V experiment V-X de-weighted
4.7 Chemical Trends
4.7.3 Comparison with Trends for BCC Transition Metals
We can compare the present results to previous observations for BCC transition metals. It has been shown that the phonon dispersions of BCC transition metals exhibit systematic trends correlating with their number of d-electrons [87, 88]. The BCC lattice exhibits inherent instabilities for deformations toward the HCP lattice and the ω structure, which leads to the occurrence of both types of martensitic transformations in Sc, La, Ti, Zr and Hf [89, 90, 91, 92, 93]. The instability toward the ω transformation appears as a pronounced dip in the dispersions at the L 2/3[111] point, orω-point. This weakness is understood as a geometric feature of the BCC lattice [87]. The displacement of the ions in this phonon mode corresponds to motions of dense rows of atoms in the [111] directions that do not alter the 1NN distance between ions. Because the 1NN interactions are strongest and they do not come into play in this displacement pattern, the corresponding energy is particularly low. This feature is also present in BCC alkali metals and was reproduced in theoretical calculations [94]. The BCC to HCP martensitic transformations in the elements of groups 3 and 4 are also explained in terms of precursor modes corresponding to a mechanical weakness of the BCC lattice. In this case, the precursor mode involved is the N-point
temperatures.
The dip at the ω-point in the L[111] phonon branch is progressively suppressed as the number of d-electron increases. Petry has related this observation to the strengthening of directional bonding with the filling of the d-band, which leads to “cross-locking” of [111]
rows of atoms [87, 88]. Directional bonding also stabilizes theT[1¯10][110] transverse phonon branch. As a result, the BCC phase is mechanically stabilized at lower temperatures for more filling of the d-band. This strong stiffening of the phonons is observed as one goes from group 5 to group 6 elements, with an increase in nominald-band filling from 3 to 4 (or number of valence electrons per atom between 5e/at and 6e/at). For example, the cutoff energy of the phonon DOS increases from 21 meV to 28 meV between Ta and W [95], from 27 meV to 33 meV between Nb and Mo [96], and from 33 meV to 40 meV between V and Cr [36] (magnetism also intervenes in the latter case, but it is believed to yield a softening, not a stiffening).
The d-band filling has also been invoked to explain the composition dependence of the elastic moduli in BCC transition metal alloys [61, 62, 60, 65, 66, 67, 63]. In the simplest of approximations, that of a free-electron metal, one can relate the bulk modulusB to the electron densitynthrough (see ref. [49])
B = 2π4/3
31/3m n5/3 . (4.9)
In the transition metals, the electronic structure is more complex. In the Friedel model, one considers a rectangular electronic DOS to approximate thed-band, while thespband is neglected altogether. The only free parameters are then the bandwidthW and the number ofd-electronsNd (the height of the rectangular DOS is fixed to 10/W, since there are 10d states to accomodate). In this model, the cohesive energy is proportional to Nd(10−Nd);
it is an inverted parabola peaking at Nd = 5. The bulk modulus exhibits an essentially similar behavior, but the modulation by the change in lattice parameter (and thus electron
density) across thed-band gives it a more linear dependence with NdforNd<5 [97]. The predicted shape ofB(Nd) reproduces well the experimental values of bulk modulus for the pure elements, independently of their crystal structure.
Measurements on Ti-V-Cr and Zr-Nb-Mo alloys have found a generally linear depen- dence of the bulk modulus with e/at [61, 62, 59, 60] in the range of 4e/at to 5.8e/at, in good agreement with the simple model of Friedel. For more Cr-rich V-Cr alloys, the magnetism yields a sudden decrease in B. The shear modulus G and single crystal shear elastic constantc44 present a more complex behavior, however. At room-temperature, the dependence of G and c44 on e/at goes from weak to much stronger, with a pronounced cusp at around 5.4e/at [61, 62, 60, 67]. This behavior is even more pronounced at low temperatures, wherec44goes from a decrease withe/at to a strong increase, at around this composition. This feature has been attributed to the crossing of the Γ25 point in the band structure at a band-filling of about 5.4e/at, corresponding to a topological change of the Fermi surface [64]. More will be said about this in chapter 5.
The rigid-band model is rather successful at describing both the temperature and electron-concentration dependence of the elastic constants of Ti-V-Cr and Zr-Nb-Mo al- loys, as well as some features of their phonon dispersions. However, it is expected that the rigid band model is limited to “homogeneous” alloys and will not be valid for alloys with components that are not neighbors in the periodic table. In the case of V-6.25%X alloys, the rigid band model describes well the elastic constants for Ti and Cr impurities. But for later transition metal impurities such as Fe, Co, Ni, Pd, and Pt, it does not work so well. We observe a stronger dependence of the elastic constants and phonons on the type of solute than would be predicted by a mere band-filling. One should keep in mind that with 6.25% solutes, the range of electron concentration sampled is small, between 4.9375e/atand 5.3125e/at. We observed a downward curvature of B and Gfrom Co to Ni impurities, for example, which would not be expected based on rigid-band filling only. Most prominently, the behavior ofGand the phonons for solutes changing from Ni to Pt (at constant e/at) is unaccounted for.
The emerging picture is thus as follows. For solutes having a similar number of d- electrons as the vanadium atoms of the host crystal, the alloy is rather homogeneous,
concentration in predicting the change in the vibrational and elastic properties. We have shown that the difference in electronegativity between the impurities and the host atoms is a good parameter to predict the sign and magnitude of the change in vibrational entropy, in the case of semi-dilute vanadium alloys.