Influence of Lithium Vacancy Defects on Tritium Diffusion in β-Li₂TiO₃

Abstract

Lithium metatitanate, Li₂TiO₃, is a leading candidate for application as a tritium breeding material in a future fusion reactor. Following transmutation of lithium, the tritium must escape the crystal in order to be extracted for use in the fusion plasma. The rate-limiting step to release tritium from the Li₂TiO₃ pebbles is diffusion through the crystal grains. In this work, the activation barriers for tritium diffusion have been calculated using density functional theory. The results show that tritium can diffuse as an interstitial with a barrier of 0.52 eV. However, when a tritium ion becomes bound to a lithium vacancy defect, the energy required to either detrap the tritium from the vacancy or for the cluster to diffuse increases to >1 eV. Overall, these results suggest that the introduction of lithium vacancies due to Li burn-up may lead to an increase in tritium retention in the pebbles.

Introduction

Future fusion reactors will employ the reaction between deuterium and tritium to release energy for electricity generation. Because there is very limited naturally occurring tritium, it is expedient to generate it in situ from the transmutation of lithium using the high-energy neutron ejected from the D–T reaction. This will be done by surrounding the plasma chamber with a blanket region containing a lithium breeder material. A number of different blanket concepts have been proposed, mostly based on either lithium ceramic pebbles or a liquid lithium lead eutectic.¹

To ensure that the fusion process is sustainable, it is essential that at least one tritium is recovered from the blanket for each fusion reaction in the plasma. Because of the inability to completely surround the plasma with the blanket and the permeation of tritium into other reactor components, a tritium breeding ratio (TBR) greater than 1.1 is necessary.² In the case of a solid breeder, the high-energy neutrons will penetrate the pebbles, leading to tritium being generated throughout the ceramic matrix. Therefore, it is essential that all generated tritium be extracted from the breeder material to ensure the maximum possible TBR. For recovery, this tritium must diffuse to the surface, as illustrated in Figure 1. Following transmutation, the tritium must be accommodated in the ceramic’s microstructure before diffusing through the crystal grains until it reaches the grain boundary. It will then diffuse along the boundary until it reaches the pebble surface. Once at the surface, the tritium will undergo isotope exchange reactions with hydrogen in the He purge gas before being recovered.

Figure 1.

Schematic of the tritium release process from lithium ceramic pebbles in the breeder blanket.

The rate-limiting step for tritium recovery is diffusion in the fuel grains. Multiple peaks in thermal desorption spectra of neutron-irradiated lithium ceramics show that there are a number of distinct mechanisms contributing to the tritium release process, with their relative importance depending on temperature.^3,4 Previous studies have used these experimental observations to identify the specific release mechanisms and what defects are facilitating them; however, there is no consensus on what these may be. Experimental activation energies for tritium diffusion vary from 0.63 to 1.5 eV for Li₂TiO₃.⁵⁻⁷ The experimental data show a significant variation that is likely due to the different experimental conditions, including burn-ups, and it does not allow the isolation of the individual mechanisms for tritium diffusion and their associated energy.

Previous atomistic simulations have shown that the mode of tritium accommodation in Li₂TiO₃ depends on the stoichiometry of the underlying crystal.⁸ Under conditions where there is a lithium excess, the tritium is predicted to sit as an interstitial that bonds to an oxygen ion, forming a hydroxyl group. The excess positive charge is then compensated by a variety of different mechanisms, depending on the conditions. Under conditions of lithium deficiency, such as those anticipated as the ceramic ages, these hydroxyl groups become bound to lithium vacancy defects, forming the {T_i¹⁺:V_Li}⁰ complex.⁸

As the availability of point defects changes, so does the tritium diffusion mechanism. Shi et al. used density functional theory (DFT) to examine the diffusion of charged tritium interstitials in Li₂TiO₃.⁹ They observed that the activation energy for diffusion in the xy-plane is 0.33 eV, while for tritium to migrate in the z direction, the barrier is much higher at 1.00 eV. This high degree of anisotropy in tritium mobility arises because of the complex layered structure exhibited by Li₂TiO₃. Li et al. have also employed DFT to study tritium diffusion in Li₂TiO₃ in the presence of lithium vacancy defects and determined that the energy barrier for a tritium to escape from the {T_i¹⁺:V_Li}⁰ complex is 0.76 eV.¹⁰ This result implies that as the fuel ages and lithium is burnt up, the activation energy for tritium release will increase. However, these works have neglected the mobility of the lithium vacancy itself and the possibility that the lithium vacancy may assist tritium diffusion through a cooperative mechanism. Therefore, the aim of the present work is to re-examine the complex tritium diffusion mechanisms in Li₂TiO₃, considering the role of the lithium vacancy.

Crystallography

The β-phase of Li₂TiO₃ is stable across a wide temperature range of 300–1215 °C, as is seen from the Li₂O–TiO₂ phase diagram,^11,12 while the cubic α- and γ-phases are present at temperatures above and below this range, respectively. The structure of β-Li₂TiO₃ was first studied by Lang¹³ and later refined by Kataoka et al.¹⁴ using X-ray diffraction of large single crystals. These works described Li₂TiO₃ as a distorted rock salt structure with alternating (111) planes of Li₆, O₆, Li₂Ti₄, and O₆. In the mixed cation layer of Li₂Ti₄, six Ti ions surround a Li ion, forming a hexagon. At low sintering temperatures, a number of different stacking sequences, corresponding to different space groups, viz. C2/m, C2/c, and P3₁12, are observed.¹⁵ However, in the present work, it is assumed that sintering took place at a high temperature, and so, only the monoclinic C2/c space group is present. An illustration of β-Li₂TiO₃ is presented in Figure 2. The experimental lattice parameters of β-Li₂TiO₃ according to Kataoka et al. are a = 5.0623 Å, b = 8.7876 Å, c = 9.7533 Å, and β = 100.212°.¹⁴

Figure 2.

Unit cell of β-Li₂TiO₃ showing the different ions and their types. The Li, Ti, and O ions are shown as green, yellow, and red spheres, respectively. The Li₆ and Li₂Ti₄ layers have been highlighted, and the O ions lie above or below these layers. Also illustrated are the Ti–Ti bonds in the Li₂Ti₄ layers with a Li ion in the center to illustrate the hexagonal structure.

As shown in Figure 2, there are three symmetrically distinct lithium lattice sites in β-Li₂TiO₃. Li1 and Li2 are found in the pure lithium layer, while Li3 is found at the center of the Ti hexagons in the mixed cation layer. Therefore, the different positions have different nearest-neighbor cation environments, as illustrated in Figure 3. In fact, the Li1 and Li2 positions are remarkably similar and have almost identical vacancy formation energies.¹⁶ In the pure lithium layer, a Li1 ion is surrounded by an equal number of Li1 and Li2 ions. By contrast, a Li2 ion is surrounded by Li1 ions only. Because of the relative location of Ti ions above and below the xy-planes, Ti–Ti and Li–Li bond distances are distorted.

Figure 3.

Local environments surrounding the lithium sites in Li₂TiO₃. Note that the oxygen ions have been removed to highlight the cation environments around the different lithium sites.

As β-Li₂TiO₃ exhibits a distorted rock salt structure, the lithium ions are octahedrally coordinated with oxygen, as illustrated in Figure 4. These oxygen ions have been designated A–F, as illustrated in Figure 4.

Figure 4.

(a) Li ion is coordinated by six oxygen ions and (b) octahedron formed by the oxygen ions designated A–F around the Li vacancy.

Computational Details

All first-principles calculations in the present work were based on DFT^17,18 and were performed using the plane wave CASTEP simulation package.^19,20 The generalized gradient approximation parameterized by Perdew, Burke, and Ernzerhof was used as the exchange–correlation functional.²¹ All calculations were performed using a supercell containing 2 × 1 × 1 replications of the unit cell for a total of 96 atoms. This relatively small simulation supercell may be susceptible to finite-size effects because of the interaction of the charged defects with their periodic images.²² However, these finite-size effects cancel out for the calculation of migration barriers. Ultrasoft pseudopotentials were employed with a plane wave cutoff energy of 600 eV. A Monkhorst–Pack scheme²³ was used to sample the Brillouin zone with a density of 0.06 Å^–1, which corresponded to a 2 × 2 × 2 k-point mesh in the 2 × 1 × 1 supercell. A Methfessel-Paxton smearing width of 0.1 eV was used.²⁴ To test the efficacy of the model, an energy minimization of the perfect unit cell was performed, and the lattice parameters obtained were a = 5.1186 Å, b = 8.8809 Å, c = 9.8976 Å, and β = 100.212°, which are in good agreement with the experimental values of Kataoka et al.¹⁴

Transition states were determined by performing an energy minimization of the start and end structures under constant volume conditions until all forces fell below 0.08 eV/Å. Activation energies were calculated using the complete linear/quadratic synchronous transit (LST/QST)²⁵ method with default settings as implemented in CASTEP. It is to be noted that more stringent conditions were tested, and this did not significantly affect the results.

A complete understanding of the tritium release process from ceramic materials such as Li₂TiO₃ requires an examination of all the possible atomic scale hops that may contribute to the diffusion process. As discussed in the Introduction, this is not solely dependent on the migration barriers for tritium hops, but also on other defects that may promote or hinder tritium diffusion. Previous studies have indicated that the lithium vacancy defect will play an important role in the tritium release process as the material ages. Therefore, this study will examine the migration barriers for diffusion of the isolated lithium vacancy and tritium interstitial defects as well as the barriers to diffusion of the bound {T_i¹⁺:V_Li}⁰ cluster.

The interstitial sites between the Li and O layers were initially located by performing geometry optimization calculations with a tritium ion (T¹⁺) initially placed in the vicinity of the oxygen ions. Once the locations of the initial and final positions of the ions were identified, the LST/QST method was used to calculate the energy barrier between them. A similar approach was used to locate the tritium interstitial sites within the lithium vacancies. In this case, the positions were located by performing geometry optimization calculations of tritium ions initially positioned halfway between the vacancy and the oxygen ions.

Results and Discussion

Lithium Vacancy Diffusion

In the first instance, the activation energies for diffusion of the lithium vacancy, V_Li^–1 defects in Li₂TiO₃, were determined. Within the C2/c space group, there are three types of lithium sites. Two of these are in the pure lithium layer and the third is at the center of the Ti hexagons in the mixed cation layer, as illustrated in Figure 3. While there are only three symmetrically distinct lithium sites present in Li₂TiO₃, the arrangement of these sites is such that there are a number of different possible hops between them, as discussed by Kuganathan et al²⁶ For completeness, barriers to all possible pathways between nearest-neighbor lithium sites were calculated. The resulting barriers for Li1 and Li2 vacancy diffusion are presented in Table 1 and those for Li3 vacancy diffusion are presented in Table 2.

Table 1. Calculated Intrinsic Li1 and Li2 Vacancy Diffusion Barriers for the Nearest-Neighbor (NN) Hops as Shown in Figure 3, Assuming the Vacancy Occupies the Central Ionic Position^a.

			barrier (eV)				barrier (eV)
class	pathway	d (Å)	forward	reverse	pathway	d (Å)	forward	reverse
xy-plane above	VLi1–1 → VLi3	2.91	0.45	0.51	VLi2–1 → VLi3	2.92	0.41	0.47
same xy-plane	VLi1–1 → VLi1	3.00	0.65	0.65	VLi2–1 → VLi1	2.99	0.67	0.67
	VLi1–1 → VLi1	2.79	0.50	0.50	VLi2–1 → VLi1	2.99	0.67	0.67
	VLi1–1 → VLi1	3.00	0.65	0.65	VLi2–1 → VLi1	2.90	0.57	0.57
	VLi1–1 → VLi2	2.89	0.55	0.54	VLi2–1 → VLi1	2.90	0.57	0.57
	VLi1–1 → VLi2	2.90	0.57	0.57	VLi1–1 → VLi2	2.89	0.54	0.55
	VLi1–1 → VLi2	2.99	0.67	0.67	VLi2–1 → VLi1	2.89	0.54	0.55
xy-plane below	VLi1–1 → VLi3	2.91	0.45	0.51	VLi2–1 → VLi3	2.92	0.41	0.47

^aDistances, d, presented in the table were calculated in the perfect crystal reported by Kataoka et al.¹⁴ There are six possible hops in the same xy-plane and one each above and below the xy-plane.

Table 2. Calculated Intrinsic Li3 Vacancy Diffusion Barriers for the NN Hops as Shown in Figure 3, Assuming the Vacancy Occupies the Central Ionic Position^a.

			barrier (eV)
class	pathway	d (Å)	forward	reverse
xy-plane above	VLi3–1 → VLi1	2.91	0.51	0.45
	VLi3–1 → VLi1	2.91	0.51	0.45
	VLi3–1 → VLi2	2.92	0.47	0.41
xy-plane below	VLi3–1 → VLi1	2.91	0.51	0.45
	VLi3–1 → VLi1	2.91	0.51	0.45
	VLi3–1 → VLi2	2.92	0.47	0.41

^aDistances presented in the table were calculated from the perfect crystal reported by Kataoka et al.¹⁴ There were three possible hops each in the xy-planes above and below the xy-plane.

The calculated activation energies are between 0.41 and 0.67 eV, which is in the range of energies predicted experimentally.²⁷⁻²⁹ The lowest energy barriers correspond to the vacancy hopping from the pure Li layer into the mixed cation layer, with the hop from the Li2 site being very slightly lower in energy than from the Li1 site (0.41 eV compared to 0.45 eV). The ordering of the hops between the pure lithium and mixed cation layer is in excellent agreement with the previous DFT simulations of Islam and Bredow.³⁰ In this previous work, there was only one transition within the pure lithium considered, which had an energy of 0.45 eV for the forward process, and for the reverse process, an energy of 0.5 eV was determined. These values are slightly lower than those predicted here, which are all in the range 0.5–0.67 eV. It should be noted that the previous work examined charge-neutral vacancies, while here, the fully charged V_Li^–1 defects are considered, and this may account for the small discrepancy. Mukai et al. calculated barriers of 0.52 and 0.53 eV for the V_Li3 → V_Li1^–1 and V_Li2 → V_Li1^–1 hops.²⁹ These values are in excellent agreement with the values of 0.51 and 0.54 eV calculated here. However, for the V_Li2 → V_Li1^–1 hops, two additional values, 0.57 and 0.67 eV, were also calculated. Kuganathan et al. predict that the lowest energy process is for vacancy migration in the pure Li layer with a barrier of 0.39 eV, while the energy to transfer between the two layers is much higher at 0.65 and 0.67 eV.²⁶ Further barriers for migration of a lithium vacancy were determined by Vijayakumar et al. who found values in the range 0.2–0.3 eV.³¹ These last two studies employed empirical pair potentials that are not able to consider charge transfer during the migration process and are, therefore, likely to be less reliable than the DFT data presented here.

Overall, the data suggest that there is a slight anisotropy in the vacancy-mediated diffusion of lithium in Li₂TiO₃, with diffusion in the z-direction having slightly lower energy barriers than diffusion in the xy-plane.

Tritium Diffusion

Tritium Diffusion as an Interstitial

The next section considers tritium diffusion as an interstitial defect. As discussed by Shi et al., there are six symmetrically distinct tritium interstitial sites in Li₂TiO₃.⁹ These positions are illustrated in Figures 5 and 6, and the fractional coordinates are presented in Table 3. In all cases, the tritium ions are bonded to oxygen ions in the form of hydroxyl groups as observed in previous work.⁸

Figure 5.

2 × 1 × 1 supercell of Li₂TiO₃ illustrating the location of the tritium interstitial sites.

Figure 6.

Tritium sites near the (a) Li₆ and (b) O₆ layers.

Table 3. Calculated Fractional Coordinates of the Different Types of Interstitial Sites for Tritium in β-Li₂TiO₃.

position	site label	fractional coordinates (x,y,z)	host oxygen
Li6 layer	1	0.21310, 0.08659, 0.46836	O2
	2	0.44976, 0.26219, 0.46833	O1
	3	0.20159, 0.40395, 0.47633	O3
O6 layer	4	0.36446, 0.58707, 0.40740	O2
	5	0.26578, 0.32535, 0.59409	O1
	6	0.23561, 0.50323, 0.39167	O3

The calculated activation energies and jump distances for the hops between different tritium interstitial positions are presented in Table 4, where they are compared to the values reported by Shi et al⁹ For simplicity, the possible diffusion processes are grouped into three distinct classes, C1, C2, and C3. The first class, C1, involves tritium hopping within the layer of interstitial positions closest to the Li₆ layer encompassing the 1, 2, and 3 tritium sites (see Figures 5 and 6). The second class, C2, corresponds to the tritium ions hopping from a Li₆ to its nearest-neighbor O₆ layer. The final class, C3, involves processes where the tritium crosses the mixed cation layer and consequently has larger jump distances.

Table 4. Calculated Diffusion Path Distances, d, and Energy Barriers for Tritium Hops between Different Interstitial Sites in a Li₂TiO₃ Crystal for Different Classes of Independent Local Minimum Energy Paths as Defined by Shi et al.⁹.

		d (Å)		barrier (eV)
diffusion class	pathway	present work	Shi et al.9	present work	Shi et al.9
C1 [Li6–Li6]	1 ↔ 1	1.77	1.68	0.49	0.33
	2 ↔ 2	1.77	1.69	0.51	0.33
	3 ↔ 3	1.98	2.04	0.73	0.55
	1 → 2 (1 ← 2)	1.68	1.58	0.46 (0.45)	0.29 (0.29)
	2 → 3 (2 ← 3)	1.82	1.73	0.60 (0.45)	0.47 (0.30)
	1 → 3 (1 ← 3)	1.86	1.80	0.62 (0.46)	0.48 (0.31)
C2 [Li6–O6]	1 → 4 (1 ← 4)	1.08	1.15	0.09 (0.01)	0.10 (0.00)
	2 → 5 (2 ← 5)	1.08	1.15	0.08 (0.02)	0.09 (0.00)
	3 → 6 (3 ← 6)	1.31	1.41	0.11 (0.16)	0.11 (0.11)
C3 [O6–Li2Ti4–O6]	6 ↔ 6	2.81	2.77	1.07	0.97
	5 ↔ 5	3.11	3.08	1.05	1.00
	4 → 5 (4 ← 5)	3.15	3.12	0.98 (0.98)	0.91 (0.92)
	4 → 6 (4 ← 6)	2.99	2.96	1.00 (0.96)	0.93 (0.86)

Analysis of the data presented in Table 4 shows that while the activation energies predicted here are mostly higher than those predicted by Shi et al., the overall picture remains the same. The processes with the lowest activation energies are the rotations around the oxygen ions of the C2 class, where the barriers can be as low as 0.01 eV. In fact, these barriers are so low that Shi et al. could not calculate a barrier for the reorientations around the O1 and O2 ions.⁹ It is important to note that there must be a barrier, as otherwise the interstitial site itself would not be stable. Shi et al. deduced that tritium sites 4 and 5 were actually not energy minimums, but cusps along a very flat potential energy surface. For diffusion around the pure lithium layer (C1 class), the barriers calculated here are 0.15 eV higher in energy than those calculated by Shi et al.; however, the ordering is identical with the diffusion from the tritium interstitial site 1 → 2 having the lowest energy barrier of 0.45 eV. This process is incapable of facilitating bulk diffusion of tritium on its own. The lowest energy pathway to net tritium diffusion in the xy-plane can be achieved through a combination of 1 → 2, 2 → 2, 2 → 1, and 1 → 1 processes with an overall activation energy of 0.51 eV. The energy barriers for the hops across the mixed cation layer have the longest jump distances and, as a consequence, the highest barriers. Table 4 shows that migration from interstitial site 6 → 4 is the lowest energy process that traverses the mixed cation layer, which is in agreement with Shi et al.⁹ This is also the overall barrier to interstitial tritium diffusion in the z-direction. Overall, therefore, the results presented in Table 4 support the prediction of highly anisotropic tritium diffusion as an interstitial.

Tritium Diffusion in the Vicinity of Lithium Vacancies

As the burn-up of lithium increases, the concentration of lithium vacancy defects in the ceramic pebbles is anticipated to increase. The lithium vacancy defect is negatively charged and the tritium interstitials are positively charged; therefore, the two defects will become bound because of the Coulomb attraction between them. However, despite being bound to the vacancy defect, the tritium ions remain bonded to the oxygen ions as hydroxyls, with slightly increased bond lengths. Due to its distorted rock salt structure, all lithium ions in Li₂TiO₃ are octahedrally coordinated with six oxygen ions surrounding them, each of which could form a bond with tritium. As illustrated in Figure 3, the lithium ions in the pure lithium layers have a very similar arrangement of nearest-neighbor cations; however, the lithium in the mixed cation layer has a different local environment. This difference in the nearest-neighbor cations leads to significant differences in the behavior of tritium in the region surrounding the vacancy as described below.

Figure 7 shows the positions of the tritium interstitial sites surrounding the Li3 site and how these positions change when the lithium ion is removed. The image shows that the positions of the tritium sites that are aligned close to the z-axis are relatively unperturbed by the vacancy; however, the hydroxyl groups that lie closest to the xy-plane become reorientated such that they point toward the now vacant site. For simplicity, it is assumed that as the position of the hydroxyl groups aligned along z are unperturbed, the energy barrier to diffuse from this site away from the vacancy can be considered to be the interstitial activation energy as discussed above. A number of barriers for tritium migration to form hydroxyls on the second nearest-neighbor oxygen sites were considered; however, as their energies are significantly higher, these processes are not discussed further.

Figure 7.

Tritium interstitial site modifications for a Li3 vacancy in the mixed cation layer. The green and red spheres represent the lithium and oxygen ions, while the white spheres represent the possible tritium interstitial sites. (a) Each of the six oxygen ions bonded to a lithium ion has two possible tritium interstitial sites. (b) On removing the lithium ion, six of these sites (a′–f′) pointing away from the vacancy remain unaffected, while the other six sites (a–f) now point toward the vacancy.

Therefore, it is only necessary to calculate the energy barriers for the reorientation, which can be combined with the interstitial barriers calculated above. Migration barriers to diffusion within the vacancy itself will be considered later. The energy barriers for the reorientation, diffusion away from the reorientated site, and the overall escape are presented in Table 5. Table 5 shows that for a vacancy on the Li3 site, there are four reorientations that have an energy barrier in the range 0.54–0.56 eV, and the two remaining pathways have higher energies of 0.74 eV. As evident from the significant migration energies for the reverse process, the formation energy of the complex with the hydroxyl pointing away from the vacancy is higher than when the hydroxyl points directly toward the formerly occupied lattice site.

Table 5. Calculated Diffusion Path Distances, d, and Energy Barriers for Trapping and Detrapping of a Tritium Inside a Li3 Vacancy.

		barrier (eV)
pathway	d (Å)	forward	reverse	escape	escape pathway
a → a′	1.78	0.55	0.01	1.06	a′(1) → 2 → 2 → 1 → 1
b → b′	1.79	0.54	0.01	1.05	b′(1) → 2 → 2 → 1 → 1
c → c′	1.80	0.56	0.005	1.065	c′(2) → 2 → 1 → 1 → 2
d → d′	1.89	0.74	0.09	1.38	d′(3) → 3 → 1 → 2 → 3
e → e′	1.90	0.74	0.10	1.37	e′(3) → 3 → 1 → 2 → 3
f → f′	1.78	0.56	0.01	1.06	f′(2) → 2 → 1 → 1 → 2

Once the reorientations have taken place, the tritium ions can diffuse as if they were members of the interstitial C1 class. This is because the de-trapped sites for the Li3 vacancy remained unaffected due to the presence of the vacancy and are thus equivalent to the interstitial tritium sites of types 1, 2, and 3 as shown in Figure 6. These barriers are then combined with the re-orientation energy to give the overall barriers for escape from the Li3 vacancy for each individual pathway. Figure 8 shows the a → a′ pathway where the overall barrier for escape was calculated to be 1.06 eV. The escape barriers are in the range of 1.05–1.38 eV as tabulated in Table 5.

Figure 8.

Energy barriers corresponding to the escape route for a tritium trapped in a Li3 vacancy for the pathway a → a′(1) → 2 → 2 → 1 → 1, with a total escape barrier of 1.06 eV.

For the Li1 and Li2 vacancies, the situation is more complicated. The tritium positions surrounding an occupied and vacant Li1 site are shown in Figure 9. As was the case for the Li3 site, there are six nearest neighbors surrounding the Li site. If the Li site is occupied, there are two possible hydroxyl arrangements on each oxygen. Similar to what was observed for the Li3 site, the hydroxyl positions closest to the xy-plane reorientate to point toward the vacancy defect. However, whereas in the Li3 case, the remaining hydroxyls were relatively unaffected, for the Li1 and Li2 sites, four of these positions no longer form stable defect sites, while the remaining two stayed very close to their initial positions. The reason for the different behavior is the presence of the Ti in the mixed cation layer being on average closer to the unstable hydroxyl sites. Therefore, in addition to determining the barriers for the reorientation, the barriers to jumps to the second nearest-neighbor sites evident in Figure 9 are also considered.

Figure 9.

Tritium interstitial site modifications for a Li1 vacancy. (a) Similar to the Li3 case, each of the six oxygen ions bonded to the lithium ion has two possible interstitial sites that can accommodate the tritium. However, here, only two of these sites point away from the Li ion (a′ and c′), while two second neighbor sites (g′ and h′) lie in close proximity to the ion. (b) On removing the Li1 ion, 10 out of 12 sites collapse to the 6 sites pointing toward the vacancy, while the other 2 sites (a′ and c′) remain unaffected. Tritium trapped at sites a and c can reorient to a′ and c′, respectively, similar to the Li3 vacancy, while tritium trapped at other sites can escape via the g′ and h′ sites, whichever is nearer. The entire process is similar in the case of a Li2 vacancy.

Irrespective of the pathway taken to escape from positions directly adjacent to the vacancy, the tritium will end up in detrapped positions equivalent to interstitial sites 4 or 5. The migration barriers for tritium diffusion away from these interstitial sites are through a combination of C1 and C2 classes (see Table 4), ensuring that the effective barrier for escape is in the range of 1.1–1.34 eV as tabulated in Table 6. This range of energy barriers is similar to the ones obtained for the Li3 vacancy. These results suggest that the introduction of Li vacancies into either the pure lithium or mixed cation layers could have a significant impact on overall tritium diffusion if the vacancy defects are considered to be immobile.

Table 6. Calculated Diffusion Path Distances, d, and Energy Barriers for Trapping and Detrapping of a Tritium Inside a Li1 Vacancy.

		barrier (eV)
pathway	d (Å)	forward	reverse	escape	escape pathway
a → a′	1.66	0.66	0.00	1.10	a′(4) → 1 → 2 → 2 → 1 → 1 → 4
b → g′	2.22	1.25	0.63	1.25	g′(4) → 1 → 2 → 2 → 1 → 1 → 4
b → h′	2.27	1.34	0.75	1.34	h′(5) → 2 → 2 → 1 → 1 → 2 → 5
c → c′	1.74	0.69	0.00	1.14	c′(5) → 2 → 2 → 1 → 1 → 2 → 5
d → g′	2.28	1.25	0.63	1.25	g′(4) → 1 → 2 → 2 → 1 → 1 → 4
d → h′	2.24	1.19	0.60	1.19	h′(5) → 2 → 2 → 1 → 1 → 2 → 5
e → h′	2.19	1.28	0.67	1.28	h′(5) → 2 → 2 → 1 → 1 → 2 → 5
f → g′	2.21	1.26	0.63	1.26	g′(4) → 1 → 2 → 2 → 1 → 1 → 4

Table 7 shows the energy barriers for diffusion between the hydroxyl groups pointing toward the lithium vacancy defects compared to the previous results of Li et al¹⁰ The results in Table 7 show that the activation energies for tritium to move around the vacant site range from 0.66 to 1.0 eV. Hops between sites that involve traversing the vacant lithium site have larger jump distances and are predicted by Li et al. to have significantly higher barriers. The results presented here also predict very high barriers to traverse the vacant Li site; however, much lower barriers are predicted to traverse the Li1 and Li2 vacancies. The reason for this is that the LST/QST methodology is detecting mechanisms that make the tritium diffuse via an intermediate state that is bonded to one of the other neighboring oxygen ions. This does not affect the overall observation that tritium will preferentially diffuse around the periphery of the vacancy rather than crossing the vacant site.

Table 7. Calculated Diffusion Path Distances, d, and Energy Barriers for 15 Possible Types of Tritium Hops Around the Vacant Lithium Sites Corresponding to Li1, Li2, and Li3 Vacancies in a Li₂TiO₃ Crystal^a.

		Li1			Li2			Li3
			barrier (eV)			barrier (eV)			barrier (eV)
	pathway	d (Å)	forward	reverse	d (Å)	forward	reverse	d (Å)	forward	reverse
present work	a → c	1.27	0.66	0.66	1.55	0.66	0.68	1.59	0.89	0.89
	a → d	1.59	0.98	0.83	1.74	0.93	0.94	1.34	0.81	0.83
	a → e	1.45	0.78	0.66	1.33	0.75	0.86	1.63	0.91	0.93
	a → f	1.70	1.00	0.87	1.67	0.85	0.96	1.34	0.81	0.82
	b → c	1.58	0.82	0.97	1.73	0.92	0.94	1.34	0.81	0.82
	b → d	1.37	0.91	0.91	1.53	0.67	0.69	1.62	0.91	0.93
	b → e	1.62	0.85	0.87	1.64	0.84	0.96	1.35	0.81	0.83
	b → f	1.45	0.79	0.80	1.33	0.73	0.84	1.60	0.88	0.90
	c → e	1.71	1.00	0.87	1.69	0.84	0.94	1.35	0.90	0.91
	c → f	1.44	0.81	0.67	1.34	0.77	0.86	1.61	0.89	0.90
	d → e	1.43	0.80	0.82	1.34	0.77	0.87	1.66	0.96	0.96
	d → f	1.64	0.96	0.98	1.66	0.85	0.95	1.35	0.91	0.90
	a → b	2.06	1.02	0.88	2.30	0.90	0.89	2.08	3.15	3.15
	c → d	2.07	1.03	0.88	2.34	0.90	0.90	2.11	3.17	3.18
	e → f	2.34	0.92	0.91	1.93	1.90	1.90	2.13	3.19	3.18
Li et al.10	a ↔c	1.62	0.71					1.85	0.85
	a ↔ d	1.93	0.96					1.41	0.64
	a ↔ e	1.86	0.80					1.86	0.86
	a ↔ f	1.39	0.63					1.44	0.75
	b ↔ c	1.92	0.95					1.40	0.66
	b ↔ d	1.34	0.45					1.84	0.83
	b ↔ e	1.61	0.62					1.44	0.75
	b ↔ f	1.99	0.97					1.83	0.82
	c ↔ e	1.62	0.62					1.41	0.64
	c ↔f	1.85	0.74					1.84	0.83
	d ↔ e	1.99	0.98					1.85	0.85
	d ↔ f	1.63	0.61					1.40	0.63
	a ↔ b	2.28	3.07					2.27	2.97
	c ↔ d	2.29	3.11					2.25	2.97
	e ↔ f	2.58	3.00					2.27	2.98

^aFirst 12 of these paths are tritium hops around the periphery of the Li vacancy, while for the last 3 paths, tritium has to traverse through the vacant lithium site.

In general, the predicted energies for tritium migration around the lithium vacancy are higher than those calculated by Li et al., although it is noted that performing a direct comparison with the previous results is complicated because of difficulties in identifying the corresponding pathways. There are, however, methodological differences that may account for this discrepancy. In their calculations, Li et al. allowed the simulation supercell to relax while performing the geometry optimization of the initial and final structures; whereas in the simulations presented here, these were fixed to those determined for the perfect cell. The calculation of volume changes for charged supercells is poorly defined, leading to unrealistic volume changes.³² While the simulations of Li et al. were all charge-neutral, that is not the case here, therefore, to ensure consistency between the results, the simulation supercells remain fixed throughout.

Diffusion of the {T_i¹⁺:V_Li}⁰ Cluster

The two previous sections have examined how tritium can move as an interstitial and how it diffuses in the vicinity of a lithium vacancy. In these simulations, it is only the movement of the tritium itself that is considered, and the remaining ions in the system are effectively spectators in the process. In this final section, processes where the presence of the lithium vacancy may help to promote tritium diffusion are examined. An example of how this process may occur is presented in Figure 10, which shows a process whereby the lithium vacancy with a hydroxyl group bound to it moves to an adjacent lattice site. As clearly indicated in the diagram, the lithium ion moves to the right, while the tritium ions moves to the left. If this is combined with the diffusion around the vacancy, as discussed above, then it is possible to create a collaborative mechanism that enables macroscopic tritium diffusion without detrapping of the tritium from the Li vacancy.

Figure 10.

Example of the collaborative diffusion mechanism of the {T_i¹⁺:V_Li}⁰ cluster. In this case, the system evolves from an initial lattice configuration, (a) where a lithium on an Li1 site moves onto a neighboring Li1 vacancy containing a tritium via the transition state (b) to a final state (c) with the simultaneous diffusion of tritium in the opposite direction.

The energy barriers for the migration of the different lithium vacancy defects in the {T_i¹⁺:V_Li}⁰ cluster are presented in Table 8. From the table, it is possible to see that the lowest energy barrier is for the V_Li1^–1 → V_Li2 process with an energy of 0.98 eV for the forward jump and 1.00 eV for the reverse jump. Other barriers are larger, with the highest being for the V_Li3^–1 → V_Li1 process. Combining the V_Li1^–1 → V_Li2 process with diffusion barriers for tritium to move around within the Li1 and Li2 vacancies as shown in Table 7, where, with the exception of the diagonal jumps within the octahedron, the barriers are always smaller than 1 eV, giving an overall energy barrier to diffusion of ∼1 eV. This barrier is smaller than the barriers predicted in Tables 5 and 6 for the overall tritium escape from the lithium vacancy sites and to diffuse as an interstitial. This implies that if a tritium ion does become bound to a Li1 or Li2 vacancy in the β-Li₂TiO₃ crystal, the two will diffuse some distance together before the tritium eventually detraps.

Table 8. Calculated Diffusion Energy Barriers for Collaborative Diffusion Mechanisms with Simultaneous Jumps of Li Vacancy and Tritium^a.

	barrier (eV)
pathway	forward	reverse
VLi1–1 → VLi1, T(f)Li10 → T(c)Li1	1.22	1.36
VLi1–1 → VLi2, (f)Li10 → T(a)Li2	0.98	1.00
VLi3–1 → VLi1, T(c)Li30 → T(f)Li1	1.72	1.58
VLi3–1 → VLi2, T(c)Li30 → T(a)Li2	1.29	1.15

^aAlphabets in the brackets represent the exact tritium sites in the Li vacancy.

Conclusions

Energy barriers for the migration of lithium vacancy defects, tritium, and their clusters in β-Li₂TiO₃ have been determined using DFT. From the data presented above, the following conclusions can be drawn:

• There is a small anisotropy in the energy barriers for diffusion of the lithium vacancy defect. For diffusion in the z-direction, the lithium vacancy can undergo a series of Li2 → Li3 and Li3 → Li2 transitions, resulting in an overall barrier of 0.47 eV. This can be compared with diffusion in the xy-plane, where the minimum barrier is 0.50 eV for the Li1 → Li1 process.

• For tritium as an interstitial, the activation energy for diffusion in the xy-plane has an overall energy barrier of 0.52 eV. Diffusion in the z-direction would require the tritium to traverse the mixed cation layer. The barriers to cross this layer are predicted to be greater than 0.98 eV; therefore, the diffusivity of tritium interstitials in the xy-plane is expected to be much greater than along z.

• The barrier for a tritium ion to escape from a {T_i¹⁺:V_Li}⁰ cluster on the Li1 and Li2 sites is in the range of 1.1–1.34 eV and that for the Li3 site is between 1.05 and 1.38 eV.

• The minimum energy barrier for the cooperative mechanism whereby the V_Li^–1 and T_i defects move through the lattice without the tritium detrapping is found to be ∼1 eV for the V_Li1^–1 → V_Li2 process. This is smaller than that calculated for escape from the lithium vacancy sites. Ultimately, this suggests that if a tritium ion becomes bound to a Li1 or Li2 vacancy, the pair will travel through the lattice together.

• Once bound to a lithium vacancy, the energy barrier to further diffusion of the tritium, whether via detrapping from the vacancy or diffusing as a cluster, is ≥1 eV. This is significantly greater than the activation energy for the tritium interstitial (0.47 eV). As the concentration of lithium vacancies is anticipated to increase as the lithium burn-up is increased, this implies that tritium will become more difficult to extract, leading to increased tritium retention later in the pebbles’ lifetime.

The authors declare no competing financial interest.