Density-functional calculations for Ce, Th, and Pu metals and alloys
The phase diagrams of Ce, Th, and Pu metals have been studied by means of density-functional theory (DFT). In addition to these metals, the phase stability of Ce-Th and Pu-Am alloys has been also investigated from firstprinciples calculations. Equation-of-state (EOS) for Ce, Th, and the Ce-Th al...
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Інститут фізики конденсованих систем НАН України
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| Цитувати: | Density-functional calculations for Ce, Th, and Pu metals and alloys / A. Landa, P. Soderlind // Condensed Matter Physics. — 2004. — Т. 7, № 2(38). — С. 247–264. — Бібліогр.: 44 назв. — англ. |
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irk-123456789-1189552017-06-02T03:03:03Z Density-functional calculations for Ce, Th, and Pu metals and alloys Landa, A. Soderlind, P. The phase diagrams of Ce, Th, and Pu metals have been studied by means of density-functional theory (DFT). In addition to these metals, the phase stability of Ce-Th and Pu-Am alloys has been also investigated from firstprinciples calculations. Equation-of-state (EOS) for Ce, Th, and the Ce-Th alloys has been calculated up to 1 Mbar pressure in good comparison to experimental data. Present calculations show that the Ce-Th alloys adopt a body-centered-tetragonal (bct) structure upon hydrostatic compression which is in excellent agreement with measurements. The ambient pressure phase diagram of Pu is shown to be very poorly described by traditional DFT but rather well modelled when including magnetic interactions. In particular, the anomalous δ phase of Pu is shown to be stabilized by magnetic disorder at elevated temperatures. The Pu-Am system has also been studied in a similar fashion and it is shown that this system, for about 25% Am content, becomes antiferromagnetic below about 400 K which corroborates the recent discovery of a Curie-Weiss behavior in this system. Фазові діаграми металів Ce, Th та Pu вивчаються за допомогою теорії функціоналу густини (ТФГ). Крім того, фазова стабільність сплавів Ce-Th та Pu-Am досліджена з першопринципних розрахунків. Рівняння стану для Ce, Th та сплаву Ce-Th було розраховано аж до тиску 1 Mбар в доброму узгодженні з експериментальними даними. Дані розрахунки показують, що сплави Ce-Th приймають об’ємоцентровано-тетрагональну структуру внаслідок гідростатичного стиску, що чудово узгоджується з вимірюваннями. Показано, що фазова діаграма Pu при нормальному тиску дуже погано описується традиційною ТФГ, але досить добре моделюється при включенні магнітних взаємодій. Зокрема показано, що аномальна δ -фаза Pu стабілізується магнітним безладом при підвищених температурах. Система Pu-Am також була досліджена подібним чином та показано, що ця система, при приблизно 25% вмісту Am, стає антиферомагнітною нижче приблизно 400 K, що узгоджується з недавнім відкриттям поведінки Кюрі-Вейса в цій системі. 2004 Article Density-functional calculations for Ce, Th, and Pu metals and alloys / A. Landa, P. Soderlind // Condensed Matter Physics. — 2004. — Т. 7, № 2(38). — С. 247–264. — Бібліогр.: 44 назв. — англ. 1607-324X PACS: 64.10.+h, 71.15.Mb, 71.27.+a, 75.10.Lp DOI:10.5488/CMP.7.2.247 http://dspace.nbuv.gov.ua/handle/123456789/118955 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
The phase diagrams of Ce, Th, and Pu metals have been studied by means
of density-functional theory (DFT). In addition to these metals, the phase
stability of Ce-Th and Pu-Am alloys has been also investigated from firstprinciples
calculations. Equation-of-state (EOS) for Ce, Th, and the Ce-Th
alloys has been calculated up to 1 Mbar pressure in good comparison to
experimental data. Present calculations show that the Ce-Th alloys adopt
a body-centered-tetragonal (bct) structure upon hydrostatic compression
which is in excellent agreement with measurements. The ambient pressure
phase diagram of Pu is shown to be very poorly described by traditional
DFT but rather well modelled when including magnetic interactions. In particular,
the anomalous δ phase of Pu is shown to be stabilized by magnetic
disorder at elevated temperatures. The Pu-Am system has also been studied
in a similar fashion and it is shown that this system, for about 25% Am
content, becomes antiferromagnetic below about 400 K which corroborates
the recent discovery of a Curie-Weiss behavior in this system. |
| format |
Article |
| author |
Landa, A. Soderlind, P. |
| spellingShingle |
Landa, A. Soderlind, P. Density-functional calculations for Ce, Th, and Pu metals and alloys Condensed Matter Physics |
| author_facet |
Landa, A. Soderlind, P. |
| author_sort |
Landa, A. |
| title |
Density-functional calculations for Ce, Th, and Pu metals and alloys |
| title_short |
Density-functional calculations for Ce, Th, and Pu metals and alloys |
| title_full |
Density-functional calculations for Ce, Th, and Pu metals and alloys |
| title_fullStr |
Density-functional calculations for Ce, Th, and Pu metals and alloys |
| title_full_unstemmed |
Density-functional calculations for Ce, Th, and Pu metals and alloys |
| title_sort |
density-functional calculations for ce, th, and pu metals and alloys |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2004 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/118955 |
| citation_txt |
Density-functional calculations for Ce, Th, and Pu metals and alloys / A. Landa, P. Soderlind // Condensed Matter Physics. — 2004. — Т. 7, № 2(38). — С. 247–264. — Бібліогр.: 44 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT landaa densityfunctionalcalculationsforcethandpumetalsandalloys AT soderlindp densityfunctionalcalculationsforcethandpumetalsandalloys |
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2025-07-08T14:57:59Z |
| last_indexed |
2025-07-08T14:57:59Z |
| _version_ |
1837091186454560768 |
| fulltext |
Condensed Matter Physics, 2004, Vol. 7, No. 2(38), pp. 247–264
Density-functional calculations for Ce,
Th, and Pu metals and alloys
A.Landa, P.Söderlind
Lawrence Livermore National Laboratory, University of California,
P.O. Box 808, Livermore, CA 94550
Received April 5, 2004
The phase diagrams of Ce, Th, and Pu metals have been studied by means
of density-functional theory (DFT). In addition to these metals, the phase
stability of Ce-Th and Pu-Am alloys has been also investigated from first-
principles calculations. Equation-of-state (EOS) for Ce, Th, and the Ce-Th
alloys has been calculated up to 1 Mbar pressure in good comparison to
experimental data. Present calculations show that the Ce-Th alloys adopt
a body-centered-tetragonal (bct) structure upon hydrostatic compression
which is in excellent agreement with measurements. The ambient pressure
phase diagram of Pu is shown to be very poorly described by traditional
DFT but rather well modelled when including magnetic interactions. In par-
ticular, the anomalous δ phase of Pu is shown to be stabilized by magnetic
disorder at elevated temperatures. The Pu-Am system has also been stud-
ied in a similar fashion and it is shown that this system, for about 25% Am
content, becomes antiferromagnetic below about 400 K which corroborates
the recent discovery of a Curie-Weiss behavior in this system.
Key words: Cerium, Thorium, Plutonium, phase diagram
PACS: 64.10.+h, 71.15.Mb, 71.27.+a, 75.10.Lp
1. Introduction
Actinide physics has seen a remarkable focus the last decade or so due to the com-
bination of new experimental diamond-anvil-cell techniques and the development of
fast computers and more advanced theory. All f -electron systems are expected to
have multi-phase phase diagrams due to the sensitivity of the f -electron band to
external influences such as pressure and temperature. For instance, compression of
an f -electron metal generally causes the occupation of the f states to change due
to a shift of these bands relative to others. This can, in some cases, such as in the
Ce-Th systems, [1,2] cause the crystal to adopt a lower symmetry crystal struc-
ture at elevated pressures. Under compression, the f -electron dominance increases
in these systems and drives the phase transition. The reason for this has been dis-
cussed [3] in terms of a Peierls or Jahn-Teller distortion that favors low symmetry
c© A.Landa, P.Söderlind 247
A.Landa, P.Söderlind
over high symmetry crystal structures. On the other hand, all bands broaden under
compression and the distortion of the lattice becomes less important, while electro-
static forces tend to move atoms to higher symmetry positions, ultimately leading to
closer packed structures with higher symmetry. This interplay between competing
effects and their pressure dependence often results in interesting multi-phase phase
diagrams and this is the case for Ce, Th, Pu, and their corresponding alloys.
The first part of our manuscript is devoted to the study of phase stabilities of
Ce, Th, and Ce-Th system as a function of compression. Cerium metal has a very
interesting phase diagram with two isostructural, face-centered-cubic (fcc) phases,
namely the γ and α phase. The latter is considerably denser than the former and
there is a substantial volume collapse associated with the γ → α transition that
occurs at a moderate pressure close to 10 kbar. The nature of this transition is
currently not fully understood, but it can be described [4] as a Mott transition of
the f electron from a localized (γ-Ce) to an itinerant (α-Ce) state. Below 100 kbar
there is also a phase transition to a lower symmetry phase which is believed to be
either orthorhombic or body-centered monoclinic [5]. Above 120 kbar, however, Ce
is stabilized in a bct structure and remains in this phase up to the highest measured
pressure. In this regard, Th is similar to Ce but has a simpler phase diagram. Only
one phase transition has been seen at low temperatures: fcc → bct at about 600 kbar.
At Mbar pressures, both these metals remain in a bct crystal [6,7] with a c/a axial
ratio close to 1.65. Also, the Ce-Th alloys show a similar behavior [1]. Theoretically,
the Ce-Th alloys are rather well described within a density-functional approach,
[2,8] although a proper, disordered, alloy treatment of the Ce-Th system has not yet
been presented. In fact, it was argued that the disorder in the realistic Ce-Th alloys
could not be well modelled by an ordered compound. The theoretical low pressure
behavior of CecTh1−c was therefore erroneous [2] for c = 0.43. In section 3 we
revisit this problem by applying a more sophisticated theory, based on the coherent
potential approximation (CPA) for alloys.
Even though the Ce-Th system is well described by traditional DFT, Ce metal
at low pressures seems to be affected by electron-correlations that are generally not
included in the DFT. Phase stability of Th and subsequent actinides up to Pu,
however, are accurately predicted by density-functional theory [9,10]. Plutonium
precedes Am, which has an atomic density, crystal, and electronic structure much
different from that of Pu. These differences are attributed to the localization of
5f electrons that occurs between Pu and Am. Many facets of Pu actually suggest
that this localization has already begun in Pu. For instance, some phases in the
Pu phase diagram share properties with Am. δ-Pu, which is stable at 593 K, is
fcc with an atomic volume 25% greater than the monoclinic ground-state α phase.
The fcc crystal structure is close-packed and quite similar to the double-hexagonal
close-packed (dhcp) of Am. Also, the δ-Pu atomic density is much closer to Am
than that of the ground-state α phase. Hence, there are dramatic changes with
temperature in the Pu electronic structure and these cannot be understood from the
traditional DFT that is generally used for the light actinides [9]. This problem was
early recognized and during the last few years new models and various corrections
248
Density-functional calculations for Ce, Th, and Pu
to the DFT have been proposed to mainly deal with δ-Pu. A description of these
attempts was recently given [11]. Perhaps the most natural procedure is to consider
magnetic interactions that have often been ignored in the past for plutonium because
of the belief that these are of negligible importance. Many researchers have now,
however, concluded that magnetic effects are necessary to include in any model for
δ-Pu and δ-stabilized alloys of Pu. In section 4 we will review some recent results
for Pu including energetics of the ambient pressure phase diagram and the stability
of δ-Pu. The theory is expanded in section 5 to also include results for the δ-Pu-Am
system.
The paper is organized as follows. Our computational approach is discussed in
section 2, followed by results from the Ce-Th (section 3), Pu (section 4), and Pu-Am
(section 5) calculations. We present our conclusions in the last section 6.
2. Computational details
The calculations we have referred to as exact muffin-tin orbitals (EMTO) are
performed using scalar-relativistic, spin-polarized Green’s function technique based
on an improved screened Korringa-Kohn-Rostoker (KKR) method, where the one-
electron potential is represented by optimized overlapping muffin-tin (OOMT) po-
tential spheres [12–15]. Inside the potential spheres the potential is spherically sym-
metric and it is constant between the spheres. The radii of the potential spheres, the
spherical potentials inside the spheres, and the constant value from the interstitial
are determined by minimizing (a) the deviation between the exact and overlapping
potentials and (b) the errors coming from the overlap between spheres. Thus, the
OOMT potential ensures a more accurate description of the full potential compared
to the conventional muffin-tin or non-overlapping approach.
Within the EMTO formalism, the one-electron states are calculated exactly for
the OOMT potentials. As an output of the EMTO calculations, one can determine
the self-consistent Green’s function of the system and the complete, non-spherically
symmetric charge density. Finally, the total energy is calculated using the full charge
density technique, [15,16].
For the total energy of random substitutional alloys, the EMTO method has been
recently combined with the CPA [17] that also allows for the treatment of magnetic
disorder [18,19]. In the present work, as well as in our previous papers, [20–23] a
paramagnetic (PM) δ-Pu was modelled within the disordered local moment (DLM)
approximation [24]. In order to calculate this state, one uses a random mixture of
two distinct magnetic states, namely, the spin up and spin down configurations of
the same atomic species in the system.
The calculations are performed for a basis set including valence spdf orbitals and
the semi-core 6p state whereas the core states were recalculated at each iteration.
Integration over the irreducible wedge of the fcc Brillouin zone (BZ) is performed
using the special k-point method [25]. The Green’s function has been calculated for
40 complex energy points distributed exponentially on a semicircle with a 1.9 Ry
diameter enclosing the occupied states.
249
A.Landa, P.Söderlind
Figure 1. Equation of state for Ce. Experimental results [6] are marked with
open squares, whereas theory is given by a solid line.
Figure 2. Equation of state for Th. Experimental results [28] are marked with
open squares, whereas theory is given by a solid line.
250
Density-functional calculations for Ce, Th, and Pu
Figure 3. The c/a axial ratio for the bct structure as a function of pressure for
Ce. Experimental data [6] are marked with open squares while theoretical results
are given by a solid line and filled circles.
3. Ce, Th, and the Ce-Th alloys
Before entering the details of the crystal structure of Ce and Th under com-
pression, we compare our calculated EOS with the experimental data for these two
metals. In figure 1 we show the theoretical and measured equations of state for Ce
metal. The agreement between theory and experiment [6] is good. Calculated equi-
librium volume and bulk modulus are V0 = 27.7 Å3 and B0 = 380 kbar compares
well with room temperature data [26,27] (V0 = 28.0 Å3 and B0 = 290 kbar). Also
for Th, the EOS is in good agreement with experiment, [28] see figure 2. The Th
equilibrium volume, 33.3 Å3, and bulk modulus, 580 kbar, are very close to measured
data, [28] 32.9 Å3 and 580 kbar, respectively.
Next we study the crystal-structure behavior for Ce, and in figure 3 we plot the
calculated c/a axial ratio for bct Ce together with experimental data [6]. At pressures
beyond about 120 kbar, cerium adopts a bct structure with a c/a ratio close to 1.65.
Quantitatively, this behavior is reproduced by our calculations. We note that there
is a rapid increase of the c/a axial ratio in the calculations were there is known
to be an intermediate phase [6] in Ce (below 100 kbar). The intermediate, lower
symmetry phase, is limited to a small pressure range and is not considered in the
present calculations, but has been investigated theoretically before [9]. A similar
behavior is found for thorium metal, see figure 4. Experimentally, Th is stable in its
ambient pressure phase (fcc) up to 630 kbar. At higher compression Th transforms
continuously into the bct phase. The transition pressure (≈ 630 kbar) is considerably
higher in Th than in Ce. The fact that the fcc → bct transition occurs at a higher
pressure in Th than in Ce is a consequence of the somewhat lower f -band population
in Th metal at low pressure [2].
251
A.Landa, P.Söderlind
Figure 4. The c/a axial ratio for the bct structure as a function of pressure for
Th. Experimental data [7] are marked with open squares while theoretical results
are given by a solid line and filled circles.
As we established that elemental Ce and Th can be very well described by
our calculations we next consider the Ce43Th57 disordered fcc alloy. In figure 5
we show theoretical EOS for the Ce43Th57 disordered fcc alloy. Notice that the
Ce43Th57 alloy curve is located between the curves for pure Th and Ce, which are also
depicted in this figure. Calculated equilibrium atomic volume and bulk modulus of
the Ce43Th57 alloy are 31.4 Å3 and 460 kbar, respectively, which is in fair agreement
with experimental data of 32.9 Å3 and 281 kbar, [1].
Figure 5. Equation of state for the Ce43Th57 disordered alloy. EOS for Ce and
Th are also shown.
252
Density-functional calculations for Ce, Th, and Pu
Figure 6. The c/a axial ratio for the bct structure as a function of pressure for the
Ce43Th57 disordered alloy. Experimental data [1] are marked with open squares
while theoretical results are given by a solid line and filled circles.
Finally, figure 6 shows the calculated and measured c/a ratio as a function of
pressure for Ce43Th57. The structural behavior of this alloy was previously modelled
by an ordered CeTh compound [2]. These calculations predicted unrealistic low-
pressure structures for this system, where the axial c/a ratio first decreased with
pressure and suddenly jumped to a high value closer to the measure value at a higher
compression. It was speculated [2] that the discrepancy with experiment was due
to the failure of modelling the disordered alloy with an ordered compound. Here
we can address this question explicitly because the EMTO-CPA formalism allows
us to treat the alloy more realistically. The EMTO-CPA calculations confirm that
the fcc → bct phase transition begins between 100–200 kbar, which is close to the
corresponding transition in Ce metal (120 kbar), but considerably lower than for Th
(630 kbar). Our calculations thus reproduce the experimental observation [1] that
the fcc → bct transition pressure is a strongly nonlinear function of Th concentration
in the Ce-Th alloy system.
4. Pu metal
Pu metal and related alloys and compounds are studied intensively experimen-
tally and theoretically, partly because of their richness of fascinating and counter
intuitive properties [29]. Theoretically, the first reliable total energies for Pu were
calculated not so long ago [30] and when comparing a number of competing phases,
the monoclinic α phase proved to be the ground state in agreement with experiment.
The α-Pu atomic volume was somewhat small and the bulk modulus too large, but
a more serious deficiency of the calculations [30] seemed to be the poor description
of some other phases, such as the δ phase. In figure 7 we show total energies [31] of
253
A.Landa, P.Söderlind
Figure 7. Total energies of α, β, γ, and δ plutonium, obtained from nonmagnetic
calculations [30].
α, β, γ, and δ plutonium. Notice that β, γ, and especially δ plutonium have total
energies so high that their existence in the Pu phase diagram cannot be explained.
Moreover, the predicted atomic densities for these phases are close to that of α-Pu
but in severe disagreement with the measured atomic volumes shown as vertical
lines in figure 7. The δ phase is not even mechanically stable because the calculated
elastic constant C′ is strongly negative.
Figure 8. Total energies as a function of atomic volume for nonmagnetic (NM),
ferromagnetic (FM), antiferromagnetic (AF), and disordered magnetic (D) con-
figurations of δ-Pu [20].
The remarkable failure of the customary DFT treatment for some of the Pu
phases, and most notably the δ phase, was early recognized but theoretical efforts
to improve the model has only recently been suggested [31–34]. A more detailed
254
Density-functional calculations for Ce, Th, and Pu
Figure 9. Total energy as a function of c/a axial ratio for antiferromagnetic (AF)
and disordered magnetic (D) configurations of Pu [20].
discussion of these efforts has been presented elsewhere [11]. Because we are mainly
interested in structural stability, which requires accurate and consistent total-energy
calculations, most of the proposed models for Pu cannot be applied. Inclusion of mag-
netic interactions into the DFT, however, seems to provide a theoretical framework
in which reliable total energies can be obtained [31].
Figure 8 shows total energies as a function of atomic volume for nonmagnetic
(NM), ferromagnetic (FM), antiferromagnetic (AF), and disordered magnetic (D)
configurations of δ-Pu [20]. Clearly, the total energies are lowered when the magnetic
spin and orbital contributions are included. A direct comparison to the α phase is
done below. Notice also in figure 8 that the atomic volumes for the FM, AF, and
D calculations are all in good agreement with the experimental value (vertical line)
while the NM calculation predicts much too small atomic volume. This plot sug-
gests that AF is the zero temperature ground-state configuration for δ-Pu, whereas
in reality this phase is stable above 593 K. The relatively small energy difference
between the AF and D configurations is therefore not sufficient to rule out either
one as a good model for the true δ phase. Next, we compare the mechanical stabil-
ity between the AF and D δ-Pu. In figure 9 we show the calculated so-called Bain
transformation path for these configurations. The energy curvature at c/a =
√
2
corresponds to the tetragonal shear constant C′. Clearly, AF spin ordering provides
an unstable situation (C′ negative) while the magnetic disorder does not. Hence, it
is tempting to suggest that spin entropy may favor the disordered state over the
antiferromagnetic state at sufficiently high temperatures. If this is the case, the γ →
255
A.Landa, P.Söderlind
δ phase transition could be understood in terms of an order → disorder transition:
δ-Pu is mechanically stable in the disordered state but upon cooling a magnetic or-
dering destabilizes δ-Pu mechanically, thus driving the structural phase transition.
Figure 10. Total energy E and its temperature derivative dE/dT as a function
of temperature in the MC simulations of δ-Pu [21].
Figure 11. Comparison of spectra [35] (dashed line) to calculated DOS [20] (solid
curve). The theoretical curve has been convoluted with life time broadening.
In order to quantify these ideas, Monte Carlo (MC) simulations were performed
to study the transition from a disordered → ordered magnetic states of δ-Pu [21].
The effective Ising cluster interactions were obtained from first-principles calcula-
tions incorporated within the structure-inverse and general perturbation methods.
Figure 10 shows the total energy per atom and its temperature derivatives in these
256
Density-functional calculations for Ce, Th, and Pu
MC simulations. The calculated transition temperature (≈ 548 K) is in good agree-
ment with the temperature measured at the γ → δ transition of plutonium (593 K).
Once we established that a paramagnetic (disordered moments) approach pro-
vides a viable model for δ-Pu, direct comparisons with experiments become very
important. First, the electronic structure has been studied [35] by means of photoe-
mission (PE). PE is a probe of the occupied part of the electron density of states
(DOS) and can be directly compared to the calculated electronic structure when
lifetime broadening and instrumental resolution are accounted for. In figure 11 we
show calculated DOS [20] and photoemission [35] for δ-Pu. Notice that the sharp
peak just below the Fermi energy, at zero energy, is very well reproduced in the
calculations. Also the shoulder, sometimes referred to as a “Kondo resonance”, at
about −0.3 eV is evident in the calculations although somewhat exaggerated. Over-
all, the comparison between theory and experiment is very impressive. It should
be noted that when magnetic interactions are ignored, PE and DOS are in great
disagreement with each other, [35].
Table 1. EMTO equilibrium volume (Å3), bulk and elastic moduli (kbar), and
zone-boundary phonons (THz) for δ-Pu.
Method V B C′ C44 XL XT LL LT
EMTO 25.5 380 81 810 7.5 2.7 2.9 1.3
Experiment 25.01,2 3001, 2902 481, 492 3401, 3102 3.12 1.72 3.12 0.482
Recently the elastic constants and phonon dispersions for δ-stabilized plutonium
were measured by x-rays [36]. The zone-boundary phonons (ZBP) are relatively
straightforward to calculate from the total energy response of a movement of an
atom corresponding to the phonon mode [37]. Results of EMTO calculations for ZBP
and elastic constants are presented in table 1 together with available experimental
data. It is clear from the table that fundamental bonding properties are captured in
the calculations. Theoretical equilibrium volume as well bulk modulus compare very
favorably with experiments. Also the anomalously large anisotropy ratio (C44/C′)
is reproduced by theory. The magnitude of the calculated elastic constants and
the ZBP are consistently larger than the measured data, however. We attribute
this systematic discrepancy, at least partly, due to the fact that our calculations
reflect these properties at zero temperature while the measurements are performed at
room temperature. It is known that the elastic constants have a strong temperature
dependence and stiffen considerably upon cooling [38] and it seems plausible that
this is true also for the phonons. Another possible source of the discrepancy might
be that the measured sample is δ-stabilized alloy whereas the theory deals with pure
δ-Pu.
Our theory so far suggests that magnetic interactions are important in Pu and
that δ-Pu is stabilized at 550–600 K due to magnetic disorder. Several properties of
1Moment R.L. [42]
2Wong J.et al. [36]
257
A.Landa, P.Söderlind
Figure 12. Calculated total energies for the six known polymorfs of plutonium as
a function of atomic volume [39]. Note that the y axis is boken to better display
the ε-Pu energies.
δ-Pu, some being quite anomalous, are indeed very well reproduced by this assump-
tion. Higher temperature phases have presumably also disordered local magnetic
moments, whereas for the lower temperature phases (α, β, and γ) the magnetic
configurations are ordered. In fact, an analysis of magnetic ordering in low tem-
perature Pu suggests antiferromagnetic order to be the prevalent configuration [39]
in agreement with first-principles calculations. In figure 12 we plot calculated total
energies for α, β, and γ plutonium (antiferromagnetic) together with δ, δ′, and ε
plutonium (disordered). The disordered magnetic configuration was modeled in a 8
atom super cell by the so-called special quasi-random structure [20]. Notice that the
total energies are in order of α, β, γ, δ, δ′, and ε. This is the actual order at which
they do occur in the ambient pressure phase diagram [40]. The numerical order of
the atomic volumes is slightly different: α, β, γ, ε, δ′, and δ. This is again in exact
agreement with the known phase diagram and that is quite remarkable when com-
paring to the failures of the theory when magnetic interactions are not accounted
for (see figure 7). From the total energies one can obtain the bulk modulus as well,
and for α-Pu it is calculated to be 500 kbar. This compares very favorably with the
experimentally suggested data ranging from 400 to 660 kbar [41]. Bulk moduli for
the β, γ, δ′, and ε phases are not known, but the present theory predicts that they
are all within 230–590 kbar. Notably, it is 410 kbar for δ-Pu that compares well with
the measured 300–350 kbar, [42].
5. Pu-Am system
Following the idea presented in [22], we conclude that an increase of the larger
element (Am) content in the Pu-Am system stabilizes the disordered magnetic state
at lower temperatures. To quantify this hypothesis we undertook MC simulations
258
Density-functional calculations for Ce, Th, and Pu
Figure 13. The DLM → AF transition temperature for δ−Pu100−cAmc alloys
[23].
similar to those presented earlier [21]. Results of the MC calculations of the DLM
→ AF transition temperature in the Pu-Am system are shown in figure 13. By
introducing Am into the Pu-Am system one could drop the magnetic DLM → AF
transition temperature from ≈ 548 K (Pu) to ≈ 372 K (Pu70Am30). From the Pu-
Am phase diagram, [43] however, it is known that already ≈ 6 at. % of Am could
stabilize δ-Pu phase to room temperature.
As one can see from figure 13, the δ-Pu75Am25 alloy will have an AF order at and
below ≈ 400 K, whereas above this temperature disordered magnetism is expected.
This magnetic transition was predicted to occur also for pure δ-Pu but at a consid-
erably higher temperature (548 K). In the case of δ-Pu the magnetic DLM → AF
transition was suggested [20–22] to drive the δ → γ transition due to a structural
instability of the AF phase. For the Pu-Am alloy, however, no such structural phase
transition has been found and this suggests that the AF configuration remains me-
chanically stable. Theoretically, this hypothesis can be corroborated by calculating
elastic constants or relevant deformation energies for the AF Pu-Am alloy.
In figure 14 we show relative energies for AF Pu3Am and AF δ-Pu as a func-
tion of c/a axial ratio. For c/a =
√
2 the fcc symmetry is recovered. Notice that
for δ-Pu the AF configuration is strongly unstable with respect to the tetragonal
distortion, whereas the Pu3Am system remains mechanically stable, with a mini-
mum in the total energy for c/a =
√
2. Hence, there is a fundamental difference
between δ-Pu and Pu3Am in that both undergo a magnetic DLM → AF transition
that destabilizes δ-Pu but not the Pu3Am. This is important because our theory
thus predicts the possibility for an AF order in a Pu-Am alloy system. In order to
verify this experimentally, one should perform magnetic susceptibility measurements
that can detect a Curie-Weiss behavior when AF order is present. Dormeval [44] has
performed such experiments that in fact confirm our theoretical picture. Her inves-
259
A.Landa, P.Söderlind
Figure 14. Relative energy as a function of c/a axial ratio for Pu and Pu3Am
[23]. For c/a = 1.414 the fcc crystal structure is recovered.
tigation shows that at about 24–26 at. % Am in δ-Pu, an unambiguous Curie-Weiss
behavior develops. In δ-Pu, where AF order is not expected theoretically, magnetic
susceptibility is known to be nearly temperature independent.
The question we deal with in the remaining part of this section is why alloying
Pu with Am stabilizes the antiferromagnetic state in this alloy. Both Dormeval’s
measurements [44] and theoretical picture presented here support the idea that δ-Pu
stabilized by ≈ 25 at. % of Am is antiferromagnetic at low temperatures. Söderlind
et al., [20] however, rule out this magnetic order for pure δ-Pu due to its mechanical
instability. It is well known that Pu and other actinides with itinerant 5f states
tend to crystallize in low symmetry and open structures and that the reason for this
is due to high density of 5f states at the Fermi level (EF) that efficiently rules out
high symmetry structures [3]. It is tempting to also associate the destabilization of
antiferromagnetic δ-Pu at low temperatures to a similar phenomenon. We therefore
show, in figure 15, the calculated DOS for antiferromagnetic Pu and Pu3Am. The
plot focuses on the DOS behavior in the vicinity of the EF located at zero energy and
marked with a dashed vertical line. Notice that for pure Pu, there is a strong peak
intersecting the EF with its maximum just below. This is an inherently unfavorable
situation due to the large contribution of this peak to the band energy [3]. For Pu3Am
(bold line), however, this peak is shifted mostly below EF, which is now located close
to a minimum in the DOS. The DOS at the Fermi level is correspondingly much
lower (≈ 25) in Pu3Am compared to Pu (≈ 45). This shift of the EF in Pu3Am
relative to pure Pu is a consequence of the additional 5f electrons provided by the
americium in this compound. We speculate that this more stable situation in Pu3Am
260
Density-functional calculations for Ce, Th, and Pu
Figure 15. Lifetime broadened [35] electronic density of states for Pu and
Pu3Am [23].
is responsible for the mechanical stability in this system.
Our electronic structure calculations predict that the DOS has a pronounced
peak at the EF for Pu that is largely diminished when alloyed with about 25 at. %
americium. Photoemission is capable of detecting such a difference and for that
purpose we also show, in figure 16, calculated DOS where lifetime broadening has
been taken into account, as described by Arko et al [35]. We know from figure 11
(section 4) that for δ-Pu the DOS compares rather well with photoemission [35]
results. Figure 16 suggests that for the Pu-Am system, the peak is shifted to lower
binding energies and that the shoulder at about −0.5 eV in Pu is fully suppressed.
The photoelectron spectroscopy measurements by Dormeval [44] show a quantitative
agreement with this finding.
6. Conclusions
We have presented accurate electronic-structure calculations for several f elec-
tron systems including Ce, Th, Pu, and Am. Generally the theory reproduces exper-
imental data very well. The structural pressure dependence of the Ce-Th system is
well understood and driven by the increased presence of f electrons under pressure.
Ce, Th, and the Ce-Th alloys behave rather similar, although Ce has intermediate
phases in the phase diagram at about 100 kbar that do not exist in the Th and Ce-
Th alloys. Consequently, the fcc → bct transition is of first order in Ce but not in
Th and the Ce-Th alloys. For the near fifty-fifty concentration of Ce-Th disordered
alloy, a CPA treatment is necessary to reproduce the correct structural behavior.
261
A.Landa, P.Söderlind
Figure 16. Total electronic density of states for Pu and Pu3Am [23]. The energy
scale is shifted so that the Fermi level is positioned at zero energy.
Many properties of Pu have not been reproduced in the past by theory and
therefore been illusive to material scientists. Here we present a magnetic model that
describes Pu remarkably well. The very complex phase diagram of Pu is shown to be
a result of strong magnetic interactions in this metal. In particular, δ-Pu is believed
to be a disordered magnet. When alloyed with Am, our theory predicts that δ-Pu is
an ordered antiferromagnet above about 25% Am concentration below 400 K. This
is entirely consistent with recent susceptibility measurements of this system.
7. Acknowledgments
We would like to thank Dr.A.Ruban and Dr.L.Vitos for helpful discussions. This
work was performed under the auspices of the U.S. Department of Energy by the
University of California Lawrence Livermore National Laboratory under contract
W–7405–Eng–48.
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Розрахунки методом функціоналу густини для
металів Ce, Th та Pu та сплавів
А.Ланда, П.Сюдерлінд
Лоуренс Лівермор Національна Лабораторія, Університет
Каліфорнії, п.о. 808, Лівермор, СА 94550
Отримано 5 квітня 2004
Фазові діаграми металів Ce, Th та Pu вивчаються за допомогою тео-
рії функціоналу густини (ТФГ). Крім того,фазова стабільність сплавів
Ce-Th та Pu-Am досліджена з першопринципних розрахунків. Рівнян-
ня стану для Ce, Th та сплаву Ce-Th було розраховано аж до тис-
ку 1 Mбар в доброму узгодженні з експериментальними даними.
Дані розрахунки показують, що сплави Ce-Th приймають об’ємо-
центровано-тетрагональну структуру внаслідок гідростатичного стис-
ку, що чудово узгоджується з вимірюваннями. Показано, що фазова
діаграма Pu при нормальному тиску дуже погано описується тради-
ційною ТФГ, але досить добре моделюється при включенні магнітних
взаємодій. Зокрема показано, що аномальна δ -фаза Pu стабілізує-
ться магнітним безладом при підвищених температурах. Система
Pu-Am також була досліджена подібним чином та показано, що ця
система, при приблизно 25% вмісту Am, стає антиферомагнітною
нижче приблизно 400 K, що узгоджується з недавнім відкриттям по-
ведінки Кюрі-Вейса в цій системі.
Ключові слова: церій, торій, плутоній, фазова діаграма
PACS: 64.10.+h, 71.15.Mb, 71.27.+a, 75.10.Lp
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