Reply to the comment on “Spin- and charge-ordering in oxygen-vacancy-ordered
mixed-valence Sr4 Fe4 O11 ”
P. Ravindran,∗ R.Vidya, H.Fjellvåg, and A.Kjekshus
Center for Materials Science and Nanotechnology, Department of Chemistry,
University of Oslo, Box 1033 Blindern, N-0315 Oslo, Norway
(Dated: September 5, 2007)
Based on the interpretation of experimental structural and Mössbauer data Adler concludes
that our previous theoretical results from density functional theory disagree with experiments on
the assignment of oxidation states to Fe in the square pyramidal and octahedral environments in
Sr4 Fe4 O11 . From analyses of structural information for oxides with Fe in different oxidation states
and the Mössbauer parameters such as isomer shift, quadrupole splitting, and hyperfine field we show
that, the charges residing on the different constituents can not be directly derived from structural
or Mössbauer measurements. From additional analysis of charge density, charge transfer, electron
localization function, crystal orbital Hamilton population, Born effective charge, and partial density
of states we substantiate our previous observation that the calculations unambiguously show that
the formal Fe3+ ions reside at the square pyramidal sites and the formal Fe4+ ions take the octahedral sites in Sr4 Fe4 O11 . Effects if covalency are very important to explain the electronic structure,
magnetism, and chemical bonding in oxygen vacancy ordered systems and on ignoring covalency
one can be mislead in oxidation state assignments.
PACS numbers: 75.50.Ee, 76.80.+y
Keywords: oxidation states, charge ordering, Mössbauer parameters
I.
INTRODUCTION
The assignment of actual oxidation state to ions in
mixed valent systems is a difficult task due to the uncertainties in establishing the total charge at different
sites. Recently, based on the results from first principles
calculations, we have reported [1] that formal Fe3+ ions
reside at the square pyramidal site and Fe4+ ions in the
octahedral site in Sr4 Fe4 O11 . By comparing the experimentally measured structural and Mössbauer parameters
for Sr4 Fe4 O11 with that for reference systems, Adler [2]
argued that Fe3+ and Fe4+ ions reside at the octahedral
and square pyramidal sites, respectively. We have analyzed the basis for the correlation between the charge
state with the bond length, hyperfine field, quadrupole
splitting, and isomer shift. In order to substantiate our
previous findings we have made additional calculations
and reconfirm our previous report.
Sr4 Fe4 O11 contains Fe(1)s and Fe(2)o ions which in an
ordered manner occupy alternative position in the lattice. Hence, it has proved difficult to distinguish which
of the Fe(1)s or Fe(2)o site that order antiferromagnetically. Two detailed experimental reports are available
on the assignment of oxidation state to the Fe sites and
the magnetic structure of Sr4 Fe4 O11 . However, the conclusion are contrary to each other. Hodges et al. [3] concluded that Fe(1) and Fe(2) take 4+ and 3+ oxidation
states respectively whereas Schmidt et al. [4] concluded
oppositely. Likewise, Hodges [3] et al. concluded that
the Fe(2)o moment exhibits long-range antiferromagnetic
∗ Electronic
address: [email protected]
order whereas the Fe(1)s moments are magnetically frustrated, however, Schmidt et al.[4] arrived at the opposite
conclusion. The results from our theoretical calculations
show that the Fe(1)s and Fe(2)o sites have 3+ and 4+ oxidation states, respectively, with the Fe(2)o moments ordered antiferromagnetically. It shall be emphasized that
this conclusion does not contradict the experimental results but rather the interpretation of the experiments advocated by Hodges et al. [3] on assignment of oxidation
state of the two different ions and by Schmidt et al. [4] on
assignment of ion that has long range magnetic ordering.
Hodges et al. [3] used a bond strength model proposed
by Ziólkowski [5] to assign the oxidation states 4+ and
3+ to the pyramidal and octahedral sites, P
respectively.
In simple oxides the bond strength sum ( s) around
a cation should exactly match its valence (z). In complex oxides such sums are expected to be different from
z. [5] Therefore, for a complex oxide such as Sr4 Fe4 O11
the bond-strength sum for crystallographically different
atoms will not be meaningful. So, Schmidt et al.[4] compared the sums over all cation sites of the unit cell derived using two different models; one with Fe3+ at Fe(1)s
and Fe4+ at Fe(2)o sites and the other vice versa. The
calculated average sum per iron site based on these two
models are 3.37 and 3.465, respectively. As difference
between the two models is very small, it is not possible
to assign valence state of Fe(1)s and Fe(2)o sites based
on bond strength calculations. So, Schmidt et al.[4] used
the chemistry and crystal structure information of Sr-FeO compounds and arrived at the same conclusion as ours
regarding oxidation state assignment.
2
II. CHANGE IN THE FORMAL OXIDATION
STATE OF Fe BY INTRODUCTION OF OXYGEN
VACANCIES
Let us now take the elementary chemical view point
to analyze how the oxidation state of Fe would evolve on
introduction of oxygen vacancies into the pristine SrFeO3
lattice. It is generally accepted that Fe will donate electrons and O will accept electrons in an Sr-Fe-O lattice.
According to an ideal ionic picture and a fully oxygenordered system Fe is surrounded by six oxygens in the
2− state. As Sr correspondingly prefers the 2+ state,
one can formally assign a charge of 4+ for all Fe atoms in
SrFeO3 . On the other hand, if one removes alternately
one apical oxygen from alternate FeO6 structural subunits of SrFeO3 , the Sr4 Fe4 O11 system will develop with
equal number of Fe with square-pyramidal [s ; Fe(1)O5 ]
and octahedral [o ; Fe(2)O6 ] coordinations. As the local environment of the octahedra will be approximately
same as that in SrFeO3 , one would expect that the charge
state of Fe(2)o will remain 4+ also after the introduction
of the oxygen vacancies. In contrast, there occur drastic changes in the chemical environment of the Fe(1)s
species upon the conversion from octahedral to squarepyramidal coordination. Hence it seems natural to expect changes in the oxidation state of Fe(1)s from 4+ to
3+ to maintain charge neutrality in the system. Moreover, as oxygen draws charge from the neighboring atoms
due to its larger electronegativity, it is natural to expect
that a reduction in the number of coordinating oxygen
would decrease the charge state of Fe(1)s , viz. convert
it from 4+ to 3+ state. From this simple chemical picture one can infer that the charge state of Fe(1)s and
Fe(2)o in Sr4 Fe4 O11 is in 3+ and 4+, respectively. This
observation is in agreement with the Pauling’s electrostatic valence rule [6] which states that the electrostatic
charges in an ionic crystal are balanced locally around
every ion as evenly as possible.
Consistent with the above view point, our further analyses of Fe, Co, Ni, and Cu based oxides show that, in oxides with mixed valence ions
the lower oxidation state ion generally prefers the
lower coordination number (CN) and vice versa. Examples involving Fe are Fe3 O4 [Fe2+ (4); Fe3+ (6)
where CN is given in bracket][7] and Na9 Fe2 O7
[Fe2+ (3); Fe3+ (4)][8], for Co-based oxides Co3 O4
[Co2+ (4), Co3+ (6)][9], Co2 RuO4 [Co2+ (4); Co3+ (6)][10],
CoMn2 O4 [Co2+ (4); Co3+ (6)][11], Rb5 Co2 O4 [Co1+ (2);
Co2+ (3)][12], and MnCo2 O4 [Co2+ (4); Co3+ (6)][13],
for Ni-based oxides K9 Ni2 O7 [Ni2+ (3); Ni3+ (4)][14],
for Cu-based oxides Cu4 O3 [Cu1+ (2); Cu2+ (4)][15],
TlCu2 O2 [Cu1+ (2); Cu2+ (4)][16], YBa2 Cu3 O6 [Cu1+ (2);
Cu2+ (5)][17], LiCu2 O2 [Cu1+ (2); Cu2+ (5)][18], LiCu3 O3
[Cu1+ (2); Cu2+ (5 and 6)][18], and YPb2 Ba2 Cu3 O8
[Cu1+ (2); Cu2+ (5)][19].
III.
CORRELATION BETWEEN OXIDATION
STATE AND Fe − O BOND LENGTH
TABLE I: Formal valence state (valence), number of oxygen
ions coordinated to Fe (coordination number; CN), number
of compound entries in the ICSD database [20] (entries), and
calculated average Fe-O bond length (in Å) (bond length; in
Å) for Fe based oxides.
Valence
CN
Entries
Bond length
5+
4+
4+
3+
3+
3+
3+
3+
3+
3+
2+
4
6
4
2
4
5
6
7
8
9
6
1
44
4
1
177
19
218
2
5
4
6
1.720
1.925
1.807
2.125
1.875
2.031
2.032
2.026
1.984
2.006
2.162
The idea behind a correlation between bond length and
oxidation state is that ions with lower oxidation states
should generally have larger ionic radii and consequently
occupy a relatively larger volume in a crystal. But the
opposite is apparently true for Fe in oxides. For example, Ref. 21 lists the ionic radii of Fe in the 2+ and 3+
oxidation state as 1.734 and 1.759 Å, respectively.
The average Fe−O bond length (dF e−O ) for the Fe(1)s
and Fe(2)o in Sr4 Fe4 O11 is 1.864 and 2.008 Å, respectively. The dF e−O for Fe(1)O5 is shorter than that
for typical Fe4+ ions (see Table I), whereas dF e−O
for Fe(2)O6 fits well to that for Fe3+ ions. However,
for Sr3 Fe2 O6 with Fe3+ in the square pyramidal arrangement, the average dF e−O is 1.961 Å (4×1.980 Å
and 1×1.886 Å) [22] which is considerably larger than
1.864 Å. This has apparently misled Adler to believe
that the Fe(1)s is in 4+ state. As mentioned by Schmidt
et al. [4] no reference compound with Fe4+ in squarepyramidal coordination with oxygen is available for comparison. The dF e−O for square-pyramidal (1.864 Å) coordination in Sr4 Fe4 O11 fit well with that for 3+ ions in
tetrahedral (1.875 Å) sites. On the contrary, the Fe(1)s O and Fe(2)o -O bond length are longer than that for 4+
ions in tetrahedral and octahedral coordination, respectively.
The actual oxidation state of an ion is decided by the
charge residing at the ion. This information can not
be directly derived from experimental structural data.
Moreover dF e−O depends not only on the oxidation state
but also by various other factors as discussed below. The
prominent external factors those influence dF e−O for a
particular oxidation state are temperature and pressure.
For example, in GdFeO3 , though Fe is octahedrally coordinated with formal 3+ state, the dF e−O varies between
2.027 and 2.050 Å between 4 and 230 K [23] and shrink
to 1.986 Å when subjected to a pressure of 80 kbar at
room temperature.[24]
Further, dF e−O also depends on the character of the
3
cation coordinated to the FeOn polyhedra. For example, if the cation contains lone pair, such as Bi in
Bi2 Fe4 O9 [25] the dF e−O in the Fe3+ O6 unit becomes
smaller (1.952 Å) than that in a system without such a
lone pair. Moreover, if the cation possesses d0 ions such
as Ti4+ , Nb5+ or W6+ such constituents participate in
covalent bonding which shortens dF e−O e.g. dF e−O in
WFe2 O6 is only 1.919 Å.[26] The average dF e−O also
depends on the size of the additional cation. For example, the cation radius for Ca2+ , Sr2+ , and Ba2+ in
an octahedral coordination is 1.14, 1.32, and 1.49 Å,
respectively.[27] In CaFeO3 , SrFeO3 , and BaFeO3 one
can explicitly assign the formal oxidation state for Fe to
4+. If dF e−O really should reflects a common value for
the oxidation state 4+, all these three compounds should
have exhibited the same dF e−O value. However, the measured dF e−O is 1.921, 1.925, and 1.995 Å for CaFeO3 ,
SrFeO3 , and BaFeO3 , respectively [28, 29] which clearly
reflects the important influence of the cation radii on
dF e−O .
Cation radius not only depends on the valence of the
ions but also on their spin state. Generally the ionic radius will be larger in the high-spin (HS) state than in
the intermediate (IS) or low-spin (LS) state due to the
magneto-volume effect. [30] Correspondingly the bond
length will also depend on the spin state of the ions. For
example, Fe2+ is octahedrally coordinated with sulphur
in both FeS and FeS2 where Fe is in HS and LS state,
respectively. Correspondingly the calculated Fe-S bond
length in FeS is much longer than that in FeS2 (2.453 vs.
2.255 Å). Partial covalence of the Fe-O bonds can also significantly reduce dF e−O . For example, due to covalency
effect some of the Fe-O bond lengths in the Fe3+ O6 structural subunit of FeAlO3 and FeGaO3 become as short as
1.847 and 1.868 Å, respectively.[31, 32].
The derived average bond length for Fe ions in different
oxidation states are given in Table I. From this compilation it is seen that among the different valence states, Fe
prefers to take the 3+ state in oxides. The formal Fe5+
and Fe2+ states are the least preferred oxidation states
(K3 FeO4 is the only compound with formal Fe5+ ions and
FeO, Rb6 Fe2 O5 , K6 Fe2 O5 , Cs6 Fe2 O5 , and Rb4 K2 Fe2 O5
with Fe2+ ions). Also, only a few compounds with Fe4+
ions are so far reported. A first glance at Table I suggests
that materials with higher oxidation state have shorter
dF e−O (2.162 for Fe2+ vs. 1.720 for Fe5+ ).
However, no direct correlation can be established between dF e−O and oxidation state, because for compounds
within the 3+ oxidation state category dF e−O varies between 1.875 to 2.125 Å depending upon the coordination
number (see Table I). Interestingly dF e−O for Fe3+ in
the tetrahedral coordination is much shorter than that
in octahedral or square pyramidal coordination. So, the
coordination number of the ions also plays a role in deciding dF e−O . From these analyses it is clear that dF e−O
depends not only on the oxidation state of Fe but also
on various other factors discussed above. So, one can not
use dF e−O alone to deduce the charge state of Fe.
IV.
COMPUTATIONAL DETAILS
The relativistic correction to the Schrödinger equation
affects the electronic wave function primarily in the vicinity of the nucleus and hence it may have impact on hyperfine interactions even for relatively light elements. So,
we have taken in to account all relativistic effects including spin-orbit coupling into our calculations. Further the
calculations correspond to the experimental situation at
0 K and the absence of external fields and pressure.
We have performed calculations with single-electron
eigen-value splitting which takes into account intraatomic Coulomb repulsion between electrons in 3d states
with different magnetic quantum number ml using the
so-called orbital-polarization correction.[33] Thus the orbital moment was calculated by including spin polarization, orbital polarization, and spin-orbit interaction corresponding to including Hund’s first, second, and third
rules, respectively. More details about the computational
details of the present study can be found in Ref. 1.
The BEC calculations were performed using the Vienna ab initio simulation package (VASP) [34] within the
projector augmented wave (PAW) method [35] as implemented by Kresse and Joubert [36]. The Kohn-Sham
equations [37] were solved self-consistently using an iterative matrix diagonalization method. This is based on
a band-by-band preconditioned conjugate gradient [38]
method with an improved Pulay mixing [39] to efficiently obtain the ground-state electronic structure. The
forces on the atoms were calculated using the HellmannFeynman theorem and these were used to perform a conjugate gradient relaxation. Structural optimizations were
continued until the forces on the atoms had converged to
less than 1 meV/Å and the pressure on the cell had minimized within the constraint of constant volume.
V.
OXIDATION STATE OF Fe FROM
CHEMICAL BONDING
In order to gain more insight into the bonding interaction between the constituents and their oxidation
state additional analyses of charge density, charge transfer, electron localization function, Born effective charges,
partial density of states, and crystal orbital Hamilton
population were made.
The electron densities calculated by VASP are displayed in Fig. 1a for a plane containing the Fe(1)s , Fe(2)o ,
and O atoms clearly shows that the bonding interaction between Fe and O is not purely ionic. The directional nature of the charge density distribution between
Fe and O indicates the presence of covalent character in
the bonding. The charge transfer distribution in Fig. 1b
also shows that electrons are transfer from both Sr (not
shown) and Fe to the O sites consistent with the ionic picture. If the bonding interaction between Fe and O had
been purely ionic, one would expect an isotropic charge
transfer distribution. The anisotropic distribution of the
4
O3
O3
O3
s
Fe1
O3
O3
O3
o
Fe2
1.0
o
Fe2
O3
s
Fe1
1.0
0.25
O3
O3
O3
o
Fe2
Fe2o
0.8
O3
s
Fe1
O3
O3
O3
O3
s
Fe1
s
Fe1
s
Fe1
O3
O3
O3
0.8
0.125
O3
O3
0.6
O3
O3
O3
O3
O3
0.0
s
Fe1
s
Fe1
O3
O3
O3
0.2
s
Fe1
s
Fe1
0.4
O3
O3
s
Fe1
s
Fe1
0.4
−0.125
O3
O3
O3
0.0
(a)
0.6
O3
O3
0.2
O3
0.0
−0.25
(b)
(c)
FIG. 1: (Color online) Calculated (a) charge density (in e/Å3 ), (b) charge transfer (in e/Å3 ; red and black color indicates
electron depleted and accepted regions), and (c) electron localization function along the base of the coordination polyhedra of
Sr4 Fe4 O11 . Fe(1)s and Fe(2)o are Fe in square-pyramidal and octahedral coordinations, respectively.
s
0.5
Fe(1) − Oa
s
Fe(1) − Ob
0.5
−0.5
−1.5
−1.5
ICOHP (eV)
−0.5
COHP
charge transfer thus confirms the presence of finite covalent component in the bonding.
ELF can distinguish different bonding interactions in
solids [40]. The negligibly small value of ELF between
the atoms in Fig. 1c confirms the presence of dominant
ionic bonding in Sr4 Fe4 O11 . The ELF distribution also
shows maxima at the O sites and minima at the Fe and
Sr sites (not shown) once more confirming the charge
transfer interaction from Sr/Fe to O sites. Moreover polarization of ELF at an O site toward other O sites indicate hybridization interaction. The conclusion from the
charge density, charge transfer, and ELF analyses is accordingly that the bonding interaction between Sr and
O as well as Fe and O have dominant ionic character
with finite covalent component. This is consistent with
the conclusions arrived at from the Born effective charge
analysis discussed below. These results also clearly establish the mixed iono-covalent character of the bonding
in Sr4 Fe4 O11 .
The crystal orbital Hamiltonian population (COHP)
is the density of states weighted by the corresponding Hamiltonian matrix element is calculated using
the tight binding linear muffin-tin orbital program
(TBLMTO) [41]. This is as an indicator of the strength
and nature of bonding (negative COHP) or antibonding (positive COHP) interaction [42, 43]. The calculated COHP for the Fe(1)s -Oa , Fe(1)s -Ob , Fe(2)o -Oa ,
and Fe(2)o -Ob interactions in Sr4 Fe4 O11 (’a’ denotes apical and ’b’ basal plane interactions) are shown in Fig. 2.
The COHP curves reveal that all bonding states for
the Fe−O interactions are within the valence band from
around −2 eV below Fermi level. The FeO6 octahedra
are highly distorted and hence the Fe(2)−Oa bonding
energy (−1.103 eV; defined as the integrated COHP up
to Fermi level) is stronger than the Fe(2)−Ob interactions
(−0.923 eV). The Fe(1)−O bonding interaction is along
the base of the square pyramid (−1.339 eV) is stronger
o
Fe(2) − Oa
o
Fe(2) − Ob
0.5
0.5
−0.5
−0.5
−1.5
−1.5
−2.5
−6
−4
−2
0
2
4
−2.5
Energy (eV)
FIG. 2: (Color online) Crystal orbital Hamilton population
(COHP) and the integrated COHP (ICOHP) for Fe–O in
Sr4 Fe4 O11 . Fe(1)s and Fe(2)o are Fe in the square-pyramidal
and octahedral coordination, respectively. Oa and Ob refer to
oxygen atoms in the apical and base of the polyhedra.
than along the apex (−1.123). The considerable difference in the ICOHP values for the Fe–O bonding interactions indicate that the Fe ions indeed are in two different
valence states.
In simple, high-symmetry oxides, the oxygen Born effective charge (Z ∗ ) is isotropic and is close to −2 [44].
Owing to the site symmetry, the diagonal components of
Z ∗ are anisotropic for all ions in Sr4 Fe4 O11 (Table II). If
5
10
TABLE II: Calculated diagonal elements of Born-effective∗
charge tensor (Zij
) for Sr4 Fe4 O11 in the ground-state G-AF
configuration.
8
ions have closed-shell-like character, each ion will carry
an effective charge close to its nominal ionic value (according to a rigid-ion picture). On the contrary, large
amounts of non-rigid delocalized charge flow across the
lattices during displacements of the ions owing to covalence effect [45, 46]. Consequently, one will obtain effective charges much larger than the nominal ionic values.
The charges on Sr and O are larger than they would have
been according to a pure ionic compound. This reveals
the presence of a large dynamic contribution superimposed on the static charge. Similar giant values of Z ∗
have been reported using quite different technical ingredients for other perovskite-related oxides [46, 47].
The Born effective charge (BEC) is indeed a macroscopic concept [48, 49], involving the polarization of the
valence electrons as a whole, while the charge “belonging” to a given ion is an ill-defined concept. The high
BEC values in Table II indicate that relative displacements of neighboring ions against each other trigger high
polarization. Roughly speaking, a large amount of nonrigid, delocalized charge is responsible for higher value
of BEC than the nominal charges. From the Table II
it is again clear that both Sr and Fe donate electrons
and O accepts electrons, consistent with the traditional
ionic picture. It can be recalled that for a pure ionic system one could expect more isotropic character of Born
effective charges. Considerable anisotropy in the diagonal components of BEC (Table II) and noticeable offdiagonal components at the oxygen sites (not given) confirm the presence of covalent bonding between O-2p and
Fe-3d orbitals. The covalent bonding between Fe and
O could explain the presence of anomalous contributions
(defined as the additional charge compared with the ionic
value) to the effective charge at the Fe and O sites. The
calculated average diagonal component of the BEC in Table II is less than 4 at Fe(1)s and larger than 4 at Fe(2)o
indicating the justification of assigning the formal valence
of Fe(1)s and Fe(2)o as 3+ and 4+, respectively.
The calculated total density of states for the Fe(1)s and
Fe(2)o sites are shown in Fig. 3. The DOS for the Fe(1)s
site is higher than that for the Fe(2) site throughout the
entire valence band. The integrated density of state (i.e.
the number of states) up to the Fermi level yield the total
charge within each atomic sphere and the higher value of
electrons at the Fe(1)s site (6.44) compared with that in
8
6.44
6
6
−1
Average
2.533
3.544
5.861
−3.087
−3.153
−2.861
5.67
4
4
2
2
0
−8
−6
−4
−2
0
Energy (eV)
2
4
NOS (electrons atom )
∗
Zzz
2.127
1.422
4.667
−4.718
−4.275
−1.504
−1
∗
Zyy
2.919
4.669
6.006
−2.368
−2.361
−3.252
Fe(1) DOS
o
Fe(2) DOS
s
Fe(1) NOS
o
Fe(2) NOS
−1
∗
Zxx
2.554
4.540
6.909
−2.174
−2.841
−3.827
s
DOS (states eV atom )
Atom
Sr
Fe(1)s
Fe(2)o
O1
O2
O3
10
0
FIG. 3: (Color online) The calculated site projected density
of states (DOS) and the number of states (NOS) at the Fe(1)s
and Fe(2)o in Sr4 Fe4 O11 . The Fermi level is set to zero.
Fe(2)o site (5.67) provide additional indication for Fe in
formal 3+ and 4+ states at the Fe(1) and Fe(2) sites,
respectively (see Fig. 3).
VI.
OXIDATION STATE FROM MöSSBAUER
DATA
Mössbauer spectroscopy provides an extremely local
probe for mapping changes in the charge and spin density
around aan atom and thus offers the possibility of getting information about individual spatial spin configurations. Both the electron contact density and the electric
field gradient are rather singular quantities, which measure properties of the electron gas at an extreme point
(the position of the nucleus) which is far from the region
where chemical bonding takes place. But, these quantities are significantly influenced by the chemical bond.
However, the extraction of the Mössbauer parameters
from experimental measurements for complex materials
with different crystallographic sites is often difficult as
the information from these experiments are rather difficult to interpret and far from transparent. Therefore,
reliable theoretical calculations are highly needed in order to provide a theoretical basis for the understanding
of the experimentally measured Mössbauer parameters.
The availability of high speed computers and the develop-
6
ment of new techniques for electronic structure calculations it has during recent years become possible to supply
first-principles description of the Mössbauer parameters.
Now let us provide a brief description of Mössbauer parameters from first principles calculations.
A.
Hyperfine field
The application of a magnetic hyperfine field BHF as
a local probe of magnetism is based on the empirical
fact that BHF is, to a good approximation, proportional
to the local magnetic moment. The magnetic moment
of the two inequivalent Fe sites in Sr4 Fe4 O11 varies by
0.673 µB due to the changes in the local environment.
This has important consequences for the distribution of
the hyperfine field (HF) in different sites within the unit
cell of Sr4 Fe4 O11 . The HF can be expressed as a sum
of four contributions: (1) The Fermi contact term (BFC )
arising from the nonzero spin density at the nucleus. (2)
The dipolar (BDip ) magnetic field produced by the onsite spin density. (3) The orbital term (BOrb ) from the
magnetic field produced by the current flowing around
the nucleus and the magnitude of which being proportional to the orbital moment. (4) The lattice contribution (BLat ), which is a dipolar field originating from the
moments on the other lattice sites.
The terms other than the Fermi-contact HF some times
decide the net HF of a particular site. This makes it difficult to assign the measured HF to a particular site for
systems with more than one magnetic sites. In such cases
theory may be useful for the unambiguous assignment.
As the theoretical knowledge about the different contributions to the BHF is important to understand the development of the hyperfine field at different sites in mixed
valent systems, we have calculated the various contributions to the HF.
The calculated hyperfine fields are listed in Table III,
separately for different contributions, together with the
local spin and orbital moments at each Fe sites. Compared with other compounds, for Sr2 Fe2 O5 and Sr3 Fe2 O6
the calculated HF is much deviating from the experimentally reported values. As the calculated isomer shift (see
Table IV) at the Fe site in these two compounds are in
good agreement with experiment, the deviation in the HF
is associated with the assumption of simple antiferromagnetic structure into the calculations that expected to give
incorrect BVal contribution. BFC is the most prominent
contribution to the total HF in Sr4 Fe4 O11 and hence we
will analyze it in detail in the following.
According to relativistic limit, the BFC (in T) is proportional to the spin density averaged over the Thomson
spheres centered at the Fe nuclei [50] reads
Z rT
2
(ρ↑s (r) − ρ↓s (r))dr
BFC = µ0 ge µB s
3
0
where ge is the electron gyromagnetic constant and
s=1/2. The radius of the Thomson sphere is given by the
formula rT = Ze2 /4π²0 me c2 and is much larger than the
radius of the nucleus (for the Fe atom, rT =72×10−15 m
while rN =4×10−15 m).
For simplicity, BFC can be split into core-polarization
contribution (BCore ) and the valence contribution (BVal ).
The dominant interaction determining the BCore at the
Fe site is the exchange coupling of the inner core s shells
with the 3d orbitals of the probe atom. Because of the
shielding effect, the influence of the magnetic moments
of neighboring atoms on BCore is negligible. So, BCore is
proportional to the local magnetization. In Fe the 1s and
2s electrons lie spatially inside the 3d electrons, while the
4s electrons are outside the 3d shell. Therefore, the spin
polarization of 1s and 2s will be opposite to the polarization of 4s, leading to orientation of BCore opposite to
the 3d moment for 1s and 2s electrons and a 4s contribution that tends to cancel 1s and 2s contributions. The
2s and 4s contributions are larger than the 1s contribution, because these orbitals are spatially closer to the 3d
and hence have stronger exchange interaction. The 3s
contribution (will be smaller and can be neglected in a
qualitative discussion) due to overlap of the 3s and 3d orbitals which leads to competing and mutually cancelling
tendencies.[51]
The BCore arising from the intra-atomic s-d exchange
is expected to vary linearly, but opposite in sign, with the
local spin moment. This is indeed confirmed by our calculations. Importantly the total HFF is decided by the
BCore . However, the local-density approximation does
not allow an accurate calculation of the hyperfine field,
e.g. in the case of Fe the core contribution seems to be
underestimated by about 30%. [50, 52] It seems especially
important to improve the description of the core states,
in contrast to many other problems in condensed matter
physics for which the valence states and the nature of the
chemical bonding are of prime interest. As generalized
gradient approximation (GGA) is based on the expression of the exchange-correlation energy for the homogeneous electrons gas (depending on the density as ∝ ρ1/3
at high electron density), it fails to treat the exchange
splitting caused by the deformation of the wave function
due to the s-d exchange interaction (core polarization).
Thus the calculated BCore is always smaller than experimental values. The self-interaction correction [53] is especially important for the bound core states in order to have
a reliable value for the core hyperfine field. Several approaches have been attempted to meet this challenge and
improvement has been made.[54] However, we adopted an
ad hoc procedure by scaling the calculated BCore by 1.3
in order to reproduce the experimentally measured HF
values. The linear fit of BCore versus local moment gives
the conversion constant −12.5 and −12.95 T/µB for the
Fe(1)s and Fe(2)o sites, respectively. These values are
close to the theoretical values in the range from −10.0 to
−12.5 T/µB reported for Fex T1−x with T = Co, Cr, and
Ni by Ebert et al.[55, 56]
Although the contribution of BVal to the hyperfine field
of the magnetic atom is normally small, it is very impor-
7
TABLE III: The calculated spin (µspin ) and orbital (µorb ) moment (in µB /atom) and calculated and experimental hyperfine
field parameters (in T) for Sr4 Fe4 O11 and other related oxides.
Compound
Sr4 Fe4 O11
SrFeO3
Sr2 Fe2 O5
Sr3 Fe2 O6
LaFeO3
Atom
Fe(1)s
Fe(2)o
Fe
Fe(1)
Fe(2)
Fe
Fe
µspin
2.86
3.53
2.89
3.21
3.38
3.33
–
µorb
0.01
0.04
0.15
0.04
0.02
0.05
–
BVal
18.782
16.262
10.225
26.110
28.584
25.559
8.903
BFC
BCore
−35.825
−45.778
−36.119
−42.723
−43.229
−43.970
−47.269
tant for the understanding of the local electronic and
magnetic properties of a system due to its sensitivity
to the neighboring environments and interactions with
neighbors. It is suggested [57] that the BVal can be denoted as
BFC,Val = Aµ4s (i) +
X
Dij nij µ(j)
j
where µ4s (i) is the on-site 4s magnetic moment of the ith atom, nij and µ(j) are the number and moment of the
neighboring j-site magnetic atoms, and A and Dij are
hyperfine coupling constants related to the hybridization
interaction and moment at both Fe and O sites. The
loc
first term (BVal
) is positive and comes from the local valence contribution, depending on the 3d moment at the
probe site. The second term (BVal,Tr ) is due to the transtr
ferred contribution is either positive or negative. As BVal
is due to the conduction electrons which are polarized
via the RKKY interaction, it depends upon the moment,
magnetic coupling, and the configuration of the neighboring atoms. It increases with an increasing number
of ferromagnetically coupled surrounding Fe atoms and
decreases with the number of AF coupled Fe atoms.[58]
The large positive value of BVal indicate (Table III) that
loc
the BVal
and hence the local 3d moment plays an important role in deciding the BVal . In general, if the Fe-O
distances are larger the hyperfine coupling between the
tr
atoms will be weaker, leading to a smaller BVal
when the
neighboring magnetic moment remains constant. So the
transferred HF at the Fe(1)s site will be larger than that
at the Fe(2)o site. The coupling is of AF character and
leads to the negative partial 4s magnetic moment of the
probe atom.
The net total magnetic moment from the oxygen
ions surrounding Fe(1)s and Fe(2)o sites are 0.363 and
0.218 µB , respectively. The average Fe(1)-O distance is
shorter (1.874 Å) than the Fe(2)-O distance (2.032 Å).
Further, our chemical bonding analysis shows that the
hybridization interaction between Fe(1)s and O is relatively stronger. The positive growth of BVal,Tr for Fe(1)s
is an effect of the increase of the magnetic polarization
of the oxygen atoms which (via repopulation) enhances
the spin polarization of the 4s states of the Fe(1)s atoms
BTot,FC
−17.043
−29.508
−25.894
−16.613
−14.645
−18.410
−38.367
BOrb
1.44
−1.86
0.82
2.62
−1.57
3.54
–
BDip
5.55
−1.52
0.13
−1.24
−2.77
4.45
–
Btheo.
Tot
−10.04
−32.88
−24.94
−15.23
−18.98
−10.42
−38.36
BTot,adhoc
−20.79
−46.62
−35.77
−28.05
−31.95
−23.61
−52.54
Bexpt.
–
45
33
54
45
52
56
despite small local moment at this site compared to the
Fe(2)o site. Effectively, the positive BVal,Tr added to the
Val
positive Bloc
contribution produces relatively large positive values of BVal at the Fe(1)s site. As a result, the
BFC,Tot at the Fe(1)s site is much smaller than that at the
Fe(2)o site (see Table III). The local s magnetic moments
at the Fe(1)s and Fe(2)o site are only 0.018 and 0.017 µB .
Generally speaking, the bonding states induced by sd hybridization produce a negative contribution to BFC,Val
while the antibonding states lead to a positive contribution to the same. It may be inferred that the larger
number of d electrons at Fe(1)s have enhanced the sd hybridization, resulting in a larger specific spin polarization
of the valence s electrons at the nucleus, in spite of the
fact that magnetic moment of Fe(1)s is smaller than that
of Fe(2)o .
The calculated BFC,Val for both Fe sites are positive
despite the moments at the considered sites are positive
indicating that the hyperfine coupling constants are positive resulting from the antibonding Fe-O states.[59] As
BVal and BCore are of opposite sign, the sum represents
a decreases in the magnitude of the total HF.
For the calculation of BOrb and BDip the method of
Blügel et al. [50] was employed. The dipolar term is
defined as
1
R
R
[σ.µN − 3(σ. )(µN . )]|ψi.
r3
r
r
where µN is the magnetic dipole moment, σ is the spin
index, and ψ is the solution of the Dirac equation solved
for a potential V (R), 0 ≤ R ≤ ∞. First, the dipolar HF
does not depend on the direction of motion of an electron, such that ±m orbitals yield the same dipolar HF.
Second, BDip depends explicitly on the the electron spin,
such that an electron with opposite spin in the same orbital m yields an opposite field. The spin dipole contributions at the Fe(1) and Fe(2) sites are 5.553 and −1.521 T
respectively.
The orbital contribution to the magnetic moment and
to the hyperfine fields are caused by an unquenching of
the orbital moment due to spin-orbit coupling. The orbital contribution to the HF is very sensitive to the local
symmetry. Because the hyperfine interaction takes place
in the vicinity of the nucleus, where relativistic effects
BDip = µB hψ|
8
such as the mass velocity enhancement, the Darwin-term,
and the spin-orbit coupling have their strongest influence
on the electronic wave functions, it is expected that these
effects are also quite important for the hyperfine interaction. The orbital contribution to the HF is defined as
e
L
hψ| 3 |ψi
mc
r
where L is the orbital momentum. Orbitals with opposite quantum number m yield opposite BOrb . Because,
in orbitals with opposite m, the electrons move in opposite directions due to different orientations of the angular
momenta. In some cases BOrb is positive and comparable with BFC and hence the resulting HF is smaller than
that originating from the FC term. [60] In some cases
BOrb is much larger than BFC and hence the net HF will
be decided by the orbital moment.[61] The BOrb can be
obtained by including spin-orbit coupling into the calculation in addition to the spin polarization. Because
the spin-orbit-induced orbital moment of iron is generally affected by electron-electron interactions, such calculated orbital moments usually come out much smaller
than the experimentally measured values.[62] By including Hund’s second rule (through the orbital-polarization
correction [33]) one can remedy this deficiency. The calculated orbital moment at the Fe(1)s and Fe(2)o sites
are 0.007 and 0.041 µB , respectively and these values are
much smaller than that found for elemental Fe (0.08 µB
experimentally.[63] The proportionality between the orbital moment and the contribution to HFF is much larger
than that between the spin moment and the core polarization part of the FC term (BCore ). Hence, even though
the calculated orbital moment at the Fe(1)s and Fe(2)o
sites are small the BOrb takes noneligible value of 1.435
and −1.859 T for Fe(1)s and Fe(2)o , respectively.
The lattice contribution, which is typically small in
ferromagnets and even smaller in an antiferromagnets.
Hence the calculated lattice contribution is 8×10−3 and
9×10−3 T for Fe(1)s and Fe(2)o , respectively.
In conclusion the detailed analysis of the various contributions to the HF it is clear that the BCore is the deciding factor for the HF at both Fe sites and indeed directly
related to the magnetic moment at the corresponding Fe
sites. The quantity which decides the charge state of the
ion is the total charge at each site. However, the measured HF reflects the spin density at each site which is
the difference between the majority and minority spin
electrons at the site concerned. As the spin density is independent of the total charge density and dependent on
the exchange interaction, one can not obtain information
about valence state of ions from the measured HF of the
Fe in Sr4 Fe4 O11 .
BOrb = −µN
B.
Isomer shift
The nucleus and its electrons interact in several ways,
the most obvious being the electrostatic attraction. If
the Fe57 nuclear charge distribution were the same for
the I = 1/2 ground state and the I = 3/2 excited state,
then the electrostatic energy of the system comprising
electrons plus nucleus would be the same for both states.
In fact the excited Fe57 nucleus is 0.1% smaller in radius than the ground state nucleus, which causes the
Mössbauer transition energy to depend on the electron
density at the nucleus. This effect produces a so-called
isomer shift (∆IS ) of the Mössbauer spectrum. The ∆IS
reflects the s-electron probability density at the nucleus
since only the s partial waves extend into the nuclear
regime. Mainly two terms contribute to the IS; one is
the contact density coming from 4s-like electrons in the
conduction band [ρ4s (0)] and the other is the contribution from the shielding effect on the 3s density [ρ3s (0)]
by the modified 3d-like band electrons.
It is often problematic to assign the experimentally observed ∆IS for the respective atomic sites in mixed valent
systems [64]. Hence we have used density functional calculations to estimate ∆IS of Fe in various compounds.
The isomer shift of a nuclear transition energy is given
by
∆IS = α[ρa (0) − ρs (0)],
(1)
where ρa and ρs are the electron charge densities at the
nuclear positions (the contact densities) of absorber (a)
and the source (s) materials, respectively. If ∆IS is measured as a relative velocity of the absorber and source
materials, the calibration constant, α is given by
α = β.∆hr2 i
where β is a numerical constant and ∆hr2 i is the difference between the mean square radius of the Mössbauer
nucleus in its excited and ground states. The calibration constant depends only on the probe nucleus and can
be determined from the comparison between results from
band structure calculations and actual experimental determinations of the isomer shift. In order to calculate
∆IS from the difference in electron contact density between the source and absorber, we have calculated the
calibration constant using the calculated charge density
at the Fe nucleus [ρ(0)] for several compounds and established the linear relation between ρ(0) and the experimentally measured ∆IS values. The electron contact
density was calculated as the average electron density in
the nuclear volume corresponding to a sphere of radius
R = 1.2A1/3 fm.
The ∆IS values given in Table IV are relative to α-Fe
and refer to low temperature. The experimental accuracy is not quoted, but is typically of the order 0.01–0.05
mm/s. However, the calculated values are subject to systematic errors, which are difficult to assess, for example,
the validity of GGA. The degree of agreement between
experimental and theoretical values may be judged from
plots of the experimental ∆IS versus theoretical electron
contact densities as shown in Fig. 4. The straight line
9
FeZr3
FeZr2
13215
YFe2
−3
Density at the nucleus (ea0 )
through the points in Fig. 4 confirm the linear relationship, whereas the scatters around the line reflect the combined experimental and theoretical uncertainties. The
calibration constant derived from the best linear fit of
Eq. 1 is α = −0.360 mm/s. It can be seen that ∆IS of
Fe(1)s is less than that of Fe(2)o in Table IV.
The shorter Fe(1)-O bonds allow more spreading of the
3d electrons in to the oxygen site which reduces the 3d3s shielding effect. The calculated s, p and d electron
density at the atomic sphere are 0.471, 0.743, and 6.019
for Fe(1)s and 0.383, 0.578, and 5.867 for Fe(2)o . The
increased number of Fe(1)s d electrons causes a shielding
of the s component of the wave function away from the
nuclear region with a resulting lower contact density.[65]
Another factor contributing to the difference between
the ∆IS for Fe(1)s and Fe(2)o is the overlap interaction
between 3d and 4s orbitals. Marshall [66] has pointed
out that the overlap of the Fe inner shells with ligandwave functions produces a significant change in ρ(0). The
mechanism for this change is called the overlap distortion.
Due to the spherical symmetry of the 4s orbitals, their
overlap interaction with the ligands will be larger than
that of 3d electrons. So, the back donation provides a
possible mechanism for an explanation of the smaller ∆IS
at the Fe(1)s site than usually found for Fe3+ ion. The
present study suggests that the covalency of Fe(1)-O is
stronger than that of the Fe(2)-O bonds, evidenced by
the smaller ∆IS for the Fe(1)s site than for the Fe(2)o
site. This is consistent with our bonding analysis which
shows strong covalency for the Fe(1)-O bond. The 1s
and 2s electron density at the nucleus is independent of
the chemical environment of the Fe ion. However, the 3d
electrons shield the 3s electrons and hence reduces their
presence at the nucleus. However this picture is made
more complicated by the involvement of Fe 4s character
in the bonding orbitals. The 4s character also reduces
the ∆IS , and as usual it is not clear which of the two
effects is dominant.
Let us now analyze the factors influencing ∆IS in more
detail. The ∆IS may in principle be used to determine
the charge state of iron, but in practice this parameter
is generally not very sensitive for iron compounds with
strong covalent bonds. [67] It is well known that on going from Ge to C(diamond) the covalency increases and
correspondingly the ∆IS decreases from −0.11 to −0.39
mm/s. [68] There exists a correlation between the covalency of Fe and ∆IS (see the ∆IS -ionicity phase diagram
in Ref. 86). From a detailed analysis of various influencing factors on ∆IS , Ingalls et al.[70] concluded that
one should not compare the ∆IS for compounds having
covalent bonds with those having distinct ionic character. Ohashi [71] suggested that increase of covalent bonding between iron cations and oxygen shortens the bond
length and takes away electrons from the Fe site and this
results in low ∆IS . Moreover, for partially empty 3d orbitals, covalency indirectly influences the isomer shift via
the screening of the 3s and eventually the 4s electrons.
The value of the ∆IS gives a hint of the degree of
13214
bccFe
FeS2(pyrite)
FeS2(marc)
13213
13212
13211
FeF2
−0.25
0
0.25
0.5
0.75
1
Experimental isomer shift (mm/s)
1.25
FIG. 4: The calculated contact density at the Fe site versus experimentally measured isomer shift. The linear fit is
represented by the line.
delocalization of electrons which lowers the 3d density
compared to high-spin compounds. If the delocalization
becomes stronger the the ∆IS will be smaller. This is
clearly demonstrated in the decrease of ∆IS for Fe2+
in the sequence FeTe2 (0.47), FeSe2 (0.39) and FeS2 (0.27
mm/s),[72] and for Fe4+ ions FeSb2 (0.45), FeAs2 (0.31),
and FeP2 (0.09 mm/s)[73].
According to a pure ionic picture, Fe4+ should be characterized by a low electron contact density, since a large
fraction of the 4s electrons are dragged from the Fe atoms
in more ionic compounds. This gives smaller ∆IS for
Fe4+ ion than that for Fe3+ ion in oxides. This may have
lead Adler to believe that Fe4+ resides at the squarepyramidal site as the estimated ∆IS is smaller in this
site. However, for compounds with partial covalency, the
4s electrons will participate more efficiently in the covalent bonding because of their spherical symmetry. This
indicates that if the Fe ions in two different sites have
the same formal valence states, ∆IS will be drastically
reduced for the species which has more noticeable covalent bonding with its neighbors.
The ∆IS value depends also on external factors such
as pressure and temperature, for example for the Fe3+
high-spin case the ∆IS change from 0.36 mm/s at room
temperature to 0.48 mm/s at 4.2 K. [74] The temperature dependence of ∆IS is usually due to the second order Doppler shift. Generally ∆IS also decreases with increase of pressure.[75–78] In Sr4 Fe4 O11 the smaller Fe(1)O bond length exerts ’pressure’ on the Fe(1)s sublattice
resulting in a more compressed p electron cloud on the
surrounding O atoms which causes the partial s wave
to diminish its value at the nucleus through shielding.
On lattice compression the energy levels of 4s electrons
10
of transition metals increase faster than the 3d electrons
due to the larger overlap of the 4s orbital. As a consequence, the 4s → 3d electron transfer will be energetically more favorable with compression which will also reduce the s electron density at the nucleus. We have also
found a smaller moment at the Fe(1)s site even though
it has more valence electrons than the Fe(2)o site. This
indicates that the s electrons are more spread out due
to more delocalized nature of the electrons and this results in relatively weaker exchange interaction and hence
smaller ∆IS than that of Fe(2)o .
For the estimation of ρ(0) the knowledge of the valence
s, p, and d occupation is not only sufficient but one should
indeed also consider the radial and angular perturbations
of each occupied valence wave function including direct
effects on the core due to the neighbors. The contact
density also depends on the spin state of the ion, for the
high-spin state it is larger than the low-spin state.[70]
In addition ∆IS depends on the coordination number
of Fe. For example, Fe3+ in trigonal planar (Cu5 FeS4 )
and tetrahedral coordinations (CuFeS2 ) are 0.37 and 0.23
mm/s, respectively.[64] The spread in ∆IS corresponding
to Fe in a particular valence states is so large that in
some cases it more or less overlaps with the ∆IS of Fe
in other valence states. For example, the ∆IS for Fe3+
in the oxides spread over a range of about 0.4 mm/s
which covers a large portion of the ∆IS values for Fe2+
ions.[70] Also, for Fe in the same valence state, ∆IS in
the tetrahedral coordination in garnets is always smaller
than ∆IS for the octahedral coordination [79] and it is
related to the coordination number change and also the
Fe-O distance for tetragonal sites is about 7% shorter
than that of octahedral sites indicating the importance
of bond length.
In conclusion, the ∆IS value is determined by the s
electron density at the nucleus which depends on the degree of localization of the electrons at a particular site
(i.e. localized electrons have large contact density and
hence larger ∆IS ). However the charge state of an Fe
ion is decided by the valence electrons that comprise of
not only the s electrons but also the p and d electrons.
So, not only the charge states but also the coordination
number, bond length, spin state, nature of bonding interaction with neighbors etc. are responsible for the actual
size of ∆IS . The changes in the shape of the s electrons
distribution by shielding and hybridization effects are the
main reasons for variation in ∆IS between the different
Fe atoms in Sr4 Fe4 O11 . From a detailed analysis of the
origin of isomer shift, it is clear that one can not obtain
the information about the total charge density at the
probe site which decides the valence state. Based on our
band structure results Fe(1)s can formally be assigned as
Fe3+ . However the presence of strong covalency at Fe(1)s
reduces ρ(0) and hence ∆IS becomes lower than usually
expected for Fe3+ ion. So, ∆IS of pure ionic compounds
can not be taken as references for assigning oxidation
state for constituents in compounds with partial covalency. In summary, it can be said that the magnitude of
the IS in two different Fe sites in Sr4 Fe4 O11 rather reflects the strength of the covalent bonding of them with
oxygen.
C.
Quadrupole splitting
The experimentally observed quadrupole splitting
(∆Q ) reflects the quadrupole-quadrupole interaction between the nucleus and the surrounding electron cloud. It
can be written in terms of the nuclear quadrupole moment and the electric field gradient (EFG), produced by
the electrons at the position of the nucleus.[80, 81] The
quadrupole splitting may provide a rather indirect indication of the valence state, however it is usually impossible to draw unambiguous valence state conclusions
from ∆Q . [67] Since the EFG tensor is directly related
to the asphericity of the electron density in the vicinity of the probe nucleus, the quadrupole splitting allows
the estimation of covalency or ionicity of the chemical
bonds in the solid, provided the quadrupole moment is
known. Unfortunately, if a system contains more than
one equivalent site, experiment does not reveal explicitly which environment corresponds to which EFG.[78]
Hence, a theoretical estimation of the EFG in different
sites will complement to the experimental measurements.
For an I = 3/2 to I = 1/2 transition, as in Fe, the
quadrupole splitting is given by
1
∆Q = e|QVzz |
2
r
1+
η2
,
3
(2)
where e is the elementary charge, Q is the quadrupole
moment of the excited Mössbauer nucleus, and Vzz and
η are the EFG and asymmetry parameter, respectively.
The asymmetry parameter η is a measure of the deviation
from uniaxial symmetry and hence ∆Q is a measure of
the deviation from cubic symmetry.
The EFG is a traceless symmetric tensor of rank two,
defined as the second derivative of the Coulomb potential
at the position of the nucleus, viz. V ij = ∂ 2 V ∗ /∂xi∂xj,
where V ∗ is the potential due to electrons in non-s states.
The Coulomb potential in the crystal can be determined
from the total charge density by solving the Poisson’s
equation. The EFG can be computed directly from the
charge density using a method developed by Schwarz et
al. [82] This approach is characterized by the component with the greatest modulus Vzz and the asymmetry
V −V
parameter η = xxVzz yy where the principal components
have been chosen in such a way that |Vzz | > |Vyy | > |Vxx |.
Vzz can be written as
r
Vzz = 2
4π
5
Z
ρ(0)
Y20 (b
r)dr.
r3
with the spherical harmonic Y20 (b
r). Note that the spherically symmetric part of ρ(r) does not contribute to the
EFG.
11
TABLE IV: Calculated and experimental Mössbauer parameters (given in brackets) for Sr4 Fe4 O11 and other related oxides.
Compound
Sr4 Fe4 O11
SrFeO3
Sr2 Fe2 O5
Sr3 Fe2 O6
LaFeO3
∆IS (mm.sec−1 )
0.136 (-0.03)
0.572 (0.47)
0.291 (0.15)
0.649 (0.49)
0.361 (0.29)
0.522 (0.48)
0.620 (0.47)
Atom
Fe(1)s
Fe(2)o
Fe
Fe(1)
Fe(2)
Fe
Fe
EFG (1021 .Vm−2 )
1.623
-5.201
1.617
4.779
9.363
3.787
-0.583
The calculated EFG can be very sensitive to small
changes in the charge distributions, especially near the
nucleus, hence highly accurate calculations are needed.
As the EFG reflects the asphericity of the charge density
distribution near the probe nucleus, it is directly related
to the electron density distribution in the crystal, the
nature of the chemical bond, and symmetry of the nearest environment of the chemical bond. The EFG has
three main contributions [83] First one is the asymmetric distribution of the valence electrons of the atom under
consideration defined as,
Vzz,Val = h
ρ(3z 2 − r2 )
i
r5
Second, the lattice contribution arising from the charges
on the lattice surrounding the Mössbauer atom in a lattice of noncubic symmetry by the effect of the crystal
“residue” (external with respect to the site selected).
Third, the contribution of point charges QV and polarization of inner shells (core polarization). It is clear that
this contribution is a consequence of the influence of the
two former effects on the core electrons, that are otherwise spherically symmetric and produce no EFG. This
contribution is defined as
Vzz,nucl =
X Qv (3z 2 − r2 )
v
v
rv2
v
,
where the summation is performed over the atoms of the
calculated fragment.
The positive Vzz,Val values at the Fe(1)s site (Table IV)
show that more charges are accumulated in the xy plane
and the contributions of dx2 −y2 , dxy , px , and py orbitals
of Fe(1)s dominate over those of dz2 , dxz , dyz , and pz
orbitals (oriented along the z axis). This is associated
with the creation of the oxygen vacancy at the apical site
which alters both charge density and (hence) the EFG.
On the other hand the negative contribution at Fe(2)o reflects charge accumulation along the z axis as well as the
strong covalent bonding between Fe(2)o and apical oxygen atoms. The covalency effect [84] which involves the
Fe 3d and 4p valence orbitals plays an important role for
the size of EFG and ∆Q . It is worthy to note that COHP
analysis also shows stronger Fe(1)-Ob bonding while the
η
0.706
0.089
0.001
0.452
0.773
0.000
0.622
∆Q (mm.sec−1 )
0.146 (0.35)
-0.434 (-0.67)
0.135
0.411
0.854
0.315
-0.052
strongest Fe(2)-O bonding interaction is perpendicular to
the base of the octahedra i.e. between Fe(2)o and apical
oxygen (i.e. Fe(2)o -Oa ; see Fig.2).
Moreover, from Fig. 1a it is obvious that the valence
charge density is relatively more spherical symmetric
around Fe(2)o than that around Fe(1)s . This indicates
that the bonding between Fe(2)o and O is more ionic than
that between Fe(1)s and O, consistent with our finding
that the formal Fe4+ ion is at the Fe(2)o site. In contrast, the charge density at the Fe(2)o nuclear site (see
the charge transfer plot in Fig. 1b)is more deformed than
that around the Fe(1)s site, as can be seen by the significant deviation from the spherical symmetry (displaying a
fourfold rotational symmetry) close to the Fe(2)o nuclei.
While the Fe(2)o -O bonds are dominated by ionic contributions, additional significant aspherical distortions occur very close to the Fe(2)o nuclei. This results in a large
EFG compared to that in Fe(1)s site ( a considerable part
of this anisotropy stems from a region very close to the
nucleus and hence gets amplified by 1/r3 ).
Let us further analyze EFG in more detail. EFG is
more precisely determined by the l = 2 components of
the lm-projected charge density ρ2m (r). Such l = 2 components arise from the anisotropy of the p- and the dcharge densities, since s-electrons do not contribute owing to their spherical symmetry and mixed s-d and p-d
terms are very small. To a good approximation the EFG
is therefore determined by the anisotropy of the local p
and d charge [85], viz.
Vzz = a∆Np + b∆Nd
(3)
where ∆Np and ∆Nd are the anisotropic p and d charges,
which for the considered symmetry with the z axis in the
(001) direction are given by
Z
EF
∆Np =
0
Z
∆Nd =
0
1
dE[ [npx (E) + npy (E)] − npz (E)]
2
(4)
EF
dE[ndx2 −y2 (E) + ndxy (E) −
1
[nd (E) + ndyz (E)] − ndz2 (E)].
2 xz
(5)
12
The calculated ∆Np and ∆Nd values for the Fe(1)s site
are −.0082 and 0.0094 respectively and those for Fe(2)o
site are 0.0046 and 0.0346. The EFG at the Fe(2)o site
is more than three times larger than that at the Fe(1)s
site as reasoned below. The Fe4+ ion in high-spin state
has a singly-occupied eg orbital which is purely Y2,0 ∝
3z 2 − 1 (with respect to the c axis). This orbital has the
most nonspherical character among the d orbitals (see
the orbital ordering scheme in Fig.6 of Ref.[1]) and this
leads to large EFG.
In short we have accordingly been able to reproduce
the experimentally reported ∆Q for both Fe(1)s and
Fe(2)o sites. The oxygen vacancies play an important
role in determining the distribution of charge density
at the Fe(1)s site and in particular, the redistribution
of the electron density around the Fe(1)s nucleus, in a
manner that changes ∆Q from negative to positive. The
more detailed analysis shows that ∆Q does not depend on
the total charge at each site, instead depends on the the
anisotropy in the charge distribution at the nucleus. It
may be noted that for the Fe3+ ion in a same structural
frame work, ∆Q increases with increase in tetrahedral
distortion.[69] In addition the ∆IS value for two different
Sn sites having the same valence state but with different
site symmetry in SnO is almost the same but the EFG difference is almost double.[87] Even if two sites have same
total charges (i.e. same valence state) the anisotropy
in the charge distribution will be different. Hence, ∆Q
is decided by the site symmetry of the atom, character
of the electrons involved in the bonding interaction with
the neighbors, coordination number, interatomic distance
etc. Even if two sites obtain same total charges (i.e. same
valence state) the anisotropy in the charge distribution
and hence ∆Q will be different. Therefore the value of
∆Q obtained from Mössbouer measurements is not appropriate to assign the oxidation state of ions unambiguously.
VII.
CONCLUSION
Bond length depends on coordination number, spin
state, charge state, neighboring ions, bond character,
radii of the ions, temperature, pressure etc. Hence, the
bond length alone can not be used to characterize the
oxidation states of ions.
We have demonstrated that it is possible to evaluate Mössbauer parameters such as hyperfine field, isomer
shift, and quadrupole moment with reasonable accuracy
using density functional calculations. Using this we have
analyzed the origin of these parameters and found that
the hyperfine field reflects the spin density at the Fe sites
in this compound, whereas oxidation state of an ion is decided by the total charge density at each site. The spin
density only provides information about the difference in
charge density between the majority and minority spin
channels, influenced by the exchange splitting and not by
the valence. The quadrupole splitting reflects the EFG
which in turn reveals the asymmetry in the charge density distribution of electrons at the nucleus. Ions with
same valence state can have different EFGs depending
upon the coordinating atoms, the character of electrons
(s, p or d etc.), and chemical bonding with the neighbors
(directional bonding or not). EFG does not provide information about total charge at each site and hence valence
state of the ions. The isomer shift bestows the amount of
charge density at the nucleus and depends on the degree
of localization/delocalization of charges. However it does
not provide information about the total charge at a given
site. Moreover, even if two ions have the same valence
state, the degree of localization may be different depending upon the hybridization interaction with the neighbors
(a short F e − O distance will increase the delocalization
and decrease IS), nature of the electrons (s electrons will
be more spatially spread than other electrons), and especially how the electrons are distributed within the site
(i.e. if same amount of electrons distributed almost uniformly then the isomer shift will be smaller and in contrast, the distribution is such that more electrons at the
nucleus and less in the outer region then the isomer shift
will be larger). So, the isomer shift reflects the shape of
electron distribution and degree of localization of electrons, but does not provide information about the total
charge at a given site and hence oxidation state of the
ion concerned.
For pure ionic cases deduction about oxidation state
from Mössbauer parameters of reference systems may
work though there is no direct correlation. The atom
specific nature of the Mössbauer measurements can be
used to distinguish ions with different valence states, spin
density, asymmetry in charge distribution, degree of localization of electrons at different sites etc. however, this
approach can not be used to quantify the valence state
of the ions. The hither to assumed correlation between
valence states and Mössbauer parameters does not have
physical basis and such deliberation should not be used
alone to establish oxidation state especially in system
with mixed bonding involved.
Acknowledgments
The authors are grateful for the financial support from
the Research Council of Norway. Parts of these calculations were carried out at the Norwegian supercomputer
facilities. PR wish to thank P. Dederichs (Forschungszentrum Jülich), H. Akai (Osaka University), S. Blügel (
Forschungszentrum Jülich) for useful discussions and P.
Novák (Academy of Sciences of the Czech Republic,) and
S. Cottenier (Katholieke Universiteit Leuven) for helpful
communications.
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