Dilatometry measuring length changes of your sample thermal expansion magnetostriction
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Dilatometry Measuring length-changes of your sample thermal expansion, magnetostriction, …. Vivien Zapf NHMFL-LANL. Heron of Alexandria (~ 0 B.C.). Today: Applications too numerous to list.

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DilatometryMeasuring length-changes of your samplethermal expansion, magnetostriction, …

Vivien Zapf

NHMFL-LANL


Heron of Alexandria (~ 0 B.C.)


Today: Applications too numerous to list

  • We still use thermal expansion for everything from car engines to nuclear power plant cooling regulation

  • Affects design of sidewalks, bridges, cryostats, …

Thermal expansion within a solid phase is much smaller

but can be an invaluable tool for probing fundamental physics


NiCl2-4SC(NH2)2: an antiferromagnetic quantum magnet

Hc2

Magnetostriction

DL/L (%)

Hc1

Lc

La

c

H

a


Dilatometry

  • T: thermal expansion: a=1/L(dL/dT); β = dln(V)/dT

  • H: magnetostriction: λ = ΔL(H)/L

  • P: compressibility: κ = dln(V)/dP

  • E: electrostriction: ξ = ΔL(E)/L

  • etc.


How to measure Dilatometry

  • Mechanical (pushrod etc.)

  • Optical (interferometer etc.).

  • Electrical (Inductive, Capacitive, Strain Gauges).

  • Diffraction (X-ray, neutron).

  • Others (absolute & differential)


Capacitive Dilatometer(Cartoon)

C

Capacitor Plates

Cell Body

D

D

Sample

L

Extra credit question:

Why don’t we put the sample between the capacitor plates?


George Schmiedeshoff, Occidental College

Rev. Sci. Instrum. 77, 123907 (2006)

Stationary capacitor plate

Moveable capacitor plate

Spring

(CuBe plate)

Sample

Sample screw

Use of needle instead of plate on top of sample means sample faces don’t need to be perfectly parallel


Why a capacitive dilatometer?

  • Fantastically sensitive

    • Sub-Angstrom resolution of length changes on a mm-sized sample

  • Versatile: wide range of signal sizes, sample sizes and shapes

  • Recall Albert’s talk on noise: no intrinsic noise in a capacitance measurement

  • Useful for the ranges of T and H at the magnet lab(20 mK to ~30 K, 0 to 45 T)


H

Dilatometer works at various orientations to the magnetic field.

Rotators available at LANL and Tallahassee


Capacitance measurement

Two shielded, grounded coax cables

Capacitance bridge (e.g. AH 2300 bridge or GC 1615)


Rev. Sci. Instrum. 77, 123907 (2006)


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


Cell Effect

High magnetic fields: Use e.g. titanium instead of Cu body to create less eddy currents in magnetic fields

High temperatures: Use quartz/sapphire (see work of John Neumeier)

Slide courtesy G. Schmiedeshoff


Other backgrounds:

Dielectric constant of liquid helium between capacitor plates

Magnetic impurities in commercial titanium

These effects are small compared to some samples (but not all!)

G. M. Schmiedeshoff, “Thermal expansion and magnetostriction of a nearly saturated 3He-4He mixture”, accepted Phil Mag. 2009.


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

    Pott & Schefzyk (1988).

  • 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. (1998).

Slide courtesy G. Schmiedeshoff


Kapton Bad (thanks to A. deVisser and Cy Opeil)

  • Replace Kapton washers with alumina.

  • New cell effect scale.

  • Investigating sapphire washers.

Slide courtesy G. Schmiedeshoff


Torque Bad

  • The dilatometer is sensitive to magnetic torque on the sample (induced moments, permanent moments, shape effects…).

  • Manifests as irreproducible/hysteretic data

  • Solution

    • Glue sample to platform (T<20 K)

    • Grease the sample screw -> grease freezes at low temperatures

    • Choose a good sample shape

Good

Bad

Ugly


Thermal gradients bad

You are measuring the difference between thermal expansion of cell and sample. Temperature of cell is important!

Dilatometer cell originally designed to be immersed in liq.uid helium

Sample is mounted on a screw that is not well-thermalized to the body of the cell

Workarounds:

Control temperature of both top and bottom of dilatometer

Connect thermalization wires from top to bottom

Immerse in liquid helium

This part relatively thermally isolated.

At LANL, we made a modified screw that contains heater, thermometer, and attachment points for thermalization wires


Bubbles are bad

Liquid helium bubbles as it boils, especially while you are pumping on it.

Bubbles cause big jumps in the capacitance.

Dilution fridge, immersed in liquid: no bubbles (but beware of field-dependence of helium dielectric constant, and of the He3-He4 boundary line crossing the capacitor)

Dilution fridge, vacuum: No bubbles, but need to thermalize the cell, sample.

Liquid helium 3: Lots of bubbles. Don’t do this.

Liquid helium 4: Ok below 2.2 K (superfluid helium has no bubbles)

Helium gas: Works if you thermalize the cell.


Mounting mechanism

Cu bracket

All titanium


20 T – Dilution fridge Dilatometer in vacuum

NHMFL – LANL

Mixing

Chamber

Zero field region

Thermometre 1

(20 mK – 4 K)

Heater

Ti

dilatometry

cell

Sample

Thermometer 2

(20mK – 4 K)

Thermal links to

the mixing chamber

Field center


How to get good dilatometry data

Avoid torque: Choose non-torquey sample shape, glue sample to dilatometer, grease the screw

Thermalize the dilatometer, put a thermometer near the sample

Calibrate & Measure the cell background

Stick to low temperatures (unless you have a quartz dilatometer)

Avoid kapton

Avoid heliumbubbles

Correct for dielectric constant of medium between capacitor plates (about 5%)

Mount dilatometer so as to avoid thermal contraction/expansion stresses by mounting mechanism on dilatometer.


Origins of thermal expansion

What creates length-changes in samples?

First theories: effects of thermal vibrations

  • Mie (1903): First microscopic model.

  • Grüneisen (1908): β(T)/C(T) ~ constant

    • A fundamental thermodynamic propertythat is often proportional to the specific heat


Grüneisen Theory

Write down Free energy of the vibrations of a solid (a set of harmonic oscillators)

Use this free energy to compute the specific heat…. Or the thermal expansion

Debye theory: assume a max. cutoff frequency of the vibrations

Grüneisen parameter

Thermal pressure due to vibrations

Thermal expansion

compressibility


Grüneisen TheoryApplies to other thermal vibrations

e.g.: phonon, electron, magnon, CEF, Kondo, RKKY, etc.

Electronic Grüneisen parameter probes effective mass

Examples: Simple metals:


Example (Metals):

Gruneisen parameter

Gold

Silver

After White & Collins, JLTP (1972).

Also: Barron, Collins & White, Adv. Phys. (1980).

(lattice shown.)

Copper


Example (Heavy Fermions):

HF(0)

After deVisser et al. (1990)


Probing Phase Transitions

Phase Transition: TN

2nd Order Phase Transition, Ehrenfest Relation(s):

1st Order Phase Transition, Clausius-Clapyeron Eq(s).:


Limitations of Grüneisen Theoryand other thermodynamic approaches to thermal expansion

  • Isotropic thermal expansion only

  • Only treats vibrational effects

  • Limited treatment of elastic effects


An anisotropic, elastic example:

Hc2

Magnetostriction

DL/L (%)

Hc1

Lc

La

c

H

a


c

a

a

Organo-metallic Quantum Magnet:

NiCl2-4SC(NH2)2

Metal

Ni2+ S=1

Superexchangecoupling:

AFM

Organic: thiourea provides structure

Ni S = 1

Cl

Jchain/kB = 2.2 K

12

14

Jplane/kB = 0.18 K


1.2

1

0.8

Bose-Einstein Condensation of Ni system

Boson number controlled by magnetic field

0.6

0.4

0.2

0

0

2

4

6

8

10

The Quantum Part

XY AFM/BEC

Magnetocaloric effect

Specific heat

12

14

H (T)

3D BEC: a = 3/2

3D Ising: a = 2

2D BEC: a = 1


We have a pretty good understanding of this material:


H

T = 25 mK

H || c

Hc2

Lc

DL/L (%)

Hc1

La

But a complete understanding requires including the spin-lattice coupling

c

a

Capacitance

CuBe spring

Titanium Dilatometer

(design by G. Schmiedeshoff)

V. S. Zapf et al, Phys. Rev. B 77, 020404(R) (2008)


Modeling the Magnetostriction (to First Order)

Origin of Magnetic stress

sM (H)

Magnetic stress

Ni

e = DL/L

Strain along c-axis

J(e)

c

Ni

Young’s Modulus:E = s/ e

JS1•S2

Assume: Lattice has linear spring response with Young’s modulus E

Assume: Zero temperature (measurements at T = 25 mK)

Neglect:Crystal field effects changing with pressure

Neglect:Magnetic effects along a-axis


Minimize the energy

Energy density: lattice and magnetic

e - dependence

Magnetic Hamiltonian:

Magnetic energy/volume

Lattice energy/volume

Minimize the energy:

sM (H)

e = DL/L

Young’s Modulus:E = s/ e


DL/L (%)

Quantum Monte Carlo simulations

H || c

T=25mK


DL/L (%)

Significance

We can measure the spin-spin correlation function!

Can extract the spatial dependence of J resulting fromthe Ni-Cl-Cl-Ni superexchange bond

H || c

T=25mK


NiCl2-4SC(NH2)2: an antiferromagnetic quantum magnet

Hc2

Magnetostriction

DL/L (%)

Hc1

Lc

La

c

H

a


Resonant Ultrasound

Cristian Pantea, Jon Betts, Albert Migliori,

NHMFL-LANL

Paul Egan, Oklahoma State

ESR

Sergei Zvyagin, Jochen Wosnitza,

Dresden High Magnetic Field Lab

Jurek Krzystek, NHMFL-Tallahassee

NHMFL-LANL

Diego Zocco, Marcelo Jaime, Neil Harrison,

Alex Lacerda

NHMFL-Tallahassee

Tim Murphy, Eric Palm

Crystal growth and magnetization

Armando Paduan-Filho

Universidade de Sao Paulo, Brazil

Inelastic Neutron diffraction

M. Kenzelmann, B. R. Hansen, C. Niedermayer,

Paul Scherrer Institute and ETH, Zürich, Switzerland

Magnetostriction

Victor Correa, Stan Tozer,

NHMFL-Tallahassee

Quantum Monte Carlo

Mitsuaki Tsukamoto, Naoki Kawashima

University of Tokyo

Theory

Pinaki Sengupta, Cristian Batista, LANL

Acknowledgements (DTN)

NSF NHMFL DOE


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