##
###
Elastic effects in torsional oscillators containing solid helium

J. R. Beamish, A. D. Fefferman, A. Haziot, X. Rojas, S. Balibar

2012
*
Physical Review B
*

A number of recent experiments have used torsional oscillators to study the behavior of solid helium. The oscillator frequencies increased at temperatures below 200 mK, an effect attributed to decoupling of a fraction of the helium mass-the signature of a "supersolid" phase. However, helium's shear modulus also increases below 200 mK and the frequency of a torsional oscillator depends on its elastic properties, as well as on its inertia. In many experiments helium is introduced via a hole in
## more »

... torsion rod, where its shear modulus contributes to the stiffness of the rod. In oscillators with relatively large torsion rod holes, changes in the helium's shear modulus could produce the entire low temperature frequency shifts that have been interpreted as mass decoupling. For these oscillators we also find that the known elastic properties of helium in the torsion rod can explain the observed TO amplitude dependence (which has been interpreted as a critical velocity) and the TO dissipation peak. However, in other oscillators these elastic effects are small and the observed frequency changes must have a different origin. In its simplest form, a torsional oscillator (TO) consists of a rigid "head" (with moment of inertia I ) attached to a stationary base by a torsion rod (with torsional stiffness K). Its resonant frequency is given by f = (1/2π ) √ K/I and can be measured very precisely for a high Q oscillator. If the torsion rod's stiffness is constant, this provides a direct and sensitive technique to measure the moment of inertia. Such oscillators have been widely used to study superfluidity in liquid 4 He and 3 He, by confining the helium in narrow channels or small pores in the TO head. 1-3 In appropriate geometries, the zero viscosity superfluid fraction decouples from the walls of the cavity, reducing the effective moment of inertia. The increase in the TO frequency is then a direct measurement of the superfluid density ρ s . The TO technique has recently been used to study the behavior of solid 4 He. 4-18 At temperatures below 200 mK, the TO frequency increases and, in analogy to measurements with liquid helium, this has usually been interpreted in terms of mass decoupling-the "nonclassical rotational inertia" (NCRI) which would characterize a supersolid. However, the behavior of a torsional oscillator can be sensitive to a number of effects in addition to the solid helium's inertia, for example, the pressure dependence of the TO background, possible slip at the walls, dissipation in the helium and, most importantly, the solid helium's shear rigidity. [19] [20] [21] Any increase in the shear modulus of the helium will stiffen the oscillator and raise its frequency, an effect which could be misinterpreted as mass decoupling. Recent low frequency measurements showed that the shear modulus of solid 4 He, μ He , increases significantly below 200 mK, with the same dependence on temperature, 3 He concentration, and frequency as the TO anomaly. 22-25 This behavior has been attributed to dislocations, which are mobile and soften the crystal at high temperatures but are pinned by 3 He impurities below 200 mK. The shear modulus also has an amplitude dependence 26 which closely resembles 7 that seen in TO measurements. These similarities raise the possibility that the TO behavior is an artifact of elastic changes which mimic mass decoupling in a supersolid. These elastic effects are not expected to be large, since the shear modulus of solid helium is much smaller than that of typical TO materials such as beryllium copper (μ He /μ BeCu ≈ 2.8 × 10 −4 ). However, the frequency changes attributed to NCRI can also be quite small so it is important to compare the two. The magnitude and frequency dependence of the elastic effects of helium in a TO depend on the design of the oscillator. If the oscillator head is not completely rigid, e.g., if it is a cylinder with relatively thin walls, then solid helium can increase the head's torsional stiffness and raise the TO frequency. Elastic effects can be even larger in heads with thin annular sample spaces where the solid helium acts as a "glue" between the inner and outer walls of the annulus. 19,20 Even if the oscillator head is rigid, solid helium is very soft and some of it will "elastically decouple," i.e., will oscillate with larger amplitude than the walls of its container. Any stiffening of the helium reduces this overshoot, raising the TO frequency and mimicking mass decoupling. This effect is reduced when the helium is confined in a narrow annulus and has a characteristic f 2 frequency dependence 21,27 that distinguishes it from a true change in inertia. Simple estimates, as well as detailed numerical modeling of particular TO geometries, suggest that this effect is too small to explain the apparent NCRI in most experiments. 27 The most direct way in which solid helium can raise the TO frequency is through its contribution to the stiffness of the torsion rod. 28-31 Most oscillators introduce helium into the sample space via a hole through this rod. When this helium freezes, its shear modulus will stiffen the torsion rod, as will any subsequent increase in the solid helium's shear modulus. This effect is independent of frequency and so is difficult to distinguish from mass decoupling. If the torsion rod's outer radius is r o and its center hole has radius r i , the shear modulus of solid helium in the rod raises the TO frequency by an amount 32 180501-1 1098-0121/2012/85(18)/180501 (5)

doi:10.1103/physrevb.85.180501
fatcat:fv4iwbx4lvcmvhlqslp6y75js4