Generalized density-of-states and anharmonicity of the low-energy phonon bands from coherent inelastic neutron scattering response in the pyrochlore osmates $A{\text{Os}}_{2}{\text{O}}_{6}$ $(A=\text{K},\phantom{\rule{0.3em}{0ex}}\text{Rb},\phantom{\rule{0.3em}{0ex}}\text{Cs})$

Generalized density-of-states and anharmonicity of the low-energy phonon bands from coherent inelastic neutron scattering response in the pyrochlore osmates $A {Os}_{2} O_{6}$ $(A = K, Rb, Cs)$

Hannu Mutka, Michael Marek Koza, Mark Robert Johnson, Zenji Hiroi, Jun-Ichi Yamaura, and Yohei Nagao

Phys. Rev. B 78, 104307 – Published 30 September 2008

Abstract

Inelastic neutron scattering on powder samples has been employed to investigate the lattice dynamics of alkali-metal osmates. We have obtained the neutron weighted generalized phonon density-of-states from momentum-transfer averaged coherent inelastic response. We have paid special attention to the low-energy modes that have been assigned to the Einstein term of the lattice specific heat and from which a “rattling” behavior of the alkali atoms has been inferred. In all of the three compounds our results reveal a split band with two prominent peaks in the low-energy range associated with the vibrational modes of $A$ atoms. The temperature dependence of these bands has been followed in the range from $1.5 K < T < 300 K$ . The strongest temperature effect is visible in the case of ${KOs}_{2} O_{6}$ , the split band peaking at $ℏ ω \approx 5.5 meV$ , and 6.8 meV at $T = 300 K$ softens on cooling resulting in a main band at about $ℏ ω \approx 3.4 meV$ at $T ≲ 50 K$ , while a smaller softening of $Δ (ℏ ω) < 1 meV$ is seen in the other two compounds. In agreement with earlier work the temperature dependent modifications of the phonon spectrum can be attributed to the weak potential felt by the $A$ atom in the oversized cage where it is located. At low $T$ the characteristic energies for each of the compounds are well correlated with the specific-heat observations published earlier. The powder averaged phonon response shows $Q$ dependence indicating coherent coupling of the alkali atom vibrations with the rest of the lattice.

Received 22 July 2008

DOI:https://doi.org/10.1103/PhysRevB.78.104307

Authors & Affiliations

Hannu Mutka^*, Michael Marek Koza, and Mark Robert Johnson

Institut Laue Langevin, 6 Rue Jules Horowitz, B.P. 156, 38042 Grenoble Cedex 9, France

Zenji Hiroi, Jun-Ichi Yamaura, and Yohei Nagao

Institute for Solid State Physics, University of Tokyo, Kashiwanoha, Kashiwa, Chiba 277-8581, Japan

^*mutka@ill.fr

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Vol. 78, Iss. 10 — 1 September 2008

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Figure 1
(Color online) Panels (a) and (b): $G (ω)$ of ${KOs}_{2} O_{6}$ , ${RbOs}_{2} O_{6}$ and ${CsOs}_{2} O_{6}$ over the full and intermediate energy ranges, based on IN6 upscattering (anti-Stokes) data. In panel (a) the data points of the Cs and Rb compounds have been shifted up by 0.3 and 0.6, respectively, for clarity of presentation. Panel (c): $G (ω) / ω^{2}$ highlighting the low-energy double peaks. $G (ω)$ has been normalized to 27 modes (3/atom of the formula unit) over the full energy range, see Eq. (A5).Reuse & Permissions
Figure 2
(Color online) $S (Q, ω)$ maps of ${KOs}_{2} O_{6}$ and ${RbOs}_{2} O_{6}$ . For each panel the top part shows the Stokes data obtained at IN4 at $T = 1.5 K$ , the one below the anti-Stokes data obtained at IN6 at $T = 300 K$ . The split band at $5 meV ≲ | ℏ ω | ≲ 8 meV$ at $T = 300 K$ is softer at $T = 1.5 K$ (as shown by the white arrows). In ${KOs}_{2} O_{6}$ the low- $T$ response is peaking at $ℏ ω \approx 3.4 meV$ . The [111] zone center $(Q = 1.07 Å^{- 1})$ is marked with the vertical yellow arrow. The horizontal dashed lines indicate the constant energy cut positions of Fig. 3, the vertical ones those of the constant $Q$ -cut positions of Fig. 4.Reuse & Permissions
Figure 3
(Color online) Constant energy $Q$ dependence of ${KOs}_{2} O_{6}$ , ${RbOs}_{2} O_{6}$ , and ${CsOs}_{2} O_{6}$ at $T = 300 K$ along the positions of the low-energy bands at the values indicated on the plot, see Fig. 2. The vertical scale is arbitrary and not the same for all plots.Reuse & Permissions
Figure 4
(Color online) Constant- $Q$ energy dependence of $g_{exp} (Q, ω)$ , Eq. (A4), for ${KOs}_{2} O_{6}$ and ${RbOs}_{2} O_{6}$ at $T = 300 K$ , at the $Q$ positions of the [111] zone center $(Q = 1.1 \pm 0.05 Å^{- 1})$ , and of the $[\frac{3}{2} \frac{3}{2} \frac{3}{2}]$ zone boundary $(Q = 1.6 \pm 0.05 Å^{- 1})$ , see Fig. 2. The vertical scale is arbitrary. The high values seen at $Q = 1.1 Å^{- 1}$ close to zero energy are due to the elastic scattering from the 111 Bragg peak.Reuse & Permissions
Figure 5
(Color online) $T$ dependence of the maxima of the low-energy peaks for the Cs, Rb, and K compounds, lines are guides for the eye. For ${KOs}_{2} O_{6}$ we plot the upper peak position only at $T = 300 K$ as the higher energy peak becomes poorly defined on cooling, see Fig. 6. The left panel with $log (T)$ scale highlights the level of the lower band positions (only those are shown for clarity) at low $T$ , including a comparison with the theory of Dahm and Ueda (Ref. 19) shown by the dashed lines calculated with the anharmonicity parameters $β = 0.1$ , 1, $β = 0.1$ , and $β = 0.025$ for K, Rb, and Cs compounds, respectively.Reuse & Permissions
Figure 6
(Color online) $T$ dependence of the momentum-transfer integrated low-energy anti-Stokes response in ${KOs}_{2} O_{6}$ . In spite of the increasing damping the softening on cooling of the lower energy maximum is rather well defined, while the position of the higher energy maximum is less clear and a shoulder persists with an almost temperature independent position at about $ℏ ω \approx 7 meV$ . The position at $T = 50$ and 300 K of the $T_{2 g}$ mode observed in the Raman study (Ref. 21) is indicated by the vertical arrows.Reuse & Permissions

Physical Review B

covering condensed matter and materials physics

Generalized density-of-states and anharmonicity of the low-energy phonon bands from coherent inelastic neutron scattering response in the pyrochlore osmates $A {Os}_{2} O_{6}$ $(A = K, Rb, Cs)$

Hannu Mutka, Michael Marek Koza, Mark Robert Johnson, Zenji Hiroi, Jun-Ichi Yamaura, and Yohei Nagao

Phys. Rev. B 78, 104307 – Published 30 September 2008

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Generalized density-of-states and anharmonicity of the low-energy phonon bands from coherent inelastic neutron scattering response in the pyrochlore osmates AOs2O6 (A=K,Rb,Cs)

Hannu Mutka, Michael Marek Koza, Mark Robert Johnson, Zenji Hiroi, Jun-Ichi Yamaura, and Yohei Nagao

Phys. Rev. B 78, 104307 – Published 30 September 2008

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Generalized density-of-states and anharmonicity of the low-energy phonon bands from coherent inelastic neutron scattering response in the pyrochlore osmates $A {Os}_{2} O_{6}$ $(A = K, Rb, Cs)$