1. Low temperature specific heat of solids James C. Ho

2. For some 200 articles on low temperature specific heat and superconductivity, I have many coauthors including:
Members of the U.S. National Academy of Sciences Ching-Wu Chu, Theodore Geballe, John Hulm, Bernd Matthias, Maw-Kuen Wu Members of the U.S. National Academy of Engineering Karl Gschneidner, John Hulm, Robert Jaffee Fellow of British Royal Society Sir John Enderby Members of the Academia Sinica (Taiwan) Ching-Wu Chu, Maw-Kuen Wu Member of Chinese Academy of Science Weiyan Guan

3. Low temperatures Specific heat and entropy in chemical thermodynamics Low temperature specific heat in solid state physics Phonon (lattice) Conduction electrons Nuclear magnetic moments Critical phenomena Superconductivity Magnetic transitions Some applications to materials science Magnetic clusters Electron localization Non-uniform superconductors Nano-size effect

4. How low is low? How low can we go? How low do we need to go?
Low temperatures
How low is low?
How low can we go?
How low do we need to go?

6. 17th-century depiction of Leiden University Library

7. Walther Nernst (Nobel prize in Chemistry, 1920) developed in the 3rd law of thermodynamics (zero entropy at zero degree)

8. Specific heat and entropy in chemical thermodynamics
Entropy change determination relies on specific heat measurements:
ΔS = δQ/T = ∫(C/T) dT
______________________________________________
U = Q – W dU = δQ – PdV
(∂U/∂T)V = (∂Q/∂T)V ≡ CV
H = U + PV dU = δQ +VdP
(∂H/∂T)p = (∂Q/∂T)p ≡ Cp
Cp – CV = VT(α2/βT)
α: Coefficient of linear thermal expansion
βT: Isothermal compressibility

9. Early models of specific heat of solids
The Dulong-Petit law (1819): CV = 3Nk (harmonic oscillators) _______________________ The Einstein model (1907) (quantized harmonic oscillators)
When Nernst learned of Einstein's paper, he was so excited that he traveled all the way from Berlin to Zurich to meet with Einstein.

10. Heike Kamerlingh Onnes (Leiden University, ) first liquefied helium in1908 and discovered superconductivity in Nobel Prize in Physics

11. Einstein and Kamerlingh Onnes

12. Specific heat and entropy in chemical thermodynamics Low temperature specific heat in solid state physics Phonon (lattice) Conduction electrons Nuclear magnetic moments Critical phenomena Superconductivity Magnetic transitions Some applications to materials science Magnetic clusters Electron localization Non-uniform superconductors Nano-size effect

13. Einstein model (individual atomic vibrations) Debye model (phonon modes)
Peter Debye ( ) Nobel prize in chemistry , President, Deutsche Physikalische Gesellschaft , Cornell University

14. Debye model of phonon (lattice) specific heat at low temperatures at T << ϴD, Cph = Nk(12π4/5)(T/ϴD)3 ≡ βT3 SiC, ϴD = 990 K (Fe-477, Al-433, Cu-347, Au-162, In-112, Pb-105)

15. Free electron model of electronic specific heat Ce = (π2/3)D(EF)k2T ≡ γT at T << TF , For metallic materials at low temperatures, C = Ce + Cph = γT + βT3 or C/T = γ+ βT2

16. Schottky anomaly in specific heat for multi-energy level systems (from a homework problem in C. Kittel: Introduction to Solid State Physics)

17. Nuclear specific heat .
In magnetically ordered materials, interactions between
s-electrons (non-zero density at nucleus) and the nuclear moment μ behave as an extremely large effective field He. It causes nuclear energy splitting, and in turn yields a Schottky-type specific heat.
At temperatures T ˃˃ μHe/k (often near or less than 1 K),   CN = Nk[(I+1)/3I](μHe/kT)2
– [(I+1)(2I2+2I+1)/30I3](μHe/kT)4 + ….
= A/T2 – A’/T4 + ….

18. Specific heat of manganese. C = 0. 055T3 + 9. 20T + 0
Specific heat of manganese C = 0.055T T /T2 with I = 5/2 and μ = μN, He = 65 kOe

19. Effective hyperfine field at the nuclei of Pt dissolved in Fe C = 8
Effective hyperfine field at the nuclei of Pt dissolved in Fe C = 8.25x10-2T x10-3/T2 – 1.31x10-6/T4

20. Specific heat and entropy in chemical thermodynamics Low temperature specific heat in solid state physics Phonon (lattice) Conduction electrons Nuclear magnetic moments Critical phenomena Superconductivity Magnetic transitions Some applications to materials science Magnetic clusters Electron localization Non-uniform superconductors Nano-size effect

21. Superconducting transition BCS theory: Ces/γT at Tc ≈ 1
Superconducting transition BCS theory: Ces/γT at Tc ≈ 1.43; Ces/γTc = a exp(-bTc/T)

22. BCS theory: Tc ≈ ϴD exp[-1/D(EF)V] V: electron-phonon interaction parameter Superconducting transitions in Ti-Mo alloys (V = constant?)

23. An antiferromagnetic transition at 3. 8 K in bulk CeAl2 S = Rln2 = 5
An antiferromagnetic transition at 3.8 K in bulk CeAl2 S = Rln2 = 5.76 J/mol K

24. Two sets of erbium ions in erbium sesquioxide -- Er3+-I and Er3+-II in 1:3 ratio Entropy associated with the observed anomaly, ΔS = ∫(Cm/T)dT = 4.14 J/mol K, a value about 72% of Rln2 = 5.76 J/mol K for all Er3+ ions (ground state doublet), indicating that only Er3+-II undergoes the magnetic transition at 3.3 K.

25. Specific heat of GdBa2Cu3O6+δ revealed a magnetic ordering of Gd3+ at 2.2 K, independent of whether superconducting transition occurs above 90 K (δ = 0.7) or not (δ = 0.5). J.C. Ho, P.H. Hor, R.L. Meng, C.W. Chu and C.Y. Huang, Solid State Commun. (1987) J.C. Ho, C. Y. Huang, P.H. Hor, R.L. Meng and C.W. Chu, Mod. Phys. Lett. (1988)

26. President Ronald Reagan gave the keynote address at the Conference on “Superconductivity: Challenge for the future”, Washington, D.C., July

27. Specific heat and entropy in chemical thermodynamics Low temperature specific heat in solid state physics Phonon (lattice) Conduction electrons Nuclear magnetic moments Critical phenomena Superconductivity Magnetic transitions Some applications to materials science Magnetic clusters Electron localization Non-uniform superconductors Nano-size effect

29. Physical Review (1893) Physical Review B -- Solid State Physics (1970) Physical Review B – Condensed Matter (1978) Physical Review B -- Condensed Matter and Materials Physics

30. Magnetic clusters (classical harmonic oscillators) Eδ = N(kT), δ = Nk C = δ + γT + βT3 or (C – δ)/T = γ + βT2 Nickel-base MAR-M200 superalloy: N ≈ 3x1020/mol ≈ 0.5x10-3NA

31. Professor Sir John Enderby Kt, CBE, PhD (London), DSc (Lough), FInstP, FRS Vice President of Royal Society, President of Institute of Physics,

32. Ordering effect on the electronic structure and subsequently the mechanical behavior (stronger but brittle) of titanium-aluminum alloys

33. Light and strong Al-Li alloys Large electronegativity difference: 1
Light and strong Al-Li alloys Large electronegativity difference: 1.47 for Al and 0.97 for Li Intermetallic compound AlLi has its m.p (700oC) extruding into the liquidus.

34. Detection of superconducting transitions
Transport property measurements (qualitative): Electrical resistivity (impurity forming a continuous network?) Thermodynamic property measurements (quantitative): Magnetic susceptibility (shielding effect?) Specific heat (qualitative) fraction of superconducting component in sample (BCS theory: (Ces/γT)Tc ≈ 2.43)

36. Superconducting transition in an non-uniform sample

37. Specific heat and entropy in chemical thermodynamics Low temperature specific heat in solid state physics Phonon (lattice) Conduction electrons Nuclear magnetic moments Critical phenomena Superconductivity Magnetic transitions Some applications to materials science Magnetic clusters Electron localization Non-uniform superconductors Nano-size effect

38. An antiferromagnetic transition at 3
An antiferromagnetic transition at 3.8 K in bulk CeAl2 is completely suppressed in 8 nm particles, yielding to a heavy Fermion behavior. The dashed line represent the sum of phonon and crystal-field contributions (open circles), based on LaAl2.

39. In conclusion, measurements of low temperature specific heat, as a thermodynamic quantity, provide some fundamental information about a solid, as well as a relatively simple and effective evaluation technique in materials science.