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