Figure 2 shows the variation of lattice parameters a, c and the axial ratio c/a of the b-Sn matrix with valence electron concentration VEC. The variation in a and c occurs in such a way to keep v decreases by increasing VEC as shown in Figure 3(a). The axial ratio c/a has a maximum value at VEC = 3.917 and minimum value at 3.852.

Figure 3(a) shows the variation of volume of unit cell v with VEC. It is found that the volume of unit cell decreases by increasing VEC (see also Table 1). Figure 3(b) shows the variation of the intensity ratio from (101) to (200) planes I₁₀₁/I₂₀₀ of b-Sn matrix with VEC. It is found that the ratio decreases to a minimum value 0.341 at 3.917 and then increases to maximum value about 1.8 at 3.983. The value of I₁₀₁/I₂₀₀ for pure Sn was found to

be 1.1 [11]. The addition of 0.5 Ag raises this value to 1.8; this may be attributed to that the Ag atoms are trapped in Sn lattice in excess of equilibrium concentration. The decrease in I₁₀₁/I₂₀₀ to 0.341may be due to the precipitation of Ag to form the intermetallic compound Ag₃Sn.

The relationship between the diameter of the Brillouin zone K_B and the diameter of the Fermi sphere 2k_F is shown in Figure 4. A linear correlation is found between K_B and 2k_F. As 2k_F increases, K_B increases in agreement of the condition of stability (see also Table 1). The value of K_B is calculated for (211) planes. This is direct evidence on the interaction between Fermi sphere and Brillouin zone and verification of Hume-Rothery condition of phase stability. From Table 1 it is clear that as the Ag concentration increases, VEC decreases since the valency of Ag is +1. All of the Fermi parameters decrease by decreasing VEC and vice versa. The decrease in v with increasing VEC means that the volume of the first Brillouin zone increases by increasing VEC since they are inversely proportional to each other. This is also indicated by the increase in K_B with increasing VEC. There-

fore by increasing VEC the diameter of Fermi sphere increases which leads to the increase of the diameter of Brillouin zone which arises from the decrease in volume of the unit cell.

3.2. Thermal Analysis

The DTA curves obtained for as-quenched melt-spun Sn-xAg alloys (x = 0.5, 1.5, 2.5, 3.5, 4.5 and 5.5 wt%) are shown in Figure 5. There is double peak in case of Sn-0.5Ag and Sn-1.5Ag alloys. This double peak may be due to the decomposition of the supersaturated solid solution of Ag in Sn which is formed due to rapid solidification. The variation of the enthalpy ∆H of fusion with valence electron concentration VEC is shown in Figure 6" target="_self"> Figure 6(a). ∆H decreases by increasing VEC i.e., ∆H increases with Ag concentration. A maximum value 76.82 × 10³ J·kg^–1 is observed for VEC = 3.85 and then ∆H decreases with VEC to a minimum value 28.81 × 10³ J·kg^–1 for VEC = 3.95. The increase in enthalpy is due to the increase in the proportion of Ag₃Sn intermetallic compound and this increases the incoherent interfaces which in turns increase the interfacial energy and hence increasing the enthalpy. The solidus T_s and liquidus T₁ temperatures are shown in Figure 6(b). It is evident that the Sn-3.5Ag alloy is the eutectic alloy because it has the lowest melting point. Also the eutectic reaction is found to occur at about 494 K.

3.3. Electrical Properties

The temperature dependence of the electrical resistivity (r) of as-quenched melt-spun alloys is shown in Figure 7 (a). The resistivity increases linearly with temperature however; there is a phase transition before melting in case of Sn-0.5Ag alloy in agreement with DTA results. Also a rapid increase in resistivity observed for 3.5 Ag alloy at about 494 K which is due melting of alloy. The variation of the resistivity at room temperature with valence electron concentration (VEC) is shown in Figure 7 (b). The resistivity increases with VEC from about 9.9 × 10^–8 to 12 × 10^–8 ohm·m. The resistivity is given by; 

where m is the effective mass, n is the number of electrons per unit volume, e is the electron charge, and t is relaxation time. The relaxation time is given by;

, where l is the mean free path and v_F is the Fermi velocity. From Table 1 it is clear that v_F increases with VEC, so that t decreases with VEC. Therefore the resis-

tivity increases with VEC which is observed experimentally.

Figure 7(c) shows the variation of temperature coefficient of resistivity a with VEC. It is found that a increases linearly with increasing VEC. It is well known that the relaxation time is inversely proportional to the temperature which leads to the linear dependence of the resistivity with temperature. Therefore as the relaxation time decreases with VEC, a increases with VEC.

3.4. Mechanical Properties

Figure 8(a) shows the variation of Young’s modulus (E) with the variation of the axial ratio (c/a) of the b-Sn matrix. It is evident that E increases by increasing c/a from 47.56 GPa at 0.5453 to 54.22 GPa at 0.5456. This result is in agreement with the result obtained by [4]. The increase in c/a ratio means the stretching of the unit cell along the c-axis, this modification in the shape of unit cell of Sn matrix may results in an increase in the bond strength which results in an increase of Young’s modulus.

Figure 8(b) shows the variation of internal friction Q^–1 with the axial ratio (c/a). Q^–1 decreases from 0.2138 at 0.5453 to 0.0962 at 0.5456. There are several mechanisms by which internal friction results such as thermoelastic effect, motion of interstitial and substitutional atoms, intercrystalline thermal current and the motion of dislocations. Which of these mechanisms is operative

depends on the frequency. In the frequency range from 0.1 to 20 Hz used in the present work the mechanism inherent here is the motion of interstitial atoms. The decrease in internal friction with c/a may be due to the stretching of the unit cell along c-axis resists the movement of Ag atoms within the unit cell.

Figure 8(c) shows the variation of Vickers microhardness number HV with VEC. It is found that HV decreases with increasing VEC from 167 MPa to a minimum value 93 MPa and then increases to 117 MPa at 3.9835. This means HV increases by increasing Ag concentration. The increase in HV due to the addition of Ag is attributed to the dispersion hardening of small particles

of Ag₃Sn intermetallic compound which increases by increasing Ag concentration.

4. Conclusion

In this study the interaction between Fermi sphere and Brillouin zone and Hume-Rothery condition of phase stability have been verified. By increasing valence electron concentration VEC the diameter of Fermi sphere 2k_F increases which leads to the increase in the diameter of Brillouin zone which arises from the decrease in volume of the unit cell. The electrical resistivity increases by increasing VEC due to the decrease in the relaxation time t with increasing VEC. Also it has been confirmed that the correlation between Young’s modulus and the axial ratio c/a of b-Sn unit cell. In conclusion, valence electron concentration is the most important factor for structure and properties changes of Sn-Ag alloys.

REFERENCES