Phonons in Vanadium Alloys at High Temperatures
6.2 High-Temperature Phonon DOS
6.2.2 Vanadium Alloys
The phonon DOS of vanadium alloys with a few percent of Co, Pt, and Nb impurities at elevated temperatures were investigated. These measurements reveal a trend with the nature of the impurity.
The phonon DOS for the alloy V-7%Co was measured at elevated temperatures by Bogdanoff, Fultz et al. (unpublished results). These measurements were conducted with the same triple-axis neutron spectrometry technique used to measure pure V [77]. Results are shown in Fig. 6.2. Contrasting with the anomalous behavior of pure V, the phonon DOS of V-7%Co undergoes a gradual softening between 293 K and 1273 K. The softening is clearly seen in low-energy transverse modes as well as high-energy longitudinal modes, in particular at the cutoff energy. The high-temperature DOS also exhibits a broadening, as can be seen on the tail-like shape of the cutoff. The longitudinal peak appears to gain in intensity at elevated temperatures, however, which could result from a differential softening along independent longitudinal branches in the dispersion curves, or a flattening of the dispersion in a particular direction.
The phonon DOS for V-6.25%Pt and V-6.25%Nb at high temperature were measured with the Pharos time-of-flight spectrometer at LANSCE. Results are shown in figures 6.3(b) and 6.4(b). The temperature-dependence of the phonon DOS for V-6.25%Pt is very similar to that observed in V-7%Co, with a gradual temperature-softening. The softening is larger than in the case of Co impurities, however. The longitudinal peak undergoes an increase in intensity with temperature, similar to V-7%Co. We conclude that Pt and Co impurities
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Figure 6.2: Phonon DOS of V-7%Co at 293, 873 and 1273 K, measured with the triple-axis spectrometer HB2. Temperatures are as labeled.
influence the temperature-dependence of the V phonon DOS in the same fashion, with a strong disruption of the anomalous temperature-dependence. We recall that these impurities also had comparable effects on the phonon DOS of V at room-temperature, both causing a large stiffening of the vibrations (see chapter 4). On the other hand, Nb impurities only had a small effect on the phonon DOS at room-temperature and their effect on the temperature- dependence of the phonon DOS is also small, as seen in figure 6.4. The transverse modes are almost unaffected by the Nb solutes, both at 300 K and 1273 K, while the longitudinal modes show only a small excess in softening at 1273 K in the presence of the solutes. A trend in the effect of impurities on the phonon-softening emerges, as Nb is isoeletronic to V, while Co and Pt are both late transition metals.
The anharmonic effects observed in the V phonon DOS can be studied more quantita- tively, and contrasted to the behavior of V-Co, using an entropy analysis.
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Figure 6.3: (a) Phonon DOS of pure V at 300, 973 and 1323 K, measured with Pharos. (b) Phonon DOS of V-6.25%Pt at 300, 973 and 1323 K, measured with Pharos (LANSCE).
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Figure 6.4: (a) Phonon DOS of pure V at 300 K and 1273 K, measured with Pharos. (b) Phonon DOS of V-6.25%Nb at 300 K and 1273 K, measured with Pharos (LANSCE).
their phonon DOS measurements [77]. These authors also compared their results to the pre- dictions of Eriksson, Wills, and Wallace [56], who analyzed the experimental heat capacity and thermal expansion and inferred the phonon entropy after subtracting an electronic con- tribution obtained from electronic structure calculations. The results of both studies are in good agreement and attribute the anomalous temperature of the phonon DOS to phonon anharmonicity.
The anharmonic entropy analysis adopted by Bogdanoff et al. is essentially similar to that described in chapter 2, although the notation used by those authors is slightly differ- ent. We illustrate their results with the help of Fig. 6.5, taken from [77]. The crosses in this figure correspond to the anharmonic phonon entropy calculated from the experimental phonon DOS, as in the left-hand side of Eq. 2.68. The thick upward line is the quasihar- monic phonon entropy ΔSphqh, obtained from thermal expansion data and corrected for the electronic contribution as in Eq. 2.61, using the electronic structure calculations of Eriksson et al. [56]. The difference between the line and the crosses, in the approximation that the phonon modes at 300 K are harmonic, is the contribution of explicit anharmonicity in the interatomic potentials to the entropy, ΔSph(3,4). It is plotted as black circles in Fig. 6.5.
This contribution is negative and almost exactly cancels out the positive term correspond- ing to thermal expansion. It represents a stiffening of the phonon modes with increasing temperature at fixed volume.
Eriksson, Wills, and Wallace performed calculations of the electronic entropy Sel and used heat capacity data, thermal expansion data, as well as harmonic phonon data from the literature to deduce the phonon anharmonicity from the relationship
Stot =Sel+Shar+Sph(2)+Sph(3,4) , (6.1) whereStot is the total entropy and the terms on the right-hand side are as defined in chap- ter 2. Their result for the explicit anharmonicity in the entropy, Sph(3,4), is plotted as the
Figure 6.5: Anharmonic contributions to the entropy of vanadium. The bold curve is as computed from Eq. 2.61. Crosses are computed from the phonon DOS of vanadium using Eq. (2.68). The solid circles are the difference between the bold curve and the crosses (the solid circle at 1673 K is obtained by extrapolating the bold curve to higher temperatures).
The solid curve is the anharmonic entropy obtained by Erikssonet al. [56].
black curve in Fig. 6.5. This curve and the black dots are in good agreement, although the anharmonic component of the entropy was derived by different means, strengthening the proposition that the anomalousT-dependence of the phonons in V stems from anharmonic- ity in the displacement potentials.