Page 13 - Spring 2010

Fig. 5. Shear elastic modulus for (a) Ba8Ga16Ge30, (b) Sr8Ga16Ge30, and (c) Eu8Ga16Ge30. The inset shows the nuclear density plot for Ba, Sr, and Eu. Adapted from Reference 10.
rattlers leave a distinct fingerprint on the elastic response of the
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material. This is shown in Fig. 5 where the temperature-
dependence of the shear modulus c44 is plotted for Ba8Ga16Ge30, Sr8Ga16Ge30, and Eu8Ga16Ge30, compounds known as clathrates. Ga and Ge form the host structure, while Ba, Ga, and Eu are the rattlers, residing in the oversized cage. The data were taken on single crystals, using RUS in a temperature region from 3 K to 300 K. The figures illustrate that these three materials—with
very similar structure—behave quite differently. The Ba- clathrate shows an almost normal behavior, the Sr-clathrate shows a small but relevant “dip” at low temperatures, and the Eu-clathrate displays a very unusual temperature-dependence, reaching a maximum at about 100 K, before plunging down. The difference in behavior between the three compounds can be related to the size of the rattler and its position in the cage. The insets in Fig. 5 show what is known as the nuclear density, i.e., the probability of finding the Ba, Sr, or Eu nucleus at differ- ent positions near the center of the cage. For Ba, a relatively large atom, the probability is clearly centered at the center of the cage, but somewhat smeared, consistent with the picture of an atom that is rattling around its equilibrium position. The Sr nuclear density is even broader than that for the Ba atoms and indicates a substantial probability for the Sr atom—which is smaller than the Ba atom—to move off the site center to one of four crystallographically equivalent positions. The Eu nuclear density distribution unmistakably shows the tendency of the Eu atoms to move away from the site center. Four separate peaks are resolved in the nuclear density maps, indicating four equiv- alent positions for the tiny Eu atom to reside inside its oversized cage. Instead of being a simple rattler, moving around its cen- tral position, the rattling Eu atom is more adequately described as traveling between the four equivalent positions in the cage, resulting in a dramatic decrease in the elastic constants at low temperatures.
Unusual thermal response
Resonances can also help to unravel puzzling behavior of solids with strange thermal behavior that are of increasing importance as micro-scale silicon-based devices come into use. Historically, solids with negative thermal expansion coefficients and elastic stiffnesses with a temperature- dependence that is opposite to “regular behavior” have been essential in solving the “longitude problem,”11 i.e., the prob- lem of determining one’s longitude while at sea for many months. Noting that most solids expand and soften on warm- ing, new materials with zero thermal expansion or zero change in stiffness on warming were needed to construct well-temperature-compensated mechanical clocks. In 1897, Guillaume discovered an amazing alloy of Fe containing 35% Ni. That alloy—now known as Invar—exhibited the remark- able property of zero thermal expansion and was of such great technological importance that its discoverer received
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the 1920 Nobel Prize in physics. Poorly understood, the
effect’s origins became especially puzzling when Grüneisen in 1912 proposed a way of understanding the general prob- lem of thermal expansion by relating the variation of elastic stiffness with pressure to variation in the volume with tem- perature in an elegant general way. His theory worked for a remarkable variety of materials but not for Fe65Ni35.
Today, the promise of micro-scale devices for applica- tions where a precise temperature-independent mechanical response is needed requires that the mainstay-semiconduc- tors of lithographically produced micro machines have mechanical responses compensated by materials with “back- ward” thermal properties. From liquid helium temperatures to just above room temperature, ZrWO8 has a negative ther- mal expansion coefficient.13,14 That is, this cubic-structure
Listening to materials 9