Heat Capacity Measurements. Scott Hannahs NHMFL Summer School 2016

Transcription

1 Heat Capacity Measurements Scott Hannahs NHMFL Summer School

2 Why Heat Capacity Fundamental Quantity R Ef(E)g(E)dE Capacity to hold energy Equipartition in states Thermodynamic, bulk measurement 2

3 In Metals For solids (metals) two components C = C electron + C phonon = T + AT 3 C T = + AT 2 At low temperatures ignore phonon contribution Simple metals, free electron model Heavy Fermions, correlated elections, magnetic f-electron interactions 1 1 3

4 Spin Systems Cs2CuBr4 Insulators Latent Heat Transition order 4

5 Outline Introduction Measurement Techniques Cryogenics Thermometry Other Issues 5

6 We re Done??? Temperature C p = lim T!0 ( Q T ) p Heat Input Time 6

7 Not quite. Real World Effects! - Heat diffusion into sample interior - Heat leak to outside world - Temperature control of sample - Measurement of temperature Addenda of heater Addenda of thermometer - Measurement of heat pulse 7

8 Typical Setup 8

9 9

10 Thermal Relaxation T = P 0 apple SH e tapple SH/(C s +C ad ) Two Time constants, Sample to platform, Platform to bath Cp from time constant and from T C p = apple 10

11 Heat Pulse Calorimetry Heat Pulse Q in short time Decays to platform Temp tc = RL Ct Ct = Ca + Cs RL = Thermal resistance of Link Ct = T / Q 11

12 Dual Slope Relaxation Can be wide range in Temp Simple Calculation Assume external effects cancel (as function of temp) High temp stability of block needed C(T h (T P h (T Can t change field Need warming and cooling Noisy derivative 12

13 AC Calorimetry Fast, Continuous Measurement Small sample Can sweep field, temperature Hard to get absolute accuracy Needs good thermometry 13

14 Rsh relatively small, fast recovery Drive heater(h) at V=cos(½ ω t) Ph = Rh V 2 Measure ω Need DC current Ist to measure Ts 14

15 T ST (t) = T 0 + T dc +T ac cos(ωt+φ) T ac = P 0!C [1 + 1 (! e ) 2 + f( i )] 1 2 Frequency greater! e 1 than conduction time through wires f( i ) 1 Function depends on internal time constant of sample T ac ' P 0!C T ac =( dt dr )R ac = T R(T ) V ac I st 15

16 Comments Drive frequency sweeps to find sweet spot Phase Shifts amplitude not uniform in sample Adjust frequency as shift field to stay in sweet spot Phase Shifts amplitude not uniform in sample Can use triangular or square wave Vsh Small sample < 1mG, thin heaters and thermometers as much as possible Rotate! No Copper! 16

17 AC Cal Cell No pumping line Can rotate Indium seal, compression Ag platform, with heater thermometer Sapphire electrical isolation Heater/Sample/Thermometer sandwich on wires 17

18 Top Loading Rotatable Calorimeter: 0.1 K - 10 K, 45 T Top-loading small-sample calorimeters for measurements as a function of magnetic field angle N.A. Fortune and S.T. Hannahs, Journal of Physics: Conference Series 568 (2014)

19 19

20 Why Rotate? Anisotropic Materials Layered, 1D Alignment 20 Field is a vector! C P /T [mj/mol-k 2 ] K 2.03 K 1.58 K 0.58 K 0.30 K 0.18 K Field [T] 20

21 Magnetic Field-Orientation Dependent High-Field Phase Transition Within Superconducting State of CeCoIn5 Top-loading small-sample calorimeters for measurements as a function of magnetic field angle N.A. Fortune and S.T. Hannahs, Journal of Physics: Conference Series 568 (2014) Magnetic enhancement of superconductivity from electron spin domains, H.A. Radovan, N.A. Fortune, T.P. Murphy, S.T. Hannahs, E.C. Palm, S.W. Tozer and D. Hall, Nature 425 (2003) 51 21

22 Bonus! Magnetocaloric Effect in the Swept-Field Limit Thermodynamics in the high-field phases of (TMTSF)2ClO4, U.M. Scheven, S.T. Hannahs, C. Immer, P.M. Chaikin, Phys. Rev. B. 56 (1997) 7804 Thermodynamics of the up-up-down phase of the S = 1/2 triangular-lattice antiferromagnet Cs2CuBr4 H. Tsuji, C.R. Rotundu, T. Ono, H. Tanaka, B. Andraka, K. Ingersent, and Y. Takano, Physical Review B 76 (2007) Cascade of Magnetic-Field-Induced Quantum Phase Transitions in a Spin-1/2 Triangular-Lattice Antiferromagnet N.A. Fortune, S.T. Hannahs, Y. Yoshida, T.E. Sherline, T. Ono, H. Tanaka, and Y. Takano, Phys. Rev. Lett. 102 (2009) ΔT sample - reservoir? Energy Conservation TdS! =!"# κ ΔTdt + C! $ dt sample " #$ from system to reservoir Thermodynamics to sample TdS = T S T H dt + T S H T dh Maxwell Relation C H = T S T H S H T = M T H Substituting C H dt T M T H dh = κ ΔTdt + C sample dt Solving for ΔT in swept-field + short relaxation time limits ΔT = T κ M T H dh dt ( C S + C ) H κ dt dt T κ M T H dh dt 22

23 Magneto-Caloric Effect b Temperature [K] a 20 a b T/min - 2 T/min 30 b a b T/min -1 T/min 30 Magnetic Field [T] 23

24 Remember. C = P o R(T )! T I st V ac Pesky thermometry and sensitivity! Need R(B, T) Need η(b, T) There is no resistance type (<1%) field independent thermometer < 1K Goal, to calibrate field dependence 24

25 Frac Temp Err Resistance [Ohm] x10 3 Thermometer Calibration - PT Fortune/Hannahs Jan 2016 B = 10.0 tesla Tchebyshev Fit Order = Fit to Tchebyshev Polynomials Orthogonal! Temperature [K] Sensitivity is function of same parameters Extract coefficients as function of B Fit C i(b) as Padé approximate 25

26 Padé Approximates Thermometer Calibration June 2015 Tchebyshev Coeficient 2 CT ST PT C(B) =C 0 + apple 1B+apple 2 B B+ 2 B Ratio of two power C series Can fit functions with rapid changes and smooth sections Field [T] Watch out for nasty poles 26

27 3ω Technique Heater = Thermometer Resistance changes give 3ω response Thin film samples 27