Low temperature specific heat
of solids
James C. Ho
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
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
Low temperatures
How low is low?
How low can we go?
How low do we need to go?
17th-century depiction of Leiden University Library
Walther Nernst (Nobel prize in Chemistry, 1920)
developed in 1906-1912 the 3rd law of thermodynamics
(zero entropy at zero degree)
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
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.
Heike Kamerlingh Onnes (Leiden University, 1853-1926)
first liquefied helium in1908 and discovered superconductivity in 1911.
1913 Nobel Prize in Physics
Einstein and Kamerlingh Onnes
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
Einstein model (individual atomic vibrations)
Debye model (phonon modes)
Peter Debye (1884-1966)
1936 Nobel prize in chemistry
1937-1939, President, Deutsche
Physikalische Gesellschaft
1940-1950, Cornell University
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)
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
Schottky anomaly in specific heat for multi-energy level systems
(from a homework problem in C. Kittel: Introduction to Solid State Physics)
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 + ….
Specific heat of manganese
C = 0.055T3 + 9.20T + 0.264/T2
with I = 5/2 and μ = 3.4532μN, He = 65 kOe
Effective hyperfine field at the nuclei of Pt dissolved in Fe
C = 8.25x10-2T + 1.40x10-3/T2 – 1.31x10-6/T4
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
Superconducting transition
BCS theory: Ces/γT at Tc ≈ 1.43; Ces/γTc = a exp(-bTc/T)
BCS theory: Tc ≈ ϴD exp[-1/D(EF)V]
V: electron-phonon interaction parameter
Superconducting transitions in Ti-Mo alloys (V = constant?)
An antiferromagnetic transition at 3.8 K in bulk CeAl2
S = Rln2 = 5.76 J/mol K
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.
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)
President Ronald Reagan
gave the keynote address
at the Conference on
“Superconductivity:
Challenge for the future”,
Washington, D.C., July
1987.
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
Physical Review (1893)
Physical Review B -- Solid State Physics (1970)
Physical Review B – Condensed Matter (1978)
Physical Review B -- Condensed Matter and Materials Physics
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
Professor Sir John Enderby
Kt, CBE, PhD (London), DSc (Lough), FInstP, FRS
Vice President of Royal Society, 1999-2004
President of Institute of Physics, 2004-2006
Ordering effect on the electronic structure and subsequently the mechanical
behavior (stronger but brittle) of titanium-aluminum alloys
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.
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)
Superconducting transition in an non-uniform sample
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
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.
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.