In recent years, a number of surprisingly complex crystal structures have been discovered in the elements at high pressures, in particular incommensurately modulated structures and incommensurate host-guest composite structures (see [1] for a review). The crystal structure of the high-pressure phase rubidium-IV shown in Figure 11 belongs to the group of incommensurate host-guest structures that have also been observed in the elements Na, K, Ba, Sr, Sc, As, Sb, and Bi. The structure comprises a framework of rubidium host atoms with open channels that are occupied by linear chains of rubidium guest atoms, and the periodicities of the host and guest subsystems are incommensurate with each other (i.e., they have a non-rational ratio). Although considerable progress has been made in determining the detailed crystal structures of the complex metallic phases at high pressure, little is known about their other physical properties, and the mechanisms that lead to their formation and stability are not yet fully understood.

Fig. 11: Inelastic X-ray scattering spectrum of Rb-IV at 17.0 GPa, with the scattering vector q = (0 0 3.2)h referring to the host lattice. The inset shows the composite crystal structure of Rb-IV with the rubidium host and guest atoms in blue and red, respectively.

We investigated the lattice dynamics in incommensurate composite Rb-IV by inelastic X-ray scattering (IXS) on beamline ID28. The focus was on the longitudinal-acoustic (LA) phonons along the direction of the incommensurate wavevector (parallel to the guest-atom chains). Calculations on simpler model systems predict these phonons to reflect the incommensurability most clearly. Phase IV of Rb is stable at pressures of 16 to 20 GPa at room temperature, and a high-quality single crystal of Rb-IV was grown in a diamond anvil high pressure cell. In the IXS experiment, the incident radiation was monochromatised at a photon energy of 17.8 keV, and two grazing-incidence mirrors focussed the X-rays onto the sample with a focal size of 25 x 60 µm. The spectrum of the scattered radiation was analysed by a high-resolution silicon crystal analyser to yield an overall energy resolution of 3 meV.

Figure 11 shows a typical IXS spectrum of Rb-IV along with its decomposition into the elastic line, the phonon excitation peaks and a constant background, which were obtained by least-squares fitting using the FIT28 software. From a series of IXS spectra collected for different momentum transfers Q, phonon dispersion curves were obtained as shown in Figure 12a. A central result of this study is the observation of two well-defined longitudinal-acoustic (LA)-type phonon branches along the chain direction. They are attributed to separate LA excitations in the host and the guest sublattices, which is a unique feature of an incommensurate composite crystal.

Fig. 12: a) Dispersion relations of longitudinal-acoustic phonons of the host and guest subsystem in Rb-IV. b) Pressure dependences of the host and guest LA sound velocities along the chain direction and results of the monatomic linear chain model. The dashed line indicates the pressure dependence obtained with the linear chain model.


A series of dispersion curves was measured at different pressures, and from this the sound velocities of the host and guest excitations and their pressure dependences were determined (Figure 12b). While the absolute values of the sound velocities in the host and the guest are rather similar, their pressure dependences differ notably. A simple ball-and-spring model of Rb-IV with only one spring constant reproduces these observations semi-quantitatively. This suggests that the difference in the pressure dependences is determined largely by geometrical factors, i.e., by the spatial arrangement of the atoms rather than differences in the chemical bonding in the two subsystems.

There is only very weak coupling between the incommensurate host and the guest in Rb-IV, which raises a rather interesting question. Can the 1D chains of guest atoms in Rb-IV be considered a manifestation of the “monatomic linear chain” treated in solid-state physics textbooks to introduce the concepts of crystal lattice dynamics? The pressure dependence of the interatomic spacing in the guest-atom chains was measured in earlier structural studies and enables the spring constant in the linear chain model to be determined, and also its pressure dependence. On this basis, the sound velocity in the linear chains and its pressure dependence were modelled as shown in Figure 12b. The results are in excellent agreement with the IXS data for the guest-atom chains in the composite Rb-IV structure, which can thus be regarded as a manifestation of the monatomic linear chain model with regard to the LA phonons.


Principal publication and authors

I. Loa (a), L.F. Lundegaard (a), M.I. McMahon (a), S.R. Evans (a), A. Bossak (b), and M. Krisch (b), Phys. Rev. Lett. 99, 035501 (2007).
(a) The University of Edinburgh (UK)
(b) ESR


[1] M.I. McMahon and R.J. Nelmes, Chem. Soc. Rev. 35, 943 (2006)