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Dilatometry George Schmiedeshoff NHMFL Summer School May 2010

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Dilation: ΔV (or ΔL) T: thermal expansion: β = dln(V)/dT H: magnetostriction: λ = ΔL(H)/L P: compressibility: κ = dln(V)/dP E: electrostriction: ξ = ΔL(E)/L etc. Intensive Parameters (dont scale with system size):

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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.

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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.

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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.

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Most dilatometers measure one axis at a time If V=L a L b L c then, one can show:

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Nomenclature (just gms?) thermal strain thermal expansion magnetic strain magnetostriction Add volume or linear as appropriate.

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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)

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Classical vibration (phonon) mechanisms > 0 Longitudinal modes Anharmonic potential < 0 Transverse modes Harmonic ok too After Barron & White (1999). kBTkBT

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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

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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

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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.

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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.

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Example (Noble Metals): After White & Collins, JLTP (1972). Also: Barron, Collins & White, Adv. Phys. (1980). ( lattice shown.)

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Example (Heavy Fermions): After deVisser et al. Physica B 163, 49 (1990) HF (0)

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Aside: Magnetic Grüneisen Parameter Specific heat in field Magnetocaloric effect Not directly related to dilation… See, for example, Garst & Rosch PRB 72, (2005).

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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).

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Piezocantilever Dilatometer (cartoon) Substrate Piezocantilever Sample L D

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Piezocantilever (from AFM) Dilatometer After J. –H. Park et al. Rev. Sci. Instrum 80, (2009)

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Capacitive Dilatometer (cartoon) D L Capacitor Plates Cell Body Sample

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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).

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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).

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Data Reduction I, the basic equations. (1) (2) (3) (4) so (5)

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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)

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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

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Cell Effect After gms et al., Rev. Sci. Instrum. 77, (2006).

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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

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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).

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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).

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Kapton Bad (thanks to A. deVisser and Cy Opeil) Replace Kapton washers with alumina. New cell effect scale. Investigating sapphire washers.

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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.

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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.

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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).

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Thermal expansion of helium liquids

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Things to study: Fermi surfaces After Budko et al. J. Phys. Cond. Matt (2006). After gms et al., Rev. Sci. Instrum. 77, (2006).

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Phase Diagrams

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After Garst & Rosch PRB 72, (2005). sign change in Quantum critical points

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Structural transitions After Lashley et al., PRL 97, (2006).

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Novel superconductors After Correa et al., PRL (2007).

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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 ?).

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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.

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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).

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