Effects of Alloying on Phonons in Vanadium
4.4 Phonon DOS
4.4.1 Pure Vanadium
The results of our measurements for the phonon density of states of pure vanadium are shown in figure 4.4, where they are compared with the previous measurement of Sears [76]
and the measurement of Bogdanoff et al. [77]. As can be seen in this figure, the results of our different measurements for pure vanadium on Pharos and LRMECS yielded phonon DOS curves that are in very good agreement with each other, as well as with the previous measurement of Sears, carried with a triple-axis spectrometer in constant-Qmode [76]. The phonon DOS for pure V measured by Bogdanoffet al. [77] using the HFIR-HB2 triple-axis spectrometer at Oak Ridge National Laboratory shows more scatter, but it is nevertheless in good agreement with our time-of-flight results.
4.4.2 Trend Across the 3d-Series
Our result for the vanadium phonon DOS and the neutron-weighted phonon DOS of V- 6.25%X, with X a 3d-series solute, are shown in figure 4.5. The average phonon frequencies
a maximum shift of about 2 meV, on the lower energy side of the transverse peak. The longitudinal peak and the cutoff frequency appear shifted down by about 1 meV. In the case of Cr, Fe, Ni, and Co impurities, the phonon DOS is stiffer than in pure vanadium and this stiffening is gradually increasing along the 3d-series. The upward energy shift affects all the modes in these alloys and the cutoff shift goes from about 1 meV in the case of Cr to about 2.5 meV in the case of Ni. Another observed trend is that impurities that induce the larger shifts of the phonon DOS also affect its shape the most and in particular the transverse and longitudinal peaks appear to coalesce for the impurities with higher atomic number (Fe, Co, Ni).
4.4.3 Trend Down the Ti Column
Our measurements for the alloys of vanadium with Ti, Zr, and Hf are shown in figure 4.6. All these solutes induce a softening in the phonon DOS, compared with pure V. The softening increases when going down the column from Ti to Hf. This is best seen at low energies, between 8 and 17 meV and around the cutoff energy. Since these impurities are isoelectronic, this trend can be associated with the increase in mass of the impurity down the column, as well as the increase the size of the impurity, which induces a larger average lattice parameter for the alloy (0.3% increase in the case of Ti, see table 4.1). Also, in the case of Zr and even more so for Hf, the partial phonon DOS for the impurities is likely to exhibit a resonance mode at low energies, due to the high impurity/host mass ratio (1.8 and 3.5, respectively). We discuss such resonance modes for Pd and Pt impurities (mass ratios of 2.1 and 3.8, respectively) below.
4.4.4 Trend Down the V Column
Figure 4.7 shows our results for the phonon DOS of the isoelectronic alloys V-Nb and V- Ta. Niobium and tantalum impurities have a remarkably small effect on the vanadium
Figure 4.5: Neutron-weighted Phonon DOS for impurities across the 3d-series measured on LRMECS. Broken traces: pure vanadium, solid lines: V-6.25%X. Ei = 50 meV, except for top curves whereEi = 55 meV.
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DOS (1/meV)
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Energy (meV) V-6.25%Hf
Figure 4.6: Neutron-weighted Phonon DOS for impurities down the Ti column measured on LRMECS,Ei = 50 meV.
phonon DOS. Nb induces a slight softening of about 0.5 meV at the cutoff as well as on the low-energy side of the transverse peak. On the other hand, Ta solutes seem to have no detectable effect on the V phonon DOS, within our experimental error bars. This is surprising considering the impurity/host mass ratio in excess of 3.5 in this case. However, it is possible that this is an artifact due to the stronger neutron-weighting for Ta impurities, with Ta modes scattering neutrons three times more weakly than vanadium modes (see the σscat/M ratios in Table 4.1). Nevertheless, the overall effect of Ta solutes on the phonon DOS appears to be very weak.
4.4.5 Trend Down the Ni Column
Ni, Pd and Pt impurities have the strongest effect on the V phonon DOS. The curves plotted in figure 4.8 are the neutron-weighted DOS curves for V-6.25%Ni, V-6.25%Pd, V-6.25%Pt and V. Ni, Pd and Pt impurities induce a large stiffening of the vanadium phonon DOS, which is gradually increasing from the case of Ni to that of Pd and Pt. The shift of the cutoff is about 2.5 meV in the case of Ni and almost 4 meV in the case of Pt. The pure V
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Energy (meV) V
V-6.25%Nb V-6.25%Ta
Figure 4.7: Neutron-weighted Phonon DOS for impurities down the V column measured on LRMECS,Ei = 50 meV.
and V-6.25%Pt measurements using Pharos are shown in figure 4.9. As can be seen on this figure, the agreement between the LRMECS and Pharos measurements is excellent. In all the alloys down the column, the shift affects the full DOS rather uniformly, although there is also some restructuring of the phonon spectrum, with the transverse and longitudinal peaks coalescing in the alloys. Also, we show below using a Born–von K´arm´an lattice dynamics model that Pd and Pt impurities exhibit resonance modes around 12 meV. However, these modes only affect a small portion of the total DOS for impurity concentrations of 6.25%.
Hence the neutron-weighting does not affect the general shape of the total phonon DOS significantly and has a minor effect on the calculated vibrational entropy, as discussed in the following.
4.4.6 Concentration Dependence in V-Co
We investigated the dependence of the phonon DOS on the concentration of Co impurities.
The V-Co system is particularly suitable for this type of study, as it is almost free of neutron- weighting, and Co impurities induce a significant distortion of the phonon DOS. The alloys
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DOS (1/meV)
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Energy (meV) V-6.25%Pd
V-6.25%Pt
Figure 4.8: Neutron-weighted Phonon DOS for impurities down the Ni column measured on LRMECS,Ei = 55 meV.
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Phonon DOS (1/meV)
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V Pharos
V-6.25%Pt Pharos V LRMECS
V-6.25%Pt LRMECS
Figure 4.9: Neutron-weighted Phonon DOS for V and V-6.25%Pt measured on LRMECS (Ei = 55 meV) and Pharos (Ei = 75 meV).
V-6.25%Co and V-2.0%Co were measured on the LRMECS time-of-flight spectrometer, to- gether with the reference V sample, while the alloy V-7.0%Co and a pure V reference were measured with the HB-2 triple-axis spectrometer at HFIR (Oak Ridge National Labora- tory). The triple-axis data are unpublished results measured by P.D. Bogdanoff, B. Fultzet al. with the experimental conditions described in [77]. The phonon DOS for the alloys and control samples are shown in figure 4.10. As seen in this figure, the addition of Co impurities in V causes a gradual stiffening of the phonons. The data from LRMECS show that the stiffening increases in proportion to the impurity concentration. The data for V-7.0%Co show a much bigger stiffening than those for V-6.25%Co. Although we cannot rule out the possibility of stronger impurity dependence of the DOS from 6.25% to 7.0% impurity, this seems rather unlikely considering the small difference in overall concentration. Besides, the random solid-solutions will present concentration fluctuations that will make the two materials locally very similar. It is possible that a small amount of a second phase is present in the V-7.0%Co alloy, but we think that the difference comes in part from the different sampling of reciprocal space between the time-of-flight and triple-axis measurements.