Mechanical and Thermo-physical Properties of Rare-Earth Materials

Abstract

The ultrasonic nondestructive technique (US-NDT) is the mostly preferred technique for material characterization. It has gained much attention in the area of materials science as there is no impairment of properties and does not affect the future usefulness of the sample under testing. The ultrasonic techniques are the versatile tools to investigate the material properties such as microstructure, elastic modulus, and grain size and the mechanical properties, velocity, attenuation, etc. This field has evolved mainly in the last few decades and is involved in developing new materials and modifying available materials by gaining a better understanding of characteristics under different physical conditions. These materials are thus referred to as condensed materials which have proven their potentials in many applications. Out of emerging materials, the exceptional structural, elastic, electrical, magnetic, phonon, and thermal characteristics of rare-earth compounds have attracted attention, as they are of technological significance. These materials crystallize into a basic NaCl structure, making them interesting samples for experimental and theoretical study. One of the most important reasons for the theoretical investigation of rare-earth compounds was the existence of a single crystal of chosen materials. After it was demonstrated that some rare-earth compounds can be grown epitaxially on III-V semiconductors, the interest in these materials has expanded even more recently. It has paved the path for their use in the manufacturing industry, infrared detectors, optoelectronic devices, spintronics, and several research domains.

Appendices

Appendix A

Table A1 Non-linearity parameters along different crystallographic directions

Direction	<100>	<110>	<111>
K2	C11	$\frac{1}{2}\left({C}_{11}+{C}_{12}+2{C}_{44}\right)$	$\frac{1}{3}\left({C}_{11}+2{C}_{12}+4{C}_{44}\right)$
K3	C111	$\frac{1}{4}\left({C}_{111}+3{C}_{112}+12{C}_{166}\right)$	$\frac{1}{9}\left({C}_{111}+6{C}_{112}+12{C}_{144}+24{C}_{166}+2{C}_{123}+16{C}_{456}\right)$

Appendix B

Table B1 Equations for Grüneisen numbers along <100> for longitudinal waves

Type of wave	No. of modes	Equations for <${\upgamma}_i^j$ >
Longitudinal	1	$\frac{3{C}_{11}+{C}_{111}}{2{C}_{11}}$
Shear	2	$\frac{C_{11}+{C}_{166}}{2{C}_{44}}$
Longitudinal	2	$\frac{C_{11}+{C}_{112}}{2{C}_{11}}$
Shear	2	$\frac{2{C}_{44}+{C}_{12}+{C}_{166}}{2{C}_{44}}$
Shear	2	$\frac{C_{12}+{C}_{144}}{2{C}_{44}}$
Longitudinal	2	$\frac{2{C}_{12}+{C}_{112}+2{C}_{144}+{C}_{123}}{2\left({C}_{11}+{C}_{12}+2{C}_{44}\right)}$
Shear	2	$\frac{2{C}_{12}+{C}_{112}-{C}_{123}}{2\left({C}_{11}-{C}_{12}\right)}$
Shear	2	$\frac{C_{12}+2{C}_{44}+{C}_{166}}{2{C}_{44}}$
Longitudinal	4	$\frac{2\left({C}_{11}+{C}_{12}+{C}_{44}\right)+{C}_{111}/2+3{C}_{112}/2+2{C}_{166}}{2\left({C}_{11}+{C}_{12}+2{C}_{44}\right)}$
Shear	4	$\frac{2{C}_{11}+\left({C}_{111}-{C}_{112}\right)/2}{2\left({C}_{11}-{C}_{12}\right)}$
Shear	4	$\frac{C_{11}+{C}_{12}+{C}_{144}+{C}_{166}}{4{C}_{44}}$
Longitudinal	4	$\frac{5{C}_{11}+10{C}_{12}+8{C}_{44}+{C}_{111}+6{C}_{112}+2{C}_{123}+4{C}_{144}+8{C}_{166}}{6\left({C}_{11}+2{C}_{12}+4{C}_{44}\right)}$
Shear	4	$\frac{4{C}_{11}+5{C}_{12}+{C}_{44}+\left({C}_{111}+3{C}_{112}\right)/2+\left({C}_{144}+5{C}_{166}\right)/2-2{C}_{123}}{6\left({C}_{11}-{C}_{12}+{C}_{44}\right)}$
Shear	4	$\frac{2{C}_{11}+{C}_{12}+{C}_{44}+\left({C}_{111}-{C}_{112}\right)/2+\left({C}_{144}+{C}_{166}\right)/2}{2\left({C}_{11}-{C}_{12}+{C}_{44}\right)}$