1. George Schmiedeshoff NHMFL Summer School May 2010
Dilatometry
George Schmiedeshoff
NHMFL Summer School
May 2010

2. 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

3. 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???

4. 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.

5. 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.

6. Most dilatometers measure one axis at a time
If V=LaLbLc then, one can show:

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

8. 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)

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

10. 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.

11. 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.

12. 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.

13. Grüneisen Theory (multiple energy scales: Ui each with Ci and i)
e.g.: phonon, electron, magnon, CEF, Kondo, RKKY, etc.
Examples: Simple metals: ~ 2

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

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

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

17. 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).

18. Piezocantilever Dilatometer (cartoon)
Sample L
Substrate

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

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

21. 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).

22. 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).

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

24. 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).

25. 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….

26. Cell Effect Cell Effect vanishes below about 2K.
Feature near 240K: Kapton?
After gms et al., Rev. Sci. Instrum. 77, (2006).

27. 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).

28. 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).

29. 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).

30. 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!

31. 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.

32. 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.

33. 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).

34. Thermal expansion of helium liquids
We’re screwed: can’t see the temperature dependence of most solids above the dielectric background…

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

36. Phase Diagrams

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

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

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

40. 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 ?).

41. 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.

42. 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).