**3.1.12. Thermal analysis**

AsO4

3− and VO4

environment. Bradger's equation [128],[129],

relationship may be expressed approximately by the equation:

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

of vibrating atoms, all other terms being invariant.

where *X* = P5+, As5+ of V5+ [128].

3−) is primarily a function of the ionic radium of X atom. Since Pb, in this case,

<sup>0</sup> 1.86 10 / *ij k* =× - *R d* (2)

(3)

3− ions,

is always the dominant externally coordinated cation, for various members, there is no significant change in the interaction between the molecular vibration and the external

( )<sup>3</sup> <sup>5</sup>

although specifically applicable to internuclear distances in diatomic molecules, reflects generally the inverse relationship between the force constant *k*0 and the internuclear dis‐ tance *R*. Symbol *d*ij denotes the constant the values of which depend on the nature of bond‐ ed atoms. The molecular vibration frequency *v* is dependent on the restoring forces, measured in terms of *k*0, between participating atoms as well as on the masses of these atoms. The

where the vibration frequency *v* is a function of the force constant *k* and the reduced mass *u*

The spectral frequency differences between pyromorphite, mimetite and vanadinite are explicable and to a considerable degree predictable in terms of these parameters. On com‐ plete substitution of As or V for P the effect of reduced force constants is reinforced by increases in mass, thereby shifting *ν*3 and *ν*1 to lower frequencies. Because of opposing mass and forceconstant effects and perhaps also because of dissimilarities in orbital configurations, the relative positions of absorption bands are less predictable for mimetite and vanadinite than for pyromorphite and mimetite. The theoretical frequency trends are depicted in **Fig. 10** [128].

**Fig. 10.** Theoretical effect of change in mass and ionic radius on infrared vibration frequency of tetrahedral XO4

1 2 *<sup>k</sup> v c <sup>u</sup>* <sup>=</sup> p

Thermal analysis (TA) refers to a group of techniques10 in which the property of a sample is monitored against time or temperature while the temperature of the sample, in a specified atmosphere11 is programmed. These methods study the relationship between sample proper‐ ty and its temperature as the sample is heated or cooled in a controlled manner. The individ‐ ual techniques are divided according to the measured property12 [131],[132], [130] as is introduced in **Table** 1.


10 The definition of terms in thermal analysis was developed by ICTAC (Confederation for Thermal Analysis and Calorimetry).

11 Gaseous environment of the sample, which may be controlled by the instrumentation or generated by the sample [130].

<sup>12</sup> Resulting dependence, i.e. any graph of any combination of property vs. time or temperature derived from a thermal analysis technique, should be termed as thermal curve, which is a simplified form of more correct term thermoanalytical curve. The first mathematical derivation of any curve with respect to temperature or time leads to the derivative thermoanalytical curve [130]. Since the name thermogram has medical usage, the thermal analysis curve should not be termed as thermogram [131].


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

The measurement should be performed as follows:


The sample-controlled method where the feedback used to control the heating is the rate of transformation is termed as controlled-rate thermal analysis (CRTA) [130].

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