Thermal analysis heat capacity. Sergey L. Bud ko

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1 Thermal analysis heat capacity 590B F18 Sergey L. Bud ko

2 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

3 relaxation (QD PPMS) Heat capacity HOW?

4 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 t = C total /K w

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

6 relaxation (QD PPMS) Heat capacity HOW? Addenda - simple model Sample simple model

7 relaxation (QD PPMS) Heat capacity HOW? Sample 2-tau model

8 relaxation (QD PPMS) Heat capacity HOW? - Good for flat relatively thin samples with reasonable thermal conductivity - Liquids: need thin-walled sealed container made out of material with high thermal conductivity and low heat capacity. personally did not try (yet?) - Powders: - press a pellet (if has structural integrity); - mix with some metallic powder and press a pellet; - seal powder in aluminum DSC container (can be used for airsensitive powder samples as well, if assembled in a glove box) Remember addenda!

9 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

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

11 ac modulation Heat capacity HOW?

12 Heat capacity HOW? ac modulation after Andreas Rydh

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

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

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

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

17 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

18 Heat capacity HOW? ac modulation under pressure

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

20 Heat capacity HOW? ac modulation under pressure

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

22 after Andreas Rydh

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

24 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)]

25 Heat capacity WHY?

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

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

29 Lattice heat capacity Einstein mode Peak associated with the Einstein ~ Q E /20

30 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

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

32 Heat capacity WHY? To learn something about superconductors Strong coupling RMP 62, 1027 (1990)

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

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

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

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

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

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

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

40 Heat capacity WHY? To learn something about phase transitions below superconducting T c T N T WFM? T c

41 Heat capacity Bulk, thermodynamic property In case of not so single phase samples results: -do not suffer from M&M effect in case of superconductors -do not suffer from impurity SC or FM signal in magnetization (although for FM need to remember about entropy)

42 Resistance SC normal normal bulk, superconducting shell V I Will get R=0 below T c of the shell

43 Low field ZFC susceptibility SC normal normal bulk, superconducting shell Will get complete flux expulsion below T c of the shell

$C p at T c of the shell proportional to the fraction of the SC phase, e.g.$

44 Heat capacity SC normal normal bulk, superconducting shell Will get complete jump of C p at T c of the shell proportional to the fraction of the SC phase, e.g. very small

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

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

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

48 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 ΔS m = R ln N

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

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

51 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

52 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 S14 Sergey L. Bud ko Detour phase transitions of 2 ½ order Heat capacity 2 ½ order phase transition Paul Ehrenfest: First-order phase transitions exhibit a discontinuity