Our accurate measurements of alpha (α)-plutonium's low-temperature elastic constants yield insights into α-plutonium's unusual physical properties, into delta (δ)-plutonium's properties, and into actinide properties in general. Delta-plutonium possesses a four-atom face-centered cubic unit cell (see the article on page 9). Alpha-plutonium possesses a sixteen-atom monoclinic unit cell. (Roughly, we can describe monoclinic as a brick-shaped object sheared along one brick surface.) We focus our discussion on three elastic constants: the bulk modulus *B* (which reflects resistance to volume change caused by pressure), the shear modulus *G* (which reflects resistance to shape change caused by a shearing force), and Poisson ratio (a simple function of the *G/B* ratio), which, better than any elastic constant, reflects interatomic-bonding type. The bulk modulus joins the binding energy and volume to form the triumvirate of cohesion properties of a crystalline solid.

*Alpha (α)-plutonium's sixteen-atom monoclinic unit cell consists of two eight-atom layers, both highly distorted hexagonal networks. With only four symmetry elements, this may be the least-symmetrical unit cell among the elements. Delta (δ)-plutonium's face-centered cubic unit cell possesses forty-eight symmetry elements, the highest-possible symmetry. The crystallographic mechanism for the δ-α transformation remains disputed.*

Although at least eleven measurements were reported of α-plutonium's ambient-temperature bulk modulus, only one very-low-temperature report exists. That study reported not the bulk modulus, but the Young and shear moduli, from which the bulk modulus was computed. The eleven previous measurements, excluding outliers, averaged 50 ± 5 gigapascals (GPa). We found 54.6 ± 0.2.

To determine these three fundamental elastic properties, we used resonant-ultrasound spectroscopy (RUS) to measure the normal vibrational modes of a polycrystalline (quasi-isotropic) parallelepiped-shape specimen. Crudely speaking, we look for the natural macroscopic vibration frequencies of a simple-shape object, analogous to ringing a bell.

The figure below shows an example of each type of mode, symmetric or antisymmetric, across the three spatial directions. The RUS method is extremely accurate, convenient, and it provides all the elastic moduli in one measurement, but requires complex computations to extract the elastic constants.

*These eight figures are computer-generated examples of vibration normal-modes of a rectangular parallelepiped. Mode shapes depend on specimen shape, mass density, and anisotropic (fourth-order-tensor) elastic coefficients, as well as on the relationship between the crystal axes and the macroscopic specimen axes. In principle, we can use any regular shape: parallelepiped, cylinder, sphere, or ellipsoid. We require sophisticated algorithms to extract the elastic-stiffness coefficients from the normal-mode frequencies.
*

Some principal results of our measurements appear in the two diagrams on the next page. The curves represent a function derived from the energy of an ensemble of thermally populated quantized vibrations (the phonon frequencies). Because elastic moduli are second derivatives of the energy with respect to deformations, an elastic modulus C(T) changes with temperature T according to C/T = C(0)[1 – K < E>], where

*C(T) = C(0) – s / *[exp*(t / T)* – 1]. (1)

This function and approximation are derived from fundamental physics of elasticity and apply to a wide material variety, ranging, for example, from diamond to the negative-thermal-expansion insulator zirconium tungstate to the metal copper. For the three parameters needed to fit this function to our measured bulk modulus, we obtained C(0) = 70.9 GPa, s = 11.3 GPa,
and *t =* 158.8 kelvins (K). For this model,
*t* ≈ ΘE = (3 / 4)ΘD; ΘE and ΘD being the Einstein and Debye characteristic temperatures.

Our measurements reveal that α-plutonium, the stiffest allotrope, shows larger elastic-stiffness changes with temperature than the much softer allotrope, δ-plutonium, which is an unexpected result. Stiffer materials usually show the weakest softening upon warming. Our measurements also reveal that a familiar, but usually ignored, elastic parameter--the dimensionless Poisson ratio (ν)--shows temperature independence: an exceptional event.

*The variation of several elastic constants with temperature is shown for alpha (α)-plutonium. The figure includes four elastic constants: longitudinal modulus *C11*, Young (extension) modulus *E,* bulk modulus *B,* and shear modulus *G.* All show smooth behavior consistent with an Einstein-oscillator model. The + and × symbols denote values from older measurements by others.*

*The relative bulk and shear moduli, *B* and *G,* versus temperature are shown for alpha (α)-plutonium. Unusually large (30 percent) changes occur, which we attribute to itinerant-to-localized electron transitions during warming. The unusual same percentage changes mean a constant *B/G* ratio and therefore a constant Poisson ratio ν; again, results expected for itinerant-to-localized electron transitions. The slight additional softening above about 325 kelvin (K) may represent premonitory behavior for the α-β transition near 400 K.*

A simple electron-gas viewpoint accounts for the unusual softening of the bulk modulus and shear modulus and the invariance of the Poisson ratio ν. For this model, elastic stiffnesses decrease as electrons change from itinerant to localized, from the free-electron gas to electrons fixed at ion positions, as we describe below. Applying this viewpoint predicts approximate divalency for plutonium, implying a 5f ^{6} localized-electron configuration. (The most advanced theories suggest either 5f ^{4} or 5f ^{5}.)

For the bulk modulus at zero temperature, we obtained 70.9 ± 0.05 GPa, much higher than all previous measurements except one. About a dozen theoretical estimates give an average value of 93 ± 48 GPa, numbers much too high. From the high-temperature slope dB/dT, we can estimate a Gruneisen parameter γ using the following relationship:

dB /dT = −3kγ (γ + 1) / V_{a}. (2)

Here, k denotes the Boltzmann constant and V_{a} atomic volume. Substitution gives γ = 5.1. The Gruneisen γ comes from an important approximation in condensed-matter physics and can be thought of as the volume dependence of the elastic stiffnesses. In diamond, there is little pressure dependence of the stiffnesses; so γ is small; in lead, a "soft" material, γ is large. There exists an alternative approach to estimating the Gruneisen parameter. We use the following relationship:

B– B_{0} = 3kΘ_{E} γ( γ + 1)/2V_{a}. (3)

Here B denotes the harmonic bulk modulus obtained by extrapolating the *B(T)* curve linearly to zero temperature. Substitution gives
γ = 5.1. The handbook value from lattice-
specific heat is 6.8. However, as we describe
later, we believe the best estimate of Gruneisen's parameter equals ~3.5. The higher apparent values from our measurements arise from electron localization-delocalization during temperature change.

Zero-temperature elastic constants provide the best estimate of the Debye characteristic temperature ΘD. Since Einstein's seminal lattice-vibration studies, many authors calculated the Debye temperature from the bulk modulus. However, ΘD depends much more on the shear modulus. We use a relationship given by Kim and Ledbetter:

Θ_{D} = 2933.22ν_{m} / V_{a} ^{1/3}. (4)

Here, νm denotes mean sound velocity obtained from the average <1 / ν3i > . This approach gives Θ_{D} = 205 K. In Equation (4), units on νm are cm/µs and on Va , A3. Although Θ_{D} (elastic) should equal Θ_{D} (specific-heat) at zero temperature, our value exceeds considerably most specific-heat values. Specific-heat values can be highly inaccurate if the material departs significantly from a low-temperature
Debye model.

The large apparent γ arising from the very large changes in elastic constants with temperature that we observe, and the near-identical fractional temperature dependence of both bulk and shear moduli lead to some new insights into itinerant-electron density. A main theme in explaining actinide-element physical-property oddities is the density of itinerant (free) electrons. Electron density changes systematically as atomic number increases in the actinide series (actinium, thorium, pro-tactinium, uranium, neptunium, plutonium, americium, . . . ).

*In this photo of a resonant-ultrasound-spectroscopy (RUS) measurement cell, the specimen (a parallelepiped in this case) sits sandwiched between two lead zirconate titunate (PZT) transducers. No bonds or adhesives are used to attach the specimen; it simply rests on the lower transducer while the weight (a few grams) of the upper transducer holds it in place. One transducer sweeps through frequency. The second transducer detects normal-mode (standing-wave) resonance frequencies manifested as vibration-amplitude maxima.*

A common view is that lighter actinides possess more itinerant, bonding electrons that increase elastic stiffness (through the degeneracy pressure), and heavier actinides possess more localized electrons that contribute less to bonding and less to elastic stiffness. Plutonium sits between these limits and possesses mixed itinerant-localized electron character--a balance that depends on temperature, pressure, alloying, and atomic coordination.

In 1936, Fuchs used extended Wigner-Seitz theory to show how itinerant-electron density affects the shear modulus. Gilman (1971) gave a similar relationship for the bulk modulus, using Heisenberg's principle to estimate the electron-gas kinetic energy.

Combining the Fuchs and Gilman equations gives *G/B* = 5/8 and ν = 7/29. Thus, both *G* and *B* soften considerably as electrons change from itinerant to localized. The Poisson ratio remains unchanged, independent of ionic charge and ionic radius.

Such similar *B, G* behavior (and invariant ν behavior) is exceptional. In most inorganic materials, elastic stiffnesses, absent phase transition, change by up to ten percent from
0 to 300 K. Ordinary materials show substantial Poisson-ratio changes. However, if a single physics driver produces larger changes in elastic stiffness with temperature, then these larger changes will overwhelm usual processes, revealing behavior associated primarily with whatever drives the large changes--in this case electron localization during warming.

The figure at right shows our ν(T) measurements for α-plutonium (monoclinic) and for δ-plutonium (face-centered cubic) alloyed with aluminum or gallium. To show typical ν(T) behavior, we show handbook measurements for aluminum (a typical face-centered cubic metal), lead (a heavy metal often compared with plutonium), and α-uranium (a heavy-metal early actinide). Lead shows an expected relatively large positive slope, reflecting its low melting temperature. Alpha-uranium shows an even larger (and convex) positive slope, arising probably from its well-known magnetic interactions, which always alter elastic responses. Alpha-plutonium shows small, nearly temperature-invariant, changes.

*The relative Poisson ratio/temperature for six materials. Aluminum (Al) represents a typical face-centered cubic metal. Lead (Pb), also face-centered cubic, represents a heavier metal, more comparable with plutonium (Pu). Compared with aluminum, lead shows a larger change because of its lower melting point. Uranium (U) (orthorhombic) represents a heavy, actinide metal. It shows a large convex-shape change that reflects its magnetic-state transitions. Alpha (α)-plutonium shows a temperature-invariant Poisson ratio, as predicted for the case where itinerant-to-localized electron transitions dominate the interatomic bonding. The more-dilute delta (δ)-plutonium-gallium (Ga) alloy behaves like α-plutonium, that is, it has a temperature-invariant Poisson ratio, also suggesting itinerant-to-localized electron transitions in this material. The more-concentrated δ-plutonium-aluminum alloy behaves like a typical metal, suggesting that higher alloying completes the itinerant-to-localized transition. The kink near 180 kelvin (K) shows the onset of the δ-α transformation, estimated to be 5 percent α-phase at zero temperature. *

These changes agree with the prediction made by the above simple electrostatics model, suggesting that the physics driver is the degeneracy pressure of the itinerant electrons and that the number of these electrons changes with temperature.

Using the Fuchs-Gilman equations, we can estimate the number of itinerant electrons. Substituting the observed bulk modulus
*B* = 71 GPa for α-plutonium at 0 K, and the atomic radius ro = 1.34 Å (obtained from
handbook unit-cell dimensions), we obtain
an ionic charge q = 2.0, meaning that six of
α-plutonium's eight outer electrons are localized. We assume that these are the six f-electrons, the ground-state electron configuration being [Rn]5f ^{6}7s^{2}. (A theory by Savrasov and Kotliar for δ-plutonium gave a 5f 5 ground state, and one by Wills et al. gave 5f 4.) Recognizing some arbitrariness in these electron-orbital assignments, we note that plutonium
(as evidenced by its low bulk modulus) behaves more like an sp-electron metal than like a
d-electron metal; more like lead than like iron.

From the Fuchs-Gilman equations, we can derive a relationship connecting ionic charges at different temperatures:

q_{2} / q_{1} = (B_{2} / B_{1})^{1/2}. (5)

Considering the α-phase bulk-modulus change from 0 to 300 K (71 to 55 GPa), we can estimate that, for this temperature interval, q_{2} / q_{1} = 0.88. Thus, warming from 0 to 300 K converts 0.24 electrons from an itinerant state to a localized state. Invoking this itinerant-localized electron-state change explains the unusually large bulk-modulus decrease with increasing temperature and the invariant Poisson ratio. Here, we neglected the smaller bulk-modulus decrease that arises from usual lattice softening. We assume that the itinerant-localized transition continues to occur as plutonium is warmed above 300 K, perhaps accounting for plutonium's low melting
temperature and multiple (six) crystal structures. Sufficient alloying with gallium or aluminum produces a Poisson ratio with near-typical temperature dependence. All this suggests that electron localization stabilizes the face-centered cubic δ-plutonium phase in the maximally localized state and that delocalization is arrested in cooled gallium-stabilized δ-plutonium.

The figure on page 7 shows a second remarkable result: the Poisson ratios for face-centered cubic high-aluminum-content δ-plutonium-aluminum alloys show near-typical temperature behavior. Thus, sufficient alloying completes localization, blocking changes with temperature. This conclusion finds confirmation in the observation that the bulk modulus of the plutonium-aluminum alloy changes only 13 percent between 0 and 300 K, versus a 30-percent change for unalloyed α-plutonium. Support comes also in the conclusion from theory that the degree of f-electron localization depends strongly on chemical environment, on the interaction between bound f-electrons and band electrons of non-f origin.

Our measured elastic constants, especially the Poisson-ratio results, and our electron-gas interpretation yield the following conclusions: in plutonium all six 5f electrons are nominally localized; as plutonium is warmed, localization increases; in face-centered cubic δ-plutonium, localization reaches completion, whether through increased temperature or through alloying; and theories that include a large itinerant-electron fraction lack support in the observed elastic constants. We can also conclude that the α-δ phase transformation does not result from abrupt electron localization; that for both phases, the ionic charge *q* ≈ 2; that plutonium behaves more like an sp-electron metal than like a d-electron metal; and that the elastic-constants temperature dependences show no phase-transformation evidence (electronic, magnetic, or structural).

*This article was contributed by Albert Migliori of the Materials Science and Technology Division-National High Magnetic Field Laboratory and the Seaborg Institute; Jonathan Betts of MST-NHMFL; and Hassell Ledbetter, David Miller, Michael Ramos, Franz Freibert, and David Dooley of the Nuclear Materials Technology Division.*

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