674
Z. Kristallogr. 223 (2008) 674–689 / DOI 10.1524/zkri.2008.1030
# by Oldenbourg Wissenschaftsverlag, München
Crystal chemistry and metal-hydrogen bonding in anisotropic
and interstitial hydrides of intermetallics of rare earth (R)
and transition metals (T), RT3 and R2T7 1
Volodymyr A. Yartys*, I, Ponniah VajeestonII, Alexander B. RiabovI, Ponniah RavindranII, Roman V. DenysI,
Jan Petter MaehlenI, Robert G. DelaplaneI and Helmer FjellvågII
I
II
Institute for Energy Technology, P.O. Box 40, 2027 Kjeller, Norway
Department of Chemistry, P.O. Box 1033, University of Oslo, Blindern, 0315 Oslo, Norway
Received May 16, 2008; accepted November 7, 2008
Metal hydrides / Crystal structures / Bonding mechanism /
Nickel / Rare earth metals
Abstract. Hydrides of the RNi3- and R2Ni7-based (R ¼
light rare earth element) intermetallics exhibit novel structural features. Structures of these hydrides, including
CeNi3D2.8, La2Ni7D6.5, LaNi3D2.8, and Ce2Ni7D4.7, are
formed via a huge volume expansion occurring along a
single crystallographic direction. Unique structural features
during the formation of the hydrides include: (a) The lattice expansion proceeds exclusively within the RNi2 slabs
leaving the RNi5 slabs unmodified. Such expansion,
60% along [001] for the Laves layers, is associated with
occupation of these slabs by D atoms; (b) New types of
interstitial sites occupied by D are formed; (c) An ordered
hydrogen sublattice is observed. In the present work we
give (a) a review of the crystal chemistry of the conventional, interstitial type hydrides formed by RT3 and R2T7
intermetallic compounds (R ¼ rare earths; T ¼ Fe, Co, Ni)
as compared to the unusual features of the crystal chemistry of anisotropic hydrides formed by the RNi3 and R2Ni7
intermetallics and (b) studies of the interrelation between
structure and bonding in anisotropic hydrides by performing density functional calculations for CeNi3 and Ce2Ni7
intermetallic alloys and their corresponding hydrides.
These studies provide an understanding of the bonding
mechanism in the hydrogenated compounds which causes
a complete anisotropic rebuilding of their structures. From
DOS analysis, both initial intermetallics and their related
hydrides were found to be metallic. Bader topological analysis for the non-hydrogenated intermetallics showed that
Ce atoms donate in average of almost 1.2 electrons to the
Ni sites. Hydrogenation increases electron transfer from
Ce; its atoms donate 1.2–1.6 electrons to Ni and H. The
Charge Density Distribution and Electron Localization
Function for the Ce2Ni7D4.7 phase clearly confirm that the
interaction between the Ce and Ni does not have any sig1
Presented at the International Symposium on Metal-Hydrogen
Systems MH2008. Fundamentals and Applications. Reykjavik, Iceland. June 24–28, 2008. Lecture F4-O-4.
* Correspondence author (e-mail: Volodymyr.Yartys@ife.no)
nificant covalent bonding. Ni is bonded with H via forming spatial frameworks ––H––Ni––H––Ni–– where H atoms
accumulate an excess electron density of 0.5e . Thus,
the tetrahedral or open saddle-type NiH4 coordination observed in the structures of these hydrides is not associated
with the formation of [Ni0H41]4 complexes containing a
hydrido-ion H1. In the structural frameworks there are
terminal bonds Ni––H, bridges Ni––H––Ni, and the bonds
where one H is bound to three different Ni. These spatial
ordered frameworks stand as the principal reason for the
anisotropic changes in the structural parameters on hydrogenation. Another unique feature of anisotropic hydrides is
the donation of electrons from nonhydrogenated RNi5 parts
to hydrogen in RNi2 slabs stabilising these fragments.
Introduction
Hydrogen absorption by intermetallic compounds (IMC)
results in a storage of atomic hydrogen in the metal lattice frequently reaching a high ratio of H/M (>1) and a
high volume density of the stored hydrogen compared to
compressed or liquefied hydrogen gas. Intermetallic hydrides exhibit a close interrelation between crystal chemistry and hydrogen sorption behaviour allowing alteration
and optimisation of their H storage performance. Hydrogen accommodation by the metal lattice is typically accompanied by modest (few percent) changes of the interatomic metal–metal distances. H atoms enter the
interstices, which are originally available in the virgin
intermetallics. However, this “typical” case does not cover a large group of very interesting and so far insufficiently studied compounds, the so-called “anisotropic”
hydrides. In such hydrides, a huge expansion proceeds in
a specific crystallographic direction upon hydrogenation
and leads to a dramatic differentiation of the properties
of the hydrides along the direction of the expansion and
normal to it. This paper will review recent data on the
“anisotropic” hydrides of intermetallic compounds in
comparison with conventional interstitial intermetallic hydrides formed by chemically related intermetallic com-
Crystal chemistry and metal-hydrogen bonding
Fig. 1. Stacking of the RT2 Laves-type and RT5 Haucke-type layers
(coloured) in the hybrid crystal structures of CeNi3, LaNi3 and
Ce2Ni7 types (ac projection). The structures contain nets if three
types stacking along [001] including RT2 (plain, inside the RT5 type),
R2T (buckled, inside the RT2 type) and T3 (plain, on the borders between RT5 and RT2 and RT5 and RT5).
pounds of rare earth elements with Ni/Co/Fe. Results of
electronic structure calculations will be presented for anisotropic hydrides, formed by CeNi3 and Ce2Ni7 intermetallic compounds and discussed as related to their unusual structural features.
The intermetallics in the systems of rare earth metals
(R) with Ni, Co or Fe (T), are frequently formed between
compositions RT2 (Laves compounds) and RT5 (Haucke
phases). Their composition RTa (2 < a < 5) can be achieved
from a combination of RT5 ( n) and R2T4 ( m) units.
These include, for example, R2T4 + RT5 ¼ 3 RT3 and
R2T4 + 2 RT5 ¼ 2 R2T7 [1, 2]. The intermediate compounds crystallise with several types of structures, which
are built from the slabs of Laves and Haucke types stacking along the hexagonal/trigonal c-axis. Consequently,
their structures are considered as hybrid ones. Particular
focus of this paper will be on a review of the data obtained
for R(Ni,Co,Fe)3 and R2(Ni,Co)7 compounds. These hybrid
structure types include PuNi3, CeNi3, Ce2Ni7 and Gd2Co7
[1]. As example, crystal structures of the CeNi3, PuNi3 and
Ce2Ni7 types are shown in Fig. 1 as an alteration along
[001] of the RT5 CaCu5-type (coloured) and RT2 Lavestype slabs. Hydrogen interaction with these hybrid intermetallic structures has been studied rather extensively (see
Table 1). These studies include also neutron powder dif-
675
fraction investigations, thus allowing to determine the structure of the hydrogen sublattice in the deuterated materials.
The known crystal structures of isotropic and anisotropic
hydrides of such compounds are presented in Table 2.
Hydrogenation of intermetallic compounds RT3 and
R2T7 proceeds via two or three different mechanisms. A
two-step hydrogen uptake occurs for the RNi3, RCo3 and
R2Co7 intermetallics leading to the formation of lower hydrides (dihydrides RNi3H1.2-2.0, RCo3H1.3–2.1 and trihydrides R2Co7H1.5–2.7) prior to the formation of saturated
tetrahydrides RNi3H3.4–4.3, RCo3H3.6–4.6 and hexahydrides
R2Co7H5.8–6.6. Continuous increase of the H content in the
RFe3-based hydrides proceeds within a single phase area
and gives hydrides with H/RFe3 changing from 1.5 to 4.8.
An alternative mechanism of the formation of saturated
hydrides is observed for the compounds of La and Ce,
where a single-step hydrogen absorption process leads to
the formation of R(Ni,Co)3H2.7–6.0 and R2(Ni,Co)7H4.1–6.5.
From a crystallographic point of view, these schemes
have distinct differences in the way the intermetallic structures are transformed into the hydrides and can be classified in the following three groups:
Type I. Lower hydrides RNi3H1.2–2.0, RCo3H1.3–2.1 and
R2Co7H1.5–2.7. Moderate expansion of the original lattices
proceeds anisotropically, along [001]. Linear expansion of
6.5–11.5% corresponds to the volume expansion of 6.9–
12.2%. DV/at. H ¼ 2.1–4.6 A3.
Type II. Higher hydrides RNi3H3.4–4.3, RCo3H3.6–4.6
and R2Co7H5.8–6.6 are formed by H uptake by the lower
hydrides RNi3H1.2–2.0, RCo3H1.3–2.1 and R2Co7H1.5–2.7.
Nearly similar lattice expansion proceeds along [001] and
in the basal plane. Thus, lattice expansion is rather
isotropic and gives volume increase of 11.8–26.6%. DV/
at. H ¼ 2.3–4.1 A3 .
For the isostructural RFe3 intermetallics, higher values
of the maximum hydrogen storage capacity are achieved,
up to 4.8 at. H/f.u. in the case of YFe3. The tetrahydrides
RFe3H4 are similar to the RCo(Ni)3H4 hydrides in their
crystallographic characteristics; lattice expansion proceeds
both in basal plane and along [001]. Hydrogenation leads
to a more pronounced enlargement of c compared to a;
thus, c/a increases to 4.83–5.02 from the original 4.73–
4.90 for the intermetallic compounds. If hydrogenation
temperatures exceed a certain critical temperature, continuous phase transformations from hydride with H content of
1.5–1.8 at. H in DyFe3 and ErFe3-based hydrides to the
saturated values of 4.0–4.2 take place. In such a case a
change in hydrogen content in RFe3Hx (M ¼ Dy, Er) [20,
21] is accompanied by a continuous increase in the unit
cell dimensions. This does not lead to significant changes
in c/a or specific volume of H, DV/at. H, which remains
in the window 2.3–3.8 A3/at. H. This range of values is
lower than for RNi3H3.4-4.2 and RCo3H3.6-4.0 tetrahydrides
(3.0–4.1 A3/at. H), as the unit cell volumes of the initial
intermetallics are much larger for RFe3 compared to the
corresponding compounds Ni or Co.
Type III. Hydrides of La and Ce compounds,
R(Ni,Co)3H2.7–5.2 and R2(Ni,Co)7H4.1–6.5, are formed by
anomalously high linear expansion along [001] reaching
35.8%. Since the basal plane remains practically unchanged, this gives similar values for the volume expan-
676
V. A. Yartys, P. Vajeeston, A. B. Riabov et al.
Table 1. Crystallographic characteristics of RT3 and R2T7 compounds and their hydrides.
IMC
Type*
a
Alloy
c
c/a
Hydride
c
H/f.u. a
Da/a, %
Dc/c, %
DV/V, %
DV/at. H, A3
Ref.
c/a
RNi3 with PuNi3 structure type
CaNi3
YNi3
YNi3
LaNi3
II
I
II
III
5.030
4.977
4.973
5.082
24.27
24.44
24.37
25.09
4.83
4.91
4.90
4.94
4.6
1.6
4.0
2.8
GdNi3
GdNi3
TbNi3
DyNi3
II
III
II
I
II
I
II
I
I
II
I
I
I
II
II
II
I
III
II
5.009
5.009
4.979
4.980
24.57
24.57
24.48
24.44
4.91
4.91
4.92
4.91
4.953
24.21
4.89
4.954
24.33
4.91
4.951
4.948
24.27
24.29
4.90
4.91
4.943
24.28
4.91
4.935
4.971
5.034
24.26
24.54
24.51
4.92
4.94
4.87
2.0
3.0
4.2
2.2
3.4
1.8
3.6
1.3
1.8
3.7
1.9
1.23
1.97
3.75
4.0
5.0
2.0
2.57
4.27
HoNi3
HoNi3
ErNi3
ErNi3
ErNi3
TmNi3
CeY2Ni9D7.7
LaY2Ni9D12.8
5.444
4.987
5.267
8.6408
b ¼ 4.9281
b ¼ 90.85
5.182
4.914
5.302
5.039
5.280
4.991
5.305
4.986
5.027
5.258
4.980
4.972
5.046
5.240
5.271
5.294
4.981
4.872
5.396
26.56
26.82
26.57
32.774
4.88
5.38
5.05
6.59
6.65
8.2
0.2
5.9
––1.8
––3.0
9.4
9.7
9.0
30.6
28.2
13.2
22.3
24.3
3.6
4.6
3.2
5.4
[3]
[4]
[5]
[6]
24.81
31.41
26.78
26.67
26.74
26.12
26.70
26.08
26.24
26.71
25.85
25.90
26.16
26.61
26.65
26.7
25.85
31.312
26.885
4.79
6.39
5.05
5.29
5.07
5.23
5.03
5.23
5.22
5.08
5.19
5.21
5.18
5.08
5.06
5.04
5.19
6.43
4.98
3.5
––1.9
6.5
1.2
6.0
0.8
6.9
0.63
1.45
6.1
0.6
0.5
2.0
5.9
6.6
7.1
0.9
––2.0
7.2
1.0
27.8
9.4
9.1
9.4
7.9
10.0
7.2
7.9
9.8
6.5
6.6
7.7
9.5
9.8
10.0
6.6
27.6
9.7
8.1
23.0
24.0
11.9
23.0
9.6
20.4
8.6
11.0
23.6
7.8
7.7
12.0
22.8
24.8
26.1
8.6
22.6
26.0
2.4
4.5
3.3
3.1
3.9
3.0
4.1
3.8
3.5
3.7
2.3
3.6
3.5
3.5
3.5
3.0
2.4
5.1
3.6
[7]
[4]
[7]
[4]
[4]
[8]
[4]
[9]
[9]
[9]
[7]
[10]
[10]
[10]
[11]
[11]
[7]
[12]
[12]
7.7
8.7
6.2
8.3
6.5
10.5
9.8
32.1
30.8
31.7
11.2
10.1
10.9
10.3
11.5
9.7
9.7
18.1
9.6
9.8
7.8
8.7
6.5
8.3
7.7
7.4
7.4
7.2
20.0
6.2
18.1
6.7
9.8
21.1
32.1
29.5
32.7
12.2
24.1
10.9
23.9
10.8
21.0
9.2
20.9
9.0
20.0
8.1
19.8
6.6
19.3
18.3
7.8
18.1
2.1
3.2
3.7
2.8
3.0
2.9
2.7
4.7
4.3
3.2
4.2
3.9
3.7
3.6
3.2
3.4
2.7
3.3
2.5
3.2
2.3
3.0
2.1
3.1
2.6
3.3
2.8
[13]
[13]
[14]
[15]
[16]
[16]
[16]
[14]
[11]
[11]
[13]
[13]
[13]
[13]
[13]
[13]
[13]
[13]
[13]
[13]
[13]
[13]
[13]
[13]
[13]
[17]
[17]
RCo3 with PuNi3 structure type
YCo3
YCo3
YCo3
YCo3
CeCo3
CeCo3
PrCo3
NdCo3
GdCo3
TbCo3
DyCo3
HoCo3
HoCo3
ErCo3
ErCo3
I
II
I
II
I
I
II
III
III
III
I
II
I
II
I
II
I
II
I
II
I
II
I
II
II
I
II
5.013
24.35
4.86
5.013
5.018
5.015
24.35
24.38
24.38
4.86
4.86
4.86
4.955
4.960
24.75
24.80
5.00
5.00
5.068
24.79
4.89
5.055
24.70
4.89
5.037
24.51
4.87
5.016
24.43
4.87
5.007
24.27
4.85
4.985
24.22
4.86
4.977
24.26
4.87
4.980
24.25
4.87
2.0
3.7
1.0
3.8
1.3
2.0
4.6
4.0
4.0
6.0
1.8
3.7
1.8
4.0
2.0
3.7
2.0
3.8
2.1
3.7
2.0
3.8
1.8
3.6
4.1
1.37
3.71
5.000
5.268
5.013
5.241
5.0209
4.9992
5.2666
4.956
4.936
4.98
5.091
5.380
5.055
5.357
5.021
5.291
5.005
5.262
4.992
5.233
4.990
5.233
4.981
5.221
5.217
4.987
5.222
26.23
26.46
25.86
26.401
25.9569
26.9295
26.7753
32.69
32.45
32.65
27.57
27.30
27.40
27.24
27.34
26.88
26.80
26.85
26.61
26.66
26.12
26.33
25.83
26.27
26.123
26.057
26.055
5.25
5.02
5.16
5.04
5.17
5.39
5.08
6.60
6.57
6.56
5.42
5.07
5.42
5.09
5.45
5.08
5.36
5.48
5.33
5.10
5.23
5.03
5.19
5.03
5.01
5.22
4.99
––0.3
5.1
0.0
4.4
0.1
––0.3
5.0
0.0
––0.5
0.4
0.5
6.2
0.0
6.0
––0.3
5.0
––0.2
4.9
––0.3
4.5
0.1
5.0
0.1
4.9
4.8
0.2
4.9
677
Crystal chemistry and metal-hydrogen bonding
Table 1. Continued.
IMC
Typea
a
Alloy
c
c/a
Da/a, %
Hydride
H/f.u. a
c
Dc/c, %
DV/V, %
DV/at. H, A3
Ref.
c/a
RFe3 with PuNi3 structure type
YFe3
II
5.137 24.61 4.79
4.8
5.375
26.46
4.92
4.6
7.5
17.9
2.3
[14]
SmFe3
II
5.187 24.91 4.80
4.2
5.40
27.09
5.02
4.1
8.8
17.9
2.7
[18]
GdFe3
II
5.167 24.71 4.78
3.1
5.38
27.01
5.02
4.1
9.3
17.5
3.8
[19]
TbFe3
II
5.143 24.64 4.79
4.2
5.355
26.71
4.99
4.1
8.4
17.5
2.6
[19]
DyFe3
II
5.116 24.55 4.80
1.8
5.26
25.54
4.86
2.8
4.0
10.0
3.4
[20]
2.5
5.34
25.80
4.83
4.4
5.1
14.5
3.6
[20]
II
4.2
5.36
26.40
4.93
4.8
7.5
18.0
2.7
[20]
DyFe3
II
II
5.130 24.52 4.80
3.0
5.31
26.59
5.01
3.5
8.4
16.2
3.3
[19]
HoFe3
II
5.177 24.48 4.73
3.6
5.316
26.39
4.96
2.7
7.8
16.4
2.8
[19]
ErFe3
II
5.096 24.48 4.80
ErFe3
1.5
5.20
25.17
4.84
2.0
2.8
7.1
2.9
[21]
II
2.7
5.26
25.68
4.88
3.2
4.9
11.8
2.7
[21]
II
4.0
5.30
26.40
4.98
4.0
7.8
16.7
2.5
[21]
2.7
5.267
26.16
4.87
3.2
6.5
13.4
3.0
[19]
21.590 4.43
––1.8
30.7
27.7
5.9
[22]
4.37
––0.5
II
5.104 24.56 4.80
CeNi3 structure type
CeNi3
III
4.964 16.52 3.33
2.8
4.8748
b ¼ 8.5590
CeNi3
III
4.964 16.53 3.33
III
3.3
4.934
21.73
4.40
––0.6
31.5
29.9
5.3
[11]
5.2
4.938
22.44
4.54
––0.5
35.8
34.3
3.9
[11]
R2Ni7 with Ce2Ni7 type of structure
La2Ni7
III
5.059 24.68 4.89
6.5
4.9534
29.579 5.97
––2.1
19.9
14.9
3.1
[23]
Ce2Ni7
III
4.941 24.51 4.96
4.4
4.9146
29.629 6.03
––0.5
20.9
18.9
5.5
[24]
29.773 6.05
––0.3
21.5
21.1
5.87
[24]
6.07
––0.8
20.9
19.1
6.0
[25]
b ¼ 8.4651
III
4.7
4.9251
b ¼ 8.4933
Ce2Ni7
III
4.939 24.50 4.96
4.1
4.8845
29.607 6.07
––1.3
6.02
––0.3
b ¼ 8.507
La1.5Mg0.5Ni7
II
5.029 24.22 4.82
II
4.45
5.3854
26.437 4.91
7.1
9.1
25.2
3.8
[26]
4.55
5.3854
26.437 4.91
7.1
9.1
26.3
3.8
[26]
[14]
R2Co7 with Ce2Ni7 structure type
Ce2Co7
III
4.940 24.46 4.95
6.0
4.949
29.69
6.00
0.2
21.4
21.8
4.7
Pr2Co7
I
5.058 24.51 4.85
2.5
5.081
26.30
5.18
0.5
7.3
8.3
4.5
[27]
5.8
5.312
26.01
4.90
5.0
6.1
17.1
4.0
[27]
II
Nd2Co7
I
5.053 24.43 4.84
II
2.7
5.069
26.29
5.19
0.3
7.6
8.3
4.1
[27]
6.2
5.268
25.92
4.79
4.3
6.1
15.3
3.3
[27]
R2Co7 with Gd2Co7 structure type
Y2Co7
I
5.002 36.15 7.23
II
Gd2Co7
I
5.017 36.31 7.24
II
Tb2Co7
I
5.007 36.27 7.24
II
Dy2Co7
I
II
4.988 36.15 7.25
1.5
4.988
37.72
7.56
––0.3
4.3
3.8
3.3
[27]
3.0
5.138
38.45
7.48
2.7
6.4
11.6
5.0
[27]
2.6
5.012
39.04
7.79
––0.1
7.5
7.3
3.7
[27]
5.9
5.199
38.62
7.43
3.6
6.4
14.2
3.2
[27]
2.7
5.011
38.96
7.78
0.1
7.4
7.6
3.7
[27]
6.6
5.175
38.48
7.44
3.4
6.1
13.3
2.7
[27]
2.6
4.984
38.70
7.77
––0.1
7.1
6.9
3.4
[27]
6.4
5.169
38.31
7.41
3.6
6.0
13.8
2.8
[27]
a: Types of hydrides: conventional interstitial hydrides: I – lower, II – higher; III –anisotropic hydrides.
678
V. A. Yartys, P. Vajeeston, A. B. Riabov et al.
Table 2. Crystal structure data for the deutrerides of RT3 and R2T7 compounds from the powder neutron diffraction.
Hydride
Space
group
a, A
c, A
CeNi3D2.8
Pmcn
4.8748
b ¼ 8.5590
21.590 58.1
CeNi3D3.3
P63/mmc
4.890
21.78
CeNi3D5.2
P63/mmc
4.902
22.34
DV=V DV=V Distance, A
(RT2) (RT5) R––D T––D
D=RT2 D=RT5 Types of occupied
interstices (Fig. 3)
Ref.
CeNi3-type of IMC structure
––1.8
2.19
1.48
4.04
0.22
1þ2þ4þ7
[22]
48.0
9.0
2.20
1.40
4.82
0.87
1þ2þ8
[11]
46.3
18.9
2.10
1.43
7.23
1.68
1þ2þ8
[11]
Ce2Ni7-type of IMC structure
La2Ni7D6.5
P63/mmc
4.9534
29.579 56.6
0.0
2.39
1.52
5.00
1.50
1þ2þ7
[23]
La1.5Mg0.5Ni7D8.9
P63/mmc
5.3854
26.437 29.9
24.5
2.08
1.51
3.78
5.32
3þ4þ5þ6þ7þ8
[26]
La1.5Mg0.5Ni7D9.1
P63/mmc
5.3991
26.543 29.6
22.9
2.06
1.48
3.53
5.32
3þ4þ5þ6þ7þ8
[26]
Ce2Ni7D4.7
Pmcn
4.9251
b ¼ 8.4933
29.773 62.1
––0.6
2.07
1.53
4.08
0.58
1þ2þ3þ7
[24]
Ce2Ni7D4.4
Pmcn
4.9146
b ¼ 8.4651
29.629 59.8
––1.0
2.20
1.53
3.96
0.46
1þ2þ3þ7
[24]
Ce2Ni7D4.1
Pmcn
4.8845
b ¼ 8.507
29.607 58.2
––2.6
2.03
1.38
3.23
0.38
1þ2þ3þ4þ7
[25]
1þ2þ4þ7
[6]
PuNi3-type of IMC structure
LaNi3D2.7
C2/m
8.6392
32.776 44.2
––3.0
48.8
0.2
b ¼ 4.9265
2.16
1.47
3.2
0.25
4.2
0.28
b ¼ 90.850
R
3m
m
R3
5.396
26.885 27.4
24.7
1.88
1.57
5.26
1.27
3þ4þ5þ6þ7þ8
[12]
4.872
31.312 47.0
––2.7
2.04
1.55
3.86
0.17
1þ2þ4þ7
[12]
R
3m
m
R3
4.961
32.69
63.9
0.4
2.21
1.51
5.81
0.47
2þ7
[11]
5.03
32.98
58.5
15.7
2.13
1.48
7.33
2.9
2þ7þ8
[11]
R3m
R
3m
4.9718
25.901 15.1
1.0
2.26
1.64
1.85
0
5þ6
[10]
5.0456
26.157 18.4
6.4
2.16
1.50
2.75
0.42
5þ6þ7þ8
[10]
R
3m
R
3m
5.2398
26.605 24.3
22.4
1.72
1.47
4.93
1.51
4þ5þ6þ7þ8
[10]
5.184
26.27
22.4
16.0
2.23
1.43
5.82
0.40
3þ8
[11]
R
3m
R
3m
5.21
26.45
23.0
19.5
2.38
1.53
7.18
1.08
3þ8
[11]
4.9837
26.057 16.1
––1.0
2.20
1.65
2.05
0.00
5þ6
[17]
R
3m
m
R3
5.2218
26.046 24.3
11.9
2.01
1.64
5.17
0.83
3þ5þ7þ8
[17]
5.217
26.123 18.5
––1.3
2.05
1.50
4.52
1.16
5þ7þ8
[15]
R3m
m
R3
4.9887
26.097 13.2
4.6
2.07
1.63
1.91
0.00
5þ6
[9]
4.991
26.12
0.5
2.13
1.55
2.37
0.63
6þ7
[28]
R3m
R
3m
5.0351
26.297 16.1
–0.63 2.10
1.48
2.65
0.12
5þ6þ7þ8
[9]
5.0209
25.957 13.2
–2.0
2.22
1.68
1.99
0
5
[16]
4.9992
26.93
21.0
13.1
2.33
1.69
3.00
0
5
[16]
YCo3D3.8
R
3m
R
3m
5.241
26.401 28.8
24.7
2.35
1.62
5.04
1.51
5þ7þ8
[15]
YCo3D4.6
R
3m
5.2666
26.775 27.4
1.0
2.27
1.51
6.47
1.00
5þ7þ8
[16]
LaY2Ni9D12.8
CeY2Ni9D7.7
CeCo3D4
CeCo3D6
ErNi3D1.23
ErNi3D1.97
ErNi3D3.75
ErNi3H3.7
ErNi3H4.9
ErCo3D1.37
ErCo3D3.71
ErCo3D3.4
HoNi3D1.27
HoNi3D1.8
HoNi3D1.81
YCo3D1.3
YCo3D2
22.4
sion of the unit cells. The named hydrides possess distinct
crystal chemistry features and belong to the group of anisotropic hydrides. Anomalously high specific volume expansion per absorbed atom H, 5.3–6.0 A3 significantly exceeds these values for the isotropic hydrides.
Crystal chemistry of the hydrides belonging to groups I
or II contrasts to the group III, which exhibits a principally different behaviour. Groups I and II are conventional, interstitial type hydrides, despite the hydrogenation
normally leads to an uneven expansion of the unit cells in
different directions. However, formation of the hydrides
preserves the coordination characteristics of the metal
atoms with hydrogen atoms filling the interstitial sites in
the metal sublattice; thus, from a crystallographic point of
view they can be classified as formed via an isotropic mechanism of hydrogenation. In contrast, a principally different mechanism of crystallographic transformation takes
place for the hydrides belonging to the type III where a
679
Crystal chemistry and metal-hydrogen bonding
huge anisotropic change in structural parameters takes
place during the hydrogenation process.
ErNi3D4.0-5.0 36i1 Er2Ni2 þ 36i2 ErNi3 þ 18h3 Er2Ni2
[11] ;
ErCo3D4.1 36i1 Er2Co2 þ 18h2 Er2Co2 þ 36i2 ErCo3
[15] ;
Isotropic hydrides
ErCo3D1.37 36i1 Er2Co2 þ 6c1 ErCo3 [17] ;
Crystal structures of the hydrides belonging to the group
of isotropic hydrides were solved for the following materials:
YCo3D1.34.6 [15, 16] ;
ErCo3D3.71 36i1 Er2Co2 þ 36i2 ErCo3 þ 18h2 Er2Co2
þ 18h3 Er2Co2 [17] ;
HoNi3D1.8 18h1 Ho2Ni2 þ 6c1 HoNi3 [28] ;
ErNi3D1.234.9 [10, 11] ;
ErCo3D1.374.1 [17],
HoNi3D1.271.8 [9, 17] ;
HoNi3D1.3 36i1 Ho2Ni2 þ 6c1 HoNi3 [9] ;
LaY2Ni9D12.8 [12] ;
La1.5Mg0.5Ni7D8.99.1 [26].
HoNi3D1.8 36i1 Ho2Ni2 þ 6c1 HoNi3 þ 36i2 HoNi3
þ 18h2 Ho2Ni2 [9] ;
All of the R(Ni,Co)3 hydrides mentioned here are formed
by IMC belonging to the PuNi3 type of structure. In total,
the initial structure contains twelve types of tetrahedral sites
(R2T2, RT3 and T4) (Fig. 2) and one type of octahedral
(R2T4) site. The hydrogen sublattice in these hydrides is
formed by a partial filling of nine from 13 available interstices, including 36i1, 36i2, 18h1, 18h2, 18h3, 18h5, 18h6,
6c1 and 6c3. Particular structures include the following list
and can be presented as filling of the sites mentioned below:
YCo3D1.3-2.0 36i1 Y2Co2 [16] ;
YCo3D3.8-4.6 36i1 Y2Co2 þ 18h2 Y2Co2 þ 36i2 YCo3
[15, 16] ;
ErNi3D1.23 36i1 Er2Ni2 þ 6c1 ErNi3 [10] ;
ErNi3D1.97 36i1 Er2Ni2 þ 6c1 ErNi3 + 36i2 ErNi3
þ 18h2 Er2Ni2 [10] ;
ErNi3D3.75 36i1 Er2Ni2 þ 36i2 ErNi3 þ 18h2 Er2Ni2
þ 18h3 Er2Ni2 + 6c3 Ni4 [10] ;
LaY2Ni9D12.8 6c3 Ni4 þ 18h2 R2Ni2 þ 18h3 R2Ni2 þ
18h6 RNi3 þ 36i1 R2Ni2 þ 6c4 Ni4 þ 18h5 RNi3
þ 36i2 RNi3 [12] .
The only representative of isotropic hydrides formed by
IMC with other than PuNi3 type structure studied so far is
La1.5Mg0.5Ni7D8.99.1 (Ce2Ni7 type) [26]. Nine types of
sites are filled by D in total, including tetrahedral
(La,Mg)2Ni2, (La,Mg)Ni3, Ni4, tetragonal pyramidal
La2Ni3 and trigonal bipyramidal (La,Mg)3Ni2 interstices.
Hydrogen is nearly equally distributed between the Lavesand Haucke-type slabs. The overall hydrogen content
can be presented as LaMgNi4D7.56 (Laves-type) þ
2 LaNi5D5.22 (Haucke-type) ¼ 2 La1.5Mg0.5Ni7D9.
Types of coordination of the hydrogen by metal atoms
in the structures of both isotropic and anisotropic hydrides
are shown in Fig. 3. Observed coordination includes octahedron R3Ni3 (1), tetrahedra R3Ni (2), R2Ni2 (3), Ni4 (4),
RNi3 (6) and R2Ni2 (7), trigonal bipyramid R3Ni2 (5) and
octahedron R2Ni4 (8).
Ni2
MgZn2
CaCu5
Ho1
36i2
6c4
Ho2
6c3
36i1
18h3
Ni3
Ni1
18h6
CaCu5
6c1
18h2
6c2
18h5
18h1
18h4
z
x
y
Fig. 2. Potential sites for the accommodation of H atoms in the trigonal crystal structure of the PuNi3 type presented for R ¼ Ho. 12 types
of the available tetrahedral interstices are shown as belonging to the
two types of the stacking along [001] layers, Laves-type MgZn2 and
Haucke-type CaCu5. These include 4 Ho2Ni2 sites (18h1, 18h2,
18h3 and 36i1), 5 HoNi3 sites (36i2, 18h4, 18h5, 18h6 and 6c1) and
3 Ni4 sites (6c2, 6c3, and 6c4). Part of the unit cell from z ¼ 0 to
z ¼ 1=3 is shown.
Fig. 3. Hydrogen coordination in the crystal structures of anisotropic
and interstitial type hydrides.
680
V. A. Yartys, P. Vajeeston, A. B. Riabov et al.
Fig. 4. Hydrogenation-induced transformations of the 12-vertex coordination polyhedron R6T6 around the Co(Ni)1 atoms inside the MgZn2-type
layers in trigonal YCo3 (PuNi3 type) and hexagonal Ce2Ni7 structures. A two-step hydrogenation process leads to the formation of the “conventional” YCo3D2.0 and YCo3D4.6 hydrides; in contrast, an anomalously expanded “anisotropic” Ce2Ni7D4.7 is formed via a different mechanism.
The figure shows characteristics of the lattice expansion illustrating that YCo3D2 hydride is formed with moderate expansion proceeding solely
along the [001] direction, whereas during the formation of the higher YCo3D4.6 hydride further expansion proceeds in the basal plane. In the
Ce2Ni7D4.7 hydride an anomalously high expansion along the z axis takes place (>60%), with basal plane practically unchanged. The relevant
distances from the central atoms Co1/Ni1 to Y/Ce/Co/Ni atoms in their 12-vertexes coordination polyhedra are given in insets. For Co-containing
hydrides, a rather modest elongation of the interatomic separations, up to 17.2% in maximum, does not change the original coordination characteristics. However, in case if Ce2Ni7D4.7 the situation is completely different. 3 from 6 Ni1––Ni distances undergo anomalously large increase
upon hydrogenation (by 71.3–72.6%), from the original 2.519 A to 4.315–4.347 A in the hydride phase, thus breaking the Ni––Ni interaction.
Thus, coordination polyhedron charges to a 9-vertex Ni3Ce6 as a result of moving of three Ni atoms far away from the central atom Ni1.
Hydrogen coordination of Co1 by six H atoms leads to the formation of octahedron Co1D6 with dCo1 . . . D ¼ 1.69–1.70 A. For Ni1, an open,
saddle-type coordination Ni1D4 is observed with dNi1 . . . D ranging from 1.52 to 1.77 A. This coordination is shown in the figure for both Co- and
Ni-containing compounds.
The octahedron R3Ni3 (1) and tetrahedron R3Ni (2) are
formed only in the structures of anisotropic hydrides, during their rebuilding. Other five types of interstitial sites (3,
4, 6–8) can be occupied both in anisotropic and isotropic
hydrides; these sites already exist in the structures of the
original intermetallic alloys. Sites 1–6 are located inside
the Laves-type slabs; sites 7 and 8 belong to the CaCu5type slabs.
Occupancies by hydrogen atoms of different sites in
the metal matrices are presented in Table 2.
During hydrogenation, interatomic Me––Me distances
in the conventional R(Ni,Co,Fe)3-based and R2(Ni,Co)7based hydrides moderately increase. To illustrate this, we
will present the data describing transformations YCo3 !
YCo3D2.0 (type I) ! YCo3D4.6 (type II) [16]. Here
expansion of the unit cell proceeds to the extent of
allowing H atoms to reach equilibrium Me––H separations; however, not changing significantly the coordination polyhedra of the metals: 16-vertex polyhedron
RR4(Ni,Co)12 for R atoms and 12-vertex polyhedron TT12
for Ni or Co (see Fig. 4). This expansion drastically
weakens but does not break the bonding between transition metal atoms compared with that between R and T; as
example, the Y––Co distances increase by 3.2–7.3%,
while Co––Co ones are extended more significantly, by
10.9–17.2%.
Thus, we conclude that hydrogenation does not significantly change the bonding mechanism which is dominated
by the metal-metal interactions with metal-hydrogen interactions playing much less significant role.
From analysis of the crystal structure data for the
ErNi3- and HoNi3-based hydrides [9, 10] it becomes evident that these hydrides belong to the group of interstitial
hydrides with H atoms filling 6 different types of interstices. Despite that the experimental diffraction pattern for
all Er- and Ho-containing hydrides were satisfactorily described in the original group of symmetry R3m, nevertheless, for some of these hydrides an alternative description
was suggested with loss of inversion symmetry and transition to the space group R3m [9, 10]. Corresponding refinements yielded equally satisfactory results for both groups;
however, description in the group R3m allowed avoiding
short H––H separations as a result of H ordering on the
split sites. On further increase of the H content to 3.75 at.
H/f.u. the formation of coordinated Ni atoms was observed. The H atoms in the vertices of these tetrahedra are
shared with other neighbouring Ni having less than 4 H
neighbours. The proposal of ordering is reasonable; however, absence of direct experimental proof calls for a single-crystal study of the hydrogenated materials to answer
the question concerning the symmetry of the unit cells and
possible H ordering.
681
Crystal chemistry and metal-hydrogen bonding
Interatomic distances R––H and T––H
Typical ranges of observed shortest interatomic distances
(
A) are: Y––D (2.20–2.35); Ce––D (2.03–2.21); La––D
(2.06–2.39); Er––D (2.04-2.26); Ho––D (2.10–2.13);
Ni––D (1.43–1.63); Co––D (1.48–1.69). These values satisfy a well known criterion dMeH rMe þ 0.25 A, where
rMe is the metallic radius of the metal atom.
Figure 4 illustrates the deformation of the metal sublattice
that occurs inside the CeNi2 slabs of the Ce2Ni7 structure
during the D uptake [24]. Together with a huge expansion
along the [001] direction, substantial shifts of both the
central Ni1 atom and surrounding Ce atoms are observed.
As a result, the distances from the Ni1 atom to some of
the formerly neighbouring Ni atoms increase by >70%
(see Fig. 4). This dramatically changes its coordination
from 12 (Ce6Ni6) to 9 (Ce6Ni3).
Anisotropic hydrides
Decrease of symmetry
The hydrogenation mechanism is principally different for
the RT3 and R2T7 compounds of light rare earth elements,
La and Ce, with Ni and Co formed via an anisotropic
mechanism (type III). Crystal structure data for the anisotropic hydrides are available on the basis of the NPD experiments for CeNi3D2.8 [22], CeNi3D3.3–5.2 [11],
LaNi3D2.8 [6], La2Ni7D6.5 [23], Ce2Ni7D4.7 [24],
Ce2Ni7D4.4 [24], Ce2Ni7D4.1 [25], CeY2Ni9D7.7 [12] and
CeCo3D4–6 [11].
Hydrogenation behaviour of RT3 and R2T7 compounds,
formed by R ¼ La and Ce, substantially differs from that
of other isostructural hybrid intermetallics. It is characterised by an extremely strong anisotropic expansion of
the unit cells proceeding along the [001] direction. Such
an expansion reaches values of Dc/c ¼ 30.7% for RT3
compounds and Dc/c ¼ 21.5% for R2T7 compounds; at the
same time the basal plane of the unit cells remains unchanged or even slightly contracts (see Table 1). Despite
differences in types of the original structures (CeNi3,
Ce2Ni7 and PuNi3-types) and type of T-element (Co or
Ni), similar anisotropic behaviour of lattice expansion is
observed for the compounds of light rare earth elements,
La and Ce (also Y when alloyed with Ce [12]). The only
exception is GdNi3H3.0 which also belongs to the type III
hydrides [4].
Anisotropic lattice expansion in RT2/RT5
From Table 2 it is evident that in most cases anisotropic
expansion of hybrid structures proceeds within the RT2
slabs only leaving RT5 slabs without changes. This
scheme is observed for all known systems, including RNi3
(R ¼ La, Ce, Y0.67Ce0.33), CeCo3, R2Ni7 (R ¼ La, Ce), studied under ambient hydrogen pressures. However, when hydrogenation pressure increases to the level exceeding
1 kbar, as in studies of the CeCo3 ––D2 and CeNi3 ––D2 systems [11], further hydrogen uptake, despite keeping preference in expansion of the RT2 slabs, also involves a much
smaller yet significant increase in the volumes of the RT5
layers. The values of the linear expansion are anomalously
large, reaching 35.8% for CeNi3D5.2 [11].
Changes in the metal-metal separations
and coordination polyhedra
A huge linear expansion along [001] causes a drastic
change in the metal sublattice within the RT2 slabs.
Shifts of the R and T atoms inside the MgZn2-type slabs
and hydrogen ordering cause deformation of the unit cells
and lowering of the initial hexagonal/trigonal symmetry to
the orthorhombic (CeNi3 and Ce2Ni7 types) or monoclinic
one (PuNi3 type). Decrease of symmetry is manifested by
the appearance of extra, not allowed by the original hexagonal/trigonal structures, peaks in the SR XRD and PND
patterns. From group–subgroup relations and observed extinctions the symmetry of the hexagonal CeNi3 and
Ce2Ni7 was concluded to be reduced to an orthorhombic
one in corresponding hydrides (P63/mmc ! Cmcm !
Pmcn). Similarly, for the LaNi3-based hydride, a monoclinic unit cell was found (R3m ! P3m1 ! C2/m). Deviation from the hexagonality
is more pronounced for the
pffiffiffi
CeNi3D2.8 (borth/ 3 a
)/a
pffiffiffiorth orth 1.4% [22]), compared
to Ce2Ni7D4.7 (borth
/
pffiffiffi 3 aorth)/aorth 0.5% [24]) and
Ce2Ni7D4.1 ((borth/ 3 aorth) / aorth 1.0% [25]). In the
latter system negative orthorhombic distortion observed for
the Ce2Ni7H4 þ x sample
pffiffiffi saturated with hydrogen at pressure of 30 bar (borth/ 3 aorth)/aorth 0.7% [25]) was
changed to a positive distortion reaching up to 1.0%,
when the sample was in contact with air in anpopen
glass
ffiffiffi
capillary. In case of LaNi3D2.8, amon/ 3 amon)/
bmon 1.2%; b ¼ 90.85 [6]. As mentioned earlier in this
paper, a decrease of the symmetry from the space group
R3m to the noncentrosymmetric R3m was suggested for
some of the ErNi3- and HoNi3-based hydrides including
ErNi3D1.23 [10], HoNi3D1.27 and HoNi3D1.81 [9].
Hydrogen sublattice
The distribution of hydrogen in the structure of anisotropic
hydrides is very uneven – hydrogen atoms are accommodated only inside the MgZn2-type layers and on the boundary between Laves- and Haucke-type layers; the bulk of
the Haucke-type slab remains empty. As a result, expansion of the anisotropic hydrides occurs only in MgZn2-type
slabs, whereas CaCu5-type slabs remain unchanged. The
expansion of the RT2-slabs reaches 63.9% resulting in
huge anisotropic changes in the metal sublattice.
Because of such rearrangements, hydrogen atoms, in
sharp contrast to the known crystal structures of other intermetallic hydrides, instead of filling initially existing interstices, attract cerium atoms into their surrounding and
form new D-occupied sites, R3T3 octahedra and R3T tetrahedra. This is illustrated by Table 3 where the data for
CeNi3D2.8 and Ce2Ni7D4.7 hydrides are given. In addition,
682
V. A. Yartys, P. Vajeeston, A. B. Riabov et al.
Table 3. Types of interstices occupied in the structures of CeNi3D2.8
and Ce2Ni7D4.1.
Intermetallic compound Deuteride
CeNi3D2.8 Ce2Ni7D4.7
New interstices formed via the rearrangement of existing interstices
Ni4
Ce3Ni3
D1
D2
CeNi3
Ce3Ni3
D2, D7
D1, D3
Formation of new interstice due to “buckling” of the MgZn2-type
layer
No interstice available
Ce3Ni
D3, D5
D5, D6
Ce2Ni2 tetrahedra on the boundary between CaCu5- and MgZn2-type
slabs and inside the latter one
R2Ni2
R2Ni2
Ce2Ni2
Ce2Ni2
D4, D8
D7, D8, D9
D4
Filling of Ni4 tetrahedra in the MgZn2-type layer
Ni4
Ni4
D6
three conventional interstitial type positions are occupied;
these are tetrahedra R2Ni2 (2 types) and Ni4.
Hydrogen ordering
One of the important features of anisotropic hydrides is
the ordering of hydrogen atoms in the lattice with all
H . . . H separations exceeding 1.8 A. CeNi3D2.8 and
La2Ni7D6.5 are completely ordered whereas in Ce2Ni7D4.7
the part of the structure within the Laves-type slab is also
ordered. It is convenient to present the way the hydrogen
sublattice is organized by stacking of the coordination
polyhedra formed by H atoms around Ce or La. In the
case of CeNi3D2.8, 12- and 7-vertex polyhedra of two
types were identified [22] while for La2Ni7D6.5 a 15-vertex
LaD15 polyhedron was formed. Stacking of these polyhedra allows building of H sublattice as layers filling the
RT2 slabs.
Ni––H and Co––H interactions
Tetrahedral and open saddle-like NiH4 coordination, and
CoH6 octahedra were observed in the crystal structures of
CeNi3D2.8 [22], Ce2Ni7D4.1 [25, 29], Ce2Ni7D4.7 [24],
ErNi3D3.7 [10] and YCo3D2.0/4.6 [16]. Coordination of T by
H increases from Ni (CN ¼ 4) to Co (CN ¼ 6, see Fig. 4).
Interestingly, a different shape of Ni-H 4-fold coordination
was reported in the same system, Ce2Ni7 ––D2, such as tetrahedron NiD4 [25] or open saddle-type coordination NiD4
[24] (see Fig. 4). Later in this paper we will present the
results of the structural optimization based on density
functional total energy calculations performed for
Ce2Ni7D4.7 and Ce2Ni7H4.1. These calculations have confirmed both the saddle-type and tetrahedral NiH4 coordinations and have shown that Ce2Ni7D4.7 is thermodynamically the more stable exhibiting the saddle type NiH4
environment.
Unusual behaviours of anisotropic hydrides and complexity of the metal-hydrogen interactions in these systems
raise questions about the nature of such interesting transformations from intermetallic alloy to a corresponding hydride.
Electronic structure and thermodynamic
properties
Total energies were calculated using the projected augmented plane-wave [30, 31] implementation of the Vienna ab
initio simulation package [32, 33]. The generalized-gradient approximation [34–36] was used to obtain accurate
exchange and correlation energies for a particular configuration of atoms. Ground-state geometries were determined by minimizing stresses and Hellman-Feynman
forces with the conjugate gradient algorithm, until forces
on all atomic sites were less than 103 eV A1. Experimentally known structural parameters were taken as a
starting point and cell volume, cell shape, and atomic positions were relaxed simultaneously in a series of calculations with progressively increasing precision. A final high
accuracy calculation of the total energy was performed
after completion of the relaxations with respect to k-point
convergence and plane-wave cut-off. Brillouin zone integrations are performed with a Gaussian broadening of
0.1 eV during all relaxations. From the various sets of calculations it was found that for the CeNi3 and Ce2Ni7
phases 18 18 6 and for CeNi3D2.8, Ce2Ni7D4.7 and
Ce2Ni7D4.0 phases 24 12 6 k-point mesh in the whole
683
Crystal chemistry and metal-hydrogen bonding
the CeNi3D2.8, in Ce2Ni7D4.7 hydride the H sublattice is
mostly ordered. It contains six completely occupied sites,
H1––H6 (from nine H sites in total) and three partially occupied sites (H7––H9). For simplicity we have assumed that H
fully occupies the H7 site with an experimental occupancy
factor of 0.47 and ignored the H8 and H9 (experimental
occupancies <0.4) sites in the calculations. Because of the
assumed vacancy of the two last sites, there is a small difference in the hydrogen stoichiometry between experimental
NPD data, D/Ce2Ni7 ¼ 4.65, and calculations, H/
Ce2Ni7 ¼ 4.50. The calculated atomic positional parameters
agree very well with the experimental findings. The calculations confirm highly anisotropic lattice distortion with huge
expansion along [001] (Dc/c ¼ 30.7%) and a small contraction in the basal plane (Da/a ¼ 1.8). Also, in agreement
with the experimental data, crystal structure calculations
correctly predicted anomalously large volume expansion on
hydrogenation, 5.9 A3/atom H for both CeNi3- and Ce2Ni7based hydrides.
The two structurally characterised hydrides with lower
hydrogen content, Ce2Ni7D4.4 [24] and Ce2Ni7D4.0 [25],
show different structures as compared to the saturated hydride Ce2Ni7D4.7 [24]. However, we have made theoretical
calculations only for Ce2Ni7H4.0. These studies showed
that agreement between the experimental and theoretical
structural data is less satisfactory compared to that for the
saturated Ce2Ni7D4.7 and CeNi3H2.8 hydrides at normal
conditions. Indeed, the difference in the calculated volume
of the unit cell, 3.3% (Ce2Ni7H4.0) compared to the experimentally observed value, noticeably exceeded such divergences for CeNi3H2.8 and Ce2Ni7D4.7 (2.7 to 2.8%).
Brillouin zone with a 600 eV plane-wave cut-off are sufficient to ensure optimum accuracy in the computed results. The density of states calculations were performed
using the tetrahedron method with the Blöchl corrections
[37].
Structural optimizations were carried out in order to
understand the reasons for the anisotropic expansion effect
during incorporation of H in the CeNi3 and Ce2Ni7 phases.
Experimental structural information was used as input
(model 1). In order to verify whether the experimentally
known phase is the correct ground state structure, the original starting structures presented in the P1 symmetry
(model 2) and additional structural relaxation calculations
were performed. All structures were fully relaxed (minimization of force and stress); no constraints on the atomic
positions and unit cell parameters were applied. The optimized structures were subjected to the symmetry analysis
which showed a successful convergence of the calculation
results to the experimental crystal structure. As it can be
seen from the data presented in Table 4 (only the data for
the CeNi3-based hydride is given), the optimized atomic
positions and lattice parameters are in very good agreement with the experimental findings. In the hydrogenated
CeNi3D2.77 phase H atoms completely fill seven different
types of sites; in addition one extra position [H8 (4c) site]
is partially, 30%, occupied by H. In order to simplify the
calculations, we have limited our considerations to the
fully occupied sites only and excluded the last position
H8; thus, stoichiometry of the calculated compound corresponds to a slightly lower H/CeNi3 ratio of 2.67 compared
to the experimentally determined value of 2.77. Similar to
Table 4. NPD-based data and theoretically optimised crystal structure data for CeNi3D2.77.
Space group Pmcn (No. 62).
Lattice parameters: a ¼ 4.8748(3); b ¼ 8.5590(5); c ¼ 21.590(2) A. T ¼ 300 K. Data recorded on the D1A diffractometer. Theory: a ¼ 4.750,
b ¼ 8.645, c ¼ 21.344 A. H/CeNi3 ¼ 2.67.
Atoms
Sites
x
Neutron powder diffraction data
y
Theory
z
x
y
z
4c
1
0.430(3)
0.2514(9)
1
0.4216
0.2504
Ce2
Ce3
4c
4c
1
=4
1
=4
0.378(2)
0.087(4)
0.0575(6)
0.9364(8)
1
=4
1
=4
0.3800
0.0459
0.0559
0.9263
Ni1
4c
1
0.755(1)
0.5297(4)
1
0.7527
0.5250
Ni2
Ni3
4c
4c
1
0.929(2)
0.748(1)
0.3348(6)
0.2461(6)
1
0.92
0.7453
0.3481
0.2422
Ni4
4c
1
0.086(1)
0.2555(5)
1
0.0905
0.2592
Ni5
Ni6
4c
8d
1
0.938(1)
0.8219(10)
0.1564(5)
0.8392(3)
1
=4
0.003
0.9214
0.8288
0.1566
0.8398
Ni7
8d
0.497(2)
0.1774(11)
0.3532(3)
0.499
0.1662
0.3580
D1
D2
4c
4c
1
=4
1
=4
0.726(2)
0.080(1)
0.8894(8)
0.1120(5)
1
=4
1
=4
0.7542
0.0823
0.8888
0.1100
D3
4c
1
0.916(1)
0.4962(7)
1
0.9106
0.4857
D4
D5
4c
8d
1
0.233(1)
0.169(1)
0.6498(5)
0.0094(4)
1
0.2133
0.1555
0.6571
0.0103
D6
4c
1
0.771(3)
0.1007(5)
1
0.7775
0.1023
4c
4c
1
0.971(2)
0.478(1)
0.4180(10)
0.1514(8)
1
0.9812
0.4098
Ce1
D7
D8a
=4
=4
=4
=4
1
=4
=4
0.005(2)
=4
=4
––0.030(2)
=4
=4
1
=4
a: Occupancy 0.30 (3).
=4
=4
=4
=4
1
=4
=4
=4
––0.025
=4
=4
Vacant
684
Evaluations of the heat of formation for Ce2Ni7H4.0 and
Ce2Ni7H4.7 (thermodynamic data will be presented later in
this paper) showed a disagreement of theoretical calculations with experimental stability of the Ce2Ni7H4.0 hydride
as compared to Ce2Ni7H4.7 (lower thermal stability for
Ce2Ni7H4.0 from theoretical study instead of experimentally observed increase of the thermal stability with decrease of the H content in the hydride). A detailed study
is in progress aimed on comparison of the Ce2Ni7-based
hydrides. The results of this work will be published elsewhere. One reason for the observed disagreements for
Ce2Ni7H4.0 is the possible thermodynamically nonequilibrium state of the studied sample which had been exposed
to air, thus affecting its properties by partial oxidation.
Nevertheless, common features were observed in the
electronic structures of all three theoretically studied hydrides, CeNi3H2.8, Ce2Ni7H4.7 and Ce2Ni7H4.0. These similarities will be presented and discussed later in the paper.
From the characteristic features of the DOS one may
be able to rationalize (see, e.g., Ref. [38]) the chemical
bonding in CeNi3 and Ce2Ni7 and changes introduced in
the metal-metal bonding upon hydrogenation. To the best
of our knowledge no electronic structure calculations have
apparently hitherto been undertaken for the considered
phases. In general, both initial intermetallic alloys and all
three hydrogenated phases have a finite number (see
Fig. 5) of electrons at the Fermi level (EF), which classifies them as metals. The metallic character of all these
phases mainly originates from the finite contributions to
the DOS at EF from the Ce-5d and Ni-3d electrons.
Fig. 5. Calculated total density of states for CeNi3, Ce2Ni7,
CeNi3H2.8, Ce2Ni7H4.0 and Ce2Ni7H4.5. The Fermi level is set to zero.
V. A. Yartys, P. Vajeeston, A. B. Riabov et al.
In this review we present only the total DOS for
CeNi3H2.7, Ce2Ni7H4.5 and Ce2Ni7H4.0 (Fig. 5) as representative with typical data for the PDOS for Ni and H in
CeNi3H2.7 (Fig. 6) and Ce2Ni7H4.5 (Fig. 7). The detailed
data on the partial DOS will be published in the forthcoming publications. Normally, introduction of hydrogen
modifies the electronic structure of the host alloy by creation of metal-hydrogen bonding states, shift of the Fermi
level, and change in the width of bands and/or modification of the lattice symmetry. One common feature in the
electronic structures of these hydrides is the occurrence of
the H states at the bottom of the valence band (VB) (see
Figs. 6 and 7). The inclusion of the additional bonding H-s
states in the energy range 9 to 3 eV changed not only
the corresponding portion of the DOS but also systematically shifted the EF towards the unoccupied states in the
non-hydrogenated phases (see Fig. 5). It is interesting to
note that in the non-hydrogenated intermetallics the site
projected DOS is almost similar. On the other hand, after
hydrogenation, they become significantly different from
each other (please, compare PDOS of Ni1 (which is
bonded with H) in Fig. 6 with that of Ni2 and Ni3 (those
are not bonded with H) in Fig. 7), with a component at
low energies, around 9 to 8 eV present only for the Hbound Ni. Further to that, the VB widths for the Ni states
also change depending on presence or absence of their H
in their surrounding. Indeed, in CeNi3 for all Ni states the
VB widths are around 7 eV. Though in the hydrogenated
Fig. 6. Partial DOSs of Ni1, H3, H5, and H6 forming the NiH4 tetrahedra belonging to the . . . H––Ni––H––Ni . . . chains in the structure of
CeNi3H2.8. The Fermi level is set at zero energy and marked by the
vertical dotted line; s-states are shaded. PDOS for Ni3 which is not
bound with H are shown for comparison.
685
Crystal chemistry and metal-hydrogen bonding
Fig. 7. Partial DOSs of Ni1, H4, H5, and H6 forming an open saddle-type hydrogen configuration around Ni1 in the structure of
Ce2Ni7H4.5 and belonging to the spatial . . . H––Ni––H––Ni . . . chains.
The Fermi level is set at zero energy and marked by the vertical
dotted line; s-states are shaded. PDOS for Ni2 which is not bound
with H are shown for comparison.
CeNi3H2.8 phase the VB width for Ni1, Ni2, Ni5, Ni6,
and Ni7 atoms those are bound with H are almost the
same (around 9 eV); in contrast with them, VB widths are
drastically reduced to ca. 6.1 eV for Ni3 and Ni4 (not
bound with H).
From PDOS data for H atoms, a clear interrelation between type of the coordination of the H sites (see Table 3)
and its electronic configuration is evident. In CeNi3H2.8
the H atoms have 4 different coordination characteristics,
including two types of octahedra, Ce2Ni4 (H6) and Ce3Ni3
(H1 and H2), and two types of tetrahedra, Ce2Ni2 (H4)
and Ce3Ni (H3, H5, H7).
As example, PDOS data for the H atoms bound to Ni1
are shown in Fig. 6. From this figure, we conclude that
the most strongly bound are H atoms with the largest
amount of Ni in their coordination. Indeed, for H6 with
4 Ni/H, filled energy levels span from 9 to 5 eV with a
maximum at 8 eV. For H1 and H2 with 3 Ni/at. H, the
filled levels are in the same range, 9 to 5 eV; however,
the peaks on the electronic density of states spectra shift
towards the higher energies (see Fig. 6).
When number of Ni neighbours decreases to 2 Ni/at. H
(for H4), further shift of the peak in the energy spectrum
towards higher energies takes place (peak is observed at
6 eV); once again the same range of the energies is covered, from 9 to 5 eV.
For H3 and H5 with just one Ni/at. H the peaks of the
DOS shift further towards higher energy appearing higher
than 5 eV. However, for H7 with a similar, Ce3Ni, environment, peak from its DOS is at lower energies, between
7 and 6 eV.
From the DOS data for Ce2Ni7H4.5 (see Fig. 7) it is
clear that, similar to CeNi3H2.8, the bonding energy is also
related to the number of Ni atoms in the environment of
H; the strongest bonding takes place for 3Ni/at. H
(H1––H3); the weakest bonding is observed for 1 Ni/at. H
(H5 and H6), with 2 Ni/at. H (H4, H7) lying in between.
Figure 7 presents the PDOS data for H6, H5, and H4
forming an open saddle type configuration around Ni1. A
clear similarity between the structure of PDOS of H3 and
H5 in CeNi3H2.8, H5 and H6 in Ce2Ni7H4.5, all with the
same, Ce3Ni, environment is evident from comparison of
the Figs. 6 and 7.
The electron localization function (ELF) is considered
as a useful tool to distinguish different bonding interaction
in solids (for more details about ELF see Refs. [39–42]).
The value of ELF spans the range 0 to 1. A high value of
ELF corresponds to a low Pauli kinetic energy, as can be
found in covalent bonds or lone electron pairs. The large
values of the ELF at the H site indicate strongly paired
electrons with dominant s-electron character. From calculations, it appears that ELF distribution on the H sites is not
spherical indicating a finite covalent interaction of H with
neighbours. We note that due to the presence of delocalized metallic Ni-d electrons, the calculated ELF on the Ni
site is low. Table 5 presents summary of the PDOS characteristics of H atoms in the crystal structures of
Table 5. Summary of the PDOS characteristics of H atoms in the crystal structures of Ce2Ni7H4.7, CeNi3H2.8 and Ce2Ni7H4. Three features
reviewed include position of the center of the H band, its width, and integrated charge on the H atom.
Ce2Ni7H4.7
Center of the
Width
band (eV)
(eV)
Integrated
(e)
CeNi3H2.8
Center of the
Width
band (eV)
(eV)
Integrated
(e)
Ce2Ni7H4,1
Center of the
Width
band (eV)
(eV)
Integrated
(e)
H1
H2
––7.5
––7.44
3
2.75
0.53
0.53
––6
––6.6
5.5
5.1
0.48
0.56
––7
––6.6
4.45
4.3
0.52
0.49
H3
––6.5
2.65
0.56
––5.4
2.48
0.55
––4.7
5.2
0.54
H4
H5
––6.7
––4.5
5
3.2
0.55
0.54
––7
––4.8
3.5
2
0.61
0.58
––6
––5
5.2
2.72
0.49
0.56
H6
––4.7
2.88
0.52
––7.7
3.62
0.54
––8
4.3
0.53
H7
––6.6
4
0.56
––4.5
5.3
0.45
––6
6.0
0.47
686
V. A. Yartys, P. Vajeeston, A. B. Riabov et al.
Ce2Ni7H4.7, CeNi3H2.8 and Ce2Ni7H4. The absence of differences in the behaviour of H forming a tetrahedral or an
open saddle-type coordination around Ni and other types
of H in the materials worthy to mention. It should be also
mentioned that an integrated charge on the H atom varies
in a rather narrow range from 0.47 to 0.61 e and never
approaches value of 1 necessary for the formation of
hydrido-ion H1. For comparison, the Bader effective
charges are calculated at the H sites in nearly pure ionic
cases such as LiH and MgH2 and are -0.84 and 0.92,
respectively. The much smaller value of the Bader effective charge at the H sites for the systems considered in the
present study clearly indicates that they do not reach a
pure ionic case of H1.
We note that similar data concerning the Ni-H bonding
in the clusters [Ni2H7] found during the studies of the
crystal structure of LaMg2Ni2H8 [43] were reported in
[44]. Electronic configuration of these clusters [44] was
assumed as [Ni2H7]7 and showed absence of the formation of H1 and Ni0 and a partial negative charge on both
Ni and H corresponding to the formula [Ni20.95H70.73]7.
In order to understand the microscopic origin of the
anisotropic volume changes during hydrogenation and
large variation in lengths of the Ni––H, and Ce––H bonds
in the system, we performed valence charge density analyses in different crystal planes for the nonhydrogenated
as well as the hydrogenated phases. The electronegativity
difference between Ce and Ni is 0.8, which indicates that
ionic interaction between these atoms is most probable.
This was indeed confirmed by the charge density analysis,
which shows that charges in CeNi3 and Ce2Ni7 intermetallics are distributed in a spherically symmetric manner and
there is no finite electron density present between Ni and
Ce. In contrast, in the hydrides CeNi3D2.67 and Ce2Ni7H4.5
(charge density distribution is shown in Figs. 8a and b,
respectively) Ni is bonded with H in a directional manner
indicating covalent type interaction. In contrast to the
[Ni0H41]4 complex, where hydrogen exists in a hydrido-
form holding a charge of 1, results from electronic structure calculations show that a) further to a partial negative
charge on H, 0.5 to 0.6 e , Ni is also carrying negative
charge reaching a maximum value of 0.3 and does not
have an 18-electron configuration formed in case of
[Ni0H41]4; and b) a partial positive charge on Ce is
rather low, smaller than 1.5 and, obviously, far away from
Ce3+ or Ce4+ configurations. From these observations we
conclude that [Ni0H41]4 complexes and Ce3þ/4þ ions are
not observed in CeNi3H2.8, Ce2Ni7H4.7 and Ce2Ni7H4.0. In
addition to the small difference in electronegativity values
between the Ni and H (only 0.3), the spatial and energetic
degenerate nature of electrons contributes to the covalent
Ni––H bonding interaction. Consequently, structural chains
. . . H––Ni––H––Ni––H . . . are formed in the CeNi3H2.8,
Ce2Ni7H4.7 and Ce2Ni7H4.0 hydrides. This feature is significantly different from the behaviours of the RNiInH1.33
(R ¼ La, Ce, and Pr) phases where the formation of
dumbbell-like H––Ni––H structural subunits is observed
caused by strong Ni––H bonding; this also results in formation of very short H . . . H separations, around 1.6 A
[38]. Interestingly, interaction between Ni atoms is significantly different between the CeNi3 and CeNi3H2.8 phases.
In the CeNi3 intermetallic alloy all Ni have metallic bonding and almost similar behaviour; in contrast, in CeNi3H2.8
hydrogen induces a difference in the bonding interaction
between various Ni sites leading to shorter Ni––Ni contacts, stronger bonding part of these Ni atoms and consequent breaking of the bonds with the remaining Ni atoms
causing huge anisotropic changes in the lattice.
In an effort to quantify the bonding interaction between
atoms and estimate the amount of electrons on and between the participating atoms we have made a Bader topological analysis. Although there is no unique definition to
identify how many electrons are associated with an atom
in a molecule or an atomic grouping in a solid, it has
nevertheless proved to be useful in many cases to perform
such analyses [45–47]. In the Bader charge (BC) analysis
Fig. 8. Calculated charge density distributions in
CeNi3H2.8 (a) and Ce2Ni7H4.5 (b) along the 101
plane. The parts of the . . . H––Ni––H––Ni . . .
chains are seen.
a
b
687
Crystal chemistry and metal-hydrogen bonding
Table 6. Calculated average Bader charges
(given in e) for CeNi3, Ce2Ni7, CeNi3H2.8,
Ce2Ni7H4 and Ce2Ni7H4.5.
Atom
CeNi3
1.2
þ0.4
Ce
Ni
H
Ce2Ni7
1.1 to 1.4
þ0.2 to þ0.4
each atom in a compound is surrounded by a surface
(called Bader regions) that run through minima of the
charge density and total charge of an atom is determined
by integration within the Bader region. The calculated
BC for the non-hydrogenated phases shows that Ce always donated almost 1.2 electrons to the Ni sites (see
Table 6). Similarly, in the hydrogenated phases Ce donates 1.2 to 1.6 electrons to the host lattice, i.e. to Ni
and H sites. In the CeNi3H2.8 phase Ni at the Ni5-site
does not donate or accept electrons from the Ce sites
and, hence, calculated change in Bader effective charge
(BEC, defined as the difference between atomic charge
and BC) for the Ni5 atoms between hydrogenated and
nonhydrogenated phases is almost zero. The present calculations show that, in general, H always accepts 0.4 to
0.6 electrons from Ni and Ce. The calculated BC indicates also that in the CeNi3H2.8 phase charges are transferred from the CeNi5 layer to CeNi2 structural subunits
increasing concentration of electrons in the latter units
and, consequently, making it possible to accommodate
more H atoms in the Laves-type slabs.
Spatial chains ––H––Ni––H––Ni–– are formed in both
Ce2Ni7H4.7 (Fig. 9a) and CeNi3H2.8 (Fig. 9b). These networks contain terminal bonds Ni––H, bridges Ni––H––Ni,
and the bonds where one H is bound to three different Ni.
Central Ni atoms have the same CN 4, but different type
of coordination by hydrogen in these two materials viz.
open saddle-type coordination NiH4 in Ce2Ni7H4.7
(Fig. 7a) and a tetrahedron NiH4 in the structure of
Ni
Ni1
Ce
D
a
Ni
Ni1
Ce
D
b
Fig. 9. . . . H––Ni––H––Ni . . . chains in the crystal structure of
Ce2Ni7H4.7 (a) and CeNi3D2.8 (b).
CeNi3H2.8
1.3 to 1.6
0 to þ0.3
þ0.4 to þ0.5
Ce2Ni7H4
1.1 to 1.42
0 to þ0.28
þ0.42 to þ0.46
Ce2Ni7H4.5
1.1 to 1.5
0 to þ0.25
þ0.4 to þ0.45
CeNi3H2.8. This differentiates anisotropic hydrides from
complex hydrides containing tetrahedral [Ni0H41]4 ions,
for example, from Mg2NiH4. However, such a difference
is not surprising as anisotropic hydrides are built via formation of spatial ordered frameworks between covalently
bonded Ni and H, whereas the dominant ionic bonding
interaction prevails in complex hydrides. Formation of
these frameworks during the hydrogenation process causes
rebuilding of the metal sublattice and this is a principal
reason for the anisotropic changes in the structural parameters on hydrogenation.
Finally, the heats of formation for the studied intermetallic hydrides were theoretically calculated from the total
energies obtained for the optimized systems. For the
CeNi3- and Ce2Ni7-based hydrides, in spite of significant
differences in the Ce/Ni and CeNi5/CeNi2 ratio of slabs,
the experimentally determined heat of formation, DHH, is
very close, 22.4 and 22.6 kJ/mol H, respectively [24].
Our calculations showed that theoretical values for
CeNi3H2.8, 23.6 kJ/mol H and for Ce2Ni7H4.5, 27.4 kJ/
mol H, are close to each other and well agree with the
experimental values. For Ce2Ni7H4.0, the calculated value
of enthalpy of formation is 5.6 kJ/mol H higher compared
to that in Ce2Ni7H4.5, indicating a lower stability of the
tetrahydride Ce2Ni7H4.0 compared to the saturated hydride
Ce2Ni7H4.7. This conflicts with the result anticipated from
the PCT dependence [24] behaviour of the Ce2Ni7 ––H2
system, where stability of the hydride increases in
Ce2Ni7H4:7x (x ¼ 0–0.7) with lowering of the H content
compared to the saturated hydride Ce2Ni7H4.7. In addition,
the PCT diagram does not show a predicted first order
phase transition in the desorption isotherm of the
Ce2Ni7 ––H2 system [25], thus raising a question of influence of oxygen on the hydrogen evolution from the saturated hydride Ce2Ni7H4.7 thus modifying the behaviour of
the whole system.
A very unusual alteration of the characteristics of the
hydrogen interaction with the related Co- and Ni-containing binary intermetallics, respectively, CeNi3 and CeCo3,
Ce2Ni7 and Ce2Co7, was noted in [24]. This alteration is
obviously caused by the preferential accommodation of
hydrogen by the RNi(Co)2 layers in the materials studied
here which differs from the conventional interstitial-type
intermetallic hydrides. In the latter case Co-containing systems are characterised by a higher thermal stability of the
hydrides and, correspondingly lower values of enthalpies
of the hydrogenation DHH (see [24] where the data for
such interstitial types, conventional CeNi5- and CeCo5based hydrides are given). In contrast, an opposite behaviours are observed for the Ni and Co equiatomic compounds forming anisotropic hydrides, Ce2Ni7 –Ce2Co7 and
CeNi3 –CeCo3. These data clearly reflect the unusual behaviours of the anisotropic hydrides. Further structural, theoretical, and thermodynamic studies of the anisotropic hy-
688
drides will be of high importance to better understand
these unusual materials.
Conclusions
Hydrogenation of intermetallic alloys RT3 and R2T7
formed by rare earth elements R and transition metals
T ¼ Fe, Co, Ni proceeds via three different schemes of
interaction and yields conventional interstitial hydrides,
isotropic hydrides (types I and II), and anisotropic hydrides (type III). From crystallographic point of view,
these schemes have distinct differences in the way the intermetallic structures are transformed into the hydrides:
Type I. Lower isotropic hydrides. Moderate expansion
of the original cells proceeds along [001] and gives
hydrides RNi3H1.2–2.0, RCo3H1.3–2.1 and R2Co7H1.5–2.7
based on the RT3 and R2T7 intermetallics;
Type II. Higher isotropic hydrides. RNi3H3.4–4.3,
RCo3H3.6–4.6 and R2Co7H5.8–6.6 are formed via H uptake by the lower hydrides. Nearly similar lattice expansion proceeds along [001] and in the basal plane.
Type III. Anisotropic hydrides. Saturated hydrides of
La and Ce, R(Ni,Co)3H2.7–5.2 and R2(Ni,Co)7H4.1–6.5,
are formed by anomalously high linear expansion proceeding exclusively along [001] and reaching 35.8%
and yielding record high specific volume expansion
(5.3–6.0 A3) per absorbed H atom. Anisotropic hydrides possess distinct crystal chemistry features differentiating them from isotropic ones.
Unique structural features observed during the formation
of the anisotropic hydrides include: (a) Exclusive lattice
expansion within the RNi2 slabs. Such expansion, 60%
along [001] for the Laves layers, is associated with occupancy by D atoms of these slabs; (b) D atoms induced
rearrangements of the metal sublattice and formation of
the new types of interstitial sites; (c) Formation of an ordered hydrogen sublattice.
DOS analysis of both Ce-containing intermetallics
CeNi3 and Ce2Ni7 and their corresponding hydrides
showed that these materials have common features in their
chemical bonding. This bonding in anisotropic hydrides of
Ce with Ni may be described as having a mixed covalentionic type (Ce atoms donate 1.2–1.6 electrons to Ni and
H), with H bonded to Ni by covalent bonding and H
bonded to Ce via ionic bonding. The bonding energy of
hydrogen atoms is clearly related to the Ni environment.
The strongest bonding takes place for 4 Ni/at. H; the
weakest bonding is observed for 1 Ni, with 2 Ni/at. H and
3 Ni/at. H showing an intermediate behaviour. CDD and
ELF do not confirm the formation of the [Ni0H41]4
complexes in the hydrogenated phases. Instead, Ni is
covalently bonded with H resulting in the formation of
H––Ni––H––Ni–– spatial frameworks with different local
coordination of Ni by H (1, 2, 3 and 4 atoms). As a result, local 4-fold coordination of Ni by H to form tetrahedra or a saddle-coordination Ni––4 H does not lead to a
formation of the specific NiH4 units. This makes anisotropic hydrides completely different from the complex hydrides containing [NiH44] anions, where each atom H has
a bond with one Ni only.
V. A. Yartys, P. Vajeeston, A. B. Riabov et al.
The formation of ––H––Ni––H––Ni–– spatial chains is
the principal reason for the anisotropic changes in the
structural parameters on hydrogenation. The donation of
electrons from nonhydrogenated RENi5 parts to hydrogen
in RENi2 slabs stabilises these latter fragments.
Acknowledgments. We are grateful to the Research Council of Norway for the financial support and the computer time at the NOTUR
supercomputer facilities.
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