Behaviour of ultrasonic velocities and elastic constants in Ag-Zn alloys

The ultrasonic properties of the hexagonal closed packed structured Ag-Zn alloys have been studied at room temperature for their characterization. For the investigations of ultrasonic properties, I have also computed second order elastic constants using Lennard-Jones Potential. The velocities V1 and V2 have minima and maxima respectively with 45 0 with unique axis of the crystal, while V3 increases with the angle from unique axis. The inconsistent behaviour of angle dependent velocities is associated to the action of second order elastic constants. Debye average sound velocities of these alloys are increasing with the angle and has maximum at 55 0 with unique axis at room temperature. Hence when a sound wave travels at 55 0 with unique axis of these alloys, then the average sound velocity is found to be maximum. Ag0.2Zn0.8 alloy is more suitable for industrial and other uses, as it having highest elastic constants and lowest attenuation in comparison to other Ag-Zn samples. The mechanical and ultrasonic properties of these alloys will not be better than pure Ag and Zn due to their low SOEC and high ultrasonic attenuation. Achieved results have been discussed and compared with available experimental and theoretical results. Copyright © 2011 VBRI press.


Introduction
Ultrasonic offer the possibility to detect and characterize microstructral properties as well as flaws in materials, controlling materials behaviour based on physical mechanism to predict future performance of the materials. Various investigators have shown considerable interest on ultrasonic properties of metals and alloys [1][2][3][4]. Wave propagation velocity is key parameter in ultrasonic characterization and can provide information about crystallographic texture. The ultrasonic velocity is directly related to the elastic constants by the relationship V=(C/), where C is the relevant elastic constants and  is the density of that particular material. The elastic constants of a solid provide valuable insight into nature of atomic bonding forces and also related hardness [5,6].
The crystal structure gives the key to many properties of solid materials. Unusual structural properties that can be subtly tuned by chemical composition or external conditions are of great interest. Most elemental metals crystallizing in the hexagonal closed packed (hcp) lattice have an axial ratio c/a that is close to the "ideal" value 1.633 obtained in a stacking of rigid spheres [7].
Yoshie Matsuo [8] has been experimentally investigated elastic behaviour and phase stability in the hcp structure of Ag-Zn alloys using an ultrasonic pulseecho overlapping method. An X-ray diffraction pattern obtained from the pink color film showed CsCl-type β phase (bcc) structure, and the pattern from silver color film showed ζ phase (hcp) structure of Ag-Zn alloys films [9].
Zener alloys (Ag-Zn alloys), which show Zener relaxation, have a great interest for scientists since last four decades [10][11][12].Low temperature internal friction and sound velocity in Zener alloys have been measured by Raychaudhuri and Pohl [13]. Magyari-Kope et al. [7] investigated lattice parameter and elastic constants in Zener alloys using ab-initio calculation. Interdiffusion coefficients in Ag-Zn  phase in the temperature range between 823 K and 1023 K have been determined by Suer and Freise analysis using solid diffusion couples consisting of pure silver and a 20 at % Zn-Ag alloys [14].
There are three types of acoustic mode of lattice vibrations: one longitudinal acoustical and two transverse

Second order elastic constants
The elastic energy density (U) is function of the strain components. U = F (e xx ; e yy ; e zz ; e yz ; e zx ; e xy ) = F (e 1 ; e 2 ; e 3 ; e 4 ; e 5 ; e 6 ) ---------- (1) where e ij (i or j = x, y, z) is component of strain tensor. The second (C IJ ) order elastic constants of material are defined by following expressions.
The elastic energy density is well related to interaction potential φ(r) between atoms. Let the interaction potential be the Lennard-Jones Potential or many body interaction potential, which is formulated as: where a 0 , b 0 are constants and m, n are integers. The definition of higher order elastic constants (Eqn.2) with this potential (Eqn.3) under equilibrium and symmetric condition leads six second order elastic constants (SOEC) for the hexagonal closed packed structured materials [15,16]  ---------- (4) where p = c/a: axial ratio; The rest second order elastic constants have zero value because under 180° rotation they have equal and opposite value for the same stress. The harmonic and anharmonic parameters (χ and ψ) can be calculated using one experimental SOEC [15,16]. In the present study, I have expanded the theory for theoretical evaluation of parameters χ and ψ. The potential energy can be expanded in the powers of changes in the squares of distances. The expansion up to cubic term can be written as: According to the Eqn. (5), χ and ψ can be written as: In solving Eqns. (6) and (7) for hexagonal closed packed structured materials: The parameters χ and ψ can be calculated using Eqs. (8)- (9) with appropriate values of m, n and b 0 so that the calculated values of elastic constants justify the experimental data.

Ultrasonic velocity
The anisotropic behaviour of the material can be understood with the knowledge of ultrasonic velocity because the velocity is related to the second order elastic constants [16]. On the basis of mode of atomic vibration, there are three types of velocities (longitudinal, quasi shear and shear) in acoustical region [17]. These velocities vary with the direction of propagation of wave from the unique axis of hexagonal structured crystal [18,19]. The ultrasonic velocities as a function of angle between direction of propagation and unique axis for hexagonal structured materials are [20]: where V 1 , V 2 and V 3 are longitudinal, quasi shear and pure shear wave ultrasonic velocities. Variables  and  represent the density of the material and angle with the unique axis of the crystal respectively. The Debye temperature (T D ) is an important physical parameter for the characterization of materials, which is well related to the Debye average velocity (V D ).
where  is quantum of action and is equal to Planck"s constant divided by  2 ; k B is Boltzmann Constant; n a is atom concentration.
The above formulae have been used for the evaluation of ultrasonic velocity and related parameters for the selected materials. All the relevant mathematical formulations are given as: Appendix-1.

Results and discussion
The unit cell parameters "a", axial ratio "p" and density "" of the chosen alloys are presented in Table 1 [21]. The value of m and n for these alloys are 6 and 7. The value of b 0 is 1.0 × 10 -67 erg cm 7 for all alloys. The second order elastic constants and bulk modulus of these alloys are presented in Table 2. Table1. Lattice parameter "a" (in Å), "p" and density "" (in 10 2 gm/cm 3   The elastic constants of the material are important, since they are related to hardness parameter. Also, the second order elastic constants are used for the determination of the ultrasonic parameters. The calculated values of C 12 , C 13 are few different than some other experimental [8] and theoretical [21] results for alloys: 8. Actually Magyari-Kope et al. [21] have based ab-initio exact muffin-tin orbitals total energy theoretical approached to evaluate elastic constants, which is quite different from present approach. Also Y. Matsuo [8] was experimentally investigated elastic behaviour and phase stability in the hcp structure of Ag-Zn alloys using an ultrasonic pulse-echo overlapping method. Although obtained order of SOEC are of the same as others hexagonal structured materials [5,6,12]. Relative magnitude of C 11 , C 33 , C 44 are well presented by our theoretical approach. The bulk modulus (B) for these alloys can be calculated with the formula B= 2(C 11 + C 12 + 2C 13 + C 33 /2)/9. The evaluated B for these alloys is presented in Table 2. It is obvious from Table 2 that, there is good agreement between the present and reported theoretical/experimental second order elastic constants of these alloys [8,21]. Thus our theoretical approach for the calculation of second order elastic constants for hexagonal structured alloys at room temperature is well justified. Hence applied theory for the evaluation of higher order elastic constants at room temperature is justified.
It can be also seen from Table 2, that the SOEC are found to be increasing from alloys: 1-9. Also the SOEC of these alloys are less than pure Zn metals [22] and pure Ag metals [23]. This increasing nature of SOEC shows that the alloys are more stable, ductile with more Zn amount.
The compute oriented dependent ultrasonic velocities of these alloys are visualized in Figs. 1-4. Figs. 1-3 clear that velocities V 1 and V 2 have minima and maxima respectively at 45° with the unique axis of the crystal, while V 3 increases with the angle from the unique axis. Indistinct activity of angle dependent velocities is correlated with activity of SOEC. The tendency of angle dependent velocity curves (Figs. 1-4) Lett. 2011, 2(2), 157-162 Copyright © 2011 VBRI press. structured materials GaN, AlN, InN, CdS, CdSe and laves-phase compounds [16, 18-20, 24, 25]. Hence it is conformity to the hcp structured of chosen alloys.   Figs. 1-4 depict that the magnitudes of ultrasonic velocities increase from alloy: 1 to alloy: 9 as boost up of zinc amount. Hence as we increase amount of zinc, the quality of the material is also increased. The longitudinal ultrasonic velocity along <001> direction (θ=0° with unique axis) is largest for alloy: 9, because alloy: 9 have highest zinc amount as well as highest C 33 value. As we know that shear wave is also known surface wave. Therefore the ultrasonic velocities V 2 and V 3 are the surface wave velocity. It is clear from the values of V 2 and V 3 for ultrasonic wave propagation along θ=0° are the same, while these are different for other directions as visualized in Figs. 2-3. Hence <001> direction is the direction of symmetry for chosen alloys. Debye average velocities (V D ) of these alloys are increasing with the angle and have maxima at 55° at 300 K (Fig. 4). Since V D is calculated using V 1 , V 2 and V 3 with Eqn. (14), therefore the temperature variation of V D follows the combined effect of temperature variation of V 1 , V 2 and V 3 . The maximum in V D at 55° is due to a significant increase in longitudinal and pure shear wave velocities and a decrease in quasi-shear wave velocity. It can be seen that from Figs. 1-4 alloy: 9 has maximum velocity and pure Ag has least velocity for all angles of the crystals. Since ultrasonic attenuation A 3 V   and velocity is the largest for alloy: 9 among others thus the attenuation A should be smallest and material should be most ductile. The minimum ultrasonic attenuation for alloy: 9 justify its quite stable hcp structure state. Also alloy: 9 has maximum elastic constants and bulk modulus among others. Hence alloy: 9 (Ag 0.2 Zn 0.8 ) is more ductile, stable and contain few defects in the crystal structure in comparison to other alloys.
In the present investigation the expressions in above Table is used for calculating the ultrasonic velocity for longitudinal and shear wave in hexagonal structured materials. An important application of velocity measurements for both longitudinal and shear waves is the determination of the elastic constants along the various axes of single crystals over very wide ranges of temperature. The determination of anisotropy of ultrasonic velocity and, hence elastic constants can yield useful information about other physical properties of the crystal. The degree of anisotropy of the ultrasonic velocities in a single crystal of such a material should agree with that for the lattice vibrations components of thermal conductivity.

Conclusion
Based on the above discussion is worthwhile to state that: i. Present method to evaluate second order elastic constants involving many body interaction potential for hexagonal wurtzite crystal structured materials (alloys) is correct. ii. All elastic constants and density are mainly the affecting factor for anomalous behaviour of ultrasonic velocity in these alloys. iii. When a sound wave travels at 55° with the unique axis of these crystals then the average sound wave velocity is maximum. Since the Debye average velocity is calculated using the constituent velocities V 1 , V 2 and V 3 , hence a good resemblance in V D implies that our calculated velocities are correct. iv. The <001> direction is the direction of symmetry for these alloys as they have the same quasi-shear and pure shear wave velocities. v. Ag 0.2 Zn 0.8 alloy is more suitable for industrial and other uses, as it having highest elastic constants as well as wave velocity and lowest attenuation in comparison to other chosen alloys. vi. The mechanical and ultrasonic properties of these alloys will not be better than pure Ag and Zn due to their low SOEC and high ultrasonic attenuation. vii. The preliminary results obtained in this work can be used for further experimental investigation with pulse echo overlap (PEO) technique for ultrasonic measurements and with conventional analytic techniques such as polarizing microscopy, X-ray diffraction (XRD), surface tension, solid state nuclear magnetic resonance (NMR), scanning electron microscopy (SEM) and transmission electron microscopy (TEM).