Dilatometry Measuring lengthchanges of your sample thermal expansion, magnetostriction,

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DilatometryMeasuring length-changes of your
samplethermal expansion, magnetostriction,

• Vivien Zapf
• NHMFL-LANL

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• Heron of Alexandria ( 0 B.C.)

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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
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NiCl2-4SC(NH2)2 an antiferromagnetic quantum
magnet
Hc2
Magnetostriction
DL/L ()
Hc1
Lc
La
c
H
a
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Dilatometry

• T thermal expansion a1/L(dL/dT) ß
  dln(V)/dT
• H magnetostriction ? ?L(H)/L
• P compressibility ? dln(V)/dP
• E electrostriction ? ?L(E)/L
• etc.

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How to measure Dilatometry

• Mechanical (pushrod etc.)
• Optical (interferometer etc.).
• Electrical (Inductive, Capacitive, Strain
  Gauges).
• Diffraction (X-ray, neutron).
• Others (absolute differential)

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Capacitive Dilatometer(Cartoon)
Capacitor Plates
Cell Body
D
Sample
L
Extra credit question Why dont we put the
sample between the capacitor plates?
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George Schmiedeshoff, Occidental College
Rev. Sci. Instrum. 77, 123907 (2006)
Use of needle instead of plate on top of sample
means sample faces dont need to be perfectly
parallel
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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 Alberts 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)

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H
Dilatometer works at various orientations to the
magnetic field. Rotators available at LANL and
Tallahassee
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Capacitance measurement
Two shielded, grounded coax cables
Capacitance bridge (e.g. AH 2300 bridge or GC
1615)
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Rev. Sci. Instrum. 77, 123907 (2006)
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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
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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
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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.
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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
• 
  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
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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.

Slide courtesy G. Schmiedeshoff
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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/hysteretic data
• Solution
• Glue sample to platform (Tlt20 K)
• Grease the sample screw -gt grease freezes at low
  temperatures
• Choose a good sample shape

Good
Bad
Ugly
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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
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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. Dont do this.
Liquid helium 4 Ok below 2.2 K (superfluid
helium has no bubbles) Helium gas Works if you
thermalize the cell.
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Mounting mechanism
Cu bracket
All titanium
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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
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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 helium bubbles 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.
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Origins of thermal expansion
What creates length-changes in samples? First
theories effects of thermal vibrations

• Mie (1903) First microscopic model.
• Grüneisen (1908) ß(T)/C(T) constant
• A fundamental thermodynamic propertythat is
  often proportional to the specific heat

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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
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Grüneisen Theory Applies to other thermal
vibrations
e.g. phonon, electron, magnon, CEF, Kondo, RKKY,
etc.
Electronic Grüneisen parameter probes effective
mass
Examples Simple metals
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Example (Metals)
Gruneisen parameter
Gold
Silver
After White Collins, JLTP (1972). Also Barron,
Collins White, Adv. Phys. (1980). (?lattice
shown.)
Copper
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Example (Heavy Fermions)
?HF(0)
After deVisser et al. (1990)
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Phase Transition TN
Probing Phase Transitions
2nd Order Phase Transition, Ehrenfest
Relation(s)
1st Order Phase Transition, Clausius-Clapyeron
Eq(s).
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Limitations of Grüneisen Theoryand other
thermodynamic approaches to thermal expansion

• Isotropic thermal expansion only
• Only treats vibrational effects
• Limited treatment of elastic effects

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An anisotropic, elastic example
Hc2
Magnetostriction
DL/L ()
Hc1
Lc
La
c
H
a
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Organo-metallic Quantum Magnet
NiCl2-4SC(NH2)2
Metal Ni2 S1
Superexchangecoupling AFM
Organic thiourea provides structure
Ni S 1
Cl
Jchain/kB 2.2 K
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Jplane/kB 0.18 K
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The Quantum Part
XY AFM/BEC
Magnetocaloric effect
Specific heat
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H (T)
3D BEC a 3/2 3D Ising a 2 2D BEC a 1
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We have a pretty good understanding of this
material
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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)
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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
Youngs ModulusE s/ e
JS1S2
Assume Lattice has linear spring response with
Youngs modulus E Assume Zero temperature
(measurements at T 25 mK) Neglect Crystal
field effects changing with pressure Neglect
Magnetic effects along a-axis
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Minimize the energy
Energy density lattice and magnetic
e - dependence
Magnetic Hamiltonian
Lattice energy/volume
Magnetic energy/volume
Minimize the energy
sM (H)
e DL/L
Youngs ModulusE s/ e
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Quantum Monte Carlo simulations
H c
T25mK
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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
T25mK
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NiCl2-4SC(NH2)2 an antiferromagnetic quantum
magnet
Hc2
Magnetostriction
DL/L ()
Hc1
Lc
La
c
H
a
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Acknowledgements (DTN)

• 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

NSF NHMFL DOE