Journal of Alloys and Compounds 368 (2004) 62–74
Temperature-dependent defect properties from ion-irradiation in Pu(Ga) M.J. Fluss∗ , B.D. Wirth, M. Wall, T.E. Felter, M.J. Caturla, A. Kubota, T. Diaz de la Rubia Lawrence Livermore National Laboratory, P.O. Box 808, East Avenue, Livermore, CA 94577, USA Received 30 May 2003; received in revised form 25 July 2003; accepted 27 August 2003
Abstract We report the measured decrease of electrical resistivity during isochronal-annealing of ion irradiation damage accumulated at lowtemperature (10 or 20 K), and the temperature dependence of the resistance of defect populations produced by low-temperature damageaccumulation and annealing in a stabilized ␦-phase plutonium alloy, Pu(3.3 at.% Ga). The normalized change in resistivity is compared for a specimen that was either self-irradiated (from Pu ␣-decay and the associated uranium-recoil) or 3.8 MeV proton-irradiated with a Pelletron electrostatic accelerator. Modeling of the annealing data through combined molecular dynamics (MD) and kinetic Monte Carlo (KMC) methods describes the defect populations as a function of irradiation type and annealing temperature. It is observed that interstitial clustering is extant for the self-irradiation, but that the corresponding vacancies from the uranium damage cascade appear to be more point defect-like, as exhibited by their subsequent annealing behavior and comparison with the experimental annealing properties from the proton-irradiation. We also report the temperature dependence of the resistance of defects resulting from low-temperature damage accumulation and subsequent annealing at three temperatures: 30, 150, and 250 K. For the two defect populations dominated by vacancies and vacancy clusters (150 and 250 K), we observe a temperature-dependent defect population resistance of the form −a[ln(T)] + b suggestive of a Kondo impurity. A discussion of possible causes leading to this observation and their effects, as it might relate to the nature of the ␦-phase of Pu, are presented. © 2003 Elsevier B.V. All rights reserved. PACS: 71.27.+a; 61.72.Cc; 61.80.Jh; 61.80.Az Keywords: Pu; Radiation damage; Annealing; Vacancies; Kondo impurity; Quantum criticality
1. Introduction The physical metallurgy of ␦-phase plutonium, particularly those aspects related to radiation damage and associated vacancy and self-interstitial properties, is technologically important, and yet still requires a deeper fundamental understanding. Low-temperature damage-accumulation and subsequent isochronal-annealing provides an experimental methodology for determining important mass transport parameters based on chemical rate equations, e.g., the interstitial migration energy and the vacancy migration energy [1]. The starting point for deducing these parameters is the ability to perform high fidelity annealing experiments.
∗
Corresponding author. E-mail address:
[email protected] (M.J. Fluss).
0925-8388/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.jallcom.2003.08.080
Although annealing experiments have been reported for ␦-stabilized Pu(Al) [2,3], the fidelity of the data was limited, and hence the analyses were only qualitative. In this paper, we report high-fidelity isochronal-annealing curves of the damage-based electrical resistance for self- and proton-irradiated Pu(Ga). The annealing properties were measured on a ␦-stabilized alloy, Pu(3.3 at.% Ga), an fcc ␦-phase of plutonium which was resistant to the martensitic phase transformation to the ␣ -phase at all temperatures of these experiments. An important element of the present work is the experimental determination of the temperature for the highest temperature-annealing stage (Stage V), which is identified with the dissolution of vacancy clusters. There is experimental evidence in the literature (albeit largely unnoticed) that the specific resistivity of defects in ␦-stabilized Pu may exhibit increasing resistivity with decreasing temperature [2]. Further, it has recently been
M.J. Fluss et al. / Journal of Alloys and Compounds 368 (2004) 62–74
reported that the low-temperature, anomalously large, magnetic susceptibility of ␦-stabilized Pu is affected by radiation damage-accumulation, and exhibits a non-linear increase in magnetic susceptibility below 70 K [4]. We have measured, as near to the dilute limit as was practical, the temperature dependence of the resistance of radiation damage of defect ensembles or populations produced by low-temperature damage-accumulation and subsequent annealing. This study has led to the discovery of a spectacular violation of Matthiesson’s rule [5,6] for vacancy-defects in ␦-stabilized Pu(3.3 at.% Ga) and confirmation of the increasing specific resistivity with decreasing temperature dependence of the vacancy-dominant defect populations.
2. Experimental description and results 2.1. Low-temperature damage-accumulation and isochronal-annealing The specimen was an ∼10-m thick Pu(3.3 at.% Ga) foil. It was prepared from a 20-year-old alloy by first melting (to remove accumulated radiogenic He), and then homogenizing the Ga by thermal treatment. The specimen was rolled, and re-homogenized at 450 ◦ C for ∼100 h. The foil was cut to 10 mm × 10 mm and clamped between two alumina frames, each of which had a 5 mm × 5 mm opening to allow for particle transmission from subsequent 3.8 MeV proton-irradiations. One frame had a built-in, spring-loaded, Kelvin probe. The sense-electrodes were 7 mm apart, and the current electrodes were 8 mm apart. A calibrated silicon-diode temperature sensor was located ∼1 mm from the edge of the specimen attached to the alumina frame with a thermally conductive epoxy. The specimen assembly was thermally well coupled to a gold-coated copper-frame. After all electrical connections were made the entire assembly was coated with 1 m of polyimid (to contain Pu recoils and avoid contamination) and then attached to the nose of a liquid-helium flow-thru cryostat located in a vacuum chamber (base pressure, 3 × 10−9 Torr) connected to the ion-beam-line of a 4 MeV Pelletron. As a practical matter, these annealing studies are difficult to execute because they involve differences of a few in ∼130 m, leading to experiments where the data is the result of a small difference between experimentally measured large numbers. Temperature measurement precision and resistance measurement precision were very important in this work. A digital controller with 100 watts of heating power achieved accurate temperatures (T) and a reproducibility of 10−2 K over the range of 5–550 K. Such precision was required to achieve high quality isochronal-annealing data because of the high specific resistivity (ρ(T)) and large resistivity-temperature derivative (dρ/dT, 4 K 0.94. We posit that such a linear fit is suggestive of a local, non-cooperative magnetism, or Kondo-like impurity behavior [11]. It is understandable that such behavior is not observable as a resistance minimum, considering the usual Kondo impurity textbook example, because of the large base resistance of the annealed specimen, Ra (Tj ), its large temperature derivative, and the corresponding relatively small value of the damage resistance. However, the Kondo-like nature of the data becomes visible in the present work when one focuses on the resistance of the defect populations alone, Rself (Tj , Ti ) versus temperature Tj . The data sets, Rself (Tj , Ti ) = 150 K and Rself (Tj , Ti ) = 250 K, are seen in the inset of the top panel of Fig. 2 to exhibit a common intercept on the ln(T) axis, suggesting that the defect sites (vacancies and vacancy clusters, which are discussed below) vary primarily in number density, but are
Table 1 Empirical estimates for Stages I, III, and V transitions as a function of the melting temperature Tm [10]
Pb Al Pu(Ga) 3.8 MeV p+ Pu(Ga) ␣-decay Ag Au Cu Ni
Tm (K)
TI (K) measured
TI (K) (0.02 ± 0.015) Tm predicted
601 933 953 953 1235 1337 1356 1726
4 37 ∼35 45 ± 5 28 – 38 56
12 19 19 19 25 27 27 35
± ± ± ± ± ± ± ±
9 14 14 14 20 20 20 26
TIII (K) measured
TIII (K) (0.2 ± 0.02) Tm predicted
160 220 180 ± 5 180 ± 5 230 290 250 340
120 187 191 191 247 267 271 345
± ± ± ± ± ± ± ±
12 19 19 19 25 27 27 35
TV (K) measured
TV (K) (0.45 ± 0.03) Tm predicted
290 – 310 ± 5 310 ± 5 540 530 605 750
270 420 429 429 556 602 610 777
± ± ± ± ± ± ± ±
18 28 29 29 37 40 41 52
The annealing data for Pu(3.3 at.% Ga) with a 300 s isochronal anneal is shown. Stage I for the p+ irradiation of the Pu(Ga) is not clear but appears to be ∼35 K. For the other Stages, statistical errors are indicated. Longer anneal times will shift the transitions to lower temperatures, but no more than ∼10 K.
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Table 2 Results of MD–KMC simulation are shown as the percentage of single vacancies (v) and single self-interstitials (I) remaining relative to irradiation at 10 K, in clusters of a given size for proton-irradiation (PI) and self-irradiation (SI) following annealing to 30, 150 and 250 K T(K)
30
150
250
Ion
defect
Cluster size 1–2
3–5
6–10
SI PI SI PI
v v I I
97.00 99.94 7.8 82.6
2.5 0.06 3.2 16.5
0.5
SI PI SI PI
v v I I
52.4 54.1 4.4 0.0
1.1 0.03 1.1 12.4
0.2
SI PI SI PI
v v I I
0.0 0.0
18.2 12.9
22.9 20.9
of a similar nature with regards to the temperature dependence of their resistance. A more satisfying analysis would be to know the specific resistivity per vacancy and its decreasing value with cluster size. However, the analysis presented later in Table 2, which describes increased vacancy clustering from 150 to 250 K for the self-irradiated specimen, is qualitatively consistent with the experimentally observed decrease in resistance from 150 to 250 K, about a factor of 4. Whether the temperature dependence of the defect population and its specific resistivity arise from the vacancies alone, or are associated with neighboring Pu atoms, at this time, is an open and interesting question. Thus, it is reasonable to conclude that the increased atomic spacing near vacancies results in a measurable Kondo behavior as evident in the temperature dependence of the vacancy-defect populations’ resistance in Pu(3.3 at.% Ga). This compels us to consider that local defects and their associated disorder may be important in determining global electronic structure properties in these highly correlated electron systems. Evidence that vacancies and not interstitials are the source of the observed temperature dependence is found in the self-irradiated Ti = 30 K data. The temperature dependence of the Ti = 30 K defect population seen in the top panel of Fig. 2, although it is also described by a linear fit over a relatively small temperature range, does not share, in extrapolation, the common intercept with the 150 and 250 K defect populations. Indeed, the shallower slope suggests that the violation of Matthiesson’s rule for this defect population is much less, and we assert that this is an indication that some fraction of the resistance of the defect population is temperature independent. Unlike the 150 and 250 K anneals, where the defects are dominated by vacancies and small vacancy clusters, the 30 K data is the result of a specimen that has undergone limited annealing recovery, and still has a significant resistivity-component associated with the interstitial-defect population. If we assume that the tempera-
8.1 0.9
11–20
21–50
50–100
>100
7.3
61.8
10.2
1.6
9.4 0.04 4.8 3.2 8.6
ture dependence of interstitial defects is much less sensitive to temperature, then a shallower slope would result. The temperature-dependent resistance of the proton- and self-irradiation defect populations following an anneal to 150 K (shown in Fig. 2b for the 40 nA proton-irradiation with a 150 K anneal (䉱), and the self-irradiation with a 150 K anneal (䊉) both exhibit a similar Kondo-like [11] behavior. Again the linear fits to the data exhibited χ2 > 0.94 and with the removal of the data point at 10 K, χ2 > 0.99 For a more detailed comparison, the self-irradiation R(T) data in Fig. 2, including the error bars, is also shown normalized to the proton-irradiation R(T). The similarity of the proton-irradiation and self-irradiation vacancy populations at 150 K, is understood based on our molecular dynamics (MD)–kinetic Monte Carlo (KMC) modeling presented in Table 2, and described below. That is the defect populations existing in the temperature range of Stages III and V are so similar as to reveal little difference with regards to the temperature dependence of their resistance. It is important to emphasize that in all cases studied here, subsequent annealing at 550 K for 1 h was found to fully anneal the defect related resistivity, thus confirming the structure–property connection of the resistivity to accumulated defect populations. We are able to estimate the Kondo temperature for vacancy defects in this specimen. The data sets, Rself (Tj , Ti = 150 K) and Rself (Tj , Ti = 250 K), show a negative deviation from the linear fit for the lowest temperature point at 10 K, in both the self-irradiated as well as the proton-irradiated, Rproton (Tj , Ti = 150 K), data. Indeed, refitting the data without the 10 K point improves the statistical quality of all fits yielding χ2 > 0.99. In the context of a Kondo impurity model this suggests a Kondo temperature of TK < 10 K, which is considerably less than that deduced recently, 200 K < TK < 300 K, for the analogous ␦-phase Pu(Al) [12]. This suggests that a broad-distribution
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of Kondo temperatures, TK , may be extant in this solid-state system, each TK possibly associated with a different defect or disorder type.
4. Discussion 4.1. Molecular dynamics and kinetic Monte Carlo modeling of low–T irradiation and isochronal-annealing As mentioned, the two lowest T recovery stages are associated with individual self-interstitial (Stage I) and self-interstitial cluster migration, along with interstitialimpurity interactions (Stage II) [8,10]. Remarkably, the measured annealing spectra (Fig. 1) from the proton-irradiation and self-irradiation specimen appear very similar in Stages III–V, but are distinctly different in the region of Stages I–II recovery. When considering that the present specimens are significantly alloyed (3.3 at.% Ga) and contain an additional 20 years worth of radiolytic impurities of Am and U, as well as a wide variety of other impurities, one expects the shape of these curves to be analogous to many other dilute fcc alloys. That is, with increasing solute content, the Stages I and II (annealing of interstitials) in dilute fcc alloys become less defined due to interstitial–solute interactions (trapping and clustering) [8,10]. Notably, this is the case for the proton-irradiated specimen, but not the self-irradiated specimens, in Stage I/II. We have combined molecular dynamics and kinetic Monte Carlo simulations to model the annealing recovery following proton-irradiation or self-irradiation. This modeling approach reveals the details of the annealing process and defines the defect populations at each stage. The spatial arrangement of defect production from MD simulations of low (∼150 eV) and high (85 keV) energy atomic collision cascades provide the primary damage source term appropriate for proton-irradiation and self-irradiation, respectively. KMC simulates the subsequent defect evolution in three dimensions, including defect diffusion, clustering, and interaction with impurities (note, not included for vacancies) and dislocations, leading to ultimate annihilation, which results from either self-interstitial vacancy-recombination or defect absorption at system sinks during isochronal-annealing. The primary damage source term for the case of protonirradiation was obtained by simulating the interaction and stopping of the 3.8 MeV proton within the binary collision approximation, using transport of ions in matter (TRIM) [13]. The TRIM calculations provide the depth dependence of the kinetic energy transferred to lattice atoms from the high-energy protons. This data, with an average energy transfer of 150 eV per proton-plutonium collision, is used to establish a primary damage production database of the spatial position of vacancy and self-interstitial atoms from MD simulations of the relevant low energy displacements, at energies provided by the TRIM calculations, in plutonium and other high-Z, fcc metals. For self-irradiation, vacancy, and
self-interstitial defects are produced from both the ∼86 keV uranium recoil and the 5.04 MeV He ion. In this case, we track point defect production from the alpha particle using the same approach as described for the proton-irradiation, although it is important to note that we do not include the helium atom as a chemical impurity in these simulations. The defect production from the uranium recoil is obtained from an MD database of displacement cascade evolution for 85 keV recoils in Pb. We have used lead as an fcc surrogate metal, because of its low melting temperature, high-Z and a presumed strong electron-phonon coupling. In addition to the appropriate primary damage source term, the KMC simulations require parameters describing the kinetics of diffusional transport, defect clustering and dissolution, and the interactions between primary defects, defect clusters, and extended defects in the materials. Where possible, this parameter list has been obtained from our atomistic simulations, other experiments on plutonium alloys, and the general defect clustering/migration behavior of other fcc metals [10]. This compendium provided an initial parameter set and was modified in an iterative manner to provide self-consistent agreement with the annealing recovery of the proton- and self-irradiation annealing data shown in Fig. 1. The modified embedded atom method (MEAM) potential of plutonium, developed by Baskes [14], has been used to investigate the properties of vacancy and self-interstitial defects, as well as the evolution of 1 and 10 keV displacement cascades responsible for defect production. Note, that this potential is approximately 10 times more computationally intensive than standard embedded atom method (EAM) potentials and, we have not completed a simulation of an 85 keV displacement in MEAM plutonium. This also partly explains and justifies the use of Pb as a surrogate material for the 85 keV cascade simulations. The results derived from the Baskes MEAM potential are in partial, but not complete agreement with the primary damage source term and defect clustering/migration kinetics required for self-consistent agreement with the experimental data. The key characteristics of the primary damage state, along with the defect clustering and migration kinetics, that are required to capture the distinct differences in Stage I/II recovery and the similarity in Stages III–V recovery for proton-irradiation and self-irradiation are summarized here. A complete discussion of the modeling methodology and results will be the subject of a future publication [15]. Table 2 shows the size dependence of the relative vacancy and self-interstitial defect populations, normalized to their initial production from the MD simulations at 10 K, which are predicted to exist following annealing to a temperature of 30, 150 and 250 K for both the proton-irradiation and self-irradiation. Again, the primary damage source term used for proton-irradiation consists predominately of isolated vacancies and self-interstitials, while for self-irradiation
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it consists of both isolated point defects (from the ␣-ion track) and a dense, spatially correlated region of high defect density, including a significant population of self-interstitial clusters (from the U-recoil) and a largely isolated vacancy rich core. The self-interstitial size populations that exist following annealing to 30 K exhibit the most pronounced source term differences. In the proton-irradiation annealing model, 99% of the self-interstitials are in the form of isolated single self-interstitials, or in small interstitial clusters with a size less than five interstitials which have formed as a result of interstitial-interstitial interactions during initial interstitial transport. However, in the self-irradiation annealing model, 89% of the self-interstitials are in clusters containing more than six interstitials per cluster. According to our MD simulations, these clusters are directly produced in the displacement cascade event. Another key aspect of the self-interstitial defect’s property is that clusters of self-interstitials are believed to behave like small, perfect dislocation loops with a high mobility in one-dimension along the cluster’s Burger’s vector orientation. We believe that this easy one-dimensional transport of self-interstitial clusters, directly produced in displacement cascades, in combination with a presumed stronger interaction between solute and impurity species with isolated self-interstitial atoms are responsible for the differences in Stage I/II recovery between proton-irradiation and self-irradiation-induced damage. Remarkably, the vacancy size distributions, at 30, 150, and 250 K predicted from the coupled MD–KMC proton-irradiation, and self-irradiation annealing simulations, are very similar throughout the annealing protocol, and we believe that this explains the similarity of the Stages III–V recovery. Although not significant to the discussion below, it is important to note that even though the vacancy population within the 85 keV cascade (self-irradiation) simulation is largely isolated (not clustered), a much larger local density of vacancies exists than for the proton-irradiation. This local spatial correlation of vacancies produced in self-irradiation results in a somewhat larger probability for vacancy clustering and the formation of a slightly larger vacancy-cluster size-distribution at 250 K, which produces a modestly greater temperature for Stage V recovery. The key characteristics of our coupled MD–KMC simulations that produce an annealing recovery for proton- and self-irradiation in Pu(Ga) consistent with the experimental results are: (i) the formation of self-interstitial clusters directly in high-energy displacement cascades; (ii) the lack of vacancy clustering in high-energy displacement cascades; (iii) the one-dimensional migration of self-interstitial clusters with low activation energy (∼0.1 eV) for migration; (iv) a stronger interstitial–solute/impurity interaction and trapping for isolated and small self-interstitial clusters
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(N < 5), which migrate three dimensionally, than for larger clusters which migrate one dimensionally; (v) a vacancy migration energy of ∼0.55 eV; (vi) vacancy clustering in the form of three-dimensional void-like structures which have relatively weak binding energies. Notably, the MEAM potential of Pu [14] reproduces some, but not all of these key damage production and defect characteristics. The MD calculations with this MEAM potential reveal a high mobility for the plutonium self-interstitial and small self-interstitial clusters. The activation energy of 0.19 eV for self-interstitial migration is reasonably consistent with the values of ∼0.1 eV used in the KMC simulations of annealing recovery. For the vacancy, we calculate a self-diffusion (sum of vacancy formation and migration) energy of 1.5 eV using the MEAM potential, which is in reasonably good agreement with experimental values of ∼1.3 ± 0.3 eV obtained from tracer diffusion and creep studies [16]. However, the distribution between formation and migration energy is not consistent with the experimental results presented here. The Baskes MEAM Pu potential gives a calculated vacancy formation energy of 0.5 eV, and a migration energy of ∼1.0 eV. Clearly, the value of 1.0 eV for vacancy migration energy is high compared to the Stage V annealing recovery temperatures of 310 K, and the activation energy of ∼0.55 eV used in the KMC simulations to reproduce the experimental behavior. Further, the average behavior of a wide range of fcc metals [10] indicates a distribution of the self-diffusion energy as approximately 60% vacancy formation and 40% vacancy migration. The U-recoil cascade evolution features obtained to date using the Baskes MEAM Pu potential, are consistent with the characteristics of self-interstitial, but not vacancy clustering in high-energy displacement cascades required by the KMC model to reproduce the experimental results. MD simulations of both 1 and 10 keV displacement cascades [17] reveal the formation of a vacancy-rich core region, with essentially no vacancy clustering in the initial 400 ps of cascade evolution. Substantial interstitial clustering was observed in the 10 keV cascades. These cascade simulations are continuing, and they will be reported in the future, with a focus on simulating the high-energy 85 keV uranium cascades of interest for ␣-decay. 4.2. Temperature dependence of defect resistivity There is no complete formalism currently available to describe fully the unusual temperature dependencies of the vacancy resistance in Pu(Ga) reported here, nor the possible consequences of this defect physics on the properties of Pu. However recent theoretical work by Millis et al. [18–20] have shown that the physics of defects in highly correlated electron systems, systems near a quantum critical point, could be both interesting and important to our fundamental
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understanding of the amplification of the consequential effects of small perturbations in these systems. In [19], Millis develops a ‘botany’ of materials exhibiting unusual sensitivity to disorder and describes a taxonomy of mechanisms for the various cases. Specifically, the ‘small parameter’ case, where some scale in the material is very small and where even a weak external perturbation may change the system properties dramatically is described. The ‘Kondo disorder’ effect in heavy Fermion materials is sighted as an example, the small parameter being the Kondo temperature. Millis remarks, that for heavy electron metals the Kondo temperature, TK = EF e−EF /J (EF is the Fermi energy and J is the exchange coupling), can be extremely small when the relevant energy ratio EF /J is not much larger than unity. Hence, a modest variation in the exchange coupling J will lead to an amplified variation, through the exponent, in TK , with correspondingly dramatic effects on the low-temperature properties. It appears that in the present work, vacancies possess a small TK in the fcc Pu alloy and the large inverse temperature dependence of their resistance surely qualifies as a dramatic and till now undiscovered effect. The novel single-impurity physics and proximity to quantum critical points are discussed in [20] where a theory for local defects in a metallic system near a quantum critical point is developed. It has been suggested [21] that the many anomalous properties of Pu, including the six solid-state phases, and the very existence of the ␦-phase, may be manifestations of plutonium’s proximity to a quantum critical point. Additionally, magnetic susceptibility data, suggestive of Kondo-like behavior linked to radiation damage-accumulation at room temperature, is found in the recent thesis of Dormeval [4]. In this work, the author examined five specimens of Pu(Am) with Am concentrations ranging from 4.9 to 24%. The specimens were annealed and held at room temperature for periods inversely related to their damage rates; 9–1.4 months, respectively, and hence, all the specimens were ostensibly equivalent in radiation damage, as measured by displacements/atom. A study of the low-temperature magnetic susceptibility revealed a damagedependent non-linearity in addition to the characteristic large susceptibility, at low fields for T < 70 K at a field of 1 T, but this non-linear increase was suppressed at a higher field of 4 T. Annealing restored the specimens to a temperature independent magnetic susceptibility. We believe that these results describe a system that has accumulated vacancy defects that exhibit enhanced susceptibility in the presence of a weak field while the Kondo effect is suppressed at higher fields. We conclude from our results, that small (three-dimensional) vacancy clusters or perhaps single vacancies, in the ␦-Pu matrix are the structure property basis for a Kondo-like effect [11] that, in this case, must be attributed to a “vacancy Kondo impurity”. It is worthwhile to explore further the relationship of this observation to the existence or stabilization of the ␦-phase itself.
5. Consideration of plutonium ␦-phase stabilization To begin, Pu is the clearest example of a highly correlated-electron element, evidenced by its anomalous resistivity, enhanced low-temperature specific heat, and large magnetic susceptibility. Of the six Pu allotropes, fcc ␦-Pu (320–463 ◦ C) is the most anomalous, and its relationship to the ␦-stabilized binary alloys remains a mystery. The transition from delocalization to localization of the 5f-electrons in the actinides takes place within the plutonium phase diagram [22]. The anomalously low-density fcc ␦-phase of Pu exhibits this by straddling the light- and heavy-actinides, its properties indicating a partial transition towards the first of the heavy-actinides, Am. It has recently been shown, through an analysis of the magnetic susceptibility and electrical resistivity data of ␦-stabilized plutonium [12] that the hypothesis of a Kondo effect [11] in ␦-plutonium is extant. The investigators deduced a Kondo temperature of TK =∼200–300 K, thus implying that “5f electrons are localized as in the case of other known concentrated Kondo systems” [12]. However, although explaining the origin of the anomalous resistivity in ␦-stabilized Pu(Al), the first principle origin of the pure Pu ␦-phase remains as an unsolved problem. The relationship between the pure ␦-phase and the alloy stabilized ␦-phase is complex, for example increasing Ga concentration changes the coefficient of expansion from its anomalous negative value to a positive one. Recently, an Invar model for Pu(Ga) was reported [23] that systematically describes the anomalous thermal expansion behavior of Pu(Ga) alloys via an Invar effect, which assumes the thermal transfer of Pu atoms from a lower-energy higher-volume state to a higher-energy, lower-volume state. The Invar model enables a precise evaluation of the coefficient of thermal expansion and of the Grüneisen constant. The addition of Ga is described as forcing a stabilization of the lower volume state and suppressing the negative thermal expansion observed in pure ␦-Pu. Vacancies may be another example of such a lower-energy high-volume state. From the viewpoint of theory, the ␣-phase of Pu has been considered as well understood within the formalism of local density approximation (LDA). However, based on dynamical mean field theory (DMFT) [24], it has been demonstrated that the ␦-phase and the ␣-phase of plutonium are two sides of the delocalization–localization knife-edge, total energies differing by only a few 10ths of an eV. In this description, the ␣-phase is not a weakly correlated phase; it is just slightly on the delocalized side of the localization–delocalization transition [24]. Here, the ␦-phase is intrinsically stable while the stabilized fcc binary alloys (e.g., PuAl or PuGa) are the consequence of a “destabilization” of the ␣-phase by small amounts of impurities. We tentatively suggest an alternative mechanism for ␦-phase stability, defect-based spin mediation, predicated on our discovery of Kondo-like impurity behavior for vacancies in ␦-phase Pu(Ga).
M.J. Fluss et al. / Journal of Alloys and Compounds 368 (2004) 62–74
EXAFS experiments have shown Ga to be substitutional in Pu(3.3 at.% Ga) with 3.7% contraction of the nearest neighbor Pu atoms towards the Ga [25]. Atomic relaxation to minimize the forces on atoms in Pu32 and Pu31 Ga supercells calculated with the full-potential LMTO method accounts for only 1.04% of this contraction [26]. Recently, Allan et al. have measured elementally specific Debye temperatures in Pu(Ga) [27], and determined that the lattice stiffness for Pu–Ga is approximately two times greater than for Pu–Pu. The nature of the structure surrounding the Ga sites and the dynamical properties of the Pu lattice suggests that the Ga sites (along with the surrounding Pu atoms) are consistent with idea of a localized Kondo impurity. Our estimate for this “free volume” between first and second neighbors is about one-third the volume of a lattice vacancy. Qualitatively this points to the possibility that the structure–property aspects of the physics leading to Kondo-like behavior of vacancies may be related to the Kondo properties observed earlier in reference [12]. A two-electron picture proposed by Cooper et al. [28] states that “5f-electrons on some sites become localized and the strong scattering from these “impurity sites” can disrupt the coherence of the 5f-band states and drive the whole system towards localized 5f-states”. Such a theory will ultimately need to explain the simultaneous localization of the impurity sites and their distribution in time. It would be satisfying, however, if the stabilization of the pure ␦-phase and the stabilized binaries were both to be understood from the basis of similar, if not the same physics. Hence, the consequences of the experimental observations reported here could be profound. It suggests, and so we conjecture, that the implied 5f localization in the neighborhood of vacancies and other dilations in the lattice results in localized magnetic behavior with a low Kondo temperature TK < 10 K and this may be compared to that determined earlier for bulk ␦-phase Pu(Al) of TK =∼200–300 K, the later suggesting an intermediate-valence system. This conjecture raises an intriguing question. Is this the origin of reduced metallic bonding? This may be the structure–property relationship, or impurity-site, that stabilizes the ␦-phase through a global electronic structure change. There are additional observations that point to the role of vacancies and negative pressure as a structure property basis for ␦-phase stabilization or the appearance of magnetic or localized spin properties. It has been reported that inspection of the failure surface of pure ␣-Pu under hydrostatic tension, reveals a so-called “ductile-dimple” failure ahead of the crack tip [29]. This was interpreted as evidence for an ␣- to ␦-phase transformation during the application of high local stress although no remnants of the ␦-phase were observed. Theoretical considerations of the phase diagram appear to justify this interpretation [30]. The addition of hydrogen into Pu, produces a tensile stress that spreads out the Pu atoms, and results in ferromagnetic behavior, suggesting that Pu may be a weak itinerant magnetic system [31] with consequential lattice expansion leading to localization.
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Recent experiments, have discovered unconventional superconductivity in PuCoGa5 [32] with a critical temperature of Tc ∼ 18.5 K. The authors assert that this appears to be spin mediated superconductivity, as opposed to phonon mediated superconductivity. Finally, recent photo-emission studies of thin film Pu have yielded strongly localized electronic structure behavior (high density of states at EF with a narrow spectral width) from a pure delta phase apparently stable at room temperature [33]. This later result may be indicative of the stabilizing effect of a free surface (a layer of vacancies) on the low-density phase of Pu. This suggests that thin-films of ␣-Pu in tension might transform to ␦-phase spontaneously. Given all of the above, we posit the following structure– property mechanism for the stabilization of pure Pu in the ␦-phase. The transition to the ␦-phase is at T = 593 K, while the melting point for Pu is 913 K. At two-thirds the melting point, we estimate an equilibrium vacancy population of at least ∼10−4 . Indeed, given the low value of the EXAFS Debye–Waller factor measured for Pu–Pu bonds in stabilized ␦-PuGa [27], a higher concentration of vacancies (a lower formation enthalpy) might even be expected. Thus, it is reasonable, but remains to be proven, that a small concentration of equilibrium vacancies gives rise to a small concentration of the local spin-sites that stabilize the ␦-phase of pure plutonium. What formalism might support the idea that local spin sites lead to global change in electronic structure? Such a structure–property mechanism has been described recently by Si et al. [34], in their analysis of 2nd order phase transformation of the electronic structure in strongly correlated systems at T = 0 K. The mechanism is quantum criticality, by which it is possible to have a localized spin site propagating in time throughout the lattice and whereby such a critical point might manifest itself at temperatures hundreds of degrees above T = 0 K. Millis et al. have recently proposed a theory for local defects in a metallic system near a quantum critical point [20]. The notion that plutonium may be an example of a new class of a quantum phase transition where, as has been stated, correlations between the magnetic spins at the quantum critical point have an infinite range in time, but a finite range in space [21,34], is indeed interesting. The low-temperature phase diagram of plutonium and its compounds needs extensive experimental investigation and may very well hold the answer to the origins of the many conflicting anomalies exhibited by plutonium, provided we can manipulate or perturb the system so as to approach the implied quantum critical point.
6. Conclusion We report the measured response of electrical resistivity during annealing of low-temperature (10–20 K) accumulated-ion irradiation damage in a stabilized ␦-phase plutonium alloy, Pu(3.3 at.% Ga). Isochronal-annealing curves of the accumulated damage resistivity ( ρ/ ρ0 ) are
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compared for specimens that were either self-irradiated (Pu ␣-decay) or 3.8 MeV proton-irradiated. Modeling of the experimental results using combined molecular dynamics and kinetic Monte Carlo methods describes the defect populations as a function of irradiation type and annealing temperature. The Stages III–V annealing characteristics are similar for the proton and self-irradiation irradiations, but quite different in Stages I and II. We interpret these results to indicate that interstitial clustering is extant for the self-irradiation, but that the corresponding vacancies from the uranium damage cascade appear to be more like isolated point defects. The annealing measurements reveal a Stage III recovery temperature of 180 ± 5 K and a Stage V recovery at 310 ± 5 K. Additionally, and perhaps more importantly, we have discovered a large negative dependence of the resistance of vacancy defects Pu(3.3 at.% Ga). The resistance follows the form −a[ln(T)] + b; apparently a Kondo-impurity behavior mediated by non-equilibrium vacancy-defects in ␦-stabilized Pu. The present work suggests that regional site-specific electron-localization may be induced in Pu(Ga) by non-equilibrium vacancy defects. This compels us to consider that thermally generated equilibrium vacancies may be a potent site in pure Pu that leads to high T delta “stabilization”, resulting in the 5f electrons localizing. Local impurity models, or site-specific localization, appear to be relevant to the present results for non-equilibrium vacancies in ␦-Pu, suggesting a cause and effect for stabilization that requires additional consideration and which may be a consequence of the delicate balance of the electronic phase stability of the system. Although unproven, taken in total the results suggest a ground-state description based on the ideas of quantum criticality [34]. If so, the properties of Pu may be “emergent” and not derivable from simple microscopic theories. Hence, defects and local disorder may play an unusually important role in determining plutonium’s solid-state properties [21]. Acknowledgements We acknowledge the professional work of D. Johnson (data acquisition), M. Lawrence (accelerator operations), and L. Walkley (engineering). We also acknowledge the helpful comments and suggestions of many colleagues on this manuscript before submittal. Work performed under the auspices of the US Department of Energy by Lawrence Livermore National Laboratory under Contract W-7405-Eng-48. References [1] A.C. Damask, G.J. Dienes, Methods of Analysis of Annealing Curves, Gordon and Breach, New York, London, 1963, Chapter 3, p. 145. [2] R.O. Elliot, C.E. Olsen, G.H. Vineyard, Acta Metall. 11 (1963) 1129. [3] D.A. Wigley, Proc. R. Soc. A 284 (1964) 344. [4] M. Dormeval, Electronic structure of Pu–Ce(–Ga) and Pu–Am(–Ga) alloys stabilized in the d-phase, PhD Thesis, Universite de Bourgogne, Dijon, France, 2001.
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