Dilatometry
George Schmiedeshoff
NHMFL Summer School
May 2010
Dilation: ΔV (or ΔL)
Intensive Parameters (don’t scale with system size):

T: thermal expansion:
H: magnetostriction:
P: compressibility:
E: electrostriction:

etc.



β = dln(V)/dT
λ = ΔL(H)/L
κ = dln(V)/dP
ξ = ΔL(E)/L
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:
Q dQ
C

T dT
“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:
V/V 1 dV d ln V



T
V dT
dT
“Coefficient of volume thermal
expansion.”
The fractional change in volume
per unit temperature change.
Most dilatometers measure one axis at a time
L / L 1 dL d ln L



T
L dT
dT
If V=LaLbLc then, one can show:
   a  b   c
Nomenclature
(just gms?)
L L(T )  L(Tref ) L(T )  L(0)


L
L(Tref )
L(0)
L L( H )  L(0)

L
L(0)
thermal strain
magnetic strain

 ( L / L )
T
thermal expansion

 ( L / L )
H
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).
TbNi2Ge2
(Ising Antiferromagnet)
after gms et al. AIP Conf. Proc. 850, 1297 (2006). (LT24)
Classical vibration (phonon) mechanisms
>0
Longitudinal modes
kBT
Anharmonic potential
<0
Transverse modes
Harmonic ok too
After Barron & White (1999).
Phase Transition: TN
Aside: thermodynamics of phase transitions
2nd Order Phase Transition,
Ehrenfest Relation(s):
dTN2
 c
 VM TN2
dp c
C p
1st Order Phase Transition,
Clausius-Clapyeron Eq(s).:
( LL )
dTN1 V

 VM
dpc
S
S
Uniaxial with  or L, hydrostatic with  or V.
Phase Transition: TN
Aside: more fun with Ehrenfest Relations
 T   T  -1  C p 
i       VM   
T 
 pi   H 
H
 i

T p i
Magnetostriction relations tend
to be more complicated.
Slope of phase boundary in
T-B plane.
Grüneisen Theory (one energy scale: Uo)
 ln U o
VM  (T )


 const.
 ln V  (T )C p (T )
Uo V

e.g.: If Uo = EF (ideal ) then:
IFG 
2
3
Note: (T) const., so “Grüneisen ratio”, /Cp, is also used.
Grüneisen Theory (multiple energy scales: Ui each with Ci and i)
 C (T )

 C (T )
i
i
i
e.g.: phonon, electron,
magnon, CEF, Kondo,
RKKY, etc.
 (T )
i
i
VM  (T )
eff 
 eff (T )
 (T )C p (T )
Examples: Simple metals:
~ 2
e 
2 d ln(m*)

3 d ln(V )
Example (Noble Metals):
After White & Collins, JLTP (1972).
Also: Barron, Collins & White, Adv. Phys. (1980).
(lattice shown.)
Example (Heavy Fermions):
HF(0)
After deVisser et al.
Physica B 163, 49 (1990)
Aside: Magnetic Grüneisen Parameter
(M / T ) H 1 T
H  

CH
T H
Specific heat in field
Magnetocaloric effect
Not directly related to dilation…
See, for example, Garst & Rosch PRB 72,
205129 (2005).
S
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).
Piezocantilever Dilatometer (cartoon)
D
Piezocantilever
Sample
Substrate
L
Piezocantilever (from AFM) Dilatometer
After J. –H. Park et al. Rev. Sci.
Instrum 80, 116101 (2009)
Capacitive Dilatometer (cartoon)
Capacitor
Plates
Cell
Body
Sample
D
L
Rev. Sci. Instrum. 77, 123907 (2006)
“D3”
 Cell body: OHFC Cu
or titanium.
 BeCu spring (c).
 Stycast 2850FT (h)
and Kapton or sapphire
(i) insulation.
 Sample (d).
(cond-mat/0617396 has fewer typos…)
Capacitive Dilatometer
3He
cold
finger
LuNi2B2C single
crystal 1.6 mm high,
0.6 mm thick
15 mm
C  εo
A
D
to better
than 1%
After gms et al., Rev. Sci. Instrum. 77, 123907 (2006).
Data Reduction I, the basic equations.
A
C  εo
D
dL  dD
(1)
dL / dT  dD / dT
so
L L(T )  L(Tref )

L
L(Tref )
 ( L / L )

T
(3)
(4)
dD
1  Di 1  Di Di  Di 1  (5)
 


dT i 2  Ti 1  Ti
Ti  Ti 1 
(2)
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).
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….
Cell Effect
 Sample 
1 dL
L dT
Sample  Cell 
1 dL
L dT
Cell  Cu
After gms et al., Rev. Sci. Instrum. 77, 123907 (2006).
  Cu
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).
Calibration
Operating
Region


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.
CMAX  65 pF
After gms et al., Rev. Sci. Instrum. 77, 123907 (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 → CMAX as D → DSHORT (plates touch) and that
εoA   C
1  
D
C   CMAX




2



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).
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 gms et al., Rev. Sci. Instrum. 77, 123907 (2006).
After Bud’ko et al. J. Phys. Cond. Matt. 18 8353 (2006).
Phase Diagrams
Quantum critical points
After Garst & Rosch PRB 72, 205129 (2005).
→ sign change in 
Structural transitions
After Lashley et al., PRL 97, 235701 (2006).
Novel superconductors
After Correa et al., PRL
98 087001 (2007).
Capacitive Dilatometers: The Good & The Bad

Small (scale up or down).

Cell effect (T ≥ 2K).

Open architecture.

Magnetic torque effects.

Rotate in-situ (NHMFL/TLH).

Thermal and magnetic hysteresis.


Vacuum, gas, or liquid
(magnetostriction only?).
Sub-angstrom precision.

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 Zapf’s dilatometry lecture from last summer.
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/DilatometryInfo.htm