid surface [143].

<sup>14</sup> For example, the Blaine method, where fixed volume of air passes through the bed at steadily decreasing rate, which is controlled and measured by the movement of oil in a manometer, the time required being measured. The method is widely used for the determination of specific surface area of cements [143].

Pore size distribution and permeability15 are a very desirable quantities for the characteriza‐ tion of structure of porous solids, which can be determined by the following [145],[146],[147]:

**i.** Stationary fluid (static) method

<sup>3</sup> *<sup>S</sup>* r

132 Apatites and their Synthetic Analogues - Synthesis, Structure, Properties and Applications

sorption methods were used to determine the specific surface area [142],[143].

among agglomerates and pores among primary particles (**Fig. 13**(**c**)) [144],[145].

isotherm (predominantly Langmuir and BET13

Besides the calculation of specific surface from its geometry and PSD curve, the adsorption

The porosity is defined as the ration of pore volume to total volume. Porous material is defined as solids containing pores **Fig. 13**(**a**), which are classified into two major types: open and closed pores. Penetrating open pores (interconnected pores) are permeable for fluid and therefore are important in applications such as filters. Pores accessible from only one end are referred to as dead-end pores. Noninterconnected (closed) pores are not accessible at all. The classification of pores according to their size is shown in **Fig. 13**(**b**). Pores can be also classified as the pores

**Fig. 13.** Schematic illustration of different morphology of pores (a) and classification of porous materials based on pore

<sup>13</sup>**BRUNAUER**–**EMMETT**–**TELLER** (BET) theory of multilayered physical adsorption of gas molecules on a solid surface [143]. <sup>14</sup> For example, the Blaine method, where fixed volume of air passes through the bed at steadily decreasing rate, which is controlled and measured by the movement of oil in a manometer, the time required being measured. The method is

size (b) and schematic illustration of pores among agglomerates and primary particles (c) [144].

widely used for the determination of specific surface area of cements [143].

*<sup>r</sup>* <sup>=</sup> (10)

and monolayer

), air permeability methods,14


The surface area and the porosity of apatites have effect on the floatability (**Section 8.7**) of apatites of different type and origin [148]. The surface area, the porosity and the pore size distribution are properties of great importance for the preparation of biological apatites in tissue engineering (**Section 10.9**) and tailoring their mechanical properties, solubility and bioactivity [149],[150],[151],[152]. Ionic surfactants, such as decyltrimethylammonium bromide (CH3(CH2)9N(CH3)3(Br), C(10)TAB), hexadecyltrimethylammonium bromide (CH3(CH2)15N (Br)(CH3)3, C(16)TAB), as well as nonionic surfactant, can be used to control the pore (pore size and volume) and surface characteristics of mesoporous apatite materials under maintained pH [153]. The porosity also affects electrical properties of oxyapatites (**Chapter 5**) [154],[155],[156].
