Lattice instabilities in metallic elements

Most metallic elements have a crystal structure that is either body-centered cubic (bcc), face-centered close packed, or hexagonal close packed. If the bcc lattice is the thermodynamically most stable structure, the close-packed structures usually are dynamically unstable, i.e., have elastic constants violating the Born stability conditions or, more generally, have phonons with imaginary frequencies. Conversely, the bcc lattice tends to be dynamically unstable if the equilibrium structure is close packed. This striking regularity essentially went unnoticed until ab initio total-energy calculations in the 1990s became accurate enough to model dynamical properties of solids in hypothetical lattice structures. After a review of stability criteria, thermodynamic functions in the vicinity of an instability, Bain paths, and how instabilities may arise or disappear when pressure, temperature, and/or chemical composition is varied are discussed. The role of dynamical instabilities in the ideal strength of solids and in metallurgical phase diagrams is then considered, and comments are made on amorphization, melting, and low-dimensional systems. The review concludes with extensive references to theoretical work on the stability properties of metallic elements.

Figure 1

The total energy of Si at 0 K and ambient pressure calculated ab initio for various lattice structures by 390.

Figure 2

The essential parts of the Pt-W phase diagram. From 107.

Figure 3

Elements from the third, fourth, and fifth rows in the periodic table. The ambient-condition ground-state structure is shown in each box, for the sequence of elements in the top row (69; 392). Generally, the bcc structure is dynamically unstable if the equilibrium structure is close packed (fcc or hcp), or the fcc structure is dynamically unstable if the equilibrium structure is bcc. For Pt and the noble metals this trend is uncertain or less pronounced. Manganese has a complex bcc-related ground-state structure with 58 atoms per unit cell and is unstable in the conventional bcc structure.

Figure 4

At ambient pressure, fcc tungsten is dynamically unstable for all long-wavelength transverse modes and for some Brillouin zone boundary modes, but there are also regions of $\mathbf{q}$ where all modes are stable. From 73.

Figure 5

Cubic crystal structures are dynamically stable when the values of $C_{11}$, $C_{12}$, and $C_{44}$ are such that $(s_2,s_3)$ falls inside the stability triangle bounded by the thick lines, corresponding to $s_3=(5/3)s_2-1/6$, $s_3=(2/3)s_2-2/3$, and $s_2=1$, respectively. Data points are shown for two very stable structures (bcc W, fcc Os), two very unstable structures (fcc W, bcc Os), and one structure that is weakly unstable (bcc Hf); cf. Table .

Figure 6

The entropy increase per atom in a Debye model of a monatomic solid, plotted as $\Delta S/(3k_{\mathrm{B}})$, when either $C_{44}$ or $C^{\prime }$ varies linearly with the concentration $c$ and becomes zero at $c_{\mathrm{crit}}=0.5$, while the other shear constant is kept constant. At concentration $c=0$, our model system is assumed to be isotropic with $C^{\prime }=C_{44}=C_{11}/3$. Note that $\Delta S$ does not diverge at $c=c_{\mathrm{crit}}$.

Figure 7

The fcc lattice structure results when a bcc lattice (central part with short-dashed sides in the figure) is stretched in one direction by a factor $c/a=\sqrt{2}$.

Figure 8

The body-centered tetragonal structure.

Figure 9

Bain paths illustrating cases where either the fcc or the bcc structure is stable. From 322.

Figure 10

The bcc-fcc volume-conserving tetragonal Bain path for tungsten at different volumes, where $V_0$ is the volume at pressure $P=0$. The fcc structure gradually goes from being dynamically unstable to metastable and finally to the thermodynamical equilibrium structure. From 73.

Figure 11

Extended Bain path for Pt. From 223.

Figure 12

Extended Bain path for W. From 223.

Figure 13

The two-parameter $\mathrm{fcc}\to \mathrm{sh}$ transformation in Si. The shear parameter $\gamma =(c/a{)}^{3/2}$. From 276.

Figure 14

The variation of the unrelaxed (triangles) and relaxed (circles) energy $E$ when fcc Cu is sheared along $⟨112⟩$ in the $\{111\}$ plane. The engineering strain is defined as ${ϵ}_{\mathrm{eng}}=(c/a{)}^{2/3}-1$. Adapted from 301.

Figure 15

The bcc lattice structure (dashed lines) can be represented by rhombohedral primitive cells (solid lines) with rhombohedral angle $\alpha =109.47°$.

Figure 16

The energy $E-E_{\mathrm{fcc}}$ along the bcc-fcc trigonal Bain path for W. The $c$ and $a$ lengths refer to the [111] direction and a perpendicular direction, respectively [cf. 164]. Adapted from 73.

Figure 17

Calculated phonon dispersion curves for bcc Hf, with arrows marking modes relevant for displacive transformation paths. From 278.

Figure 18

The energy $E-E_{\mathrm{bcc}}$ along the $\mathrm{bcc}\to \omega$ transformation path in Ti. $\delta$ is the displacement associated with the LA[$2/3$, $2/3$, $2/3$] phonon mode; cf. 123. From 278.

Figure 19

Calculated phonon dispersion curves for selected branches of bcc W at the ground-state volume $V_0$ (top curves) and four compressed volumes $(0.91V_0,0.85V_0,0.66V_0,0.44V_0)$, scaled to a common maximum frequency; 18.9 THz for longitudinal and 15.7 THz for transverse modes, respectively. For each Brillouin zone direction, the polarization ($L$ or $T$) with the strongest pressure-induced softening is plotted. The lowest curve corresponds to the lowest volume, and so forth. From 73.

Figure 20

Calculated phonon frequencies in bcc W at different volumes, corresponding to the pressures 1200, 300, 60, and 30 GPa, respectively. $V_0$ is the equilibrium volume of bcc W at zero pressure. From 73.

Figure 21

Calculated phonon frequencies in fcc W at four different volumes. $V_0$ is the equilibrium volume of fcc W at zero pressure. From 73.

Figure 22

An elastic instability may be localized to a certain pressure range, with stability at lower and higher pressures. The figure, based on data from 293, shows smoothed $C_{44}$ in bcc vanadium, as a function of pressure $P$. This system has the less common feature that $C_{44}<0$ while $C^{\prime }>0$.

Figure 23

A tentative $P\mathrm{\text{−}}T$ phase diagram for Mg, from 101 and based on calculations by 249.

Figure 24

$P\mathrm{\text{−}}T$ phase diagram obtained in a simple model for $G(P,T)$; see Appendix .

Figure 25

A double well used by 376 to model the instability of the $T_1$ $N$-point phonon in bcc Zr. The displacement $d$ is in units of the lattice parameter $a$.

Figure 26

Calculated phonon dispersion curves for bcc Ti, Zr, and Hf at $T=0\mathrm{K}$ and at high temperatures. Filled circles are experimental data (115; 279; 350). From 336.

Figure 27

The changes in the free energy and the entropy as calculated by 264 at $T=2500\mathrm{K}$, as a function of the Bain distortion in W.

Figure 28

Schematic dependence on the wave vector of the $[\xi \xi 0]$ phonon dispersion in fcc W at high temperatures.

Figure 29

The electron density of states $N(E)$ for fcc and bcc Mo, calculated at high temperatures in a static lattice and in an average thermally disordered lattice. From 7.

Figure 30

The entropy Debye temperature ${\Theta }_S$ for hcp and bcc phases of Ti and Zr, calculated from phonon frequencies obtained in neutron scattering experiments. From 101.

Figure 31

The elastic shear constant $C^{\prime }$ for the fifth row elements. Data from Table .

Figure 32

The elastic shear constant $C_{44}$ for the fifth row elements. Data from Table .

Figure 33

The elastic constants $C^{\prime }$ and $C_{44}$ in the fcc structure, calculated in the fifth-moment approximation, as a function of the number of electrons $n_d$. Adapted from 254.

Figure 34

The measured elastic constants $C^{\prime }$ and $C_{44}$ of binary bcc Zr-Nb-Mo alloys at room temperature, plotted vs the number $n$ of electrons per atom ($n=4$, 5, and 6 for Zr, Nb, and Mo). Data from the compilation by 78.

Figure 35

Data as in Fig. 34 but now normalized as $C^{\prime }/B$ and $C_{44}/B$, where $B$ is the bulk modulus.

Figure 36

The room-temperature elastic shear constant $C^{\prime }$ in solid solutions of Ga in Fe. Adapted from 49.

Figure 37

The elastic constants $C^{\prime }$ and $C_{44}$ for hypothetical random bcc solid solutions in the Ag-Zn system, obtained in ab initio electronic structure calculations. From 208.

Figure 38

Elastic constants $C^{\prime }$ and $C_{44}$ as in Fig. 37, but for the fcc lattice structure.

Figure 39

The total energy per atom, as a function of the engineering strain, for uniaxial loading of tungsten along the [001] and [111] directions (perpendicular dimensions of the lattice are relaxed). The arrows show the inflection points, corresponding to the maximum theoretical strength. From 323.

Figure 40

The stress $\sigma$ as a function of the engineering shear strain ${ϵ}_{\mathrm{eng}}$ in bcc W for $⟨111⟩$ shear in the $\{110\}$ (circles), $\{112\}$ (squares), and $\{123\}$ (triangles) planes. Adapted from 168.

Figure 41

The stress $\sigma$ as a function of the engineering shear strain ${ϵ}_{\mathrm{eng}}$ calculated in ferromagnetic bcc iron for uniaxial tension (squares) and uniaxial tension plus a biaxial tension (circles), with the largest tensile stress in the [001] direction. Arrows indicate the location of the tetragonal-to-orthorhombic bifurcation instability. From 51.

Figure 42

The energy $U(ϵ)$ per atom as a function of the relaxed [001] tensile strain $ϵ$, calculated for Mo and Nb along the tetragonal Bain deformation path (circles) and along an orthorhombic path that is branching off. The maximum of the orthorhombic path has the same energy as the local minimum of the tetragonal path, since they correspond to the same tetragonal saddle-point structure. From 204.

Figure 43

The stress-strain relation for the same loads as in Fig. 42. From 204.

Figure 44

Transverse phonon dispersion curve in the [111] direction of aluminum, for $[11\overline{2}](111)$ shear at four different amounts of shear strain $ϵ$ (given in %). The onset of the shear instabilities appears very suddenly above $ϵ=0.14$. Adapted from 53.

Figure 45

The energy difference $\Delta H_{\mathrm{bcc}\mathrm{\text{−}}\mathrm{fcc}}$ between the bcc and fcc structures for $4d$ metals, as calculated ab initio (lines) and as obtained at that time by CALPHAD methods (circles). Corresponding results for the hcp structure are also shown. From 314.

Figure 46

The Gibbs free energy $G_{\alpha }$ and $G_{\beta }$ for two phases that are dynamically stable at all concentrations $c$ (solid curves) and also for the case that phase $\alpha$ becomes dynamically unstable at $c_{\mathrm{crit}}$ (short-dashed curve). $G_{\alpha }$ is not a thermodynamically defined quantity when $c>c_{\mathrm{crit}}$. The common-tangent construction (long-dashed curve) gives the concentrations $c_1$ and $c_2$ of the end points of the two-phase field in the phase diagram. Model assumptions: $H(T=0)$ varies with $c$ as $H_{\alpha }=H_{A,\alpha }+c{H}_{\alpha }^{\prime }$ and $H_{\beta }=H_{A,\beta }+c{H}_{\beta }^{\prime }$. The temperature-dependent part of $H$ is described by the classical result $3k_{\mathrm{B}}T$ per atom, and cancels in $G_{\alpha }-G_{\beta }$. There is a random disorder entropy (per mole) $S=R[c\ln (c)+(1-c)\ln (1-c)]$. The vibrational entropy difference $S_{\alpha }-S_{\beta }=0$. At the considered $T$, $H_{A,\alpha }/(RT)=0$; $H_{B,\beta }/(RT)=0.75$; $H_{\alpha }^{\prime }/(RT)=0.5$; and $H_{\beta }^{\prime }/(RT)=1.5$.

Figure 47

The elastic constants $C^{\prime }$ and $C_{44}$ in fcc Pb (78) do not extrapolate to zero as $T$ approaches $T_m=601\mathrm{K}$.

Figure 48

The mixing Gibbs free energy $G_{\mathrm{mix}}$ as a function of composition $c$ for a system that shows spinodal decomposition.

Figure 49

The energy $U$ as a function of volume $V$ for two systems, where the right graph implies a spinodal instability.

Figure 50

Simple model geometry, with two planes sliding over each other.