Interactions Between Ferrihydrite and CoO Nanoparticles
1. Interactions between Nanoparticles
of Antiferromagnetic Materials
An investigation of interactions between ferrihydrite and CoO.
Master Thesis
by
Britt Rosendahl Hansen
Niels Bohr Institute for Astronomy,
Physics and Geophysics, University of
Copenhagen
Department of Physics, Technical Univer-
sity of Denmark
2.
3. Abstract
It is shown that CoO nanoparticles can be prepared from Co3O4 by
ball milling and subsequent reduction by heating in H2. From refinement
of XRD spectra the size of the CoO nanoparticles is found to be ∼ 10
nm.
The CoO nanoparticles were mixed with 6-line ferrihydrite and the
effect on the superparamagnetic relaxation of ferrihydrite was studied us-
ing M¨ossbauer spectroscopy. The measurements showed that interaction
with CoO nanoparticles lead to a suppression of the superparamagnetic
relaxation of ferrihydrite.
Interaction between the nanoparticles was also seen in magnetization
measurements, when comparing the data for the two nanoparticles with
a mixed sample. A modified Langevin function and modified Curie law
were fit to the magnetization data for ferrihydrite to obtain the particle
magnetic moment and the N´eel temperature. The validity of the modified
functions is questioned.
i
4.
5. Preface and
Acknowledgements
This thesis is submitted in partial fulfillment of the requirements for obtaining
the degree of Master of Science at the University of Copenhagen (KU). The
work detailed in the thesis was carried out at the Technical University of
Denmark (DTU) and in collaboration with Risø National Laboratory in the
period from September 2003 to November 2004. Supervisors were Professor
Steen Mørup of the Nanostructured Materials group at the Department of
Physics, DTU, Senior Scientist Kim Lefmann from the Materials Research
Department, Risø, and Associate Professor Morten Bo Madsen at the Center
for Planetary Science, KU.
Special thanks are given to Professor Steen Mørup for his guidance and
initial suggestion that I do my thesis work in his group - it has been a wonderful
and educational time. Many thanks are given to Christian Robert Haffenden
Bahl for his help with experiments, for discussions, for transmission electron
microscopy images and for creating a pleasant atmosphere in the office.
I would like to acknowledge Bente Lebech who provided valuable advice re-
garding Rietveld refinement of XRD spectra. Also, Associate Professor Mikkel
Fougt Hansen at the Department of Micro and Nanotechnology (MIC), DTU,
was kind enough to help me with the LakeShore vibrating sample magnetome-
ter and gave me valuable input on the interpretation of the data. Likewise,
Associate Professor Leif Gerward from the Nanostructured Materials group,
DTU, is acknowledged for instructing me in the operation of the x-ray diffrac-
tometer at the Department of Physics. I am grateful for the opportunity to use
the above-mentioned instruments. I would also like to acknowledge Cathrine
Frandsen, who has been helpful with information and also contributed to the
atmosphere in the group. Many thanks are given to Thomas Pedersen and
Peter Gath Hansen for proofreading the thesis.
Also many thanks to Lis Lilleballe and Helge Rasmussen for their help in
the preparation of the studied nanoparticles and for their help with miscella-
neous tasks in the chemistry and M¨ossbauer laboratories.
Britt Rosendahl Hansen
November 2004
iii
8. List of Figures
List of Figures
1.1 Areal density trend of IBM hard disk drives . . . . . . . . . . . . . 1
2.1 The Langevin function and Curie-Weiss law. . . . . . . . . . . . . 9
2.2 Ferromagnetic domain structure. . . . . . . . . . . . . . . . . . . . 9
2.3 Ordered magnetic moments in an antiferromagnet. . . . . . . . . . 10
2.4 Temperature dependence of antiferromagnetic susceptibility. . . . . 11
2.5 Magnetic anisotropy energy in a crystal with uniaxial symmetry. . 12
2.6 Magnetic anisotropy constants K1 and K2 of Fe. . . . . . . . . . . 13
2.7 Antiferromagnetic nanoparticle. . . . . . . . . . . . . . . . . . . . . 14
2.8 Reversal of the magnetic moment in a nanoparticle. . . . . . . . . 14
2.9 Dipole and exchange interaction. . . . . . . . . . . . . . . . . . . . 15
3.1 Geometric derivation of Bragg’s equation. . . . . . . . . . . . . . . 20
3.2 Scattering planes given by their Miller indices (hkl). . . . . . . . . 20
3.3 Transitions and x-ray spectrum for Cu. . . . . . . . . . . . . . . . 21
3.4 In-house x-ray powder diffraction setup. . . . . . . . . . . . . . . . 22
3.5 Effects of isotropic and anisotropic strain on a scattering peak. . . 22
3.6 Refinement of XRD pattern of Si standard. . . . . . . . . . . . . . 24
3.7 The Jeol JEM-3000F TEM. . . . . . . . . . . . . . . . . . . . . . . 25
3.8 Resonant absorption of a γ-quantum. . . . . . . . . . . . . . . . . 27
3.9 The M¨ossbauer spectroscopy setup. . . . . . . . . . . . . . . . . . . 28
3.10 The decay scheme of 57Co. . . . . . . . . . . . . . . . . . . . . . . 29
3.11 The isomer shift. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.12 Quadrupole splitting. . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.13 Magnetic hyperfine splitting. . . . . . . . . . . . . . . . . . . . . . 33
3.14 Octahedral and tetrahedral coordination. . . . . . . . . . . . . . . 36
3.15 Angular part of 3d orbitals. . . . . . . . . . . . . . . . . . . . . . . 37
3.16 eg and t2g orbitals in octahedral coordination. . . . . . . . . . . . . 38
3.17 Relative energies of the e and t2 orbitals. . . . . . . . . . . . . . . 38
3.18 High-spin and low-spin states. . . . . . . . . . . . . . . . . . . . . . 39
3.19 The basic setup of a VSM. . . . . . . . . . . . . . . . . . . . . . . 41
3.20 Output of demodulator in a lock-in amplifier. . . . . . . . . . . . . 42
3.21 Hysteresis loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.1 Ferrihydrite in a ferriferous spring in Iceland. . . . . . . . . . . . . 45
vi
9. 4.2 XRD spectra of ferrihydrite. . . . . . . . . . . . . . . . . . . . . . . 46
4.3 Idealized models of hematite and goethite. . . . . . . . . . . . . . . 47
4.4 Molecular model of ferritin . . . . . . . . . . . . . . . . . . . . . . 49
4.5 The cubic, trigonal and monoclinic Bravais lattices. . . . . . . . . . 49
4.6 Crystal and magnetic structure of CoO . . . . . . . . . . . . . . . 50
5.1 Refined XRD spectrum of as prepared Co3O4. . . . . . . . . . . . 54
5.2 Fritsch pulverisette 5 planetary mill. . . . . . . . . . . . . . . . . . 55
5.3 Broadening of Co3O4 XRD peak. . . . . . . . . . . . . . . . . . . 56
5.4 XRD spectra of as prepared Co3O4 reduced in H2. . . . . . . . . . 57
5.5 Refinement of XRD spectrum of sample CoO1. . . . . . . . . . . . 58
5.6 Refinement of XRD spectrum of sample CoO6. . . . . . . . . . . . 59
5.7 XRD spectrum of ferrihydrite sample. . . . . . . . . . . . . . . . . 60
5.8 TEM image of ferrihydrite showing agglomeration of the particles. 61
6.1 M¨ossbauer spectra and hyperfine field distributions of ferrihydrite. 66
6.2 M¨ossbauer spectrum of sample CoO6. . . . . . . . . . . . . . . . . 67
6.3 TEM image of ground ferrihydrite. . . . . . . . . . . . . . . . . . . 68
6.4 XRD spectra comparing untreated and ground ferrihydrite. . . . . 69
6.5 M¨ossbauer spectra of treated ferrihydrite. . . . . . . . . . . . . . . 70
6.6 Hyperfine field distributions of treated ferrihydrite. . . . . . . . . . 71
6.7 XRD spectra comparing untreated and heated ferrihydrite. . . . . 72
6.8 M¨ossbauer spectra of samples mixed in aqueous solutions . . . . . 74
6.9 Hyperfine field distributions of samples mixed in aqueous solutions. 75
6.10 M¨ossbauer spectrum of mixed sample FHCoO 2250 RTD . . . . 76
6.11 M¨ossbauer spectra of samples heated in a reflux water-condenser. . 78
6.12 M¨ossbauer spectrum of sample mixed with ultrasound. . . . . . . . 79
6.13 M¨ossbauer spectrum of mixed sample FHCoO6 Mix. . . . . . . . 80
6.14 M¨ossbauer spectrum of mixed sample FHCoO6 GH. . . . . . . . 81
6.15 M¨ossbauer spectra of samples mixed and heated in H2. . . . . . . 82
6.16 M¨ossbauer spectra of mixed samples heated in H2. . . . . . . . . . 84
6.17 Hyperfine field distributions of samples heated in H2. . . . . . . . 85
6.18 M¨ossbauer spectrum of ferrihydrite at room temperature. . . . . . 86
6.19 Room temperature spectra of ferrihydrite and two mixed samples. 87
7.1 Field sweep of empty sample cup. . . . . . . . . . . . . . . . . . . . 90
7.2 Low-field part of ZFC and FC hysteresis loops for ferrihydrite. . . 91
7.3 Field sweeps showing the progression in transformation. . . . . . . 92
7.4 M¨ossbauer spectra of transformed ferrihydrite. . . . . . . . . . . . 93
7.5 Comparison of FC hysteresis curves. . . . . . . . . . . . . . . . . . 94
7.6 Comparison of ZFC hysteresis curves. . . . . . . . . . . . . . . . . 95
7.7 Magnetic moment at maximum field (±16 kOe). . . . . . . . . . . 96
7.8 FC and ZFC hysteresis loops of mixed sample. . . . . . . . . . . . 96
7.9 The moment shift. . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
7.10 Exchange bias of sample CoO1. . . . . . . . . . . . . . . . . . . . 98
7.11 Exchange bias of mixed sample FHCoO RTD. . . . . . . . . . . . 99
vii
10. List of Figures
7.12 Diagram showing the calculation of the coercivity. . . . . . . . . . 100
7.13 Coercivity of FC and ZFC ferrihydrite. . . . . . . . . . . . . . . . . 100
7.14 Coercivity of CoO1 showing a dip at 40 K. A line is drawn at 40
K to show this clearly. . . . . . . . . . . . . . . . . . . . . . . . . . 101
7.15 Superposition of two curves and the effect on coercivity. . . . . . . 101
7.16 Coercivity of mixed sample FHCoO RTD. . . . . . . . . . . . . . 102
7.17 Temperature dependence of M0, µp and χa. . . . . . . . . . . . . . 103
7.18 Thermoinduced magnetization. . . . . . . . . . . . . . . . . . . . . 104
7.19 Field sweeps at 300 K with cryostat and oven mounted. . . . . . . 106
7.20 Initial susceptibility and inverse initial susceptibility of ferrihydrite. 106
7.21 Extrapolation of M0 to T = 0 K. . . . . . . . . . . . . . . . . . . . 107
7.22 Plot of the susceptibility and result of TN estimate. . . . . . . . . . 108
viii
11. List of Tables
2.1 SI and CGS Gaussian units. . . . . . . . . . . . . . . . . . . . . . . 16
3.1 Transition probabilities for 57Fe. . . . . . . . . . . . . . . . . . . . 35
3.2 Typical values of isomer shift and electric quadrupole splitting. . . 39
6.1 Overview of samples. . . . . . . . . . . . . . . . . . . . . . . . . . . 64
6.2 Parameters of room temperature spectra of ferrihydrite. . . . . . . 86
7.1 Temperature ranges of fields sweeps made on three samples. . . . . 90
D.1 Ball milling timetable for the sample CoO1. . . . . . . . . . . . . . 120
ix
12.
13. One
Introduction and Motivation
Magnetic nanoparticles have received much attention during the last two
decades. One reason for this is that the computer hardware industry utilizes
magnetic nanoparticles in magnetic storage devices. These particles must be
able to retain a magnetic field direction as this is how information is stored.
If the magnetic properties are not stable, the information coupled to the par-
ticles is lost. As the demand for storage space is high the industry seeks to
increase the areal density of stored information and one way to do this is to
make the particles smaller. This has been a trend for many years and, in
addition to improvements to the read and write heads, is the reason for the
far greater storage capacity of hard disk drives today compared to the 1950s,
when hard disk drives were invented, see Fig. 1.1. When magnetic particles
Figure 1.1: Areal density trend of IBM hard disk drives [1].
1
14. 1. Introduction and Motivation
become very small an effect known as superparamagnetism may be significant
and can cause the magnetic moment direction of the nanoparticles to fluctu-
ate or even flip, which constitutes a loss of data in magnetic storage media.
The computer hardware industry has reached the so-called superparamagnetic
limit and will in the future require innovative ideas if the current growth rate
in areal densities is to continue. As magnetic storage such as a hard disk
drive is still the most important storage device in computers, the research in
magnetic nanoparticles is of great economic interest.
Another interesting application of magnetic nanoparticles is in biomedicine
[2]. Here, magnetic nanoparticles are used for separation of specific biological
entities. By coating the magnetic nanoparticles with biocompatible molecules
tagging of the biological entities becomes possible and they may be separated
from their native environment by application of a magnetic field gradient.
Other uses of magnetic nanoparticles in biomedicine being studied are drug
delivery and hyperthermia. By attaching a cytotoxic1 drug to a magnetic
nanoparticle carrier it is possible to target the therapy, which requires a lower
dosage and reduces side effects. Hyperthermia refers to the treatment of cancer
by dispersion of magnetic particles throughout the target tissue and causing
them to heat up by applying an AC magnetic field. Again, the force lies in
the targeting possible with the magnetic particles.
For the scientist, magnetic nanoparticles offer a way to study small-scale
effects. The magnetic properties of nanoparticles often differ markedly from
those of bulk materials allowing new properties and thereby possibly new
devices and applications to be explored.
Thus, besides the obvious economic interests for the computer hardware
industry, magnetic nanoparticles represents an area of basic science with
promises of technological spinoffs.
Interactions between magnetic nanoparticles affect their magnetic behavior
and makes it possible to manipulate their magnetic properties. For instance, it
has been shown that nanoparticle mixtures of the mineral hematite (α-Fe2O3)
and the transition metal oxides CoO and NiO affect the superparamagnetic
relaxation of hematite [3].
This thesis examines the effect on the superparamagnetic relaxation of the
mineral ferrihydrite by interactions with CoO nanoparticles. Ferrihydrite is a
naturally occurring antiferromagnetic ferriferous2 mineral. The particles are
always of nanometer size, i.e. bulk ferrihydrite does not exist. The super-
paramagnetic relaxation time, which is the time between flips of the magnetic
moment, is exponentially dependent on the factor V
T , where V is the volume
of the particle and T is the temperature. When studying the magnetic prop-
erties of nanoparticles the results obtained are conditional upon the time scale
of the measurement compared to the relaxation time. On the time scale of
M¨ossbauer spectroscopy measurements ferrihydrite shows superparamagnetic
1
Of, relating to, or producing a toxic effect on cells.
2
iron containing
2
15. relaxation at room temperature and some samples are superparamagnetic at
temperatures as low as 23 K [4]. The fast superparamagnetic relaxation of
ferrihydrite makes it difficult to study its magnetic properties. If it is possible
to suppress the superparamagnetic relaxation, this would allow one to study
the magnetic properties of ferrihydrite at higher temperatures.
The suppression of the superparamagnetic relaxation is due to a stabilizing
effect via so-called magnetic exchange interaction by the CoO nanoparticles
on the otherwise relaxing magnetic moments. In this study various recipes
for mixing the ferrihydrite and CoO nanoparticles are used in order to ex-
amine what can be done to enhance the interaction leading to suppression of
the superparamagnetic relaxation. Part of the thesis will also deal with the
production of the CoO nanoparticles as this was no trivial task.
This study is part of the continued research on magnetic nanoparticles in
the M¨ossbauer Group at the Department of Physics at the Technical Univer-
sity of Denmark (DTU). The research focuses on the magnetic behavior of
nanoparticles and inter-particle interactions.
Outline of thesis
The chapter headings should be self-explanatory and the thesis hopefully eas-
ily navigated. The thesis is essentially in two parts: theory and experiment.
The theoretical part, chapters 2, 3 and 4, describes magnetism and magnetic
nanoparticles, experimental methods used and gives a short description of fer-
rihydrite and CoO. In the experimental part, chapters 5, 6 and 7, describes
the production of the nanoparticles, the M¨ossbauer spectroscopy study and vi-
brating sample magnetometry study of the nanoparticles. Finally, conclusions
and thoughts on further studies are summarized in chapter 8.
A few common abbreviations used in the text are defined on first usage
and listed below.
Abbreviations used in the text
FC Field Cooled
TEM Transmission Electron Microscope, Transmission Electron Microscopy
VSM Vibrating Sample Magnetometer, Vibrating Sample Magnetometry
ZFC Zero Field Cooled
XRD X-Ray Diffraction
3
16.
17. Though this be madness, yet
there is method in’t.
William Shakespeare
- Hamlet(II, ii, 206)
Theory and Characterization
Methods
5
18.
19. Two
Theory of Magnetic Nanoparticles
This chapter describes the theory of magnetic nanoparticles needed in order
to understand terminology and experimental findings of later chapters.
2.1 Magnetism
All materials display an extremely weak form of magnetism called diamag-
netism when a magnetic field is applied. Diamagnetism arises from the effect
of the applied magnetic field on the atomic orbits of the electrons and is
usually a negligible effect if the atoms have a magnetic moment, which will
cause a stronger response to the applied field. Atoms with unfilled electron
shells possess a magnetic moment arising from the total orbital angular mo-
mentum and total spin angular momentum of its electrons. In a material
containing magnetic atoms the magnetic moments may be correlated or un-
correlated. If uncorrelated, the magnetic moments will be randomly oriented
and the material is said to be paramagnetic. If correlated, long-range ordering
of the moments exists and the orientation of a single moment is dependent
on the orientation of its neighbors. Such ordered moments are the basis of
the permanent magnets. All magnetically ordered materials have a transition
temperature above which the ordering disappears and the material becomes
paramagnetic. Materials with long-range ordering may be ferromagnetic, an-
tiferromagnetic or ferrimagnetic. When a material is magnetically ordered
preferred directions of magnetization exist, which are given by the magnetic
anisotropy described in section 2.4.
Magnetization and susceptibility
Magnetization and susceptibility are two important quantities, when studying
the magnetic properties of a compound. The terms are used extensively in
later chapters and the definitions and symbols given are here.
7
20. 2. Theory of Magnetic Nanoparticles
The magnitude of the magnetic moment, µ, per unit volume is the mag-
netization, M. Thus,
M =
µ
V
(2.1)
This quantity may also be characterized by the mass magnetization, σ, given
by
σ =
µ
M
(2.2)
where M is the mass. Susceptibility is a measure of how easily a material
responds to an applied magnetic field, H, i.e. how magnetizable the material
is. The mass susceptibility, χ, is given by
χ =
∂σ
∂H
(2.3)
For small applied fields this reduces to
χ =
σ
H
(2.4)
2.2 Paramagnetism
In the paramagnetic state the magnetic moments are independent. A small
applied magnetic field will cause some of the magnetic moments to align with
the field, but when the field is removed the moments will again orientate
themselves randomly. The amount of alignment in applied field depends on
the strength of the field, the size of the magnetic moment and the temperature.
Classically, the average magnetization, M , of a paramagnet is given by
M = M0L
µB
kBT
(2.5)
where M0 is the magnetization at saturation, i.e. when all the moments are
aligned with the field, B is the magnetic flux density and L(y) is the Langevin
function, see Fig. 2.1,
L(y) = coth(y) −
1
y
(2.6)
For a paramagnet and for small applied fields, the susceptibility follows the
Curie-Weiss law
χ =
C
T − θ
(2.7)
where θ is a constant. If θ > 0 the material is a ferromagnet above its
transition temperature, whereas θ < 0 indicates an antiferro- or ferrimagnet.
8
21. Magnetic Ordering
Figure 2.1: Left: The Langevin function used to describe the magnetization of an
ideal paramagnet when a magnetic field is applied. Right: Plot of the Curie-Weiss
law of susceptibility shown for the three cases θ < 0, θ = 0 and θ > 0.
2.3 Magnetic Ordering
Ferromagnetism
A ferromagnet has a magnetization even at zero applied field and is said to
have a spontaneous magnetization. The moments tend to align in parallel, but
the magnetization is dependent on the history of the ferromagnet. When a
ferromagnet is produced the moments are arranged in ferromagnetic domains,
see Fig. 2.2, inside which all the magnetic moments are aligned in the same
direction. When a weak magnetic field is applied, the domains with moments
Figure 2.2: Ferromagnetic domain structure of a single crystal platelet [5].
9
22. 2. Theory of Magnetic Nanoparticles
pointing in the same direction as the field will grow at the expense of other
domains. If a strong field is applied all the magnetic moments will align
with the field. Between domains are domain walls, which are several hundred
lattice constants thick, where the moments gradually turn. Having domains
is energetically favorable for bulk ferromagnets. In particles, however, there
is a critical size below which they have but a single domain.
The transition temperature above which a ferromagnet becomes paramag-
netic is known as the Curie temperature.
Antiferromagnetism
In antiferromagnetic materials the magnetic moments are of equal magnitude
and antiparallel, see Fig. 2.3. Thus, a perfectly structured bulk antiferromag-
net has no net magnetization below its ordering temperature, which is known
as the N´eel temperature, TN . The antiferromagnetic lattice may be thought
Figure 2.3: Arrangement of magnetic moments in the antiferromagnet MnO below
its ordering temperature [6].
of as two interpenetrating lattices each of which consists of spins that are ori-
ented in parallel to the rest of the spins on the same lattice. Thus, one way to
visualize antiferromagnet ordering is as two ferromagnetic lattices canceling
each other out, but one should remember that this is just for visualization.
The moments on the two sublattices are not independent but interacting.
As seen in Fig. 2.4 the antiferromagnetic susceptibility in a single crys-
tal below the ordering temperature is dependent on whether the applied field
is parallel or perpendicular to the spin axes. In a polycrystalline antiferro-
magnet the susceptibility is a combination of the parallel and perpendicular
susceptibility. The decrease in inverse susceptibility with temperature is a
distinguishing feature of bulk antiferromagnets. Antiferromagnetic materials
10
23. Magnetic Anisotropy
Figure 2.4: The temperature dependence of antiferromagnetic susceptibility [7].
may have a net magnetization due to lattice defects or, in nanoscale particles,
uncompensated spins.
Ferrimagnetism
A ferrimagnet has magnetic moments arranged antiparallel or at an angle
with each other. If the moments are arranged antiparallel, what distinguishes
it from an antiferromagnet is that the magnetizations of the sublattices are
unequal. This can occur if there are more moments in one sublattice than in
the other or if the moments are of unequal magnitude. Thus, a ferrimagnet
has a net magnetization.
2.4 Magnetic Anisotropy
Magnetic anisotropy means that the magnetic energy of the system is not
invariant with respect to the direction of the magnetization. In all magnetic
materials preferred directions of magnetization exist, so-called easy directions
of magnetization, which are defined by among other factors the crystal lattice
and the shape of the sample. This is known as magnetocrystalline anisotropy
and shape anisotropy, respectively. In the absence of an external magnetic
field the direction of spontaneous magnetization will arrange itself according
to the magnetic anisotropy so as to obtain a state with the lowest energy.
The directional effect of the magnetocrystalline anisotropy arises from the
spin-orbit coupling. The orbital wavefunctions are affected by the symmetry
of the crystal lattice and the spins are affected through the coupling. Shape
anisotropy is caused by dipole interaction between the stray field of the sample
11
24. 2. Theory of Magnetic Nanoparticles
and the individual magnetic moments inside the sample. In elongated particles
or thin films shape anisotropy is significant.
An increase in anisotropy energy occurs when the magnetization is rotated
out of a preferred direction by an applied magnetic field. Thus, the anisotropy
energy is a function of the angle, θ, by which the magnetization deviates from
a preferred direction. In a crystal with uniaxial symmetry1 the magnetocrys-
talline anisotropy energy is conventionally represented by a power series in
sin θ with only even terms as a reversal of the direction of magnetization does
not change the anisotropy energy [7]. For a ferromagnet the anisotropy energy
E(θ) is given by
E(θ) = K1V sin2
θ + K2V sin4
θ + . . . (2.8)
where Kn are magnetocrystalline anisotropy constants. For practical purposes
it is often sufficient to use the first term in Eq. 2.8, so that E(θ) = KV sin2
θ.
KV represents the energy barrier between the two easy directions of magne-
tization, see Fig. 2.5.
Figure 2.5: The magnetic anisotropy energy as a function of the angle θ by which
the magnetization deviates from a preferred direction in a crystal with uniaxial
symmetry.
In analogy with the ferromagnetic case the magnetocrystalline anisotropy
energy for an antiferromagnet with uniaxial symmetry is given by [8]
E(θ) =
1
2
KV (sin2
θA + sin2
θB) (2.9)
where θA (θB) is the angle between the sublattice magnetization MA (MB)
and the easy direction.
1
Symmetry about a single axis.
12
25. Antiferromagnetic Nanoparticles
The magnetocrystalline anisotropy constants are temperature- dependent
and they must clearly be zero at the transition temperature where the mag-
netic ordering disappears. In Fig. 2.6 is shown how the anisotropy constants
of a ferromagnet fall off with temperature much more rapidly than does the
magnetization.
Figure 2.6: The magnetic anisotropy constants K1 and K2 of Fe as a function of
temperature [7].
In magnetic nanoparticles, V is small and thermal excitation can cause
the magnetization to cross the energy barrier KV . This leads to superpara-
magnetic relaxation, where the magnetization of the nanoparticles is no longer
fixed in direction but fluctuates as described in subsection 2.5.
2.5 Antiferromagnetic Nanoparticles
As described in section 2.3 there is a minimum size below which magnetic
particles have but a single domain. This critical size is, for typical material
parameters, 10 − 100 nm in diameter. The moment distribution may be more
complex as studies of nanoparticles below the critical size show that they often
have disordered surface moments. These surface moments interact with the
core moments and give rise to a variety of moment distributions even though
the size of the nanoparticles is within the single domain regime [9]. This is not
explored further in this thesis and the assumption is that the nanoparticles
described here are single domain particles.
Bulk antiferromagnets have a net magnetization of zero as the magnetic
moments cancel each other out. Due to the finite number of spins in an-
tiferromagnetic nanoparticles, however, this cancelation may not be perfect
leading to a net magnetization. A significant ratio of spins are located at the
13
26. 2. Theory of Magnetic Nanoparticles
surface and a simple picture is one where the net magnetization arises from
uncompensated surface spins as illustrated in Fig. 2.7.
Figure 2.7: Rough sketch of an antiferromagnetic nanoparticle with an imperfect
cancelation of surface spins leading to a net magnetization.
Studies have shown that the N´eel temperature of antiferromagnetic nanopar-
ticles is sometimes less than that of a bulk sample, e.g. disc-shaped NiO
nanoparticles [10] and goethite nanoparticles (α−FeOOH) [11].
Superparamagnetic relaxation
In two articles in 1949 [12, 13] Louis N´eel theorized that if a single-domain par-
ticle is small enough, thermal fluctuations may cause a reversal of its magnetic
moment.
Figure 2.8: Reversal of the
magnetic moment in a
nanoparticle with uni-
axial symmetry.
This effect in nanoparticles is different from the sta-
ble magnetic behavior of bulk magnetic materials.
The articles are referenced often, but are in French.
In his Nobel Lecture [14], however, N´eel himself
makes reference to having shown this effect known
as superparamagnetism as early as 1942.
The reversals take place between easy directions
of magnetization, see Fig. 2.8. The time between
two reversals in a particle in the superparamagnetic
state is known as the relaxation time, i.e. a short
relaxation time means rapid fluctuations. For non-
interacting nanoparticles the relaxation time, τ, is
usually described by the Arrhenius relation
τ = τ0 exp
KV
kBT
(2.10)
where τ0 is of the order of 10−10 − 10−12 s [15], K
is the magnetic anisotropy constant and V is the
volume of the particle. When V becomes small, the
fluctuations of the spin direction may be so fast that
the nucleus senses a zero net magnetic field. This
will affect the M¨ossbauer spectrum as described in section 3.3.4. From Eq.
14
27. The CGS Gaussian System versus the SI
2.10 it should be clear that the spin reorientation can be slowed down by de-
creasing the temperature. As described in section 2.4 on magnetic anisotropy
KV represents the energy barrier between two easy directions of magnetiza-
tion in a crystal with uniaxial symmetry.
In the superparamagnetic regime the magnetization of the particles will
behave as a super-spin. When a magnetic field is applied the response of
a super-spin not interacting with other particles will be super-paramagnetic.
Thus, the average magnetization of a sample of superparamagnetic nanopar-
ticles is described using a Langevin function
M = M0L
µpB
kBT
(2.11)
where µp is the magnetic moment per particle.
Interactions between antiferromagnetic nanoparticles
Magnetic particles may interact via dipole interaction, whereby the dipole field
of the particles interact, or exchange interaction, whereby surface spins couple,
see Fig. 2.9. Antiferromagnetic nanoparticles have very small dipole fields
Figure 2.9: Left: Dipole interaction between magnetic nanoparticles. Right: Ex-
change interaction between magnetic nanoparticles.
and will interact via exchange coupling. Studies have shown that exchange
interaction between antiferromagnetic nanoparticles may lead to suppression
of the superparamagnetic relaxation [3, 16].
2.6 The CGS Gaussian System versus the SI
For historical and practical reasons two metric systems are used in literature
on magnetism. These are the CGS Gaussian system and the SI (Syst´eme In-
ternationale). The two metric systems use a different base unit for length and
mass, but the same base unit for time. CGS stands for centimeter, gram and
second. The SI (sometimes called the MKS system) has the meter, kilogram
15
28. 2. Theory of Magnetic Nanoparticles
and second as base units. With two different metric systems in use one must
be careful when comparing obtained results with those found in literature es-
pecially since what is sometimes used are so-called practical units which are
a mixture of the two.
The historical reason for the widespread use of the CGS system in early
literature is simply that it came first. In 1874 the British Association for the
Advancement of Science2 (BA) formally introduced the CGS system and it
was accepted by many scientists as the metric system of choice for decades to
come. A problem with the CGS system is that in time it was not one system
but several systems, because electricity and magnetism could be described in
many ways in terms of the three base units. The MKS system was introduced
by the International Bureau of Weights and Measures3 (BIPM) in 1889. For
many years the MKS system was no better than the CGS system as they
both had three base units, which differed only in size. In 1954 the Tenth
General Conference on Weights and Measures (CGPM) adopted the MKS
system and added ampere, degree Kelvin and candela as base units. The
name International System of Units (SI) was chosen in 1960 and today the
SI has the seven base units: meter, kilogram, second, ampere, kelvin, candela
and mole. Even though the SI base and derived units are recommended for
use in all instances, practicality plays a role as well in the choice of metric
system used.
SI CGS Gaussian
Magnetic field, H A/m Oersted (Oe)
Magnetic flux density, B Tesla (T) Gauss (G)
Magnetization, M A/m emu/cm3
Mass magnetization, σ A·m2/kg emu/g
Mass susceptibility, χ m3/kg emu/(Oe· g)
Table 2.1: SI and CGS Gaussian units for quantities, which are often given in CGS
Gaussian units.
In magnetic measurements the magnetic field, H, the magnetic induction
(or magnetic flux density), B, the magnetization, M, the mass magnetization,
σ and the mass susceptibility, χ, are often given in CGS Gaussian units. It
should be noted that the magnetic induction, B, is often called the magnetic
field, but as the units of B and H are different they can be distinguished in
this way. Table 2.1 shows the units used in the two metric systems. One
may wonder that the CGS Gaussian system is still in use in literature on
magnetism, when the SI is far more consistent. One reason is that magnetic
flux densities and magnetizations measured in T and A/m sometimes give
numbers which are difficult to handle. Another reason is that in the CGS
2
http://www.the-ba.net
3
http://www.bipm.fr
16
29. The CGS Gaussian System versus the SI
Gaussian system B and H are similar in that 1 G = 1 Oe in free space, which
is practical.
In this thesis magnetization measurements are given in a unit appropriate
for comparison with literature. A conversion table between the CGS Gaussian
system and the SI is given in appendix B.
17
30.
31. Three
Experimental Methods
Various experimental methods were employed to characterize and study the
magnetic nanoparticles. The theory of x-ray powder diffraction, transmission
electron microscopy, M¨ossbauer spectroscopy and vibrating sample magne-
tometry is outlined in sufficient detail for the reader to understand the exper-
iments performed in this study.
3.1 X-ray Powder Diffraction
X-ray diffraction is based on the constructive interference of x-ray waves scat-
tered by a periodic distribution of electron densities. One form of a periodic
distribution of electron densities is a crystal lattice and x-rays are well suited
for scattering by crystal lattices as the wavelength of x-rays (∼ 1˚A) is of the
same order as crystal spacings. The condition for constructive interference by
waves scattered in a crystal lattice is given by Bragg’s equation
mλ = 2d sin θ (3.1)
where m is the order of the reflection and an integer, d is the distance between
scattering planes and θ is the angle between the scattering plane and the inci-
dent beam. The formula is easily deduced using geometry and the condition
that the difference in path lengths of the waves must be an integral number
of wavelengths, see Fig. 3.1. When performing x-ray diffraction (XRD) on
a single crystal several different directions and angles of the incoming beam
will lead to constructive interference, see Fig. 3.2. In x-ray powder diffraction
the sample is a large number of small crystals oriented randomly. Thus, all
possible orientations of the crystals are irradiated by a single beam and only
the angle need be altered. An x-ray powder diffraction spectrum is therefore
a plot of reflected intensity versus angle, see Fig. 3.6 on page 24. All mate-
rials have a characteristic x-ray powder spectrum as the peak positions and
intensities depend on the structure of the material. Thus, XRD is an essential
tool for identification and purity control of samples. In addition information
19
32. 3. Experimental Methods
Figure 3.1: Geometric derivation of Bragg’s equation. The difference in path lengths
of waves scattered by different planes is marked in blue.
Figure 3.2: Scattering planes given by their Miller indices (hkl).
on crystallite size and other structure parameters may be obtained from an
XRD pattern.
3.1.1 X-ray Diffraction Setup
In-house x-ray sources can be produced in an evacuated x-ray tube, where
the principal parts are a W (tungsten) filament and an anode. A current is
run through the filament causing it to heat up and thermionic emission of
electrons takes place. The electrons are accelerated towards the anode by a
high potential and interacts with the atoms in the anode. This interaction
produces mostly heat, but a small percentage (< 1%) of the electron energy is
converted to x-rays producing the spectrum seen in Fig. 3.3. Several materials
are suited for use as an anode, but Cu is the most commonly used. Two
distinct features are visible in this spectrum, the continuous bremsstrahlung,
produced by deceleration of electrons impinging on the anode, and discrete
lines. The discrete lines are produced when an inner electron is removed
completely by ionization followed by the transition of an electron from the L
or M shell to the vacancy in the K shell, see Fig. 3.3. The discrete lines are
by far the most intense and for a Cu anode the ratio Kα1 :Kα2 :Kβ is 10:5:2.
A Ni foil filter is used to attenuate the Cu Kβ radiation, so that only the
Kα1 and Kα2 radiations are used in the x-ray diffraction scattering. With two
20
33. X-ray Powder Diffraction
Figure 3.3: Left: The spectrum produced in an x-ray tube with a Cu anode showing
the continuous bremsstrahlung radiation and the discrete lines termed Kα and Kβ
originating from electron transitions between inner shells. Right: Schematic of
the most common transitions for the K spectrum in Cu. Adapted from [17].
wavelengths impinging on the sample, two scattering peaks are seen in the
spectrum with a spacing that increases as the angle increases. The two peaks
can be distinguished when 2θ 50◦ as seen in the diffraction pattern of a Si
standard in Fig. 3.6.
Instrumentation used
All x-ray powder spectra described in this thesis were recorded on a Philips
PW 1050/25 goniometer with a PW 1965/60 proportional detector, see Fig.
3.4, using Ni-filtered Cu K radiation.
3.1.2 Crystallite Size Determination
The width of an XRD peak is inversely proportional to the size of the effective
length of coherent diffraction in the particles studied. It is not correct to think
of this length as the particle size as the particles may contain several domains,
termed crystallites, having different orientations.
The size of the crystallites is given by a formula known as the Scherrer
formula, which states that the mean size of the crystallites, D, is given by [19]
D =
kλ
β cos θ
(3.2)
where k is the Scherrer constant, λ is the wavelength of the radiation,
21
34. 3. Experimental Methods
Figure 3.4: Left: In-house x-ray powder diffraction setup. A) X-ray tube housing.
B) X-ray radiation protection shield assembly containing the specimen holder. C)
Proportional detector. Right: Scattering geometry [18].
Figure 3.5: Simplified view of
the effects of isotropic and
anisotropic strain on a scat-
tering peak [17].
β is the full width at half maximum (FWHM)
of the peak in radians corrected for instrument
broadening and θ is the scattering angle.
In the original article the Scherrer constant
was found to be 0.93. Since then a myriad of
values for this constant have been calculated
for different crystallite shapes, lattice indices
and size distributions. With D defined as the
effective length in the direction perpendicu-
lar to the scattering plane it has been shown
that a value of k in the neighborhood of 0.9 is
found [20]. When using the Scherrer formula
one should be aware of factors other than the
size of the crystallites which may cause broad-
ening of peaks. Isotropic strain will cause the
peaks to be shifted, whereas anisotropic strain
will lead to a broadening of peaks as seen in
Fig. 3.5. However, strain leads to a broad-
ening with a different dependence on θ than
crystallite size broadening, which makes it pos-
sible to differentiate between the two when a
spectrum with several peaks is available. The
broadening of a peak caused by stress is given
22
35. X-ray Powder Diffraction
by [17]
β = 4 tan θ (3.3)
where β is the broadening in radians (again corrected for instrumental broad-
ening) and is the residual strain.
It should also be remembered that the radiation produced by an x-ray
tube is not monochromatic. In an x-ray tube setup using Cu Kα radiation,
scattering of both Kα1 and Kα2 will occur. As the energies of Kα1 and Kα2
are nearly the same, the scattering peaks of the two will be very close to each
other in a spectrum. If the spectrum shows broad lines, the two peaks will
not be discernible as such, and one may mistakenly fit as a single peak what
should instead be fitted as two peaks.
3.1.3 Rietveld Refinement and Quantitative Phase Analysis
Using Rietveld refinement one may extract quantitative information about the
different phases in a diffraction pattern. Rietveld refinement is based on the
minimization of a sum of weighted, squared differences between an observed
and a calculated intensity for each step in a powder pattern. The function
minimized is [17, 21]
R =
j
wj|Ij(o) − Ij(c)|2
(3.4)
where Ij(o) and Ij(c) is the observed and calculated intensity at the jth step,
respectively, and wj is the weight. For Rietveld refinement of an XRD pat-
tern an approximate crystal structure, i.e. space group, lattice constants and
atomic positions, must be known for each phase. The program used in this
study for Rietveld refinement of obtained XRD patterns is Fullprof 2000
[22]. The program is used to determine crystallite size and the weight per-
centages of different phases in produced samples.
Instrument resolution function
A spectrum of a Si standard was obtained and fitted to determine an instru-
ment resolution function, see Fig. 3.6. The individual peaks in the Si pattern
were fitted with a pseudo-Voigt function, which is an approximation to a con-
volution of a Gaussian and a Lorentzian function. Both Kα1 and Kα2 peaks
were fitted with λKα1 = 1.54056 and λKα2 = 1.54439. Half-width constants
characterizing the instrument resolution function were determined and saved
in a separate file for use in later refinements. When an instrument resolution
file is provided, Fullprof 2000 can give an estimate of the mean crystallite
size using the Scherrer formula [21]. Strain and size broadening are discerned
in the refinement.
23
36. 3. Experimental Methods
Figure 3.6: Refinement of XRD pattern of Si standard used to determine the instru-
ment resolution function.
3.2 Transmission Electron Microscopy (TEM)
In an x-ray tube an electron gun is used to produce x-ray radiation by ioniza-
tion of inner electrons in a target anode as described in the previous section.
In TEM the electrons themselves are used as probes. There are many ways
in which electrons may interact with a sample; in TEM the electrons pass-
ing through the sample are used for imaging. Thus, the samples used must
be thin and special preparation of the sample is necessary. TEM has a very
high resolution and can provide morphological information such as the size,
shape and arrangement of particles on nanometer scale. It may also yield
crystallographic information such as lattice planes and atomic scale defects.
3.2.1 Microscope Design
A transmission electron microscope (TEM) is shown in Fig. 3.7. Basically,
an electron source produces a stream of electrons, which is focused onto the
sample by the use of condenser lenses. The image of the sample is magnified
by the objective lens before hitting a CCD camera, which generates the image.
24
37. Transmission Electron Microscopy (TEM)
The entire column is evacuated to a high vacuum to increase the mean free
path of the electrons.
Figure 3.7: Schematic and picture of the Jeol JEM-3000F TEM. Images adapted from
[23] and [24].
The wavelength, λ, of the electrons is given by the de Broglie relationship
λ =
h
p
(3.5)
Thus, the higher the energy of the electrons, the smaller the wavelength. To
find the momentum of the electrons in the potential, V , one must use the
relativistic formula,
p = m0c
qV
m0c2
2 +
qV
m0c2
(3.6)
25
38. 3. Experimental Methods
where m0 is the rest mass and q the charge of the electron. For a potential of
300 kV as used for the TEM images in this study, we find that the electrons
have a wavelength of 1.97 pm. The resolution of a TEM is very high, but still
limited by aberrations in the objective lens.
Instrumentation used
All TEM images presented in this thesis were made by Christian R. H. Bahl
on a Jeol JEM-3000F (UHR) TEM at the Materials Research Department,
Risø National Laboratory. The specifications for this instrument [25] states
a point resolution of 0.17 nm, accelerating voltages of 100 − 300 kV and a
magnification range of ×60 to ×1, 500, 000. Images are captured by a 16
Megapixel CCD camera, which gives images with 4096 × 4096 pixels. TEM
images are shown in Fig. 5.8 on page 61 and Fig. 6.3 on page 68.
Preparation of samples
The powder sample is dispersed in distilled water and/or ethanol and given
ultrasound to break up aggregates. The dilution and ultrasonic treatment is
necessary as the best results are obtained when the particles do not crowd or
gather in aggregates. A drop of the dilute suspension is then placed on a lacey
carbon grid and left to dry. The drying may be facilitated by a piece of filter
paper against the back of the grid. A lacey carbon grid is a finely meshed
metal grid coated with an amorphous carbon film, which forms a pattern of
holes of varying sizes.
Analysis of TEM images
Besides the morphological information immediately available from a TEM im-
age, lattice planes may also be seen when Bragg scattering of the electrons oc-
curs in crystalline samples. Using a program such as Gatan DigitalMicrograph
one can make a Fourier transform of the lattice planes seen in a TEM image
and obtain the lattice spacing. If more than one lattice plane is visible in a
particle it may be possible to identify the orientation of the sample from the
Miller indices (hkl) of the planes.
3.3 M¨ossbauer Spectroscopy
M¨ossbauer spectroscopy is well suited for studying magnetic interactions as it
is a nuclear spectroscopy with an energy resolution high enough to resolve the
hyperfine structure of nuclear levels. M¨ossbauer spectra were obtained of all
samples in this study and the section describes the information that can be
gained from such spectra. Part of this section is taken from a report handed
in during the three-weeks course 10322 Experimental M¨ossbauer Spectroscopy
and based on the notes used in this course [15].
26
39. M¨ossbauer Spectroscopy
The M¨ossbauer effect is the recoil-free resonant absorption of γ-quanta in
solids as described in the following. A transition in a source nucleus from an
excited state to the ground state results in the emission of a γ photon, see
Fig. 3.8. This photon can then be absorbed by another nucleus in the ground
state if the energy of the photon is near the resonance energy of the nucleus.
Figure 3.8: Resonant absorption of a γ-quantum.
During the emission of such radiation a free nucleus recoils. This reduces
the energy, Eγ, of the photon by the recoil energy, ER, which is given by
ER =
E2
γ
2Mc2
(3.7)
where M is the mass of the nucleus. A free nucleus will also be in thermal
motion at a finite temperature, so that the emitted photon is Doppler-shifted.
These phenomena make it impossible to have resonant absorption in free nuclei
as ER is much greater than the natural line width of the transition. If, however,
the nuclei are bound in a crystal lattice a certain fraction, f, of the resonant
absorption events occur without recoil as the mass in Eq. (3.7) is the mass
of the entire crystal. The thermal Doppler-broadening is also very small for
atoms in a solid and resonant absorption becomes possible.
The f-factor
The finite probability that no lattice vibration will occur when a γ-quantum
is emitted is expressed by the so-called f-factor. The f-factor of the source
(absorber) gives the probability that a γ-quantum will be emitted (absorbed)
without phonon interaction. Thus, for resonant absorption we want the f-
factor to be high in both source and absorber. The f-factor is given by
f = exp −
E2
γ
2c2
x2
(3.8)
where x2 is the mean square amplitude of the thermal motion of the atom
in the direction of emission. We see that f decreases as the temperature
increases, because thermal vibrations increase. Using the Debye model for the
phonon spectrum one finds the following approximation for high temperatures,
f exp −
6ERT
kBθ2
D
(T ≥
θD
2
) (3.9)
The Debye temperature, θD, is typically of the order 200 − 400 K. With
θD = 300 K, f = 0.64 at room temperature.
27
40. 3. Experimental Methods
3.3.1 M¨ossbauer Spectroscopy Setup
The experimental setup for producing a M¨ossbauer spectrum consists basically
of a source, an absorber and a detector. The radioactive source is oscillated
back and forth, while the absorber is kept in a fixed position, see Fig. 3.9.
The movement of the source Doppler shifts the energy of the emitted pho-
Figure 3.9: The M¨ossbauer spectroscopy setup.
tons and a sweep of energy around the emission line is created. A M¨ossbauer
spectrum is then recorded in the detector as transmission of the radiation
through the absorber as a function of velocity. Only one isotope in the ab-
sorber contributes to the spectrum as the absorbing atoms must have the same
γ−transition as the source and the spectrum reveals information about this
particular isotope. Several effects influence the nuclear energy levels of the
isotope and the M¨ossbauer spectrum can tell us much about the surroundings
of the absorbing nuclei.
57Fe M¨ossbauer spectroscopy
Only a handful of isotopes are useful for M¨ossbauer spectroscopy and the
most commonly used is 57Fe, which is also the one used in this study. For
57Fe M¨ossbauer spectroscopy the source consists of the radioactive isotope
57Co embedded in a Rh matrix. 57Co decays to an excited state of 57Fe via
electron capture. This excited state then decays to the ground state of 57Fe
mostly via an intermediate state. The transition of interest for M¨ossbauer
spectroscopy is the 14.4 keV transition from the intermediate state as seen in
the decay scheme of 57Co in Fig. 3.10.
Preparation of samples
The powder samples studied were placed in round perspex containers each
marked with the sample name. Such an absorber is characterized by the
thickness factor, t,
t = fanaσ0 (3.10)
where σ0 is the absorption cross-section at full resonance, fa is the f-factor
of the absorber and na is the surface density of the M¨ossbauer isotope in the
absorber. For a thin absorber (t 1) the absorption, and thus the quality
28
41. M¨ossbauer Spectroscopy
Figure 3.10: The decay scheme of 57
Co.
of the spectrum, increases proportionally to t, whereas for t 1 saturation
phenomena occur, so that thin absorbers with t ≈ 1 are preferable. Thick ab-
sorbers result in changes in the relative absorption line intensities as described
in subsection 3.3.3.
3.3.2 Hyperfine Interactions Affecting the M¨ossbauer
Spectrum
The surroundings of the absorbing nuclei perturb and/or split the nuclear en-
ergy levels, which affects the shape of the M¨ossbauer spectrum. Three interac-
tions have a particularly large effect and are important for the understanding
of the M¨ossbauer spectra obtained in this study. They are:
• The isomer shift.
• The electric quadrupole interaction.
• The magnetic hyperfine interaction.
The isomer shift
The s-electrons in an atom have a finite probability of being at the nucleus.
This results in an interaction between the charge distribution of the nucleus
and the density of the s-electrons inside the nucleus. As the density of the
s-electrons is affected by chemical bonding through the outer electrons, the
interaction reflects the valence state and bond formation of the atom. The
influence of the s-electrons on the nucleus results in a shift of energy levels,
29
42. 3. Experimental Methods
δE, which is different for the ground state and excited state because of the
difference in their radii. The shift in transition energy is then given by
∆E = δEE − δEG =
1
10 0
Ze2
(R2
E − R2
G)|ψ(0)|2
(3.11)
where RE and RG are the radii of the nucleus in the excited state and ground
state, respectively, and ψ(0) is the density of s-electrons at the nucleus. If the
atoms in the source and absorber have different chemical environments, ψ(0)
will be different for the two and the difference in transition energy results in
the isomer shift, see Fig. 3.11. In measurements, the isomer shift is not given
Figure 3.11: The energy of the nuclear levels are shifted due to the interaction between
the charge distribution of the nucleus and the density of s-electrons at the nucleus.
A M¨ossbauer spectrum shows a shifted singlet.
relative to the source, but rather relative to a reference material. Thus, the
isomer shift, δ, is given by
δ =
1
10 0
Ze2
(R2
E − R2
G)(|ψA(0)|2
− |ψR(0)|2
) (3.12)
where |ψA(0)|2 and |ψR(0)|2 are the densities of s-electrons at the nuclei of
the absorber and reference material, respectively.
30
43. M¨ossbauer Spectroscopy
Electric quadrupole interaction
If a nucleus has a non-spherical charge distribution, which is true for states
with nuclear spin I > 1
2 , the electric quadrupole moment, eQ, of the nucleus
will be non-zero. This quadrupole moment will interact with the electrostatic
field at the nucleus, which is described by the electrostatic field gradient (EFG)
tensor,
Vxx Vxy Vxz
Vyx Vyy Vyz
Vzx Vzy Vzz
By choosing a suitable set of basis vectors, the off-diagonal terms in the matrix
will be zero and only Vxx, Vyy and Vzz will be non-zero. The EFG can be
thought of as made up by two contributions:
1. A lattice contribution from the charges of the surrounding atoms.
2. A valence electron contribution from the M¨ossbauer atom itself.
The quadrupole interaction energy is given by
EQ =
eQ
4I(2I − 1)
Vzz[3m2
− I(I + 1)] 1 +
η2
3
(3.13)
where m is the nuclear magnetic spin quantum number and η is an asymmetry
parameter defined as η =
Vxx−Vyy
Vzz
. We see that EQ is proportional to m2 and
thus causes a splitting of levels with the same value of the nuclear spin I but
different absolute values of the magnetic quantum number m, see Fig. 3.12.
The observed quadrupole splitting, ∆, is then given by
∆ = EQ ±
3
2
− EQ ±
1
2
=
eQVzz
2
1 +
η2
3
(3.14)
If the nuclear environment has cubic or spherical symmetry, Vxx = Vyy =
Vzz = 0 and the quadrupole splitting is zero.
Magnetic hyperfine interaction
If the spin quantum number, I, of the nucleus is non-zero it will have a mag-
netic dipole moment, µ, given by
µ = gnβnmI (3.15)
where gn is the Land´e g-factor and βn the nuclear magneton. This magnetic
moment will interact with any effective magnetic induction, B, at the nucleus
and the magnetic interaction energy is given by
E = −µ · B = −gnβn(mI · B) (3.16)
We see from this equation, that the magnetic interaction lifts the degeneracy
of the 2I+1 states, see Fig. 3.13. Even though 8 transitions seem possible, only
31
44. 3. Experimental Methods
Figure 3.12: Illustration of the effect of electric quadrupole interaction on the energy
levels of 57
Fe. Also shown is the isomer shift. A doublet is seen in the M¨ossbauer
spectrum of FePO4 at room temperature due to quadrupole splitting.
6 are allowed as a selection rule for magnetic dipole radiation is ∆m = 0, ±1.
The total effective magnetic induction acting on the nucleus is given by
B = Ba + Borb + BD + BC (3.17)
Here, Ba is an applied magnetic induction, Borb is the contribution from the
orbital motion of the electrons, BD is the contribution from the spins of the
electrons outside the nucleus and BC is the contribution of the electron spin-
density at the nucleus arising from s-electrons. BC is known as the Fermi
contact field and is by far the largest contribution to B. For numbers, see the
subsection on crystal field splitting in section 3.3.4.
If the magnetic induction at the nucleus fluctuates as is the case with su-
perparamagnetic relaxation complex spectra may arise as described in section
3.3.4.
32
45. M¨ossbauer Spectroscopy
Figure 3.13: The magnetic hyperfine splitting shown in combination with quadrupole
interaction. Also shown is a M¨ossbauer spectrum of the iron oxide hematite
(α−Fe2O3) at 295 K showing a sextet.
33
46. 3. Experimental Methods
Summary of hyperfine interactions
Interaction Influence on spectrum Physical background
Isomer shift, δ Center of spectrum is shifted
away from 0 mm/s
Difference in s-electron den-
sity compared to reference
material. Gives information
on the valency and bond for-
mation, i.e. on the chemical
bonding.
Quadrupole shift, ∆EQ If no magnetic splitting, two
lines appear. Causes asymme-
try of line positions in mag-
netically split spectrum.
Reflects the symmetry of the
nuclear environment.
Magnetic splitting Six lines appear as all degen-
eracies are lifted.
Proportional to the magnetic
flux density acting on the nu-
cleus.
Calibration
A reference material is used for calibration of a M¨ossbauer spectrometer. The
isomer shift of the known standard absorber used as reference is defined as
zero such that the isomer shifts of other materials are given relative to this.
Further, the known absorption lines of the reference material are used to cal-
culate the number of mm/s per channel. In this way the relationship between
channel number and velocity is calibrated. In 57Fe M¨ossbauer spectroscopy,
the reference material is α-Fe and one mm/s, which is a convenient unit in
M¨ossbauer spectroscopy, equals 4.8 × 10−8 eV. This unit is such that it is
possible to detect the changes in the energy of the excited state caused by the
hyperfine interactions.
Instrumentation used in this report
All M¨ossbauer spectra in this study were obtained at the M¨ossbauer laboratory
at the Department of Physics, DTU. The spectrometers consist of a 57Co/Rh
source and a proportional counter in transmission geometry. The spectra are
stored in multi channel analyzers. Lorentzian line fits of the spectra were
made using the program mfit [26] with doublets and sextets constrained to
have equal widths and intensities. Distributions of hyperfine fields are fitted
using the program distfit developed by C. Wivel and C. A. Oxborrow and
made available at DTU.
3.3.3 Intensities of Lines in Magnetically Split Spectra
The relative intensities in magnetically split spectra are dependent on the
angle between the incoming photons and the magnetic field at the absorbing
nucleus. If the sample is a powder with no preferred orientation and no applied
magnetic field, the relative intensities of the magnetically split lines in an 57Fe
34
47. M¨ossbauer Spectroscopy
M¨ossbauer spectrum is 3:2:1 as seen for hematite in Fig. 3.13. This naturally
comes from the probabilities of transition between ground state and excited
state as given in Table 3.1. The relative intensities are, however, a function of
Transition ∆m Angular dependence Random orientations
+3
2 → +1
2 −1 9
4 (1 + cos2 θ) 3
−3
2 → −1
2 +1 9
4 (1 + cos2 θ) 3
+1
2 → +1
2 0 3 sin2
θ 2
−1
2 → −1
2 0 3 sin2
θ 2
−1
2 → +1
2 +1 3
4 (1 + cos2θ) 1
+1
2 → −1
2 −1 3
4 (1 + cos2θ) 1
Table 3.1: Probabilities of transition for the magnetic dipole interaction of 57
Fe, where
the ground state has nuclear spin Ig = 1
2 and the excited state Ie = 3
2 .
absorber thickness. For thick absorbers the line intensities and areas saturate.
As this happens earlier for the lines with the higher transition probability
their relative line intensities are decreased. Thus, one might obtain lines with
relative intensities (3 − x) : (2 − y) : 1, where x and y are positive.
3.3.4 Interpretation of 57
Fe M¨ossbauer Spectra
The most commonly used isotope for M¨ossbauer spectroscopy is 57Fe, which
has Fe(II) and Fe(III) as its most common valence states. In Fe the electron
configuration is [Ar]4s23d6. Ferrous iron, Fe(II), and ferric iron, Fe(III), have
six 3d and five 3d electrons, respectively, in the valence shell. The differing
number of 3d electrons affects the density of the 3s electrons. The electron
density of 3s electrons at the nucleus of Fe(II) is smaller than that of Fe(III)
as it has more 3d electrons. Thus, |ψA(0)|2 − |ψR(0)|2 in the equation for the
isomer shift,
δ =
1
10 0
Ze2
(R2
E − R2
G)(|ψA(0)|2
− |ψR(0)|2
) (3.18)
will be a larger (negative) quantity for Fe(II) than for Fe(III). However, the
factor (R2
E − R2
G) from Eq. 3.18 is negative, as the nuclear radius of 57Fe in
the excited state is smaller than that of the ground state. Thus, the isomer
shift of Fe(II) is generally larger than that of Fe(III).
Crystal field splitting
From M¨ossbauer spectra one may also determine whether the absorbing ions
are in a so-called high-spin or low-spin state. In an ion the valence electrons
35
48. 3. Experimental Methods
will arrange themselves in the orbitals according to the Pauli exclusion princi-
ple and if the ion is free also according to Hund’s rules. The latter states that
the electrons will occupy the orbitals in such a way that the ground state is
characterized by:
1. The largest total spin angular momentum S. This rule is a result of the
Coulomb repulsion between electrons.
2. The largest total orbital angular momentum L consistent with the first
rule.
3. The value of the total angular momentum, J = L+S, is equal to |L−S|
when the shell is less than half full and to |L+S| when the shell is more
than half full.
The rare earth ions follow Hund’s Rules very well even in complexes as their
4f valence shell lies deep inside the ion and is shielded by the 5s and 5p shells.
The valence shell in Fe, however, is the outermost shell. Thus, the electrons
in the partially filled 3d shell are influenced by the electric field produced by
the neighboring ions. This electric field is known as the crystal field. The
effect of the crystal field on the energy of an orbital depends on the symmetry
of the local environment and the orientation of the orbital.
Fe is commonly octahedrally or tetrahedrally coordinated and the two
symmetries have different effects on the energy of the orbitals. An ion is octa-
hedrally coordinated when the ligands1 are placed at the faces of a cube and
tetrahedrally coordinated when the ligands are placed on alternate corners of
a cube, see Fig. 3.14. By looking at the angular part of the 3d electron wave-
Figure 3.14: Left: Octahedral coordination. Right: Tetrahedral coordination. [27]
functions in Fig. 3.15, we see that they are not spherically symmetric; only
s orbitals have this property. The radial part of the wavefunctions is ignored
as it is independent of direction. As stated above, the effect of the crystal
1
An ion, a molecule, or a molecular group that binds to another chemical entity to form
a larger complex.
36
49. M¨ossbauer Spectroscopy
Figure 3.15: Angular part of 3d orbitals [27]. The notations eg and t2g refer to the two
sets of orbitals into which the five orbitals are split by an octahedral or tetrahedral
environment as described in the text.
field depends on the symmetry of the point charges surrounding the orbitals.
In crystal field theory this is described by group theory, where the symmetry
operations and elements of the local environment are given mathematically.
In this context, the octahedron belongs to the group Oh and the tetrahedron
to the group Td [28]. The splitting of the orbitals caused by the symmetry
of the environment is then found by applying the symmetry operations of the
group to the orbitals. Unless the symmetry is spherical this will cause the
otherwise degenerate orbitals to be split into sets depending on the effect of
the symmetry operations on the individual orbitals. In this way, one finds
that the five d orbitals are split into two irreducible sets by the Oh symmetry.
These sets are the triply degenerate set T2g and the doubly degenerate set
Eg. The effect of the Td symmetry on the splitting of the orbitals is the same,
but the irreducible sets are called T2 and E. The subscript g refers to the
d orbitals being symmetric to inversion, but this has no meaning in an en-
vironment that is not centrosymmetric such as the tetrahedral coordination.
Now we know that the octahedral and tetrahedral environments cause the
same splitting of the orbitals, but what of the relative energies? We see in
Fig. 3.15 that the e orbitals extend along the axes, while the t2 orbitals lie in
between the axes. Thus, in the octahedral environment the eg orbitals have a
higher energy than the t2g orbitals as they get closer to and are repelled more
37
50. 3. Experimental Methods
by the charge density of the ligands located on the axes, see Fig. 3.16. The
Figure 3.16: eg and t2g orbitals in octahedral coordination. [29].
reverse is true for a tetrahedral environment and the energy splitting of the
orbitals in the two symmetries is illustrated in Fig. 3.17. The energy splitting
Figure 3.17: The relative energies of the e and t2 orbitals due to the splitting of the
d orbitals by octahedral and tetrahedral environments [28].
between the eg orbitals and the t2g orbitals is called the crystal field splitting
energy and is denoted ∆o or ∆t for octahedral and tetrahedral coordination,
respectively. The splitting energy depends on the identity of the metal ion,
the charge on this ion, and the nature of the ligands coordinated to the metal
ion. If ∆o is large compared to the pairing energy of the electrons Hund’s rules
need to be modified as the t2g orbitals are filled before the eg orbitals and we
have a low-spin state. The high-spin state occurs when the splitting energy
38
51. M¨ossbauer Spectroscopy
is smaller that the pairing energy, see Fig. 3.18. Tetrahedral complexes are
Figure 3.18: High-spin and low-spin states of Fe(II) and Fe(III).
almost always high-spin as ∆t is small.
The spin state of the Fe ion effects all the hyperfine interactions. Typical
values of the isomer shift and quadrupole splitting are given in Table 3.2.
The quadrupole splitting of Fe(II) high-spin and Fe(III) low-spin states are
S δ mm
s ∆ mm
s Temp. dep. of ∆
Fe(II)
low-spin 0 −0.1 − 0.4 0.0 − 2.0 no
Fe(III)
low-spin 1
2 −0.1 − 0.4 0.0 − 3.0 yes
Fe(II)
high-spin 2 0.9 − 1.4 0.0 − 4.5 yes
Fe(III)
high-spin 5
2 0.2 − 0.6 0.0 − 2.0 no
Table 3.2: Typical values of the isomer shift (δ) and electric quadrupole splitting (∆)
of Fe(II) and Fe(III) in the high-spin and low-spin states.
temperature-dependent as a spin transition may occur if the electron(s) are
thermally excited. Further, the magnetic hyperfine field will depend on the
valence and spin state of the Fe ions. The Fermi contact field, see Eq. 3.17 is
roughly proportional to the total spin of the ions and it is not surprising that
Fe(III) in the high-spin state has the largest magnetic hyperfine field of the
order 50 − 60 T. The magnetic hyperfine field of Fe(II) is generally smaller
than that of Fe(III) because of the contribution from the orbital field Borb,
which is opposite in direction to the Fermi contact field. The typical value of
the magnetic hyperfine field for Fe(II) is 10−40 T. In Fe(III) the orbital field
is zero due to quenching of the orbital angular moment.
Superparamagnetic relaxation
Thermal fluctuations of the magnetic moment of a single-domain particle
is known as superparamagnetic relaxation and is described in section 2.5.
Whether the relaxation of the magnetic moment influences the M¨ossbauer
spectrum depends on the relaxation time, τ, compared to the time scale of
39
52. 3. Experimental Methods
M¨ossbauer spectroscopy, which is the Larmor nuclear precession time, τL. The
magnetic moment of a nucleus precesses in the magnetic hyperfine field acting
on it and τL is the time it takes for one precession. We have that
τL =
2π
2µnB
(3.19)
where µn is the magnetic moment of the nucleus and B is the magnetic induc-
tion acting on it. For a fully split sextet to appear in the M¨ossbauer spectrum
the hyperfine field must remain constant for at least one precession. For a
sample of nanoparticles there are three cases to consider:
• τ τL means that the nuclei ’see’ a constant magnetic hyperfine field
and a discrete sextet is seen in the M¨ossbauer spectrum.
• τ τL results in a lack of magnetic splitting as the hyperfine field
changes so rapidly that it averages to zero.
• τ ≈ τL leads to a complex M¨ossbauer spectrum where a singlet or a
doublet is superimposed on a distribution of sextets. This is due to the
spread in particle sizes so that the particles have different relaxation
times.
The time scale of 57Fe M¨ossbauer spectroscopy is about 10−8 − 10−9 s. As
τ is a function of temperature, a temperature series of M¨ossbauer spectra
will show a magnetically split spectrum at a low temperature and a gradual
collapse of the splitting with increasing temperature as seen, for instance, in
Fig. 6.1 on page 66. The temperature below which the superparamagnetic
relaxation is slow compared to the time scale of the experimental technique is
known as the blocking temperature, TB.
3.4 Vibrating Sample Magnetometer (VSM)
The VSM is a versatile instrument widely used for the characterization of
magnetic materials. It is based on Faraday’s law of induction, which states
that a change in magnetic flux density induces an electric field,
× E = −
∂B
∂t
(3.20)
In an inductor coil with cross-sectional area, A, and N number of turns Eq.
3.20 becomes
V = −NA
dB
dt
(3.21)
where V is the induced voltage. The sign in Eqs. 3.20 and 3.21 is given by a
rule known as Lenz’ law, which can be stated as: Nature abhors a change in
flux, i.e. when an electric field is induced by a time-varying magnetic field it
will be in such a direction that the flux it produces will oppose the change in
magnetic flux [30].
40
53. Vibrating Sample Magnetometer (VSM)
3.4.1 Vibrating Sample Magnetometer Setup
The principle of the VSM is illustrated in Fig. 3.19. The sample to be investi-
Figure 3.19: Left: Sketch of the first VSM built in 1955 illustrating the basic
setup. (1) loud-speaker transducer, (2) conical paper cup support, (3) drinking
straw/sample tail, (4) reference sample, (5) sample, (6) reference coils, (7) sample
coils, (8) magnet poles, (9) metal container [31, 32]. Right: The LakeShore 7407
VSM used in this work.
gated is placed in a sample holder, which is then mounted on a fiber glass rod
known as the sample tail (3). The sample tail is attached to the VSM drive
(1) and vibrated sinusoidally and perpendicularly to a uniform magnetizing
field. Between the sample and the magnet poles (8) are placed pick-up coils
(7). The oscillating magnetic field of the vibrating sample induces a voltage
in the pick-up coils. This induced current is proportional to the magnetic
moment of the sample. A second voltage is induced in the reference coils (6)
by a reference sample (4) and used by the lock-in amplifier as described in the
following subsection. The voltage V across the pick-up coils is given by
V = GAωµ cos(ωt) (3.22)
where G is a constant depending on the geometry of the pick-up coils, A
is the amplitude of the sinusoidal vibration, µ is the magnetic moment of
the sample and ω is the angular frequency with which the sample is moved.
41
54. 3. Experimental Methods
For measurements where an applied magnetic field is needed an electromagnet
supplies a uniform magnetic field within the small region containing the sample
and pick-up coils.
Lock-in amplifier
The AC signal is measured using a lock-in amplifier, which provides a DC
output proportional to the AC signal. The term lock-in refers to the fact that
the amplifier is ’locked’ to a reference voltage, which has the same frequency
as the signal. In the VSM setup this reference signal for the lock-in amplifier
is supplied by a reference sample (4) and a reference pick-up coil (6).
In the amplifier a demodulator multiplies the reference signal and the
incoming signal generating an output the mean level of which is the DC output.
When the two inputs multiplied are in phase a positive mean level is obtained,
see Fig. 3.20, whereas a mean level of zero is obtained when the two signals are
completely out of phase. In an experimental setup, the signal will not be noise-
free, but as the noise has no fixed frequency or phase relationship with the
reference it does not result in a change of the mean DC level. Mathematically,
Figure 3.20: Left: An incoming signal in phase with the reference signal will yield
a demodulator output in the form of a sinusoidal wave with twice the reference
frequency and a mean level which is positive. Right: In the case of signal with
a phase that is delayed 90◦
with respect to the reference the demodulator output
is still a sinusoidal wave with twice the reference frequency, but the mean level is
zero [33].
the output voltage, Vout is given by
Vout = A cos(ωt) · B cos(ωt + θ) =
1
2
AB cos θ +
1
2
AB cos(2ωt + θ) (3.23)
where A cos(ωt) is the signal and B cos(ωt + θ) is the reference signal with a
user-adjustable phase-shift θ. If B is kept constant it should be appreciated
that the mean DC level is proportional to the amplitude of the signal, A and
a function of the phase angle, θ, between the signal and reference. The 2ωt
component is removed in a low-pass filter before the output leaves the lock-in
amplifier as a DC signal.
42
55. Vibrating Sample Magnetometer (VSM)
Using this procedure, a high sensitivity is obtained as the measurements
are made insensitive to changes in the vibration amplitude, vibration fre-
quency, small magnetic field instabilities, amplifier gain and amplifier linear-
ity. Using a suitable configuration of the pick-up coils the measurements are
also insensitive to the exact sample position [31]. Indeed, the data sheet for
the LakeShore 7407 VSM used in this study specifies a moment measurement
range of 0.1 × 10−6 emu to 1000 emu.
Calibration
A small standard reference sample is used for calibration of a VSM. Usually,
a Ni sample is used as it has a high and well-known moment, so that a large
signal is obtained for a small sample. Other considerations are its chemical
stability, low saturation field, low cost, the availability in high purity and the
fact that it is only slightly temperature sensitive even at 300 K [32].
Instrumentation used
In this study was used the LakeShore 7407 VSM located at the Department of
Micro and Nanotechnology (MIC), DTU. Assistance with the VSM was kindly
provided by Associate Professor Mikkel Fougt Hansen.
3.4.2 Interpretation of VSM Data Obtained
A multitude of measurements can be made with a VSM. In this study hys-
teresis loops or field sweeps are made. The applied magnetic field is driven to
maximum before starting the measurement of the magnetization. The mea-
surement is started and, in incremental steps, the field is reversed and then
driven to maximum again all the while the magnetization is measured. In this
way the relation between the magnetization and applied field is determined.
If irreversibilities such as domain wall shifting happens in the sample during
the sweep of the magnetic field a hysteresis seen, see Fig. 3.21. In the fig-
ure are seen the parameters most often extracted from a hysteresis loop to
characterize the magnetic properties of the sample. The coercivity or coercive
field, denoted Hc in the figure, is the field at which the magnetization is zero
after having been saturated. The remanence or remanent magnetization, Mr,
is the magnetic moment at zero applied field. Finally, Ms is the saturation
magnetization. If the magnetization does not reach saturation at maximum
field the loop is said to be a minor loop.
A horizontal shift in the hysteresis loop may be seen when an antiferro-
magnetic and ferromagnetic phase can interact via exchange and the sample
is field cooled from above the N´eel temperature but below the Curie tempera-
ture. In this case the ferromagnets magnetization will align with the field and
as the sample is cooled through the N´eel temperature the moments of the an-
tiferromagnet will align with those of the ferromagnet. If the anisotropy of the
antiferromagnet is large it will tend to hold its alignment against the sweeping
43
56. 3. Experimental Methods
Figure 3.21: Hysteresis loop indicating the coercivity, Hc, the remanence, Mr and
the saturation magnetization, Ms. Figure from [27].
field as a hysteresis loop is recorded. Through exchange interaction the spins
of the ferromagnet will be given a preferential direction and are said to be
pinned. To turn the spins away from the easy direction a higher magnetic
field is required than to return the spins to the easy direction. Thus, there
is said to be exchange bias in the system and the hysteresis loop is shifted
horizontally in the opposite direction of the easy direction. If not all spins
rotate to the harder direction there will also be a vertical shift. Exchange
bias has been studied using thin films, but the effect was discovered in fer-
romagnetic Co nanoparticles having an antiferromagnetic shell of CoO [34].
Antiferromagnetic nanoparticles may display both coercivity and loop shifts
as the uncompensated spins couple to the antiferromagnetic core [35].
44
57. Four
Ferrihydrite and CoO
4.1 Ferrihydrite
Ferrihydrite is a poorly crystalline Fe(III) oxyhydroxide1, which typically oc-
curs in nature, see Fig. 4.1, as the result of rapid oxidation of Fe(II). It may
Figure 4.1: Image from [36] with caption: Iron oxide formation in the environment:
ferrihydrite formed by oxidation of Fe2+
in a ferriferous spring in Iceland. (Photo
courtesy: Dr. Liisa Carlson, University of Helsinki).
also occur where inhibitors, be they organics, phosphate or silicate species,
1
An oxyhydroxide contains O and OH groups
45
58. 4. Ferrihydrite and CoO
stabilize ferrihydrite and prevents it from transforming to more stable miner-
als such as hematite. Thus, the typical environments in which ferrihydrite is
found are Fe containing springs, drainage lines, lake oxide precipitates, ground
water and stagnant-water soils and river sediments [36].
The chemical composition of ferrihydrite is not well-understood, but pos-
sibilities given in literature are Fe5HO8 ·4H2O, 5Fe2O3 ·9H2O, Fe6(O4H3),
Fe2O3 · 2FeOOH2.6H2O and Fe4.5(O,OH,H2O)12 [4]. As seen from the
different suggestions part of the trouble lies in determining the water content.
Because ferrihydrite is poorly crystalline an XRD spectrum does not provide
a crystal structure to constrain the chemical formula.
Ferrihydrite comes in several different forms designated according to the
number of peaks in their XRD spectra. The most common are 2-line and
6-line ferrihydrite, see Fig. 4.2. As can be seen from the XRD spectra 6-line
ferrihydrite is the more crystalline, while 2-line ferrihydrite shows an almost
amorphous XRD spectrum. Despite the poorly crystalline structure apparent
Figure 4.2: XRD spectra (Co Kα) of 2-line and 6-line ferrihydrite [36].
in XRD, ferrihydrite is not amorphous and crystal structure is seen in TEM
images, see Fig. 6.3 on page 68.
The structure of ferrihydrite is believed to be at least partly similar to that
of hematite (α−Fe2O3). In hematite layers of edge- and face-sharing FeO6-
octahedra are stacked in the c direction, see Fig. 4.3. The iron to oxygen ratio
is lower in ferrihydrite than in hematite and one suggestion is that ferrihydrite
contains a defect hematite-like component with vacancies in some of the Fe
positions. Whereas the fundamental structure unit in hematite is the FeO6-
octahedron, the fundamental structure unit in ferrihydrite is believed to be the
Fe(O,OH)6-octahedron. Another suggestion is that ferrihydrite is closer to
46
59. Ferrihydrite
Figure 4.3: Left: Idealized model of hematite showing the edge- and face-sharing
octahedra. The Fe3+
ions are located in the center of the octahedra. Where the
octahedra are face-sharing (shading) the centers of the octahedra are closer and
the repulsion between the cations move them off-center. Right: Idealized model
of goethite showing the edge-sharing octahedra linked by corner-sharing. Again the
Fe3+
ions are located in the center of the octahedra. The double lines represent H
bonds. [36].
goethite (α−FeOOH) in structure. In goethite double bands of edge-sharing
FeO3(OH)3 octahedra are linked by corner-sharing to create tunnels, see
Fig. 4.3. It seems that the hematite- or goethite-like structure is confined to
the core of ferrihydrite as results from EXAFS2 and XANES3 spectroscopy
indicate that the surface of ferrihydrite contains a large number of Fe ions
in tetrahedral coordination [4]. Several models for the crystal structure of
ferrihydrite have been suggested, but none has so far been able to provide a
calculated XRD pattern, which has been incontrovertible.
The poorly ordered crystals of ferrihydrite are between 2 and 7 nm in size.
This leads not only to a smearing-out of XRD patterns, but also to a distribu-
tion of quadrupole splittings in the superparamagnetic state and a distribution
of magnetic hyperfine fields in the magnetically ordered state in M¨ossbauer
spectra [37]. The spread in particle size also means that no temperature exists
at which a sample of ferrihydrite orders magnetically rather the transition is
gradual with superparamagnetism and magnetic ordering co-existing over a
wide temperature range. Ferrihydrite may remain superparamagnetic to tem-
peratures as low as 23 K, but at 4.2 K all samples appear as sextets with no
superparamagnetic doublet.
In Fe(III) the orbital angular moment is quenched, so that the magnetic
moment is entirely from the spin of the electron. The electron configuration of
2
Extended x-ray absorption fine-structure
3
X-ray absorption near-edge structure
47
60. 4. Ferrihydrite and CoO
Fe is [Ar]4s23d6, so Fe(III) has five unpaired electrons in the high-spin state.
The magnetic moment, µ, of an ion given by
µ = (L + gS)µB (4.1)
where L is the orbital angular momentum, S = s(s + 1) is the spin angular
momentum and g = 2.00232 is the Land´e factor. Thus, with s = 5
2 we find
theoretically a magnetic moment of 5.92µB per Fe(III) ion.
By examining M¨ossbauer spectra of ferrihydrite at 4.2 K in applied fields
up to 9 T, it has been determined that 6-line ferrihydrite is antiferromagnetic
[38]. The N´eel temperature as determined by neutron scattering is 330(20)
K [39].
4.1.1 Ferrihydrite in Living Organisms
Ferrihydrite is part of the core of a protein called ferritin, which acts as an iron-
storage in living organisms. Ferritin is a hollow sphere inside which ferrihydrite
is attached to the inner walls. In humans, ferritin is primarily found in the
liver, spleen, and bone marrow, but a small amount is also found in the blood.
A test of the amount of ferritin in a blood sample is used as an indication of
the amount of iron stored in the body. Ferritin acts to contain iron so that it
does not react with other molecules, it acts as a buffer against iron deficiency
and as a means to release the iron in a controlled fashion. The body needs
iron in the Fe(II) oxidation state, but the Fe in ferrihydrite is in the Fe(III)
oxidation state. A reduction agent is used to change Fe(III) into Fe(II), before
it leaves the protein via a 3-fold channel, see Fig. 4.4.
4.2 CoO
The transition metal oxide CoO is antiferromagnetic with a bulk N´eel temper-
ature of TN 293 K. Neutron scattering [41] and VSM [42] measurements on
CoO nanoparticles find that their N´eel temperature is close to the bulk value.
The size of the nanoparticles in the two studies is 20 and 18 nm, respectively.
In the paramagnetic phase the crystal structure of CoO is simple cubic,
see Fig. 4.5. The transition to the ordered antiferromagnetic state is coupled
with a large tetragonal contraction along the cubic [001] direction, i.e. the c
axis is shortened so that c/a < 1. A smaller deformation along the cubic [111]
direction was inferred from a high-resolution synchrotron powder diffraction
study and found to scale with the tetragonal distortion [43]. This latter distor-
tion is controversial as the magnetic ordering would be coupled with a cubic-
to-monoclinic symmetry breaking making the paramagnetic-antiferromagnetic
phase transition of first order, which is not seen in other studies [44]. In Fig.
4.6 is shown the relation between the paramagnetic and antiferromagnetic unit
cell of CoO. Also shown in Fig. 4.6 is the magnetic structure. The electron
configuration of Co is [Ar]4s23d7 and that of O is [He]2s22p4. In CoO two of
48
61. CoO
Figure 4.4: Molecular model of ferritin. Magenta subunits are farthest away, light
blue subunits are closest and dark blue subunits are in between. The circles labeled
3-fold and 4-fold refer to hydrophilic and hydrophobic channels, respectively. After
Fe(III) has been reduced to Fe(II) it leaves the protein via a 3-fold channel [40]
Figure 4.5: The cubic, tetragonal and monoclinic Bravais lattices. Adapted from [45].
the electrons are paired with two of the 2p electrons in O. This leaves 3 un-
paired electrons in a high-spin configuration and a magnetic moment arising
purely from the spin of the Co ion would have a magnitude of 3.87µB per Co
ion. A magnetic moment of 3.98(6)µB per Co ion has been determined using
neutron scattering [43]. The larger magnetic moment measured is due to an
incomplete quenching of the orbital magnetic moment.
CoO has a high anisotropy with the value of the first anisotropy constant
calculated to be K1 2.7 × 108 erg/cm3 [46].
49
62. 4. Ferrihydrite and CoO
Figure 4.6: The relationship between the paramagnetic crystal structure and the
magnetically ordered monoclinic structure of CoO. [43].
50
63. The principle of science, the
definition, almost, is the
following: The test of all
knowledge is experiment.
Experiment is the sole judge of
scientific “truth.”
Richard P. Feynman,
Lectures on Physics
Experimental Results and
Conclusions
51
64.
65. Five
Production of Ferrihydrite and CoO
Nanoparticles
In the following the production of samples used in the M¨ossbauer and VSM
studies is described.
5.1 CoO
Two batches of CoO nanoparticles, CoO1 and CoO6, were made using a two-
step procedure. A powder of as prepared Co3O4 was ball-milled to reduce
the size of the particles and then reduced in hydrogen at a temperature of
250◦C. The reverse procedure is more difficult because CoO will transform
into Co3O4 if ball-milled in air. An attempt was made with ball-milling CoO
in an Ar atmosphere, but this unexpectedly lead to a transformation into
Co3O4. This is thought to be due to a lack of proper containment in the
grinding bowl used, so that air was introduced to the sample. As the two-step
procedure of ball-milling with subsequent reduction in H2 was successful in
producing CoO nanoparticles, the milling of CoO in an atmosphere deprived
of oxygen was not pursued further.
An XRD spectrum of as prepared Co3O4 was refined using Fullprof
2000, see Fig. 5.1, and a crystallite size of 27.6 nm was determined. Co3O4
is an antiferromagnet. Its has a bulk TN = 40 K and in 8 nm particles a N´eel
temperature of 30 K has been determined [47].
5.1.1 Fritsch Pulverisette Ball Mill
In the production of CoO nanoparticles a ball mill is used to reduce the size
of the particles. Following is a brief description of the working principle of a
planetary ball mill such as the Fritsch pulverisette 5 planetary mill used in
this study. The mill consists of a number of grinding bowls that rotate around
their axis during operation of the ball mill. These grinding bowls are situated
on a counter-rotating disc. The powder to be ground is put inside a bowl with
53
66. 5. Production of Ferrihydrite and CoO Nanoparticles
Figure 5.1: Refined XRD spectrum of as prepared Co3O4.
a number of grinding balls. When the ball mill is operating the balls will both
grind and crush the powder. The grinding takes place as powder is caught
between the inner wall of the bowl and a ball driven along the inner wall.
Due to the force from the large counter-rotating disc the balls will separate
from the wall, cross the bowl and crush the powder against the inner wall, see
Fig.5.2.
In all operations with the ball mill a WC (tungsten carbide) grinding
bowl and nine WC grinding balls were used. The WC balls have an average
diameter of 19.47 mm and a combined weight of 514.86 g. The choice of bowl
and balls should be considered not only for its suitability in terms of hardness,
but also with the consideration that material from the grinding bowl and balls
are mixed with the powder to be ground. An alternative in this study was
using a stainless steel bowl, but this would contaminate the sample with Fe.
Choosing a bowl and balls of WC, however, contaminates the sample with
Co as this is used as a hardener in the alloy. The analysis of the grinding bowl
provided by Fritsch, see appendix C, states that the actual composition of the
WC grinding bowl is 93.5% WC, 6.0 % Co and 0.5 % TaC (tantal carbide).
54
67. CoO
Figure 5.2: Left: Drawing of a Fritsch pulverisette 5 [48]. The drawing is of a newer
model than the one used in this study. Right: Drawing depicting the grinding and
crushing motion of the balls.
Thus, it cannot be avoided that some Co is introduced into the ball-milled
samples. The Co content of the WC grinding bowls and balls was not known
at the time of production.
In this study the ball mill was always run at 200 rpm.
5.1.2 Step One: Ball-Milling of Co3O4
During initial runs with the ball mill it was found that a continuous run does
not reduce particle size as much as when a cooling off period is introduced
between runs. When run continuously, the bowl heats up and this works
against the size reduction of the particles. To illustrate, the ball mill was run
with 1.0 g of as prepared Co3O4 for a total of 280 minutes with a break every
20 minutes where an XRD spectrum of the grinding material was made. The
breaks were of varying lengths with the shortest being 40 minutes. All the
spectra were made with 2θ = 34◦ −39◦, a stepping of 0.05◦ and a measurement
time of 4 seconds per step. This angle interval covers the most intense peak
of Co3O4 at 2θ = 36.85◦ and a much less intense peak at 2θ = 38.55◦.
After grinding in the ball mill the peaks recorded in the XRD spectra show
broadening. With only one peak it is not possible to discern size and strain
broadening, so even though the peak broadens visibly this broadening cannot
be translated into a crystallite size. The ball mill was then run continuously
for 220 minutes, again with 1.0 g of as prepared Co3O4. Without assuming
that the broadening of the peak at 2θ = 36.85◦ can be directly translated to a
particle size using Scherrer’s formula, we will use it to quantitatively estimate
the effect of ball-milling with and without cooling off periods. The peak is
fitted with a Lorentzian and the FWHM plotted as a function of time, see
Fig. 5.3. The FWHM of the peak at 2θ = 36.85◦ after a total ball-milling
time of 200 min disturbs what would otherwise look like a smooth broadening
55
68. 5. Production of Ferrihydrite and CoO Nanoparticles
Figure 5.3: Broadening of Co3O4 XRD peak after ball-milling with and without
cooling off periods.
of peaks and may be an artefact possibly due to the short measurement time
of the XRD spectrum. If not, it is not correct to conclude that ball-milling
with cooling off periods leads to smaller particle sizes. Still, in the production
of the samples CoO1 and CoO6 it was decided to continue ball-milling with
cooling off periods.
5.1.3 Step Two: Heating in H2
By heating Co3O4 in H2 it reduces to CoO and then to Co. To determine
if it is possible to reduce to CoO without further reducing to Co 1.0 g of
as prepared Co3O4 was reduced in H2 at various temperatures and heating
times, see Fig. 5.4. It was found that heating at a temperature of 250◦C for
45 minutes reduced the as prepared Co3O4 to CoO.
It was initially thought that the 45 minutes at 250◦C in a H2 flow needed to
reduce the sample of as prepared Co3O4 would also be correct for reducing the
ball-milled Co3O4 nanoparticles to CoO. However, after heating a ball-milled
sample of Co3O4 for 45 minutes in H2 at 250◦C, an XRD spectrum showed
that the sample contained a mixture of Co3O4 and CoO. It was decided that
higher temperatures should not be attempted as this might reduce the Co3O4
to Co and instead the Co3O4 nanoparticles were heated for a longer time.
The reason for the extended heating time required to reduce the Co3O4
nanoparticles to CoO nanoparticles could be that it is more difficult for molec-
56
69. CoO
Figure 5.4: XRD of Co3O4 reduced in H2 at the temperatures and times indicated.
The lines indicate the scattering angles for CoO. The successful production of
CoO without traces of Co3O4 or Co is shown in bold.
ular nucleation to take place. When particles transform, molecular nucleation
starts at a site and spreads from there. The smallness of the particles means
that there are fewer sites where molecular nucleation can start. There are
also far more particles and the nucleation must start at a site in each. How-
ever, molecular nucleation is more likely to start at surface defects and as
the nanoparticles have a higher surface to volume ratio, this would further
molecular nucleation.
57
70. 5. Production of Ferrihydrite and CoO Nanoparticles
5.1.4 Production Details for Samples CoO1 and CoO6
CoO1
1.7 g of as prepared Co3O4 was ball-milled with cooling off periods for a total
ball-milling time of 24 hrs and 12 minutes. The details of the ball-milling
times and cooling off periods are given in Table D.1 in appendix D. The
sample CoO1 was made before a timer was installed to ease ball-milling with
cooling off periods. After ball-milling the Co3O4 nanoparticles were heated
for 45 min at 250◦C in H2. As this was not enough to reduce the Co3O4
nanoparticles to CoO, the sample was heated twice for 5 hours at 250◦C in
H2. An XRD spectrum of the sample was refined using Fullprof 2000, see
Fig. 5.5, and the CoO particle size estimate is 11 nm. It was also determined
that the weight percentages of Co3O4 and WC in the sample are 3.3 and
6.6%, respectively. No Co is seen in the XRD spectrum.
Figure 5.5: Refinement of XRD spectrum of sample CoO1.
58
71. CoO
CoO6
2.0 g as prepared Co3O4 was ball-milled with a timer programmed to run the
ball mill for 15 min then stop and wait 45 min before the next run. The ball
mill was run in this manner for 4 days, which comes to a total ball-milling time
of 24 hours. 1.0 g of the ball-milled Co3O4 nanoparticles was heated twice for
5 hours. Again, the XRD spectrum was refined using Fullprof 2000, see Fig.
5.6. A size estimate of 9 nm is found for the CoO nanoparticles and weight
percentages of Co3O4 and WC in the sample are 0.2 and 8.3%, respectively.
No Co is seen in the XRD spectrum.
Figure 5.6: Refinement of XRD spectrum of sample CoO6.
5.1.5 Samples CoO2, CoO3, CoO4 and CoO5
Samples CoO2, CoO3, CoO4 and CoO5 contained amounts of Co visible
in the XRD spectra and were discarded. One sample was heated longer to try
and remove more of the Co3O4, but the reduction went too far. In the case
of the other three samples it was discovered that the high energy ball-milling
59
