Heat Capacity Measurements

Scott HannahsNHMFL Summer School 2016

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Why Heat Capacity• Fundamental Quantity

• Capacity to “hold energy”Equipartition in states

• Thermodynamic, bulk measurement

C = @@T

REf(E)g(E)dE

2

In Metals

• For solids (metals) two components

• At low temperatures ignore phonon contribution

• Simple metals, free electron model

• Heavy Fermions, correlated elections, magnetic f-electron interactions

C = Celectron

+ Cphonon

= �T +AT 3

C

T= � +AT 2

� ⇡ 1

� � 1

3

Spin Systems

• Insulators

• Latent Heat

• Transition order

4

Cs2CuBr4

Outline

• Introduction

• Measurement Techniques

• Cryogenics

• Thermometry

• Other Issues

5

We’re Done???He

at In

put

Time

Tem

pera

ture

Cp = lim�T!0

(�Q�T )p

6

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

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Typical Setup

8

9

Thermal Relaxation

• Two Time constants, Sample to platform, Platform to bath

• Cp from time constant and from ∆T

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Cp = ⌧

�T = P0SH

e�tSH/(Cs+Cad)

Heat Pulse Calorimetry

• Heat Pulse ∆Q in short time

• Decays to platform Temp tc = RL Ct Ct = Ca + CsRL = Thermal resistance of Link

• Ct = ∆T / ∆Q

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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

• Can’t change field

• Need warming and cooling

• Noisy derivative

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C(T ) = Ph(T )@Th(T )

@t � @Tc(T )@t

AC Calorimetry

• Fast, Continuous Measurement

• Small sample

• Can sweep field, temperature —————————————

• Hard to get absolute accuracy

• Needs “good” thermometry

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• Rsh relatively small, fast recovery

• Drive heater(H) at V=cos(½ 𝜔 t) Ph = Rh V2

• Measure Ts @ 𝜔

• Need DC current Ist to measure Ts14

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TST(t) = T0 + Tdc +Tac cos(ωt+φ)

Tac =P0!C [1 + 1

(!⌧e)2+ f(⌧i)]�

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!⌧e � 1Frequency greater

than conduction time through wires

f(⌧i) ⌧ 1Function depends on internal time constant

of sample

Tac ' P0!C

Tac = ( dTdR )Rac =T

⌘R(T )VacIst

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

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

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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) 032008

Top Loading Rotatable Calorimeter: 0.1 K - 10 K, 45 T

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Why Rotate?• Anisotropic Materials

Layered, 1D

• Alignment

• Field is a vector!

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-60

-40

-20

0

20

∆CP

/T [

mJ/

mol

-K2 ]

323028262422201816

Field [T]

3.06 K 2.03 K 1.58 K 0.58 K 0.30 K 0.18 K

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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) 032008

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

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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) 060406

Cascade of Magnetic-Field-Induced Quantum Phase Transitions in a Spin-1/2 Triangular-Lattice AntiferromagnetN.A. Fortune, S.T. Hannahs, Y. Yoshida, T.E. Sherline, T. Ono, H. Tanaka, and Y. Takano, Phys. Rev. Lett. 102 (2009) 257201

Bonus! Magnetocaloric Effect in the Swept-Field LimitThermodynamics 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

Energy Conservation

Thermodynamics

−TdSfrom system! =κ ΔTdt

to reservoir!"# +CsampledT

to sample!"$ #$

TdS = T ∂S∂T

⎛⎝⎜

⎞⎠⎟ H

dT +T ∂S∂H

⎛⎝⎜

⎞⎠⎟ TdH

Maxwell Relation CH = T ∂S∂T

⎛⎝⎜

⎞⎠⎟ H

∂S∂H

⎛⎝⎜

⎞⎠⎟ T

= ∂M∂T

⎛⎝⎜

⎞⎠⎟ H

Solving for ΔT in swept-field + short relaxation time limits ΔT = − T

κ∂M∂T

⎛⎝⎜

⎞⎠⎟ H

dHdt

−CS +CH( )

κdTdt

≈ − Tκ

∂M∂T

⎛⎝⎜

⎞⎠⎟ H

dHdt

Substituting −CHdT −T ∂M∂T

⎛⎝⎜

⎞⎠⎟ H

dH =κ ΔTdt +CsampledT

ΔT sample - reservoir?

Magneto-Caloric Effect

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0.210

0.200

Tem

pera

ture

[K]

302520Magnetic Field [T]

+1 T/min -1 T/min

a

a

b

b

0.310

0.300302520

+2 T/min - 2 T/min

b

ba

Remember….

• 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

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C = Po

!R(T )⌘T

Ist

Vac

• Fit to Tchebyshev PolynomialsOrthogonal!Sensitivity is function of same parameters

• Extract coefficients as function of B

• Fit Ci(B) as Padé approximate25

2x103

3

4

5

6789

Resis

tanc

e [O

hm]

8 90.1

2 3 4 5 6 7 8 91

2 3 4 5 6 7 8 910

Temperature [K]

-0.02-0.010.000.010.02

Frac

Tem

p Er

rThermometer Calibration - PTFortune/Hannahs Jan 2016B = 10.0 teslaTchebyshev Fit Order = 6

Padé Approximates

• Ratio of two power series

• Can fit functions with rapid changes and smooth sections

• Watch out for nasty poles

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1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

C 2

151050

Field [T]

Thermometer CalibrationJune 2015Tchebyshev Coeficient 2

CT ST PT

C(B) = C0 +1B+2B

2

1+�1B+�2B2

3𝜔 Technique

• Heater = Thermometer

• Resistance changes give 3𝜔 response

• Thin film samples

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