Post on 26-Mar-2015
Dilatometry
George SchmiedeshoffNHMFL Summer School
May 2010
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 (don’t scale with system size):
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:
dT
dQ
T
QC
“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:
dT
Vd
dT
dV
V
ln1
T
V/V
“Coefficient of volume thermal expansion.”
The fractional change in volume per unit temperature change.
Most dilatometers measure one axis at a time
dT
Ld
dT
dL
LT
LL ln1/
If V=LaLbLc then, one can show:
cba
Nomenclature (just gms?)
)0(
)0()(
)(
)()(
L
LTL
TL
TLTL
L
L
ref
ref
thermal strain
T
LL
)/( thermal expansion
)0(
)0()(
L
LHL
L
L
magnetic strain
H
LL
)/( 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
Anharmonic potential
< 0
Transverse modes
Harmonic ok tooAfter Barron & White (1999).
kBT
Phase Transition: TN
2nd Order Phase Transition, Ehrenfest Relation(s):
pN2M
c
N2
CTV
p
T
c
d
d
S
)(V
S
V
p
TM
N1
L
L
cd
d
1st Order Phase Transition, Clausius-Clapyeron Eq(s).:
Uniaxial with or L, hydrostatic with or V.
Aside: thermodynamics of phase transitions
Phase Transition: TN
Magnetostriction relations tend to be more complicated.
i
i
T
H
p
Slope of phase boundary in T-B plane.
Aside: more fun with Ehrenfest Relations
T
C
p
T p
ii
1-MV
H
T
Grüneisen Theory (one energy scale: Uo)
VUo
.)()(
)(
ln
lnconst
TCT
TV
V
U
p
Mo
e.g.: If Uo = EF (ideal ) then:3
2IFG
Note: (T) const., so “Grüneisen ratio”, /Cp, is also used.
Grüneisen Theory (multiple energy scales: Ui each with Ci and i)
)()()(
)(T
TCT
TVeff
p
Meff
)()(
)(T
TC
TC
ii
iii
Examples: Simple metals:
~ 2)ln(
*)ln(
3
2
Vd
mde
e.g.: phonon, electron, magnon, CEF, Kondo, RKKY, etc.
Example (Noble Metals):
After White & Collins, JLTP (1972).
Also: Barron, Collins & White, Adv. Phys. (1980).
(lattice shown.)
Example (Heavy Fermions):
After deVisser et al. Physica B 163, 49 (1990)
HF(0)
Aside: Magnetic Grüneisen Parameter
SH
HH H
T
TC
TM
1)/(
Specific heat in field
Magnetocaloric effect
Not directly related to dilation…See, for example, Garst & Rosch PRB 72, 205129 (2005).
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)
Substrate
Piezocantilever
Sample L
D
Piezocantilever (from AFM) Dilatometer
After J. –H. Park et al. Rev. Sci. Instrum 80, 116101 (2009)
Capacitive Dilatometer (cartoon)
D
L
Capacitor Plates
Cell Body
Sample
“D3”Rev. Sci. Instrum. 77, 123907 (2006)
(cond-mat/0617396 has fewer typos…)
Cell body: OHFC Cu or titanium.
BeCu spring (c).
Stycast 2850FT (h) and Kapton or sapphire (i) insulation.
Sample (d).
Capacitive Dilatometer
D
AεC o
3He cold finger
15 mm
to better than 1%
LuNi2B2C single crystal 1.6 mm high, 0.6 mm thick
After gms et al., Rev. Sci. Instrum. 77, 123907 (2006).
Data Reduction I, the basic equations.
D
AεC o
dDdL
)(
)()(
ref
ref
TL
TLTL
L
L
T
LL
)/(
(1)
(2)
(3)
(4)
so dTdDdTdL //
1
1
1
1
2
1
ii
ii
ii
ii
i TT
DD
TT
DD
dT
dD(5)
• 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)
• 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….
Data Reduction IIa: remove dc jumps
Cell Effect
CuCuCellCellSampleSample dT
dL
LdT
dL
L
11
After gms et al., Rev. Sci. Instrum. 77, 123907 (2006).
State of the art (IMHO): RT to about 10K.
After Neumeier et al. Rev. Sci. Instrum. 79, 033903 (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
Calibration
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
Operating Region
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
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).
2
MAX
o
C
C1
C
AεD
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 Bud’ko et al. J. Phys. Cond. Matt. 18 8353 (2006).After gms et al., Rev. Sci. Instrum. 77, 123907 (2006).
Phase Diagrams
After Garst & Rosch PRB 72, 205129 (2005).
→ sign change in
Quantum critical points
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). 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 ?).
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