Dilatometry Measuring length-changes of your sample thermal expansion, magnetostriction, …

1. DilatometryMeasuring length-changes of your samplethermal expansion, magnetostriction, … Vivien Zapf NHMFL-LANL

2. Heron of Alexandria (~ 0 B.C.)

3. 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

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

5. 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.

6. How to measure Dilatometry • Mechanical (pushrod etc.) • Optical (interferometer etc.). • Electrical (Inductive, Capacitive, Strain Gauges). • Diffraction (X-ray, neutron). • Others (absolute & differential)

7. 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?

8. 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

9. 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)

10. H Dilatometer works at various orientations to the magnetic field. Rotators available at LANL and Tallahassee

11. Capacitance measurement Two shielded, grounded coax cables Capacitance bridge (e.g. AH 2300 bridge or GC 1615)

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

13. 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

14. 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

15. 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.

16. 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

17. 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

18. 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

19. 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

20. 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.

21. Mounting mechanism Cu bracket All titanium

22. 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

23. 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.

24. 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

25. 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

26. Grüneisen TheoryApplies to other thermal vibrations e.g.: phonon, electron, magnon, CEF, Kondo, RKKY, etc. Electronic Grüneisen parameter probes effective mass Examples: Simple metals:

27. Example (Metals): Gruneisen parameter Gold Silver After White & Collins, JLTP (1972). Also: Barron, Collins & White, Adv. Phys. (1980). (lattice shown.) Copper

28. Example (Heavy Fermions): HF(0) After deVisser et al. (1990)

29. Probing Phase Transitions Phase Transition: TN 2nd Order Phase Transition, Ehrenfest Relation(s): 1st Order Phase Transition, Clausius-Clapyeron Eq(s).:

30. Limitations of Grüneisen Theoryand other thermodynamic approaches to thermal expansion • Isotropic thermal expansion only • Only treats vibrational effects • Limited treatment of elastic effects

31. An anisotropic, elastic example: Hc2 Magnetostriction DL/L (%) Hc1 Lc La c H a

32. 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

33. 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

34. We have a pretty good understanding of this material:

35. 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)

36. 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

37. 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

38. DL/L (%) Quantum Monte Carlo simulations H || c T=25mK

39. 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

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

41. 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