Elasticity of Pu and ZrW2O8-a window into
fundamental understanding
Albert Migliori NHMFL/LANL
1)Intro and review
2)ZrW2O8
3)Pu
Who took this picture and where was she?
A.Migliori, H. Ledbetter, J.B.Betts, D. Doolley, D.Miller C. Pantea, I. Mihut,, M.Ramos, F.
Drymiotis, F.Freibert, R. Ronquillo, J.P.Baiardo J.C.Lashley, F. Drymiotis, S. El-Khatib, A.C.
Lawson, F. Balakirev, B.Martinez, R. McQueeney, J.M.Wills, M.Graf, S. Rudin, J. Singleton, C.
M. Varma, G. Kotliar, E. Abrahams…..
Elasticity—we like it!
Mass and spring
moduli
Solid and Temperature
Symmetrized strains
This is the elastic energy.
We measure adiabatic elastic moduli-they really connect to physics!
For band structure guys, adiabatic means leave electron occupation
numbers fixed and deform—the very easiest thing to do!
Elasticity and entropy
The sound speeds (the dispersion curves) determine characteristic
vibrational temperature-and most of the entropy at high T.
Not the Debye Temperature
If phases with lots of entropy are accessible, they tend to become
the stable high-T phase.
Entropy goes up with:
Electronic instabilities
Structural instabilities
Soft structures
……
Anharmonicity and Elasticity
Perfectly linear models and this talk
•
The volume V is independent of
temperature T.
•
The elastic moduli cij—should be
independent of T and V and not differ
between adiabatic and isothermal
conditions
Pretty much rubbish
All materials are anharmonic.
This is why solids undergo structural phase transitions and
eventually come apart
As system energy (temperature) rises,
anharmonicity can induce new behavior
not easily predicted!
Example: classic soft-mode transition
where phonon frequencies decrease to
zero with decreasing temperature.
The “stopped” vibration is a static
distortion or new phase!—here it is cubic
to tetragonal.
Migliori’s theorem: (unproven-a not unusual situation)
no solid can be harmonic-furthermore, anharmonicity is an intrinsic
fundamental property!
Ordinary thermal expansion and specific heat
Zwikker, 1953, Physical
Properties of Solids
• In 1912, Grüneisen defined
• and then found such things as
• In the subsequent 90+ years, this concept has remained
astonishingly useful.
E. Grüneisen, Ann. Phys. 39, 257 (1912)
β= thermal expansion, Cp=specific heat per unit mass at constant pressure, B= bulk modulus,
ρ =1/V= density, ω=mode frequency
Pair potentials do not easily explain thermal expansion
It is really, really hard to compute thermal expansion from pair
potentials-all k vectors are needed!
The single-atom potential for any crystal has the crystal symmetry. It
therefore never has odd terms (like cubic).
The primary source for thermal expansion in solids is the increase of
entropy with volume--not a cubic term in the potential. (D.C. Wallace)
Elasticity and the not-intuitive:
Constant volume and constant pressure
We almost always measure at constant pressure
Difference between constant
volume and constant pressure is
a very big deal.
B almost always decreases with T at constant pressure but:
B almost always increases with T at constant volume
if not, something new is happening!
Why they pay me to look at ZrW2O8
ZrW2O8 complex, heterogeneous in important ways at
the unit cell level, but homogeneous macroscopically.
Pu Complexity in the phase diagram, doping-induced
heterogeneity, martensite?
Can we use volume and elasticity of easy-to-study
negative expansion systems to understand Pu?
RUS uses normal modes that look like this
RUS systems at LANL
Note that the sample is mounted flat on the transducers
Approximate cubes are the best shape
290mK-350K
0-15T
Aging is different in α and δ PU
300K
On short time scales, we observe very rapid changes.
ZrW2O8
ZrW2O8-Structure and Thermal Expansion
A. P Ramirez and G. R. Kowach, Phys. Rev. Letters, 80, 22 (1998)
ZrW2O8 thermal expansion
Looks like temperature and volume are the interesting
variables—perfect subject for ultrasound which gets
the second derivatives of the volume wrt strains.
T. A. Mary, J. S. O. Evans, T. Vogt, and A. W. Sleight, Science 272, 90, 1996
Ultrasound and ZrW2O8
The contracting solid gets softer,
opposite what most materials
do, and the change is an order of
magnitude too large.
It’s negative
Shear modulus change is normal= 7%
F. Drymiotis, H. Ledbetter, J. Betts, T, Kimura, J. Lashley, A. Migliori, A. Ramirez,
G. Kowach, and J. Van Duijn, Phys. Rev. Lett. 93, 025502(2004).
ZAP cell for pulse-echo ultrasound under hydrostatic pressure
Why monocrystals are important
ZrW2O8
Pressure induced phase transition at
5.8kbar
Neutron scattering studies of powder
samples published in Science got this
wrong—the stress risers in grains of
powder produce signatures of the
phase transition well below the actual
transition—the transition was reported
at 2kbar.
Pulse-echo on monocrystals gets it
right!
For this material, maybe the incipient phase transition with pressure is the source
of all weirdness
The other moduli
A cubic solid should exhibit only small changes in the shear elastic constants
as it changes volume because bond angles hardly change
Confirmation via Raman spectroscopy
Plutonium
polycrystal measurements
(Required viewgraph for any Pu talk)
Only Pu exhibits so many phases in such a small (< a factor of 2)
temperature range.
Or this?
This?
Is Pu the second most interesting element?
UNCLASSIFIE
All the science comes from electronic structure-but…
Phase diagram-with Ga
Stability-Atomic number
Phase diagram with pressure
Pu also has pressure-induced phases. Do these contribute to the strong
temperature dependences at zero pressure?
Elastic moduli: the usual temperature dependence
Varshni figured this out
We can get γ from either thermal expansion or elastic moduli
Cu: 3.5% decrease from 10K to 300K
α-Pu: 34% decrease from 10K to 350K
Bulk and shear moduli have same temperature dependence—and
it is much bigger than Cu
δ-Pu: 31% decrease from 10K to 350K
Mechanically-induced hysterisis.
Cannot recover original state on heating to 500K.
Poisson ratio independent of temperature from 10
-300K for low-Ga alloys
Poisson’s ratio determines how
much a material bulges when
stressed uniaxially.
It is typically strongly
associated with bond strengths
V=0.183 @ 300K
In Pu, moduli change an order of magnitude more than in
a typical metal, but not Poisson’s ratio. This suggests a
single new physics driver is responsible
From Poisson ratio
In alpha Pu, strong temperature variation of moduli and same
temperature dependence of B and G suggest one physical driver
overwhelms ordinary temperature dependence.
Pu single crystal measurements
(Ledbetter and Moment, 1975)
• c11
=36.3 GPa controls longitudinal sound speed
• c44 =33.6 GPa controls one shear speed
• c*
=4.8 GPa controls the other shear speed
Biggest shear anisotropy of an fcc metal
Hidden phases like ZrW2O8?
Different sign for elasticity and thermal expansion.
Grüneisen bites the dust.
Invar fit
Invar-a gradual change yields negative thermal
expansion
Fe0.65Ni0.35 invar
Volume fixed, stiffness
increases –OK????
The framework solid model: Pu and ZrW2O8
M.E. Simon and C. M. Varma, Phys. Rev.
Lett. 86, 1782 (2001)
Rigid squares, floppy bonds-remember this for Pu!
Cold—the square has the biggest area.
Hot—the average area is reduced!
This is a “thermodynamic” model-it has
enough degrees of freedom.
Pressure and constrained lattices
Bain’s path, framework solids etc.
The bcc volume is
less than the fcc
volume.
fcc(stiff)
bcc(soft)
d-Pu is very soft (softer than Pb) and has an especially soft shear mode.
Poisson’s ratio is about 0.43 along the soft direction, making Pu nearly like a
liquid when squeezed in this direction.
Pu does not care what its volume is!!!!!!!!!!!!!!!!!!
Does this make the framework solid model applicable?
Poisson ratio for fcc Pu
Today’s key question
• Pu and ZrW2O8, soften as volume
decreases. Why? What microscopic
models can make this happen
We have not succeeded in answering all of our questions. Indeed, we
sometimes feel that we have not completely answered any of them. The
answers we have found only served to raise a whole new set of questions.
In some ways we feel that we are as confused as ever, but we think we are
now confused on a higher level, and about more important things.
-Andre Malraux
Beware of the muddle puddles. -Benjamin Migliori, age 3, April 1986
I was trying to paint a jungle
scene but it came out looking like
a clown suit
-Robert Migliori, age 10, Mar
2000