Low-energy surface phonons of decagonal and icosahedral quasicrystals by inelastic He-atom scattering H. R. Sharma, 1 K. J. Franke, 1 W. Theis, 1, * P. Gille, 2 Ph. Ebert, 3 and K. H. Rieder 1 1 Institut fu ¨r Experimentalphysik der Freie Universita ¨t Berlin, 14195 Berlin, Germany 2 Sektion Kristallographie, Department fu ¨r Geo- und Umweltwissenschaften, Ludwig-Maximilians-Universita ¨t Mu ¨nchen, 80333 Mu ¨nchen, Germany 3 Institut fu ¨r Festko ¨rperforschung, Forschungszentrum Ju ¨lich GmbH, 52425 Ju ¨lich, Germany Received 17 April 2003; published 19 August 2003 Low-energy surface phonons Rayleigh mode on quasicrystals were investigated by inelastic He-atom scattering. The tenfold surface of decagonal Al 71.8 Ni 14.8 Co 13.4 and the fivefold surface of icosahedral Al 70.5 Pd 21 Mn 8.5 are found to possess well-defined Rayleigh modes with isotropic sound velocities of about 3840 m/s and 3470 m/s, respectively. The observed sound velocities are in good agreement with calculations based on the respective bulk data. The experimental phonon dispersions exhibit quasi-Brillouin-zone centers located at strong Bragg peaks. DOI: 10.1103/PhysRevB.68.054205 PACS numbers: 61.44.Br, 68.35.Ja, 68.49.Bc, 63.22.+m I. INTRODUCTION Since their discovery in 1984, 1 quasicrystals have become an interesting topic for theoretical and experimental investi- gations. Perfect long-range order in combination with the lack of translational symmetry is the most striking feature of quasicrystalline alloys. Aside from their intriguing structure and their outstanding bulk properties, quasicrystals exhibit many interesting surface characteristics. While significant progress has been made in various domains of surface sci- ence of quasicrystals such as structure and physical properties, 2,3 experimental knowledge regarding the nature of low-energy surface vibrations had still been lacking. Theoretical 4–8 and experimental investigations of bulk phonons 9–17 show that quasicrystals, as periodic crystals, ex- hibit well-defined acoustic modes in the continuum limit. Phonons in periodic crystals have a well-defined energy and wave vector due to the lattice periodicity and phonon modes can be fully characterized by a wave vector confined to the first Brillouin zone and a band index. 18 Due to the lack of periodicity a Brillouin zone cannot be properly defined in quasicrystals. However, the positions of strong Bragg peaks act as quasi-Brillouin-zone QBZ centers. 19,20 The QBZ boundaries are packed hierarchically around the zone cen- ters. The dispersion curves originate from the QBZ centers, follow a linear relation up to a certain wave vector, and fi- nally become dispersionless at the QBZ boundaries. Experi- mental studies reveal that bulk phonon peak widths are lim- ited by the instrumental resolution up to a certain wave vector and increase rapidly for larger wave vectors. 9,16 This paper reports the results of low-energy surface phonons Rayleigh mode of decagonal and icosahedral qua- sicrystals investigated by inelastic He-atom scattering. The Rayleigh waves propagate along the surface with the polar- ization vector direction of displacement of atoms lying in the sagittal plane the plane defined by the surface normal and propagation direction of the wave. In the continuum limit the displacement of atoms decays exponentially with distance from the surface into the bulk. 21 Two different qua- sicrystals, namely, decagonal d Al 71.8 Ni 14.8 Co 13.4 and icosa- hedral i Al 70.5 Pd 21 Mn 8.5 were studied. These two quasicrys- tals are the most common systems used for surface studies due to the availability of large single grain samples. The decagonal quasicrystal belongs to the class of 2D quasicrys- tals with quasicrystalline planes stacked periodically. In con- trast to two-dimensional 2D quasicrystals, the icosahedral quasicrystal has quasicrystalline order in all three dimen- sions. The high symmetry surfaces of these quasicrystals the tenfold d-Al-Ni-Co and fivefold i-Al-Pd-Mn are found to possess well-defined Rayleigh modes. II. EXPERIMENT Single grain d-Al 71.8 Ni 14.8 Co 13.4 and i-Al 70.5 Pd 21 Mn 8.5 quasicrystals were grown by the Czochralski method. 24,23,22 The i-Al-Pd-Mn was annealed for three months at 820 °C, Refs. 25 and 26. The samples were cut and polished perpen- dicular to the tenfold and fivefold axes, respectively, before surface treatment in the He-atom scattering chamber base pressure 2 10 -10 mbar). The surfaces were prepared by sputtering (Ne + , 1–5 keV and annealing at 650 °C and 850 °C, respectively. The quality of the surface was examined after each sputter-annealing cycle by monitoring the He specular reflected intensity. The cleaning processes were repeated until an optimum specular intensity was obtained. The surface phonons were measured by inelastic He-atom scattering in a time-of-flight TOF setup. Details of the TOF setup are discussed in Ref. 27. The key idea is that the He- atom beam is scattered from the surface and the time of flight of inelastically scattered He atoms is measured. The time of flight along with other experimental parameters such as total scattering angle (90°), beam energy, and angle of incidence  i determine the energy and parallel momentum transfer on the basis of energy and momentum conservation. 28 The TOF spectra were recorded at an elevated sample temperature. This has two important advantages. First, due to the low intensity of inelastically scattered He atoms, the measurement time of each TOF spectrum has to be fairly long to observe pronounced phonon peaks each of the spec- tra presented here was recorded for 3 h. However, at room temperature the surface contaminates within a few hours due PHYSICAL REVIEW B 68, 054205 2003 0163-1829/2003/685/0542056/$20.00 ©2003 The American Physical Society 68 054205-1