Thermal analysis heat capacity. Sergey L. Bud ko

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1 Thermal analysis heat capacity 590B S14 Sergey L. Bud ko Detour phase transitions of 2 ½ order Heat capacity

2 2 ½ order phase transition Paul Ehrenfest: First-order phase transitions exhibit a discontinuity in the first derivative of the free energy with a thermodynamic variable. Second-order phase transitions continuous in the first derivative exhibit discontinuity in a second derivative of the free energy.

3 Fermi surface - reminder

4 2 ½ order phase transition aka: electronic topological transition (ETT), Lifshitz transition also:

5 2 ½ order phase transition electronic DOS parameter: thermodynamic potential T=0, no scattering

6 2 ½ order phase transition Li-Mg alloy low temperature, no scattering ALSO other control parameters

7 2 ½ order phase transition New box the same good old taste QCP in heavy fermions LSCO high-t c SC

8 2 ½ order phase transition

9 Heat capacity C p = (dq/dt) p = T( S/ T) p Q heat, S - entropy C v = (dq/dt) v = T( S/ T) v C p C v = TVβ 2 /k T β volume thermal expansion, k T isothermal volume compressibility For solids C p ~ C v Usually we measure C p

10 Heat capacity WHY? * Useful knowledge for useful (structural) materials how easy to cool down

11 Heat capacity WHY? * To learn something about electrons and phonons T -> 0: C p = γt + βt 3 (no magnetism) γ ~ N(0)(1 + λ) - electronic density of states λ - measure of e-e (and e-ph) interactions, m*/m 0 = (1 + λ) γ > 400 mj/mol K 2 heavy fermions an enormous, separate research field that entertains number of us Θ D = (1944/β) 1/3 - Debye temperature (use right units) tells you something about stiffness of the lattice, helps to understand BCS superconductors [T c ~ Θ D exp(-1/n(0)v)]

12 Heat capacity WHY?

14 Lattice heat capacity Debye and Einstein Einstein single frequency Debye elastic continuum (T << Θ D ) (T >> Θ D )

15 Lattice heat capacity Debye and Einstein Debye Einstein One can also use realistic phonon DOS and calculate C p Einstein Debye Blackman realistic

16 Heat capacity WHY? To learn something about magnons Ferro (ferri) magnets isotropic anisotropic Antiferromagnets isotropic anisotropic And need to remember contributions from electrons and phonons: C p = γt + βt 3

17 Heat capacity WHY? To learn something about superconductors C s ~ exp(-1.76t c /T) s c BCS

18 Heat capacity WHY? To learn something about superconductors BCS value (3.53) phenomenological (1970s) α-model: 2 0 /k B T c fitting parameter; temperature dependence of SC gap BCS; can calculate thermodynamic properties.

19 Heat capacity WHY? To learn something about superconductors MgB 2 Two-gap superconductor

20 Heat capacity WHY? To learn something about superconductors MgB 2 Two-gap superconductor

21 Heat capacity WHY? To learn something about superconductors four gaps

22 Heat capacity WHY? To learn something about superconductors BCS superconductors with paramagnetic impurities BCS PM impurities La-Gd jump in specific heat

23 Heat capacity WHY? To learn something about superconductors Kondo effect in SC jump in specific heat

24 Heat capacity WHY? To learn something about superconductors Iron-arsenides C 3 p ~ T c Smart person might be able to make sense out of this

25 Heat capacity WHY? To learn something about crystal electric field (Schottky anomaly) maximum:

26 Heat capacity WHY? To learn something about spin glasses Scaling hypothesis

27 Heat capacity WHY? To learn something about spin glasses 1/T T max ~ 1.5 T f

28 Heat capacity WHY? To learn something about magnetic transitions HoNi 2 B 2 C Also can play entropy game and evaluate degeneracy of the ground state S m = R ln N

29 Heat capacity beware of fool s gold low temperature C = AT in insulators

30 Heat capacity beware of fool s gold low temperature C = AT in insulators glasses - distribution of two level systems

31 Heat capacity beware of fool s gold fake heavy fermions (apparently large Sommerfeld coefficient): - (really) low temperature magnetic ordering - low temperature spin glass behavior One measurement technique, whatever sophisticated, is not enough

32 relaxation (QD PPMS) Heat capacity HOW?

33 Heat capacity HOW? relaxation (QD PPMS) simple P 0 0 power of the heater sample + platform T of the bath thermal conductance of wires solution: exponent with τ = C total /K w

34 Heat capacity HOW? relaxation (QD PPMS) addenda more realistic fit with two exponents

35 relaxation (QD PPMS) Heat capacity HOW? calibrated heater and thermometer sophisticated temperature control and fitting software need to measure addenda (platform + grease) every time need to calibrate heater and thermometer in magnetic field need to shape your sample vertical 3 He platform may oscillate in magnetic field remember about torque assembly is fragile measurements take long time not good for 1 st order phase transitions

36 Heat capacity HOW? ac modulation τ 1 sample to bath; τ 2 response time of the sample if τ 2 << 1/w << τ 1

37 ac modulation Heat capacity HOW?

38 Heat capacity HOW? ac modulation after Andreas Rydh

39 Heat capacity HOW? ac modulation Commercial chips with membrane (this is a commercial thermal conductivity gauge)

40 Heat capacity HOW? ac modulation Commercial chips with membrane you are invited to play with it

41 Heat capacity HOW? ac modulation differential cell after Andreas Rydh

42 Heat capacity HOW? ac modulation differential cell need to have (at least primitive) thin film technology and lithography after Andreas Rydh

43 Heat capacity HOW? ac modulation fast scalable, good for small sample accurate (relative measurements) easy to put on rotator can use e.g. in pressure cells hard to get absolute values

44 Heat capacity HOW? ac modulation under pressure

45 Heat capacity HOW? ac modulation under pressure Bridgman cell -heater and thermometer are at ambient pressure for DAC laser heating with thermocouple as sensor

46 Heat capacity HOW? ac modulation under pressure

47 Heat capacity HOW? ac modulation under pressure need to choose frequency

48 after Andreas Rydh

49 Reading: Quantum Design Manual and refs therein E.S.R. Gopal, Specific heats at low temperatures. A.Tari, Specific heat of matter at low temperatures. T.H.K. Barron and G.K. White, Heat capacity and thermal expansion at low temperatures. Review of Scientific Instruments search on heat capacity

Thermal analysis heat capacity. Sergey L. Bud ko

Thermal analysis heat capacity 590B F18 Sergey L. Bud ko Heat capacity C p = (dq/dt) p = T( S/ T) p Q heat, S - entropy C v = (dq/dt) v = T( S/ T) v C p C v = TVβ 2 /k T β volume thermal expansion, k T