Elasticity of Pu and ZrW 2 O 8 -a window into fundamental understanding Albert Migliori NHMFL/LANL 1)Intro and review 2)ZrW 2 O 8 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 Solid and Temperature moduli 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 c ij —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, C p =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 ZrW 2 O 8 ZrW 2 O 8 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.
ZrW 2 O 8
ZrW 2 O 8 -Structure and Thermal Expansion A. P Ramirez and G. R. Kowach, Phys. Rev. Letters, 80, 22 (1998)
ZrW 2 O 8 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 ZrW 2 O 8 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 ZrW 2 O 8 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) • c 11 =36.3 GPa controls longitudinal sound speed • c 44 =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 ZrW 2 O 8 ? Different sign for elasticity and thermal expansion. Grüneisen bites the dust. Invar fit
Invar-a gradual change yields negative thermal expansion Fe 0.65 Ni 0.35 invar Volume fixed, stiffness increases –OK????
The framework solid model: Pu and ZrW 2 O 8 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 ZrW 2 O 8, 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
Recommend
More recommend
