Effect Of Grain Size On Neel Temperature, Magnetic And Electrical Properties Of NiFe2O4 Nanoparticle Synthesized By Chemical Co-Precipitation Technique

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Introduction
Introduction
to Nanomaterials: Nanoclusters are ultrafine
particles of nanometer dimensions located in the transition region between
molecules and microscopic (micron-size) structures. It is in this size regime
that many recent advances have been made in biology, physics, and chemistry .
For example,
Ø 
When the particle
dimensions of semiconductor materials become comparable to, or smaller than the
Bohr radius, an increase in the energy band gap is observed.
Ø 
In noble metals,
the decrease in size below the electron mean free path (the distance the
electron travels between scattering collisions with the lattice centers)  gives rise to intense absorption in the
visible–near-UV region .
Ø 
Metal
nanoparticles also exhibit a broad range of fascinating mechanical behavior
such as super plasticity .
Ø 
Ceramic materials
composed of powders with a particle size in the nanometric range are also
receiving attention because they may significantly enhance sintering rates or
dramatically lower sintering temperatures .
Ø 
Also, ceramic
matrix composites with dispersed nanoparticles have better  mechanical properties .
Ø 
Better biocompatibility
and successful self-heating temperature rising
characteristics specially of NiFe2O4 and Fe3O4 nanoparticles.
The
remarkable properties of the nanocrystalline magnetic materials arise from the
various magnetic phenomena occurring at the nanosize level and hence the author
is interested in understanding the effect of particle/grain size on the
magnetic properties of both soft and hard magnetic materials. In view of this,
he has chosen Mn-Zn ferrite, yittrium iron garnet and gadolinium iron garnet
which are soft magnetic materials and SmCo₇ and Nd₂Fe₁₄B/α-Fe
which are hard magnetic materials for his study.
The
present chapter gives an introduction to the techniques for the synthesis, the
structure and magnetic properties of the nanocrystalline soft magnetic
materials like ferrites and garnets and the permanent magnetic materials like
Sm₂Co₁₇ and Nd₂Fe₁₄B/ α-Fe that are
studied and reported by the author in his thesis.
  1. Effect of
particle size on the magnetic properties of nanocrystalline materials:
The
magnetic materials are classified  into
two groups such as soft and hard magnetic materials. Nanocrystalline soft
magnetic materials have low magnetocrystalline anisotropy resulting in
reduced  coercivity and high permeability
such as Finemet alloys and high curie temperature Hitperm (Fe-Si-B-Nb-Cu )  , Nanoperm (Fe-M-B-Cu) alloys and high curie
temperature Hitperm (Fe-Co-Cu-Zr-B) alloys used for high frequency
applications. The coercivity increases with grain size reduction down to the
single domain size obeying the 1/D law where D is the grain size and when
reduced further, it follows the sixth power of the grain size. However, the
stress anisotropy is found to increase the coercivity in the ball-milled Finemet
alloys . The initial permeability of the ball-milled Finemet alloy core has
been improved upon by the substitution of Al due to the reduction in the
magnetocrystalline anisotropy energy and the magnetostriction coefficient . The
permeability of the Finemet and the Co-based amorphous alloys are also found to
be superior to that of the other bulk alloys. The random anisotropy  realized in theses materials results in an
enhanced permeability and hence these alloys show a better frequency response. The
saturation magnetization for the nanocrystalline ferrites, in general, is found
to be lower compared to their bulk value, which is attributed to surface spin
effects . In some cases an enhancement in the saturation magnetization is
observed due to the change in cation distributions  which depends on the crystal field
stabilization energy of the cations. Apart from grain size, the cation
distribution, which depends on the synthesis condition, is found to play a
major rule in the observe changes in their magnetic properties . An enhancement
in the Neel temperature and spin casting  are observed in ferrites. The coercivity for
the nanocrystalline materials is  small
when they are in their multidomain state and will be maximum at the critical
single domain size and decreases further with grain size reduction as it
approaches superparamagnetism, due to the increase in the thermal energy
compared to the anisotropy energy. The superparamagnetic behavior, below a
critical size, limits them for magnetic recording applications. Recently it has
been reported that the superparamagnetic limit could be overcomed by increasing
the exchange anisotropy. However, the superparamagnetic property of ferrite
nanoparticles can be utilized in treating cancers from the heat produced by
superparamagnetic particles due to Neel and Brownian relaxations [9, 4]. The
core-shell morphology of the nanoparticles 
in an enhanced anisotropy and magnetization  is also studied by researches. The surface
anisotropy of the nanomagnetic materials increases with particle size reduction
for some soft magnetic materials .
The
high coercivity  of the hard magnetic
materials arises from the large magnetocrystalline anisotropy of the materials
with anon-cubic structure. FePt nanoparticles of f.c.t. structure with high
coercivity, synthesized using chemical methods have the potential for use as
magnetic recording medium . Apart from the structure, the mechanism for the
enhanced magnetic properties at the nanoscale could be different like domain
wall pinning or nucleation as in Sm-Co permanent magnetic materials or exchange
coupling as in nanocomposite permanent magnetic materials. Nanocrystalline
Sm-Co permanent magnetic materials with cellular microstructures are being
developed for high temperature applications. Nanocomposite Nd₂Fe₁₄B/α-Fe
exhibits enhanced energy product due to this improved exchange coupling between
the two phases when the  grain size of
the materials is reduced to a few nanometer.  
1.2. Synthesis of nanostructed magnetic
materials
The
nanophase materials can be synthesized with varying grain size using the
techniques like mechanical milling , melt spinning , chemical vapor
deposition  , inert gas condensation,
pulsed laser ablation, sputtering etc. Nanoparticles can also  be synthesized though co-precipitation ,oxidation
, reverse micelles  and  polyol process  in addition to their chemical methods . As
some of the techniques like chemical, mechanical alloying and melt spinning are
used by the author to synthesize the nanomagnetic materials.
1.2.1. Chemical methods
A
variety of chemical methods like co-precipitation, oxidation, sol-gel, reverse
micelle,  polyol process etc. are
available for the synthesis of nanomagnetic materials. The co-precipitation  technique, which results in particle size
smaller than 25 nm, is widely used in the synthesis of Nanoparticles. The co-precipitation
technique has been used in the synthesis of CdFe₂O₄  , ZnFe₂O₄ , Ni-ZnFe₂O₄
 and MgFe₂O₄  in addition to other spinal ferrites. The
size-dependent magnetic properties of MnFe₂O₄ with
particle size from 5 to 25 nm have been studied for the samples synthesized
using the co-precipitation method . The super paramagnetic behavior of CuFe₂O₄
has been studied for the co-precipitated samples prepared in a polymer matrix.
The super paramagnetic behavior in MgFe₂O₄ in the
particle ranging from 6 nm to 18 nm has been studied by Chen and Zhang . The Co-precipitation
technique was also used in synthesizing Co-Cu powders by Chow et al.
The
reduction of metal salts to their corresponding metals in an alcohol medium
like ethylene glycol or propylene glycol is employed in the synthesis of
metals, compounds and ferrite Nanoparticles. The thermal decomposition of the
carbonyl precursors was used to synthesize SmCo₅ through chemical
methods. However, the oxidation of the rare earth compounds prevents them from
being synthesized through chemical methods and there are no reports in the
literature on  the synthesis of the Nd₂Fe₁₄B
compound through chemical methods. Magnetic spinel ferrite nanoparticles in the
size range of 10-20 nm are synthesized through reverse micelles methods . 
Oxidation
method in the which the ferrous ions are oxidized to ferrite ions using air
oxidation or an oxidant  like KNO₃
has been used in the synthesis of spinel ferrites . The mechanism for the
size-controlled synthesis of particles was described by LaMer and Dinegar . A
modified oxidation method has been utilized to synthesize ferrites with large
particle sizes where a suitable concentration of ferric ion is used along with
precursor salts thereby separating the nucleation and growth stages by
Chinnasamy et al.
Mechanical
milling is effectively used in the reduction of bulk materials to the nanocrystalline
or amorphous form and in the synthesis of nanomaterials. Various types of mills
are used in the synthesis of nanomaterials like planetary mills, vibratory
mills and attritor mills. The repeated welding and fracturing of powder
particles in between the balls during milling changes the physical
characteristics of the material.
Hence,
the extent of the physical change depends on the mechanical properties of the
materials especially their ductile and brittle behavior. The energy imparted in
a planetary mill is larger compared to that in vibratory mills. The ball movement
is in a planetary ball mill. During the collision of balls with the powder, the
events like direct and indirect seizure and dynamic, forging and shear
fractures occur, which alter the particle morphology. Mechanical milling has
been extensively carried out for the grain size reduction  in the study of magnetic properties of
nanocrystalline spinel ferrites and rare earth permanent magnetic materials .
Ball milling has yielded interesting magnetic properties like changes in the cat
ion distribution and Neel temperature in spinel ferrites. Through ball milling
not only the changes in the magnetic properties are observed, but also the
structural properties are drastically altered like decomposition as in hex
ferrites, reduction of ferrites to metals and ions  and phase changes .
Extensive
work has been done in the study of rare earth permanent magnetic materials
using mechanical milling and alloying. Mechanical milling leads to the formation
of amorphous phase  and nanocomposites
can be synthesized using both mechanical milling and mechanical alloying .
The
rapid solidification process through melt spinning technique is widely used to synthesize
amorphous alloys. They are widely used in the synthesis of metal Finemet and
rare earth alloys . In melt spinning technique the molten alloy is allowed to
pass through a narrow orifice on to a rotating copper wheel which quenches the
molten alloy at a cooling rate greater than 10⁴ k/s. The resulting
samples will be in the form of ribbons, typically a few microns thick. The
final microstructure and phase of the ribbon sample is influenced by the
melting temperature, ejecting pressure, wheel speed, nozzle diameter, distance
between the wheel to orifice, conductivity of the wheel and the nature of the
sample . Since a lot of parameters are involved in the synthesis of nonmaterial
through melt spinning technique, the material properties naturally vary with
processing condition. Melt spinning technique is efficiently used in the synthesis
of rare earth permanent magnetic materials containing boron due to their glass
forming nature, especially Nd₂Fe₁₄B compounds . Depending
on the composition of the material chosen, amorphous or crystallized ribbon is
obtained. Nd₂Fe₁₄B permanent magnetic material with
varying grain sizes were synthesized using melt spinning technique by altering
the wheel speed. The influence of the cooling rate on the magnetic properties
of Nd-Fe-B materials has been studied and the critical cooling rate for the amorphous
phase formation is determined as 5.4×10⁵ to 7.1 ×10⁵ k/s.
A serious of Nd₉Fe_91-xB_x (x=0-9) ribbons
prepared using the melt spinning technique are found to a metastable TbCu₇
phase for x ≤ 7 . The cooling rate for the formation of Fe₃B/Nb₂Fe₁₄B
composite has been estimated to be 2 ×10⁵ k/s above which the
ribbons become amorphous . The grain sizes in the rage 10 to 500 nm can be
obtained through melt spinning technique. The magnetic properties are found to
be dependent on the processing conditions of the melt-spun ribbons .
1.3. Characterization techniques for the
nanophase materials
Characterization
of nanocrystalline materials is required in order to understand the
correlation
between the properties and the particle size. Various characterization techniques
are employed to study the structural and physical properties of nanomaterials.
Some of the nanostructured materials like line broadening in the X-ray peak are
used to estimate the average grain size using the X-ray diffraction. The
surface morphology and shape of the nanoparticles are examined using electron
microscopy. Other techniques used are atomic force microscopy, scanning
tunneling microscopy  Raman spectroscopy,
Rutherford backscattering, extended X-ray absorption fine structure (EXAFS), X-ray
photoelectron spectroscopy (XPS), X-ray magnetic circular dichroism (XMCD),
positron annihilation and muon spin rotation . The relaxation effects occurring
in the nanoparticles on size reduction are examined using Mossbauer
spectroscopy  and the changes in the
magnetic properties of nanocrystalline materials are studied using the
vibrating sample magnetometer (VSM) and superconducting quantum interference
device (SQUID). They are also used in the determination of the blocking
temperature of the small particles . The sophisticated technique like atom
probe field ion microscopy (APFIM) is also employed to understand the
correlation of the microstructure to the properties .  
In technologies where
ferrites are to be used for magnetic or electrical applications, high-density materials
are generally required and the ferrites are often prepared by high temperature
solid-state reactions between finelground powders. Although most applications
of ferrites as ceramic materials require high densities to achieve the desired
properties, there are many applications for which lower densities and high
surface area are preferred .Spinel-type oxides with a general formula of AB2O4
are important mixed oxides in gas sensors, and have been investigated for the
detection of both oxidizing and reducing gases. In particular, nickel ferrite
as a p-type semiconducting oxide has show to be a very good sensor to detect
oxidizing like chlorine , whereas nickel ferrite prepared bythehydrazine method
showed an n-type semiconducting behavior.The semiconductor gas sensors offer
good advantages with respect to other gas sensor devices (such as spectroscopic
or optic systems), due to their simple implementation, low cost and good
reliability for real-time control systems.In our previous papers we have shown
that various ferrites with various doping elements are sensitive to gases
and/or humidity. In the present study, nickel ferrite doped with small amounts
of calcium, cobalt and manganese was investigated as gas sensor. In an attempt
to improve the sensitivity and impart selectivity, Nanoparticles of nickel
ferrite have been partly replaced with Co and Mn on place of Ni and Fe, respectively.
The self combustion method was used for preparation because the followings two
advantages:
(1) heat generated in the
exothermic reaction accelerates the process and
(2) the resulting magnetic
powder is fine grained with grain size smaller than that of the starting
powders. Grain size and pore structure have a major effect on the properties in
polycrystalline materials and their full characterization should be the first
step in the study of materials. Also, the microstructure has a major role in
the performance of a ceramic sensor.
Nanotechnology
is beginning to allow scientists, engineers and physicians to work at the
cellular and molecular levels to produce major advances in the life sciences and
healthcare. Real applications of nanostructures materials in life sciences are
uncommon at the present time. However, the excellent properties of these
materials when compared with their bulk counterparts provide a very promising
future for their use in this field . Industrial applications of magnetic
nanoparticles cover a  broad spectrum such
as magnetic seals in motors, magnetic inks for bank cheques, magnetic recording
media and biomedical applications such as magnetic resonance contrast media and
therapeutic agents in cancer treatment . Each potential application requires
the magnetic nanoparticles to have different properties. For example, in dated
storage applications, the particles need to have a stable, switch able magnetic
state to represent bits of information, a state that is not affected by
temperature fluctuations. We can classify biomedical applications of magnetic
nanoparticles according to their application inside (in vivo) or outside (in
vitro) the body. In vivo applications could be further separated in therapeutic
(hyperthermia and drug-targeting) and diagnostic applications (nuclear magnetic
resonance (NMR) imaging). While for in vitro applications the main use is in
diagnostic (separation/selection, magnetorelaxometry)
 NiFe₂O₄ Nanoparticles
in  vivo applications:
(a)Therapeutic
applications  
׃ Hyperthermia therapy, a form of
cancer treatment with elevated temperature in the range of 41–45°C, has been
recently paid considerable attention because it is expected to significantly
reduce clinical side effects compared to chemotherapy and radiotherapy and can
be effectively used for killing localized or deeply seated cancer tumors.
Accordingly, various forms of hyperthermia have been intensively developed for
the past few decades to provide cancer clinics with more effective and advanced
cancer therapy techniques. However, in spite of the enormous efforts, all the
hyperthermia techniques introduced so far were found to be not effective for
completely treating cancer tumors. The low heating temperature owing to the
heat loss through a relatively big space gap formed between targeted cells and
hyperthermia agents caused by the hard to control agent transport, as well as
killing healthy cells attributed to the difficulties of cell differentiations
by hyperthermia agents, are considered as the main responsibilities for the
undesirable achievements. In a possible breakthrough, researchers in Singapore
now report the very promising and successful self-heating temperature rising
characteristics of NiFe₂O₄ Nanoparticles. Different from
conventional magnetic hyperthermia, in-vivo magnetic Nanoparticles
hyperthermia is expected to be one of the best solutions for killing tumor
cells which are deeply seated and localized inside the human body.
The recent development of magnetic Nanoparticles technology
accelerated a new form of hyperthermia treatment, so-called “in vivo
hyperthermia,” because magnetic Nanoparticles are expected to provide a great
deal of technical advantages for hyperthermia: 1) direct injection of
hyperthermia agents through blood vessel; 2) easy transport of nanoparticles to
the targeted cell by externally controlled magnetic field; 3) small heating
loss during hyperthermia due to direct heating of cell; and 4) possibility for
differentiation of tumor cells from healthy cells by using antibody-antigen
biological reaction.
For biomedical applications
the use of particles that present superparamagnetic behavior at room
temperature (no remanence along with a rapidly changing magnetic state) is
preferred. Furthermore, applications in biology and medical diagnosis and
therapy require the magnetic particles to be stable in water at neutral pH and
physiological salinity. The colloidal stability of this fluid will depend
first, on the dimensions of the particles, which should be sufficiently small
so that precipitation due to gravitation forces can be avoided.
The main reason
for considering NiFe₂O₄ nanoparticles as a hyperthermia
agent is that these are expected to have urgently required magnetic properties
for hyperthermia such as soft magnetism and small magnetic degradation at high
frequency.
Hyperthermia is a
therapeutic procedure used to raise the temperature of a region of the body
affected by malignancy or other growths. It is administered together with other
cancer treatments (multimodal oncological strategies). The rationale is based on a direct cell-killing effect at
temperatures above 41-42ºC . Modern clinical hyperthermia trails focus mainly
on the optimization of thermal homogeneity at moderate temperatures (42-43ºC)
in the target volume. The temperature increase required for hyperthermia can be
achieved, among other methods, by using fine iron oxide magnetic particles  . The physical principles for which a magnetic
material can be heated by the action of an external alternating magnetic field
are the loss processes that occur during the reorientation of the magnetization
of magnetic materials with low electrical conductivity .  The advantage of magnetic hyperthermia is that
allows the heating to be restricted to the tumors area. Moreover, the use of
the subdomain magnetic particles (nanometer-sized) is preferred instead of
multidomain (micron-sized) particles because nanoparticles absorb much more
power at tolerable AC magnetic fields .
Finally it should be
mentioned that the heating potential is strongly dependent on the particle size
and shape, and thus having well defined synthetic routes able to produce
uniform particles is essential for a rigorous control in temperature.
׃
Since the pioneering concept
proposed by Freeman et al  that fine iron
particles could be transported through the vascular system and be concentrated
at a particular point in the body with the aid of a magnetic field , the use of
magnetic particles for the delivery of drugs or antibodies to the organs or
tissues altered by diseases has become an attractive field of research. The
process of drug localization using magnetic delivery systems is based on the
competition  between forces generated
from the magnet, i.e. applied field. When the magnetic forces exceed the linear
blood flow rates in arteries or capillaries, the magnetic particles are
retained at the target site and may be internalized by the endothelial cells of
the target tissue . For this application the use of nanoparticles favor  the transport through the capillary system of
organs and tissues avoiding vessel embolism.
(b)Diagnostic
applications:
NMR imaging
: The development of the NMR imaging
technique for clinical diagnosis has prompted the need for a new class pharmaceuticals,
so-called magneto- pharmaceuticals. These drugs must be administered to a
patient in order to (1) enhance the image contrast between normal and diseased
tissue and/or (2) indicate the status of organ functions or blood flow . A
number of different agents have been suggested as potential NMR contrast
agents. Most contrast agents used in NMR imaging studies to date have been
paramagnetic. Super paramagnetic particles represent an alternative class of
NMR contrast agents that are usually referred to as T₂ or T₂^*
contrast agents as opposed to T₁(longitudinal relaxation time )
agents such as paramagnetic Gadolinium (Ш) chelates . 
The relaxation rate increase produced by magnetic
particles is a contribution of several complex mechanisms. The particles
possess very large magnetic moments in the presence of a static magnetic field,
and dipolar interactions between the super paramagnetic cores and surrounding
solvent protons result in an increase in both longitudinal and transverse
relaxation rates, especially for particles with diameters below 10 nm .  Commercial iron oxide Nanoparticles of magnetite
have been used as contrast agents in NMR imaging for location  and diagnosis of brain and cardiac in ferrets
, liver lesions or tumors, where the  magnetic Nanoparticles tend to accumulate at
higher levels due to the differences in the tissue composition and endocytotic
uptake processes. Especially promising results have been detected in the
improvement of sensitivity of detection  and
delineation of pathological structures, such as primary and metastic brain tumors,
inflammation and ischemia . For this purpose, proteins such as transferring,
peptides such as the membrane translocation tat peptide of the HIV tat protein,
and oligonucleotides of various sequences have been attached to aminated
cross-linked iron oxide Nanoparticles in order to obtain specific NMR imaging
agents .
1.5.2. In
vitro applications :
(a)
Diagnostic applications: Separation
and selection. At present, considerable is being paid to solid-phase extraction
(SPE) as a way to isolate and preconcentrate desired components from a sample
matrix. SPE offers an excellent alternative to the conventional sample
concentration methods, such as liquid-liquid extraction. The separation and
preconcentration of the substance from large volumes of solution can be highly
time consuming when using standard column SPE, and is in the field where the
use of magnetic or magnetizable adsorbents called magnetic solid-phase
extraction (MSPE) gains importance. In this procedure the magnetic adsorbent is
added to a solution or suspension containing the target. This is adsorbed onto
the magnetic adsorbent and with the adsorbed target is recovered from the
suspension using an appropriate magnetic separator. For separation and
selection the advantage of using magnetic Nanoparticles instead magnetic micro
particles is that we can prepare suspensions that are stable against
sedimentation in absence of an applied magnetic field. The applicability of
iron oxide magnetic Nanoparticles in MSPE is clearly evidenced by the fact that
already exists in the market companies (DYNAL Biotech) that commercialize these
products.
Recently,
magnetorelaxometry was introduced as a method for the evaluation of
immunoassays . Magnetorelaxometry measures the magnetic viscosity, i.e. the
relaxation of the net magnetic moment of a system of magnetic nanoparticles
after removal of a magnetic field . There are two different relaxation
mechanisms. First, the internal magnetization vector of a Nanoparticles relaxes
in the direction of the easy axis inside the core; this is called Neel
relaxation. Second, particles accomplish rotational diffusion in a carrier liquid,
called Brownian relaxation. Neel and Brownian relaxation can be distinguished
by their different relaxation times . Further, Brownian relaxation can take
place only in liquids, whereas Nell relaxation does not depend on the
dispersion of the Nanoparticles. The fact that magnetorelaxometry depends on
the core size, the hydrodynamic size and the anisotropy allows this technique
to distinguish between free and bound conjugates by their different magnetic behavior,
and therefore can be used as an analytical tool for the evaluation of
immunoassays. For this application the benefits of reducing particle size to
the nanometer-sized are similar to those described for separation and selection
applications.  
1.5.3.
Future applications 
Magnetically directed microspores containing radio
nuclides have been used for internal radiotherapy. However, little work has
been done in the use of magnetic Nanoparticles in radiotherapy. One strategy
under active investigation to improve dose localization is that of administration
of drugs, metabolites ,etc that have been labeled with radioactive isotopes in
a quantity sufficient to deactivate the tumor cells . In this way, the use of
surface-activated magnetic Nanoparticles could have tremendous impact in
improving the efficiency of the cancer treatments. 
We can even envisage a future in which magnetic
particles could be used for the repair of the human body with prosthetics or
artificial replacement parts. In this field special mention deserves the pioneering
work of Dailey et al who have reported the synthesis of a silicone based
magnetic fluid for use in eye surgery. Retinal detachment is a major cause of
vision loss in adults. It occurs when the retina separates  from the choroids, resulting  in eventual death of the retina and
subsequent loss of vision. Dailey and co-workers have developed an internal
tamponade from modified silence fluid containing statically stabilized 4-10 nm sized
metal particles, which will be held in place with an external magnetized several
buckle.
Literature Review
Review of the Work on Nanocrystalline Material:
The review of the
historical previous works of many renowned scientists on nano- materials have
been discussed in this chapter. During the last few years nanoscale spinel
ferrites have drawn a major attention because of their technological importance
in magnetic recording, magnetic fluids and catalyst. Nanoferrites are usually
prepared using various physical and chemical methods like ball milling,
microwave plasma, co-precipitation (Ferele and Baberschke 1987; Tang et al 1991;
Chatterjee et al 1993). The synthesis of ferrite nanoparticles is of great
interest for studying and tailoring of specific magnetic properties. It was
found that the doping of lanthanum impurity ions could suppress the long-range
ferroelectric order in the lead zirconium titanate (PZT) system (Li et al 1995;
Xuet al 1995). The nanostructured phase was found to be induced with increasing
La impurity content in the ferroelectric ceramic system. A similar approach is
made in the present case and a set of nanocrystalline spinel ferrites are
prepared using the conventional ceramic route with suitable doping by
aliovalent ions (Kundu and Chakravorty 1999).
The
size-dependent magnetic properties of ferrites have been investigated by many
researchers for particle sizes less than 25 nm where the saturation
magnetization decreases with particle size reduction due to surface spin
effects as explained by Kodama and Berkowitz . The model proposed by Kodama et
al. consists of ferrimagnetically aligned core spins and a spin-glass like
layer. For the MnFe₂O₄ prepared, using the co-precipitation
technique, a 50 % reduction in the saturation magnetization for the 7.5 nm
particles has been observed compared to the bulk value . Various
nanocrystalline mixed spinel ferrites synthesized using the reverse micelle
technique yielded very low saturation magnetization of the order of 5 to 25 emu/g
and the low values are attributed to the core-shell structure with a spin-glass
like surface layer .
Extensive
work by Sepelak et al. have shown that the ball-milled samples exhibit spin
canting and cation redistributions . Chinnasamy et al.  have shown that ZnFe₂O₄
and CdFe_2O4, which are antiferromagnetic in the bulk state, exhibit
ferromagnetic ordering when the grain size is reduced to the cation
redistribution. The ferrite nanoparticles are found to exhibit an enhanced
Curie temperature compared to the bulk value . Rath et al.  have observed an enhanced curie temperature in
Mn-Zn ferrite nanoparticles prepared using chemical methods, which is
attributed to a metastable cation ferrites is related to the cation
distribution among the A- and B- sites which depends on the synthesis
conditions . A good review, covering the recent literature, discusses the
various aspects of the changes in the magnetic properties of nanostructured
spinel ferrite. Smaller ferrite particles are found to exhibit
superparamagnetic behavior with their coercivity approaching zero. The critical
size for superparamagnetism has also been calculated for various ferrites with
the sizes being 14 nm, 25 nm and 50 nm for the CoFe₂O₄,
Fe₃O₄ and MnFe₂O₄ respectively .The
blocking temperature for the superparamagnetism of the nanoparticles depends on
their magnetocrystalline anisotropy. A comparative study of CoFe₂O₄
and MgFe₂O₄ with 20 nm particle sizes has suggested that
the blocking temperature of CoFe₂O₄ is higher than that
of MgFe₂O₄ by 150 K which is due to the higher
manetocrystalline anisotropy of the former . The blocking temperature not only
depends on the magnetocrystalline anisotropy, but also on the  anisotropy energy, KV compared to the thermal
energy, K_BT. However, surface anisotropy tends to increase with particle
size reduction . The anisotropy constant of CoFe₂O₄
particles of size about 3.3 nm is 3.15 × 10⁶ J /m³ which
is an order higher than that of the bulk materials .  Mn-Zn ferrites prepared through the
hydrothermal route have resulted in smaller particle sizes . The control over
particle size to obtain better magnetic properties was attempted for Mn-Zn
ferrites and a growth assisted co-precipitation had yielded magnetization as
large as 50 emu/g compared to the co-precipitated samples with an average
crystallite size of 12 nm. 
The
size-dependent magnetic properties of various ferrites have been studied for
particles with sizes less than 25 nm synthesized using aqueous methods. The
synthesis of larger particles involve heat treatment which leads to particle
agglomeration. Since the last decade quite new and interesting magnetic
properties have been reported for nanocrystalline spinel ferrites. The
observation of large magnetic moments and ferromagnetic and ferromagnetic ordering
on grain size reduction to a few nanometer in the well-known low temperature
antiferromagnetic spinels ZnFe₂O₄ and CdFe₂O₄
has been found to be quite exciting. The controversy with respect to the
type of magnetic ordering namely, whether it is ferromagnetic or ferromagnetic
ordering is yet to be resolved. The rule played by the method of synthesis of
these materials on the magnetic properties can not be ignored as materials from
different laboratories are found to exhibit different magnetic properties. The
EXAFS study by Jeyadevan et al.  has
shown that ZnFe₂O₄ undergoes a change in cation
distribution from the normal spinel structure in the bulk form to mixed spinel
structure in the nanocrystalline form. Nanocrystalline NiFe₂O₄
spinel is another such system which has been extensively studied by Mossbauer
and magnetization techniques . Recently, we have shown that nanocrystalline
NiFe₂O₄ exhibits mixed spinel structure whereas in the
bulk form it is a well- known inverse spinel with all Ni²⁺ ions on
the B-sites and Fe³⁺ions equally distributed between A  and B_{­­­­­­­­­­­­­­­­­­­­­­­­­­­ ­­­­­­­­­}
sites. Our study has shown that the ball milled NiFe₂O₄
with a grain size of 60 nm is a mixed spinel with about four percent Ni²⁺
ions occupying the A-sites. Also, the magnetization of NiFe₂O₄
spinel  with grain size of 60 nm has been
enhanced by 8% from its bulk value.
Chen
et al. have reported changes in cation distribution and a decrease in Neel
temperature in MnFe₂O₄ as grain size is decreased. The
decrease in Neel temperature has found to be consistent with the finite size
scaling . To our knowledge, this is the only spinel ferrite for which the grain
size dependent Neel temperature has been reported. In the present study, we
report the effect of grain size on Neel temperature in the case of NiFe₂O₄
spinel and also its effect on the magnetic properties such as saturation
magnetization, coercivity and B-H loop-shifts, etc. The coercivity reported in
this paper is found to be the highest of all the values reported so far for the
ultrafine particles of NiFe₂O₄. 
One
of the latest techniques in materials science is to tailor-make classical
products with controlled properties for special uses. Particular attention
should be paid to the preparation methods that allow the synthesis of particles
nearly of uniform size and shape. This goal can be achieved by precipitation
from a homogeneous solution under controlled conditions or by controlling the
particle growth in a precursor in aerosol or vapour form is decomposed.
Examples of such preparations include good colloids, sulfur sols, metal oxides
and hydrous oxides. In the case of magnetic nanoparticles for biomedical
applications we have classified the synthesis methods into those that produce magnetic
nanoparticles from solution techniques or from aerosol/ vapour phases, and
those producing composites consisting of magnetic nanoparticles dispersed in
submicron-sized organic or inorganic matrixes that usually have spherical
shape. Finally, we have also described briefly another group of methods that
use size selection principles to produce uniform nanoparticles starting from
polydisperse particles.  In general these methods allow the
preparation of magnetic nanoparticles with a rigorous control in size and shape
in a sample rather way and thus are very appropriate for their use in
biomedical applications. Uniform particles are usually prepared via
homogeneous  precipitation reactions, a
process that involves the separation of the nucleation and growth of the nuclei
. 
In
a homogeneous precipitation, a short single burst of nucleation occurs when the
concentration of constituent species reaches critical supersaturation. Then,
the nuclei so obtained are allowed to grow uniformly by diffusion of solutes
from the solution to their surface until the final size is attained. To achieve
monodispersity, these two stages must be separated and nucleation should be
avoided during the period of growth. This is the classical model proposed by
LaMer and Dinegar  first to explain the
mechanism of sulfur collides and also for a limited number of cases. However,
uniform particles have also been obtained after multiple nucleation events. The
uniformity of the final product is in this case achieved through a self-
sharpening growth process. In addition, uniform particles have also been
obtained as a result of aggregation of much smaller subunits rather than continuous
growth by diffusion . An artificial separation between nucleation and growth
processes may be achieved by seeding in which foreign particles are introduced
into the solution of monomers below the critical supersaturation . 
The
most important methods described in the bibliography to obtain uniform
iron-based nanoparticles in solution are briefly described in the following
sections: co-precipitation, microemulsions, the polyol process and
decomposition of organic precursors. Co-precipitation. There are two main
methods for the synthesis in solution of magnetite spherical particles in the
nanometer range. In the first, ferrous hydroxide suspensions are partially
oxidized with different oxidizing agents . For example, spherical magnetic
particles of narrow size distribution with mean diameters between 30 and 100 nm
can be obtained from a Fe (II) salt, a base and a mild oxidant .  The other method consists in ageing
stoichiometric mixtures of ferrous and ferric hydroxides in aqueous media,
yielding spherical magnetite particles homogeneous in size . In addition, it
has been shown that by adjusting the p^H and the ionic strength of
the precipitation medium, it is possible to control the mean size of the
particles over one order of magnitude (from 15 to 2 nm) . Both parameters
affect the chemical composition of the surface and consequently, the
electrostatic surface charge of the particles. Under these conditions,
magnetite particles are formed by aggregation of particles formed with an Fe
(OH)₂ gel. This is an ordered aggregation that gives rise to spherical
crystalline particles . The smallest particles can also be generated after
adding polyvinyl alcohol (PVA) to the iron salts . 
Pileni
and co-workers  prepared nanosized
magnetic particles with average sizes from 4 to 12 nm and standard deviation
ranging from 0.2 to o.3 using microemulsions. A ferrous dodecyl sulfate, Fe
(DS)₂, micellar solution was used to produce nanosized magnetic
particles whose size is controlled by the surfactant concentration and
temperature. Magnetic nanoparticles around 4 nm in diameter have been prepared
by the controlled hydrolysis with ammonium hydroxide of FeCl₂ and
FeCl₃ aqueous solutions within the reverse micelle nanocavities
generated by using AOT as surfactant and heptane as the continuous oil phase . 
Carpenter
and co-workers  prepared metallic iron
particles coated by a thin layer of gold via a microemulsion. The gold shell
protects the iron core against oxidation and also provides functionality,
making these composites applicable in biomedicine. The reverse micelle reaction
is carried out using cetyltrimethyl ammonium bromide (CTAB) as the surfactant,
octane as the oil phase, and aqueous reactants as the water phase. The metal
particles are formed inside the reverse micelle by the reduction of a metal
salt using sodium borohydride. The sequential synthesis offered by reverse
micelles is utilized to first prepare an iron core by the reduction of ferrous
sulphate by sodium borohydride. After the reaction has allowed to go to
completion, the micelles within the reaction mixture are expanded to
accommodate the shell using a larger micelle containing
additional
sodium borohydride.
Alivisatos
and co-workers  have demonstrated that
injecting solutions of FeCup₃ (Cup: N-nitrosophenylhydroxylamine) in
octylamine into long-chain amines at 250-300ºC yields nanocrystals of
maghemite. These nanocrystals range 4 to 10 nm in diameter, are crystalline,
and are dispersable in organic solvents.
Hyeon
and co-workers have also been able to prepare monodisperse maghemite
nanoparticles by a non-hydrolytic synthetic method. For example, to
prepare  maghemite nanoparticles of 13
nm, Fe(CO)₅ was injected into a solution containing surfactants and
a mild oxidant.  Very recently, Sun and Zeng  have been able to prepare monodispersed
magnetic nanoparticles with sizes from 3 to 20 nm by the high-temperature
(265ºC) reaction of iron (III) acetylacetonate in phenyl ether in the presence of
alcohol, oleic acid, and oleylamine. In particular, magnetic nanoparticles
around 4 nm were obtained by the thermal decomposition of the iron precursor
but to obtain diameters up to 20 nm a seed-mediated growth method was
required. 
Mann
and co-workers have been able to prepare magnetite and magnetite/ maghemite
nanoparticles  of about 6-7 nm in
diameter. The magnetite/ maghemite particles were generated by oxidation of
apoferritin with trimethylamino-N-oxide, which was loaded with various amounts
of iron (II) ions. 
Since
the pioneering work of Cannon and co-workers on the continuous production of
nanometric powders by laser-induced processes, different powders such as Si,
SiC, Si₃N₄ and a Si/ C/ N composite have been prepared
under a variety of conditions with sizes ranging from 5 to 20 nm. The method
involves heating a flowing mixture of gases with a continuous wave carbon
dioxide laser, which initiates and sustains a chemical reaction. Above a
certain pressure and laser power, a critical concentration of nuclei is reached
in the reaction zone which leads to homogeneous of particles that are further
transported to a filter by an inert gas. 
Theoretical Background
3.1. Introduction to Magnetism:
The
history of the development of magnetic materials is undeniably long and
splendid. According to the Chinese history, a legendary ruler of the ancient
kingdom, Hung-Ti, made use of the compass to direct his victorious battles
against barbarian tribesmen before 2600 BC. The history of Asia Minor, another
cradle of civilization, claims that lodestone was discovered as a natural
magnet in magnesia more than 3500 years ago . The stone was called magnets
lapis, which means magnesian stone. Magnet and Magnetism were derived from the
word magnesian Thales of Milestus, the Greek philosopher, stated that the
magnetic interaction between magnetite and iron was known before 600 BC.
Magnetic
materials also played a prominent role in the discovery of the New world and in
the development of modern technology. Without the compass Christoper Columbus
would not able have made his voyages and discoveries. The outstanding magnetic
properties and low cost of iron has made it possible to generate inexpensive
electricity on a massive scale since 1886 when Westinghouse Electric Company
built the first commercial AC generating station at Buffalo, New York. The use
of magnetic materials to perform vital functions is not limited to the utility
industry, other industries consuming appreciable amounts of magnetic materials
include communications, computer, audio-visual, home appliances, to name just a
few. From the scientific viewpoint, it is understandable that magnetic
materials have contributed to the history of Civilization and technology. In
the first place, all substances, whether solid, liquid or gas, display certain
magnetic characteristics at all temperatures. Hence magnetism is one of the
basic properties of materials. Secondly, although only three elements (Fe, Co,
and Ni) out of the total of 106 are ferromagnetic at room temperature, we are
fortunate that iron constitutes approximately 3.3 % of the earth’s crust, and
hence, it is abundant in natural resources and relatively inexpensive. Thirdly,
metallic meteorites are known to consist largely of pure iron. It is logical to
assume that, by the time ancient man appeared on the earth, pure iron had
already existed for millions of years. The natural occurrence of Fe₃O₄
as lodestone is just as old as meteorites and probably more widespread than the
latter. Thus It is not surprising that the magic stone and pure iron were used
so early in our history.
Soft
magnetic material is more a technique term than a scientific one. The word
“Soft” means temporary in the sense that the ferromagnetism merges only when a
magnetic field is applied. In contrast, hard or permanent magnets display
ferromagnetism in the absence of an external field. All magnetic elements in
the pure form are soft; whereas magnetic solid solutions and compounds can be
either soft or hard, depending on the composition and metallurgical treatments.
3.2 Magnetic phenomena:
When
a magnetic field intersects a real material, the magnetizing field H creates or
aligns magnetic dipoles similar to the electric dipole chains. As a result the
total magnetic flux density B, which is analogous to the total electric flux
density D, is the summation of the magnetizing field and the total effect of
the magnetic dipoles:
B
= µ₀H + µ₀M = µ′H………………………………………3.1
Where
M is the magnetization of the material, µ₀ the permeability of free
space, and µ′
is
the effective permeability of the material.
In
a rationalized mks system of units, the field strength H is measured in units
of amperes per meter. For a vacuum
E
= √(µ₀/ ε₀) H = 120π H
V/m……………………………..3.2
And
the magnetic flux density in a vacuum is given by
B
= µ₀H = 4π ×10^-7 H V/sec m² = Wb/m²
………………….3.3
Then
E/B = 1/(√ε₀µ₀) = light velocity = 3 × 10⁸ m/s
………………3.4
The magnetic dipole moment per unit volume is the
product of the number of elementary magnetic dipoles per unit volume n and
their magnetic moment P_m:
M
= nP_m = nα_mH ……………………………………………3.5
Where
α_m is the magnetizability of the elementary constituents. The
magnetic moment is proportional to the magnetizing field strength. Magnetic
properties, in parallel with dielectrics, can also be measured as the ratio of
the magnetization to the applied field, called the magnetic susceptibility:
χ=
M/H………………………………………………………3.6
when
a current I circles an area a, a magnetic dipole moment is created, m = ia,
where m
is
a vector normal to the plane of the enclosed area. This is the situation when
an electron
circles
a proton υ times per second in an orbit of radius r, producing a magnetic
moment – e υπr²; simultaneously, it has the quantized angular
momentum. From Bhor’s theory of
the
atom, when the azimuthal quantum number l is equal to 1, the combination of
magnetic moment and angular momentum leads to an elementary magnetic moment and
angular momentum leads to an elementary magnetic moment of
P_m
= µ_B = eh /(4πm) = 9.27 ×10^-24 Am²/ electron
where
h is plank’s constant and µ_B is defined as one Bohr magneton, the
orbital contribution to the magnetic moment of an atom by one electron when l =
1. This is not the only mechanism which contributes to the magnetic moment of
an atom. The electron itself has an intrinsic angular momentum which gives rise
to an electron-spin contribution of approximately 2s Bohr magnetons, where s is
the spin quantum number (±1/2).
According to the Pauli exclusion principle, only two
electrons can fill any energy level; these have opposite spin directions (s = +
½, s = – 1/2), and their magnetic moments cancel. Permanent magnetic moments
arise in systems in which unpaired electrons are present.
The
onset of magnetic order in solids has two basic requirements:
 
Individual atoms
should have magnetic moments (spins),
Exchange
interactions should exist that couple them together .
Magnetic
moments originate in solids as a consequence of overlapping of the electronic
wave functions with those of neighboring atoms. This condition is best
fulfilled by some transition metals and rare earths. The exchange interactions
depend sensitively upon the interatomic distance and the nature of the chemical
bonds, particularly of nearest neighbor atoms. When the positive exchange
dominates, which corresponds to parallel coupling of neighboring atomic moments
(spins), the magnetic system become ferromagnetic below a certain temperature T_c
called curie temperature. The common spin directions are determined by the
minimum of magneto-crystalline anisotropy energy of the crystal. Therefore,
ferromagnetic substances are characterized by spontaneous magnetization. But a
ferromagnetic material in the demagnetized stage displays no net magnetization
in zero field because in the demagnetized state a ferromagnet of macroscopic
size is divided into a number of small regions called domains, spontaneously
magnetized to saturation value and the directions of these spontaneous
magnetization of the various domains are such that the net magnetization of the
specimen is zero. The existence of the domains is a consequence of energy
minimization. The size and formation of these domains is in a complicated
manner dependent on the shape of the specimen as well as its magnetic and
thermal history. When negative exchange dominates, adjacent atomic moments
(spins) align anti- parallel to each other, and the substance is called
anti-ferromagnetic below a characteristic temperature, T_N, called
the
Neel
temperature. In the simplest case, the lattice of an antiferromagnet is divided
into two sub lattices with the magnetic moments of these in anti-parallel
alignmemt. This results in zero net magnetization. A special case of anti-ferromagnetism
is ferrimagnetism. In ferrimagnetism there are also two sub-lattices with
magnetic moments in opposite directions, but the magnetization of the
sub-lattices are of unequal strength resulting in a non-zero magnetization and
therefore has net spontaneous magnetization. At the macroscopic level of domain
structures, ferromagnetic and ferrimagnetic materials are similar. The Curie
and Neel temperatures characterize a phase transition between the magnetically
ordered and disordered (paramagnetic) states. From the simple cases of magnetic
ordering, various types of magnetic order exist, particularly in metallic
substances. Because of long rage order and oscillatory nature of the exchange
interaction, medicated by the conduction electrons, structures like helical,
conical and modulated patterns might occur. Examples of different types of
magnetic order are shown in figure 3.1 .

Figure 3.1: Examples of different types of magnetic order using a
linear array of
localized
moments, including paramagnet.
3.4. Fundamental
Quantities of Magnetism :
The
materials which are commonly referred as “magnetic materials “are known as
ferromagnetic materials .While all materials are magnetic to some degree many
of them fall under the  heading of
paramagnet or diamagnetic materials possessing only a very weak and relatively
undetectable magnetic property. As such, they are not used to any extent in any
practical devices but have great scientific importance. The cooperative effect
called “ferromagnetism” leads to materials that have magnetic forces many
orders of magnitude larger than the aforementioned materials. Ferromagnetic
materials may exist as conductors (metals), insulators (ceramics), or as
semiconductors. Let us a brief review on fundamental quantities of magnetism.
 
Magnetic
Poles:
Experiment
shows that a natural magnet in the shape of a long and thin bar sets up a
magnetic field at each end. This field is radially outward at one end (N or
positive) and radially inward at the other end (S or negative). The magnet may
then be considered as possessing two equals and opposite poles at the ends. The
strength of each pole m is defined in the coulomb interaction that the force
that exerted on a magnetic pole of strength m₂ by another pole of
strength m₁at a distance r is
F=K (m₁m₂/r²) r_o
………………………………………………..3.8
Where
r_o­ is a unit vector along r. In the SI, the magnitude of F is
expressed in neutron m, a scalar quantity, in A.m when the constant of
proportionality K takes the rationalized value of 1/μ_oc²,
where μo=4π×10^-7 henry/meter is the magnetic permeability of free
space
and
c=2.998×10⁸ meter/second is the speed of light in a vacuum.
 
Magnetic
Dipole and Magnetic Moment(μ):
Magnetic
poles have never been observed in isolated form, but occur in pairs. Such a
pair is called a magnetic dipole. The magnetic moment of a dipole is defined as
μ=ml……………………………………………………….3.9
Where
l is the vector pointing from the S pole to the N pole and μ is expressed in
A.m².
 
Magnetic
Field(H):
The strength of the magnetic field H induced by either
pole of a bar magnet of pole strength m₁, at a distance r, is
defined as the magnetizing force acting on a unit test pole (m₂=1)
placed at that position.
Accordingly,
H=F/m₂=K(m₁/r²) r_o………………………………………………………….3.10
The popular oersted (Oe) is the unit of H
in the CGS system. The SI unit of H is based on the fact that a magnetic field
is often produced by a current of electricity. For a long solenoid having N
turns per meter and carrying a current of I amperes, the magnetic field
produced inside the solenoid is given by
H=NI………………………………………………..3.11
Hence the SI unit of H is the ampere-turn per meter,
or simply, the ampere per meter(A.m^-1). One ampere per meter is
equivalent to 0.0126 oersted and one oersted is roughly 80 ampere per meter.
3.4.4 Various Kinds of Energies in the
Formation of Magnetic
Domains: Magnetic materials are normally divided into several
domains inside the materials. Domains are formed basically to reduce the
magnetostatic energy which is the magnetic potential energy contained in the
field lines connecting north and south poles outside the material. When a
domain is split into n domains, the energy of the new structure is about 1/ nth
of the single domain structure. In a ferromagnetic domain, there is parallel
alignment of the atomic moments. Each domain becomes a magnet composed of
smaller magnets. Domains contain about 10¹² to 10¹⁵ atoms
and their dimensions are on the order of microns (10^-4cm). Their
size and geometry are governed by certain considerations.
The
primary reason for the existence of domains within a crystal is that their
formation reduces the magnetic free energy. In the simplest case for such a
crystal, the energy, E, is the sum of the several free energy terms
E
= E_ex+ E_k + E_λ +E_D +E_H
…………………………………………3.12
Where
E_ex is the exchange energy, E_k is the magnetocrystalline
anisotropy energy, E_λ is the magnetoelastic energy, E_D
magneto-static energy, and E_H is the energy of the domains in the
presence of an applied field. There is also a wall energy E_w as a
separate term in the equation.
Magnetostatic energy: This is the energy contained in the magnetic field. It
increases with the volume of the field. A piece of material consisting of a
single domain would have a large magnetic field around it. The material could
reduce its energy by splitting into domains oriented to allow the field lines
to exist as loops inside the material, with little field outside. This is
essentially the state of an unmagnetized ferromagnetic material.
Magnetostrictive energy: This energy is due to the effect of magnetostriction,
a slight change in the dimensions of the crystal when magnetized. This causes
elastic strains in the lattice, and the direction of magnetization that
minimizes these strain energies will be favored.
Zeeman energy: Energy resulting from an externally applied field
. 
Magnetization
(M) and Magnetic Induction (B):
Magnetization, or more completely, the intensity of
magnetization M, is the total magnetic moment of dipoles per unit volume in
units of A.m² per m³ or A.m^-1. considering a
bar magnet of pole strength m, length l and cross-sectional area A, the vector
quantity M has its direction pointing from the S pole to the N pole and its magnitude
is  given by
M = μ/V = m/A………………………………………………….3.13
So magnetization is also the pole strength per unit
area in units of A.m^-1, which is equivalent to 10^-3 gauss
(G) in the CGS system.
Magnetic induction or magnetic flux
density B is the flux per unit area and expressed in units of Wb.m^-2
or tesla (T). By flux is meant the number of induction crossing a given area at
a right angle. The Weber is the magnetic flux that, linking a circuit of one
turn, uniform rate in one second. In free space a magnetic field produces a magnetic
induction given by B = μ₀H. If the space is filled with any magnetic
substance in which the induced magnetization is μ₀M, the total
induction now becomes
B = μ₀(H+M)……………………………………………………3.14
Thus both the magnetizing field and the magnetization
contribute to the induction. For ferromagnetic materials, however, the
contribution of M usually dominates B.
In the CGS system, since μ₀ = 1, equation
(1.7) becomes
B = H + 4πM…………………………………………………….3.15
The factor 4π arises from the fact that a
unit pole produces a unit field everywhere on the sphere of unit radius (1 cm) enclosing the pole and
the surface area of this sphere is 4π cm². The unit of B in the CGS
system is the popular gauss. One gauss is equal to one line of induction per cm².
In the SI, as indicated by eqation (1.7), H and M have the same unit in A.m^{–1. In the CGS system, since B = H in free space,
gauss is frequently used to H in place of oersted. For conversion it is well to
remember that one tesla is equivalent to 104} gauss.
3.4.6
Susceptibility (χ) and permeability(μ):
To compare the magnetic response of various materials,
we define the volume susceptibility, or susceptibility, χ and the
absolute permeability, μ by
χ
= M/H and μ =B/H………………………………3.16
Where
H is the applied magnetic field .
3.5. Various kinds of Magnetism:
We
have stated in the outset that all the substances display certain magnetic
properties at all temperatures, regardless of their composition and state. Now
we shall review the various kinds of magnetism that have been observed in
solids. Broadly speaking, there are five basic kinds of magnetism; namely, (1)
Diamagnetism, (2) Paramagnetism, (3) Ferromagnetism, (4) Ferrimagnetism, (5)
Antiferromagnetism. In this listing Metamagnetism, in which antiferromagnetism
is replaced by ferromagnetism upon the application of a strong magnetic field
or a decrease in temperature. Nor are superparamagnetism, parasitic ferromagnetism
and mictamagnetism included.
In
simple terms, a solid is said to be diamagnetic if it is repelled by a
permanent magnet and to be paramagnetic, ferromagnetic or ferromagnetic if it
is attracted. The classification of magnetism can be made more precisely. A
convenient way to define the first four kinds of magnetism is to use the
susceptibility or relative permeability as follows:
Criteria for
Susceptibility,
χ   < 0  ≥ 0   » 0
Relative
Permittivity,
µ^′ < 1 ≥
1 » 1
For
instance, inert gases and noble metals (Cu, Ag, and Au) are diamagnetic, alkali
metals (Li, Na, K, and Rb) and all transition metals except Fe, Co, Ni are
paramagnetic, iron group transition metals and heavy lanthanide metals are
ferromagnetic, etc.
The magnetic effect which corresponds to the induced
dielectric
effect is called diamagnetism. The induced magnetization M is a linear function
of
the magnetic field strength H; χ_m is a constant and independent of
the field, so that  χ_m  is negative. The effect is weak, and the
relative permeabilityµ^′/µ₀ is only slightly less than
unity.
If we consider the classical picture of the atom as a nucleus with electronic
charges circulating in definite orbits, we can gain a picture of the physical
origin of diamagnetism. In a manner similar to the way that we derived Eq. 3.7
we consider how the angular velocity of the electron is altered if a magnetic
field is slowly applied, assuming that the radius of the electron orbit is
unchanged. The change in the angular velocity gives rise to a net magnetic
moment µ_D of magnitude
µ_D
= – (e²µ₀r²H)/4m………………………………………………3.17
where
e is the charge of the electron , m is the mass, r is the radius of the orbit,
and the magnetic field H is applied normal to the plane of the orbit.
3.5.2 Paramagnetic materials: Ions from the transition series and rare earth series
possess a net magnetic moment because the ion contains an odd number of
electrons. In the absence of a magnetic field these moments usually point in
random directions, producing no macroscopic magnetization. However, in the
presence of a magnetic field, the moments tend to line up preferentially in the
field direction and produce a net magnetization. When the unpaired electrons
are acted on individually with no mutual interaction between them, the effect
is called paramagnetism. The paramagnetic susceptibility χ_m is
positive because the moments line up in the same direction as the field and
thus enhance the magnetic flux density.
3.5.3 Ferromagnetic and Ferrimagnetic
Materials: In some materials the
magnetic moments of the individual ions are strongly coupled, and thus there
are regions in the solid in which the spins are aligned parallel even in the
absence of a magnetic field. This 
results in a large macroscopic magnetic moment for the small regions,
called weiss domains, even in the macroscopically demagnetized state. In the
weiss domains in a ferromagnetic material, the system energy is lowered by a
parallel alignmemt of all the electron spins. 

The
exchange interactions between electron spins in a ferromagnetic material is
positive; that is , all spins align in the same direction. However, in some
solids the exchange between the unpaired electrons causes antiparallel
alignment of spins. Several of the transition metal monoxides (MnO, FeO, NiO,
and CoO) exhibit this behavior. Thus the spins of the d electrons of adjacent
iron ions in FeO are aligned in opposite directions. We call this behavior
antiferromagnetism. In an FeO crystal having the rock salt structure , ions on
any (1 1 1) plane have parallel spins, but ions on adjacent (1 1 1 ) planes
have antiparallel spins. The aligned moments of the ions in the two directions
cancel, and the FeO crystal as a whole has no magnetic moment.
Antiferromagnetism with no net magnetic moment is a special case in which the
number of spins aligned in opposite directions is just equal. In the general
case in which ions with unpaired electrons are arranged on two sublattices with
antiparallel spin alignment, we must sum the net moment for each sublattice. A
ferromagnetic material is one in which these net moments for the two
sublattices are unequal, which results in a net macroscopic magnetic moment.
That is, we have incomplete cancellation of the antiferromagnetically arranged
spins. This class of materials is the most important group of magnetic oxides.
A
ferromagnetic or ferrimagnetic material is divided up into many small regions
or
domains
each of which is fully magnetized; that s, all the moments within each domain
are aligned in the same direction. When the bulk material is unmagnetized, the
net magnetization of these domains is zero. The way that the magnetization
vectors, that is, these net magnetic moments, sum to zero is important in
understanding magnetic oxides. The two opposing magnetic domains in Fig. 3.2a
sum to zero; however, the energy of the material is lowered by the successive
breakup of the domains shown in Fig. 3.2b and 3.2c. In each of these latter two
cases the sum of the magnetization is also zero. The pie-shaped domains on the
ends of material are called closure domains and complete the magnetic flux path
within the solid; when the magnetic flux is kept mostly in the solid, Fig. 3.2b
and c, the energy of the system is lower. 
Because of anisotropy in the ligand field, there are preferred
low-energy crystallographic directions of spontaneous magnetization. The boundary
regions between these domains consist of a gradual transition in spin
orientation, as shown in Fig. 3.3. The thickness of this transition region, the
domain wall, is a balance between adjacent spins, requiring a thick domain
wall, and a tendency for the spins to have a particular crystallographic
orientation, requiring a thin wall. Typically the domain wall has a thickness
of about 1000Ǻ and an energy about 2×10^-8 cal/cm².  Just as dielectrics change their length when
polarized, magnetic materials change their length when magnetized. The
fractional change in length associated with a change in magnetization from zero
to saturation is the magnetostriction. For example, NiFe₂O₄
contracts in the direction of the direction of the magnetization by about 45
parts per million at the saturation magnetization. The amount of dimensional
change observed is a function of both the magnetic field strength and the
crystallographic orientation.

Fig.3.2 Several domain structures of a solid, each having
zero net magnetization.

Fig. 3.3 change in atomic-dipole orientation through a domain
wall. All moments lie in the plane of the wall. The N’s represent poles which
are formed on the surface of the material.
The
state of magnetization of a solid is a function of the strength and direction
of the
magnetizing
field. If we consider a ferrimagnetic material which contains many small
magnetic domains but no net magnetization, we can examine what happens to the
domains as the field strength is increased (Fig. 3.4). As the field is
increased from zero, the effect on the solid is to displace domain boundaries
in a reversible fashion. If the magnetic field is switched off, the domain
boundaries return to their starting positions. Thus the initial part of the B-H
curve results from reversible domain boundary displacement, and the slope is
called initial permeability µ_i. As 
the magnetic field strength is increased, there is an irreversible
boundary displacement, and at first the induced magnetization increases more
rapidly than the field strength and gives a maximum slope µ_max.
Finally in the upper part of the magnetization curve all domain boundaries have
been displaced, and further increases in the magnetic field cause rotation of
the domains in the direction of the applied field. At this point the material
is saturated; higher fields cannot induce more magnetization.
As
the magnetic field is decreased to zero, the induced magnetization does not
decrease to zero, but the alignment of most of the domains during magnetization
results in a remanent magnetization or remanence B_r. When the
direction of the magnetic field is reversed, the induced magnetization
decreases and finally becomes zero at a value of the magnetic field strength
called the coercive force H_c. Further increased magnetic field
strength in this opposite direction eventually causes magnetic saturation in
the reverse direction and produces a saturation B_s and remanence B_r
values of the same magnitude as in the first quadrant. As an applied field is
cycled from one direction to the other direction, the hysteresis loop is
followed. Since the area of the hysteresis loop represents the energy or work
to bring about changes in the magnetic domain structure, the product B.H,
called the energy product, represents a net loss in the system, usually in the
form of

Fig. 3.4 Magnetization
characteristic and Hysteresis loop caused by domain action.
heat.
In applications in which the magnetic material is cycled around the
magnetization curve many times per second, hysteresis losses are critical, and
soft magnetic materials are required. In addition to the energy loss due to the
hysteresis curve there are energy losses resulting from electrical currents,
eddy currents, induced in the material. The changing in magnetic flux produces
a power loss in the system proportional to Φ²/R, where Φ is the
locally induced voltage and R is the resistance of the material. For this
application, magnetic oxides which have a high electrical resistivity have
small eddy-current losses and a distinct advantage over metals.
For
permanent magnets, hard magnetic materials are required. For permanent magnets
a high coercive force is also required, so that the material is not easily
demagnetized. A single quality, the energy product, is commonly used to
describe the quality of a permanent magnetic material. This is usually the
maximum value of B.H product. High-quality permanent magnetic materials have an
energy product of about 1 cal/cm³. From the hysteresis curve, Fig.
3.4, one notes that high values of (BH)_max require high values of
both remnant magnetization and coercive field.
3.8 Coercively:
In
materials science, the Coercivity, also called 
the coercive field, of a ferromagnetic material is the intensity of the
applied magnetic field required to reduce the magnetization of that material to
zero after the magnetization of the sample has been driven to saturation. Coercivity
is usually measured in oersted or ampere/ meter units and is denoted by H_C
.When the coercive field of a ferromagnet is large, the material is said to be
a hard or permanent magnet. Permanent magnets find application in electric
motors, magnetic recording media .
3.8
Theory of permeability:
Permeability is defined as the proportionality
constant between the magnetic field induction B and applied intensity H;
B
= μ H………………………………………3.18
This
definition needs modification when magnetic material is subjected to an ac
magnetic
field
given bellow;
H =H₀e^iωt
…………………………………….3.19
In
such a field the magnetic flux density experiences a delay with respect to H.
The delay is used due to the presence of various of various loses and thus
expressed as,
B = B₀e^{i
(ω t – δ)} …………………………………3.20
Where
δ is the phase angle and marks the delay of B with respect to H. The
permeability
is
given by
μ = B/H = (
B₀e^{i (ω t – δ)}) / (H₀e^iωt)
= (B₀e^-iδ)
/H_{0
(B0 / H0)cosδ – i(B0 / H0)sinδ}
= μ^΄-i
μ^΄΄………………………………………3.21
Where,
μ^΄= (B₀/H₀)cosδ……………………………………………….3.22
And,
μ^΄΄= (B₀/H₀)sinδ………………………………………………….3.23
The real part μ^΄ of complex permeability μ
as expressed in equation (1.13) represents the component of B, which is in
phase with H, so it corresponds to the normal permeability. If there is no
loses we should have μ= μ^΄. The imaginary part μ^΄΄
corresponds to the part of B which is delayed by phase angle 90^º from
H. The presence of such a component requires a supply of energy to maintain the
alternating magnetization, regardless of the origin of delay. The ratio of
gives
μ^΄΄
/ μ^΄=(B₀/H₀)sin δ/( B₀/H₀)
cos δ =tan δ ………………..3.24
The
tan δ is called the loss factor. The Q-factor or quality factor defined as the
reciprocal of this factor, i.e., Q
= 1/tanδ……………………………………….3.25
3.9
The Mechanism of Permeability:
The
demagnetized material is divided into number of Weiss domains separated by
block walls. In each domain all the magnetic moments are oriented in parallel
and have its saturation value M_s. In the walls the magnetization
direction changes gradually from the direction of magnetization in one domain
to that in the next. The equilibrium positions of the walls result from the
interactions with the magnetization in neighboring domains and from the
influence of pores; crystal boundaries and chemical in homogeneities which tend
to favor certain wall positions.
The
mechanism of wall permeability arises from the displacement of the domain walls
small fields. Let us consider a piece of material in the demagnetized state,
divided into Weiss domains with equal thickness L by means of 180^º.
Bloch walls as shown in figure 3.5. The walls are parallel to the Y-Z plane.
The magnetization M_s in the domains is oriented alternately in the
+Z or –Z direction. When the field H with a component in the +Z directions is
applied, the magnetization in this direction will be favored. A displacement dx
of the walls in the direction shown by the dotted lines will decrease the
energy density by an amount: 2M_sH_zdx)/L (. This can be
described as a pressure 2M_sH_z, exerted on each wall. The
pressure will be countered by restoring force, which for small deviation may be
assumed to be dx per unit wall surface. The new equilibrium position is given
by
d = (2M_sH_zdx)/L………………………………………………3.26
From
the change in the magnetization,
ΔM = (2M_sd)/L…………………………………………………3.27
The
wall susceptibility χ_w may be calculated. Let H make the angle θ
with Z direction. The magnetization in the direction θ becomes
and d = (2M_sH_z)/K………………………………………………3.29
We
obtain, χ_w  = (ΔM)_θ/H
= (4M²_s cos²θ)/KL……………………………………3.30

Figure 3.5 Magnetization by wall motion and spin rotation.
3.9.2. Rotational Susceptibility:
The
rotational permeability mechanism arises from rotation of the magnetization in
each domain. The direction of M can be found by minimizing the magnetic energy
E as a function of the orientation. Major contributions may be due to the
stress and shape anisotropy. The stress may influence the magnetic energy via
the magnetostriction. The shape anisotropy is caused by the boundaries of the
sample as well as by pores, nonmagnetic inclusions and inhomogeneities. For small angular deviations α_X
and α_Y or M, where
α_X = M_X/M_s
and α_Y = M_Y/M_s …………………………………………………3.31
From
the equilibrium Z-direction may be expressed as
E = E₀ + 1/(2α²_XE_XY)
+ 1/(2α²_YE_YY)…………………………………………3.32
Where
it is assumed that X and Y are the principal axes of the energy minimum.
Instead
of
E_XX and E_YY, the anisotropy field H_X^A
and H_Y^A are often introduced. Their magnitude
is given by
H_X^A and H_Y^A
represent the stiffness with which the magnetization is bound to the
equilibrium direction for deviations in the X and Y direction, respectively.
The rotational susceptibilities χ_r,x and χ_r,y , for
fields applied along X and Y directions, are χ_r,x = M_S/H_X^A
 
and
χ_r,y = M_S/H_Y^A respectively. 
For
cubic materials it is often found that H_X^A and H_Y^A
are equal. For H_X^A = H_Y^A = H^A
and a field of H makes an angle θ with Z-direction, the rotational
susceptibility, χ_r,c in one crystallite becomes
χ_r,c
= M_S/ (H^Asin² θ)…………………………………………3.34
A
polycrystalline material consisting of a large number of randomly oriented  grains of different shapes, with each grain
divided into domains in a certain way the rotational susceptibility χ_x
of the material has to be obtained as a average of  χ_r,c of each crystallite, where
the mutual influence of neighboring crystallites has to be taken into account.
If the crystal anisotropy dominates other anisotropies, the H^A will
be constant throughout the material. So only the factor sin²θ has to
be averaged. Snoek assuming a linear averaging of χ_r,c , found the
following relation:   
χ_r = 2M_S/ (3H⁴)……………………………………..3.35
The
total internal susceptibility 
χ
= χ_ω + χ_r 
=
(4M_S²cos² θ) /(KL)  + 2M_S/ (3H⁴)………………3.36
If
the shape and stress anisotropies cannot be neglected, H⁴ will be
larger. Any estimation of χ_r  will rather uncertain as long as the domain
structure and the pore distribution in the material are not known. A similar estimate of χ_ω
would require the knowledge of the stiffness parameter k and the domain width
L. These parameters are influenced by the factors like, imperfection, porosity,
crystallite shape and distribution of pores.
3.10. Ferrites:
Ferrites
are complex magnetic oxides that contain the ferric oxide (Fe₂O₃)
and their basic magnetic component. The general chemical composition can be
written as MO. Fe₂O₃, where M represents a divalent metal
ion such as Ni, Mn or Zn. The outstanding property of ferrites, which makes
them suitable for many applications, is their high electrical resistivity
compared to those metals. Their specific resistivity ranges from 10²
to 10¹⁰ Ω-cm which is up to 15 orders of magnitude higher than that
of metals like iron. High frequency application thus demands extensive implication
of ferrites industry.
3.11. Types of ferrites:
The
crystallography of fall in a natural manner into three types: (a) the cubic
ferrites of spinel type, (b) the cubic ferrites of the garnet type and (c) the
hexagonal ferrites. The magnetic ferrites fall into two groups of different
crystal structure as shown below.
These ferrites are also called ferrospinels because
they crystallize in the same crystal structure as the mineral spinel and they
derive their general formula MFe₂O₄ from the mineral
spinel having composition MgAl₂O₄. In this formula, M
represents a divalent ion of metal. Besides the divalency, another condition to
qualify a metal in ferrospinels is its ionic radius, which should fall between
0.6 and 1.0ºA [9]. Mg, Fe, Co, Ni, Cu, Zn and Cd, all satisfy these two
conditions and thus form various single cubic ferrites. Magnetic, which
contains one ferrous ion and two ferric ions in each formula unit is a typical
ferrite.
(b)The cubic ferrites of the garnet
type: The mineral garnet refers to a
group of mixed oxides, of which the widely known one has the chemical formula
Mn₃Al₂Si₃O₁₂, or equivalently,
3MnO.Al₂O₃.SiO₂. Simple magnetic garnets have
the general formula 3M₂O₃.5Fe₂O₃ =
2M₃¹¹¹Fe₅¹¹¹O₁₂. Note
that, in magnetic garnets, the 24 positive charge units are divided unequally
between the ferric ions (15 units) and another species of trivalent ions (9
units). Technically useful garnets are those with M= Sm, Eu, Gd, Tb, Dy, Ho,
Er, Tm, Yb, or Y. They are known as the rare- earth garnets. A code system has
been adopted to name them: REG stands for the rare-earth garnets, GdIG for the
gadolinium-iron garnet (Gd₃Fe₅O₁₂), YIG for
the yttrium-iron garnet (Y₃Fe₅O₁₂), etc.
Garnets crystallize in the cubic system with –fifths of the ferric ions forming
a body-centered cubic lattice. Like the ferrospinels, the garnets, too, pack a
large number (160) of ions in eight formula units in a unit cell. The lattice
constant is approximately 12.5 Ǻ, about 50% larger than the ferrospinel. Also,
the crystal structure of the garnets is more complicatd than the spinel
structure because of the size (0.85-1.10 Ǻ) of the M¹¹¹ions. They
are too large to accommodate at the interstitial sites between the oxygen ions.
Hence, the oxygen are prohibited from forming a close-packed structure as in
the spinel. In each unit cell, which contains eight formula units , there are
three kinds of cation sites, of which 16 octahedral [a] sites are occupied by
Fe¹¹¹ions, 24 tetrahedral sites are also occupied by Fe¹¹¹
ions and 24 dodecahedral {c} are occupied by M ions . 
(c) The
hexagonal ferrites: The third type of
ferrites is often called the barium ferrites these compounds usually contain
BaO, in addition to Fe₂O₃, as the basic component oxide.
They are also known as the magnetoplumbites. The common chemical formula of barium
ferrites is l (BaO). m (MO). n (Fe₂O₃), or Ba_l¹¹M_m¹¹Fe_2n¹¹¹O_l+m+3n
where l, is much more complex than the previous two in both composition and
crystallography. There arefour ways in which the composition of barium ferrites
may be changed. One is to vary the M¹¹ ions. Mg, Mn, Fe, Co, Ni, Cu
and Zn are found  suitable for the
formation of hexagonal ferrites. Another way is to alter the values for l, m,
n. Basic combinations are found at 1-0-6 (the ferrite is designated M ), 1-2-8
(M₂W), 2-2-6 (M₂Y) and 3-212(M₂Z). Compounds
formed under these combinations are termed classical hexagonal ferrites. Still,
another way to substitute for Ba with Pb or Sr and substitute for Fe with Al, Ga,
Cr or Mn. A fourth to vary the composition of the barium ferrites is to mix two
or more of the classical hexagonal ferrite in different proportions. 
Depending on the method of magnetization, ferrites are
also classified into two types; hard and soft. 
Hard
ferrites: When the ferrites are
placed in a strong magnetic field and they are easily magnetized, then these
ferrites are called hard ferrites. They have high saturation magnetization as
well as high coercive forces; that is they are not easily demagnetized. Therefore
hard ferrites are used to make permanent magnets.
Soft
ferrites: Soft ferrites are
magnetized due to low H-values and low coercive field. This type of ferrite is
typically used where reversal of magnetization is required, such as electromagnet,
transformer cores and relays.
3.12.
Crystal structure of Spinel Ferrites:
Ferrites have the cubic
structure, which is very close to that of the mineral spinel MgO.Al₂O₃
and are called cubic spinel. Analogous to the mineral spinel, magnetic spinel
have the general formula MeO.Fe₂O₃ or MeFe₂O₄
where Me is the divalent metal ion. Nickel ferrite (NiFe2O4) is basically an
inverse spinel ferrite in which the tetrahedral (A) sites are occupied by ferric
ions, and the octahedral (B) sites by ferric and nickel ions. Thus, the
compound can be represented as