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Remanent Magnetisation in
Hemo-Ilmenite
Master Thesis
Morten Sales
Niels Bohr Institute, University of Copenhagen
Helmholtz-Zentrum Berlin für Materialen und Energie
April 27, 2014
Resumé
Den naturlige remanente magnetisering i en krystal fra Sydnorsk hemo-ilmenit
klippe er blevet undersøgt ved brug af neutronspredning for at opnå viden om
prøvens oprindelsessteds bidrag til magnetiske anomalier i Jordens magnetfelt.
Prøven består af en ilmenit vært (paramagnetisk) med exsolution lameller
af hematit (canted antiferromagnetisk), hvilket skaber et stort kontakt område
mellem værten og lamellerne, som er i gennemsnit omkring 2 µm store. Der er
fremsat en hypotese om, at den naturlige remanente magnetisering skyldes lamellar magnetisering i kontaktlagene, og ved brug af polariseret neutrondiffraktion,
har vi målt den gennemsnitlige vinkel, hvorved spinnene i prøven orienterer sig
i forhold til et påtrykt magnetfelt. En tæt-på-90◦ vinkel antyder, at de cantede antiferromagnetiske momenter er årsagen til den remanente magnetisering,
hvorimod en tæt-på-0◦ vinkel vil betyde, at forklaringen er at finde i lamellar magnetisme. Vores resultater viser en vinkel på ∼61◦ , hvilket antyder, at
forklaringen er en kombination af begge eller flere bidrag.
Yderligere undersøgelser af prøven inkluderer Néel faseovergangen, hvor overgangstemperaturen blev fundet til at være ∼41.3 K. Den kritiske eksponent, β,
blev fundet til at være 0.22, hvilket antyder, at systemet ikke kan modeleres
udelukkende ved brug af enten en 3D Heisenberg eller en Ising model.
3
Abstract
The natural remanent magnetisation of a crystal of South Norwegian hemoilmenite rock has been studied using neutron scattering to obtain knowledge on
the sample’s origin-deposit’s contribution to magnetic anomalies of the Earth’s
magnetic field.
The sample consists of an ilmenite host (paramagnetic) with exsolution lamellae of hematite (canted antiferromagnetic), thereby having a large area of contact
layer between the bulk and the lamellae, which have an average size of around
2 µm. The natural remanent magnetisation has been hypothesised to be caused
by lamellar magnetism in the contact layers, and using polarised neutron diffraction we have measured the average angle by which the spins in the sample align
with an external magnetic field. A close-to-90◦ angle would suggest canted antiferromagnetic moments to be the origin of the remanent magnetisation, whereas
an angle of 0◦ would imply that the explanation is to be found in lamellar magnetism. Our results show an angle of ∼61◦ , which suggests that a combination
of both or more explanations are needed.
Further investigations of the sample include the Néel transition, where the
transition temperature was found to be ∼41.3 K. The critical exponent, β, was
found to be 0.22, which suggest that the system cannot be modelled using only
either a 3D Heisenberg or an Ising model.
4
Contents
1 Introduction
7
2 Magnetism
2.1 Total angular momentum . . . . . . . .
2.2 Hund’s rules . . . . . . . . . . . . . . . .
2.3 Interactions between magnetic ions . . .
2.3.1 Direct exchange interaction . . .
2.3.2 Indirect exchange interaction . .
2.3.3 Dzyaloshinski-Moriya interaction
2.3.4 Zeeman interaction . . . . . . . .
2.4 Ordered magnetic structures . . . . . . .
2.4.1 Ferromagnetism . . . . . . . . .
2.4.2 Antiferromagnetism . . . . . . .
2.4.3 Ferrimagnetism . . . . . . . . . .
2.4.4 Canted Antiferromagnetism . . .
2.4.5 Paramagnetism . . . . . . . . . .
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8
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3 Neutron Scattering Basics
3.1 Neutron properties and production . .
3.2 Scattering . . . . . . . . . . . . . . . .
3.2.1 Wave description of scattering
3.3 Diffraction . . . . . . . . . . . . . . . .
3.3.1 Scattering by several nuclei . .
3.3.2 Bragg’s Law . . . . . . . . . . .
3.3.3 Crystal structures . . . . . . .
3.3.4 Scattering from a crystal . . .
3.3.5 Nanoparticles . . . . . . . . . .
3.4 Magnetic neutron diffraction . . . . .
3.4.1 Polarised Diffraction . . . . . .
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15
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4 The Sample
4.1 Origin . . . . . .
4.2 Sample Content .
4.2.1 Ilmenite .
4.2.2 Hematite
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25
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5
4.3
4.4
4.2.3 Hemo-Ilmenite . . . . . . . . . . .
Previous Studies . . . . . . . . . . . . . .
4.3.1 Impurities . . . . . . . . . . . . . .
4.3.2 Quality of Sample Crystal . . . . .
Hypothesis for Hemo-Ilmenite magnetism
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5 Diffraction Experiment
5.1 Diffraction at IN12, ILL . . . . . . . . . . . . . .
5.1.1 Instrumental set-up . . . . . . . . . . . .
5.1.2 Alignment . . . . . . . . . . . . . . . . . .
5.2 Fitting Procedures . . . . . . . . . . . . . . . . .
5.3 Flipping Ratio Correction . . . . . . . . . . . . .
5.4 Hematite spin orientation in applied field . . . .
5.5 Nano-size effects . . . . . . . . . . . . . . . . . .
5.6 Antiferromagnetic order, transition temperature .
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30
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36
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6 Conclusion
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References
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6
1 Introduction
Approximately one billion years ago a crystal of the minerals hematite (α-Fe2 O3 )
and ilmenite (FeTiO3 ) was being formed in cooling magma in what is now the
south-western part of Norway[32]. During this slow cooling it formed microscopic exsolution structures resembling aligned layers.
This project aims to investigate this crystal – in particular it aims to analyse
the magnetic properties of the natural intergrowth structure of hemo-ilmenite
exsolution. Our investigations have been done with the use of neutron scattering, which is a powerful experimental technique for examining both the atomic
an magnetic structures of a system such as our hemo-ilmenite cystal.
Crystals of naturally intergrown hemo-ilmenite possesses a Natural Remanent
Magnetisation (NRM) of such strength and stability that it cannot be explained
by separately taking the composing minerals into account. The aim of this
project is to provide a basis for an explanation of the NRM of hemo-ilmenite
by examining the two-mineral crystal structure as a whole with the use of neutron scattering. Chapter 2 will give an introduction to the relevant concepts
in magnetism and this will be the foundation for concepts used throughout the
project. Chapter 3 will present a explanation of the basics of neutron scattering
– a investigation technique well suited for examining magnetic properties such
as those of our sample.
After this, a description of our sample (Chapter 4) and its know properties
will follow, including a review of the hypotheses of the cause behind the NRM.
Chapter 5 will deliver a detailed explanation of the data collection and treatment from the polarised neutron diffraction experiment.
The final chapter (Chapter 6) will discuss the findings, present the results and
draw possible conclusions from these[7].
7
2 Magnetism
Magnetism is one of the fundamental forces of nature. A point charge moving
in a electromagnetic field will experience a force, called the Lorentz force. It is
given by:
F = q(E + (v × B))
(2.1)
Where E and B is the electric and magnetic field, respectively. q and v is
the charge and velocity of the particle. The strength of a magnetic field is
proportional to the magnetic moment of the object that is causing the field.
This can e.g. be a current loop with current, I, and vector area a having a
magnetic moment µ of:
µ = Ia
(2.2)
In our case the magnetic moments we investigate are caused by electrons in the
sample. An electron has a moment related to its angular momentum, written
µl , and a moment related to its spin (intrinsic angular momentum), written µs .
These moments are given by:
µl = γe l
(2.3a)
µs = −gµB s
(2.3b)
Where l and s is the angular momentum and spin of the electron and
−e
= 1.7609 · 1011 rad s−1 T−1
2me
g = 2.0023
eh̄
= 9.274 · 10−24 JT−1
µB =
2me
γe =
(2.4a)
(2.4b)
(2.4c)
are constants called the electrons gyromagnetic ratio, the g-factor and the Bohr
magneton.
2.1 Total angular momentum
An atom or ion usually contains several electrons, and to describe its total angular momentum we need to look at how the electrons occupy the different electron
8
energy levels. The Pauli exclusion principle dictates that a quantum state cannot be simultaneously occupied by two identical fermions (spin-1/2 particles).
Consider an electron in orbital 1s2 (having n = 1, l = 0 and therefore ml = 0).
For another electron to occupy the other 1s2 level, the two electrons need to differ in their last principal quantum number, which is the spin quantum number
parameterising the electrons’ spin. We choose the the z-axis as the quantisation
axis, so one electron has to be in the up-state (sz = +1/2) while the other is in
the down-state (sz = −1/2). Here units of h̄ is used, which will be the default
case when nothing else is stated.
Another effect influencing the energy levels of the electrons is the spin-orbit
interaction. This describes the coupling between the electron spin and its motion
around the nucleus and how this gives rise to shifts of the atomic energy levels.
2.2 Hund’s rules
To determine the total angular momentum of a atom or ion, a set of rules called
Hund’s rules can in some cases be used to describe how the electrons fill up the
shells. These rules are driven by Coulumb repulsion, the Pauli principle
and the
P
spin-obit coupling.
The total orbital angular momentum is L = i li , the total
P
spin is S = i si , and the total angular momentum is J = L + S. Hund’s rules
then commands[11, sec. 7.2.4.1]:
Firstly:
Maximise S
Secondly: Maximise L
If less-than-half-filled shell:
Thirdly:
If more-than-half-filled shell:
Minimise J
Maximise J
After the rules have been applied, the effective magnetic moment, µeff , of the
atom/ion can be calculated as:
p
µeff = gJ µB J(J + 1),
(2.5)
where gJ = 32 + S(S+1)−L(L+1)
is the Landé g-factor, which I will not go further
2J(J+1)
into.
The Hund’s rules apply when calculating the magnetic ground state of the
ions in hematite which consists of Fe3+ and O2− . O2− has outer shell structure
2p6 so L = S = J = 0 making Fe3+ the only magnetic ion in hematite. It has
5
an outer shell which is 3d5 , so according to Hund’s rules L =
√ 0, S = /2 and
3+
Fe
J = 5/2. Which gives a effective magnetic moment of µeff = 35µB .
In the case of ilmenite which consists of Fe2+ , O2− and Ti4+ , only the Fe2+
ions have a net magnetic moment with its outer shell structure of 3d6 , since Ti4+
has 3p6 as its outer shell. The magnetic ground state of the 3d ions cannot be
9
calculated using Hund’s rules1 because of an effect called orbital quenching[3,
p. 49], which is the effect of the crystal field from the neighbouring ions being
larger than the effect of spin-orbit coupling in such a way that it is more energetic
favourable to choose a ground state such that L = 0, and therefore J =
√S. So
2+
for ilmenite we get S = 2 and an effective magnetic moment of µFe
=
24µB .
eff
3+
2+
The measured effective magnetic moments of Fe and Fe are 5.9µB and
5.4µB respectively [18, sec. 12-106].
2.3 Interactions between magnetic ions
Now that we know the moments of the magnetic ions, we discuss how they
interact to create magnetic ordered structures. This section is based on chapter
4 in [3].
The magnetic field from one dipole moment can exert a force on another
magnetic dipole and vice versa. This dipole-dipole interaction, however, is so
small that it can be neglected [17, sec. 9.1.2] at temperatures above a few
Kelvin. The interactions we need to consider are exchange interactions, which
are basically the effects that arises from two electrons that Coulomb repels each
other, thus saving energy by being apart.
The Hamiltonian of the exchange interaction between many electrons is described by the Heisenberg model:
X
ĤH = −
Jij Si · Sj .
(2.6)
<i,j>
The summation is over nearest neighbours with non-zero net spin or angular
momentum (magnetic atoms) and Si and Sj are the spins on site i and j. Jij is
the exchange integral for electron i and j. It is defined as:
1
J = (ES − ET ).
(2.7)
2
ES and ET are the energies of the so called single- and triplet states. As mentioned, electrons are fermions, and can therefore not occupy the same quantum
state at the same time. What this means, is that if two electrons have symmetric spatial wave functions then their spins have to be antiparallel resulting
in singlet state (S = 0). On the other hand, if they have antisymmetric spatial
waves functions then their the spins can also be parallel, which gives us a triplet
the state (S = 1). So their arrangement depend on the energies ES and ET and
what configuration is most energetically favourable.
The interactions that governs the magnetic orderings are called exchange interactions. For our case we will consider three different exchange interactions:
direct exchange, indirect exchange and anisotropic exchange.
1
Except for 3d5 and 3d10 where Hund’s rules gives L = 0.
10
2.3.1 Direct exchange interaction
In direct exchange the orbitals with free electrons on neighbouring atoms overlap.
The effect of the direct exchange interaction is a situation, where it is more
favourable to lower the Coulomb repulsion by having the spins point in the
same direction and therefore not overlap due to the Pauli principle. This is,
however, not something that usually controls the magnetic ordering, e.g. as in
our case where the 3d orbitals are too localised to have overlap between the iron
ions.
2.3.2 Indirect exchange interaction
In ionic solids, magnetic ground states can occur when non magnetic ions acts
as mediators between the magnetic ones. In our case of hematite it would be
the p electrons in O2− overlapping with the multiple neighbouring d electrons in
Fe3+ . The d electrons can now – through the oxygen ion – ’hop’ between metal
ions thereby lowering its kinetic energy by becoming more de-localised, if they
are anti-parallel. If the electrons had had parallel spins, this could not happen.
Therefore the anti-parallel configuration is energetically favourable when indirect
exchange interaction rules.
2.3.3 Dzyaloshinski-Moriya interaction
When indirect exchange interaction involves an electron ’hopping’ from the
ground state in one magnetic ion through a non magnetic ion to an excited
state in another magnetic ion, it can favour a so called canting2 of the metal
ion moments, and this is called anisotropic exchange interaction. It was first
described in detail by Dzyaloshinskii[15] and Moriya[26], who described lowsymmetry crystals while taking spin-orbit coupling into account when looking
at indirect exchange interactions. The Hamiltonian of this interaction, which is
also called the Dzyaloshinski-Moriya interaction, is:
X
ĤDM =
D · Si × Sj
(2.8)
<i,j>
This will favour an 90◦ degree angle between the two spins in a plane perpendicular to D. The effect can cause canted anti-ferromagnetism, see section 2.4.4
below.
2
see section 2.4.4
11
2.3.4 Zeeman interaction
Magnetic moments in an applied field will through the Zeeman interaction tend
to align with the field:
X
ĤZ = −gµB
Sjz
(2.9)
j
HZ is the Zeeman Hamiltonian, and the z-direction is here the direction of the
field.
2.4 Ordered magnetic structures
2.4.1 Ferromagnetism
In ferromagnets all the spins are aligned as shown in figure 2.1. This gives a net
magnetic moment in the direction of the spins. Ferromagnetism usually comes
from direct exchange interaction.
Figure 2.1: Ferromagnetic ordering.
2.4.2 Antiferromagnetism
In antiferromagnets the spins are antiparallel to each other as shown in figure 2.2. The gives zero net magnetic moment, since all the spins cancel out.
Antiferromagnetism (AFM) usually comes from indirect exchange interaction.
Antiferromagnets have magnetic unit cells with at least twice the volume of the
structural unit cell.
2.4.3 Ferrimagnetism
Ferrimagnets have spins aligned as antiferromagnets, but they consists of two
ferromagnetic magnetic sublattices with moments of different size as shown in
12
Figure 2.2: Antiferromagnetic ordering. Here shown with the magnetic unit cell.
figure 2.3. The net moment is in this case parallel to one sublattice and antiparallel to the other.
Figure 2.3: Ferrimagnetic ordering.
2.4.4 Canted Antiferromagnetism
Anisotropic exchange interaction can make it favourable for antiferromagnets
not to have directly opposing spins. Instead the spins are tilted slightly in the
same direction as shown in figure 2.4. This is called canted antiferromagnetism
(CAF) and the net moment is perpendicular to the average direction of the
antiparallel aligned spins. CAF is e.g. found in hematite[27] which is of interest
for this thesis.
2.4.5 Paramagnetism
In systems where the moments on different sites do not interact, they will orient themselves randomly, resulting in no net magnetisation as shown in figure
13
Figure 2.4: Canted antiferromagnetic ordering.
2.5. This can also occur in ferromagnetic and antiferromagnetic materials being
heated to a temperature where the spin fluctuations because of thermal energy
are so strong that they do not align. The spins in paramagnets can align in the
presence of an external field.
Figure 2.5: Paramagnet with randomly oriented spins.
14
3 Neutron Scattering Basics
In this chapter a basic introduction to the theory behind using neutrons a a probe
for studying ordered structures such as our hemo-ilmenite sample is introduced.
3.1 Neutron properties and production
The neutron was discovered in 1932 and it is one on the most common particles
in existence. Together with the proton it is what makes up the cores of mostly
all atoms. It has a rest mass of mn = 1.675 · 10−27 kg, it is charge neutral and
it has a magnetic moment of µ = γµN = γeh̄/(2mp ) = −1.913·5.051·10−27 J/T.
Neutrons for scattering can be produced in several ways. They can be produced by radioactive decays in laboratory sources, e.g a Radium-Beryllium
source. However, such a source is not powerful enough to produce the amount of
neutrons needed for most scattering studies, so stronger sources are needed for
these purposes. Experimental reactor sources produce neutrons in high amounts
from controlled nuclear reactions in for example radioactive uranium. These
sources demands a lot of security measures and maintaining and facilities producing neutrons by fission are large with many users and applications. Another
way to produce neutrons at high flux (neutrons/second/cm2 ) is through spallation where protons are accelerated into a dense target wherein highly excited
nuclei are created, subsequently decaying and in the process sending out a burst
of nuclei. Fission and spallation neutron production methods are shown in figure
3.1.
Common for the different neutron production methods described above is
that the resulting neutrons have too high energies and are unsuited for scattering purposes. Therefore a moderation of the neutrons is needed. This is done by
surrounding the neutrons with moderating material such as water or liquid hydrogen. Within this material the neutrons bounce around and are being slowed
down by giving of energy to the atoms in the moderator.
In quantum mechanics particle-wave duality gives a relationship between the
energy, E, and wavelength, λ, of a particle moving with constant velocity. The
15
Figure 3.1: Principle of fission and spallation neutron production. From [36].
so-called de-Broglie wavelength is given by:
λ=
2πh̄
mn v
(3.1)
Where h̄ = 1.0546 · 10−34 Js is the Planck constant and v is the neutron velocity.
The wave number is k = 2π/λ and from this we get the energy (using the
energy-momentum relation E = p2 /(2mn )):
E=
h̄2 k 2
2mn
(3.2)
So by moderating the neutrons to lower energies we get neutrons with larger
wavelengths. When investigating matter on length scales in the Ångstrom and
nanometer range, neutron scattering can be a powerful tool. One of the reasons
for this is that the wavelength of neutrons is of the same order as the length
scales investigated.
3.2 Scattering
Waves interact with matter through scattering, and the basic principle of neutron scattering is that by directing a neutron beam at a sample and measuring
how the neutrons scatter, knowledge of the inner workings of the sample can
be obtained. In basic neutron scattering, the neutrons interact with the nuclei
of the sample through the strong nuclear force, whereas in magnetic neutron
scattering, they interact with the internal magnetic field of the sample. This
field is most often (as in our case) caused by unpaired electrons in the outer
16
shells of atoms in the sample.
The description of neutron scattering in this chapter is based on the description in [17].
The magnitude of an incoming neutron beam can be described by the aforementioned flux, which is the number of neutrons on a given surface area perpendicular to the incoming neutron beam within a given time. The flux is
denoted by Ψ. Neutron scattering works because different structures and materials have different abilities of scattering neutrons. This is quantified by the
neutron scattering cross section which can be visualised as the effective area of
the the scattering nuclei ’seen’ by the incoming neutron. It is given by:
σ=
1
(# of neutrons scattered per second)
Ψ
(3.3)
Knowing the direction of the scattered neutron is essential to neutron scattering
and this is described by the differential cross section:
1 (# of neutrons scattered per sec. into solid angle dΩ)
dσ
=
dΩ
Ψ
dΩ
(3.4)
In inelastic neutron scatting the incoming and the outgoing neutrons have
different energies and it is necessary to describe the change in energy as well.
This is done by the partial differential cross section:
d2 σ
1 (# of neutrons scattered per sec. in dΩ with energies [Ef ; Ef + dEf ])
=
dΩdEf
Ψ
dΩdEf
(3.5)
Where Ef is the energy of the outgoing scattered neutron.
3.2.1 Wave description of scattering
The incoming neutron can be described as a plane wave with wavevector ki :
ψi (r) = exp(iki · r)
(3.6)
From eq. (3.1) we get the speed of the plane wave, and from this we get the flux
of incoming neutrons:
h̄ki
mn
Ψi = |ψi |2 v = v
v=
(3.7)
(3.8)
17
The plane wave hits a fixed nucleus at the origin and is scattered as a spherically
symmetrical wave:
−b
exp(ikf r)
(3.9)
r
Where b is the the scattering length, which varies from isotope to isotope. The
density of outgoing neutrons is:
2
b
2
(3.10)
|ψf | =
r
ψf (r) =
The number of outgoing neutrons passing through a small area, dA, is v|ψf |2 dA
and a solid angle is given by dΩ = dA/r2 . Now the differential cross section can
be written as (3.11), since the nucleus is assumed to be fixed and therefore the
neutron wave number (and energy) is preserved:
dσ
vb2 dΩ
=
= b2
dΩ
Ψi dΩ
(3.11)
A single nucleus scatters in all directions, and it therefore has the total scattering
cross-section:
σ = 4πb2
(3.12)
3.3 Diffraction
Neutron diffraction deals with elastic scattering, which is when the neutron has
the same energy going in to as coming out of the scattering event.
3.3.1 Scattering by several nuclei
In a real experiment, the samples investigated are composed of many nuclei,
from which the scattered neutron waves interact in ways that can be measured
and used to obtain knowledge on the inner structure of the sample.
Consider two identical nuclei at positions rj and rj 0 , being hit by a incoming
plane wave ψi (r). the outgoing wave is then:
ψi (rj 0 )
ψi (rj )
ψf (r) = −b
exp(ikf |r − rj |) +
exp(ikf |r − rj 0 |)
(3.13)
|r − rj |
|r − rj 0 |
Choosing an origin close to the nuclei, an observer far away compared to nuclei
distance (|r − rj | r) so that the denominators can be considered equal, and
using (3.6), we get:
ψf (r) =
18
−b exp(iki · rj ) exp(ikf |r − rj |) + exp(iki · rj 0 ) exp(ikf |r − rj 0 |)
r
(3.14)
Separating the nuclei co-ordinates into components parallel
and perpendicular to
q
r and using Pythagoras’ theorem, we get |r − rj | = |r − rjk |2 + |rj⊥ |2 , where
the second term in the square root can be left out, since it is insignificant. This
gives us:
kf |r − rj | = kf · (r − rjk )
(3.15)
Where the wavevector kf is parallel to r. The outgoing wave can now be written
as:
ψf (r) =
−b
exp(ikf · rj ) exp(i(ki − kf ) · rj ) + exp(i(ki − kf ) · rj 0 )
r
(3.16)
As in (3.11), we can now calculate the differential cross-section to be:
dσ
= b2 | exp(iq · rj ) + exp(iq · rj 0 )|2 = 2b2 [1 + cos(q · (rj − rj 0 )]
dΩ
(3.17)
Where the scattering vector q is defined as q ≡ ki − kf . It can be seen that for
some values of q the cosine term becomes 1, and differential cross section is then
four times larger than for a single nucleus, see (3.11). For other q-values the
differential cross section completely vanishes. This is the effect of interference
between waves from multiple scatterers. For more than just two scatterers the
elastic differential cross-sections is:
2
X
dσ
=
bj exp(iq · rj )
(3.18)
dΩ j
3.3.2 Bragg’s Law
Consider a number of nuclei arranged in a periodic structure as in figure 3.2.
A plane wave hits the structure and in a certain direction there is constructive
interference. The distance between the atomic layers is d. The wavelength of the
neutron wave is λ, and it enters at an angle, θ, with respect to the atomic planes.
The neutron ray taking the path where it is being reflected by the second layer,
has a travelled a distance 2d sin θ longer. If this path length difference is equal
to a whole number, n, of wavelengths, constructive interference will occur[5][24].
This is the principle of Bragg’s Law:
nλ = 2d sin θ
(3.19)
What this means is that by measuring a sample consisting of periodically arranged scatterers and varying θ and λ you can find the values that result in
constructive interference, and from this calculate the characteristic distances, d,
in the sample.
19
d
θ
d sinθ
Figure 3.2: Bragg’s Law. nλ = 2d sin θ
3.3.3 Crystal structures
In a single crystal the nuclei are arranged periodically in a crystal lattice, meaning that the crystal can be viewed as a stack of identical unit cells. The simplest
form of arrangement is a Bravais lattice, where only one atom is situated in each
unit cell.
Such a lattice is pictured in figure 3.3. The red coloured unit cell is expanded
by the three lattice vectors: a, b and c, and its volume is V = a · b × c. α is
the angle between b and c, β is between a and c, and γ is between a and b.
Three-dimensional crystal structures can be categorised into different space
groups according to the symmetry properties of the unit cell. Symmetries can
be through e.g. rotation, glide planes, mirroring and screw axes. In three
dimensions there is a total of 230 unique combinations. The notation used to
classify a given space group is most commonly the international short symbol
notation found in [14].
It is often useful to use work in reciprocal space, where the reciprocal unit cell
is spanned by the reciprocal lattice vectors a∗ , b∗ and c∗ :
a∗ = 2π
b×c
,
a · (b × c)
b∗ = 2π
c×a
,
a · (b × c)
c∗ = 2π
a×b
a · (b × c)
(3.20)
Each point in reciprocal space given by τ hkl = ha∗ + kb∗ + lc∗ , where h, k
and l are integers; denoted Miller indices, corresponds to a plane of nuclei in
real space. The characteristic distances, d, from equation (3.19) can then be
calculated as: dhkl = 2π/τ hkl .
20
Figure 3.3: Bravais lattice. The red volume is the unit cell, containing only one
atom (the atoms in the other 7 corners belong to neighbouring unit
cells.)
Reflections from a crystal can be labelled by its Miller indices, (hkl).
3.3.4 Scattering from a crystal
For a crystal equation (3.18) can be written as:
2
2
X
X
dσ
2
=
bi exp[iq · (rj + ∆i )] = |FN (q)| exp(iq · rj )
dΩ j
i,j
(3.21)
Where rj points to the unit cell and ∆i points to the
P nuclei inside the cell. FN
is the nuclear structure factor defined as: FN (q) ≡ i bi exp(iq · ∆i ).
It can be shown that equation (3.21) can be written as:
X
(2π)3
dσ
=N
|FN (q)|2
δ(q − τ )
dΩ
V
τ
(3.22)
if assuming an infinite lattice, but still counting the number of unit cells, N . The
delta function leads to the so called Laue condition: q = τ , which corresponds
to Bragg’s Law.
3.3.5 Nanoparticles
For small systems you can not assume an infinite lattice and a distribution of
q-values around q = τ will now result in scattering. This broadening effect happens in small nano-sized crystals, and in our case we use a Lorentzian profile1
1
The Lorenzian distribution will be convoluted with a Gaussion profile, that is coming from
the instrumental resolution
21
to describe it. The Full Width Half Maximum (FWHM) (of the Lorentzian
distribution) as a function of scattering angle, 2θ, is given by the Scherrer
formula[33][28] (in radians):
B(2θ) =
Kλ
L cos θ
(3.23)
Where L is the relevant size of the small sized particle. K is a dimensionless
shape factor, that depends on the shape of the nano structure. It has a value
close to 1, and when the shape is not precisely known, it is typically set to
around 0.9 [9], which is close the the value for monodisperse spherical particles.
3.4 Magnetic neutron diffraction
In the previous section the scatterers were referred to as the nuclei, however
neutrons can diffract from magnetically structured materials as well, since neutrons have magnetic moments[2]. Neutrons interact with a magnetic scatterer
through the Zeeman interaction (see eq. (2.9)). We will here only consider
elastic magnetic neutron scattering. The elastic differential magnetic scattering
cross-section is given as (with unpolarised neutrons):
dσ γre 2
=
|q̂ × (hM(q)i × q̂)|2
dΩ magn.
2µB
(3.24)
Here M(q) is the Fourier transform2 of the magnetic moment distribution, and
re = 2.818 · 10−15 m is the classical electron radius. q̂ is a unit vector in the
direction of q, and the brackets, h i, denotes thermal average. Here it should be
noted that only the part of M(q) that is perpendicular to q contributes to the
scattering, and one normally defines the magnetic interaction vector as:
M⊥ (q) ≡ q̂ × (hM(q)i × q̂)
(3.25)
3.4.1 Polarised Diffraction
In polarised neutron scattering the initial spin orientation of the neutrons is
known. If polarisation analysis[25] is used, the outgoing spin orientation is
measured as well. As mentioned, the neutron has a magnetic moment. This
is caused by the fact that the neutron has a spin. As was the case for the
electron, the neutron is a spin-1/2 particle (a fermion). Considering the spin
projected along the z-axis a spin-1/2 can only have two values: either sz = 1/2
or sz = −1/2. These two spin states are called the spin-up state (|↑i) and the
2
The Fourier transform can be said to relate the direct and reciprocal spaces to each other.
22
spin-down state (|↓i). The incoming polarisation vector, Pi , has the length ±1
when the beam is fully polarised, and 0 when the beam is unpolarised.
Without considering the nuclear spin, the total cross section (both magnetic
and structural) is (from [31, sec. 2]):
σ = FN (q)FN∗ (q) + M⊥ (q) · M∗⊥ (q) + Pi · [M⊥ (q)FN∗ (q) + M∗⊥ (q)FN (q)]
+ iPi · [M∗⊥ (q) × M⊥ (q)]
(3.26)
The last term is the chiral term, which is zero in our case and will not be
considered in this context. The first term gives purely nuclear scattering, the
second term gives purely magnetic scattering, and the third term is called the
nuclear-magnetic interference term.
A neutron can before the scattering event be either in the |↑i-state or the
|↓i-state. After the scattering event, it can either be in the same spin state as
before, or have flipped to the other state. This is respectively called Non Spin
Flip (NSF) and Spin Flip (SF) scattering.
It can be shown that the nuclear scattering does not alter the spin state of
the neutron (NSF), meaning that an incoming neutron in the |↓i-state (|↑i) will
after undergoing a nuclear scattering event still be in the |↓i-state (|↑i) state.
The pure magnetic scattering can result in both NSF scattering, where the
neutron keeps its spin state, or in SF scattering, where the spin state is flipped
from one to the other. A magnetic scatterer with moment parallel to Pi will
only give rise to NSF scattering, and moments perpendicular to Pi will only
give rise to SF scattering. This a very important feature in polarised neutron
scattering, where the incoming and outgoing neutron spin orientations can be
measured. It is shown in [34, sec. 9.3]:
h↑| σ̂ · M⊥ |↑i = M⊥z ,
(3.27a)
h↓| σ̂ · M⊥ |↓i = −M⊥z ,
(3.27b)
h↑| σ̂ · M⊥ |↓i = M⊥x − M⊥y
(3.27d)
h↓| σ̂ · M⊥ |↑i = M⊥x + iM⊥y ,
(3.27c)
0 1 , σ ≡ 0 −i and σ ≡
σ̂ is the
Pauli
spin
operator[13,
p.
186]:
σ
≡
x
y
z
1
0
i
0
1 0 . (3.27a) and (3.27b) show NSF scattering with cross-section:
0 −1
2
σNSF = KM⊥z
(3.28)
and (3.27c) and (3.27d) show SF scattering with cross-section (choosing q along
x, and remembering that only magnetic moments perpendicular to q contribute
to the magnetic scattering):
2
σSF = KM⊥y
(3.29)
23
K is a constant.
The nuclear-magnetic interference term is only non-zero, when a reflection has
both magnetic and nuclear scattering.
Consider the scattering geometry in figure 3.4. By measuring both the NSF
and the SF intensities, the angle, θ, by which the spins in the sample are oriented,
can be calculated:
M⊥2 cos2 θ
σ SF
= tan2 θ
=
σ NSF
M⊥2 sin2 θ
(3.30)
z
Pi
M┴
MT,Z
θ
y
MT ,y
q
x
Figure 3.4: Pi is the incoming polarisation vector, q is the scattering vector and
M⊥ is the magnetic
interaction The
vector.scattering
Figure from [6,
sec. 5.3.2].
5.3: Experimental
geometry.
vector
q is
Figure
along the
x - axis, theInincident
polarisation
Pi is along
the z beam
- axis
and
M⊥ is in the
a real life experiment
the polarisation
of the neutron
is not
perfect,
more on
and how
correct
for it can
found
section 5.3.
y-z plane with
anthisangle
θ to
with
respect
tobethe
z in
- axis.
The geometry in our experiment is shown in figure 5.3. The scattering vector
q lies in the horizontal plane and is for simplicity placed along the x - axis.
The polarisation direction Pi of the incident beam is along the positive z
- direction, and M⊥ is in the y-z plane describing an angle θ with respect
to the z - axis. This geometry makes it possible to obtain the components
of the spin directions perpendicular to the scattering vector directly from
the NSF and SF intensities. For a purely magnetic peak the NSF and SF
cross-sections are (from eqs. (5.9) and (5.13)):
24
2
σ N SF = CM⊥,z
= CM⊥2 cos2 (θ),
2
σ SF = CM⊥,y
= CM⊥2 sin2 (θ),
(5.14)
(5.15)
where the constant C is the product of the prefactors in (5.9) and (5.13).
The angle θ can be found from the NSF and SF intensities as:
σ SF
sin2 (θ)
=
= tan2 (θ).
N
SF
2
σ
cos (θ)
(5.16)
4 The Sample
This chapter describes our hemo-ilmenite sample – where it came from, what
it is made of, how it has been studied and possibly why it has a remanent
magnetisation.
4.1 Origin
The sample was dug out from the Pramsknuten/Jerneld dike in the SouthWestern part of Norway (location shown in figure 4.1).
It was formed by nature around 932 million years ago [32] under conditions
that has so far not been possible to recreate in the laboratory. This makes our
sample a unique specimen with interesting characteristics to investigate. Figure
4.1d shows the rock from which the sample was carved[30] and figure 4.2a shows
the sample crystal itself.
Figure 4.2c shows a transmission electron microscopy image of a rock of similar composition to ours, with a clear display of exsolution lamellae of hematite
crystal inside bulk ilmenite. The lamellae are shaped like flattened ellipsoids and
are arranged with their major axes aligned perpendicular to the c-axis. Different
size populations of lamellae seem to exist. A further investigation shows that
not only are the major axes aligned but the lamellae are also flattened in parallel planes, causing a high anisotropy of the lamellae orientation (in addition
to the the magnetic anisotropy that restricts the hematite spins to the plane
perpendicular to the c-axis).
The area from where the sample was taken has been investigated for magnetic anomalies to the earth’s magnetic field[22] as shown in figure 4.3. The
red X marks the position of the Pramsknuten dike, and it can be seen that the
area has a negative remanent magnetisation, so that net field measured is below the average. When the sample was formed nearly a billion years ago, the
earth’s magnetic poles were pointing in the opposite direction of what they are
today, and the sample’s net magnetic moment ’froze’ when the sample solidified
thus remembering the earth field at the time, giving the sample its remanent
magnetisation.
25
(a) Southern Norway
(b) South Rogaland
(c) Håland-Helleren massif
(d) Hemo-Ilmenite rock with sawed surface
Figure 4.1: The sample origins from the southern part of Norway (a), where it
was dug out from the Pramsknuten/Jerneld dike (c) in the HålandHelleren massif anorthosite (b). Several crystals were found, with
ours being from the fifth chip (d). (b) and (c) is from [10] and (d) is
from [30].
26
(a) Our hemoilmenite sample crystal.
896
(b) Microscopy of sample
surface.
Figure 4.2: Picture and microscopy of our sample crystal and a transmission
electron microscopy of a similar rock (from [20], white scale bar
is 100 nm). The exact size of the lamellae in our sample is not
know, though from investigations of similar samples and from the
microscopy images of the sample surface, the thickness of the lamellae is estimated be from a few nanometers all the way up to the
micrometer range.
Figure 4.3: Aeromagnetic map of South Rogaland. From [23]. The X marks the
spot where our sample was found.
27
S. A. McEnroe et al.
2.2.3 High-resolution elemental maps
coherent with the host, shown by the dark patches of strain contrast surrounding the lamellae and the absence of dislocations at the
precipitate–host interface. The ∼4 nm thick ilmenite precipitates are
coherent and relatively straight, but several do appear to be kinked
(see the magnified region of Fig. 6). In regions where no discrete
lamellae are visible the haematite has developed a very fine-scale
mottling, indicative of chemical heterogeneity on a unit-cell length
scale.
Immediately adjacent to the large haematite lamellae, the ilmenite
host appears to be homogeneous (Fig. 5a). The ilmenite develops a
finely mottled texture with increasing distance from the haematite
lamellae (Fig. 5b), which in turn develops into discrete fine-scale
precipitates of haematite (Fig. 5c). A more detailed image is shown
in Fig. 7. The lower part of Fig. 7 consists of well-defined discrete haematite precipitates with thicknesses of the order 10 nm
(0.01 µm) surrounded by homogeneous ilmenite host. The precipitates are fully coherent and oriented parallel to (001) of the ilmenite
host. The upper part of Fig. 7 illustrates the gradual transition from
discrete lamellae with thicknesses of the order 4 nm to fine-scale
mottling.
Figure 7. Bright-field TEM image of the ilmenite host containing fine-scale
haematite precipitates. Scale bar = 100 nm.
(c) TEM of similar system.
The TEM observations reveal several generations of exsolutions
with sizes down to the nanoscale level. This suggests that the microprobe measurements are overlap analyses of ilmenite host or
coarse haematite lamellae with corresponding nanoscale exsolutions. Such overlap analyses can be avoided when employing a
TEM equipped with an energy-dispersive X-ray detector. To obtain such pure quantitative analyses of the oxides, we thus performed
measurements with a ThermoNORAN spectrometer, attached to the
PHILIPS CM20 FEG scanning TEM at the Bayerisches Geoinstitut,
University of Bayreuth. The EDX spectrometer is equipped with an
ultrathin NORVAR window and a germanium detector. This analytical configuration allows the simultaneous detection of k lines of
light (e.g. oxygen) and heavy elements.
To quantify the mineralogical compositions, we calibrated the
kX/Fe factors for the elements X contained, and performed an absorption correction (Langenhorst et al. 1995). The kX/Fe factors were
determined according to the parameterless correction method (Van
Cappellen 1990), using homogeneous, well-characterized standards
such as garnet, pyroxene and perovskite. The relative errors in k
factors are 1–3 per cent and represent systematic errors in the quantification. Relative statistical errors expressed as a 1σ deviation are
0.5–1 per cent for major elements (O, Fe, Ti in ilmenite) and 2–
3 per cent for minor elements (Mg in ilmenite, Ti in haematite).
Trace elements with a concentration smaller than 1 at per cent are
3.2 TEM–EDX and TEM–EELS operating conditions
Compositional data were collected on a Cameca SX-50 electron microprobe at the Bayerisches Geoinstitut set at an accelerating potential of 20 keV, a beam current of 15 nA, and a typical beam diameter
of 1 µm. Counting times of 30 s per element were used. Corrections for differential matrix effects were done using the Cameca online PAP correction routine. The analytical precision is estimated at
±0.1 weight per cent for oxide components present at the 1 weight
per cent level. Analytical precision on typical values of 0.5 weight
per cent V2 O3 and Cr2 O3 is estimated to be ±0.06 weight per cent
at the 95 per cent confidence level. In addition there is believed to
be a systematic overestimate of V2 O3 in ilmenite of ∼+0.1 weight
per cent caused by Ti Kβ-V Kα interference.
3.1 Electron microprobe operating conditions
3 M I N E R A L A N A LY S E S
precipitates within the ilmenite host from Fig. 7. Compositional heterogeneities on a length scale of 2 nm and below are clearly visible
in Fig. 8(b). To put this in perspective, the unit cell parameter of
ilmenite perpendicular to (001) is 1.4 nm (corresponding to just
six individual cation layers). The presence of compositional heterogeneities on this scale has been shown to have a profound impact
on the magnetic properties of the solid solution (Harrison & Becker
2001; Robinson et al. 2002a).
4.2 Sample Content
The sample weighs 1.977 g[30], has a size of about 8×8×12 mm3 and is composed
of the two minerals hematite and ilmenite (plus some impurities).
4.2.1 Ilmenite
Ilmenite is a mineral composed of iron, titanium and oxygen: FeTiO3 . It is
the bulk component of the sample. The space group is R3̄ and the lattice
parameters are a = b = 5.088 Å and c = 14.086 Å. α = β = 90◦ and γ =
120◦ [37]. The structure can be seen in figure 4.4. As mentioned in section
2.2, there is an excess of uncompensated spin in the iron atoms. These spins
align antiferromagnetically (see section 2.4.2) along the c-axis below the Néel
temperature without the presence of strong counteracting external fields.
The Néel temperature, TN , is the temperature above which the material becomes paramagnetic (see section 2.4.5) with no net magnetic moment. For a
single crystal of pure ilmenite it is 58 K [16]. The change in magnetic ordering
around TN happens as a second order phase transition, where the net magnetisation is zero above TN . For such a phase transition the magnetisation as a
function of temperature is given by:
TN − T 2β
M∝
(4.1)
TN
close to and below TN . The critical exponent β is depending on the dimensionality of the system, the symmetry of the order parameter (the magnetisation)
and the range of the forces involved[3, p. 120].
4.2.2 Hematite
Hematite is a mineral composed of iron and oxygen: α-Fe2 O3 . The α indicates
that it is hematite, which belongs to space group R3̄c (corundum structure)
with lattice parameters: a = b = 5.038 Å and c = 13.772 Å, α = β = 90◦ and
γ = 120◦ [37]. At room temperature the spins of hematite are aligned canted
antiferromagnetically (see 2.4.4) with the spins perpendicular to the c-axis. The
hematite structure is shown in figure 4.4.
Under normal circumstances, hematite undergo a so called Morin transition
when cooled below a certain temperature, TM . This is similar to a so called a
spin-flop transition (though not induced by an external field), and below TM the
hematite is no longer ordered CAF with spins perpendicular to the c-axis, since
the spins flips to be AFM aligned with the c-axis instead. For pure bulk hematite
TM ≈ 263 K [1], however, in our sample the transition is suppressed and does
not occur[20]. This can be caused be internal strain, nano-size structures or
impurities (e.g. a low amount of Ti substitution)[4].
28
Figure 4.4: Structure of hematite (left) and ilmenite (right). From space groups
R3̄c and R3̄, respectively. The iron ions present (Fe3+ in hematite
and Fe2+ in ilmenite) have a net magnetic moment.
29
4.2.3 Hemo-Ilmenite
The combined structure of hematite and ilmenite is called hemo-ilmenite with
the chemical formula (Fe2 O3 )1−x (FeTiO3 )x where 0.5 < x < 1.
The lattice parameters of the two minerals are close to each other, thereby
allowing the two to be in registry with each other. However, since the parameters
differ a bit, there is bound to be some strain in the system.
As a result of the aforementioned lamellar structure of the sample, there is a
lot of contact layer between ilmenite and hematite, which can be of importance
when explaining the remanent magnetisation of hemoilmenite.
4.3 Previous Studies
Previously studies of our sample include Electron backscatter diffraction (EBSD),
X-ray Fluorescence Spectroscopy (XRF), Electron Microprobe analysis (EMP)
and neutron scattering.
• The previous EBSD, from [30, sample 5-1 T(A)], was done to find the
orientation of the crystallographic axes inside the sample. This data was
used for our alignment.
• The XRF and EMP measurements produced the weight fractions of hematite
and ilmenite and the impurity information in table 4.1.
• The NRM of the sample is 2.613 · 10−3 Am2 /kg (from [30]).
• The saturation magnetisation, MS , of 0.19 Am2 /kg of the sample is from
personal communication between Erik Brok and Richard Harrison [6, table
3.1].
• The previous neutron scattering studies have been presented in great detail
by Erik Brok in his Master thesis[6], and they have been the basis for all
measurements done in this project. Among the things described in the thesis were indications of a possible explanation for the NRM found through
a polarised neutron diffraction experiment, though an instrumental resolution that was too low and other experimental obstacles left the results
inconclusive, which led to the suggestion of our new optimised experiment.
4.3.1 Impurities
Besides from hematite and ilmenite the sample contains different impurities,
which has been found in previous studies (see below). The impurities are shown
in table 4.1. The most prominent impurity is the Geikielite (MgTiO3 ) in the
ilmenite group.
30
Table 4.1: Mineral composition of sample in percentage of end members. From
bulk XRF measurements[10][30].
Ilmenite
FeTiO3
MgTiO3
MnTiO3
ZnTiO3
NiTiO3
sum
group
64.0961
19.2258
0.4741
0.0252
0.0016
83.8228
Hematite group
Fe2 O3 15.5810
Al2 O3
0.8605
Cr2 O3
0.4137
V2 O3
0.3220
sum
16.1772
4.3.2 Quality of Sample Crystal
Though our sample is of unique quality, there is still evidence that it is not a
perfect single crystal but rather two slightly misaligned crystallites. This was
found in the previous investigations of the same sample by Erik Brok et al.[6]
where they mapped out the (003) diffractions peaks (see Section 5.1) using the
MORPHEUS diffractometer at the Paul Scherrer Institute in Switzerland. The
result is shown in figure 4.5, where the shoulders on the two diffraction peaks
indicate that two crystallites are present in the sample.
In our investigations also a Small Angle Neutron Scattering (SANS) experiment was performed at the D22 instrument at the Institut Laue-Langevin. The
description here will only deal with the obtained evidence for two crystallites
being present in our sample. SANS is a neutron technique where a highly collimated neutron beam is scattered by the sample at small angles to reveal information on the sample structures that are much larger (∼nm size) than the
wavelength of the neutrons, which mean that we in our case probe the lamellar
structure of sample, which is the source of inhomogenities in scattering length
density in the sample (which is the average of the scattering lengths, b, in a
unit volume, see eq. (3.9)). The SANS signal was recorded using a position
sensitive detector, that outputs intensity as a function of scattering vector. It
can be seen in figure 4.6a that the lamellar structure is observed as a lack of
spherical symmetry since the lamellae are aligned and flattened in the horizontal
direction.
The measurement was done three times times with the neutron beam penetrating the sample through the bottom, centre, and top as shown in figure 4.6.
Here an azimuthal average of a part of the detector has been calculated as it
can be seen that peak locations, which corresponds to the orientation of the
lamellae, are not the same for the three different parts of the sample. Since the
lamellae align with the crystal lattice, this shows that two crystallites seem to
31
Figure 4.5: From [6], 2θ is the scattering angle and ω is the rotation angle of the
sample. The peak at 2θ ≈ 60◦ is the structural ilmenite reflection
and the peak at 2θ ≈ 62◦ is the magnetic hematite reflection (see
Section 5.1). The shoulders present on both peaks indicate that two
slightly misaligned (by ∼ 0.6◦ ) crystallites are present in the sample.
32
Beam through sample b ottom Beam through sample centre
Beam through sample top
3
2
1
Counts (logscale)
Detector pixel
4
0
Detector pixel
Azimuthal average
Top of Sample
Centre of Sample
Bottom of Sample
Normalised Intensity
1
0.8
0.6
0.4
0.2
0
220
240
260
280
Angle [degrees]
300
320
Figure 4.6: SANS results showing the presence of two slightly misaligned crystallites. Top are 2D SANS data with the neutron beam passing
through different parts of the sample. Bottom shows corresponding
azimuthal averages.
be present in our sample.
4.4 Hypothesis for Hemo-Ilmenite magnetism
The first idea to come to mind when trying to explain the remanent magnetisation of hemoilmenite might be looking at magnetic impurities. However, previous
studies have found no magnetite or other ferro- or ferrimagnetic minerals in the
sample[10].
The second thought might be to explain the NRM by the CAF hematite spins.
The saturation magnetisation of CAF hematite is 0.404 Am2 /kg, and since only
16% of the sample is hematite, the CAF moments of hematite (0.065 Am2 /kg)
33
can only explain around 15% the sample’s measured saturation magnetisation
of MS = 0.43 Am2 /kg.
As mentioned previously there is a lot of hematite-ilmenite contact layers
within the sample, and it might be here that the origin to the NRM shall be
found. The idea is, that somehow there are excess spins that can co-align in
the contact layers to produce a net magnetic moment. This is called lamellar
magnetism, and its detailed presentation by McEnroe et al. in [20] is the basis
of the description in this section.
Figure 4.7 shows a Monte Carlo simulation of how the ions arrange in and
around hemo-ilmenite contact layers[21]. A contact layer between hematite and
ilmenite consists of a mix of Fe2+ and Fe3+ and it is coupled antiferromagnetically to the neighbouring hematite, such that the Fe2+ and Fe3+ in the layer are
ferromagnetically aligned. The hematite lamellae with an odd number of layers
will have a net moment in the opposite direction of the moments in the contact layer, and the two contact layers will be aligned with moments in the same
direction. Assuming a model with 50:50 ratio of Fe2+ and Fe3+ in the contact
layers (as in [21]), the Fe3+ moments in the two contact layers surrounding a
odd-number layered hematite lamellae will cancel out the moment coming from
the uncompensated Fe3+ layer inside the lamellae. The leaves a net moment
coming from the aligned Fe2+ moments in the contact layers. These moments
will be perpendicular to the c-axis within the contact layer. When the crystal
was created, the lamellae with c-axis perpendicular to the earth’s magnetic field
would have its excess contact layer spins align with it. This could in theory produce a net moment that might explain the maximum saturation magnetisation
measured in our sample[29].
34
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5 Diffraction Experiment
5.1 Diffraction at IN12, ILL
The diffraction experiment was performed at the IN12 triple axis spectrometer
(TAS) at the Institut Laue-Langevin (ILL) in Grenoble, France in August 2010.
The findings described here have also been published in an article in Physical
Review B [7].
5.1.1 Instrumental set-up
The instrument is angle dispersive, meaning that a fixed wavelength is used,
and that the angle, θ, is scanned to fulfill Bragg’s Law (see section 3.3.2). An
incoming wavevector of 1.55 Å−1 (λ = 4.05 Å) was selected by adjusting the
incoming angle from the monochromator crystal. This crystal uses the principle
of Bragg’s law to reflect only neutrons within a small wavelength band, thereby
monochromising the beam on the sample. The analyser crystal works in the
same way. The layout of IN12 is shown in figure 5.1. Not shown in figure 5.1b
is the cryostat that was installed at the sample position in order to vary the
sample temperature between 2 K and 290 K. The set-up also included a magnet
that was used to apply fields between -0.75 T and 2.5 T.
To measure magnetic order we used a polarised incoming beam. This was
done with a supermirror bender that utilises the fact that the ferromagnetic
material, it is composed of, has different scattering length densities for different
spin orientations of the incoming neutrons, thereby reflecting one state (in our
case the up-state) while transmitting the other state to an absorbing layer in
the neutron guide. The Heusler analyser is composed of aligned ferromagnetic
crystals, and is only accepting neutrons in one spin state. In our case, the polariser and analyser were set up in such a way, that the analyser was selecting
spin-states anti-parallel to the ones the polariser was reflecting – so-called spinflip scattering. To also be able to measure the non-spin-flip (see section 5.3) a
neutron π-flipper was installed between the sample and the analyser. An outer
coil cancels out the guide field, that keeps the neutrons polarised, and an inner coil creates a horizontal field, around which the neutrons precess. The spin
flipper was calibrated so that passing neutrons with the wavevector of 1.55 Å−1
had their spins flipped 180 degrees, when the flipper was on. For most purposes,
including ours, the flipper can be assumed to be 100% effective and flipping all
36
passing neutrons when turned on.
(a) IN12 Instrument Layout
(b) Picture of part of IN12
Figure 5.1: Layout and picture of IN12, ILL. The layout shows the instrument
after the instrument was moved to the end of the beamline. This
was done after our experiment was performed, however the layout
was almost identical to what it is now. Layout image from [38].
Further components included in the instrumental set-up were:
Monitors, that measures the neutron flux, so that measured intensities can be
normalised properly even when the source flux is varying.
Collimators, that limits the beam divergence in order to get the necessary
spatial resolution.
A filter to remove higher order reflections from the monochromator. The filter
is usually made of beryllium, which has high transmission for the neutrons above
3.9 Å, and high absorption cross-section for the higher energy neutrons.
The detector, that works by capturing a neutron in 3 He, which is then converted to tritium, a proton and excess energy. The proton is accelerated in a
high voltage potential so it has a kinetic energy enough to ionise a stopping gas
kept inside an electric field. The resulting current is then amplified and measured.
37
5.1.2 Alignment
The first thing to do was to align the sample, so that the orientation of the
crystal planes within the sample is controlled. Some test scans were done, and
it turned out that we needed a new sample holder. The reason for this, is that
the previous experiment done on the sample had been a Small Angle Scattering
experiment, where only the c-axis needed to be in the horizontal scattering
plane. So in order to get the a-axis horizontal as well, the sample was rotated
90◦ before being placed in the new holder thereby making it possible to probe
all (h0l) reflections with the sample only having to be rotated around a vertical
axis. Since the sides of the sample crystal are not completely parallel to the
crystal planes, the sample was placed at a slight angle of ∼10◦ as shown in
figure 5.2 (see also figure 4.2a).
Figure 5.2: Sampleholder. Made from aluminium at the ILL and placed upside
down in the cryostat.
The structural (003) ilmenite peak and the structural (102̄) hematite peak
were used to check alignment. It was noticed that a tilt of ∼1.2◦ was need for
alignment, which was acceptable since the cryostat had a limit of 5◦ . The sample
was then mounted inside the cryostat, realignment was done and it was noted
that some polarisation was lost with zero field, thereby making a minimum field
of ±0.05 T necessary.
The reflections that were measured are shown in table 5.1:
38
Table 5.1: Reflections from hematite (hem) and ilmenite (ilm). Type refers to
structural peak (s) or magnetic peak (m). q is the scattering vector
(see section 3.3.1). The reflections are obtained from [6], and have
been calculated by Bente Lebech.
Miller indices
(003)
(003)
(101/¯2)
(101)
(101)
(102̄)
(102̄)
Mineral
ilm
hem
ilm
ilm
hem
ilm
hem
Type
s
m
m
s
m
s
s
q [Å−1 ]
1.34
1.37
1.44
1.49
1.51
1.66
1.70
5.2 Fitting Procedures
The wavelength of the incoming neutrons is never a single value, but rather a
distribution of wavelengths. This distribution comes partly from the mosaicity
of monochromator, which is made up of slightly misaligned crystals sheets. Since
there is a limited divergence (the neutrons paths are not completely parallel),
there is also a uncertainty in the angles by which they hit the sample, and in
the angle they exit by. The reason for this is to get higher counting statistics,
by allowing more neutrons to be reflected. In a measurement with a number of
independent random sources of error, the outcome will be distributed normally
around the true value [35, p. 235]. In our case we have scanned q-values to
find the one that gives the highest intensity of reflected neutrons. The resulting
normal distribution can then be described by a Gaussian function, with centre
at the mean (true) q-value, amplitude proportional to the maximum neutron
intensity, and width related to the variance of the distribution. The width
of this gauss function is only caused by the instrument itself, however other
broadening effects can come from the sample.
When having done a measurement resulting in data like the ilmenite peak in
figure 5.3, the next step is to find the values for the parameters of the Gaussian
function that best fits the data. This has been done in MATLAB[39] with the
Spec1d[38] and iFit[12] fitting routines and tools. These programmes uses the
least squares method to find the function with minimum difference between
observed and fitted values. The errorbar on each measured data point is taken
as the square root of the count numbers.
For further data analysis, other functions than Gaussians have been used to
fit the data; e.g in the case of the (003) hematite peak, where the peak was
39
further broadened by nano-size effects (see section 3.3.5), a Voigt profile1 was
used. Figure 5.3 shows examples, where data has been fitted with a Gaussian
plus a Voigtian profile. The reason for this choice is that the data scan contains
a signal from both an ilmenite peak and a hematite peak. The ilmenite is the
bulk mineral of the sample, and it is reasonable to assume that there is no nanosize broadening of the ilmenite peak, therefore a Gaussian distribution is chosen.
The hematite peak on the other hand shows sign of broadening on top of the
Gaussian distribution. Since the hematite is situated in nano-size lamellae, a
Voigtian profile is used for this peak. The two peaks are fitted at the same time,
and the sum of the two distributions is used for the fit.
5.3 Flipping Ratio Correction
In a real life experiment like ours, the polarisation of the beam is never perfect.
This can be adjusted for by performing a flipping ratio correction. As described
in section 3.4.1, the neutron can either be in an up-state |↑i or down-state |↓i.
The number of neutrons in either state are denoted by n↑ or n↓ . For a neutron
beam with all neutrons in the |↑i-state (|↓i-state) the polarisation of the beam, p,
is then 1 (-1). The probability of having a neutron in the up-state (down-state)
is p↑ (p↓ ), and p↑ + p↓ = 1. From the definition:
p=
n↑ − n↓
n↑ + n↓
(5.1)
we get:
1+p
,
2
1−p
p↓ =
,
2
p↑ =
(5.2a)
(5.2b)
For each measured point in q-space two data points were taken. One with the
flipper on (polariser and analyser reflecting neutrons with parallel spins) and one
with the flipper off (polariser and analyser reflecting neutrons with anti-parallel
spins). These are respectively called the Non Spin Flip (NSF) and the Spin Flip
(SF) measurements, since NSF refers to scattering where the spin state of the
neutron is not altered in the sample, and SF is scattering where the neutron
is flipped by interacting with a magnetic ordering in the sample. A structural
peak should only be seen in the NSF data, whereas a magnetic peak can give
rise to both NSF and SF scattering.
As explained in section 3.4.1, let the true cross-sections of NSF and SF scattering be denoted as σ N SF and σ SF respectively. The incoming neutron beam
1
A Voigt is a convolution of a Gaussian and a Lorentzian profile
40
is a combination of up-polarised neutrons, p↑ , and down-polarised neutrons, p↓ .
The measured intensities are then given by:
N SF σ
=
(5.3)
σ SF
N SF N SF p p and we can define: I ≡ II SF , P ≡ p↓↑ p↑↓ and σ ≡ σσSF . We measure
I and we want σ, so we need to find P−1 to correct for imperfect polarisation.
The ratio between the number of |↑i-state and |↓i-state neutrons is called the
Flipping Ratio (F R):
I N SF
I SF
p↑ p↓
p↓ p↑
FR =
n↑
,
n↓
(5.4)
and it is seen that
p=
FR − 1
FR + 1
(5.5)
Finding F R, in order to calculate P−1 , is done by measuring a structural reflection where σ SF = 0 so that:
FR =
I N SF
I SF
(5.6)
In our measurement, the flipper was turned on or off between each collected data
point. It turned out, however, that this didn’t work properly, and as a result
some scans of a peak is missing one to three data points, for a particular point
in q-space, it would mostly only be a NSF or a SF point missing, and not both.
So to be able to calculate F R an interpolation was made to get the few missing
points.
Figure 5.3 shows a scan of the structural (003) ilmenite peak (and the magnetic
(003) peak). It can be seen that there is a signal at the structural peak position
even in the SF measurement. This is mainly caused by an imperfect beam
polarisation.
The intensities of the (003) ilmenite peak gives a flipping ratio of 33.3 corresponding to a polarisation of 94.2%. The peak intensity has been calculated
peak amplitude times peak width, since this is proportional to the area under
the peak and thereby the intensity of the peak, as is the case for a Gaussian
distribution. Figure 5.4 shows the FR corrected scans. There still seems to
be a small structural signal left, which should not be there in a perfect set-up.
However, it is likely that the leftover signal was caused by the sample consisting
of two slightly misaligned crystallites (see Section 4.3.2).
In addition to the FR values obtained from the structural ilmenite peak,
the flipping ratio has been calculated from the (102̄) structural hematite peak
41
Intensity [cnts/mon]
(003) Reflections. Struct. Ilm., Mag. Hem.
9
NSF
Data
SF
before FR correction
Fit of Gauss + Voigt
Gauss part
6
Hematite
Voigt part
Hematite
3
0
Ilmenite
Ilmenite
2.9
3
3.1
2.9
q l [r.l.u.]
3
3.1
Figure 5.3: Non Spin Flip and Spin Flip data of the (003) structural ilmenite
peak and the (003) magnetic hematite peak. Before FR correction.
(B=0.5 T, T=250 K).
42
Intensity [cnts/mon]
(003) Reflections. Struct. Ilm., Mag. Hem.
9
NSF
Data
SF
after FR correction
Fit of Gauss + Voigt
Gauss part
6
Hematite
Voigt part
Hematite
3
0
Ilmenite
2.9
3
3.1
2.9
q l [r.l.u.]
3
3.1
Figure 5.4: Non Spin Flip and Spin Flip data of the (003) structural ilmenite
peak and the (003) magnetic hematite peak. After FR correction.
It can be seen that the (003) ilmenite structural reflection is close to
only being present in the NSF signal. This is to be expected from a
structural peak (see section 3.4). The (003) magnetic hematite peak,
on the other hand, is seen in both the NSF and the SF signal. This
is what is expected from such a magnetic peak. This shows that the
FR correction seems to work well in isolating the magnetic signal.
The tiny leftover structural ilmenite peak in the SF signal is most
likely caused by two slightly misaligned crystallites being present in
the sample (see Section 4.3.2). (B=0.5 T, T=250 K).
43
measurements (see page 45 for more information on this peak). Figure 5.5 shows
the polarisation values for all the different measured combinations of reflection,
applied field and sample temperature.
Polarisation calculated from structural reflections
1
290
250
200
150
( 003) s t r uc t ur al ilme nit e r e fle c t ion
( 10 2̄) s t r uc t ur al he mat it e r e fle c t ion
0.99
K
K
K
K
100 K
65 K
35 K
2K
0.98
0.97
Polarisation
0.96
0.95
0.94
0.93
0.92
0.91
0.9
−1
−0.5
0
0.5
1
1.5
2
2.5
Applied Field, [T]
Figure 5.5: Beam polarisation calculated from measurements of structural peaks.
It is seen that the stronger the field, the better the polarisation is
kept, which is most likely caused by the higher field functioning
as a better guide field. The data also seem to suggest that higher
temperatures give rise to higher polarisation. Note the narrow range
of the y-axis.
It can be seen from figure 5.5 that the polarisation of the beam is not constant
with varying temperatures, fields and reflections. As mentioned, the polarisation
is lost at zero applied field and the measurements suggest that the larger the field
is, the better the polarisation is kept. Also the temperature of the sample seems
to have an influence, albeit a small one. The reason for this, however, is not
immediately clear. Perhaps it is caused be a higher depolarisation in the sample
at low temperatures due to ordering of the ilmenite host when the temperature
is lower than TN , or it could be caused by a small systematic error in the fitting
44
3
procedure. However, the biggest variation in polarisation comes from the crystal
orientation. It should be noted though, that the beam polarisation fluctuation
is rather small. The reason for the polarisation values calculated from the (102̄)
peaks are lower can likely arise from a SF signal caused by a small canting of the
hematite spins. This signal will in the FR calculation be regarded as a ’false’
signal caused by imperfect polarisation. If only the (102̄) ilmenite peak intensity
is used to calculate the polarisation, the values are a bit lower than the (003)
ones.
For the FR correction of the (003) magnetic hematite reflection, a polarisation
value is calculated from each scan of the adjacent (003) structural ilmenite peak.
To FR correct the (101) magnetic hematite peak, a single polarisation value has
been calculated from the data shown in figure 5.5, where the (102̄) data have
been weighed equally to the (003) data.
p(003)
mean = 0.949
(5.7a)
2̄)
p(10
mean = 0.930
(5.7b)
This gives us the polarisation value, that will be used to correct the (101) reflection data:
p(101) = 0.940 ± 0.014
(5.8)
The longitudinal scans of the (102̄) structural hematite reflection (example in
figure 5.6) clearly show another peak in addition to the Voigt shaped hematite
peak. This is assumed to be the (102̄) structural ilmenite reflection, even though
the peaks seem to be a little closer to each other than what would be expected
(see table 5.1).
As seen in the figure, the intensity profile have been fitted with the sum of a
Voigtian and a Gaussian profile, and to calculate the polarisation, the intensity
was found by trapezoidal integration of the fit. To properly fit the peak at lower
ql -values, the Voigtian profile was used as was the case with the (003) magnetic
hematite peak.
The scans of the (101) magnetic hematite reflection also showed the (101)
structural ilmenite reflection, however after FR correction only the magnetic
peak should remain in the SF signal. This is seen in figure 5.7, where the
polarisation calculated in (5.8) was used.
5.4 Hematite spin orientation in applied field
To learn about the spin orientation of the hematite spins, we applied different
vertical fields on the sample at different temperatures. From these measurements, we use the relationship between the intensities of the hematite peak in
45
Intensity [cnts/mon]
3 NSF
2.5
(10 2̄) Structural Reflections
Data
2
Hematite
SF
before FR correction
Fit of Gauss + Voigt
0.25
Gauss part
0.2
Voigt part
1.5
0.15
Ilmenite
Hematite
1
Ilmenite
0.1
0.5
0
0.3
0.05
−2.04
−2
−1.96
−2.04
q l [r.l.u.]
−2
−1.96
0
Intensity [cnts/mon]
(a) Before FR correction.
3 NSF
2.5
(10 2̄) Structural Reflections
Data
2
Hematite
SF
after FR correction
Fit of Gauss + Voigt
0.25
Gauss part
0.2
Voigt part
1.5
0.15
Ilmenite
1
0.1
0.5
0
0.3
0.05
−2.04
−2
−1.96
−2.04
q l [r.l.u.]
−2
−1.96
0
(b) After FR correction.
Figure 5.6: Non Spin Flip and Spin Flip data of the (102̄) structural hematite
and ilmenite peaks. Note that the scale of the NSF intensities is ten
times the scale of the SF intensities. The purely structural peaks
should give rise to no SF signal after FR correction, which is also
what is observed. (B=2 T, T=150 K).
46
Intensity [cnts/mon]
6
5
(101) Reflections. Struct. Ilm., Mag. Hem.
NSF
Data
Ilmenite
4
SF
after FR correction
0.6
Fit of Gauss + Voigt
0.5
Gauss part
0.4
Voigt part
3
Hematite
0.3
Hematite
2
0.2
1
0.1
0
0.97
1
1.03
0.97
q l [r.l.u.]
1
1.03
0
Figure 5.7: (101) magnetic hematite and (101) structural ilmenite reflections.
The field was 0.05 T and the temperature was 65 K. As would be
expected, only the magnetic peak is seen in the NSF signal after
FR correction, again showing that the FR correction can be used to
eliminate structural peaks. Note the difference in y-axis scaling.
47
the NSF and the SF signal to calculate the average in-plane spin direction with
respect to the external field. This is done using equation (3.30). The results
obtained from the (003) measurements are shown in figure 5.8. The intensity of
the hematite peaks (see e.g. figure 5.4) was calculated by trapezoidal integration
of the fitted Voigtian profile. The errors on the Voigtian parameters given by
the fitting procedure were used to perform Monte Carlo simulations to obtain
the error on the intensity. For each fitting parameter, 104 values2 from a normal
distribution of pseudorandom numbers, with mean equal to the fitting parameter and standard deviation equal to the error on the fitting parameter, were
calculated in MATLAB. From these distributions the intensities and therefrom
the in-plane spin angle with respect to the applied field was calculated. The
results are shown in figure 5.8, where the errorbars are smaller than the data
points.
It can be seen that the spins responsible for the magnetic moment turn away
from the applied field when the field magnitude increases. The higher the temperature, the lower the field needed to turn the spins. This result is backed up
by examining the (101) reflection. Since the hematite spins are strongly bound
to be perpendicular to the c-axis, the information from (101) magnetic hematite
scans indicate how the spins are orientated in the (003) plane. Figure 5.9 shows
the intensity of the Voigt fit to the (101) magnetic hematite peak for different
temperatures and applied fields. Our obtained result for the in-plane spin orientations (within the field range we have measured on the (003) reflection) are
supported by the measured NSF intensities of the (101) magnetic hematite peak
shown in figure 5.9. The SF intensities are less conclusive, though also more
noisy.
Weather or not the maximum spin angle of ∼61◦ with respect to applied field
in figure 5.8 is the saturation angle was investigated in an experiment done
by students[8] from the University of Copenhagen under supervision by Kim
Lefmann. They measured on the same sample at the RITA-II instrument at the
Paul Scherrer Institute using unpolarised neutron diffraction. They measured
the (101) hematite peak in fields from 0 T to 11 T applied in the (003) plane.
The spins lie perpendicular to the c-axis and therefore their orientation in the
(003) plane is connected to their scattering intensity in (101) direction, since
only the part of M(q) that is perpendicular to q contributes to the scattering.
Their result at temperatures of 2 K and 150 K is shown in figure 5.10.
It is seen, that there is a saturation around 2 T to 2.5 T. However, at around
6 T there seems to be a rise in the intensity in the measurements at 2 K, which is
caused by the spins turning back towards alignment with the field. The reason
for this might be some form of a spin-flop transition, possibly in the ilmenite,
since it is most clearly seen below TN . The applied field is perpendicular to the
2
Done a couple of times to check for consistency.
48
In−Plane−Spin−Angle [degrees]
61
57
53
49
45
41
61
57
53
49
45
41
61
57
53
49
45
41
290 K
250 K
200 K
150 K
100 K
65 K
−0.5 0 0.5 1 1.5 2 2.5
35 K
2K
Coming from
higher field
Coming from
lower field
−0.5 0 0.5 1 1.5 2 2.5 −0.5 0 0.5 1 1.5 2 2.5
B−field [T]
Figure 5.8: In-plane spin orientation with respect to different applied fields and
for eight different temperatures for the (003) measurements. The
results suggest a saturation angle of around 61◦ , which is seen
most prominently at higher temperatures. Errorbars were calculated through Monte Carlo simulations based on the errors from the
fitted Voigtian profiles. Note that that this result differs slightly
from the saturation angle of ∼ 56◦ reported in [7]. The reason for
this is a different approach to the locking of the Gaussian widths in
the fitting procedure.
49
(101) Magnetic Hematite reflection
NSF
Intensity of voigt fit (normalised)
1
NSF T=290 K
1
SF
SF
T=290 K
NSF T=150 K
SF
T=150 K
NSF T= 65 K
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0
1
0.5
2
0
Field [T]
SF
1
T= 65 K
2
Figure 5.9: Voigt fit intensities of (101) magnetic hematite peak for different
temperatures and applied fields. The intensity is related to the spin
orientation (see eq. (3.30)). It can be seen that the NSF intensity
drops. This supports the results shown in figures 5.8 and 5.10.
50
1.55
Scanning from 11 T to 0 T
Scanning from 0 T to 11 T
2K
1.5
1.45
1.4
1.35
Intensity [a.u.]
1.3
1.25
Scanning from 11 T to 0 T
Scanning from 0 T to 11 T
150 K
1.4
1.35
1.3
1.25
1.2
1.15
0
1
2
3
4
5
6
Field [T]
7
8
9
10
11
Figure 5.10: Unpolarised nuclear diffraction measurement at 2 K and 150 K of
the (101) peak, for fields between 0 T and 11 T. Green 4 points
taken when field is ramped up, red points 5 taken when field is
ramped down. Data recorded by [8].
51
c-axis, along which the ilmenite spins lie. They might then flip at the observed
∼6 T to minimise the energy by aligning more with the field.
5.5 Nano-size effects
As mentioned earlier the Lorentzian broadening causing the Voigtian profile for
the magnetic hematite peaks is likely caused be nano-size effects in the lamellae,
see equation (3.23). The width of the Lorentzian is inversely proportional to the
average particle size in the sample. Figure 5.11 shows the particles sizes calculated from the widths of the (003) magnetic and the (102̄) structural hematite
peaks.
(003) Magnetic Hematite reflection
(10 2̄) Structural Hematite reflection
2800
2800
2600
2600
NSF T=290 K
SF
2400
2400
2200
2200
NSF T=200 K
2000
2000
SF
SF
Particle Size [ Å]
T=290 K
NSF T=250 K
T=250 K
T=200 K
NSF T=150 K
1800
1800
SF
1600
1600
NSF T=100 K
1400
1400
NSF T= 65 K
1200
SF
SF
1200
1000
1000
800
SF
800
NSF
600
T= 65 K
0
1
2
0
0.5
1
1.5
T= 35 K
NSF T=
2K
SF
2K
600
2
Field [T]
Figure 5.11: Particle sizes calculated from Lorentz widths using the Scherrer
formula (eq. (3.23)).
As previously described, the NSF (SF) signal is caused by spins orientated
parallel (perpendicular) to the applied field. The lamellae might be thought of
as a thick plate of hematite between two thinner discs of contact layer, which
would then describe how the CAF hematite moments ((003) SF widths) are
found in larger particles than the lamellar magnetism ((003) NSF widths) in the
contact layers. The measurements on the (102̄) structural hematite reflection
estimates the particle size to be around 2400 Å, which is larger than the size
obtained from the (003) NSF widths, since the particles are thinnest in the
direction of the c-axis. It is important to note that other factors such as strain
and imperfections of the lattice can contribute to the broadening of the peaks
52
T=100 K
NSF T= 35 K
SF
NSF
−1
T=150 K
T=
as well, so the result from figure 5.11 should be taken as a lower boundary on
the particle size.
It should be noted that the fitting procedure of fixing the Gaussian widths to
obtain the different Lorentzian widths is somewhat fragile, so the results should
be taken with a grain of salt, e.g. is a thickness of the contact layers equal to
around 700 Å seemingly larger than what would be expected, though the shape
of the contact layer might play a role here as well.
5.6 Antiferromagnetic order, transition temperature
As described by equation (4.1), the ilmenite spins align antiferromagnetically
below a transition temperature called the Néel temperature, TN . To find this
transition temperature, the intensity of the (101/¯2) magnetic ilmenite peak was
measured for different temperatures. Two scans of the peak are shown in figure
5.12. Only the SF data is shown and FR correction has not been done, since it
is the relative intensity for the different temperatures that is relevant, and FR
is close to being independent of temperatures. It is here clearly seen that more
than one peak is present.
(10 21̄ ) magnetic ilmenite p eak. T = 30K, B = 0T
SF data
Fit of sum of two Gaussians
Gauss 1 of fit
Gauss 2 of fit
0.8
0.6
0.4
0.2
SF data
Fit of sum of two Gaussians
Gauss 1 of fit
Gauss 2 of fit
1
Intensity [cnts/mon]
Intensity [cnts/mon]
1
0
−0.515
(10 21̄ ) magnetic ilmenite p eak. T = 40K, B = 0T
0.8
0.6
0.4
0.2
−0.51
−0.505
−0.5
ql [r.l.u.]
(a) T = 30 K
−0.495
−0.49
−0.485
0
−0.515
−0.51
−0.505
−0.5
ql [r.l.u.]
−0.495
−0.49
(b) T = 40 K
Figure 5.12: Scan of the (101/¯2) magnetic ilmenite peak. SF signal. As expected
the signal drops as the temperature increases.
Above TN there should be no magnetic ilmenite peak, and the intensity of
the peak should follow a power-law in temperature region between ∼ TN /2
53
−0.485
to a couple of degrees below TN . Only three full scans of the spin flip peak
were done in this temperature region, however several other measurements were
done, where only one data point was measured for each temperature. These
data points measured at the intensity at approximately the peak centre of the
most intense peak. The data points were not guaranteed to have been taken
exactly at the centre, so the variation in intensity around the measured q-values
for the three full scans were used to estimate the uncertainty of the single data
point measurement. Then the amplitude was plotted and fitted as a function
of temperature as seen in figure 5.13, where data outside the fitted area is also
shown. The errorbars shown are the estimated standard deviations.
Ampl i tude of (10 1̄2 ) magne ti c i l me ni te p e ak
1
Amplitude [arb. unit]
0.8
0.6
0.4
0.2
Pe ak Ampl i tude
µ
¶2 · 0 . 2 2
41. 3 − T
Fi t, I = 1. 30
41. 3
E x trap ol ati on of fit
0
0
5
10
15
20
25
30
Temperature [K]
35
40
45
Figure 5.13: Temperature variation of the (101/¯2) magnetic ilmenite peak, showing the second order phase transition. Both black and hollow points
are peak amplitude measurements, however, the fit is to the black
points only, since the power law behaviour is only valid within approximately this range.
54
50
The Néel temperature for the ilmenite in our sample is found to be ∼ 41.3 K.
The critical exponent, β, was found to be 0.22. This less than for the 3 dimensional Ising (β = 0.326) and Heisenberg (β = 0.367) models, and it is higher
that the 2 dimensional Ising model (β = 1/8) [3, table 6.2]. The Ising models
describe a system where the spins can be in one of two discrete states (+1 or
−1) along a given direction (e.g. the z-direction). In the Heisenberg model
not only the z-component of the spins interact, since they can take arbitrary
directions. Our result from figure 5.13 suggest that these simple models cannot
explain the spin behaviour in antiferromagnetic ilmenite, probably because ilmenite behaviour is predominately 2 dimensional in the Fe2+ planes, but order
3D with weak couplings between planes.
55
6 Conclusion
We set out to try to explain the somewhat mysterious remanent magnetisation
of naturally nano-structured hemoilmenite, and through a high quality polarised
neutron diffraction experiment at the ILL, we got some steps further to the goal.
The data from IN12, showed a resolution high enough to separate the reflections with peaks lying close together such as the 1.34 Å−1 structural (003)
hematite peak and the 1.37 Å−1 magnetic (003) hematite peak. This gave us
the possibility to use the structural (003) ilmenite peak to flipping ratio correct
the (003) magnetic hematite with individual polarisation values for each scan.
The FR correction proved to highly effective, which could be seen in the spin
flip signal, where only magnetic peaks showed up as expected. For magnetic
reflections without a neighbouring structural peak, such as the magnetic (101)
hematite peak, a single polarisation value was used based on the polarisation
measurements from the structural (003) hematite and (102̄) ilmenite peak.
By measuring the magnetic signal of the (101/¯2) ilmenite peak at different
temperatures around the phase transition from antiferromagnetic structure to
paramagnetic structure, we determined the Néel temperature of the ilmenite
in our sample to be ∼41.3 K. The same measurement gave us a value for the
critical exponent of 0.22. What this specifically implies needs to be further
investigated, though it shows that the system cannot be explained by the simple
Ising or Heisenberg models.
Our main result came from the measurements of the (003) magnetic hematite
peak. It was shown in figure 5.8 that at close to zero external field, the spins
start out an an average angle of 45◦ with respect to the vertical upwards pointing field. As the field strength is increased the spins turns away from the field,
meaning that the angle increases with stronger field. Within our field strength
range, the angles seem to saturate after turning a little more than 15◦ , ending
up at 61◦ . The magnitude of the field necessary for saturation is around 2 T to
2.5 T.
If the the NRM was only caused by canted antiferromagnetism, the spins
would align at an angle close to 90◦ with the applied field. If it was lamellar
moments, that are responsible, the spins would align parallel with the applied
field. Since the case is neither perpendicular nor parallel, but rather ∼61◦ , the
explanation for the NRM, is a combination of both or more contributions.
56
Acknowledgements
I would like to thank my supervisor Kim Lefmann from the University of
Copenhagen (KU), without whom I could not have done this project. Also a
I owe many thanks to Erik Brok from the Technical University of Denmark
(DTU), who through his experience with the content of this project has been
a source advice for me. Kim and Erik together with Luise Theil Kuhn (DTU),
Richard J. Harrison (from University of Cambridge) and Sonja Lindahl Holm
(KU) have conducted the neutron scattering experiments with me – a task I
could never have done alone. Suzanne McEnroe from Geological Survey of Norway (NGU) is the lender our unique sample, which is the pivot for this entire
investigation. Both Richard Harrison and Suzanne McEnroe plus Peter Robinson (NGU) and others have written papers with information and background
vital to me. Thanks also go out to Wolfgang F. Schmidt who was the instrument
responsible and local helpful contact at the Institut Laue-Langevin instrument
IN12, where our experiments were performed.
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