About thirty elements occur as face-centered cubic crystal structures, the closest packing of hard spheres in an infinite space. Common examples include aluminum, copper, silver, gold, and lead. Face-centered cubic elements possess a useful feature: high mechanical ductility arising from easy dislocation movement. The material's elastic constants provide interest in this regard because dislocations move in an anisotropic elastic-strain field.
Besides relating to plastic deformation, elastic constants also relate to many other solid-state properties, including atomic-vibration amplitudes, characteristic vibration frequencies (Einstein, Debye), creep, diffusion, fracture strength, hardness, melting, phase transformations, phonon spectra, point-defect energies, sound velocities, stacking-fault energy, theoretical strength, superconducting-transition temperatures, and thermal conductivity. In principle, the elastic constants relate to any lattice-vibrational property.
Face-centered cubic crystal structure of delta (∂)-plutonium.
Among all the face-centered cubic elements, delta (δ)-plutonium shows the most remarkable elastic constants. This extreme behavior appears particularly in the Zener anisotropy ratio A =2C44 /(C11–C12 ), the average value being 2.5 ± 0.5 for face-centered cubic elements. For δ-plutonium this ratio is 7.0. This means that δ-plutonium's various elastic constants, which are fourth-order tensors, change strongly with crystallographic direction. Delta-plutonium's other elastic oddities include unusually high C12/C11 and C44/C11 ratios.
A material has many elastic constants; the more common include the Voigt elastic stiffness Cij, Young modulus Ehkl, shear modulus Ghkl,lmn, bulk modulus B, and Poisson ratio vhkl,lmn. The bulk modulus lacks indices because, being the negative ratio of pressure change to fractional volume change, it is a rotation-invariant scalar. The Poisson ratio is not a tensor, but instead the dimensionless ratio of elements of two fourth-order tensors.
At Los Alamos, we focus on plutonium's elastic constants because they provide accurate core physical properties that relate directly to theory and to other core physical properties (enumerated above). We measure them by various methods, especially resonance-ultrasound spectroscopy (RUS), which is described in the preceeding article.
Young moduli of delta (∂)-plutonium and aluminum. Aluminum shows near isotropy. Plutonium represents the most anisotropic cubic element, a property not yet well understood. Maxima occur along <111>, minima along <100>.
Torsion moduli of delta (∂)-plutonium and aluminum. Plutonium's maxima occur along <100>, the minima directions for the Young modulus. The plutonium figure, reflecting elastic compliance more than elastic stiffness, presents a surprise.
The diagrams above compare δ-plutonium, the most anisotropic known element, with aluminum, a more-typical material. Plutonium's Young modulus shows eight lobes along the four <111> directions with minima along the six <100> directions, the maximum/minimum ratio being 5.4. The contrast with aluminum is enormous where this ratio is 1.3. Plutonium's torsion modulus shows opposite behavior to the Young modulus with maxima along the <100> directions and minima along the <111> directions. The maxiumum/minimum ratio is 5.0; for aluminum this ratio is 1.2.
Plutonium's Poisson ratio provides particular interest because it shows different profiles in different principal crystallographic directions. And, in some directions, it shows large negative values. A negative Poisson ratio means that pulling in one direction causes expansion in the transverse plane, a phenomenon associated traditionally with cork and network structures.
Results for plutonium are shown with those for aluminum in the illustrations at right. The illustrations are for extensional strains in the three crystalline directions <100>, <110>, and <111>. For plutonium, an extensional strain e in the <100> direction results in an isotropic contraction of 0.42e in the perpendicular direction. For an extension e in the <111> direction, the perpendicular isotropic contraction is only 0.09e. For an extension e in the <110> direction, the lateral strain is highly anisotropic. In fact, the extension e in the <110> direction results in an extension 0.48e in the <-110> and <1-10> directions, but a contraction of 1.09e in the <001> direction.
Much current research proceeds on negative-Poisson-ratio materials, a material class called auxetic. The diagrams for plutonium dispel any notions of elastic similarity, isotropic behavior, always-positive Poisson ratios, and simple crystallographic-direction/stiffness relationships.
Poisson ratios for delta (∂)-plutonium and aluminum along three principal crystallographic directions. Along , plutonium shows negative (red) values. This means unusual (auxetic) behavior: opposite sign for the transverse strain.
Plutonium's extreme high elastic anisotropy reveals interesting features of its interatomic bonding. Max Born's lattice-dynamics model (1943) for face-centered cubic crystals predicts A = 2. Invoking farther-neighbor interactions or volume forces cannot increase this by much. Thus, we need to invoke angular (three-body) forces, which impose difficult computational problems. Fuchs' (1936) extended Wigner-Seitz calculations showed that for face-centered cubic lattices, considering only the electron-electron-ion electrostatic terms, one obtains A = 9.0, thus providing a promising possible explanation for δ-plutonium's high elastic anisotropy.
The high anisotropy also means that δ-plutonium lies near a Born mechanical instability boundary. Well known are the Bain lattice correspondence δ–δ'–ε phase interrelationships among face-centered cubic, face-centered tetragonal (body-centered tetragonal), and body-centered cubic phases. The high elastic anisotropy and the low C11–C12 value mean that these three crystalline phases interconvert easily, a fact first appreciated by Zener for the alkali metals. (Body-centered cubic at ambient temperatures, only the alkali-metal elements rival plutonium's high elastic anisotropy.) Zener also realized that high elastic anisotropy means high vibrational entropy, lower free energy, and a tendency toward phase transformation.
Clearly, for most purposes, aluminum can be treated as a nearly isotropic material. Plutonium's enormous elastic anisotropy implies strong anisotropy in related mechanical-physical properties, and it provides a keen quantitative check (constraint) on the many emerging ab initio theories of plutonium.
Our current research focuses not only on plutonium's peculiar anisotropic elastic properties, but also on how these properties depend on composition, temperature, and pressure, where another set of peculiarities appear. Recently, we succeeded in relating the elastic constants to 5f-electron itineracy localization, a problem affecting the physical properties of all the actinides.
"Elastic properties of face-centered cubic plutonium," Hassel Ledbetter and Roger Moment, Acta Metallurgica 24 (1976), pages 891-899.
Elasticity and Anelasticity of Metals, Clarence Zener (University of Chicago Press, Chicago, 1948), especially Chapters II-IV.
"Lattice theory of mechanical and thermal properties of crystals," Gunther Leibfried, in Handbuch der Physik (Springer, Berlin, 1955), pages 104-324.
This article was contributed by Hassel Ledbetter of the Nuclear Materials Technology Division; Albert Migliori of the Materials Science and Technology Division-National High Magnetic Field Laboratory and the Seaborg Institute; and Robert G. Leisure, Physics Department, Colorado State University, and visiting scientist at the NHMFL.
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