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**George Schmiedeshoff NHMFL Summer School May 2010**

Dilatometry George Schmiedeshoff NHMFL Summer School May 2010

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Dilation: ΔV (or ΔL) Intensive Parameters (don’t scale with system size): 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. Volume is a quintessential extensive parameter

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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. Fahrenheit: First precision thermometric instrument. Gruneisen: All the original papers are in German. Translators???

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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=LaLbLc then, one can show:

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**Nomenclature (just gms?)**

thermal strain magnetic strain thermal expansion 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). TbNi2Ge2 (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 kBT < 0 Transverse modes Harmonic ok too After Barron & White (1999).

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**Aside: thermodynamics of phase transitions**

Phase Transition: TN Aside: thermodynamics of phase transitions 2nd Order Phase Transition, Ehrenfest Relation(s): Pc = uniaxial pressure along c-axis 1st Order Phase Transition, Clausius-Clapyeron Eq(s).: Uniaxial with or L, hydrostatic with or V.

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**Aside: more fun with Ehrenfest Relations**

Phase Transition: TN Aside: more fun with Ehrenfest Relations Magnetostriction relations tend to be more complicated. Pc = uniaxial pressure along c-axis Slope of phase boundary in T-B plane.

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**Grüneisen Theory (one energy scale: Uo)**

Gruneisen Parameter. Einstein Solid, Schottky system, ideal Fermi gas. e.g.: If Uo = EF (ideal ) then: Note: (T) const., so “Grüneisen ratio”, /Cp, is also used.

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**Grüneisen Theory (multiple energy scales: Ui each with Ci and i)**

e.g.: phonon, electron, magnon, CEF, Kondo, RKKY, etc. Examples: Simple metals: ~ 2

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**Example (Noble Metals):**

After White & Collins, JLTP (1972). Also: Barron, Collins & White, Adv. Phys. (1980). (lattice shown.) T-dependence finite but small.

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**Example (Heavy Fermions):**

1- Significant temperature dependence. 2- Big at T=0. HF(0) After deVisser et al. Physica B 163, 49 (1990)

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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)**

Sample L Substrate

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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)**

Capacitor Plates D Cell Body L Sample

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Rev. Sci. Instrum. 77, (2006) (cond-mat/ has fewer typos…) “D3” 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**

3He cold finger LuNi2B2C single crystal 1.6 mm high, 0.6 mm thick 15 mm to better than 1% After gms et al., Rev. Sci. Instrum. 77, (2006).

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**Data Reduction I, the basic equations.**

(1) so (2) (3) (4) (5)

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**Data Reduction II (gms generic approach)**

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).

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**Data Reduction IIa: remove dc jumps**

Jumps associated with strain relief in dilatometer are localized, they don’t 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….

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**Cell Effect Cell Effect vanishes below about 2K.**

Feature near 240K: Kapton? After gms et al., Rev. Sci. Instrum. 77, (2006).

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**State of the art (IMHO): RT to about 10K.**

Kapton bad Thermal cycle Yikes! fused quartz glass: very small thermal expansion compared to Cu above ~10K. Operates in low-pressure helium exchange gas: thermal contact. After Neumeier et al. Rev. Sci. Instrum. 79, (2008).

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**Calibration Operating Region CMAX 65 pF**

Use sample platform to push against lower capacitor plate. Rotate sample platform (θ), measure C. Aeff from slope (edge effects). Aeff = Ao to about 1%?! “Ideal” capacitive geometry. Consistent with estimates. CMAX >> C: no tilt correction. Ao is the physical, machined, or directly measured area. Change Cmax w/ trivial rebuild. CMAX 65 pF 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 → CMAX as D → DSHORT (plates touch) and that after Pott & Schefzyk J. Phys. E 16, 444 (1983). For our design, CMAX = 100 pF corresponds to an angular misalignment of about 0.1o. 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. “enormous feature” equivalent to ΔL of about 10 microns!

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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**

We’re screwed: can’t see the temperature dependence of most solids above the dielectric background…

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**Things to study: Fermi surfaces**

After gms et al., Rev. Sci. Instrum. 77, (2006). After Bud’ko et al. J. Phys. Cond. Matt (2006).

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Phase Diagrams

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**Quantum critical points**

After Garst & Rosch PRB 72, (2005). → sign change in

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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 Zapf’s dilatometry lecture from last summer.

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Recommended: Book: Heat Capacity and Thermal Expansion at Low Temperatures, Barron & White, 1999. Collection of Review Articles: Thermal Expansion of Solids, v.I-4 of Cindas Data Series on Material Properties, ed. by C.Y. Ho, 1998. Book (broad range of data on technical materials etc.): Experimental Techniques for Low Temperature Measurements, Ekin, 2006. Book: Magnetostriction: Theory and Applications of Magnetoelasticity, Etienne du Trémolet de Lacheisserie, 1993. Book: Thermal Expansion, Yates, 1972. Review Article: Barron, Collins, and White; Adv. Phys. 29, 609 (1980). Review Article: Chandrasekhar & Fawcett; Adv. Phys. 20, 775 (1971).

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