A (very) Brief History Heron of Alexandria (0±100): Fire heats air, air expands, opening temple doors (first practical application…). Galileo (1600±7): Gas thermometer. Fahrenheit (1714): Mercury-in-glass thermometer. Mie (1903): First microscopic model. Grüneisen (1908): β(T)/C(T) ~ constant.
Start at T=0, add some heat ΔQ: Sample warms up: T increases by ΔT: Heat Capacity for a specific bit of stuff. Or: Specific Heat per mole, per gram, per whatever.
Start at T=0, add some heat ΔQ: …and the volume changes: Coefficient of volume thermal expansion. The fractional change in volume per unit temperature change.
Most dilatometers measure one axis at a time If V=L a L b L c then, one can show:
Nomenclature (just gms?) thermal strain thermal expansion magnetic strain magnetostriction Add volume or linear as appropriate.
C and β (and/or ) - closely related: Features at phase transitions (both). Shape (~both). Sign change ( ). Anisotropic (, C in field). TbNi 2 Ge 2 (Ising Antiferromagnet) after gms et al. AIP Conf. Proc. 850, 1297 (2006). (LT24)
Classical vibration (phonon) mechanisms > 0 Longitudinal modes Anharmonic potential < 0 Transverse modes Harmonic ok too After Barron & White (1999). kBTkBT
Phase Transition: T N 2 nd Order Phase Transition, Ehrenfest Relation(s): 1 st Order Phase Transition, Clausius-Clapyeron Eq(s).: Uniaxial with or L, hydrostatic with or V. Aside: thermodynamics of phase transitions
Phase Transition: T N Magnetostriction relations tend to be more complicated. Slope of phase boundary in T-B plane. Aside: more fun with Ehrenfest Relations
Grüneisen Theory (one energy scale: U o ) e.g.: If U o = E F (ideal ) then: Note: (T) const., so Grüneisen ratio, /C p, is also used.
Grüneisen Theory (multiple energy scales: U i each with C i and i ) Examples: Simple metals: ~ 2 e.g.: phonon, electron, magnon, CEF, Kondo, RKKY, etc.
Example (Noble Metals): After White & Collins, JLTP (1972). Also: Barron, Collins & White, Adv. Phys. (1980). ( lattice shown.)
Example (Heavy Fermions): After deVisser et al. Physica B 163, 49 (1990) HF (0)
Aside: Magnetic Grüneisen Parameter Specific heat in field Magnetocaloric effect Not directly related to dilation… See, for example, Garst & Rosch PRB 72, (2005).
Dilatometers Mechanical (pushrod etc.) Optical (interferometer etc.). Electrical (Inductive, Capacitive, Strain Gauges). Diffraction (X-ray, neutron). Others (absolute & differential). NHMFL: Capacitive – available for dc users. Piezocantilever – under development. Two months/days ago: optical technique for magnetostriction in pulse fields. Daou et al. Rev. Sci. Instrum. 81, (2010).
Piezocantilever Dilatometer (cartoon) Substrate Piezocantilever Sample L D
Piezocantilever (from AFM) Dilatometer After J. –H. Park et al. Rev. Sci. Instrum 80, (2009)
Capacitive Dilatometer (cartoon) D L Capacitor Plates Cell Body Sample
D3 Rev. Sci. Instrum. 77, (2006) (cond-mat/ has fewer typos…) Cell body: OHFC Cu or titanium. BeCu spring (c). Stycast 2850FT (h) and Kapton or sapphire (i) insulation. Sample (d).
Capacitive Dilatometer 3 He cold finger 15 mm to better than 1% LuNi 2 B 2 C single crystal 1.6 mm high, 0.6 mm thick After gms et al., Rev. Sci. Instrum. 77, (2006).
Data Reduction I, the basic equations. (1) (2) (3) (4) so (5)
Measure C(T,H) - I like the Andeen-Hagerling 2700a. A from calibration or measure with micrometer. Use appropriate dielectric constant. Calculate D(T,H) from (1) and remove any dc jumps. Measure sample L with micrometer or whatever. Calculate thermal expansion using (2). Integrate (4) and adjust constant offset for strain. Submit to PRL…unless there is a cell effect (T > ~2 K dilatometer dependent). Data Reduction II (gms generic approach)
Jumps associated with strain relief in dilatometer are localized, they dont affect nearby slopes. Take a simple numerical derivative, (5). Delete δ -function-like features. Integrate back to D(T). Differentiate with polynomial smoothing, or fit function and differentiate function, or spline fit, or…. Data Reduction IIa: remove dc jumps
Cell Effect After gms et al., Rev. Sci. Instrum. 77, (2006).
State of the art (IMHO): RT to about 10K. After Neumeier et al. Rev. Sci. Instrum. 79, (2008). fused quartz glass: very small thermal expansion compared to Cu above ~10K. Operates in low-pressure helium exchange gas: thermal contact. Yikes! Kapton bad Thermal cycle
Calibration Use sample platform to push against lower capacitor plate. Rotate sample platform (θ), measure C. A eff from slope (edge effects). A eff = A o to about 1%?! Ideal capacitive geometry. Consistent with estimates. C MAX >> C: no tilt correction. C MAX 65 pF Operating Region After gms et al., Rev. Sci. Instrum. 77, (2006).
Tilt Correction If the capacitor plates are truly parallel then C as D 0. More realistically, if there is an angular misalignment, one can show that C C MAX as D D SHORT (plates touch) and that after Pott & Schefzyk J. Phys. E 16, 444 (1983). For our design, C MAX = 100 pF corresponds to an angular misalignment of about 0.1 o. Tilt is not always bad: enhanced sensitivity is exploited in the design of Rotter et al. Rev. Sci. Instrum 69, 2742 (1998).
Kapton Bad (thanks to A. deVisser and Cy Opeil) Replace Kapton washers with alumina. New cell effect scale. Investigating sapphire washers.
Torque Bad The dilatometer is sensitive to magnetic torque on the sample (induced moments, permanent moments, shape effects…). Manifests as irreproducible magnetostriction (for example). Best solution (so far…): glue sample to platform. Duco cement, GE varnish, N-grease… Low temperature only. Glue contributes above about 20 K.
Hysteresis Bad Cell is very sensitive to thermal gradients: thermal hysteresis. But slope is unaffected if T changes slowly. Magnetic torque on induced eddy-currents: magnetic hysteresis. But symmetric hysteresis averages to zero: Field-dependent cell effect Cu dilatometer.
Environmental effects immersed in helium mixture (mash) of operating dilution refrigerator. cell effect is ~1000 times larger than Cu in vacuum! due to mixture ε(T).
Thermal expansion of helium liquids
Things to study: Fermi surfaces After Budko et al. J. Phys. Cond. Matt (2006). After gms et al., Rev. Sci. Instrum. 77, (2006).
After Garst & Rosch PRB 72, (2005). sign change in Quantum critical points
Structural transitions After Lashley et al., PRL 97, (2006).
Novel superconductors After Correa et al., PRL (2007).
Capacitive Dilatometers: The Good & The Bad Small (scale up or down). Open architecture. Rotate in-situ (NHMFL/TLH). Vacuum, gas, or liquid (magnetostriction only?). Sub-angstrom precision. Cell effect (T 2K). Magnetic torque effects. Thermal and magnetic hysteresis. Thermal contact to sample in vacuum (T 100mK ?).
How to get good dilatometry data at NHMFL: Avoid torque (non-magnetic sample, or zero field, or glue). Ensure good thermal contact (sample, dilatometer & thermometer). Well characterized, small cell effect, if necessary. Avoid kapton (or anything with glass/phase transitions in construction). Avoid bubbles (when running under liquid helium etc.). Use appropriate dielectric corrections (about 5% for liquid helium, but very temperature dependent), unless operating in vacuum. Mount dilatometer to minimize thermal and mechanical stresses. Adapted from Lillian Zapfs dilatometry lecture from last summer.
Recommended: Book: Heat Capacity and Thermal Expansion at Low Temperatures, Barron & White, Collection of Review Articles: Thermal Expansion of Solids, v.I-4 of Cindas Data Series on Material Properties, ed. by C.Y. Ho, Book (broad range of data on technical materials etc.): Experimental Techniques for Low Temperature Measurements, Ekin, Book: Magnetostriction: Theory and Applications of Magnetoelasticity, Etienne du Trémolet de Lacheisserie, Book: Thermal Expansion, Yates, Review Article: Barron, Collins, and White; Adv. Phys. 29, 609 (1980). Review Article: Chandrasekhar & Fawcett; Adv. Phys. 20, 775 (1971).