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ISSN: 2052-5206
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The inverse perovskite BaLiF3: single-crystal neutron
diffraction and analyses of potential ion pathways
Dennis Wiedemann, Falk Meutzner, Oscar Fabelo and Steffen Ganschow
Acta Cryst.
(2018). B74, 643–650
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Acta Cryst.
(2018). B74, 643–650 Dennis Wiedemann
et al.
·The inverse perovskite BaLiF3
research papers
Acta Cryst. (2018). B74, 643–650 https://doi.org/10.1107/S2052520618014579 643
Received 25 July 2018
Accepted 15 October 2018
Edited by R. C
ˇerny
´, University of Geneva,
Switzerland
Keywords: fluoroperovskite; high-temperature
neutron diffraction; bond-valence energy land-
scape; maximum-entropy methods; topological
analysis.
CCDC references:434514; 434515; 434516;
434517
Supporting information:this article has
supporting information at journals.iucr.org/b
The inverse perovskite BaLiF
3
: single-crystal
neutron diffraction and analyses of potential ion
pathways
Dennis Wiedemann,
a
* Falk Meutzner,
b,c
Oscar Fabelo
d
and Steffen Ganschow
e
a
Institut fu
¨r Chemie, Technische Universita
¨t Berlin, Strasse des 17. Juni 135, Berlin, 10623, Germany,
b
Institut fu
¨r
Experimentelle Physik, TU Bergakademie Freiberg, Leipziger Strasse 23, Freiberg, 09596, Germany,
c
Samara National
Research University, Moskovskoye Shosse 34, Samara, 443086, Russian Federation,
d
Institut Laue Langevin, 71 Avenue
des Martyrs, CS 20156, Grenoble Cedex 9, 38042, France, and
e
Leibniz-Institut fu
¨r Kristallzu
¨chtung, Max-Born-Strasse 2,
Doped barium lithium trifluoride has attracted attention as component for
scintillators, luminescent materials and electrodes. With lithium and fluoride, it
contains two possibly mobile species, which may account for its ionic
conductivity. In this study, neutron diffraction on oxide-containing BaLiF
3
single-crystals is performed at up to 636.2C. Unfortunately, ion-migration
pathways could not be mapped by modelling anharmonic ion displacement or by
inspecting the scattering-length density that was reconstructed via maximum-
entropy methods. However, analyses of the topology and bond-valence site
energies derived from the high-temperature structure reveal that the anions can
migrate roughly along the edges of the LiF
6
coordination octahedra with an
estimated migration barrier of 0.64 eV (if a vacancy permits), whereas the
lithium ions are confined to their crystallographic positions. This finding is not
only valid for the title compound but for ion migration in all perovskites with
Goldschmidt tolerance factors near unity.
1. Introduction
Since the early 2000s, fluoride perovskites have attracted
attention because of their prospective application in semi-
conductors, optoelectronics and chemosensorics. More speci-
fically, the inverse cubic perovskite BaLiF
3
is contemporarily
discussed, e.g. as a component for scintillators (Kurosawa et
al., 2017), luminescent materials (Qiang et al., 2016) or
specialized arc-welding electrodes (Wang, 2014).
BaLiF
3
is of the perovskite CaTiO
3
-type and crystallizes in
the space group Pm
33mwith a’3.992 A
˚(Ludekens & Welch,
1952). The fluoride ions form octahedra around the lithium
and cuboctahedra around the barium ions, whereas they
themselves are coordinated in the fashion of a compressed
square bipyramid. The term ‘inverse perovskite’
1
signifies that,
in contrast to the situation in a normal perovskite, the cation
of the higher charge occupies the position with the higher
coordination number (Roy, 1954). The Goldschmidt tolerance
factor (Goldschmidt, 1926), calculated herein with ionic radii
according to Shannon (1976), illustrates why this is the case: t=
0.995 for the inverse configuration shows an almost ideal radii
relation, whereas t= 0.594 (calculated using the radius of a
lithium ion in eightfold coordination) is clearly out of range
ISSN 2052-5206
#2018 International Union of Crystallography
1
Inverse perovskites should not be confused with antiperovskites, in which the
role of cations and anions are interchanged with respect to normal perovskites.
electronic reprint
even for a distorted normal perovskite. Aware of this, Lude-
kens & Welch (1952) assigned BaLiF
3
the correct structure
when first reporting its successful synthesis.
A variety of structural bulk defects in BaLiF
3
have been
computed and discussed, amongst them LiF pseudo-Schottky
defects as the thermodynamically most stable (Jackson et al.,
1996). However, more recent studies indicate that cation
antisite defects, according to equation (1) (Zahn et al., 2011;
Du
¨vel et al., 2018), and fluoride–oxide substitution (from
water vapour or carbon dioxide), according to equation (2)
(Jackson & Valerio, 2002; Qiao et al., 2009), are the most
common intrinsic and extrinsic defects, respectively:
Ba
Ba þLi
Li ¼Ba
Li þLi0
Ba;ð1Þ
2F
FþH2O¼O0
FþV
Fþ2HF:ð2Þ
To our knowledge, the only neutron investigation on
BaLiF
3
so far has been a study of inelastic scattering to record
phonon dispersion curves (Boumriche et al., 1994). Our own
research interest in BaLiF
3
is due to the presence of two
potentially mobile species, Li
+
and F
–
, and their possible
interplay. Based on high-temperature neutron diffraction, we
will herein show that defects are indeed abundant in
Czochralski-grown single crystals. Using topological and
bond-valence methods, we will derive the preferred pathways
for anion and (hypothetical) lithium-ion diffusion, as well as
estimate the associated energy barriers.
2. Experimental
2.1. Sample preparation
A BaLiF
3
single crystal was grown using a Czochralski
technique with automatic diameter control. The raw material,
a mixture of x(BaF
2
) = 43 mol% (single-crystalline chunks,
Korth Kristalle GmbH) and x(LiF) = 57 mol% (optical grade
from GFI Advanced Technologies, Inc.), was premelted in a
platinum boat under flowing HF gas according to the proce-
dure described in more detail by Baldochi et al. (1996). The
crystal was pulled from an inductively heated platinum
crucible along the h111idirection at a rate of 2.0 mm h
1
.A
variable crystal rotation between 10 and 25 min
1
was set to
improve mixing of the melt and to avoid local occurrence of a
constitutionally supercooled melt. The growth chamber was
rinsed with a mixture of argon and CF
4
(’= 10 vol.%) during
the whole process. The crystal obtained was 18 mm in
diameter and 75 mm long, and was colourless, transparent and
free of macroscopic defects.
For neutron diffraction, a {100}-oriented cube of 5 mm
5mm5 mm was cut from a slice of the crystal. A shard-like
fragment of the same slice with a suitable size was subjected to
X-ray diffraction (for details, see xS1 in the supporting infor-
mation).
2.2. Elemental analyses
The oxygen content was measured using a Leco EF-TC 300
N
2
/O
2
analyser (hot-gas extraction). The absence of hydro-
xides was confirmed using a Thermo Finnigan FlashEA 1112
NC analyser. The content of barium, fluorine and lithium was
quantified after alkali fusion with sodium and potassium
carbonates (= 1:1) in a platinum crucible. The barium
content was measured gravimetrically following a standard
protocol (Bracher et al., 2003) and corrected against a refer-
ence specimen. The contents of fluorine and lithium were
measured using a Dionex DX-120 ion chromatograph and a
Thermo Fisher Scientific iCAP 6300 duo optical emission
research papers
644 Dennis Wiedemann et al. The inverse perovskite BaLiF
3
Acta Cryst. (2018). B74, 643–650
Table 1
Experimental details.
For all structures: BaLiF
2.8
O
0.1
,M
r
= 199.1, cubic, Pm
33m,Z=1,= 0.05 mm, crystal size (mm): 5.00 5.00 5.00. Experiments were carried out with neutron
radiation, = 0.83860 A
˚using the hot-neutron four-circle diffractometer D9. Extinction was corrected for using a SHELXL-type expression (Sheldrick, 2015),
F
c*
=kF
c
[1 + 0.001xF
c
2
3
/sin(2)]
1
4
.
26.8C 412C 555C 636.2C
Crystal data
a(A
˚) 3.9978 (2) 4.0406 (2) 4.0581 (3) 4.0763 (4)
V(A
˚
3
) 63.90 (1) 65.97 (1) 66.83 (2) 67.73 (2)
Data collection
T
min
,T
max
0.975, 0.978 0.975, 0.978 0.976, 0.979 0.976, 0.979
No. of measured, independent
and observed [I>2(I)] reflections
364, 82, 81 286, 68, 67 362, 82, 74 343, 78, 70
R
int
0.019 0.020 0.023 0.021
(sin /)
max
(A
˚
1
) 1.008 0.909 0.993 0.989
Refinement
R[F
2
>2(F
2
)], wR(F
2
), S0.012, 0.046, 1.14 0.011, 0.038, 0.85 0.015, 0.054, 1.02 0.011, 0.045, 0.98
No. of reflections 82 68 82 78
No. of parameters 6 7 7 7
u0.03324 0.03616 0.03942 0.03224
max
,
min
(fm A
˚
3
) 0.35, 0.25 0.13, 0.12 0.12, 0.20 0.08, 0.13
Extinction coefficient 6.6 (4) 5.0 (3) 5.3 (4) 4.6 (3)
Weighting scheme based on measured s.u.’s, w=1/[
2
(I)+(uI)
2
]. Computer programs: NOMAD (ILL, 2016), RAFD9 (Filhol, 1988), RACER (Lehmann & Larsen, 1974; Wilkinson et al.,
1988), SUPERFLIP (Palatinus & Chapuis, 2007), JANA2006 (Petr
ˇı
´c
ˇek et al., 2014), VESTA (Momma & Izumi, 2011).
electronic reprint
spectroscope with an inductively coupled plasma, respectively.
For the latter, the emission line at 610.3 nm was evaluated.
Anal. calcd for BaLiF
2.8
O
0.1
: Ba, 68.99; F, 26.72; H, 0.00; Li,
3.49; O, 0.80%. Found: Ba, 68.33; F, 26.73; H, 0.00; Li, 3.40; O,
0.14 (5)%.
2.3. High-temperature neutron diffraction
Diffraction data of a single crystal in vacuum glued to a
titanium pin with high-temperature alumina-filled ceramic
adhesive (Ceramabond 552) were collected at the hot-neutron
four-circle diffractometer D9 at ILL (Grenoble, France) with
Cu(220)-monochromated radiation (take-off angle 2
M
=
56.30) using !and !–2scans. Sample and beam stability
were ensured by monitoring the (111) reflection in intervals of
80 reflections. Data sets were acquired at 26.8 (3), 412 (3),
555 (3) and 636.2 (9)C (Wiedemann, Fabelo et al., 2017). Data
reduction was performed using RACER (Lehmann & Larsen,
1974; Wilkinson et al., 1988). An analytical absorption
correction using Gaussian integration was employed with
JANA2006 (Petr
ˇı
´c
ˇek et al., 2014). Structures were solved with
SUPERFLIP (Palatinus & Chapuis, 2007) using a charge-
flipping algorithm.
2.4. Refinement details
Structures were refined with JANA2006 against F2
odata
using the full-matrix least-squares algorithm. A SHELXL-
type extinction correction (Sheldrick, 2015) was applied. A
weighting scheme based on standard uncertainties with a fixed
instability factor (as derived from data merging) was
employed. All ions were refined with fixed occupations in the
final stage (see Table 1 for details).
Cationic displacement was modelled isotropically
(constrained by site symmetry), whereas anharmonic displa-
cement of the anions was modelled using the Gram–Charlier
formalism. We attempted to refine parameters of the fourth
and sixth order (those of the third and fifth order had to be
zero for reasons of site symmetry) keeping only significant
ones [D
ijkl
>3(D
ijkl
)] in the refinement.
2.5. Maximum-entropy reconstruction
For the visualization of scattering length density (SLD)
distributions, they were reconstructed from the final structure
factors put out by JANA2006 using maximum-entropy
methods (MEM) as implemented in Dysnomia 0.9 (Momma et
al., 2013). The unit cell was divided into 192 192 192
voxels and set to contain the formula unit BaLiF
2.8
O
0.1
.
Starting from a uniform intensity prior, the limited-memory
Broyden–Fletcher–Goldfarb–Shanno (L-BFGS) algorithm
(Nocedal, 1980) was employed with uncertainties augmented
by E= 0.5 and relative weights
2
,
4
, and
6
for the gener-
alized constraints chosen to give final central moments near
unity (
n
= 0 for orders n> 6; for details, see Table S2).
2.6. Topological analyses
Analysis of the procrystal void surface under exclusion of
different ions was performed on ordered supercells using
CrystalExplorer3.1 (Wolff et al., 2012). For Voronoi–Dirichlet
partitioning (VDP) analysis, ToposPro5.3 (Blatov et al., 2014)
was employed.
2.7. Bond-valence site energies (BVSE)
Bond-valence site energies (BVSE) were calculated using a
beta version of the program softBV with a resolution of 0.1 A
˚
based on published data (Adams & Rao, 2011). Substitu-
tionally disordered crystals are best handled as statistically
equivalent ordered supercells. Because of this, we computed
the BVSE for lithium and fluoride ions using a randomly
created 4 44 supercell incorporating 180 F
(F
F
), six
O
2
(O
F
0) and six vacancies (V
F) to mimic the composition
determined via diffraction (see xS2). This resulted in the sum
formula LiBaF
2.8125
O
0.09375
(see Fig. S2).
3. Results and discussion
3.1. Structure and composition
Unlike with X-ray diffraction, the position of a lithium ion
(scattering factor as high as
1
3of fluorine nucleus) can be
clearly established using neutrons. In the present case, the
models derived from structure solution (see Fig. 1) were
identical to the established ones. However, we found that
allowing occupancy reduction for the anion position improved
the fit drastically (e.g. by R
1
= 0.0114, wR
2
= 0.0441 and
S= 1.08 for all reflections at 26.8C). At the same time, we
observed neither a reduction of cation occupancies nor
interstitial chunks of difference density that we could refine as
additional fluoride positions. Furthermore, there is no indi-
cation of ample antisite defects, which would lead to a
reduction of observed scattering length densities at both
cation positions. We therefore concluded that oxide substitu-
tion (instead of Schottky or Frenkel defects) was the only
appropriate model in our case. This view is corroborated by
the results from barium, lithium and fluoride analyses (see
x2.2): There is no significant deficiency in cations, which
excludes ample Schottky defects, whereas the fluoride content
is clearly lower than expected. Test refinements resulted in
compliant fluoride/oxide occupancies (see xS2) so that we
decided to fix the composition at BaLiF
2.8
O
0.1
. We attribute
research papers
Acta Cryst. (2018). B74, 643–650 Dennis Wiedemann et al. The inverse perovskite BaLiF
3
645
Figure 1
Crystal structures at different temperatures according to neutron
diffraction (grey: barium, pink: lithium, green: fluoride/oxide ions;
displacement ellipsoids at 90% probability, isosurfaces for probability
density of 0.1 A
˚
3
; unit cell in black). Caveat: for 412C and above, the
anharmonically displaced anions are represented by a probability density
isosurface that is not directly comparable to harmonic displacement
ellipsoids.
electronic reprint
the shortfall of the oxygen analysis to inaccuracies at very low
contents.
Test refinements showed no significant anharmonic displa-
cement of the lithium ion and only one significant anharmonic
parameter (fourth order) for the anions at 412C and above,
which leads to an ever so slightly diamond-like shape of the
probability-density isosurface (see green anions in Fig. 1). This
observation is in accordance with the direction of optical
phonon modes involving fluoride ions in perovskites (Naka-
gawa et al., 1967) and with the statement that ‘BaLiF
3
seems to
be the least anharmonic [ ...] of the AMF
3
fluo[ro]per-
ovskites’ (Boumriche et al., 1989). This means the small
anharmonicity observed herein can be satisfactorily explained
with lattice vibrations alone, i.e. not involving ion migration.
This is further substantiated by the fact that, even at very low
isovalues, the probability density surfaces describing the
anionic displacement do not connect to continuous pathways.
An interesting peculiarity was observed in the plot of the
unit-cell volume and the equivalent atomic displacement
parameters against temperature (see Fig. 2). Although they
grow in a monotonic fashion, the increase between 555 and
636.2C is substantially larger/smaller than expected from
linearity for the volume/displacement, respectively. The
melting point of 826C (Neuhaus et al., 1967) is well above the
experimental temperatures and cannot account for this irre-
gularity. Thermodilatometric analysis on supposedly oxide-
free single crystals grown in CF
4
atmosphere did not show
such an anomaly (Bensalah et al., 2003). A linear expansion
coefficient of 3.33 10
5
C
1
between 100 and 500C was
reported, whereas we estimate 2.83 (6) 10
5
C
1
from
neutron diffraction between 26.8 and 555C (linear approx-
imation).
3.2. MEM-reconstructed scattering-length density (SLD)
To corroborate the anharmonic model of displacement, we
had a closer look at the SLD (i.e. the neutron analogue of the
electron density probed with X-ray diffraction) that we
reconstructed using MEM. These methods yield the maximum
variance of calculated structure factors within standard
deviations of observed structure factors, in this way producing
density maps, which contain less artefacts and are less prone to
misinterpretation (Wiedemann et al., 2016; Wiedemann, Islam
et al., 2017). These maps (see Fig. 3 for examples) show that
the assessment is indeed correct: the pieces of SLD associated
with the atomic positions do not connect to each other above
noise level, thus indicating vibration instead of migration. The
diamond-like anion distribution is also reproduced. In addi-
tion, we found small chunks of positive SLD at (1
2, 0, 0) for
isovalues lower than 3.95 (26.8C) and 0.85 fm A
˚
3
(636.2C).
Their associated integral scattering lengths were, however, so
small that a refinement as additional anion led to insignificant
and/or negative occupation.
3.3. Procrystal-void analysis
As we were unable to map ion migration in BaLiF
3
single
crystals experimentally, we have subjected the model derived
from neutron diffraction at high temperature to state-of-the-
art topological methods. In this way, we want to identify and
assess the most probable pathways. One of the most easy-to-
use method is the procrystal-void analysis, which maps the
‘voidest’ parts of the structure according to a procrystal model
(spherically averaged, diffuse electron density).
The results show that the pathways accessible to anions run
between adjacent positions roughly along the edges of the
LiF
6
coordination octahedron [see Fig. 4(a)]. Around the
equilibrium positions, empty space allows vibration in a
diamond-like shape without strong distortion. Hypothetical
lithium pathways between adjacent positions [see Fig. 4(b)]
would run along (1
4,1
4,1
4) and (1
2, 0, 0). However, the high
isovalue of 0.0135 a.u., at which the pathways connect, indi-
cates that they are in fact too narrow to conduct lithium ions.
The only viable path for these ions would be a linear one
through an anion vacancy [see Fig. 4(c)], leading to localized
jumps instead of long-range transport.
research papers
646 Dennis Wiedemann et al. The inverse perovskite BaLiF
3
Acta Cryst. (2018). B74, 643–650
Figure 3
Isosurface representation of the SLD of 1fmA
˚
–3
at (a) 26.8C and (b)
636.2C with (100) section through a Ba–F plane (blue: positive, pink:
negative SLD; section in greyscale, white: maximum, black: near-zero
SLD; ions with arbitrary radii; unit cell in black).
Figure 2
Unit-cell volumes Vand equivalent displacement parameters U
eq
for all
ions at different temperatures (dotted and dashed lines merely guides to
the eye, error bars smaller than symbols).
electronic reprint
3.4. Voronoi–Dirichlet partitioning (VDP)
VDP offers a deeper insight into the conduction pattern and
allows geometrically rationalizing the viability of different
pathways. For the tentative lithium conduction pattern, three
types of elementary voids were detected [see Fig. 5(b) and
Table 2], all of which should be slightly too small to accom-
modate lithium ions: they are characterized by spherical-
domain radii R
sd
< 1.30 (1) A
˚– a size which would be typical
for lithium ions in fluoride environments (Blatov, 2004). As
the centre of the smallest void type is in fact the crystal-
lographic lithium position, it can actually host a lithium ion.
Accordingly, we may not use this criterion strictly in our case.
Nevertheless, all voids apart from the crystallographic lithium
position exhibit at least one strong interaction with a barium
ion (solid angle > > 10% 4) and are thus unfit to host a
lithium ion. All elementary channels connecting the voids are
too small for conduction: their radii R
ch
are well outside of
R
ch
1.76 A
˚(90% of typical Li–F distance
2
). In summary, the
elementary voids and channels are found to be probabilistic at
best; the crystal is not a lithium-ion conductor. If, however, a
lithium ion were to migrate, a path from one position Li1 to an
adjacent one Li10via the VDP-determined void centres ZA2
and ZA1 would be the least disfavoured.
In contrast to this, only one type of elementary void is found
in the anion conduction pattern. Interestingly, it does not
coincide with the crystallographic anion position, but four
voids are symmetrically arranged around the latter. They are
fit to accommodate fluoride [R
sd
= 1.59 (9) A
˚]oroxideions
[R
sd
= 1.55 (12) A
˚] (Blatov & Serezhkin, 2000). The elemen-
tary channels are of two types: those connecting to other voids
associated with the same anion position and those connecting
to voids associated with an adjacent anion position. The
former are determined by two interactions with lithium ions
and one interaction with a barium ion, the latter vice versa.To
assess the accessibility of the channels between neighbouring
anions, using a weighted average of the anion–cation distances
(F–Li: 1.76, O–Li: 1.85, F–Ba: 2.39, O–Ba: 2.48 A
˚; 90% of the
typical distances
2
) is in order. The interaction-weighted
distances of 2.18 and 2.27 A
˚for F
and O
2
, respectively, are
only slightly greater than the channel radius. Thus, the crystal
can be considered a moderately viable anion conductor.
research papers
Acta Cryst. (2018). B74, 643–650 Dennis Wiedemann et al. The inverse perovskite BaLiF
3
647
Table 2
Size and suitability of elementary voids and channels at room
temperature and 636.2C.
Elementary
void Position
Wyckoff
position
R
sd
(A
˚)
(r.t., 636.2C)
N†
(> 10% 4)
Cation substructure
ZA1 1
4,1
2,0 12h1.574, 1.605 0
Ba–F substructure
ZA1 1
2,0,0 3d1.240, 1.264 2
ZA2 1
4,1
4,1
48g1.264, 1.288 1
ZA3 1
2,1
2,1
2(Li1) 1b1.240, 1.264 0
Elementary
channel
R
ch
(A
˚)
(r.t., 636.2C)
Cation substructure
ZA1–ZA1 2.120, 2.162
Ba–F substructure
ZA1–ZA2 1.632, 1.664
ZA2–ZA3 1.632, 1.664
† Number of interactions with ions with charge of the same sign. Criterion is the
percentage of the full solid angle subtended by the respective ion.
Figure 4
Procrystal-void surfaces at 636.2Cof(a) the [LiBa]
3+
framework at
pro
=
0.0081 a.u., the [BaF
3–2x
O
x
]
framework in (b) an averaged cell at
pro
=
0.0135 a.u. and (c) an ordered 2 21 supercell at
pro
= 0.0049 a.u.
(grey: barium, pink: lithium, green: fluoride, red: oxide ions; ellipsoids of
99% probability, harmonic contribution only).
Figure 5
Most probable pathways of (a) anion diffusion in the [LiBa]
3+
framework
and of (b) lithium diffusion in the [BaF
3–2x
O
x
]
framework according to
VDP analysis (grey: barium, pink: lithium, green: fluoride/oxide ions,
black: major voids and channels; ions with ionic radii; unit cell in black).
2
In the literature, these values are derived inconsistently. While mostly atomic
radii according to Slater (1964) are employed [e.g. Li–O distance by Anurova
et al. (2008)], they are sometimes calculated from ionic radii according to
Shannon (1976) [e.g. Na–O distance by Blatov et al. (2006)]. We stuck to the
former procedure for reasons of comparability.
electronic reprint
3.5. Bond-valence energy landscape (BVEL)
In addition to the topology of preferred pathways, we were
interested in an estimate of the associated migration barriers –
a piece of information that bond-valence methodology offers.
It describes the empirical mathematical relationship between
bond strength and bond length (Brown, 2009). The bond
valence sof a pair of atoms is most commonly represented by
an exponential term with the statistically evaluated (and
tabulated) bond-valence parameters R
0
and band the
measured interatomic distance R:
s¼exp ðRR0Þ=b
:ð3Þ
By summing up all bond valences of an atom and its neigh-
bours, the bond-valence sum (BVS) of the respective atom is
calculated. In an ideal case, the BVS is equal to its oxidation
state. This concept works best with ionic materials but also
reasonably well with covalent compounds.
Besides the verification of crystal structures and the
prediction of positions of light elements such as lithium and
hydrogen, the analysis and prediction of ionic conduction is
one of the main applications of bond-valence methodology.
For this purpose, the concept was successfully expanded to
implement outer-coordination-sphere atoms (Adams, 2001)
and introduce bond-valence sum mismatch (BVSM) between
measured and ideal values (Adams, 2006). More recently, this
BVSM was translated into bond-valence site energies (BVSE)
using Morse-like potentials (Adams & Rao, 2011). The BVSE
are calculated for every point in the unit cell. Their difference
to the minimum, which is usually found at a crystallographic
position of the ion type under investigation, is an estimate of
the energy necessary to displace it to that point (i.e. the
migration energy). Together, the BVSE form the BVEL that
shows the way of preferred migration as path of lowest energy
difference between atomic sites.
To model anion vacancies and oxygen doping, we have
performed calculations in a representative supercell (see
Fig. S2) with the following results:
(a) The barrier for linear lithium migration through an
anion vacancy is 2.08 eV [see Fig. 6(a), right side]. It is lowered
to 2.07 eV if an oxide ion is coordinated to a neighbouring
lithium ion or even 2.00 eV if an oxide ion is directly adjacent
to the path [see Fig. 6(a), left side]. Accordingly, this could
lead to local hopping at very high temperatures.
(b) With 3.04 eV, the barrier for lithium migration in the
ideal structure via (1
4,1
4,1
4)and(
1
2, 0, 0) is too high to overcome
in reality [see Fig. 6(b)].
(c) The barrier for fluoride migration via edges of LiF
6
octahedra is 0.64 eV [see Fig. 6(c)], enabling this process at
medium temperatures.
However, these energies must be considered estimates as
they account for neither defect formation nor relaxation near
defects. These shortcomings can only be remedied using
quantum-chemical computation.
4. Conclusion
Neutron diffraction indicated ample anion vacancies to be
present in Czochralski-grown single crystals of the tentative
composition BaLiF
3
. We attribute this to oxide substitution,
which is abundant in fluoride perovskite single crystals
(Chadwick et al., 1983) and can as well happen after growth
(Jackson & Valerio, 2002). While the anomaly in the
temperature dependence of cell volume and atomic displace-
ment parameters might be associated with processes like the
dissociation or healing of defect pairs [e.g. V
FO0
For V
FF0
i
(Ziraps et al., 2001)], measurements with more temperature
points have to be conducted first to rule out an artefact.
A model of the displacement of lithium ions and anions and
a direct inspection of MEM-reconstructed SLD show anhar-
monicity for the anions at high temperature. However, this is
adequately explained with lattice vibrations; ion migration, as
would be indicated by connection of smeared SLD to
continuous paths, was not observed. This means that thermally
activated ion migration in the (in spite of the defects) highly
ordered single-crystalline state is too sluggish to be mapped
using this method, i.e. jumps are performed by too few indi-
vidual ions during the experiment.
A variety of studies has shown that there is ionic conduction
in powders as well as single crystals of BaLiF
3
. Our analyses of
the procrystal voids, the VDP, and the BVSE in the high-
temperature structure
3
agree in the assessment that anions are
potentially mobile in this structure, whereas lithium ions
should be static. This corroborates the assumption that BaLiF
3
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648 Dennis Wiedemann et al. The inverse perovskite BaLiF
3
Acta Cryst. (2018). B74, 643–650
Figure 6
Details of the BVEL for the lithium ions at potential energies of (a)2.1
and (b) 3.2 eV as well as (c) for the fluoride ions at a potential energy of
0.7 eV (grey: barium, pink: lithium, green: fluoride, red: oxide ions, blue:
BVSE isosurface; ions with ionic radii).
3
It should be noted that these methods are invariant to the amount of anion
vacancies and substitution as they only depend on atomic positions and
crystallographic symmetry.
electronic reprint
is an anion conductor (Kamata et al., 1998; Rush et al., 2001)
and precludes lithium ions from playing a role in long-range
transport. Any mobility of these (Du
¨vel et al., 2010, 2018)
would be confined to anion vacancies, antisite defects or
surfaces (Kunkel et al., 2014). As these findings are founded on
purely geometrical considerations (including ionic radii), they
are valid not only for BaLiF
3
but also for all perovskites with
similar ratios of ionic radii, i.e. for those with a near-ideal
Goldschmidt tolerance factor t’1.
All three methods account for the diamond-like shape of
anharmonic anion displacement asserted using neutron
diffraction. This means that the directions of vibration corre-
late with the ‘least occupied’ parts of the crystal. Furthermore,
the same pathway for anion migration is predicted: nearly
linear and roughly along the edge of the LiF
6
octahedron.
Inspection of the BVSE gives an estimate of 0.64 eV as the
associated migration barrier. This is within reasonable range
from the activation energy of 0.5 eV attributed to intrinsic
conduction in BaLiF
3
single-crystals (Rush et al., 2001; Du
¨vel
et al., 2010). The difference indicates, on one hand, that defect
formation plays indeed no role in the measured activation
energies (else, they would have to be considerably greater)
and, on the other hand, that local relaxation account for the
lower barrier compared to the static BVSE model.
These considerations make it evident that the formation,
existence, and healing of defects (especially anion vacancies)
have a major influence on ion migration in the tightly packed
perovskites. For this reason, neutron diffraction experiments
on powders are ongoing.
Acknowledgements
We thank Professor Martin Lerch (Technische Universita
¨t
Berlin) for productive discussion, Dr Adelheid Hagenbach
and Professor Ulrich Abram (Freie Universita
¨t Berlin) for the
collection of X-ray diffraction data, Ms Martina Rabe
(Leibniz-Institut fu
¨r Kristallzu
¨chtung, Berlin) for crystal
growing, Ms Claudia Kuntz, Ms Ines Pieper and Ms Juana
Krone (Technische Universita
¨t Berlin) for lithium, fluorine
and hydrogen analyses. Furthermore, we gratefully acknowl-
edge support from Dr Va
´clav Petrı
´c
ˇek (Institute of Physics,
AV C
ˇR, Prague) in using the program JANA2006 and from
Professor Stefan Adams (National University of Singapore)
for providing a beta version of softBV.
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