Dilatometry

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Dilatometry

• George Schmiedeshoff
• NHMFL Summer School
• May 2010

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Dilation ?V (or ?L)
Intensive Parameters (dont scale with system
size)

• T thermal expansion ß dln(V)/dT
• H magnetostriction ? ?L(H)/L
• P compressibility ? dln(V)/dP
• E electrostriction ? ?L(E)/L
• etc.

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A (very) Brief History

• Heron of Alexandria (0100) Fire heats air, air
  expands, opening temple doors (first practical
  application).
• Galileo (16007) Gas thermometer.
• Fahrenheit (1714) Mercury-in-glass thermometer.
• Mie (1903) First microscopic model.
• Grüneisen (1908) ß(T)/C(T) constant.

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Start at T0, add some heat ?Q

• Sample warms up T increases by ?T

Heat Capacity for a specific bit of stuff.
Or Specific Heat per mole, per gram, per
whatever.
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Start at T0, add some heat ?Q

• 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 VLaLbLc 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 ß (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

• gt 0
• Longitudinal modes
• Anharmonic potential

kBT

• lt 0
• Transverse modes
• Harmonic ok too

After Barron White (1999).
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Phase Transition TN
Aside thermodynamics of phase transitions
2nd Order Phase Transition, Ehrenfest
Relation(s)
1st Order Phase Transition, Clausius-Clapyeron
Eq(s).
Uniaxial with ? or L, hydrostatic with ? or V.
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Phase Transition TN
Aside more fun with Ehrenfest Relations
Magnetostriction relations tend to be more
complicated.
Slope of phase boundary in T-B plane.
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Grüneisen Theory (one energy scale Uo)
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.)
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Example (Heavy Fermions)
?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, 205129 (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, 033909 (2010).

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Piezocantilever Dilatometer (cartoon)
D
Piezocantilever
L
Sample
Substrate
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Piezocantilever (from AFM) Dilatometer
After J. H. Park et al. Rev. Sci. Instrum 80,
116101 (2009)
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Capacitive Dilatometer (cartoon)
Capacitor Plates
D
Cell Body
L
Sample
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D3
Rev. Sci. Instrum. 77, 123907 (2006) (cond-mat/061
7396 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
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, 123907
(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 PRLunless there is a cell effect (T gt
  2 K dilatometer dependent).

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Data Reduction IIa remove dc jumps

• Jumps associated with strain relief in
  dilatometer are localized, they dont affect
  nearby slopes.
• Take a simple numerical derivative, (5).
• Delete d-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
After gms et al., Rev. Sci. Instrum. 77, 123907
(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,
033903 (2008).
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Calibration
Operating Region

• Use sample platform to push against lower
  capacitor plate.
• Rotate sample platform (?), measure C.
• Aeff from slope (edge effects).
• Aeff Ao to about 1?!
• Ideal capacitive geometry.
• Consistent with estimates.
• CMAX gtgt C no tilt correction.

CMAX ? 65 pF
After gms et al., Rev. Sci. Instrum. 77, 123907
(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.

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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 e(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. 18 8353
(2006).
After gms et al., Rev. Sci. Instrum. 77, 123907
(2006).
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Phase Diagrams
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Quantum critical points
After Garst Rosch PRB 72, 205129 (2005). ? sign
change in ?
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Structural transitions
After Lashley et al., PRL 97, 235701 (2006).
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Novel superconductors
After Correa et al., PRL 98 087001 (2007).
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Capacitive Dilatometers The Good The Bad

• Cell effect (T 2K).
• Magnetic torque effects.
• Thermal and magnetic hysteresis.
• Thermal contact to sample in vacuum (T 100mK
  ?).

• Small (scale up or down).
• Open architecture.
• Rotate in-situ (NHMFL/TLH).
• Vacuum, gas, or liquid (magnetostriction only?).
• Sub-angstrom precision.

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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, 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).
• http//departments.oxy.edu/physics/gms/Dilatometry
  Info.htm