Optimization of structure-property relationships in nickel ferrite.pdf

Journal of Physics and Chemistry of Solids 151 (2021) 109928
Available online 7 January 2021
0022-3697/© 2020 Elsevier Ltd. All rights reserved.
Optimization of structure-property relationships in nickel ferrite
nanoparticles annealed at different temperature
Sanjeet Kumar Paswan a
, Suman Kumari b
, Manoranjan Kar b
, Astha Singh a
, Himanshu Pathak c
,
J.P. Borah d
, Lawrence Kumar a,*
a
Department of Nanoscience and Technology, Central University of Jharkhand, Ranchi, 835205, India
b
Department of Physics, Indian Institute of Technology Patna, Bihta, Patna, 801106, India
c
School of Engineering, Indian Institute of Technology Mandi, Kamand, Mandi, 175075, India
d
Department of Physics, National Institute of Technology Nagaland, Chumukedima, 797103, Nagaland, India
A R T I C L E I N F O
Keywords:
Spinel ferrite
Citrate sol-gel
X-ray diffraction
Rietveld
Elastic parameter
Optical properties
Magnetic characteristics
A B S T R A C T
In this report, a detail analysis of the impact of annealing temperature on the structural, elastic, morphological,
optical, and magnetic behavior of NiFe2O4 nanoparticles prepared by the citrate sol-gel method is presented.
Analyzing the XRD patterns by the Rietveld method confirms that all the annealed samples have been crystallized
to cubic spinel structure belonging to Fd3
−
m space group with a single phase. Rietveld analysis demonstrates the
change in structural and microstructural parameters and movement of cations from tetrahedral to octahedral
sites and vice-versa upon annealing. The quantitative estimation of Ni2+
& Ni3+
and Fe2+
& Fe3+
has been carried
out using XPS analysis. Decreases in peak broadening and shift of five Raman active peaks towards higher fre
quency upon annealing have been analyzed using the phonon confinement model. The variation in elastic pa
rameters with annealing temperature has been assessed by FTIR analysis. The UV analysis reveals the increase of
the optical energy band gap and the decrease of Urbach energy with annealing temperature enhancement. A
noticeable sharp absorption band at 748 nm in UV spectra is attributed to 3
A2g(3F)→3
T1g(3F) electronic tran
sition. Room temperature magnetic hysteresis loops exhibit an increase of saturation magnetization upon
annealing which is discussed with reference to finite size effects and disorderly surface spins. The estimated value
of magnetocrystalline anisotropy constant by Law of Approach to saturation (LAS) theory as well as coercivity
value elucidates the annealing effect in changing the magnetic single domain state of the particle to a multi
domain state. Analysis of ZFC and FC magnetization curve measured at 100 Oe in the temperature range 400
K–60 K reveals the significant impact of annealing temperature on magnetic anisotropy, inter-particle interac
tion, and blocking temperature. Exploring the magnetic hysteresis loop measured in the temperature range
60–400 K over field strength of ± 3 T demonstrates the significant role of annealing on magnetic exchange
interaction. Temperature dependent behavior of saturation magnetization and coercivity has been analyzed
using modified Bloch’s law and Kneller’s relation. The magnetic heating efficiency examined by the induction
heating system reveals that the sample has enough potential for hyperthermia application.
1. Introduction
In recent years nanostructured spinel ferrites possessing general
formula MFe2O4 (M represents a divalent metal cation like Mg, Mn, Ni,
Co, Cu, and Zn) has drawn significant attention because of their tech
nological applications in diverse fields and very helpful in understand
ing the fundamental of magnetism at the nanometer scale [1]. Among
the nanostructured spinel ferrite family, nickel ferrite (NiFe2O4) is a
versatile and thoroughly investigated material owing to its several
interesting characteristics like high electrical resistivity, low magnetic
coercivity, low magnetostriction, high Curie temperature, low magnetic
anisotropy, moderate saturation magnetization, low eddy current loss,
high permeability in RF region, high electrochemical and thermal sta
bilities [1,2]. These properties make it appropriate for wide applications
in many fields including, electronic devices, ferrofluid technology,
magnetocaloric refrigeration, magnetic guided drug delivery,
* Corresponding author.
E-mail address: lawrencecuj@gmail.com (L. Kumar).
Contents lists available at ScienceDirect
Journal of Physics and Chemistry of Solids
journal homepage: http://www.elsevier.com/locate/jpcs
https://doi.org/10.1016/j.jpcs.2020.109928
Received 24 October 2020; Received in revised form 20 December 2020; Accepted 24 December 2020

2
hyperthermia, spintronics, energy storage devices, photocatalyst, sensor
technology, wastewater treatment, magnetic resonance imaging, power
transformer, and so on [1]. NiFe2O4 is n-type semiconducting and soft
ferrimagnetic material. Bulk NiFe2O4 shows ferrimagnetic ordering (Tc)
up to 860 K [2]. It crystallizes to complex spinel structure having
face-centered cubic lattice with space group Fd3
−
m. The general struc
ture formula of NiFe2O4 could be written as
(Ni2+
1− xFe3+
x )Tet[Ni2+
x Fe3+
2− x]OctO4. Here x denotes the inversion degree,
which is expressed as a percentage of the tetrahedral sites occupied by
Fe3+
cations. It depends on heat treatment and synthesis methods and
conditions [1]. If in this structure formula x = 1, then NiFe2O4 is cate
gorized as an inverse spinel structure. If x lie between 0 and 1, then it is
said to be in a mixed spinel structure [1]. Bulk NiFe2O4 is said to be in a
complete inverse spinel structure [1]. However, it is reported to be in
mixed spinel structure at the nanometer length scale [2–4]. The entire
different magnetic properties of NiFe2O4 nanoparticles from its bulk
counterpart is attributed to finite size effect and surface effect, which
includes the occurrence of a single domain, superparamagnetism,
reduced magnetization, spin glass like characteristics of surface spin,
canted spins, frustration, enhanced magnetic anisotropy and magneti
cally dead layer at the surface [2–5]. In its bulk form, the magnetic
properties are strongly dependent on cations distribution over tetrahe
dral and octahedral sites. On the contrary, the physical behavior of
NiFe2O4 at the nanometer length scale is affected not only by cations
distribution over tetrahedral and octahedral sites, but it depends upon
some other factors like synthesis technique, morphology, and size of the
particle [4,6–8]. The literature survey reveals that the properties of the
nickel ferrite system at the nanometer length scale strongly depend on
both cation distribution and size of the particle, which could be tuned
effectively by varying the annealing temperature [8–13]. Different
research groups have reported the impact of annealing temperature on
various properties of NiFe2O4 nanoparticles [4,6,8–13]. The extensive
literature survey suggests that the physical characteristics of NiFe2O4
nanoparticles could be modified strongly through variation of annealing
temperature.
To understand the physical properties of spinel ferrite in a better
way, it requires complete knowledge of its crystal structure, distribution
of cations over interstitial sites, and oxidation state of cations. However,
limited efforts are made for detail structural analysis of NiFe2O4 nano
particles with reference to annealing temperature. In the present work,
Rietveld refinement of the XRD pattern has been performed for detail
structural analysis along with the estimation of quantitative cation
distribution over interstitial sites. The variation of structural parame
ters, including cation distribution, bond length, as well as the bond angle
and lattice parameter with reference to annealing temperature, have
been analyzed in detail. Further, the line profile broadening analysis for
the determination of microstructural parameters like average crystallite
size and micro-strain using the Rietveld method has not been reported so
far for the NiFe2O4 system. One could extract the elastic parameters of
the sample by analyzing its FTIR spectra. The extensive literature review
divulges the lack of study on the behavior of elastic parameters of
NiFe2O4 nanoparticles with reference to annealing temperature.
Furthermore, there is a lack of detail analysis of annealing temperature
impact on the optical properties of NiFe2O4 nanoparticles. Limited ef
forts are made for estimation of magnetocrystalline anisotropy constant
by Law of approach to saturation (LAS) theory for NiFe2O4 nano
particles, especially temperature dependent magnetic anisotropy study
is meager in literature. In this work, a slight high value of magnetic
anisotropy constant has been observed. Attempt has been made to
establish the structure-property relationships in sample so as to explore
its potential application. Herein, we present the detailed analysis of the
structural, elastic, morphological, optical, and magnetic characteristics
of the NiFe2O4 system with reference to annealing temperature. It has
been observed that annealing temperature has produced a remarkable
impact in modifying the properties. Beside this, the present sample has
also been examined for their prospective in magnetic hyperthermia
heating.
2. Experimental section
2.1. Materials
Ni(NO3)2.6H2O (Nickel (II) nitrate hexahydrate) and C6H8O7.H2O
(citric acid monohydrate) were purchased from Fisher Scientific with
99% purity. Fe(NO3)3.9H2O (Iron (III) nitrate nonahydrate) was pro
cured from Merck with 99% purity. These chemicals were directly used
in the synthesis of nanocrystalline NiFe2O4 materials. The aqueous so
lutions of citric acid, iron nitrates, and nickel nitrates were prepared
employing deionized water (Milli-Q grade).
2.2. Synthesis
The synthesis of nickel ferrite nanoparticles was carried out by
standard citrate sol-gel method [14]. An aqueous solution of
Ni(NO3)2.6H2O and Fe(NO3)3.9H2O was mixed with a solution of citric
acid in a 1:3 M ratio. The synthesis method and role of citric acid in a
metal-citrate solution for ferrite powder synthesis has been reported
elsewhere [15]. The obtained powder was annealed for 3 h under an air
atmosphere at a temperature of 500 ◦
C, 600 ◦
C, 700 ◦
C, 800 ◦
C, and 900◦
to get nanoparticles of different size with the desired crystalline phase.
The annealing temperature was selected following the TGA analysis.
The chemical reaction for the formation of nickel ferrite powder
could be expressed as
Ni(NO3)2.6H2O(aq) + 2Fe(NO3)3.9H2O(aq) + 3C6H8O7.H2O (aq)
+
7
2
O2(g)→NiFe2O4(s) + 39H2O + 4N2(g) + 18CO2(g)
2.3. Characterization
Thermogravimetric analysis (TGA) of the sample was performed in a
30◦
C–900 ◦
C temperature range with a 10 ◦
C per minute heating rate by
using the NETZSCH thermal analyzer instrument (Model STA 449 F1
Jupiter) under nitrogen atmosphere. The crystal structure and phase
purity of all the annealed samples were investigated by recording
powder X-ray diffraction pattern using Cu-rotating anode based Rigaku
TTRX-III X-ray diffractometer with Cu-Kα radiation (λ = 0.154 nm)
operating in the Bragg-Brentano geometry. The Rietveld refinement of
the XRD pattern for all the annealed samples was carried out employing
FullProf suite to extract the structural and microstructural parameters.
The oxidation states of Ni and Fe elements were examined by X-ray
Photoelectron Spectroscopy (XPS) using Thermofisher Scientific (Model:
Nexa base). The XPS spectrum was recorded using Al Kα source (1486.6
eV). The room temperature Raman spectrum of all the annealed samples
was measured using a confocal micro-Raman spectrometer (Seki Tech
notron Corp Japan) to study the vibrational modes. The Raman spec
trometer was operated in the backscattering geometry. Ar+
ion laser
with 514.5 nm was taken as excitation source. The room temperature
FT-IR (fourier transform infrared) spectrum of all the annealed samples
was measured by the PerkinElmer spectrometer (model 400). Analysis of
morphological of samples was done by field emission scanning electron
microscopy (FESEM, Zeiss Gemini SEM 500) and transmission electron
microscope (TEM, JEM-F200). The compositional analysis of the sample
was performed using Energy dispersive spectroscopy (EDS) attached
with FESEM. The SAED (selected area electron diffraction) pattern of the
sample was studied by a transmission electron microscope. The UV–Vis
absorption spectrum of all the annealed samples was measured by
UV–Visible spectrophotometer (UV 3600 Plus, Shimadzu, Japan) to
investigate the optical properties. Magnetization measurement of all the
annealed samples was carried out using Quantum Design (VersaLab)
vibrating sample magnetometer (VSM). The measurement of magnetic
S.K. Paswan et al.

3
hysteresis loops was performed in the temperature range 60 K–400 K
over a magnetic field strength of ±3 T. Magnetization measurement with
ZFC (zero field cooled) and FC (field cooled) mode were carried out in
temperature range 60–400 K under 100 Oe applied field. In ZFC mode,
first cooling of the samples were performed from 400 K to the 60 K in
applied magnetic field absence. Afterwards magnetic field of strength
100 Oe was applied and the magnetization curve of the samples was
measured as the temperature increases from 60 K to 400 K. In field
cooled (FC) mode, the samples were cooled under 100 Oe magnetic field
from 400 K to 60 K, and then magnetization of the samples were
measured by raising the temperature from 60 to 400 K under an external
applied field of 100 Oe. The potential of the present sample for hyper
thermia application was investigated by AC magnetic induction heating
system (Easy Heat 8310, Ambrell, UK) with magnetic field strength and
frequency 12.97 kA/m (161 Oe) and 336 kHz respectively.
3. Results and discussion
3.1. Thermogravimetric analysis (TGA)
In order to determine the annealing temperature required for the
formation of a well crystalline phase, as synthesized sample has been
characterized by TGA. The TGA plot of as synthesized sample is shown in
Fig. 1(a).
It shows a weight loss of about 28% between 100 and 450 ◦
C, which
is endorsed to evaporation of amount of water vapour adsorbed on the
sample surface, removal of the residual organic components, and
decomposition of nitrates of metal into their corresponding oxides as
well. Above 450 ◦
C, the sample does not show any weight loss (as the
TGA curve is nearly flat), which implies the completion of the decom
position reaction, free of carbonaceous matter, residual reactants, and
formation of a stable phase. Considering the above thermal behavior, the
annealing temperature for the present sample has been selected above
450 ◦
C. To compute the required activation energy for the thermal
decomposition process leading to the formation of spinel nickel ferrite,
the TGA data has been analyzed using Coats–Redfern method [16].
According to Coats-Redfern method, the mathematical relation for
first-order reaction is expressed as [16,17].
log
[
− log(1 − α)
T2
]
= log
AR
βEa
[
1 −
2RT
Ea
]
−
Ea
2.303RT
(1)
where T denotes the absolute temperature, β is the symbol of linear
heating rate, A corresponds to the frequency factor, Ea stands for acti
vation energy, R represents the gas constant, and α denotes the fraction
of decomposed sample at time t represented by α = Wo− Wt
Wo− Wf
, Wo is the
initial sample weight (before the start of decomposition reaction), Wt
represents sample weight at any given temperature and Wf stands final
sample weight after completion of the reaction.The activation energy is
estimated by linear fitting to log
[
− log(1− α)
T2
]
versus 1000
T plot, as illustrated
in Fig. 1(b). The activation energy Ea is found to be ~24 kJ/mol, which
is near to the previous reported value for NiFe2O4 nanoparticle [16]. The
calculated value of activation energy also agrees well with previously
reported spinel ferrite system [17].
3.2. XRD analysis
Fig. 2 illustrates the powder XRD patterns of NiFe2O4 annealed at
500 ◦
C to 900 ◦
C. The observed diffraction patterns are in agreement
with the ICDD data of NiFe2O4 (ICDD no PDF 74–2081).
The intense diffraction peak centered at 2θ ~35◦
along with other
peak at 18◦
, 30◦
, 35◦
, 37◦
, 43◦
, 53◦
, 57◦
, and 63◦
corresponding to
reflection planes (311), (111), (220), (222), (400), (422), (511) and
(440) provides a clear sign for the formation of a NiFe2O4 system. The
XRD patterns of annealed samples do not show any additional phase
within the detection limit of XRD. Hence present samples are in a single
phase face-centered cubic spinel structure with Fd3
−
m space group (No
227). As the annealing temperature is raised from 500 ◦
C to 900 ◦
C, the
peaks in the XRD pattern are becoming sharper with a reduction in their
FWHM (full width at half maxima). This indicates a decrease of strain at
Fig. 1. (a) TGA curve of as prepared nickel ferrite sample. (b) Coats and Redfern plot for nickel ferrite.
Fig. 2. XRD patterns of NiFe2O4 annealed at various temperatures.
S.K. Paswan et al.

4
the lattice site and an increase in average crystallite size. The observed
behavior points toward the nanocrystalline nature of the present sample
[9]. Refinement of XRD pattern by the Rietveld method for all the
annealed samples has been performed in cubic phase with space group
Fd3
−
m using Full Prof program for determination of both structural pa
rameters like fractional coordinates of atoms, lattice parameters, ther
mal parameters, cations site occupancy and microstructural parameters
such as average size of the crystallite and micro-strain. The cubic
NiFe2O4 system with spinel structure (Fd3
−
m space group) contains three
atoms per asymmetric unit with Ni2+
I /Fe3+
I cations occupying the
Wyckoff 8(a) sites at
(
1
8,1
8,1
8
)
, Ni2+
II /Fe3+
II cations occupying the Wyckoff
16(d) sites at
(
1
2, 1
2, 1
2
)
and oxygen anions occupying Wyckoff 32(e) po
sitions at (u,u,u). The oxygen positional parameter u is free parameters
during refinement. The origin has been placed at a vacant octahedral site
with 3
−
m point symmetry for structural refinement [18,19]. Background
intensity has been modeled using six coefficient polynomial. The XRD
peak profiles have been modeled by Thompson-Cox-Hastings (TCH)
pseudo-Voigt profile functions to refine the shape and FWHM parame
ters. So as to extract microstructural parameters by the Rietveld method
using Full Prof program, it is essential to provide instrumental resolution
function (IRF) in the refinement for the deconvolution of the broadening
of peak contributed by sample from the wholepeak broadening. For this
purpose, Rietveld refinement of the standard LaB6 sample has been
carried out in an identical condition as for the present sample. The ob
tained IRF (instrumental resolution function) has been employed in
refinement so as to separate the instrumental contribution to the overall
broadening. The methodology for the estimation of microstructural
parameters by Rietveld method using Full Prof program is discussed in
the literature [20,21]. Between the calculated and observed XRD profile,
a very good agreement has been reached for all the annealed samples
with a distribution of Ni2+
and Fe3+
cations over both the interstitial
sites. Typical Rietveld refined X-ray diffraction pattern for 700 ◦
C
annealed is depicted in Fig. 3(a). The observed diffraction pattern in
Fig. 3 (a) is shown by small circles, while the calculated diffraction
pattern is illustrated by a continuous line. The positions of the Bragg
allowed peaks are denoted by vertical bars, and the bottom curve de
notes the difference profile between the calculated and observed XRD
pattern.
The values of various R-factors like Rexp(expected profile factor),
RB(Bragg factor), Rp(Profile factor), RF(crystallographic factor),
Rwp(weighted profile factor), and goodness of fit (χ2
) for all the annealed
samples are presented in Table 1. The obtained low values of various R-
factors, as well as goodness of fit, justifies that the refined model and
experimental data are in well agreement with each other. The refined
values of isothermal parameters, unit cell volume, lattice constant, and
oxygen positional parameters, along with the error bar for all the
annealed samples, are presented in Table 1. The refined oxygen posi
tional parameter (u) values for the sample under study are in the range
of 0.25213–0.25837. It agrees well with the previous reported u values
for NiFe2O4 nanoparticles [22]. The oxygen positional parameter (u) is
regarded as measurement of lattice distortion level in spinel lattice. In an
undistorted lattice (ideal case), the reported value of u is 0.25000. The
observed value of u indicates relatively slight distortion in spinel lattice
which is always expected in the real system due to the presence of
non-negligible structural defects [23]. As expected, the oxygen posi
tional parameter u is changing with the annealing temperature. The
possible explanation for this observation is as follow: The standard ionic
Fig. 3. (a) Rietveld refined XRD pattern of NiFe2O4 annealed at 700 ◦
C. (b) Polyhedron representation of face centered cubic spinel structure of nickel ferrite sample
generated by VESTA program.
Table 1
Agreement factors Rp, Rwp,Rexp, RBragg, RF and χ2
of Rietveld structure refine
ment, isotropic thermal factors of tetrahedral site (BA) and octahedral site (BB),
lattice constant (a), unit cell volume (V), oxygen positional parameter (U), X-ray
density (ρ), dislocation density(δ), average crystallite size (t) by Rietveld, W–H
plot and modified scherrer method, micro-strain (ε), stress value (σ), hopping
length at tetrahedral (dA) and octahedral (dB) site, number of unit cell(n) and
tolerance factor (T)for nickel ferrite samples annealed at different temperatures.
Parameters 500 ◦
C 600 ◦
C 700 ◦
C 800 ◦
C 900 ◦
C
Rp(%) 12.6 13.8 8.24 19.7 18.3
Rwp(%) 16.6 18.0 11.4 25.3 15.2
Rexp(%) 12.02 12.03 7.55 10.82 4.16
RB(%) 11.1 11.7 3.17 13.1 5.37
RF(%) 15.6 14.7 3.70 12.7 3.99
χ2
(%) 1.90 2.07 2.28 3.78 6.89
BA(Å2
) 0.305 0.321 0.391 0.423 0.456
BB(Å2
) 0.623 0.648 0.712 0.756 0.783
a(Å) 8.343
(0.004)
8.341
(0.003)
8.340
(0.002)
8.338
(0.002)
8.337
(0.001)
V(Å3
) 580.885 580.389 579.615 579.576 579.575
U(Å) 0.2521
(0.0004)
0.2529
(0.0003)
0.2553
(0.0001)
0.2583
(0.0004)
0.2582
(0.0006)
ρ (g/cm3
) 5.359 5.362 5.365 5.369 5.371
δ (1016
/m2
) 0.0016 0.0012 0.0005 0.0003 0.0001
t (nm)
Rietveld 24.98 28.45 44.33 51.12 88.21
W–H Plot 29.12 31.43 51.25 56.19 92.14
Modified
Scherrer
32.15 35.21 55.26 58.17 95.32
ε
Rietveld 0.00087 0.00067 0.00039 0.00031 0.00020
W–H Plot 0.00095 0.00072 0.00051 0.00042 0.00034
σ (MPa) 138 126 112 101 95
dA(Å) 3.6127 3.6121 3.6114 3.6105 3.6102
dB (Å) 2.9498 2.9493 2.9487 2.9479 2.9477
n 12.45 ×
103
19.79 ×
103
76.91 ×
103
119 × 103
612 × 103
T 1.0164 1.0227 1.0284 1.0364 1.0371
S.K. Paswan et al.

5
radius of Ni2+
cation is found to be 0.55 and 0.69 Å in four and six co
ordination while the standard ionic radius of Fe3+
cation (high spin) in
four and six coordination is 0.49 and 0.64 Å. Hence, redistribution of
cations caused by annealing is expected to produce changes in the radius
of tetrahedral and octahedral site so that the lattice structure could be
adjusted to achieve minimum potential energy for its stability. As an
effect, the oxygen anions is expected to move toward or away from the
nearby A-site/B-site cation in the [111] direction. The size of the octa
hedron and tetrahedron is expected to increase or decrease at the
expense of each other, leading to a change in oxygen positional
parameter [24].
Table 1 show that values of lattice parameter decreases with the rise
in annealing temperature. The reason behind this observation might be
attributed to cations redistribution between tetrahedral and octahedral
sites, polyvalence of cations and structural defects [25]. Similar effect of
annealing temperature on lattice constant for nanocrystalline NiFe2O4
system has been reported in literature [8,9,25]. Fig. 3(b) illustrates the
face centered cubic spinel structure of NiFe2O4 system (700 ◦
C annealed
sample) in polyhedron representation generated by the VESTA program
using CIF file obtained through Full Prof structural refinement of
experimental data. With the help of estimated lattice constant, the X-ray
density ρx of the present samples have been estimated according to
relationship [22].
ρx =
ZM
NAa3
(2)
where NA stands for Avogadro’s number (6.02 × 1023
mol− 1
), M rep
resents the molecular weight of the NiFe2O4 ferrite sample (NiFe2O4 =
234.381 g mol− 1
), Z corresponds to the number of formula unit in the
unit cell (Z = 8) and a denotes the lattice constant. The value of the X-ray
density of NiFe2O4 annealed at different temperature is presented in
Table 1, which is in increasing trend with annealing temperature. Using
the value of lattice constant, the length of hopping between the magnetic
cations at the octahedral site and tetrahedral site in the spinel lattice
could be estimated from the following relations [11].
dA = 0.25a
̅̅̅
3
√
(3)
dB = 0.25a
̅̅̅
2
√
(4)
where dA and dB are the length of hopping between the magnetic cation
within the tetrahedral site and octahedral site. The calculated dA and dB
values of the present sample annealed at different temperature are
tabulated in Table 1. The estimated hopping lengths dA and dB are found
to be in decreasing trend with annealing temperature, which indicates
that magnetic ions are approaching close to each other. The estimated
value of hopping length in B-sites is smaller than in A-sites. It suggests
that chance of hopping of a charge carrier (electron) between cations at
octahedral sites (B-site) is more than that of tetrahedral sites (A-site). As
a result, different physical properties, especially electrical properties,
are expected to be different in A-sites and B-sites [26]. Using the esti
mated values of oxygen positional parameter (u) and lattice constant (a),
the values of shared octahedral edge length (dBE), tetrahedral edge
length (dAE), unshared octahedral edge length (dBEU), radius of the
octahedral (rB) and tetrahedral (rA) site for all the annealed samples can
be calculated according to following relations [27].
dAE = a
̅̅̅
2
√
(2u − 0.5) (5)
dBE = a
̅̅̅
2
√
(1 − 2u) (6)
dBEU = a
(
4u2
− 3u +
11
16
)1
2
(7)
rA = a
̅̅̅
3
√
(
u −
1
4
)
− R(O) (8)
rB = a
(
5
8
− u
)
− R(O) (9)
where R(O) corresponds tothe ionic radius of oxygen anions (1.32 Å). It
is to be noted that equations (5)–(9) have been discussed in the literature
for unit cell origin at 4
−
3m on A-site cation. In the present Rietveld
structural refinement, unit cell origin is at 3
−
m on an octahedral vacancy.
So as to use equations (5)–(9) for unit cell origin at 3
−
m, in the above
expressions u is replaced with u + 1
8 [28].
The evaluated dAE, dBE, dBEU, rA and rB values are presented in
Table 2. All the estimated values are consistent with the earlier reported
values of similar NiFe2O4 system [7,22,27]. Using the value of octahe
dral site radius (rB), tetrahedral site radius (rA), and oxygen ionic radius
(Ro = 1.32 Å), the tolerance factor (T) can be calculated for the spinel
structured material. It is expressed by the following expression [29].
T =
1
̅̅̅
3
√
(
rA + RO
rB + RO
)
+
1
̅̅̅
2
√
(
RO
rA + RO
)
(10)
For an ideal spinel structured material, the value of the tolerance
factor is unity [29]. The estimated tolerance factor (T) values of the
sample under study are in the range of 1.0164–1.0363, which is close to
unity. The calculated T values agree well with the previous reported
result for the NiFe2O4 system [30]. It suggests the presence of fewer
defects within the structure.
According to literature, Rietveld refined X-ray diffraction pattern
enables to roughly estimate the cations distribution over octahedral and
tetrahedral interstitial sites by analyzing the site occupancies of cations
[20,31,32]. Therefore, in the present study distribution of nickel and
iron cations over 8(a) and 16(d) crystallographic sites has been esti
mated through the refinement of site occupancies. The estimated dis
tribution of cations and inversion degree (δ) for all the annealed samples
are enlisted in Table 3.
It shows that increasing the annealing temperature yields different
Table 2
The values of tetrahedral edge length (dAE), shared octahedral edge length (dBE),
unshared octahedral edge length (dBEU), radius of the tetrahedral (rA) and
octahedral (rB) site, bond length and bond angle for nickel ferrite samples
annealed at different temperature. Errors are given in the bracket.
Bond Length (Å)
Parameters 500 ◦
C 600 ◦
C 700 ◦
C 800 ◦
C 900 ◦
C
dAE (Å) 2.999
(3)
3.017
(7)
3.073
(7)
3.143
(7)
3.141
(2)
dBE (Å) 2.900
(2)
2.880
(8)
2.823
(6)
2.752
(2)
2.754
(3)
dBEU (Å) 2.950
(4)
2.949
(7)
2.950
(3)
2.951
(2)
2.950
(9)
rA (Å) 0.516
(7)
0.527
(9)
0.562
(2)
0.605
(4)
0.603
(5)
rB (Å) 0.748
(3)
0.741
(2)
0.720
(8)
0.695
(3)
0.696
(7)
O2−
- Fe3+
/Ni2+
(A-site)
(Å)
1.835
(3)
1.841
(2)
1.870
(2)
1.869
(2)
1.828
(6)
O2−
- Fe3+
/Ni2+
(B-site)
(Å)
2.069
(2)
2.066
(3)
2.047
(7)
2.051
(3)
2.068
(2)
Fe3+
/Ni2+
(A) - Fe3+
/
Ni2+
(B) (Å)
3.458
(4)
3.459
(4)
3.456
(8)
3.457
(7)
3.456
(5)
Fe3+
/Ni2+
(B) – Fe3+
/
Ni2+
(B) (Å)
2.949
(3)
2.949
(8)
2.948
(2)
2.949
(6)
2.947
(8)
Bond Angle (Degree)
Fe3+
/Ni2+
(A) - O2−
-
Fe3+
/Ni2+
(B)
124.60 124.48 123.78 124.8 124.70
O2−
- Fe3+
/Ni2+
(B) – O2-
89.10 91.12 92.12 88.0 90.79
O2−
- Fe3+
/Ni2+
(A) - O2-
109.5 109.5 109.47 109.5 109.47
Fe3+
/Ni2+
(B) – O2−
-
Fe3+
/Ni2+
(B)
90.92 91.11 92.08 92.0 90.78
Fe3+
/Ni2+
(A) - O2−
-
Fe3+
/Ni2+
(A)
72.72 72.67 72.40 72.43 72.75
S.K. Paswan et al.

6
arrangement of Ni and Fe cations at the octahedral and tetrahedral
positions. However the same cubic symmetry is maintained for all the
annealed samples as evident from Rietveld analysis. Taking into
consideration the refined values of occupancy, the proposed structural
formula for 500 ◦
C annealed sample is (Ni0.21Fe0.79)A[Ni0.79Fe1.21]B while
the structural formula for 900 ◦
C annealed sample is
(Ni0.06Fe0.94)A[Ni0.94Fe1.07]Bwhere small bracket denote tetrahedral sites
and square bracket refers octahedral sites. The distribution of cations for
500 ◦
C annealed samples are in random manner while for 900 ◦
C
annealed sample, it is very close to that of inverse type. It is evident from
Table 3 that upon enhancing the sample annealing temperature from
500 ◦
C to 900 ◦
C, the Ni ions at tetrahedral sites continues to migrate to
the octahedral sites while Fe ions continues to move from octahedral to
the tetrahedral sites so as to minimize the overall potential energy of the
NiFe2O4 structure. The observed behaviour suggests that cation distri
bution for nanosized ferrite system is expected to be in metastable state.
The increasing occupancy of Fe ions at tetrahedral site with the
annealing temperature is expected to produce more and more Fe3+
A −
O − Fe3+
B superexchange interaction, which could give rise to enhanced
magnetization. The estimated cation distribution for the investigated
sample is consistent with the existing literature [22,31]. The annealing
induced movement of cations from A-sites to B- sites and vice-versa is
expected to produce variation in bond length and bond angles. Table 2
shows that bond length, bond angle and effective bond length changes
with the annealing temperature which could be ascribed to the cations
redistribution at octahedral and tetrahedral sites. The estimated octa
hedral bond length (RB) is larger than tetrahedral bond length (RA). The
larger values of octahedral bond length could be endorsed to a slighter
overlapping of orbitals of Ni2+
/Fe3+
and O2−
at the octahedral site. The
change in bond length, bond angle and effective bond length is likely to
play major role in deciding the overall magnetic interaction and ex
change coupling. All the estimated values are consistent with the pre
vious reported values [7].
The microstructural parameters such as average size of crystallite
and micro-strain of all the annealed samples have been estimated by
Rietveld analysis. Refinement has been carried out supposing that both
Gaussian and Lorentzian part make their contribution to size broadening
and micro-strain broadening. The following expression is used in Riet
veld method for the estimation of accurate microstructural parameters
from peak broadening [20].
H2
G =
(
U + D2
ST
)
tan2
θ + V tan θ + W +
IG
Cos2θ
(11)
HL = X tan θ +
Y
Cosθ
+ Z (12)
In the above expression, H is known as FWHM of the peak profile.
The parameters U, V, W,IG,X, Y and Z are refinable parameter. The
subscript G and L denote Gaussian and Lorentzian profiles, respectively.
The values obtained by Rietveld analysis for microstructural parameters
such as average crystallite size and microstrain are tabulated in Table 1.
The average size of the crystallite is increasing progressively as the
annealing temperature is raised from 500 ◦
C to 900 ◦
C. The observed
phenomenon confirms that thermal annealing has improved the crys
tallinity of the investigated sample. The possible explanation for
increased crystallite size with annealing temperature is as follow:
generally nanocrystalline materials have an increased volume of grain
boundary where lack of periodic structure (periodicity) prevails. Atoms
in grain boundary region are loosely bonded. Receiving the thermal
energy by annealing, the mobility of these loosely bonded atoms is ex
pected to increase. It helps atoms to move to energetically favored po
sition to merge with nearby crystallites. As a result, improved atomic
diffusion provides grain growth leading to enhancement in crystallite
size. In other words, during thermal annealing atomic diffusion leads to
solid-solid interface thereby reducing the surface area. This lowers the
overall free energy of the system giving rise to volume expansion. As an
effect, growth of crystal takes place leading to increase in crystallite size
[33]. On the contrary; estimated micro-strain follows the opposite trend.
The value of micro-strain is decreasing with annealing temperature.
Generally micro-strain is induced in nanocrystalline sample due to the
presence of crystallographic defects. The annealing minimizes the
crystallographic defects, thereby lowering the strain at the lattice site. In
addition, annealing provides atmospheric oxygen in order to complete
the terminated unit cells at surface which is expected to reduce surface
stress and strain [34].
Further the estimation of average crystallite size and microstrain has
been performed using Williamson-Hall plot so as to compare the ob
tained values of average crystallite size and micro-strain by Rietveld
method. The Williamson-Hall equation is represented as [26].
βhklCosθ =
kλ
t
+ 4εSinθ (13)
Where βhkl is the FWHM analogous to (hkl) plane, t stands for average
crystallite size, ε represents the microstrain, λ is the wavelength of CuKα
radiation (1.54 Å), and k corresponds to the correction factor taken as
0.94. In order to perform size and strain analysis by the Willamson-Hall
method, the peak broadening (FWHM) corresponding to each (hkl)
plane has been assessed by fitting the curve assuming the pseudo-Voigt
function for the peak profile. The subtraction of instrumental broad
ening from each peak has been taken into account (standard LaB6
sample). In literature, equation (13) is reported to represent the uniform
deformation model (UDM), where the nature of crystalline material is
considered to be isotropic and intrinsic strain is presumed to be uniform
in every crystallographic direction [26]. Typical UDM Williamson-Hall
plot for 700 ◦
C annealed sample is depicted in Fig. 4(a).
The average crystallite size is estimated from the intercept on Y axis,
while microstrain is calculated from the slope obtained by the fitting.
The estimated values of crystallite size and microstrainare presented in
Table 1. According to the literature, the Williamson-Hall equation could
be utilized to extract the elastic properties of the crystal [26]. The
original Williamson-Hall equation could be modified to estimate the
stress in the crystal using Hook’s law approximation, which states that
linear proportionality between stress and microstrain is maintained
within the elastic limit i.e. for a small value of strain. The estimated
value of strain is low for the investigated sample. Hence σ = Ehklε could
be incorporated in original Williamson-Hall equation where Ehkl repre
sents the Young modulus or modulus of elasticity of the set of lattice
plane (hkl) in the perpendicular direction of plane. Applying the
Hooke’s law approximation, the modified Williamson-Hall equation can
be represented as [26].
βhklCosθ =
kλ
t
+
4σSinθ
Ehkl
(14)
According to the literature, equation (14) represents the uniform
stress deformation model (USDM), where it is presumed that stress is
uniform every crystallographic direction even if the material is aniso
tropic [26]. The Young modulus Ehkl for a cubic crystal having the crystal
lattice plane (hkl) is expressed by following relation [26].
1
Ehkl
= S11 − 2(S11 − S12 − 0.5S44)
(
h2
k2
+ k2
l2
+ h2
l2
h2 + k2 + l2
)
(15)
Table 3
Cation Distribution and inversion parameter for NiFe2O4 sample annealed at
different annealing temperature.
Annealing temperature A - Site B - Site Inversion (x)
500 ◦
C (Ni0.21Fe0.79)A [Ni0.79Fe1.21]B 0.79
600 ◦
C (Ni0.18Fe0.82) A [Ni0.82Fe1.18] B 0.82
700 ◦
C (Ni0.16Fe0.84) A [Ni0.84Fe1.16] B 0.84
800 ◦
C (Ni0.11Fe0.89) A [Ni0.89Fe1.11] B 0.89
900 ◦
C (Ni0.06Fe0.94) A [Ni0.94Fe1.07] B 0.94
S.K. Paswan et al.

7
where S11, S12 and S44 represents the elastic compliances of NiFe2O4
system. These elastic compliances are calculated using elastic stiffness
constant(Cij), which are expressed as follow [26].
S11 =
C11 + C12
(C11 − C12)(C11 + 2C12)
(16)
S12 =
( − C12)
(C11 − C12)(C11 + 2C12)
(17)
S44 =
1
C44
(18)
The value of elastic stiffness constant C11C12 and C44 for NiFe2O4
system is reported to be 275 GPa, 104 GPa, and 95.5 GPa, respectively
[26]. It is to be noted that the value of elastic stiffness constants are
common for all the face centered cubic spinel ferrite. Typical USDM
Williamson-Hall plot for 700 ◦
C annealed sample is shown in Fig. 4(b).
The slope of the fit provides uniform stress. The values of estimated
stress are tabulated in Table 1 and agree well with the previous reported
literature values of similar spinel ferrite system [26,35]. The estimated
stress is decreasing with increasing annealing temperature, which is
accredited to the minimization of the strain at the lattice site due to
annealing. In recent years, several authors have reported the strain and
stress values for spinel structured CoAl2O4 system using UDM and USDM
Williamson-Hall plot method [36,37]. According to literature, the
average size of the crystallite can also be estimated using modified
Scherrer equation. It is represented as [38,39].
ln β = ln
(
kλ
t
)
+ ln
(
1
Cosθ
)
(19)
The significance of the modified Scherrer formula is to minimize the
source of errors while estimating the size of crystallite. The average size
of the crystallite is obtained by the least square method by taking all the
peaks into account in order to reduce the errors mathematically. Fig. 5
(a) shows the typical modified Scherrer plot of ln(β) against ln
(
1
Cosθ
)
for
700 ◦
C annealed sample. A single value of t is obtained by the intercept
of a least squares regression line, which passes through all of the
available peaks. It is observed that values of average crystallite obtained
by the Rietveld method, Williamson-Hall plot method, and Modified
Scherrer plot method are different from each other. It is because
different method uses a different approach for peak profile analysis.
One could also estimate the theoretical value of activation energy
required for the growth of the crystallite upon annealing using the Scott
equation, which is represented as [40,41].
t = A exp
(
− E
RT
)
(20)
where t stands for the size of crystallite, E represents the activation
energy needed for the grain growth, A is constant, T denotes the absolute
temperature, and R corresponds to the ideal gas constant. Fig. 5(b)
shows the Arrhenius plot of ln(t) against
(
1
T
)
. The crystallite size esti
mated from Rietveld analysis has been used for this purpose. The
Fig. 4. (a) Plot of βhkl cos θ versus 4 sin θ and (b) Plot of βhkl cos θ versus 4 sin θ/Ehkl for NiFe2O4 annealed at 700 ◦
C.
Fig. 5. (a) Modified Scherrer equation plot for NiFe2O4 annealed at 700 ◦
C. (b) Plot of ln(t) as a function of 1/T.
S.K. Paswan et al.

8
activation energy value estimated from the linear fit on the experimental
data is 11.87 kJ/mol. The estimated activation energy value agrees well
with the literature value [40,41]. The concentration of defect in the
sample could also be expressed in terms of dislocation density (δ), which
is expressed as the length of lines of dislocation per unit crystal volume.
Roughly it is expressed as [37].
δ =
1
t2
(21)
where t corresponds to the size of crystallite. The values of dislocation
density for all the annealed samples are presented in Table 1. It gets
lower upon increasing the annealing temperature. It appears that
annealing reduces the amount of defect during the crystal growth of the
sample. The number of unit cells (n) in the sample could be estimated by
following relation [37].
n =
π × t3
6V
(22)
where V represents a unit cell volume. The calculated values of (n) at
various annealing temperature are listed in Table 1. The number of unit
cells is found to be increased with increasing annealing temperature,
which is ascribed to an increase of crystal growth and decrease of de
fects. Furthermore, to visualize the distribution of electron density in
side the unit cell, the study of electron density mapping has been carried
out using the GFourier program in the FullProf package. The visualiza
tion of electron density is significant in identifying the positions of atoms
of the constituent elements of the compound within the unit cell. The
electron density corresponds to the Fourier transform (FT) of the
geometrical structure factor taken over the whole unit cell. It is repre
sented as [41].
ρ(xyz) =
1
V
∑
hkl
|Fhkl|exp{ − 2πi(hx + ky + lz − αhkl)} (23)
where ρ(xyz) corresponds to electron density at point x, y, z inside a unit
cell volume V, Fhkl represents the structure factor amplitude, and αhkl
stands for phase angle of Bragg plane. The electron scattering density
ρ(xyz) is presented as either a two or three dimensional Fourier map.
The two dimensional Fourier maps are typically drawn as contour to
indicate electron density distribution around each atom of the constit
uent elements of the compound. If the electron density contours are
dense and thick, then in the unit cell, it indicates the position of a
relatively heavier element. On the contrary, the three dimensional (3-D)
Fourier maps engross a chicken-wire style network, which indicates a
single electron density level. Typical two dimensional Fourier electron
densities mapping of Ni/Fe and O atoms on the yz plane (x = 0) in the
unit cell of 700 ◦
C annealed sample is depicted in Fig. 6(a). The dense
and thick circular nature of the contours around Ni/Fe might be ascribed
to mainly the distribution of valence d orbitals electrons. The contours
around the O might be attributed to the distribution of valence 2s and 2p
orbitals electrons. The black colour in Fig. 6 (a) corresponds to the zero-
level density contour, while the contour with the coloured region
around the Ni/Fe and O indicate the electron density levels. The three
dimensional Fourier electron density mapping of Ni, Fe, and O element
in the unit cell of NiFe2O4 at x = 0 is shown in Fig. 6(b).
It depicts a strong peak which corresponds to the 8a and 16d sites for
the Ni cations. The peak between the two intense peaks corresponds to
the 8a and 16d sites for Fe cations. The scattering of the X-rays banks on
the number of electrons around the atom. The Ni atoms have more
number of electrons than Fe. Therefore Ni is expected to scatter the X-
rays strongly in comparison to Fe. Hence the peaks corresponding to Ni
are intense than that of Fe. A similar study on electron density mapping
using FullProf program has been reported in recent literature by Abbas
et al. [42] and Singh et al. [43] for perovskite structured LaFeO3 and
BiFeO3.
3.3. XPS (X-ray photoelectron spectroscopy) analysis
The assessment of the chemical oxidation states of the Ni and Fe
element in the present NiFe2O4 system has been carried out using XPS.
Typical XPS survey spectrum for 700 ◦
C annealed sample is depicted in
Fig. 7(a). It demonstrates the co-existence of Ni, Fe and O element,
which are the constituent elements of the present sample. No other
elemental impurity is observed, which endorses the purity of the present
synthesized sample. The atomic % of Ni2p, Fe2p and O1s is found to be
9.69, 21.02, and 42.04 i.e. the elemental ratio of Ni, Fe and O in the
present synthesized sample is 1:2.1:4.3 which is near to the sample
chemical formula (theoretical value). The XPS binding energy peak
obtained from the sample has been standardized by the C1s binding
energy peak at 284.68eV. The high resolution core level binding energy
XPS spectra for the Ni 2p, Fe 2P and O 1s are illustrated in Fig. 7 (b), (c)
and (d). The Ni 2p XPS spectrum exhibit two main binding energy peaks
around ~854.36 and 872.27 eV, respectively, together with two distinct
satellite peaks around ~861.27 and 879.69 eV. The binding energy peak
around 854.36 and 872.27 eV corresponds to Ni 2p3/2 and Ni 2p1/2. The
Fe 2p XPS spectrum shows the main binding energy peak around 710.46
and 724.17 eV, respectively, and associated satellite peaks around 722
eV and 734 eV.
The appearance of associated satellite peaks for both Ni 2p and Fe 2p
is ascribed to the electron transition to the vacant 4s orbital from a 3d
Fig. 6. (a) 2D-Electron density map in unit cell of NiFe2O4 annealed at 700 ◦
C. (b) 3D-Electron density map in unit cell of NiFe2O4 annealed at 700 ◦
C.The electron
density is measured in e/Å3
.
S.K. Paswan et al.

9
orbital in the course of the ejection of the 2p core electron. Generally,
the transition elements in the spinel ferrite compounds show multiple
oxidation states. The octahedrally coordinated element show lower
binding energy in comparison to tetrahedrally coordinated elements
[46]. So as to study the oxidation state of Ni and Fe in the present
sample, the deconvolution of XPS peaks of both Ni 2p and Fe 2p have
been carried out using Gaussian fitting, as illustrated in Fig. 7 (b) and (c).
The deconvolution of the Ni 2p3/2 peak provides two peaks at around
854.37eV and 855.42 eV. The peak at 854.37eV is ascribed to Ni2+
states
of Ni located at the B-sites, whereas the peak at 855.42eV is accredited to
Ni3+
states of Ni located at the A-sites [47]. The appearance of Ni3+
state
to higher binding energy side might be because of additional coulombic
interaction between the photo emitted electrons and ion core. It is re
ported that XPS peak positions of Fe2+
state of Fe in the spinel ferrite
system is octahedrally coordinated and coincide with the peak position
of Fe3+
state located at the octahedral sites. It infers that the peak po
sition of the Fe2+
state of Fe coincides with the peak position of Fe 2p3/2
and Fe 2p1/2 octahedral sites (B-sites) where the Fe3+
state is located.
The deconvolution of Fe 2p3/2peak results a peak around 710.03 eV and
712.08 eV. The resolved peak observed at 710.03 eV is accredited to
Fe2+
and Fe3+
state of Fe, which is reported to be octahedrally coordi
nated. The peak observed at 712.08 eV is ascribed to the Fe3+
state of Fe
located at tetrahedral (A-sites) sites [45,48,49]. The above study sug
gests that Fe ion in octahedral sites could be present both in Fe2+
and
Fe3+
state. The existence of Fe2+
and Fe3+
state at the B- site results in
electron hopping between Fe2+
and Fe3+
ions, which could play a sig
nificant role in electronic conduction in the NiFe2O4 system [50]. The
deconvolution of O 1s XPS spectrum provides a peak at 529.72 eV,
which corresponds to metal-oxygen bond (lattice oxygen) in the sample.
Another peak at 531.38 eV refers to oxygen vacancies or adsorbed ox
ygen species at the surface of sample [49]. The peak areas of deconvo
luted Fe 2p3/2 and Ni 2p3/2peak have been used for quantitative
estimation of Ni and Fe element at A-site and B-sites along with their
oxidation state. The calculation shows that 38% of tetrahedral sites are
filled by Ni ions and the remaining 62% tetrahedral sites are occupied by
Fe ions. Similarly 62% of octahedral sites are filled by Ni ions and 38%
remaining octahedral sites are reached by Fe ions. Moreover, the results
show that 48% of Ni ions are found to be in Ni2+
state, and the remaining
52% Ni ions achieve Ni3+
state. The Fe2+
state is reached by 39% Fe ions
and the remaining 61% Fe ions observed to be in Fe3+
state. A similar
analysis for estimation of the relative proportion of oxidation states of
element for spinel ferrite system has been reported in the literature [51,
52]. It is to be noted that the occupancy value of Ni and Fe ions at A-sites
and B-sites estimated by XPS peak analysis is different from XRD anal
ysis. This discrepancy might be attributed to different sensitivity of the
XPS and XRD technique. The XPS is recognized as a surface sensitive
technique where it provides the information from the top of atomic
layers whereas XRD is regarded as a bulk method [45]. The binding
energy peak around 710.46 and 724.17 eV belongs to Fe 2p3/2 and Fe
2p1/2[44]. The 2p states of both Ni and Fe split into two components,
namely 2p3/2 and 2p1/2 which is attributable to spin-orbit coupling
interaction between the unpaired 2p core-level electrons and unpaired
3d valence shell electrons during photoelectron emission [45].
Fig. 7. (a)XPS full scan spectra (b) Ni 2p spectrum (c) Fe 2p spectrum (d) O 1s spectrum.
S.K. Paswan et al.

10
3.4. Raman spectroscopy
Room temperature Raman spectra of NiFe2O4 sample annealed at
500–900 ◦
C measured in the frequency range from 100 to 1800 cm− 1
is
depicted in Fig. 8(a).
The frequency positions of Raman peak for all the annealed samples
closely match with the earlier reported data [4,13,53]. The observed
Raman spectrum supports the spinel structure formation in the synthe
sized sample. The NiFe2O4 system belongs to Fd3m space group.
Although the nickel ferrite unit cell comprises 56 atoms, nevertheless
the asymmetric unit holds only 14 atoms. Hence, there is a possibility of
42 vibrational modes. According to group theory, spinel ferrite system
with space group Fd3
−
m has the following vibrational modes [53].
Γ(AB2O4) = A1g(R) + Eg(R) + 3T2g(R) + 4T1u(IR) + T1g(in) + 2A2u(in)
+ 2Eu(in) + 2T2u(in)
(24)
where (R), (IR), and (in) represents the Raman active vibration,
infrared-active vibration, and inactive (silent) modes. The symbol A, E,
and T stands for one dimensional, two dimensional and three dimen
sional representations. The character g and u represents the symmetric
and antisymmetric with reference to the inversion center. The subscript
1 and 2 represents symmetric and antisymmetric with reference to the
axis of rotation that is perpendicular to the principal axis [54]. Group
theory calculation envisages the first order five Raman active modes for
NiFe2O4, namelyA1g + Eg + 3T2g [53]. The Raman peak around ~192,
~322, ~478, ~565, and ~693 cm− 1
is assigned to T2g(1), Eg,
T2g(2),T2g(3), and A1g mode respectively [4,13,53]. All the observed
Raman peaks of samples are asymmetric. All five Raman modes
composed of the motion of both tetrahedral and octahedral site cations
and oxygen anions. The T2g(1) mode at 192 cm− 1
is accredited to the
translational drive of the tetrahedron, where Ni/Fe metal cations are
tetrahedrally coordinated with oxygen atoms. The Eg mode at ~322
cm− 1
relates to the bending of oxygen anion symmetrically with refer
ence to the Ni/Fe metal cation. The T2g(2) mode at 478 cm− 1
is
accredited to stretching of Fe/Ni and O bond asymmetrically in the
B-sites (octahedral sites), whereas T2g(3) mode at 565 cm− 1
is the
consequence of asymmetric bending of oxygen anion with reference to
tetrahedral and octahedral metal cations. The A1g mode belongs to the
stretching of oxygen anions symmetrically along Ni–O and Fe–O bonds
in the A-sites [55]. The Raman peaks appearing in the spectra above 600
cm− 1
corresponds to vibrational modes of the tetrahedral metal complex,
while below 600 cm− 1
is related to vibration modes of the octahedral
metal complex [4]. Further, the Raman peaks have been suitably
deconvoluted using Gaussian fitting in order to investigate the local
symmetry of oxygen polyhedra (i.e., oxygen tetrahedron and octahe
dron) because the Raman spectroscopy is structure sensitive tool and
detect any changes very effectively [56]. Typical deconvoluted Raman
spectrum for 700 ◦
C annealed sample is depicted in Fig. 8(b). The
deconvolution of A1g peak provides clear A1g(2) and A1g(1) peak at 646
and 692 cm− 1
respectively. The A1g(2) peak at 646 cm− 1
is originating
from NiO4 tetrahedron, whereas the A1g(1) peak at 692 cm− 1
is from
FeO4 tetrahedron [57]. The possible explanation for this observation is
that Ni–O bond distance and Fe–O bond distance is different (as evident
from XRD study) due to different electronic structure and ionic radii of
Ni and Fe ions. Inside the crystal structure, considerable distributions of
Ni–O and Fe–O bond distance are expected, thereby changing the local
symmetry of NiO4 and FeO4 tetrahedron [56,57]. The Raman peaks for
500 ◦
C annealed samples are broad. Upon enhancing the annealing
temperature from 500 to 900 ◦
C, the Raman peaks are becoming nar
rower and appears to shift towards to higher frequency side. A similar
observation of peak broadening and peak shift with annealing temper
ature for Raman spectra of nanocrystalline NiFe2O4 has been reported in
literature [4,13,31,53]. The above observation can be elucidated on the
basis of the phonon confinement model. The Raman peaks are expected
to follow the phonon dispersion at k = ±2π
L where k is the phonon wave
vector, and L is the length of the crystalline materials. If the length of the
crystalline material is too large, optical phonons (which are quantized
vibrations of the atoms in the crystal lattice) which are close to the
center of the Brillouin Zone (Brillouin Zone is an allowed energy region
of electrons in reciprocal space) are Raman active. It contributes to the
Raman spectrum due to the momentum conservation between the
incident light and phonons, thereby leading to sharp Raman peaks. On
the contrary, for nanocrystalline materials, the length of the crystal is
short, and there is a loss of long range order. Therefore phonons could be
confined in the region by defects or crystal boundaries. This might lead
to uncertainty in the phonon momentum. As a result, those optical
phonons which are not near the center of Brillouin Zone are also allowed
to contribute to the Raman spectrum because of the relaxation of the
momentum conservation rule. The uncertainty in the phonon mo
mentum is expected to be larger for smaller crystallite size. Conse
quently, the broadening, as well as shifting of the peak position, is
increased in the Raman spectra upon decreasing the crystallite size
[58–60]. Other factors such as an increase in sample crystallinity, par
ticle size, and cations redistribution over A-sites and B-sites might also
be responsible for the above observation [53]. According to the litera
ture, during the synthesis of the NiFe2O4 ferrite system, there might be a
probability of formation of a small extent of other phases like Fe3O4 and
γ-Fe2O3. These phases could not be detected by XRD because their
crystal structure and XRD pattern are similar to that of NiFe2O4 system
[13,56]. Raman spectroscopy provides insight to differentiate NiFe2O4
Fig. 8. (a) Raman spectra of NiFe2O4 annealed at various temperature. (b) Deconvoluted peaks and cumulative fit for the NiFe2O4 annealed at 700 ◦
C.
S.K. Paswan et al.

11
system from other phases. The Raman peaks for Fe3O4 are reported to be
sharp and well defined [13,56]. On the contrary, the investigated Raman
spectrum of the synthesized NiFe2O4 sample, as shown in Fig. 8(b),
exhibit a shoulder like feature at the lower wave number side of the
Raman peaks. The double like a feature of Raman peaks for NiFe2O4
system is reported in the literature [61]. It is reported that γ-Fe2O3 shows
strong Raman peaks at ~1146, ~1378, and 1576 cm− 1
[13,62]. How
ever, these peaks have not been appeared in any Raman spectra (as
shown in Fig. 8(a)). Even if it is assumed that these Raman peaks of
γ-Fe2O3 are present in the noisy background of the sample, the intensity
of these peaks is incomparable with A1g and T2g(2) Raman peak of the
sample [13]. Thus, Raman spectra of investigated samples indicate that
the trace of γ-Fe2O3 is negligible.
3.5. FTIR spectroscopy and elastic properties
Typical Fourier transform infrared spectra of NiFe2O4 sample
annealed at 500 ◦
C, 700 ◦
C, 800 ◦
C and 900 ◦
C in the frequency range
from 250 to 700 cm− 1
is depicted in Fig. 9.
The spectrum of all the annealed samples exhibits two distinct ab
sorption bands below 700 cm− 1,
which supports the crystallization of
ferrite samples into spinel structure, as suggested by Waldron [63]. All
the annealed samples shows an absorption band around 550-560 cm− 1
and 415-425 cm− 1
which are found to be in agreement with the reported
values [4,53]. The higher absorption band (ν1) about 550–560 cm− 1
is
apportioned as stretching vibration of the tetrahedral metal complex,
which consist of bonding between oxygen anion and A-site metal cation.
The lower absorption band (ν2) about 415–425 cm− 1
is apportioned as
stretching vibration of the octahedral metal complex, which is regarded
as bonding between oxygen anion and B-site metal cation [64]. The
emergence of two prime absorption bands below 700 cm− 1
are
accredited to change in metal-oxygen bond length (Fe3+
-O2-
) at both
tetrahedral and octahedral coordination [64]. The vibration of the
tetrahedral metal complex occurs at a higher wave number in compar
ison to the octahedral complex. It may be due to shorter metal-oxygen
(M-O) bond length at the tetrahedral site relative to octahedral one
because the tetrahedral sites have a smaller dimension than that of
octahedral sites. As the energy is proportional to its wave number, the
shorter metal-oxygen bond length at the tetrahedral site requires more
energy for stretching vibration. The cations present at tetrahedral sites
vibrate alongside the line linking the cation with nearby oxygen ions,
whereas the cations at the octahedral site vibrate in a direction
perpendicular to the line linking the tetrahedral site metalation and
oxygen anion [65]. The tetrahedral and octahedral vibrational band
positions for all the annealed samples are tabulated in Table 4. The metal
cation-oxygen anion bond length at both octahedral and tetrahedral
complexes is expected to change with annealing temperature so as to
ease the strain at the lattice site. It is reflected in Fig. 9 by the shifting of
absorption bands towards to higher wave number side [33]. In order to
estimate the strength of bonding between the metal cation and oxygen
anion at tetrahedral and octahedral metal complexes, the force constant
can be determined using FTIR spectroscopy. The force constant could be
estimated by following expression as suggested by Waldron [63].
Kt = 7.62 × MA × ν2
1 × 10− 7
(25)
Ko = 5.31 × MB × ν2
2 × 10− 7
(26)
Here Ko and Kt are the force constant corresponding to octahedral
and tetrahedral metal complex. MB and MA represents the molecular
weight of cations present at the octahedral and tetrahedral site. MA and
MB for each annealed sample have been estimated from the cation dis
tribution achieved through Rietveld analysis. The calculated values of
force constants for annealed samples are tabulated in Table 4. The
estimated values agree well with the reported literature values [64,66].
It illustrates the increasing trend with annealing temperature, which is
expected because of variation in metal-oxygen bond length at both
A-site and B-site. The calculated value of Kt is found to be higher than Ko.
The tetrahedral site metal-oxygen bond length is less than the octahedral
site metal-oxygen bond length. Hence the strength of the metal-oxygen
bond at the tetrahedral site is much stronger than the octahedral site. As
more energy is requisite to break the shorter bonds, it leads to a higher
value of Kt [64]. The vibrational frequency bands obtained from FTIR
spectra could be used in calculating Debye temperature, as suggested by
Waldron [63].
The Debye temperature is regarded as the temperature where lattice
exhibit maximum vibration. The Debye temperature of all the annealed
samples is estimated by following equation [64].
θ*
D =
hcν12
kβ
(27)
Here c represents the velocity of light (3 × 108
m/s), kB stands for
Boltzmann’s constant (1.38 × 10− 23
J/K), h denotes Plank’s constant
Fig. 9. FTIR spectra of NiFe2O4 annealed at various temperature.
Table 4
The values of tetrahedral and octahedral vibration frequency band position νA
and νB, Force constant at tetrahedral and octahedral site (Kt& Ko), average force
constant (Kav), the stiffness constant (C11&C12), elastic constant porous (B, R and
Y) – Bulk modulus (B), Rigidity modulus (R) and Young’s modulus (Y), Poission
ratio (σ), longitudinal elastic wave velocity (Vl), transverse elastic wave velocity
(Vt), mean elastic wave velocity (Vm), Debye temperature (θD) and lattice energy
(UL) of NiFe2O4 annealed at different temperature.
Parameters 500 ◦
C 700 ◦
C 800 ◦
C 900 ◦
C
νA(cm− 1
) 552 554 556 558
νB(cm− 1
) 417 419 421 423
Kt( × 102
N/m) 2.65 2.67 2.69 2.71
Ko( × 102
N/m) 1.06 1.07 1.07 1.08
Kav ( × 102
N/m) 1.855 1.87 1.88 1.895
C11 = C12 (GPa) 222 224 225 227
B(GPa) 219.78 221.76 222.75 224.73
R(GPa) 73.9996 74.6632 74.9981 75.6654
Y(GPa) 199.5974 201.3881 202.2910 204.0907
σ(GPa) 0.348638 0.348643 0.348640 0.348639
Vl( × 103
m/s) 6.4357 6.4610 6.4730 6.5009
Vt( × 103
m/s) 3.7157 3.7302 3.7372 3.7533
Vm( × 103
m/s) 4.0803 4.0963 4.1039 4.1214
θ*
D (K) (FT-IR absorption band) 685 687 690 694
θD(K) (Elastic data) 589 592 594 597
UL (eV) − 120.95 − 121.98 − 122.37 − 123.07
S.K. Paswan et al.

12
(6.624 × 10− 34
J s) and ν12 corresponds to the average wave number of
absorption bands expressed as ν12 = ν1+ν2
2
, ν1 and ν2are the frequency of
absorption bands associated to A-sites and B-sites. The values of Debye
temperature employing FTIR absorption data are presented in Table 4.
The Debye temperature is increased with annealing temperature, which
corresponds to an increase of the normal vibration mode of crystal. The
elastic behavior of the samples under study could also be evaluated
using the FTIR absorption bands frequencies [64,66]. The elastic prop
erties of the crystal are generally described by considering crystal as a
homogeneous continuum medium, assuming that the wavelength of the
elastic wave is very long in comparison to interatomic distance [67]. The
elastic behavior of present samples have been expressed by estimating
different elastic moduli such as elastic stiffness constants, different kinds
of modulus, Debye temperature, and elastic wave velocity [64,66]. The
elastic stiffness constants denoted as C11 is calculated using force con
stant by the relation [68].
C11 =
Kav
a
(28)
where Kav = Kt +Ko
2 and a is the lattice constant. The elastic stiffness
constant C11 as well as C12 are equal for ferrite material possessing cubic
symmetry [68]. The value of stiffness constant is listed in Table 4, which
depicts that elastic stiffness constant, shows an increasing trend with
annealing temperature. The observed change might be attributed to
stiffness of bonding between the atoms in the crystal lattice caused by
cations redistribution between the interstitial sites [64]. The Young
modulus (Y) is evaluated using the expression [68].
Y =
9BR
(3B + R)
(29)
Where B is the Bulk modulus and R is the modulus of Rigidity. The Bulk
modulus (B) is expressed as [68].
B =
1
3
[C11 + 2C12] (30)
The Rigidity modulus (R) is given as [69].
R = ρxV2
t (31)
where ρx the X-ray density is obtained from XRD analysis and Vt is the
transverse elastic wave velocity. The values of these elastic constants are
presented in Table 4. The elastic moduli are observed to be in increasing
trend with annealing temperature which indicates that interatomic
bonding amid various atoms in crystal is getting strengthened continu
ously due to redistribution of cation [64]. Hence deformation of the
sample is expected to be difficult and sample might have a tendency to
retain its original equilibrium position. The Poisson ratio have been
calculated using Bulk modulus (B) and Rigidity modulus (R), which is
calculated as [70].
σ =
3B − 2R
6B + 2R
(32)
The calculated values of the Poisson ratio of annealed samples using
eq (9) are presented in Table 4. As per the theory of elasticity, the values
of σ generally lie in the range of − 1 to 0.5. The value of σ is found to be
within the range from 0.2813 to 0.2816, which implies good elastic
behavior of samples, and it is in accordance with the elasticity theory
[64]. The longitudinal (Vl) and transverse elastic wave velocity (Vt) are
estimated using elastic stiffness constant, which is expressed as [69].
Vl =
̅̅̅̅̅̅̅
C11
ρx
√
(33)
Vt =
̅̅̅̅̅̅̅
C11
3ρx
√
(34)
The values of Vl and Vt are given in Table 4. It is observed that the
velocity of the longitudinal elastic wave is more than the transverse
elastic wave. It is explained as follow: when a wave travels through a
medium, the wave transfers its energy to the particle of the medium,
which force the particle to vibrate. The vibrating particle transfers its
energy to another particle of the medium, which results it to vibrate. For
longitudinal elastic wave, the particle of medium vibrates along the
direction of wave propagation. Therefore less energy is needed so as to
vibrate the neighboring particles. This enhances the energy of waves. As
a result, longitudinal wave velocity is more than that of transverse wave
velocity [69]. Further, the values of longitudinal (Vl) and transverse
elastic wave velocity (Vt) are being used to calculate the mean elastic
wave velocity, and it is expressed as [69].
Vm =
[
1
3
(
2
V3
t
+
1
V3
l
)]− 1
3
(35)
The mean elastic wave velocity (Vm) is employed to calculated Debye
temperature, and it is given as [69].
θD =
hVm
kβ
(
3ρxqN
4πM
)1
3
(36)
where h represents Planck constant (6.626 × 10− 34
J-s), kβ stands for
Boltzmann constant(1.38 × 10− 23
J-K− 1
), N corresponds to Avogadro’s
number (6.022 × 1023
mol-1
), M denotes nickel ferrite molecular weight
(234.37 g/mol), q represents the number of atoms present in one for
mula unit (7, in the present study), ρx denotes the sample X-ray density.
The values of mean elastic wave velocities and Debye temperature are
tabulated in Table 4, which shows the increasing tendency with
annealing temperature. Rise in Debye temperature point toward the
enhancement of samples rigidity [69]. The Debye temperature calcu
lated using eq (36) with the help of elastic moduli and the structural
parameter is extending from 589 K to 597 K whereas the Debye tem
perature obtained from FTIR spectra absorption band analysis using eq
(27) falls in the range of 685–694 K. The small inconsistency concerning
the two approaches might be because of the fact that obtained Debye
temperature using equation (36) employs the X-ray diffraction data
where it takes consideration of defects while equation (36) assumes that
atoms of the sample are elastic sphere as well as vibrates as a whole [70].
In this work, all the elastic parameters and Debye temperature estimated
by FTIR data agrees well with the reported literature values [64]. One
can also express the strength of bonds in compounds in terms of its
lattice energy [70]. The lattice energy is regarded as potential energy
which originates due to atomic orbitals overlapping in the crystal
structure. The lattice energy can be calculated using the expression
represented as [71].
UL = − 3.108
(
MV2
m
)
× 10− 5
eV (37)
where M stands for ferrite samples molecular weight, and Vm represents
mean elastic wave velocity. The values of lattice energy listed in Table 4
are consistent with the previously reported values [70]. The change in
lattice energy supports the elastic behavior of the present samples [71].
3.6. FESEM study
FESEM micrographs of NiFe2O4 sample annealed at 500–900 ◦
C are
shown in Fig. 10(a–e). It shows that majority of the particles are nearly
in spherical morphology. The particles are homogeneously distributed,
and their size is non-uniform. The average size of the particle for each
sample has been estimated by fitting the particle size distribution his
togram to the log-normal distribution function, which is represented as
[72].
S.K. Paswan et al.

13
f(D) =
(
1
̅̅̅̅̅
2π
√
σD
)
exp
⎡
⎢
⎢
⎣ −
ln2
(
D
D0
)
2σ2
⎤
⎥
⎥
⎦ (38)
where D corresponds to average particle size and σDis the standard de
viation. Typical fitting of log normal distribution function to particle size
distribution histogram for 700 ◦
C annealed sample is illustrated in
Fig. 10 (f). The estimated average particle size for 500 ◦
C, 600 ◦
C,
700 ◦
C, 800 ◦
C and 900 ◦
C annealed sample is found to be 35, 37, 59, 63,
and 139 nm along with a standard deviation of 1.17, 1.21, 1.25, 1.28
and 1.37 nm. The average size of the particle is found to be increased
with annealing temperature. The particles of the sample annealed within
500–800 ◦
C falls in the nanometer range, which are near to the average
crystallite size estimated by the XRD study. Whereas the average size of
particle for the sample annealed at 900 ◦
C is around 0.139 μm, which is
bigger than the size of crystallite estimated from XRD analysis. The
above observation reveals that 900 ◦
C is the sufficient annealing
temperature to transform the sample from nanocrystalline to bulk form.
The possible explanation for the increase of particle size with annealing
temperature is as follow: smaller particles possess a big surface area. By
increasing the annealing temperature, a number of nearby particles get
fuse together to agglomerate via surface melting. As a result, the size of
the particles becomes big. The qualitative chemical composition analysis
of the samples has been studied by energy-dispersive X-ray spectroscopy
(EDS). The EDS spectrum confirms the existence of Ni, Fe, and O ele
ments in the samples. Typical EDS pattern of 700 ◦
C annealed sample is
depicted in Fig. 11. The atomic % of Ni: Fe: O is 11.42:29.18:56.40, i.e.
1:2.5:5.0, which is close to the theoretical stoichiometry of the sample
with an error of 2–3%.
3.7. TEM study
Typical TEM micrograph, along with SAED (selected area electron
diffraction) pattern for 700 ◦
C annealed sample is depicted in Fig. 12(a)
and (b).
Fig. 10. FESEM image of NiFe2O4 nanoparticles annealed at (a) 500 ◦
C (b) 600 ◦
C (c) 700 ◦
C (d) 800 ◦
C (e) 900 ◦
C (f) Particle size distribution for 700 ◦
C annealed
sample fitted with a log normal distribution function.
S.K. Paswan et al.

14
The TEM micrograph illustrates that the majority of particles appear
spherical in shape as well as, to some extent, agglomerated. The esti
mated average particle size is 58 nm, along with a standard deviation of
1.23 nm, which has been estimated by fitting the particle distribution
histogram using log-normal distribution function. The SAED pattern
depicts concentric rings with spots, which indicate the polycrystalline
nature of the present sample [42]. Each ring represents the Bragg
reflection planes with different interplanar spacing. The spotty rings
correspond to (111), (220), (311), (222), (400), (422), (511), and (440)
reflection with Fd3
−
m space group. The SAED pattern of the present
NiFe2O4 system has been indexed using the C spot program.
3.8. UV absorbance study
The UV–vis optical absorbance spectra of NiFe2O4 sample annealed
at 500 ◦
C, 600 ◦
C, 700 ◦
C, 800 ◦
C, and 900 ◦
C over the wavelength range
200–800 nm is shown in Fig. 13(a).
The spectra show the sequential shift of absorbance band edge to
wards to higher wavelength as the annealed temperature is increased
from 500 ◦
C to 900 ◦
C. It clearly indicates that the optical energy band
Fig. 11. Energy-dispersive X-ray spectrum NiFe2O4 annealed at 700 ◦
C.
Fig. 12. (a) Transmission electron microscopy image, (b) Selected-area electron diffraction pattern NiFe2O4 annealed at 700 ◦
C.
Fig. 13. (a) UV–vis absorption spectra of NiFe2O4 annealed at different temperature. (b) Tanabe Sugano energy level diagram for Ni2+
cation in octahedral
environment.
S.K. Paswan et al.

15
gap of the sample could be modified by varying annealing temperature.
The absorbance spectra show a broad absorption band around at 261 nm
which is accredited to ligand-to-metal charge transfer transition where
the electrons present in O 2p valence band states makes transition to Fe
3d conduction band states [73]. A wide absorbance band is observed in
300–800 nm range. These absorption bands are result of electronic
transition within the Ni d orbitals which arise under the impact of
octahedral crystal field. Interestingly, a sharp absorption band around at
748 nm has been appeared in the absorption spectra, which is an
important observation of the present study. A very few literatures are
available where the appearance of a sharp absorption band around 748
nm for NiFe2O4 materials is reported [74–76]. In the present work, we
have analyzed the appearance of the UV–vis absorption band in
300–800 nm using crystal field theory and Tanabe Sugano (T-S) energy
level diagram [77]. The T-S energy level diagram for Ni2+
cation in an
octahedral environment, and the spin allowed possible d-d transition is
depicted in Fig. 13(b). The electronic distribution of free Ni2+
ion (d8
)
give rise to spectroscopic 3
F3
P1
G,1
D, 1
S energy terms, where 3
F and 3
P
are respectively the ground state and first excited state term. The de
generacy of 3
F term in octahedral symmetry is lifted and gets split into
three crystal field terms. These terms are obtained by group theory
analysis and expressed in terms of Mullikan symbol as 3
F→ 3
A2g+ 3
T2g+
3
T1g where 3
A2g is the ground state term. The 3
P term remains unaffected
in the octahedral field and represented in Mullikan symbols as 3
T1g [77].
The general feature of Ni2+
absorption described in terms of ligand field
theory yield three prominent absorption band in the octahedral envi
ronment, which arise due to the following spin allowed transition
3
A2g(3F)→3
T2g(3F) (t6
2ge2
g →t5
2ge3
g ), 3
A2g(3F)→3
T1g(3F) (t6
2ge2
g →t5
2ge3
g ) and
3
A2g(3F)→3
T1g(3P)(t4
2ge4
g →t4
2ge4
g ) [73,77]. The transition from 3
A2g
ground state to 3
T2g
3
T1g(3F) and 3
T1g(3P) excited states exists in
1100–1400, 600–900, and 380–450 nm wavelength region [78]. Hence
in the present study, the sharp appearance of the absorption band
around at 748 nm is attributed to 3
A2g(3F)→3
T1g(3F) electronic transi
tion. In a real semiconductor, always defects are present. The presence of
defects gives rise to addition potential energy, which leads to the
different energy distribution of density of electronic states (appearance
of a tail in the density of electronic states) in comparison to an ideal
semiconductor [79]. For the correct estimation of the energy band gap of
real semiconductor crystal using UV visible absorption spectra, one can
utilize the following relation presented by Tauc [80].
αhν = A
(
hν − Eg
)S
(39)
where Eg represents optical energy band gap expressed as Eg = hc
λ
, h
stands for Planck’s constant (6.6 × 10− 34
J-s), c denotes the velocity of
light (3 × 108
m/s), λ corresponds to the absorbed wavelength, α rep
resents the absorption coefficient, hν stands for the incident photon
energy in eV, A is band edge sharpness constant, and exponent S rep
resents different types of allowed electronic transition. If S = 1
2
then, the
system corresponds to direct allowed inter-band transition while S = 2
represents indirect allowed inter-band transition in the system.
Furthermore, the value of the absorption coefficient α is estimated using
the following relation [80].
α =
4πk
λ
(40)
where k and λ are absorbance and incident light wavelength. With
the help of equations (2) and (3) the direct and indirect allowed energy
band gap could be estimated by plotting (αhν)2
and (αhν)
1
2
versus inci
dent photon energy hν and linearly regressing the linear portion of the
(αhν)2
and (αhν)
1
2
to zero. The line meeting at the point on the incident
photon energy axis represents the direct and indirect band gap energy.
The estimation of the optical energy band gap for NiFe2O4 nanoparticles
by the Tauc plot has been reported in the literature for both direct and
indirect allowed transition [9,66,75,80,81]. In the present work, we
have analyzed the UV absorption spectra of annealed samples by Tauc
plot for both direct and indirect allowed transition. Typical Tauc plot of
(αhν)2
versus hν for the estimation of direct energy band gap for 700 ◦
C
annealed sample is illustrated in Fig. 14(a). The values of the direct
optical energy band gap for annealed samples are presented in Table 5. It
lies in the range of 3.66 eV to 3.92 eV, which is in close agreement with
the recent published literature [80].
A typical indirect transition Tauc plot of (αhν)
1
2 versus hν for 700 ◦
C
annealed sample is presented in Fig. 14(b) for the estimation of the in
direct energy band gap. The estimated indirect energy band gap values
for all the annealed sample is in the range of 1.51eV to 1.69eV, which is
reasonably well with the experimental and theoretical reported values
[9,81]. The value of the indirect energy band gap is smaller than that of
the direct energy band gap, which is usually endorsed to the involve
ment of phonon in the optical absorption process [67]. The important
observation is the finding of the reasonable value of the indirect energy
band gap of the samples. The origin of the indirect band gap in the
sample could be well supported by the XPS study. The metastable mixed
valence states of Ni2+
and Ni3+
might create intrinsic oxygen vacancies
and defect levels in materials [82]. As a result, the sample may develop
an indirect band gap. A similar investigation of the indirect band gap is
reported in literature by Pradhan et al. [82] for thin film of Ni–Zn ferrite.
The present study suggests that both direct and indirect band gap system
has been developed for NiFe2O4 sample, which are in well agreement
with the existing literature [9]. If we look at Table 5, the optical energy
band gap follows an increasing trend with the annealing temperature. It
could be analyzed as follow: The defect states are always present in the
crystalline material. Hence, one could not decline the creation of
localized states of energy in the energy bandgap region due to crystal
defects. These localized defect energy states trap the excited electrons
and prevent their direct transition to the conduction band [83]. The
smaller size of nanoparticle has a high proportion of surface to volume
atoms. As an effect, unsaturated bonds on the surface of nanoparticle put
the atoms in stressed conditions, which may lead to significant vacancies
of oxygen and other crystalline defects. These crystal defects as well as
oxygen vacancies act as exciton trapping center within the energy band
gap, forming a series of metastable energy levels, which prevents the
transition of charge carriers to the conduction band. The increase of
annealing temperature helps the atoms to arrange themselves in an
organized way leading to an increase of particle size with enhanced
crystallinity and stoichiometry. As a result, there is a lowering of oxygen
vacancies, crystal defects, and trap levels between the conduction band
and the valence band. Hence the optical band gap is increased with the
annealing temperature [10]. In summary, lattice parameter, particle
size, change in cation distribution, presence of defects, structural and
thermal disorder, and formation of sub-band gap energy levels are
responsible for band gap modification in materials [9,11,81]. Similar
observation for NiFe2O4 and other spinel ferrite system has been re
ported in the literature [9,10].
The defect states in the optical band gap region are represented by an
optical parameter known as Urbach energy. These localized states of a
defect in the band gap region are accountable for the formation of ab
sorption tail in the absorption spectrum. This tail is termed as Urbach
tail, and energy associated with it is called Urbach energy [83]. The
Urback energy can be extracted from absorption spectra, and it can be
calculated using the following relation [83].
α = αo exp
(
E
Eu
)
(41)
where α denotes the absorption coefficient, αo is constant, E denotes the
incident photon energy, which is equal to hν and Eu represents the
Urbach energy. The value of Eu(Urbach energy) is estimated from ln(α)
versus photon energy plot. Typical plot for 700 ◦
C annealed samples is
S.K. Paswan et al.

16
shown in Fig. 14(c). The reciprocal of the slope obtained by fitting the
linear part of the curve yield the value of Eu [83]. The value of Urbach
energy of all the annealed samples is presented in Table 5. It shows that
the behavior of the optical energy band gap to that of Urbach energy are
opposite to each other with annealing temperature enhancement. The
sample annealed at 500 ◦
C has a higher value of Urbach energy. It
suggests that there is a considerable structural disorder in the sample
due to the presence of surface dangling bond, which leads to a high
density of localized defect states in the band gap region. The Urbach
energy decreases upon annealing temperature enhancement, which in
dicates the tendency of the creation of less localized defects states in the
band gap region [83]. It is caused by an increase of the size of the par
ticle and a decrease in structural disorder because annealing relaxes the
structure of the material and produce a more organized structure [84].
Similar studies on Urbach energy for ferrite materials and other oxides
materials have been reported in recent literature [85]. The band edge of
the valence band (VB) and conduction band (CB) for all the annealed
samples can be calculated using the obtained values of the energy band
gap. The valence band edge and conduction band edge can be
determined by following relations [66].
ECB = χ − EC
− 0.5Eg (42)
EVB = ECB + Eg (43)
where EVB is the valence band edge, ECB represents the conduction band
edge and Eg denotes the optical energy band gap between VB and CB of
the sample. The term EC
represents the free electrons energy on the
hydrogen scale, whose value is 4.5 eV. The term χ is the absolute elec
tronegativity (Mullikan electronegativity) of the sample, which is
calculated using the following equation [86].
χ =
[
χ(A)a
χ(B)b
χ(C)c] 1
(a+b+c)
(44)
where a, b and c are the numbers of atoms in the compound. The value of
χ(A) is determined by taking the arithmetic mean of the first ionization
energy and electron affinity of atom A. Similarly, the value of χ(B) and
χ(C)have been calculated. Taking the values of electron affinity and first
ionization energy of oxygen, nickel, and iron atom from the periodic
table, the calculated value of absolute electronegativity χ of the sample
is ~5.80 eV. With the help of absolute electronegativity (χ) and direct
energy bandgap, the value of valence band edge (ECB) and conduction
band edge (EVB) of all the annealed samples have been calculated and
presented in Table 5. Our results are consistent with the previous re
ported results [66]. The estimated values of the energy band gap can be
used for the determination of the refractive index of the sample. A linear
empirical relation between energy band gap and refractive index is
represented as [66,80].
n = 4.084 − 0.62Eg
The estimated value of the refractive index of 500 ◦
C, 600 ◦
C, 700 ◦
C,
800 ◦
C and 900 ◦
C annealed samples is found to be 1.81, 1.76, 1.73,
1.70, and 1.65. The calculated values of the refractive index agree well
Fig. 14. (a) Direct transition Tauc plot (b) Indirect transition Tauc plot (c) Urbach energy plot for NiFe2O4 annealed at 700 ◦
C.
Table 5
Direct energy band gap, indirect energy band gap, Urbach energy, conduction
band and valence band parameters for NiFe2O4 annealed at different
temperature.
Annealing
temperature
Direct
energy
band gap
(eV)
Indirect
energy band
gap (eV)
Urbach
energy
(eV)
ECB(eV) EVB(eV)
500 ◦
C 3.66 1.51 0.704 − 0.53 3.13
600 ◦
C 3.74 1.53 0.492 − 0.57 3.17
700 ◦
C 3.79 1.56 0.375 − 0.59 3.20
800 ◦
C 3.84 1.66 0.367 − 0.62 3.22
900 ◦
C 3.92 1.69 0.353 − 0.66 3.26
S.K. Paswan et al.

17
with the previous reported values [66,80]. The refractive index is found
to be inversely proportional to the energy band gap.
3.9. Magnetic studies
The magnetic hysteresis loop measured at room temperature for 500,
600, 700, 800, and 900 ◦
C annealed sample is shown in Fig. 15.
The samples annealed within the range 500–800 ◦
C do not reach
saturation level even at the highest magnetic field of 30 kOe, whereas
the 900 ◦
C annealed sample reach the saturation level at the same field.
The non-saturation behavior for samples annealed up to 800 ◦
C point out
the presence of spin disorder on the nanoparticles surface [87]. The
highest magnetization at 30 kOe is stated as the saturation magnetiza
tion (Ms) throughout the discussion. The observation of narrow mag
netic hysteresis loops for all the annealed samples illustrate the soft
ferrimagnetic nature of samples. The value of coercivity for all the
annealed samples is presented in Table 6. The coercivity values of pre
sent samples agree well with the reported values [88]. Upon enhancing
the annealing temperature from 500 to 800 ◦
C, the coercivity follows an
increased trend with values increasing from 100 Oe to 180 Oe. Further
raising the annealing temperature from 800 to 900 ◦
C, the value of
coercivity is observed to be decreased from 180 Oe to 60 Oe. The
observed behavior of coercivity with annealing temperature indicates
that samples annealed up to 800 ◦
C consists of an ensemble of single
domain magnetic nanoparticles. The reduction in coercivity for 900 ◦
C
annealed sample indicates the formation of multidomain particles in the
sample [88].
The present observation reveals that a single domain structure of the
magnetic nanoparticle system could be tuned effectively by varying the
annealing temperature. A similar observation of coercivity change with
annealing temperature for the NiFe2O4 system has been observed by
Malik et al. [88]. The increase of sample coercivity with annealing
temperature (up to 800 ◦
C) in the single domain region could be
described by Stoner-Wohlfarth theory. According to the
Stoner-Wholfarth model, the magnetic anisotropy for single domain
nanoparticles is expressed as EA = KVSin2
θ where K represents the
magnetic anisotropy constant, V stands for particle volume, and θ cor
responds to the angle between the easy axis and magnetization direction
of the nanoparticle [89]. Coercivity exemplifies the magnetic field
strength required to overcome on magnetic anisotropy energy for flip
ping of spins. As the magnetic anisotropy energy is proportionate to
particle volume, the particle whose size is bigger would result in large
magnetic anisotropy energy. Hence, a higher external applied magnetic
field is needed to overcome the magnetic anisotropy energy for flipping
of moments coherently away from its easy axis, which results high value
of coercivity. The increase of the size of the particle upon annealing is
expected to increase magnetic anisotropic energy, which results in the
enhancement of coercivity. The sample annealed at 900 ◦
C is in bulk
form, which is in accordance with FESEM image analysis. The sample in
its bulk form comprises of multidomain particles where magnetic do
mains are separated by domain walls. The demagnetization in bulk
material occurs through the movement of the domain wall which is
easier as compared to single domain moment reversal. Hence low
magnetic field is required [15]. Therefore, coercivity for 900 ◦
C
annealed sample is low as compared to 800 ◦
C annealed samples. All the
annealed samples at room temperature exhibit a very narrow hysteresis
loop with a very low value of coercivity. So Arrott plots have been uti
lized for all the annealed samples in order to confirm the ferrimagnetic
character and superparamagnetic behavior [90]. The Arrott plot is a
variation of M2
as a function of H
M [90]. Typical Arrott plot for 700 ◦
C
annealed sample is shown in the inset of Fig. 15. The room temperature
Arrott plot of all the annealed samples depicts clear positive intercept on
the M2
axis at H = 0. It supports the presence of spontaneous magneti
zation, which corresponds to ferrimagnetic phase [90]. Hence Arrott
plot confirms the ferrimagnetic character and rules out the possibility of
superparamagnetic behavior in all the annealed samples at room tem
perature [90]. The value of Ms at room temperature for all the annealed
samples at 30 kOe has been estimated via fitting to the magnetization
curve above than 15 kOe using “Law of Approach to Saturation” (LAS).
The value of Ms for all the annealed samples at 30 kOe has been tabu
lated in Table 6. The estimated values of Ms are consistent with the
previous reported values [6,91]. Generally, the saturation magnetiza
tion (Ms) is believed to increase with annealing temperature or precisely
with the increase of particle size [6,88]. In the present study, the
NiFe2O4 system follows the same trend. The highest value of Ms is found
to be 48 emu/g for 900 ◦
C annealed sample, which is close to the bulk
value (55 emu/g) of NiFe2O4 [91]. The value of Ms for 500 ◦
C annealed
sample is 27emu/g, which is significantly lower than 900 ◦
C annealed
sample. The expected smaller value of Ms for ferrite nanoparticles in
comparison to its bulk counterpart could be described using the
core-shell model [91]. This model presumed that each ferrite nano
particle has a ferrimagnetic core (which is responsible for order
parameter), and it is surrounded by a magnetic dead layer [91]. The
magnetic dead layer of thickness t is expressed as [72].
Ms(d) = Ms(bulk)
(
1 −
6t
d
)
(45)
where Ms(d) denotes saturation magnetization for particles of size d and
Ms(bulk) represents saturation magnetization of the bulk sample. The
magnetic dead layer is supposed to present on the nanoparticle surface.
The magnetic dead layer is associated with the surface effect, which
includes atomic vacancies, uncompensated disorder of surface spin, non-
collinear spin structure at particle surface (spin canting with respect to
core spin), dangling bond, pinning of moments at the surface and
reduced co-ordination of atoms [5]. These surface effects mainly arise
because of breaking of the crystal symmetry at the particle surface (loss
of long range order). Hence magnetic atoms at the surface are expected
Fig. 15. Room temperature DC magnetic hysteresis loop for NiFe2O4 annealed
at different temperature. The inset (a) shows the Arrott plot for 700 ◦
C annealed
sample. The inset (b) shows enlarged view of hysteresis loop around the origin.
Table 6
The value of saturation magnetization (Ms), coercivity (Hc) and magneto
crystalline anisotropy constant (K1) for NiFe2O4 annealed at different
temperature.
Annealing temperature Ms (emu/g) Hc (Oe) K1(erg/cm3
)
500 28.38 100 7.02 × 105
600 38.33 120 11.02 × 105
700 42.59 140 13.75 × 105
800 45.61 180 14.89 × 105
900 47.84 60 4.02 × 105
S.K. Paswan et al.

18
to experience a broken exchange interaction originating from broken
exchange bonds. Therefore this magnetic dead layer is expected to affect
the saturation magnetization substantially, especially for a sample
consisting of smaller nanoparticles. The magnetic dead layer thickness
and surface spin disorder is expected to decrease on increasing the size
of nanoparticle owing to a small proportion of surface to volume atoms.
The magnetic dead layer thickness for 500 ◦
C annealed sample is rela
tively high to that of a 900 ◦
C annealed sample. Hence surface effects are
likely to be higher in 500 ◦
C annealed samples leading to reduced
magnetization. As the particle size is increasing with annealing tem
perature, the saturation magnetization is expected to increase due to the
minimization of surface effects. In the present study, we have observed
the same trend, and it is attributed to a reduction in spin disorder and a
dead layer at the surface. Another possible factor for increase in satu
ration magnetization of samples with annealing temperature might be
ascribed to cations redistribution over A-sites and B-sites i.e. exchange of
Ni2+
and Fe3+
cations from A-sites to B-sites and vice versa. The redis
tribution of cations over interstitial sites is expected to arise due to
increased mobility of cations at elevated temperature. The magnetic
ordering in NiFe2O4 is due to superexchange interaction, and this
interaction is strongest between A-sites and B-sites (AB interaction)
cations. The distribution of cations estimated from Rietveld analysis
(discussed in previous section) illustrates that increase of annealing
temperature results in increased occupancy of Fe3+
cations at tetrahedral
sites (A-sites), whereas the occupancy of Ni2+
cations are more at octa
hedral sites (B-sites). As a result, more number of Fe3+
A − O− Fe3+
B
superexchange interactions is expected. The magnetic moment of
Fe3+
cation (5μβ) is more than Ni2+
(2μβ) cation. Thus Fe3+
A − O− Fe3+
B
superexchange interaction is expected to be strongest among the avail
able A-B interaction. Hence the increase of saturation magnetization
with an annealing temperature of the present NiFe2O4 system is
accredited to the strengthening of Fe3+
A − O − Fe3+
B superexchange
interaction [92]. The increase of saturation magnetization with
annealing temperature also indicates the increase of the average size of
the magnetic domain. The average size of the magnetic domain can be
estimated using the following equations [91].
ds =
[
18kβT
πρxM2
s
(
dM
dH
)
H=0
]1
3
(46)
Here
(
dM
dH
)
represents the slope of the M-H curve at H = 0 Oe
(calculated from Fig. 15), kβ denotes the Boltzmann constant (1.38 ×
10− 16
erg/K), T is temperature (300 K), ρx corresponds to density of
nickel ferrite (5.380 g/cm3
) and Ms represents the saturation magneti
zation. The estimated average size of the magnetic domain is around 24,
28, 47, 62, and 108 nm corresponding to sample annealed respectively
at 500, 600, 700, 800, and 900 ◦
C. It implies that the average magnetic
domain size is increasing with the increase of the particle size. The
surface to volume proportion of atoms decreases on increasing the size.
Hence the moments which are pinned on the surface is expected to
decrease. Upon increasing the size of the magnetic domain, the align
ment of more and more atomic spins is expected in the field direction. As
a result, it leads to the enhancement of saturation magnetization [97]. In
summary, the existence of canted spin at particle surface with respect to
core spin, disordered surface spin, redistribution of cations, broken ex
change bonds, crystal defects, dislocation, and lattice strain could result
in a reduction of saturation magnetization, and these effects becomes
less significant with annealing temperature [93].
As the magnetic nanoparticles exist only in a single domain state, its
underlying magnetization reversal mechanism is related to magnetic
anisotropy only due to the absence of a domain wall. The magnetic
anisotropy is an amalgamation of magnetocrystalline anisotropy, shape
anisotropy, strain anisotropy, and surface anisotropy. The magnitude of
magnetic anisotropy is represented by magnetic anisotropy constant (K),
which is expressed as anisotropy energy per unit volume. For spherical
magnetic nanoparticles, the largest contribution to magnetic anisotropy
mainly comes from magnetocrystalline anisotropy, which is expressed in
terms of magnetocrystalline anisotropy constant (K1) [89]. Hence, for
spherical magnetic nanoparticles, the magnetic anisotropy constant (K)
could be approximated as magnetocrystalline anisotropy constant (K1)
[89]. The FESEM micrographs of the present sample illustrates that
particles are almost in spherical morphology. Hence, in the present
study, K1 (magnetocrystalline anisotropy constant) could be assumed to
equal to K (magnetic anisotropy constant). So as to extract the infor
mation about magnetic anisotropy, the initial magnetization curves of
all the annealed samples have been fitted to LAS (law of approach to
saturation) to estimate the magnetocrystalline anisotropy constant (K1).
LAS delineate the dependency of magnetization (M) on the applied field
(H) where applied field (H) is much greater than the coercive field (Hc).
LAS is generally used to describe the magnetization in a high magnetic
field region where the rotation of the magnetic domain plays a signifi
cant role. It is presented as [94].
M = Ms
[
1 −
a
H
−
b
H2
]
+ κH (47)
The term a
H
is associated with structural defects, and its origin is non-
magnetic. The term b
H2 represents the rotation of magnetization against
the magnetocrystalline anisotropy energy. The term κH is recognized as
forced magnetization, and it is like a paramagnetic term which is caused
by increase in spontaneous magnetization linearly with the applied
magnetic field. The forced magnetization term is generally required
when magnetic hysteresis curves are subjected to higher temperatures
and very high magnetic fields. Different research groups have estimated
the magnetocrystalline anisotropy constant for spinel ferrite materials
by fitting the high field data with LAS keeping the term b
H2 only and
neglecting the term a
H
and κH [95]. In the present case, we have neglected
the term a
H
and κH in equation (47) for estimation of magnetic anisotropy
constant. The observed magnetization data for the applied magnetic
field above 2 kOe for all the annealed samples have been fitted using
equation (47). The fitting parameter Ms and b is used to estimate the
magnetocrystalline anisotropy constant (K1), which is related as
K1 = μoMs
̅̅̅̅̅̅̅̅̅̅
105b
8
√
(48)
Here μo stands for permeability of the free space, Ms represents
saturation magnetization, and K1 denotes the magnetocrystalline
anisotropy constant with cubic symmetry. The numerical coefficient in
equation (48) applies to a polycrystalline sample with cubic anisotropy.
A typical LAS fitting curve for the 700 ◦
C annealed sample is shown in
Fig. 16. Fitting to the Law of Approach to Saturation (LAS) for NiFe2O4
annealed at 700 ◦
C.
S.K. Paswan et al.

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