Composition dependence of the lattice parameter in NicMg1-cO solid solutions

Article abstract of DOI:10.1088/0953-8984/10/36/004

We present an accurate x-ray powder diffraction study of stoichiometric Ni_cMg_1-cO solid solutions performed using an internal-standard method. The results obtained allowed us to establish for the first time that the composition dependence of the lattice parameter a(c) is not linear, as had been asserted, and interpreted within Vegard's model, in previous works. A new model, which is able to explain both the a(c) behaviour and the variation of the local structure around nickel ions, obtained previously by x-ray absorption spectroscopy (Kuzmin A, Mironova N, Purans J and Rodionov A 1995 J. Phys.: Condens. Matter 7 9357), is suggested and discussed.

Full text of DOI:10.1088/0953-8984/10/36/004

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J. Phys.: Condens. Matter 10 (1998) 7937–7944. Printed in the UK
PII: S0953-8984(98)92776-X
Composition dependence of the lattice parameter in
NicMg1−cO solid solutions
A Kuzmin† and N Mironova‡
† Institute of Solid State Physics, 8 Kengaraga Street, LV-1063 Riga, Latvia
‡ Nuclear Research Centre, Latvian Academy of Science, LV-2169 Salaspils, Latvia
Received 19 March 1998, in final form 9 July 1998
Abstract. We present an accurate x-ray powder diffraction study of stoichiometric NicMg1−cO
solid solutions performed using an internal-standard method. The results obtained allowed us
to establish for the first time that the composition dependence of the lattice parameter a(c) is
not linear, as had been asserted, and interpreted within Vegard’s model, in previous works. A
new model, which is able to explain both the a(c) behaviour and the variation of the local
structure around nickel ions, obtained previously by x-ray absorption spectroscopy (Kuzmin A,
Mironova N, Purans J and Rodionov A 1995 J. Phys.: Condens. Matter 7 9357), is suggested
and discussed.
1. Introduction
Diluted antiferromagnets represent an interesting class of materials whose crystallographic
structure is closely related to their magnetic properties [1]. Recently, a correlation between
long- and short-range order for solid solutions with face-centred-cubic (fcc) magnetic
sublattices has been reviewed for a number of zinc-blende- and rock-salt-type compounds
[2]. In these systems, depending on the type of the host-lattice packing, the cation/anion radii
ratio and the difference in size of the ions, the ordered average crystal lattice experiences
various local distortions, which are successfully probed by x-ray absorption spectroscopy
(XAS) [1]. Therefore it is of great interest to understand whether such local distortions
can also affect the values of the solid-solution lattice parameters determined from powder
diffraction data.
A series of the NicMg1−cO solid solutions provide examples of crystallographic structure
intensively studied by x-ray diffraction (XRD) [3–7] and XAS [7–9]. Due to the small
2+
difference (about 0.02 A [10]) between the ionic radii of Ni and Mg2+ ions, they
readily substitute for each other, so a continuous series of NicMg1−cO solid solutions,
with 0 < c < 1, can be prepared. Up to now, it has been believed that the lattice parameter
a(c) of the NicMg1−cO system depends linearly on the composition and follows Vegard’s
rule
˚
a(c) = caNiO + (1 − c)aMgO
(references [3–7].) However, the results of the recent Ni K-edge XAS studies [8, 9]
suggested strong deviation of the local structure around nickel ions from Vegard’s model:
it was observed that whereas the interatomic Ni–Ni2 distance (the subscript indicates
the coordination shell number) remains nearly constant at all concentrations, the Ni–O1
c
0953-8984/98/367937+08$19.50 ꢀ 1998 IOP Publishing Ltd
7937

7938
A Kuzmin and N Mironova
and Ni–Mg2 distances increase linearly upon dilution of the nickel oxide, i.e. when c
decreases. Such behaviour was explained by an off-centre displacement of nickel ions
[8]; this interpretation also agrees with the results of low-temperature optical absorption
and luminescence measurements [9, 11], for which displacements of nickel ions could
explain why the zero-phonon-line splitting occurs. Thus it becomes important to understand
the reason for the ‘disagreement’ between the XRD and the XAS data available in the
literature.
In this paper, we present for the first time an accurate XRD study of NicMg1−c
O
solid solutions performed using an internal-standard method. Our results unambiguously
show that the composition dependence of the lattice parameter a(c) in NicMg1−cO deviates
from the linear one assumed within Vegard’s model. To explain the a(c) behaviour, we
suggest a new model, which also agrees with the XAS data from our previous work
(reference [8]).
2. Experimental details
Solid solutions of NicMg1−cO (c = 0.05, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9)
were prepared from the appropriate amounts of aqueous solutions of Mg(NO3)2·6H2O
and Ni(NO3)2·6H2O salts which were mixed and slowly evaporated. The remaining dry
◦
‘flakes’ were heated up to 500–600 C to remove NO2 completely. The polycrystalline
solid solutions obtained were powdered and annealed for 100 h at a temperature of 1200 ◦C
in air and then quickly cooled down to room temperature. The pure NiO (c = 1) and
MgO (c = 0) were obtained by thermal decomposition of the above-mentioned salts. Thus-
obtained NicMg1−cO powders with c > 0 have greenish colour, whose intensity depends on
the nickel content and varies from green for c = 1 to light green for small c-values. Pure
MgO is colourless.
The greenish colour of the NicMg1−cO solid solutions can be used as a ‘fingerprint’ of
their stoichiometry [12, 13]. Additionally, the composition of our samples was probed by
instrumental neutron-activated analysis [14, 15], and it was confirmed [14] that the content
of nickel in the samples was in agreement with the stoichiometric one within ±0.01%.
Note that the NicMg1−cO samples with c > 0 were the same as had been used in XAS and
optical measurements in [8, 9]. The XAS results [8], which did not reveal any deviation in
the coordination numbers of nickel ions within the first and second coordination shells in
the series, also prove the stoichiometry of the solid solutions.
All of the samples were annealed for four hours at 500 ◦C before diffraction
measurements were made. The samples for XRD study were prepared by homogeneous
mechanical mixing of finely ground NicMg1−cO and polycrystalline silicon powder. The
mixtures, in a quantity around 1–2 g, were back-pressed into a copper sample holder.
Diffraction patterns were recorded using a Bragg–Brentano powder diffractometer (θ/2θ
scans) with a graphite monochromator in the diffracted beam for elimination of sample
fluorescence. A conventional tube with a copper anode (Cu Kα radiation) was used as
the x-ray source. The measurements were performed at room temperature over the angular
range 2θ = 25–100◦ in steps of 1(2θ) = 0.02◦. The temperature during the measurements
◦
was unchanged within ±2 C, leading to an uncertainty in the determination of the lattice
parameter of about 0.003%, or less than ±1.3 × 10−4 A, as estimated from the coefficient
˚
of linear expansion ((11.4/15.6) × 10−6 ◦C−1 for MgO and (12.0/12.6) × 10−6 ◦C−1 for
NiO [15]).

The lattice parameter of NicMg1−c
O
7939
Figure 1. Powder diffraction patterns of NicMg1−cO solid solutions mixed with crystalline
silicon powder. Only a few spectra are shown for the sake of clarity.
3. Results and discussion
The XRD patterns obtained are presented in figure 1. The observed splitting of the diffraction
lines at high angles (2θ > 45◦) is due to the Cu Kα1–Kα2 doublet. As can be seen, crystalline
silicon serves as a perfect internal standard in our case, since peaks due to silicon do not
overlap the ones from the NicMg1−cO system for angles 2θ < 90◦ and they are uniformly
distributed throughout the scattering angle range. One can notice strong variation of the
(111) peak intensity for solid solutions, which occurs due to the difference between the
x-ray scattering intensities of the nickel and magnesium sublattices. The intensity of the
(111) peak in compounds with the NaCl-type structure, which is the structure type of the
NicMg1−cO series, depends on the difference between the number of electrons of the anions
and cations [16]: this explains the smaller magnitude of the (111) peak for MgO as compared
to that for NiO.
The XRD patterns shown in figure 1 were least-squares fitted by a set of pseudo-Voigt
functions taking into account both Kα1 and Kα2 components [17]. The thus-determined peak
positions were utilized in the least-squares refinement of the NicMg1−cO lattice parameters
a(c) (table 1) by the XLAT code [18] using silicon-related peaks as the internal standard
˚
(a0(Si) = 5.430 880 A; JCPDS-ICDD PDF card No 27-1402). The fitting uncertainty was
˚
less than ±0.0002 A in all cases. Thus, the main contribution to the error of the a(c) values
was due to the uncertainty in the angle determination: it was estimated to be equal to about
˚
˚
±0.0007 A for c 6 0.7 and ±0.0009 A for c > 0.7 due to complete overlap of Si(511) and
NicMg1−cO(400) reflections for c > 0.7 (figure 1). Note that several measurements for each
˚
sample led to differences in the calculated lattice parameters of less than ±0.0005 A—the
value being half that of the total uncertainty 1a.

7940
A Kuzmin and N Mironova
Table 1. The lattice parameter a(c) in NicMg1−cO solid solutions determined using an internal-
˚
standard method. 1a takes into account the uncertainty of the angle determination (∼±0.0007 A
˚
˚
for c 6 0.7 and ∼±0.0009 A for c > 0.7), the fitting uncertainty (6±0.0002 A) and the
˚
temperature-variation effect (<±0.000 13 A).
˚
c
a(c) ± 1a (A)
Reference
MgO
4.213
4.211
JCPDS-ICDD PDF card No 4-829
[6]
0.0 (MgO) 4.2123 ± 0.0010
This work
This work
This work
This work
This work
This work
This work
This work
This work
This work
This work
This work
0.05
0.1
4.2108 ± 0.0010
4.2074 ± 0.0010
4.2030 ± 0.0010
4.1999 ± 0.0010
4.1963 ± 0.0010
4.1920 ± 0.0011
4.1891 ± 0.0011
4.1863 ± 0.0011
4.1830 ± 0.0012
4.1801 ± 0.0012
4.1773 ± 0.0012
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0 (NiO)
NiO
4.1769
4.177
4.1765
4.180
JCPDS-ICDD PDF card No 4-835
[19]
[20]
[6]
The values obtained for the lattice parameters a(c) and their total uncertainties 1a are
reported in table 1. As can be seen, our results for pure NiO and MgO are in excellent
agreement with the known values published by other authors [6, 19, 20]. Also, the values
of a for pure NiO correspond in all cases to the ideal cubic cell which in reality exists in the
paramagnetic phase above the Ne´el temperature TN = 523 K [21]. According to [20, 22],
the NiO structure at room temperature experiences rhombohedral distortion; however, this
is so small that appreciable splitting of the reflections can only be detected at high angle
values (above 129◦).
The variation of the lattice parameter a(c) versus the nickel concentration c is shown
in figure 2(a). A deviation is observed between the experimental points and Vegard’s
model (VM), especially for the intermediate concentrations. By looking at the difference
a(c) − aVM(c) in figure 2(b), one can see that it is larger than the estimated uncertainty of
˚
a(c), and its maximum value reaches about −0.002 A.
Assuming statistical distribution of nickel and magnesium ions, the lattice parameter
a(c) of the NicMg1−cO system is given by
a(c) = c2aNN(c) + 2c(1 − c)aNM(c) + (1 − c)2aMM(c)
(1)
where the first term represents the contribution from all Ni–Ni atom pairs, the second term
is due to all Ni–Mg (Mg–Ni) atom pairs and the last term is due to all Mg–Mg atom pairs
(here only the metal–metal pairs located along unit-cell edges in the h100i directions are
considered). In terms of the nearest-neighbour metal–oxygen bonds, the lattice parameters
are aNN = 2R(Ni–O), aMM = 2R(Mg–O) and aNM = R(Ni–O) + R(Mg–O). Note also that
aNN(1) = aNiO and aMM(0) = aMgO
.
In Vegard’s model, the values of the Ni–O and Mg–O bond lengths in the solid solutions

The lattice parameter of NicMg1−c
O
7941
Figure 2. (a) The variation of the lattice parameter a(c) (full circles) in NicMg1−cO. (b) The
difference between the values of the lattice parameter a(c) obtained and the prediction of
Vegard’s model aVM(c). In both panels, solid lines represent Vegard’s model, and dashed
lines correspond to the model suggested in the present work and described by equation (3).
are considered to be constant as in NiO and MgO. Therefore, one has aNN(c) = aNiO
,
aMM(c) = aMgO and aNM(c) = (aNiO + aMgO)/2, so, from equation (1), the lattice parameter
a(c) of the solid solution becomes equal to the additive sum of the lattice parameters of
individual components weighted with the concentrations of each part:
aVM(c) = caNiO + (1 − c)aMgO
.
(2)
The same expression for the lattice parameter can be obtained in the virtual-crystal
approximation (VCA) which assumes that the atoms are located on the ideal-lattice sites
of the average unit cell, and average values for the bond lengths, bond ionicity, atomic
potentials etc are considered. Therefore, in the framework of the VCA, the metal–oxygen
(MO) and metal–metal (MM) distances vary as the lattice parameter a(c) does, and one has
aNN(c) = aNM(c) = aMM(c) = caNiO + (1 − c)aMgO
.
Thus, Vegard’s model and the VCA model represent two extreme cases from the short-
range-order point of view based on the contradictory notions of the invariant radii, introduced

7942
A Kuzmin and N Mironova
by Pauling [23], and of the completely relaxed bonds, respectively. However, they both
suggest linear dependence (equation (2)) of the lattice parameter on the composition, which
is not consistent with the results for NicMg1−cO solid solutions obtained in the present work
(figure 2).
Figure 3. The variation of the Ni–Ni2 (open circles) and Ni–Mg2 (full circles) interatomic
distances between the nickel ions and the Ni/Mg ions located in the second coordination shell
according to the XAS data taken from [8]. Solid and dashed lines correspond to Ni–Ni2 and
Ni–Mg2 distances calculated according to the model suggested in the present work and described
by equation (3).
To explain the experimental behaviour of a(c), we suggest the following model based
on equation (1). It is known from the XAS data [8] that: (1) the average Ni–Ni2
interatomic distance remains nearly constant at all compositions; therefore we suppose that
aNN(c) = aNiO = constant; and (2) the average Ni–Mg2 interatomic distance varies around
the values given by the VCA model (figure 3); therefore we approximate its composition
dependence by a linear function aNM(c) = f (b1 + b2c) (note that the coefficient b2 should
be negative to achieve an increase of the Ni–Mg2 distance when c decreases). Also, taking,
˚
according to Shannon [10], the radius of the oxygen ion equal to 1.40 A, one obtains that
the ionic radius of the magnesium ion, R(Mg2+) ' 0.71 A, is larger than that of the nickel
˚
ion, R(Ni2+) ' 0.69 A. Therefore it is reasonable to assume that the larger magnesium
˚
ions are squeezed by the oxygen sublattice and, thus, that they are always located at the
regular sites of the crystal lattice. As a result, the composition dependence of the Mg–Mg
interatomic distance should follow the lattice parameter and can be approximated by the
expression used in the VCA model: aMM(c) = caNiO + (1 − c)aMgO. Finally, one gets the
expression for the lattice parameter a(c) of NicMg1−cO in the form
a(c) = c2aNiO + 2c(1 − c)(b1 + b2c) + (1 − c)2(caNiO + (1 − c)aMgO
)
(3)
with aNiO and aMgO defined in table 1 and b2 < 0. The coefficients bi were determined by
a least-squares fit of equation (3) (see the dashed lines in figure 2) to the composition
˚
dependence of a(c) obtained by XRD: they are b1 = 4.206 ± 0.001 A and b2
=
˚
−0.014 ± 0.002 A. It is important that, unlike Vegard’s model and the VCA model, the
suggested model (equation (3)) is in good agreement with our XRD data and also satisfies
the XAS results from [8] (figure 3).

The lattice parameter of NicMg1−c
O
7943
Finally, it is interesting to estimate the type of local deformation around impurity ions
in MgO and NiO host lattices. To do this, we compare the values of aNM, calculated from
the coefficients bi, with the host-lattice parameter a(c) at c = 0 and c = 1, i.e. when the
impurity Ni (Mg) ion can be considered to be isolated in the host MgO (NiO) matrix. At
˚
˚
c = 0, aNM = 4.2075 A is smaller than aMM = aMgO = 4.2123 A but it is larger than
˚
the sum (4.1948 A) of the undistorted Ni–O and Mg–O distances. This means that in
Ni-doped MgO crystals, the local environment around nickel impurities relaxes partially in
the direction of the impurity ions, i.e. the host lattice shrinks around nickel ions, as they
have ionic radii smaller than those of the magnesium ions. This conclusion is consistent
with the result of the Ni K-edge XANES analysis reported in [24] where it was observed
that the overlap between oxygen 2p and nickel 3d orbitals increases at small c-values. At
˚
˚
c = 1, aNM = 4.1925 A is larger than aNN = aNiO = 4.1773 A but it is smaller than the
sum of undistorted Ni–O and Mg–O distances. This means that in Mg-doped NiO crystals,
the local environment around magnesium impurities relaxes partially in the direction away
from the impurity ions, i.e. magnesium ions move the surrounding host lattice aside. Thus,
substituting Mg (Ni) ions induce local deformations of the host NiO (MgO) lattice, and the
type of deformation is in agreement with the one which can be expected from the relative
values of the ionic radii.
4. Summary and conclusions
The lattice parameter a(c) of stoichiometric NicMg1−cO solid solutions was determined
by x-ray powder diffraction using an internal-standard method. It was found for the first
time that the composition dependence of the lattice parameter a(c) deviates from the linear
Vegard model obtained in previous work [3–7].
To explain this result, a combination of the lattice parameter a(c) determined by
XRD in the present work with the values of the Ni–O, Ni–Ni and Ni–Mg interatomic
distances obtained by us previously using x-ray absorption spectroscopy [8, 9] was used to
derive a new structural model (see equation (3)), which takes into account the off-centre
displacements of nickel ions suggested previously in [8, 9, 11]. The proposed model is in
good agreement with the experimentally observed variations of the long-range order and
local atomic structure in NicMg1−cO as a function of composition.
Acknowledgments
One of us (AK) wishes to thank the Universita` di Trento (Professor G Dalba) and the Centro
CNR-ITC di Fisica degli Stati Aggregati (Trento) (Dr F Rocca) for hospitality and providing
the opportunity to carry out XRD measurements. This work was partially supported by the
Latvian Government Grants Nos 93.652, 93.279, 96.0670, 96.412.
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