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Research
Cite this article: Das NK, Rigby K, de Leeuw
NH. 2014 The effect of helium nano-bubbles on
the structures stability and electronic
properties of palladium tritides: a density
functional theory study. Proc. R. Soc. A 470:
20140357.
http://dx.doi.org/10.1098/rspa.2014.0357
Received: 2 May 2014
Accepted: 19 August 2014
Subject Areas:
computational chemistry
Keywords:
density functional theory simulations,
Pd defects, helium cluster, diffusion, migration
Author for correspondence:
N. K. Das
e-mail: [email protected]
The effect of helium
nano-bubbles on the
structures stability and
electronic properties of
palladium tritides: a density
functional theory study
N. K. Das, K. Rigby and N. H. de Leeuw
Department of Chemistry, University College London, 20 Gordon St.,
London WC1H 0AJ, UK
Density functional theory calculations have been
used to study the incorporation of helium in perfect
and defect-containing palladium tritides, where we
have calculated the energetics of incorporation and
the migration behaviour. Helium atoms preferably
occupy the octahedral interstitial and substitutional
sites in the perfect and Pd vacancy-containing
tritides, respectively. The energetics reveal that helium
clusters can form in the lattice, which displace the
Pd metal atoms. The defective lattice shows less
expansion compared with the perfect lattice, which
can accommodate the helium less easily. The path
from octahedral–tetrahedral–octahedral sites is the
lowest energy pathway for helium diffusion, and
the energetics indicate that the helium generated
from tritium decay can accumulate in or near the
octahedral sites. Density of states analyses shows
the hybridization between palladium d and tritium
s orbitals and repulsion between palladium d and
helium s orbitals, which can distort the lattice as a
result of generating localized stress.
1. Introduction
One contribution to a Special feature
‘New developments in the chemistry and
physics of defects in solids’.
Tritium is important in the nuclear industry and its
storage has been suggested in the form of metal
tritides [1]. Tritium is radioactive, has a half-life of about
12.3 years and decays to 3 He. During tritium decay,
helium atoms agglomerate in metals owing to their
extremely low solubility [2]. The tritium decay products
do not immediately diffuse out of the lattice, but they can
2014 The Author(s) Published by the Royal Society. All rights reserved.
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DFT calculations were performed using the Vienna ab initio simulation package [20,21].
The generalized gradient approximation (GGA) and Perdew–Burke–Ernzerhof (PBE) pseudopotentials [22] were used to describe the electronic structure of palladium, hydrogen and helium
atoms, with the palladium 4d and 4s electrons treated as valence electrons. The electronic
structures of hydrogen and tritium are almost the same, and therefore it is reasonable to use
hydrogen to simulate the electronic behaviour of tritium. Standard hydrogen and helium PBE
potentials were used, with the hydrogen and helium 1s electrons treated as valence electrons and
the hydrogen atom was manually modified to have an atomic mass of 3 instead of 1. The number
of Monkhorst–Pack k-points was determined by performing individual DFT calculations on a PdT
system with an increasing number of k-points. An appropriate number of k-points was deemed
to have been reached when a total energy convergence of 1 meV per atom was observed, and
as result a 6 × 6 × 6 k-point grid was used for 2 × 2 × 2 supercell calculations. The calculations
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2. Computational details
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be strongly trapped by interstitial sites and defects containing a free volume, particularly
vacancies and grain boundaries, where they can subsequently form helium clusters or nanobubbles [3–5]. The presence of helium nano-bubbles plays an important role in the microstructural
evolution of storage materials, and the bubbles can significantly degrade the mechanical
properties of materials.
The behaviour of helium in metal and metal tritides has been investigated using experimental
approaches and theoretical methods. For example, a nuclear magnetic resonance study has
suggested 3 He bubble formation in LiT, TiT1.9 and UT3 and has further indicated an increase
in the bubble sizes as the metal tritides age [6]. Weaver et al. [7] have shown that 3 He atoms were
initially trapped interstitially in titanium tritide, whereas older samples were found to contain
helium gas bubbles. The stability of metal tritides decreases with the increased generation of
helium and in titanium tritide, helium was shown to migrate between two neighbour interstitial
sites [8]. The effects of helium bubbles on metal tritides are clearly of considerable interest and it
is important to gain understanding of the atomic-level interactions of helium atoms and clusters
with metal tritides.
Palladium is an outstanding tritium storage material, which is of particular interest owing
to its high resistance to oxidation and poisoning and its ability to easily absorb, as well as
release, tritium and retain the helium generated in the matrix [9]. Several experimental studies
have been carried out on the helium behaviour in palladium tritide [10–12]. A transmission
electron microscopy study found bubble formation in an aged Pd-T. However, the study revealed
that grain boundaries and other extended defects do not appear to play a major role in bubble
formation [10]. Other studies have focused on helium release from aged palladium tritide,
with X-ray diffraction analysis results showing that palladium tritides retain more than 95% of
helium at room temperature for at least 9 years of ageing [13]. Additionally, density functional
theory (DFT) and molecular dynamics methods have been used to investigate the effect of
helium and helium clusters in Pd, Ti, W, Fe and Mo [14–19], which provide an atomistic
understanding of helium in these metals. Zeng et al. [14] have studied the properties of small
helium-vacancy clusters in palladium and they reported that vacancies in Pd are the preferred
sites to capture helium atoms. The formation energies of helium vacancies in metals have also
been calculated using molecular dynamics and semi-empirical methods [15], but, to date, no
computational studies have been carried out on helium in tritides. A suitable tritium storage
material would need to have a large capacity for helium and the ability to expand and withstand
mechanical stress, while still retaining its overall macrostructure. As the interaction between
helium clusters and palladium tritides plays an important role in the material’s integrity, in this
study, we have employed DFT methods to investigate the structural and electronic properties
of helium clusters in perfect and defect-containing palladium tritide lattices and calculate the
helium migration energies.
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3
Figure 1. The PdT0.22 structure. The dark blue and white spheres indicate the palladium and tritium atoms, respectively. (Online
version in colour.)
were performed using a high planewave cut-off energy of 660 eV. Total energy convergence was
achieved for each calculation when all forces on the system were lower than 0.001 eV per Å and
the atomic positions, shape and volume of the supercells were optimized. The tritium atoms were
placed in interstitial sites, and the most stable structure was selected for further calculations.
It has been observed experimentally that 60% of the octahedral interstitial sites in palladium
metal can accommodate tritium atoms and still retain the metal’s fcc crystal structure [10].
However, here, we have considered a relatively low concentration of tritium to retain more vacant
sites for helium incorporation. Seven tritium atoms were placed in octahedral interstitial sites of
the 2 × 2 × 2 palladium supercell giving a composition of PdT0.22 , as shown in figure 1. Although
this is not a realistic concentration at room temperature, where the material would phase separate
into low concentration α and high concentration β domains, the simulation cell is convenient to
study explicitly all interactions that we are interested in, not only between helium and the tritium
atoms, but also between tritium atoms themselves, and it provides a number of sites for helium
incorporation and diffusion. The unrealistic tritium concentration overall is, in reality, an artefact
of the periodic boundary conditions, which repeats this very local concentration throughout the
lattice, although the cell is big enough that there are no unrealistic interactions between the mirror
images. For the defect-containing lattice, a Pd vacancy was created through the removal of a Pd
atom in the perfect lattice. For the calculation of the interaction between helium clusters and Pd
vacancies, helium atoms were added to the interstitial and vacant sites one by one.
The formation energy of a helium cluster composed of n helium atoms in an interstitial site
and in a Pd vacancy are given, respectively, by
Ef (Hen ) = Etot (PdT0.22 Hen ) − nEHe − Etot (PdT0.22 )
(2.1)
Ef (Hen V) = Etot (PdT0.22 Hen V) − nEHe − E(PdT0.22 V),
(2.2)
and
where Etot (PdT0.22 Hen ) is the total energy of a system containing a Hen cluster, Etot (PdT0.22 Hen V)
is the total energy of a system containing a Hen and Pd vacancy, Etot (PdT0.22 ) is the energy of the
perfect tritide, Etot (PdT0.22 V) is the energy of a Pd vacancy-containing system without helium
and EHe is the energy of an isolated helium atom.
The binding energy of a helium atom to Hen clusters is calculated for the perfect and defective
PdT0.22 lattices as follows
Eb (He) = Ef (He) + Ef (Hen−1 ) − Ef (Hen )
(2.3)
Eb (He) = Ef (He) + Ef (Hen−1 V) − Ef (Hen V),
(2.4)
and
which means that a positive binding energy is an energetically preferred process.
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(a) Bulk properties
The calculated lattice constant, cohesive energy and bulk modulus of the Pd bulk are 3.96 Å,
3.76 eV and 190 GPa, respectively, in good agreement with previously obtained results [14] and
experimental values of 3.89 Å, 3.89 eV and 180 GPa [24], although the GGA–PBE functional used
leads to a small overestimation of the lattice constant and a consequent underestimation of the
cohesive energy.
The Pd bulk lattice was affected by the insertion of helium and tritium atoms into the
interstitial sites. Insertion of a single tritium atom in the octahedral site resulted in a small
(less than 0.5%) expansion of the palladium lattice volume, whereas a helium atom expanded
the structure by about 1.5%. When both the tritium and helium were present in two different
octahedral sites, the palladium lattice expanded by 2.5%. Increasing the tritium concentration in
the interstitial sites naturally increased the size of the palladium lattice. For example, the PdT0.22
structure showed a 3.5% lattice volume expansion compared with Pd metal, which is consistent
with a previous study [25]. The palladium tritide lattice was further expanded by 5.7% compared
with the pure Pd metal owing to insertion of one helium atom in the PdT0.22 lattice.
(b) Tritium distribution in Pd lattice
When hydrogen is occluded in palladium, initially, the α phase is formed, which converts to
the β phase at high hydrogen concentrations. Both are fcc structures, but a volume expansion
occurs on going from the α to the β phase [26]. We have first studied the distribution of tritium in
the bulk palladium structure, where a 2 × 2 × 2 supercell of PdT0.6875 was modelled, containing
32 palladium atoms and 22 tritium atoms. PdT0.6875 was chosen for this part of our study as it
has a similar molecular formula to the experimental β-phase of palladium tritide (PdT0.7 ). The
PdT0.6875 composition gives rise to 53 unique configurations of the Pd and T in the 2 × 2 × 2
cell, where we have calculated the total energy for each configuration. It was found that each
configuration had a similar total energy, where the largest energy difference between the highest
and lowest energy configurations was only 0.05 eV. As such, it can be assumed that the probability
of each configuration existing experimentally is relatively equal. This also supports the theory
that tritium atoms diffuse quickly through palladium and do not remain in the same octahedral
interstitial site for an extended time. However, it was found that the configurations with the
lowest energies had octahedral interstitial vacancies evenly distributed throughout the material,
whereas, in the highest energy configurations, the octahedral interstitial vacancies were clustered
together. This suggests that configurations where vacancies are spread across the material are
most stable.
A converse system can also be considered where there are few tritium atoms and many
octahedral interstitial vacancies, such as in PdT0.22 . Although for tritium concentrations up to
x = 0.6, there is a phase miscibility gap at equilibrium between the α- and β-phases, it has been
suggested that at low temperatures (less than 50 K) an ordered α phase can exist [27], whereas
the β-phase can be maintained in hydride samples through a continuous concentration gradient
from 0.03 to 0.60 [26], whereas materials with a tritium concentration of x = 0.22 might also be
expected to occur in small regions at interfaces between α- and β-phases within the miscibility
gap. As already outlined in §2, we have chosen to study the PdT0.22 composition as a model for
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3. Results and discussion
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The dimer method was used to investigate the migration mechanism and the energy barriers
of helium diffusion in the lattice, which calculations were performed in a supercell with and
without a fixed volume condition for comparison [23]. The dimer method was developed to
identify saddle points in the potential energy surface within a solid, without knowledge of the
final state of transition, and without the use of second derivatives of the potential. The final
converged transition state in each case was verified to have only one imaginary frequency.
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(a)
(b)
11
3.0
2.8
Ef /n (eV)
7
6
2.6
2.4
2.2
5
2.0
4
1.8
3
1.6
1
2
3
4
no. He atoms (n)
5
6
1
2
3
4
no. He atoms (n)
5
6
Figure 2. Formation energy of small helium clusters versus number of helium atoms in PdT0.22 (a) the total formation energy
and (b) average formation energy per helium atom. (Online version in colour.)
the system with a relatively low concentration of tritium in the lattice, which allows for explicit
consideration of T–T as well as He–He and He–T interactions. Again, we have studied all the
unique configurations of PdT0.22 , where the largest energy difference between the highest and
lowest energy configurations was 0.04 eV. The most stable configurations of PdT0.22 have tritium
atoms evenly distributed throughout the material and this structure was therefore selected for
further study of helium incorporation and diffusion.
(c) Clustering of helium atoms in perfect PdT0.22
Initially, we considered tetrahedral and octahedral interstitial sites for the incorporation of a single
helium atom in both fcc Pd and PdT0.22 systems corresponding to a helium concentration of 0.03
per Pd, which we henceforth call PdHe0.03 and PdT0.22 He0.03 , respectively. In both systems, the
octahedral interstitial site is more stable than the tetrahedral sites for helium insertion, although
the energy differences between octahedral and tetrahedral sites are only 0.12 and 0.16 eV for Pd
and PdT0.22 , respectively.
Considering the favourable octahedral interstitial site, the stabilities of helium clusters in
PdT0.22 are investigated, with a variation of cluster size from one to six atoms, corresponding
to helium concentrations of 0.03–0.19 per Pd, respectively. We next inserted helium atoms one
by one into the octahedral site of the PdT0.22 system and the energetics of the formation of these
interstitial helium clusters into the system are shown in figure 2a,b. The formation energy per
helium atom is decreasing as the number of helium atoms increases, indicating that a helium
atom in an octahedral site can attract more helium atoms and form a cluster of at least up to
six helium atoms, as calculated here. The He2 forms a pair in the octahedral site with a He–He
distance of 1.73 Å, which is very close to the value of 1.704 Å in the hcp Ti structure [16]. For
He3 , the most stable structure formed is a scalene triangle in the octahedral site with He–He
distances of 1.87, 1.88 and 1.886 Å. The ground-state structure of the He4 cluster was a planar
rhombus with an average He–He distance of 1.90 Å, whereas the He5 cluster was found to be a
trigonal bipyramidal structure. In the case of He6 , the lowest energy structure was the octahedron
structure indicating the onset of three-dimensional structures. The average He–He distance again
was 1.90 Å. The average He–He distances of the helium clusters therefore vary between 1.73 and
1.90 Å, i.e. significantly less than that of gas phase helium at 4.50 Å. The bigger clusters displayed
longer He–He distances, acquiring more space in the interstitial site. Large clusters could therefore
generate stress in the tritide lattice.
The helium binding energies in PdT0.22 have also been calculated, i.e. the energy required to
bind a helium to a cluster. The binding energy for the smallest cluster of He2 is 0.58 eV and that
energy is increasing as more helium atoms are added to the cluster, suggesting that an interstitial
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8
Ef (eV)
5
3.4
10
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It is important to understand the migration behaviour of helium in palladium tritide to assess
the mechanism of helium bubble formation. The diffusion pathways are not intuitive, but we
have investigated all major migration pathways for helium diffusion, that can reasonably be
expected in the PdT0.22 lattice, and calculated the lowest energy paths with respect to the tritium
positions in the system. In order to evaluate if the size of the simulation cell could affect the
diffusion activation energies, we have first calculated the cohesive energies and energies for
helium incorporation in the octahedral and tetrahedral interstitial sites in both (2 × 2 × 2) and
(3 × 3 × 3) simulation supercells. However, these energies do not deviate significantly and as
such the activation energies for helium diffusion are unlikely to be affected by cell size, and we
have therefore carried out all diffusion calculations in a (2 × 2 × 2) cell. From an octahedral site,
the helium can migrate to another available octahedral site following different paths, such as
directly from octahedral to octahedral site (parallel and perpendicular) or octahedral–tetrahedral–
octahedral (O–T–O), as shown in figure 3. The migration energy of helium for the path O–T–O was
lower than that of O1–O2 and O1–O3 diffusion. Previous studies have shown that the O–T–O path
is also the lowest energy path for helium diffusion in Al and Pd metals [14]. The migration of an
interstitial helium via O–T–O was determined to require about 0.48 eV to overcome the energy
barrier, with the T site being 0.44 eV higher in energy than the O sites. Figure 3, on the other
hand, shows that the diffusion along path O1–O2 for an interstitial helium requires an energy of
1.05 eV, and 1.14 eV for path O1–O3, both of which are higher than the O–T–O migration path,
and thus would only be likely to occur at high temperatures. A previous experimental study
found a value of 0.35 eV for the helium activation energy in Ni metal, whereas a hybrid molecular
dynamics method obtained a value of 0.43 eV for the same system [31,32]. Moreover, the ErT
system found a value of 0.74 eV experimentally [33], which is larger than that of the present study
on PdT0.22 . Helium bubble growth rate is slow when activation energies are high, and helium
bubble formation is thus more difficult in PdT0.22 compared with the pure Ni metal. However,
this study shows that an interstitial helium in PdT0.22 is less likely to jump directly from one
O site to another without passing through an intermediate tetrahedral site, owing to the much
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(d) Helium diffusion in PdT0.22
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helium atom in PdT0.22 could be easily trapped by another. The largest binding energy in PdT0.22
was obtained for helium addition to a He4 cluster at 2.78 eV but this decreased to 1.93 eV for the
addition to the He5 cluster to make a He6 cluster.
Previous DFT and experimental studies have found that the octahedral interstitial site is more
stable than the tetrahedral interstitial site in the Pd lattice [14,28,29], suggesting that the helium
from tritium decay would preferably occupy octahedral interstitial sites. Zeng et al. [14] have
shown that the palladium metal can accommodate up to four helium atoms in an interstitial site,
but there are no reports about the interaction between helium clusters and palladium tritides. The
calculated formation energy in the PdT0.22 system is gradually increasing with increasing cluster
size, whereas the formation energy per helium atom is decreasing as the number of helium atoms
increases, consistent with previous studies in Pd and Ti metal [14,16]. The values for the PdT0.22
system are lower than those for helium insertion in palladium metal, which are reported at 3.70–
11 eV for He1 –He4 clusters [14], and the presence of tritium in the octahedral sites thus clearly
affects the formation energies. When we compare helium binding energies with the previous
studies, we see that tritium occlusion in palladium has an effect on helium cluster formation.
Helium binding energies for the PdT0.22 system are very close to a number of metals, e.g. 0.65 eV
for Pd, 0.66 eV for Ti and 0.43 eV for Fe [4,14,16]. The helium binding energy in nickel, on the other
hand, is found to be near constant at about 2 eV for He6 and large clusters [30] and in an fcc Pd
lattice, a value of 1.33 eV was found for a He4 cluster [14]. Such differences in binding energies are
indicative of the effect of interstitial tritium. Therefore, helium from tritium decay can accumulate
in an interstitial site to form a bubble in the bulk PdT0.22 lattice. The binding energies reveal that
the cluster is formed preferentially over dispersed atoms and the helium would be trapped until
the system was exposed to very high temperatures [30].
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(a)
7
(ii)
O1
O3
(iii)
(b)
1.2
1.0
0.8
0.6
0.4
0.2
0
O1–O2
O1–T–O2
O1–O3
Figure 3. (a) Illustration of helium migration paths. The dashed teal line indicates direct octahedral to octahedral, the red
line shows octahedral to octahedral via tetrahedral site and the dashed green line indicates helium motion from octahedral to
octahedral between two layers. The dark blue atoms are palladium, tritium is not shown. The red and green spheres indicate the
octahedral and tetrahedral interstitial sites, respectively, and (b) schematic illustration of the helium activation energy. (Online
version in colour.)
higher migration energy. Moreover, the migration of helium in the tritides will also depend on
the amount of tritium decay, because the number and distribution of octahedral vacancies will
affect the diffusion.
(e) Clustering of helium atoms in a defect-containing lattice
The calculated monovacancy formation energy in a (2 × 2 × 2) bulk cell of Pd metal was 1.26 eV,
which is 0.28 eV lower than that found by thermal diffusivity experiments [34]. Tritium atoms
were again added in the octahedral interstitial sites of the vacancy-containing Pd lattice and the
structure optimized. In the vacancy-containing PdT0.22 lattice, we placed the helium atom in all
possible octahedral and tetrahedral interstitial sites as well as the vacancy site, in effect replacing
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(i)
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T
O2
O1
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(a)
(b)
11
10
8
2.8
Ef /n (eV)
Ef (eV)
7
6
2.2
2.0
5
1.8
4
1.6
3
1
2
3
4
no. He atoms (n)
5
6
1
2
3
4
no. He atoms (n)
5
6
Figure 4. Formation energy of small He-vacancy clusters versus number of helium atoms (a) the total formation energy and
(b) average formation energy per helium atom. (Online version in colour.)
20
change of lattice volume (%)
18
16
PdT0.22 perfect lattice
PdT0.22 lattice with a Pd defect
14
12
10
8
6
4
2
0
no. He atoms
Figure 5. The change in local PdT0.22 lattice volume caused by the presence of helium clusters. (Online version in colour.)
a Pd by a helium atom, to calculate the lowest energy site. The preferred site for one helium atom
was in the vacancy site, where the energy of the helium incorporation was 2.30 eV lower than
in the octahedral interstitial site. A molecular dynamics study found that the formation energy
of helium in a vacancy site in fcc Ni was 2.70 eV lower than that of the interstitial site [30], in
accordance with our PdT0.22 study. Thus, the interstitial helium will be trapped by the vacancy.
The addition of helium atoms one by one into the vacant site shows that the cluster formation
energy is again gradually increasing with increasing cluster size, as shown in figure 4a, but all
values are lower than in the perfect PdT0.22 He0.13 lattice. The cluster formation energy per atom
was gradually decreasing with increasing cluster size up to our calculated He6 cluster. The perfect
PdT0.22 lattice showed the same trend, predicting the clustering of a number of helium atoms
could be trapped. The helium binding trend indicates that one monovacancy can accommodate
at least six helium atoms and that Pd vacancies could therefore be nucleation sites for helium
bubble formation.
The helium cluster geometries are very similar to those in the perfect PdT0.22 lattice, but
the He–He distances are different, i.e. 1.62 Å for the dimer with the average He–He distances
varying from 1.70 to 1.82 Å for the clusters He3 –He6 . With less than seven helium atoms in the
monovacancy, He–He distances in vanadium are in the range of 1.75–1.85 Å, suggesting that the
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2.6
9
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(a)
140
DOS (states eV–1)
...................................................
100
s of Pd
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120
9
total
p of Pd
d of Pd
80
60
40
20
0
–10.0
–7.5
–5.0
–2.5
energy (eV)
0
2.5
20.0
total
s of Pd
p of Pd
d of Pd
s of T
DOS (states eV–1)
17.5
(b)
140
total
120
DOS (states eV–1)
15.0
12.5
enlarged
view
10.0
7.5
5.0
2.5
s of Pd
0
–10.0
p of Pd
100
5.0
d of Pd
–7.5
–5.0
–2.5
0
energy (eV)
2.5
5.0
s of T
80
60
40
20
0
–10.0
–7.5
–5.0
–2.5
0
2.5
5.0
energy (eV)
Figure 6. The calculated total density of states of (a) bulk palladium, (b) PdT0.22 , (c) PdT0.22 He0.03 and (d) PdT0.22 He0.19 . (Online
version in colour.)
interstitial clusters are in a compressed state in the cluster compared with a free cluster [35].
Helium clusters in the vacancy significantly displace the metal atoms away from their original
positions and the lattice expands as a result.
(f) The effect of helium on the structural parameters of palladium tritides
Interstitial helium atoms have a significant effect on the palladium tritide structures. If we
compare the PdT0.22 with PdT0.22 He0.03 , incorporation of a single helium atom leads to an increase
in the lattice vector by about 1.9%. Increasing the number of helium atoms in the PdT0.22 lattice
gradually expands the structure, as shown in figure 5. It is interesting to note that expansion of the
perfect PdT0.22 lattice is very small from He5 to He6 clusters, perhaps owing to the structure of the
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(c)
10
20.0
DOS (states eV–1)
100
80
12.5
enlarged
view
10.0
7.5
5.0
140
120
15.0
total
s of Pd
p of Pd
d of Pd
s of T
s of He
2.5
0
–10.0
–7.5
–5.0
–2.5
energy (eV)
0
2.5
5.0
60
40
20
0
–10.0
–7.5
–5.0
–2.5
energy (eV)
0
2.5
5.0
(d)
20.0
DOS (states eV–1)
17.5
DOS (states eV–1)
100
80
12.5
enlarged
view
10.0
7.5
5.0
140
120
15.0
total
s of Pd
p of Pd
d of Pd
s of T
s of He
total
s of Pd
p of Pd
d of Pd
s of T
s of He
2.5
0
–10.0
–7.5
–5.0
–2.5
energy (eV)
0
2.5
5.0
60
40
20
0
–10.0
–7.5
–5.0
–2.5
energy (eV)
0
2.5
5.0
Figure 6. (Continued.)
helium clusters. The structure of the tetramer is two-dimensional, but the pentamer and hexamer
clusters are three-dimensional. This result is qualitatively consistent with X-ray studies, where
it was found that after several months of ageing the metal tritide was distorted in an extremely
inhomogeneous fashion to the point that a unique lattice parameter could not be defined [36].
...................................................
DOS (states eV–1)
17.5
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total
s of Pd
p of Pd
d of Pd
s of T
s of He
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To gain more insight into the bonding and electronic structure of the systems, we have plotted
the density of states (DOS) of Pd, PdT0.22 , PdT0.22 He0.03 and PdT0.22 He0.19 , as shown in figure 6.
Figure 6a clearly shows that Pd d orbitals are highly localized at the Fermi level (EF) and the total
orbitals spread up to 7.0 eV below the Fermi energy. The PdT0.22 DOS in figure 6b shows a new
peak at around −7.5 eV, which indicates that the tritium s orbital is hybridizing with the Pd d and
s orbitals. Previously, Gupta & Gupta [39] stated that a peak below approximately −5.0 eV results
from the bonding Pd-d–T-s interactions. The insertion of one helium in an octahedral site led to
only a very small change in the electronic properties, as shown in figure 6c. With an increasing
number of helium atoms in the octahedral site, a significant contribution from the helium s
orbitals has modified the peak from −6.25 to −7.5 eV (figure 6b). This peak is mainly owing to
the admixture of helium s and Pd d orbitals with some contribution from tritium s orbitals. The
Pd d states have been pushed upward through an interaction with helium atoms, indicating the
repulsive character of the atom in the matrix [25].
4. Conclusion
This study shows that helium from tritium decay can become trapped in interstitial sites and
form clusters in the bulk PdT0.22 lattice. When Pd vacancies are present in the lattice, helium
atoms will accumulate preferentially in the vacancy site. The calculated energetics show that
clusters are thermodynamically preferred over helium dispersed in the lattice, where it should be
retained until the system is exposed to high temperatures. The activation energy of migration of
an interstitial helium is 0.48 eV for an O–T–O path, which is lower than the O–O paths. Of course,
the migration of helium in the metal tritides may have to follow a path between octahedral sites
in different layers, depending on the distribution of octahedral vacancies.
The average He–He distances in the helium clusters vary from 1.62 to 1.90 Å, significantly
less than that of gas phase helium. Even so, increasing cluster sizes require more space in the
interstitial site and the process leads to an expansion of the PdT0.22 lattice, which the DOS analysis
shows may be due to the repulsion between helium s and Pd d orbitals. Thus, the accumulation
of helium atoms in the interstitial site in the lattice induces local stress around the cluster, as
shown by the calculated bulk moduli for the pure and He-containing systems, which may cause
material failure.
...................................................
(g) Electronic structures
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The accumulation of helium atoms in a vacancy showed the same trend but the percentage of
lattice expansion of the defective PdT0.22 was much smaller than that of the perfect lattice. The
vacant site has extra space that can accommodate more helium before significant expansion of the
lattice becomes necessary. An ab initio study of the effect of helium on the peak tensile strength
of tungsten metal showed that the presence of interstitial helium led to the highest increase in
the tensile strength [37], which indicates that interstitial helium atoms are more harmful than
substitutional helium, which is consistent with our observation.
The optimized PdT0.22 and PdT0.22 Hen configurations were used to calculate the bulk moduli,
which is a material property that relates the change in volume with a change in pressure. The
bulk moduli are calculated to be 150 and 129.2 GPa for the PdT0.22 and PdT0.22 He0.03 systems,
respectively, whereas further increases in helium cluster size in the PdT0.22 system lead to a
gradual increase of the bulk modulus, for example, the bulk modulus varies from 180 to 257 GPa
for the PdT0.22 He0.06 to PdT0.22 He0.19 systems, respectively. A previous study has shown that
the bulk modulus for some compounds in inversely proportional to the lattice volume and
proportional to the lattice pressure [38]. However, here we observe that the accumulation of
helium atoms in an interstitial site increases the lattice volume, as well as the bulk modulus
of PdT0.22 structures. This means that the larger helium clusters create higher pressures in the
PdT0.22 systems as a result of more deformation of the crystal structures, which may eventually
lead to microstructural failure.
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Acknowledgements. This work made use of the facilities of HECToR, the UK’s national high-performance
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Funding statement. NHdL acknowledges the Royal Society for an Industry Fellowship and thanks AWE
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