Structural mechanisms and order-parameter symmetries for the high-pressure phase transitions in alkali metals

The structural transitions occurring between the high-pressure phases of alkali metals are described by displacive mechanisms. The different structures are associated with critical instabilities of the ambient-pressure body-centered-cubic structure. Unifying mechanisms are found for the transitions occurring in (Li and Na) and (K, Rb and Cs), respectively. The RbIII-RbIV-RbV and CsIII-CsIV phase sequences are interpreted as reflecting the hybridization process of the $s$- and $d$-wave functions, which induces a deformation of the atomic shells, allowing a closer packing of the KIII, RbIV, and RbV structures. The RbIII and CsIII structures are proposed to represent commensurate approximations of incommensurately modulated structures locking in at the RbV and CsIV phases.

Figure 1

(Color online) Phase sequences for the alkali metals as a function of pressure (Refs. 2, 3, 4, 5, 6, 7, 11, 13, 14, 15).

Figure 2

(Color online) Mechanism for the reconstructive $\mathrm{bcc}\to \mathrm{fcc}$ transition, involving a stretching of the bcc unit cell along one of the fourfold axes and a compression along the other fourfold axes. Atoms on different heights along $z$ are shown in different gray shades (colors).

Figure 3

(Color online) Mechanism of the ferroelastic deformation of the fcc structure of LiII leading to the rhombohedral structure of LiIII. The displacements are indicated by arrows. Atoms on different heights along $z$ are shown in different gray shades (colors).

Figure 4

(Color online) (a) The LiIV and NaIII structures obtained from the fcc structure by the displacements represented in (b). Atoms on different heights along $z$ are shown in different gray shades (colors).

Figure 5

(Color online) (a) Mechanism for the reconstructive $\mathrm{K}\mathrm{II}\to \mathrm{K}\mathrm{III}$ transition involving antiparallel displacements of the two layers of $(\pm 1∕4,0,0)$ related to the fcc structure of KII combined with rotations of the square atomic motifs within the two layers, indicated by curved arrows. The structure is projected along ${\left[001\right]}_{\mathrm{fcc}}$. Atoms on different heights along $z$ are shown in different gray shades (colors), representing the two-layer character of the structure. The atoms that form the guest atoms in the KIII structure are shown in light gray (light blue). (b) Atomic configuration of the tetragonal guest-host structure of KIII, projected along [001]. The same color labeling is used as in Fig. 5a.

Figure 6

(Color online) Orthorhombic structure of RbIII, described as a four-layer structure with 13 atoms per layer.

Figure 7

(Color online) Displacive mechanism associated with the $\mathrm{Rb}\mathrm{II}\to \mathrm{Rb}\mathrm{III}$ transition. In the figures on the left side the small unit cell is the initial fcc cell of RbII and the unit cell shown by thick lines represents the 13 times larger transformed cell of RbIII.

Figure 8

(Color online) Tetragonal “guest-host” structure of RbIV, projected along [001].

Figure 9

(Color online) Combined mechanism of $\text{shifting}+\text{rotation}$ transforming the structure of RbIII into RbIV. The displacements are indicated by arrows; the additional rotation of the square atomic motifs within the layers is indicated by curved arrows.

Figure 10

(Color online) (a) Tetragonal body-centered structure of RbV (Ref. 12). Atoms on different heights along $z$ are shown in different gray shades (colors), representing the four-layer character of this structure. (b) Tetragonal “guest-host” structure of RbIV, projected along [001]. Atoms belonging to different layers stacked along $z$ are shown in different gray shades (colors). The guest atoms are shown in light gray (light blue). The small unit cell indicated by thick lines represents the tetragonal unit cell of the RbV structure, projected along [001]. The displacements involved in the transformation of RbIV into RbV have the following magnitudes: $(-1∕4,0,0)$ [layer 1 (light gray arrows)], $(0,-1∕4,0)$ [layer 2 (gray arrows)], $(+1∕4,0,0)$ [layer 3 (dark gray arrows)], and $(0,+1∕4,0)$ [layer 4 (black arrows)]. Additional rotations of the square atomic motifs within the four layers of the RbIV structure are indicated by curved arrows.

Figure 11

(Color online) Displacive mechanism associated with the $\mathrm{Rb}\mathrm{V}\to \mathrm{Rb}\mathrm{VI}$ transition. The displacements can be divided into (1) specific displacements occurring along the initial ${\left[110\right]}_{\mathrm{fcc}}$ directions of $(+1∕8,+1∕8,0)$ [layer 1 (light gray arrows)], $(-1∕8,+1∕8,0)$ [layer 2 (gray arrows)], $(-1∕8,-1∕8,0)$ [layer 3 (dark gray arrows)], and $(+1∕8,-1∕8,0)$ [layer 4 (black arrows)]. Additional small displacements, indicated by the small arrows, bring the atoms into the specific positions characteristic for the RbVI structure, which is shown in Fig. 12a. The unit cell of the RbV structure is shown in gray (red), the unit cell of the RbVI structure is shown in black.

Figure 12

(Color online) Structure of RbVI, projected along (a) [100] and (b) [001].

Figure 13

(Color online) Orthorhombic structure of CsIII described as a four-layer structure with 21 atoms per layer.

Figure 14

(Color online) Displacive mechanism transforming the CsII structure into CsIII. The displacements are indicated by arrows.

Figure 15

(Color online) $\mathrm{Cs}\mathrm{III}\to \mathrm{Cs}\mathrm{IV}$ displacive mechanism. The displacements along the initial ${\left[110\right]}_{\mathrm{fcc}}$ directions are indicated by arrows. The unit cell of the CsIII structure is shown in black; the unit cell of the CsIV structure is represented by gray (red) lines.

Figure 16

(Color online) Mechanism transforming the orthorhombic structure of CsV into the dhcp structure of CsVI (right). The displacements of the layers are indicated by arrows: layer 1 (light gray arrows), layer 2 (gray arrows), layer 3 (dark gray arrows), and layer 4 (black arrows). The unit cell of the dhcp structure of CsVI is shown in gray (red) in the figure on the left-hand side.

Figure 17

Symmetry relationship between the different alkali metal structures and the bcc parent structure, deduced from Table .

Figure 18

Points of the bcc Brillouin zone involved in the transition mechanisms of alkali metals. The path followed in the $\mathrm{Rb}\mathrm{III}\mathrm{Rb}\mathrm{IV}\mathrm{Rb}\mathrm{V}\mathrm{Rb}\mathrm{VI}$ sequence of phases is indicated by a thick arrow.

Figure 19

(Color online) Sphere-packing and ellipsoid-packing models for (a) KIII, RbIV and (b) RbV, CsIV. The ellipsoid packings are approximately 5–8 % denser than the sphere packings.

Figure 20

(Color online) Variation of the interatomic distances in RbIII along the initial [100] direction of the bcc lattice indicated in the upper figure.

Figure 21

(Color online) Variation of the interatomic distances in CsIII along the initial [100] direction of the bcc lattice indicated in the upper figure.