Download Magnetism and Mössbauer spectroscopy on Mars

Magnetism and Mössbauer spectroscopy on Mars
- Lecture Notes Haraldur Páll Gunnlaugsson
Department for Physics and Astronomy
Aarhus University
Version 0.3: 29/10 2010
1
Preface
Magnetism and Mössbauer spectroscopy represent two different methods to examine material.
When applied together, they can give knowledge on the physics of a system that are out of reach for
each one of them. Magnetic methods probe bulk properties of a material, by measuring the effects
of external magnetic field, but lacks often direct information how these effects relate to the atomic
build-up of the material. Mössbauer spectroscopy can probe the atomic information, and how
atomic nuclei are affected by its nearest neighbours in a crystal, and among them the magnetic
interactions. The method on the other hand does not give directly information on bulk properties of
the material. With this base I have supervised an advanced course in experimental physics at the
Aarhus University which aim of combining the two different views of a material that can be learned
from the application of the two methods in material sciences. Understanding magnetism from the
atomic level, to planetary magnetism gives students a unique opportunity to see processes in nature
and how physics can be applied to understand the world we live in.
In the course I have used the book from Crangle on Magnetism and lecture notes from Steen
Mørup on Mössbauer spectroscopy. For introduction to application of Mössbauer spectroscopy in
geology, I have used my own notes. Though the two first were good sources, I felt that they did not
explain the matter in the way I wanted it explained, and lacked especially material related to the
experimental part and interpretation of experimental data. Due to this, I have expanded these notes,
and set them together, and for the first time hope it will form the basis of the material used in the
course.
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Content
Magnetism and Mössbauer spectroscopy on Mars ..............................................................................1
- Lecture Notes - ..................................................................................................................................1
Preface..................................................................................................................................................2
Content .................................................................................................................................................3
1 Magnetism and magnetic Material...............................................................................................6
1.1
Introduction..........................................................................................................................6
1.1.1
What is magnetic material?..........................................................................................6
1.1.2
Units in Magnetism......................................................................................................6
1.1.3
Definitions from magnetism ........................................................................................7
1.1.4
Origin of magnetism ....................................................................................................7
1.1.5
Quantifying the magnetism..........................................................................................9
1.2
Magnetism of ionic material ................................................................................................9
1.2.1
Crystal field effects and comparison to experiments .................................................14
1.3
Ferromagnetism (Weiss theory).........................................................................................15
1.3.1
Susceptibility above the Curie temperature ...............................................................17
1.3.2
Susceptibility below the Curie temperature ...............................................................18
1.3.3
Antiferromagnetism ...................................................................................................19
1.3.3.1 Susceptibility of antiferromagnetic material..........................................................19
1.3.4
Ferrimagnetism ..........................................................................................................20
1.3.4.1 Spinel ferrites: magnetite .......................................................................................20
1.3.5
Summary of susceptibility. ........................................................................................21
1.3.6
Exchange interactions ................................................................................................21
1.4
Domain magnetism ............................................................................................................22
1.4.1
Magnetocrystalline energy.........................................................................................22
1.4.2
Magnetostatic energy .................................................................................................23
1.4.3
Domain walls .............................................................................................................24
1.4.4
Single domain particles magnetisation properties. ....................................................26
1.4.5
Shape anisotropy ........................................................................................................29
1.5
Superparamagnetism..........................................................................................................29
1.6
Magnetisation measurements.............................................................................................30
1.6.1
Interpretation of the magnetisation curve ..................................................................31
1.6.2
Inversion of maghemite (γ-Fe2O3 -> α-Fe2O3) ..........................................................32
1.6.3
Magnetic properties of permanent magnets ...............................................................33
1.6.3.1 Hard magnets .........................................................................................................33
1.6.3.2 Soft magnets...........................................................................................................34
1.7
Magnetic forces and the magnetisation curve....................................................................34
1.7.1
Paperclip magnetism ..................................................................................................35
2
Mössbauer spectroscopy ............................................................................................................38
2.1
Introduction........................................................................................................................38
2.1.1
Experimental setup.....................................................................................................38
2.2
The f-factor ........................................................................................................................39
2.3
Mössbauer isotopes ............................................................................................................42
2.4
Resonance conditions and thickness effects ......................................................................44
2.5
Second order Doppler shift ................................................................................................46
2.6
Interactions between the nuclei and the electron density...................................................47
2.7
The isomer shift .................................................................................................................48
3
2.7.1
Thickness factors in multiphase systems ...................................................................50
2.7.2
Alloy broadening........................................................................................................51
2.7.3
Cosine broadening......................................................................................................51
2.8
Quadrupole splitting...........................................................................................................52
2.8.1
Quadrupole splitting in 57Fe.......................................................................................53
2.8.2
Calculating the strength of the quadrupole interaction ..............................................54
2.9
Magnetic hyperfine interactions.........................................................................................54
2.9.1
Magnetic hyperfine splitting in 57Fe ..........................................................................55
2.9.2
Line intensities and single crystal measurements ......................................................56
2.9.3
Combined magnetic hyperfine and quadrupole interactions......................................57
2.9.4
Distribution analysis ..................................................................................................58
2.10 Calibration of Mössbauer spectra ......................................................................................59
2.11 Application of Mössbauer spectroscopy ............................................................................61
2.11.1
Diffusion broadening .................................................................................................62
2.11.2
Superparamagnetism..................................................................................................64
2.11.3
Applications in metallurgy.........................................................................................69
2.12 Setups used.........................................................................................................................71
2.12.1
Transmission Mössbauer spectroscopy (TMS)..........................................................71
2.12.1.1
Optimal absorber thickness................................................................................71
2.12.2
Backscatter Mössbauer spectroscopy (b-MS)............................................................72
2.12.2.1
The NIMOS II Mössbauer spectrometer............................................................72
2.12.3
Conversion electron Mössbauer spectroscopy (CEMS) ............................................73
2.12.4
Radioactive Mössbauer spectroscopy ........................................................................74
2.12.4.1
Resonance detectors...........................................................................................75
57
2.12.4.2
Co ....................................................................................................................75
2.12.4.3
In-Beam Mössbauer spectroscopy (IBMS)........................................................76
57
Mn ...................................................................................................................76
2.12.4.4
2.13 Synchrotron Mössbauer spectroscopy ...............................................................................77
2.13.1
Setup for S-MS ..........................................................................................................77
2.13.2
Nuclear Forward Scattering (NFS) ............................................................................78
2.13.3
Nuclear Inelastic Scattering (NIS) .............................................................................80
3
Mössbauer spectroscopy of volcanic material ...........................................................................81
3.1
Introduction........................................................................................................................81
3.2
Paramagnetic minerals .......................................................................................................82
3.2.1
Olivine........................................................................................................................84
3.2.2
Pyroxenes ...................................................................................................................86
3.2.3
Ilmenite ......................................................................................................................87
3.2.4
Other important paramagnetic minerals.....................................................................88
3.3
Magnetic minerals..............................................................................................................90
3.3.1
Magnetite ...................................................................................................................90
3.3.1.1 The Verwey transition in magnetite.......................................................................91
3.3.2
Titanomagnetite .........................................................................................................91
3.3.3
Hematite .....................................................................................................................95
3.3.4
Maghemite .................................................................................................................97
3.4
The FeO-Fe2O3-TiO2 ternary diagram .............................................................................101
3.5
Other important magnetic minerals..................................................................................104
3.5.1
Goethite....................................................................................................................104
3.6
Analysis of Mössbauer spectra ........................................................................................104
4
3.6.1
Quantitative analysis of Mössbauer spectra.............................................................105
3.6.2
Examples of analysis of Mössbauer spectra: ...........................................................106
3.6.2.1 Beach sand from Skagen......................................................................................106
3.6.2.2 Mixture of components ........................................................................................108
3.7
Mössbauer spectra from Mars..........................................................................................110
3.7.1
Geology of Mars ......................................................................................................110
3.7.1.1 Magnetic field of Mars.........................................................................................112
3.7.1.2 Mineralogical Maps .............................................................................................112
3.7.2
SNC meteorites ........................................................................................................115
3.7.3
Mars Exploration Rovers .........................................................................................116
3.7.3.1 Spirit landing site .................................................................................................117
3.7.3.2 Opportunity landing site ......................................................................................123
3.7.4
Magnet results ..........................................................................................................126
3.7.5
Mars analogues ........................................................................................................132
3.7.5.1 Rocks of terrestrial olivine basalt.........................................................................132
3.7.5.2 Dust analogues .....................................................................................................136
References........................................................................................................................................138
Appendix 1: Constants .....................................................................................................................140
Appendix 2: Mössbauer parameters.................................................................................................141
Some paramagnetic iron compounds ...........................................................................................141
Some magnetic iron compounds ..................................................................................................142
Index.................................................................................................................................................143
5
1 Magnetism and magnetic Material
1.1 Introduction
1.1.1 What is magnetic material?
There are many ways to define what a magnetic material is. Often one uses the definition “It is a
piece of material that can produce observable magnetic field without the application of external
magnetic field”. This definition would apply for permanent magnets, but would on the other exclude
antiferromagnetic material and even regularly shaped iron, as pure iron does not make magnets
without help. In geosciences, a definition that makes sense is “magnetic material is a material that
can be attracted with a hand magnet”. This would include pure iron but exclude antiferromagnetic
minerals. Another way is to define magnetic material is “a material where there exists magnetic
coupling”. This last definition is more realistic to the situations used in this text, however, it turns
out that none of the above definitions are an accurate description of magnetic material. Nature has
provided much more complicated situations that can not be defined in a single sentence. The
definition of a magnetic material depends on what one is looking at, and this should become evident
from the study here.
1.1.2 Units in Magnetism
In the following chapters, vectors will always be designated with boldface letters, such as the
magnetic field H, and the length of it as italic H = |H|. However, if a direction has been introduced,
H may represent the component along that direction, and can thereby become negative.
In all the text, the SI unit system is applied consequently. Other unit systems are commonly used
in magnetism, and in these cases, one needs to be extremely careful.
Units are an important part of magnetism. They can be used to check whether calculations have
been performed correctly, and give insight into which quantities one is working with. Derived
quantities such as Joule (J) and Tesla (T) may cause confusion. Here we apply the SI units for time,
mass, length, current and temperature as seconds (s), kilograms (kg), meters (m), Amperes (A) and
Kelvin (K). Table 1 shows commonly applied units in magnetism
Table 1: Commonly used units in magnetism. The numbers show the power of the unit.
Symbol
Energy
B-field
H-field
Magnetisation
Magnetisation per unit volume
Magnetisation per unit mass
E
B
H
µ
M
Derived
unit
J
T
σ
s
kg
m
-2
-2
1
1
2
-1
-1
2
-1
2
A
-1
1
1
1
1
For example the unit for magnetisation per unit mass is [σ] = Am2/kg. Commonly used constants
can also be written in this system, and these are shown in Table 2.
6
Table 2: Common constants in Magnetism
Atomic mass unit
Avogadro’s numbera
Bohr magneton
Boltsmann constant
Electron charge
Speed of light
Permeability of vacuum
Planks constant
a
symbol
amu
NA
µB
k
e
c
µ0
h
S
-2
1
-1
-2
-1
kg
1
1
m
A
2
2
1
-1
1
1
1
1
1
2
K
-2
Value
1.66·10-27
6.02·1023
9.27·10-24
1.38·10-23
1.60218·10-19
2.99792·108
4π·10-7
1.0546·10-34
Avogadro’s number has a special unit, number/mol. Used together with the periodic system
that gives the elements molar mass in gram/mol.
Energy is often represented in different ways. Commonly in electron volts (eV) using where E
[J] = E [eV] ·e, in Kelvin’s (K), using E [J] = E [K]·k. or in wavenumbers (cm-1) where E [cm-1] =
2πe /(hc) (=8065.54) E [eV].
1.1.3 Definitions from magnetism
The behaviour of magnetic material in magnetic field is governed by the relation
B = μ 0 (H + M )
(1-1)
where H is the magnetic field, B the magnetic induction, M the magnetisation per unit volume of
the material and µ0 the permeability of vacuum µ0 = 4π·10-7 mkgs-2A-2. Outside a magnetic material
(M = 0) there is no basic difference between H and B. Instead of calling B the magnetic induction,
the designation the B-field, or the magnetic field B is used here.
The magnetic field B is the usual quantity to measure and use. Table 3 shows some of the
magnitudes of what kind of field is created by different methods:
Table 3: Different sources of magnetic field.
Method
Earths magnetic fielda
Common kitchen magnet
Usual hand magnet
Strong hand magnets
Usual electromagnets
Strong electromagnets
Superconducting magnets
Hyperfine field in ferrites
Weiss fieldb
a
µ = 10-6, bDoes not exist
B field
50 µT
10 mT
100 mT
1T
1T
2T
10 T
50 T
~300 T
1.1.4 Origin of magnetism
Magnetism originates from the current distribution and spins around an atoms. Magnetic material
can therefore be viewed as a set of many small permanent magnets. Each of them can be
represented as a current loop as illustrated in Fig. 1.
7
µ
µ
µ
µ
Iµ
Iµ
I
µ
I
I
µ
µ
I
µ
I
µ
I
µ
I
µ
I
µ
I
I
I
µ
µ
I
µ
I
I
I µ
I
I
µ
I µ
µ
µ
I
µ
I
I
I
I µ
I µ
µ
µ
I
µ
µ
µ
µ
I
I
µ
I
µ
I
µ
µ
I
µ
I
µ
I
M
I
I
I
Fig. 1: two views of magnetic material.
Inside the material, the current loops cancel each other, and the material has only surface current
density. This current density is denoted by a vector M which has direction parallel with the
magnetic moments inside the material. It has the unit of A/m which can be understood as the
surface current distribution and generates a magnetic field that can be detected outside the material.
The magnetic field B originating from this current distribution can be easily calculated using the
law of Biot and Savart as
B(r ) =
μ 0 dI × R
,
4π ∫ R 3
(1-2)
where R is a vector from the observation point, r, to the current element dI. Alternatively, one can
sum up the magnetic field from all the atoms that make up the material by using
B(r ) =
μ0
4π
⎛ 3(μ ⋅ r )r μ ⎞
− 3⎟
⎜
5
r ⎠
⎝ r
(1-3)
for each individual dipole moment µ that is given in units of Am2.
Still another view is to use the concept of magnetic poles to calculate the H field. Outside the
magnetic material, the H and the B fields are identical, except for the constant µ0. The H field is
given by
H (r ) =
M ⋅n ⋅ R
dS
3
R
poles
∫
(1-4)
where R is a vector from the magnetic pole to the observation point R and n is a normal vector to
the pole.
The importance of distinguishing between the H and B fields will become evident when dealing
with demagnetisation factors and magnetic stability, where it is of importance to be able to visualize
the difference between the two quantities. For building initiation about these quantities, it is good
idea to imagine that the H field originates from the poles, while the B field originates from the
current distribution along the sides of a magnet. For a long narrow rod, the H field, originating from
the poles, falls rapidly as 1/R3, and is relatively small inside the rod. On the other hand, the B field
that originates from the current distribution on the length is relatively larger. For a thin disk, the
8
situation is different, where the current distribution is relatively far away from the centre of the disk,
while the poles are relatively close to the interior.
1.1.5 Quantifying the magnetism
The total magnetic moment of material is usually denoted by µ, having the unit Am2. This does not
tell us about the material properties, and the usual denotations are to define magnetisation in terms
of volume M = µ/V, [M] = A/m, or by mass σ = µ/m, [σ] = Am2/kg. The relationship between these
quantities is M/σ = ρ, where ρ is the mass density.
Another important quantity is the susceptibility, χ, which is the ratio of change in the magnitude
of the magnetisation (Δμ, ΔM or Δσ) relative to the change in external magnetic field (ΔB or ΔH).
There are therefore six different ways of defining the susceptibility, and generally, reading the
literature, one will find all in use at some point of time. The natural way would be to use χ =
ΔM/ΔH, that is without a unit. Here it is on the other hand necessary to state that the SI system is
used as χSI = 4πχcgs. Most often, the mass magnetic moment over the magnetic field H is used
(Δσ/ΔH) that has the unit of inverse density m3/kg.
1.2 Magnetism of ionic material
An electron around an atomic nucleus, forms a current loop with a magnetic moment
1
μ = − eωr 2
2
(1-5)
where r is the radius of the orbit and ω is the angular velocity. This can be written in terms of the
angular momentum of the i’th electron, li = mωir2 as
μ = (−e / 2m)l i
(1-6)
The angular momentum is quantized in units of the Planks constant, ħ, and the lowest non-zero
value of µ is
μB =
eh
2m
(1-7)
which is called the Bohr magneton. This treatment applies only for the orbital part. Electrons have
additionally a spin si, where the magnetic moment becomes
μ = (−e / m)s i
(1-8)
The magnetic moment of an ion will be a sum over the contributions from individual electrons.
Each electron can be described in terms of the quantum numbers n, l, ml and ms. l can take the
values 0, 1, 2, … (n - 1). ml can take the values -l, -l + 1, … 0, … l - 1, l. The spin quantum number
ms can take the values ± ½. The total angular momentum has a maximum value given by sum over
all electron L = Σli and the total spin S = Σsi.
The electronic configuration of an atom (filled shells do not have to be considered) is usually
given in by letters that originate from spectroscopic methods.
9
L
0
1
2
3
4
Symbol
s
p
d
f
g
For example, Fe2+ has a 3d6 electronic configuration, meaning that the n = 3 shell is being filled
with 6 electrons on a d (l = 2) shell. When the electronic structure is known, one can apply the socalled Hund’s rules that are empirical rules that help in deducing the total spin and angular
momentum of an atom.
(1) Maximise the value of 2S+1.
(2) Maximize L
(3) If the shell is more than half filled J = |L+S|, less than half filled J = |L-S|, and when exactly
half filled J = S.
To illustrate how these rules are easily combined, we consider the Co2+ ion. It has an 3d7 electronic
structure, which means that the third shell is being filled with d (li = 2) electrons. One can start by
making the following table
3d7
ml
-2
-1
0
1
2
½
ms
-½
And in this table, one needs to fill in 7 electrons. Starting from the lower left, going up, and then
taking the next column automatically fullfills the Hunds rule’s, i.e.
3d7
ml
-2
-1
0
1
2
½
X
X
X
X
X
ms
-½
X
X
Now it is possible to count the angular momentum and spin contributions, giving S = + 3/2 and L
=3. According to the third Hund’s rules, one would expect J = 9/2.
The spin and the angular momentum contributions to the magnetic moment of the atom have to
be combined, and for this, one needs to form the total angular momentum vector J = L + S. From
quantum mechanics, we note that one can write
10
L2 = L( L + 1)h 2
S 2 = S ( S + 1)h 2
(1-9)
J 2 = J ( J + 1)h 2
So the contributions to the magnetic moments will be
μ L = μ B L( L + 1)
(1-10)
μ S = 2μ B S ( S + 1)
The vector J that is formed from S and L is not necessarily in the same direction as the vector µ
formed from µL and µS as the µ vector has twice the spin contribution. Since L and S can depend on
time while J does not, we can split µ into two contributions, a time independent µJ and high
frequency part µ’. As illustrated in Fig. 2.
µ'
S
J
µJ
L
µS
µL
Fig. 2: Vector model showing the combination of spins and magnetic moments of an atom.
It is now a straight forward exercise to find the magnetic moment of the atom, and one finds
μ J = g L μ B J ( J + 1)
gL = 1+
J ( J + 1) + S ( S + 1) − L( L + 1)
2 J ( J + 1)
(1-11)
where gL is the Landé g-factor. For spin-only (L = 0) it takes the value 2 and the value 1 for S = 0.
In an external magnetic field, Be, the sublevels split up, and the energy is given by
E (m) = −μ J B e = − g L μ B mBe
(1-12)
The thermal average of the magnetization μ J ↑ in the direction of the external magnetic field is
found by
11
⎛ E ( m) ⎞
⎟
kT ⎠
m=− J
= gLμB m = gLμB J
⎛ E ( m) ⎞
exp⎜ −
⎟
∑
⎝ kT ⎠
m=− J
J
μ J↑
∑ m exp⎜⎝ −
(1-13)
The result is commonly given in terms of the Brillouin function.
⎛ Jg μ B ⎞
μ J ↑ = Jg L μ B F⎜ J , L B e ⎟
kT
⎝
⎠
(1-14)
⎛⎛
1 ⎞
1 ⎞ ⎞ 1
⎛
⎛ 1 ⎞
F( J , y ) = ⎜1 +
coth⎜ ⎟
⎟ coth⎜⎜ ⎜1 +
⎟ y ⎟⎟ −
⎝ 2J ⎠
⎝ 2J ⎠
⎝ ⎝ 2J ⎠ ⎠ 2J
(1-15)
where. F(J, y) is given by
shown graphically in Fig. 3.
1
0,9
J = 5/2
J=1
J = 1/2
0,8
J=∞
F(J ,y )
0,7
0,6
0,5
0,4
0,3
0,2
0,1
0
0
1
2
3
4
5
y
Fig. 3: The Brillouin function for several values of J.
It is worth the effort to study the solution. The parameter y = Jg L μ B Be / kT is a ratio between the
magnetic ordering energy and the thermal energy. If the magnetic ordering energy dominates (y >>
1, low temperatures and/or high external magnetic fields), F approaches 1 and the magnetisation
approaches the ions saturation value μ S = Jg L μ B . For high temperatures and/or weak magnetic
field, the thermal fluctuations result in low ordering.
It should be noted that the saturation magnetisation µS = JgLµB is not the same as the total
magnetic moment of the ion due to the quantization.
For Fe3+ ions (J = 5/2, gL = 2) at room temperature, the field needed to approach saturation (y ~
1) is about 90 T. This is much higher field than is used in any reasonable experiment, and
paramagnetic saturation is only observed at low temperatures.
For low external magnetic fields, the Brillouin function can be approximated
12
y →0
F( J , y ) =
( J + 1) y
3J
(1-16)
and one finds the susceptibility
J ( J + 1) g L2 μ B2
=
3kT
d μ J↑
Be→0
dBe
(1-17)
So by measuring the susceptibility as a function of temperature, it is possible to determine the spin
state described by J ( J + 1) g L2 . The Curie constant is defined as
C=
J ( J + 1) g L2 μ B2
3k
(1-18)
so (1-17) becomes
d μ J↑
dBe
Be→0
= C /T .
(1-19)
Table 4 show magnetic susceptibilities for some common minerals.
Table 4: Susceptibilities of some diamagnetic and paramagnetic minerals at room temperature (values taken
from [Dunlop and Özdemir, 1997]
Mineral name/class
Formula
Quarts
Orthoclase feldspar
Calcite
Forsterite
Water
Pyrite
Siderite
Ilmenite
Orthopyroxenes
Fayalite
Intermediate olivine
Serpentinite
Amphiboles
Biotites
Illite
Montmorillonite
SiO2
KAlSi3O4
CaCO3
Mg2SiO4
H2O
FeS2
FeCO3
FeTiO3
(Fe, Mg)SiO3
Fe2SiO4
(Fe, Mg)2SiO4
Mg3Si2O5(OH)4
13
Magnetic
susceptibility
(10-8 m3/kg)
-0.62
-0.58
-0.48
-0.39
-0.90
30
123
100-113
43-92
126
36
~120
16-94
67-98
15
14
1.2.1 Crystal field effects and comparison to experiments
The so-called effective Bohr magneton number peff = g L J ( J + 1) , of 4f and 3d ions is shown in
Fig. 4.
4f
3d
Hund
calculation
Spin-only
Fig. 4: Experimental and calculated values of the effective Bohr magneton number
For the 4f ions (La to Lu), there is a relatively good agreement between experiment and Hund’s
rules. This is due to the fact that the 4f orbitals lie deep within the electronic structure of the atom
and are well shielded from crystal field effects. For the 3d transition metal ions in crystal field, the
situation is different. The 3d orbitals extend beyond the orbitals of other electrons, and participate in
bindings of the ion in a crystal and its valence. This means that the free ion model as described
above is not a good description for the magnetic moments of the 3d metals. For free atoms, each of
the five 3d orbitals have the same energy and electrons can move freely between them to generate
angular orbital momentum. The effect of the crystal field remove this degeneracy, and for 3d
metals, the magnetic moment can be described with a spin-only magnetic moment. This effect is
often referred to as ”quenching of the orbital momentum”. Here gL = 2 and J = S. The extent of the
3d orbitals has other effects that have to be taken into account. Fig. 5 shows a picture of the 3d
orbitals.
14
Fig. 5: Picture of the 3d orbitals (St. from
http://chemed.chem.purdue.edu/genchem/topicreview/bp/ch12/crystal.php).
Placing an atom in a crystal with octahedral symmetry, i.e. we place an atom along each of the axis
at ±x, ±y and ±z. The z2 and x2-y2 will be compressed more than each of the other orbitals resulting
in higher energy of them. The opposite is the case in tetrahedral symmetry. Fig. 6 shows the
splitting in different geometries
e2g
t2g
free ion
ΔO
ΔT
t2g
e2g
tetrahedral
oktahedral
Fig. 6: Level splitting in octahedral and tetrahedral symmetries for 3d metal ions.
The 3d orbitals are usually split up into two sets, the t2g set of dxy, dxz, and dyz orbitals, and the e2g set
of d x 2 − y 2 and d z 2 orbitals. If the splitting constants ΔO and/or ΔT are large, this could lead to the low
spin species.
1.3 Ferromagnetism (Weiss theory)
In the Weiss theory, it is assumed that there is an internal magnetic field Bm that originates from all
the magnetic polarization of the rest of the material. This so-called Weiss field or molecular field
would have to be of the order of hundreds of Tesla, which is not realistic. This is due to the fact that
the underlying polarisation does not originate from an interaction with an internal magnetic field but
15
from exchange interactions, to be described later. However, the Weiss theory is still a useful theory
to understand the general trends of ferromagnetism. We write
Bm = γ m μ J ↑
(1-20)
where γm is the Weiss constant or the molecular field coefficient. As defined here, it has the unit
T/(Am2), and can be understood as the magnetic field B produced by single unit of Am2. To find the
spontaneous magnetisation one needs to solve
⎛ Jg L μ Bγ m μ J ↑
= F⎜ J ,
⎜
Jg L μ B
kT
⎝
μ J↑
⎞
⎟
⎟
⎠
(1-21)
The left hand side of this equation, called the reduced magnetisation, is a straight line in μ J ↑ ,
while the right hand side follows the form given in Fig. 7. For very low temperatures, F() will rise
μ J↑
quickly and there exists a solution close to
= 1 . At a given temperature, TC, there exists only
Jg L μ B
the
μJ↑
Jg L μ B
= 0 solution, and this happens at the temperature
J ( J + 1) g L2 μ B2 γ m
TC =
3k
(1-22)
called the Curie temperature. Eq. (1-21) can be simplified slightly in terms of the ion saturation
magnetisation and the Curie temperature, and takes the form
μ J↑
μS
⎛
μ
3J
= F⎜ J ,
⋅ J↑
⎜ ( J + 1)(T / TC ) μ S
⎝
⎞
⎟
⎟
⎠
Fig. 7 shows the solution of as a function of T/TC and Table 5 lists some important Curie
temperatures.
16
(1-23)
1
J = 5/2
Reduced magnetization
0,9
J = 1/2
0,8
J=∞
0,7
0,6
0,5
0,4
0,3
0,2
0,1
0
0
0,2
0,4
0,6
0,8
1
T/T C
Fig. 7: Magnetisation curves in the Weiss theory
Table 5: Curie temperatures of selected compounds
Compound
Formula
iron
cobalt
nickel
gadolinium
magnetite
Fe
Co
Ni
Gd
Fe3O4
hematite
α-Fe2O3
γ-Fe2O3
δ-FeOOH
maghemite
Feroxyhite
Curie temperature
(K)
1043
1388
627
293
858
948
903
93
1.3.1 Susceptibility above the Curie temperature
By adding a small external magnetic field above the Curie temperature, where μ J ↑
μ S ≈ 0 , Eq.
(1-21) becomes
μ J↑
μS
=
J + 1 μS
Be + γ m μ J ↑
3 J kT
(
)
(1-24)
which is easily solved and gives
d μ J↑
dBe
=
C
T − TC
With C as the Curie constant defined in (1-18).
17
(1-25)
The difference between the susceptibility found here and the susceptibility found for
paramagnetic ions (Eq. (1-19)) is subtraction of the Curie temperature from the temperature.
Generally, this will result in higher susceptibilities at temperatures just above the Curie temperature
than would be expected from purely paramagnetic material. Due to experimental reasons, Eq. (1-25)
is usually written
d μ J↑
dBe
=
C
T −θ p
(1-26)
With θp slightly larger than the Curie temperature.
1.3.2 Susceptibility below the Curie temperature
At the Curie temperature, we expect high susceptibilities. This is where thermal forces that try to
randomize the magnetic moment compensate with internal forces (The Weiss field) and the effect of
a small external magnetic field is greatest. At lower temperatures, the Weiss field dominates, and
the effect of a small magnetic field may be reduced resulting in lower susceptibility. The details,
however, depend more on the domain structure of the sample, and general description is not
possible. Fig. 8 shows a typical example of the susceptibility of a natural sample containing
magnetite (Fe3O4).
Fig. 8: Typical susceptibility curve for sample containing magnetite (Fe3O4). The room temperature
susceptibility was χ = 0.148 SI.
Around the Curie temperature, one observes a spike in the susceptibility that becomes lower at
lowering temperatures. At temperatures below room temperature, magnetite has a phase transition
(Verwey transition), that is also picked up in the susceptibility.
One rule of thumb that is often applied is that the expected susceptibility is χ = CσS with C =
1/30 kg/(Am2), and χ is given in dimensionless SI units. This empirical law gives the susceptibility
in many classes of natural material within a factor of 2.
18
1.3.3 Antiferromagnetism
In antiferromagnetism one has two sublattices, A and B that couple in such a way that the spins on
the one sublattice are in opposite direction to the other. This case is possible to examine with the
Weiss theory, defining two interaction parameters q1 and q2, where q1 represents the contribution
from the same type of lattice atoms and q2 is the contribution from the other type of lattice atoms.
The field observed on the A sublattice will originate from the polarisation from both the A and B
sublattices, and the same applies for the B sublattices.
BWA = − q1 μ A − q 2 μ B
(1-27)
BWB = − q 2 μ A − q1 μ B
Where µA,B is a shorthand writing for μ JA↑, B . The convention is to have negative signs in Eq.
(1-27) and require q1, q2 > 0. Since µA = -µB, we obtain for the A site BWA = μ A (q 2 − q1 ) , that can
be inserted into Eq. (1-21) giving formally the same result as in the case of ferromagnetism with γm
= (q2 - q1). For antiferromagnetic material, one uses the term Neel temperature, TN, instead of Curie
temperature TC.
Table 6: Neel temperatures of selected compounds
Compound
Formula
Neel temperature
(K)
120
57
ulvöspinel
ilmenite
Fe2TiO4
FeTiO3
Goethite
lepidocrocite
α-FeOOH
γ-FeOOH
480
77
1.3.3.1 Susceptibility of antiferromagnetic material
Adding a small magnetic field above the Neel temperature, the magnetisations of each sublattice
become
J +1
( Be − q1 μ A − q 2 μ B )
3 JkT
J +1
μ B = μ S2
( Be − q 2 μ A − q1 μ B )
3 JkT
μ A = μ S2
(1-28)
The susceptibility per atom, is now
μA + μB
2 Be
=
C
T + C (q1 + q 2 )
(1-29)
As q1+q2 is larger than 0, this result means that one has finite susceptibility at the Neel temperature.
For Antiferromagnetic material, a qualitative picture of the susceptibility is that at T << TN, the
spins are held in place by the Weiss field, and a small external field is not able to have significant
effect on the system. Unlike the ferromagnetic case, domains are not moving around the materials.
At T >> TN, the material is governed by thermal fluctuations, and behaves as a paramagnetic
19
material. At T ~ TN, the internal Weiss field and the thermal fluctuations are of the same magnitude,
and the small external field has the best possibility of changing the magnetisation, which is
represented by a peak in the susceptibility, example wise illustrated in Fig. 9.
Fig. 9: Typical susceptibility of material with Neel temperature of 2.5 K.
1.3.4 Ferrimagnetism
In oxides, Ferrimagnetism is the more common type of magnetism. Generally, there are two or
more sublattices, that couple either antiferromagnetically or ferromagnetically. If there are unequal
numbers of atoms in the two sublattices, there may be an external magnetic moment.
1.3.4.1 Spinel ferrites: magnetite
An important group of ferrites are the spinel ferrites. Spinels have cubic crystal structure with the
general formula A[B2]O4 where A denotes atoms on tetrahedral sites and B denotes atoms on
octahedral sites. Magnetite is an inverse spinel Fe3+[Fe3+, Fe2+]O4 shown in Fig. 10
Fig. 10: Lattice structure of magnetite.
The structure can be viewed as sheets of O atoms in the <111> direction. In-between every second
sheet there are the B sites and in-between the others are both A and B sites. The magnetic properties
20
arise from the fact that the A and B sublattices are antiferromagnetically coupled. For magnetite,
this means that one has a net saturation moment of 4µB per formula unit. The saturation
magnetisation per unit mass can easily be calculated as
σS =
4μ B N A
Am 2
= 96.2
3 ⋅ Mw(Fe) + 4 ⋅ Mw(O)
kg
(1-30)
Which is very close to the experimental values. Table 7 shows experimental room temperature
values of the saturation magnetisation of selected materials.
Table 7: Saturation magnetisation of selected materials.
Material
Composition
Iron
Magnetite
Maghemite
Titanomagnetite
Hematite
Goethite
Pyrrhotite
Greigite
α-Fe
Fe3O4
γ-Fe2O3
Fe2.4Ti0.6O4
α-Fe2O3
α-FeOOH
Fe7S8
Fe3S4
σS
(Am2kg-1)
218
92
75
25
0,47
0,12
18
31
1.3.5 Summary of susceptibility.
Graphically, susceptibility as a function of temperature is often illustrated as 1/χ. For paramagnetic
material, this gives a linear dependence starting at the origo.
1/χ
antiferromagnetic
1/χ
ferromagnetic
paramagnetic
θ
TN
T
ferrimagnetic
θ
TC
θ
T
Fig. 11: Variation of reciprocal susceptibility with temperature.
1.3.6 Exchange interactions
Exchange interactions originate from the overlap of wave function is therefore a very short range
interactions. This interaction can be written as
Eex = −2 J ex S1 ⋅ S 2
21
(1-31)
The so-called exchange integral, Jex, requires complicated quantum mechanical calculations that are
beyond the scope of this text. If Jex is positive, Eq. (1-31) describes ferromagnetism and otherwise a
antiferromagnetic situation. For a single spin, Si, the exchange energy can be written as a
Eex ,i = −2 J ex ∑ S i ⋅ S j
j
(1-32)
If the sum is only taken over z nn neighbours, this can be written
Eex ,i = −2 J ex S i ⋅ S z
(1-33)
which is formally the same type of interaction as was introduced as a magnetic field in the Weiss
theory with
J ex =
g L2 μ B2 γ W
2z
(1-34)
1.4 Domain magnetism
If only the exchange interactions dominated a ferromagnetic material, we would expect all
potentially magnetic material to act like permanent magnets. However, permanent magnets have
magnetostatic energy, and this is minimized by breaking the magnetic structure up into magnetic
domains of higher width/length ratio. This interaction counteracts with the anisotropy energy that
tries to hold the spins in place along some crystal axis.
1.4.1 Magnetocrystalline energy
In solids, there may be a number of reasons that lead to a favourable direction of the magnetism.
These can be caused by the shape of the magnet, surface effects and many more. Here we discuss
the one caused by crystal effects.
In a crystalline solid, the magnetisation will be pointed toward some of the so-called easy
directions. The simplest case is a uniaxial easy axis, where one direction is the most stable direction
for magnetisation. To describe this, one can describe the energy minimum in terms of infinite series
in the angle as
ε K = K1 sin 2 θ + K 2 sin 4 θ + ... = ∑ K n sin 2 nθ
(1-35)
Where the Kn are empirically constants of unit J/m3, called the anisotropy constants. Their value
depends on the material and may vary with temperature in nanoparticles. For cubic structures, one
can write
ε K = K1 (α 12α 22 + α 12α 32 + α 22α 32 ) + K 2 (α12α 22α 32 ) + ...
(1-36)
Where αi are the direction cosines of the magnetisation direction with respect to the cubic axes of
the crystal. Often K1 is sufficient to describe the anisotropy, and often one uses K without a
subscription as effective crystal anisotropy constant. Table 8 shows some K constants for different
material.
22
Table 8: Anisotropy constants in selected materials.
Material
Iron
Cobalt
Nickel
Sm-Co magnet
Nd-Fe-B magnet
Maghemite
Hematite
Magnetite
K1 (Jm-3)
4.7·104
4.1·105
5.1·103
1.1·107
6.0·106
-4.65·103
1.2·106
-1.35·104
Composition
Fe
Co
Ni
SmCo5
Nd-Fe-B
γ-Fe2O3
α-Fe2O3
Fe3O4
1.4.2 Magnetostatic energy
Magnetostatic energy originates from the interactions of spins with the magnetic field H inside the
material. This is often called the demagnetisation field HD. The H field inside the magnetic material
can be written as
H = -NM
(1-37)
where N is the so-called demagnetisation factor, depending on the geometry of the magnet. For a
thin disk-shaped magnet N → 1 and for a long rod, N → 0. The demagnetisation field is generally
not homogeneous inside the magnet, neither in direction nor intensity, so Eq. (1-37) must be
regarded as a simplification. The only case where it is homogeneous and parallel to the
magnetisation throughout the material is in the case of ellipsoids. Fig. 12 shows the average
demagnetisation factor calculated for cylinders
1
0,9
0,8
0,7
l
N
0,6
M
0,5
w
0,4
0,3
0,2
0,1
0
0
2
4
6
8
10
Length/width ratio
Fig. 12: Average demagnetisation factor N in cylinders.
The self energy is
Es =
1
NVμ 0 M 2
2
(1-38)
Where the factor ½ comes because mutual interactions are not included. The self energy is lowered
each time one splits the magnet up into domains, that have a higher length/width ratios, and
according to this, one would expect the magnet to break up into domains until the exchange
23
interactions would play a role. However, forming a domain wall costs energy, both because of the
increase in exchange energy and due to increase in Magnetocrystalline energy.
1.4.3 Domain walls
The rotation can be described after distance as θ(x). For a uniaxial anisotropy ε = Kcos2θ, one could
set θ(0) = 0 (centre of wall) and expect that at large distances, X that θ (± X ) = ±π / 2 . The exchange
energy for the series of atoms would here be described with Eq. (1-31) as
Fig. 13: Illustration of a domain wall
N
E ex = 2∑ − 2 J ex S 2 cos(θ i −1 − θ i )
(1-39)
i =1
where N = X/a, were a is the lattice constant and the sum is taken twice. Using θ i −1 − θ i ≈ aθ&( xi ) ,
Eq. (1-39) can be interchanged with an integral
E ex = −
4 J ex S 2 X
cos aθ& dx
a ∫0
( )
(1-40)
Assuming that θ varies slowly, one obtains
E ex =
4 J ex S 2
a
⎛ (aθ&) 2
⎞
∫0 ⎜⎜⎝ 2 − 1⎟⎟⎠dx
X
(1-41)
For the anisotropy energy, we obtain
E an = K a
3
N
∑ cos
2
θ =K a
X
2
i =1
∫ cos θ dx
2
(1-42)
0
In order to solve the equation and obtain the functional form of θ(x), one can use the principle of
variation. Here the path function I
I = ∫ f ( y, y& , x)dx
24
(1-43)
is minimized if
∂f
d ⎛ ∂f ⎞
− ⎜⎜ ⎟⎟ = 0
∂y dx ⎝ ∂y& ⎠
(1-44)
Aθ&& + K sin θ cos θ = 0
(1-45)
This gives the differential equation
with
A≡
2 J ex S 2
a
(1-46)
Though the solution of (1-46) is not trivial, one can see the important result that the form of
solutions with the same A/K must be the same. The width of the wall, δw, can be estimated in
different ways, and a convenient way is to take the slope of the θ function at origo, and define the
half width where it intercepts θ = π/2. This gives the thickness parameter
δw = π A/ K
(1-47)
This relationship is often given as δ w = D A / K , where D ∈ (3 − 4) . The wall energy per unit area
is usually given as
γw = G AK
(1-48)
With G = 1. Generally one applies G ∈ (1 / 2 − 2) . It is now possible to find the critical radius for
single domain particles.
rC
M
Em :
Ew :
M
VN (1) μ 0 M 2
2
=
M
2(V / 2) N ( 2) μ 0 M 2
2
πrC2γ w
0
25
Using the values N(1) = 1/3 and N(2) = 1/6 one obtains
rC =
9γ w
μ0 M 2
(1-49)
Which is known as the single domain/multi domain critical radius.
Table 9: Theoretical and experimental values for the critical SD/MD radius of selected materials.
Material
Iron
Magnetite
Maghemite
Titanomagnetite
Hematite
Phyrotite
Critical SD/MD
radius (µm)
Theory
Exp
0,017
0,023
0,082
0,055
0,09
0,6
0,6
15
1,6
Composition
α-Fe
Fe3O4
γ-Fe2O3
Fe2.4Ti0.6O4
α-Fe2O3
Fe7S8
1.4.4 Single domain particles magnetisation properties.
Single domain particles play an important role in magnetism. First of all, they are the simplest
system to calculate the magnetic properties of, and this problem is examined below. In geology,
they are often the bearer of the remanent magnetisation as they can have remanent magnetisation
two orders of magnitude larger than multi domain particles.
We assume that we have a single domain particle with easy axis along the vector n which can be
denoted with n = [sinϕ, cosϕ]. The general magnetic moment µ = µ[cosψ, sinψ]. The external
magnetic field is along the z axis as Be = [0, Be] as is illustrated in Fig. 14.
n
µ
ϕ
ψ
Be
Fig. 14: Definition of parameters for a single domain particle.
The total energy of the particle is then written
E = KV sin 2 (ψ − ϕ ) − μBe sinψ
Starting by letting ϕ = π/2 (easy axis along z-axis), one obtains:
26
(1-50)
E = KV cos 2 ψ − μBe sinψ
(1-51)
Which is illustrated for few combination of b ≡ µBe/KV.
Energy (arb. units)
(A)
b=3
(B)
b=1
(C)
b = -1
(D)
b = -2
(E)
b = -3
−π −π/2 0 π/2
(F)
b=2
π −π/2 0 π/2
π
ψ
Fig. 15: Energy diagram for single domain particles with easy axis parallel to the external magnetic field.
Starting with a high positive field b = µBe/KV = 3, a single solution is found at ψ = π/2 (Fig. 15
(A)). The magnetisation along the field direction is hence µsin(ψ) = +µ. As the field is decreased to
b = 1, a new, less stable, solution emerges at ψ = π/2 (Fig. 15 (B)). Decreasing the field further to
negative fields (b = -1; Fig. 15 (C)), this solution becomes the most stable one, however, here it is
important to keep in mind that the magnetic moment is locked in place at the ψ = π/2 minimum and
no change happens to the magnetisation of the particle. At b = -2, the ψ = π/2 minimum becomes a
saddle point (Fig. 15 (D)), and the magnetisation moves toward the ψ = -π/2 minimum. This means
also that the magnetisation changes to -µ. Decreasing the field further (b = -3; Fig. 15 (E)), does not
change the minimum nor the magnetisation anymore. When the magnetic field is increased again,
the ψ = -π/2 minimum becomes unstable at b = 2 and the magnetisation changes direction again.
This scenario is shown in Fig. 16.
27
magnetisation/µ = sin(ψ)
(C)
(B)
(A)
1
(D)
b = µBe/KV
-3 -2 -1
1
2
3
(G)
(F)
-1
Fig. 16: Magnetisation curve of a single domain particle with easy axis along the external field direction.
In the case ϕ = 0, the energy becomes
E = KV sin 2 ψ − μBe sinψ
(1-52)
For highly positive fields, there is a single stable direction at ψ = π/2. Below b = 2 this minimum
splits up into two minima as illustrated in Fig. 17. In the range from b = 2 to b = -2, the
magnetisation of both minima is the same, sinψ = b. At b = -2 the two minima emerge again at ψ =
π/2 resulting in a single solution of magnetisation +µ.
(A)
b=3
magnetisation/µ = sin(ψ)
(B)
b=1
(A)
1
(C)
b=0
(B)
b = µBe/KV
(C)
(D)
b = -1
-3 -2 -1
1
2
3
(D)
-1
Fig. 17: Energy diagram for single domain particles with easy axis perpendicular to the external magnetic field.
In this case, the hysteresis loop has no opening. Randomly oriented single domain particles have a
wide range of solutions, and the average is shown in Fig. 18.
28
Fig. 18: Hysteresis loops for single domain particles.
Single domain particles have high saturation remanent magnetisation σSr/σS = 0.5 and relatively
high coercivity, µ0Hc = 0.958b.
1.4.5 Shape anisotropy
Another important type of anisotropy is the so-called shape anisotropy. Consider a magnetic particle
of the shape of ellipsoid without magnetocrystalline energy (K = 0). For a particle that is symmetric
along the x axis, the self energy can be written as
(
1
Es = Vμ 0 M 2 N x cos 2 ψ + N y , z sin 2 ψ
2
)
(1-53)
Where ψ is the angle between the magnetisation vector and the long axis. Shape anisotropy can be
the most important source of anisotropy for the magnet. In (some) Alnico magnets, anisotropy is
obtained by having needle shaped particles embedded in a non-magnetic matrix. In magnetic
needles made of iron, the shape anisotropy ensures that the needle has preferential direction.
1.5 Superparamagnetism
It is of interest to make magnetic particles as small as possible for e.g. magnetic memories. There
are on the other hand limits to how small they can practically become. Taking the uniaxial case, the
magnetisation can take two directions, θ = 0 and θ = π separated by a barrier of KV. When the
barrier becomes similar to the thermal energy kT, one can imagine jump or tunnelling through this
barrier. The rate of which this jump takes place, or the relaxation time, is governed by the equation
⎛ KV ⎞
⎟
⎝ kT ⎠
τ = τ 0 exp⎜
(1-54)
Where τ0 is related to the crystal vibration properties and is of the order 10-10 s-1. If the particle
volume is lowered and/or the temperature increased, the jump frequency is increased. The
29
Neodymium-iron-boron (Nd-Fe-B) permanent magnets, have anisotropy constant of 6·106 J/m3. If
one would use them as magnetic memories, one would need to ensure a magnetic memory to hold
for at least 100 years (= 3.1·109 s) at room temperature the minimum radius of particles would be 2
nm. If the radius is reduced by only 10%, the relaxation time would become few hours.
The properties of a material containing superparamagnetic particles depend significantly on the
method used. Magnetic methods have a time constant of the order of seconds to minutes, so for
system consisting of superparamagnetic particles, the information on magnetisation may be lost,
and superparamagnetic particles have usually low saturation remanence and coercivity.
In nature, the iron containing particles precipitated from aqueous solutions, are often of the order
of few nm in diameter. This is the size range that can be probed with Mössbauer spectroscopy, and
Mössbauer spectroscopy can hence be used to determine the size distribution of the crystallites.
1.6 Magnetisation measurements
The most common tool for measuring a magnetisation of a given sample is the vibrating sample
magnetometer (VSM). A diagram of such a system is shown in Fig. 19.
Fig. 19: Illustration of Vibration Sample Magnetometer.
The sample is vibrated up and down in a homogeneous external magnetic field produced by e.g. an
electromagnet. The induced magnetic flux is measured by pick up coils, that give a variable signal
which is proportional to the magnetic moment of the sample.
30
1.6.1 Interpretation of the magnetisation curve
The magnetisation curve can contain a wealth of information on a system. Fig. 20 shows a typical
magnetisation curve.
Magnetization, σ [Am2/kg]
Saturation
magnetization, σS
Saturation remanence
magnetization, σSr
"Virgin" curve
Remanence
magnetization, σSr
coercivity,
Bc = μ0Hc
Magnetic field, B [T]
susceptibility,
χ0=ΔM/ΔH
Fig. 20: Typical magnetisation curve.
There may be a small magnetisation without an external magnetic field. For volcanic samples, this
is an indicator that the sample has solidified in an external field. Both the magnitude and direction
can give information on the geological history. This remanent magnetisation is usually given in
units of A/m, and can be estimated as the product of four quantities:
M r = ρ ⋅ wt.%(mag) ⋅ (σ r / σ S ) ⋅ σ S ,mag
(1-55)
First is the density ρ, to get the unit correct. Then the weight percentage of magnetic phase, only
showing that more magnetic phase, the more remanence is possible. The ratio (σ r / σ S ) depends on
the domain structure and the external magnetic field during solidification. In a magnetic field of
terrestrial magnitude (~50 µT), SD and PSD particles this ratio is of the order 10-2, while for MD
particles, this ratio is commonly 10-4. For sediments, this ratio is of the order 10-6. A special case is
MD hematite that can acquire the value 0.5. Finally σ S ,mag is the saturation magnetisation of the
magnetic phase.
Slowly cooled basalt contains usually 2 wt.% of MD magnetite (Fe3O4), that has a saturation
magnetisation close to ~100 Am2/kg. In this case, one would estimate the remanent magnetisation
of the order 0.6 A/m, which is close to the 1-2 A/m observed.
For a small external magnetic field, typically of the order of few mT, the change in
magnetisation curve is linear and reversible. For a sample that contains a mixture of paramagnetic
and magnetic minerals (geological samples), it may be difficult to interpret this value in a clear
manner. For pure materials, it can however be used to determine the spin state of the system.
At certain magnetic field, the magnetisation curve starts to change in an irreversible manner due
to the irreversible movement of domain walls, and continues to rise until it reaches saturation value.
Determination of the saturation magnetisation is an important indicator of the presence of highly
magnetic phases such as magnetite and/or maghemite. See Table 7 for saturation magnetisation
values.
Among parameters determined from magnetisation measurements are the reduced saturation
magnetisation (σSr/σS) and coercivity (µ0Hc). It was shown earlier that for a randomly oriented
single domain (SD) particles, the reduced saturation magnetisation was of the order of 0.5. For both
superparamagnetic (SP) and multi-domain (MD) particles, this value is lower. Fig. 21 shows this for
magnetite particles.
31
Fig. 21: Reduced saturation magnetisation of magnetite particles.
The critical SD/MD radius for magnetite is close to ~0.055 µm, and in this range σSr/σS values
approaching 0.5 are observed. For particles approaching 0.2 µm, the σSr/σS value is down to 0.1.
The coercivity tells a similar story. For random SD particles, the coercivity is bc = µBe/(|K|V) =
0.958, that gives coercivity µ0Hc = 0.958 K/M = 28 mT, which is similar to what is seen in Fig. 22.
Fig. 22: Coercivity as a function of particle size for magnetite.
1.6.2 Inversion of maghemite (γ-Fe2O3 -> α-Fe2O3)
The transformation of pure maghemite to hematite is well known and takes place at T > 300oC. Fig.
23 shows the magnetisation of synthetic maghemite before and after annealing.
32
100
0,4
Hematit
80
Magnetisation (Am2/kg)
Magnetisation (Am2/kg)
Maghemit
60
40
20
-1,5
0
-0,5 -20
-40
-60
-80
0,5
B decr. Experim.
B inc. Experim.
B decr. Model
B inc. Model
1,5
0,3
0,2
0,1
0
-1,5
-0,5
-0,1
-0,2
-0,3
-100
0,5
B decr. Experim.
B inc. Experim.
B decr. Model
B inc. Model
1,5
-0,4
Magnetic field (T)
Magnetic field (T)
Fig. 23: Magnetisation measurements of 2 µm maghemite particles, and hematite particles obtained after
annealing of the maghemite at 600oC over night.
The data has been analysed using the empirical formula
σ± =
σ S ( B m µ0 H c )
B m µ0 H c + γ
(1-56)
And the results are shown in Table 10
Table 10: Results from analysis of the magnetisation measurements of synthetic maghemite.
µ0Hc
σS
γ
(T)
(Am2/kg)
(T)
maghemite 79(2) 0.040(3) 0.019(1)
Hematite
0.39(6) 0.41(9) 0.15(3)
Sample
σr/σS
0.32(3)
0.27(8)
The saturation magnetisation of the maghemite is in rough agreement with table values. For 2 µm
particles, one would expect low saturation remanent magnetisation, but the particles are needle
formed and have significant shape anisotropy. After annealing, the coercivity increases significantly
due to the higher K/M ratio of hematite.
1.6.3 Magnetic properties of permanent magnets
1.6.3.1 Hard magnets
Permanent magnets that can be bought from stores have many different properties, depending on
the application. This involves e.g. the coercivity and the saturation remanent magnetisation, and in
all cases one needs to bear in mind the thermal stability of these quantities. VACUUMSCHMELZE
in Germany has many different types of magnetic material, and one can view their properties online at http://www.vacuumschmelze.de. Fig. 24 shows the magnetic properties of their VACODYM
722 HR magnetic material.
33
Fig. 24: Magnetic properties of VACODYM 722 HR from VACUUMSCHMELZE
At room temperature, the magnets produce surface magnetic field of ~1.5 T. If we were to make a
thin disk of such material, the internal demagnetization field would be of the order of -1200 kA/m,
leading to demagnetization of the magnetic material. The magnetic properties of VACODYM 225
HR magnet are shown in Fig. 25.
Fig. 25: Magnetic properties of VACOMAX 225 HR from VACUUMSCHMELZE
Thin disk would have internal field of -870 kA/m meaning that it would loose the magnetic strength
significantly. Both magnets are based on SmCo5 alloy material. This particular magnet has excellent
thermal stability and has been used in magnetic targets on Mars missions.
1.6.3.2 Soft magnets
Material with soft magnetic properties (no remanent magnetisation or coercivity) are of importance
in transformers. The ideal material has a high saturation magnetization, and no coercivity. The loss
in transformers is proportional to the opening of the hysteresis loop.
1.7 Magnetic forces and the magnetisation curve
The force on a magnetic particle in a magnetic field can be written
F = mσ∇B
34
(1-57)
The force is caused by the magnetic field gradient and not the magnetic field as such. However, for
paramagnetic material, the magnetic field is needed to magnetise the material, in order for the
magnetic field gradient to have an effect. In such case, one needs to describe the magnetisation as σ
σ B
= χB or generally for magnetic samples as σ ( B) = S .
B+γ
1.7.1 Paperclip magnetism
Paperclips are basically MD particles with low coercivity. In a weak magnetic field, it’s
magnetisation can be described with σ = χBe. Lets hypothetically place a cylindrical magnet
(diameter 20 mm, height 10 mm) and lets assume it has internal magnetisation of µ0M = Bwp = 1.0
T) as shown in Fig. 26.
M
(0,0,0)
x
Fig. 26: Definition of paperclip magnetism
The magnetic field as a function of distance along the x direction is illustrated below.
0.5
B (T)
0.4
0.3
0.2
0.1
0
-40
-30
-20
-10
0
10
20
30
x (mm)
The magnetic field is highest inside the magnet, but falls fast outside the magnetic material. The
magnet is too short for the internal magnetic field to reach the work-point value (1 T). The magnetic
field gradient dB/dz is shown below.
35
40
30
dB/ dx (T/m)
20
10
0
-10
-40
-30
-20
-10
0
10
20
30
-20
-30
-40
x (mm)
The magnetic force is proportional to B·dB/dx, which is illustrated below
15
2
B ×dB/ dx (T /m)
10
5
0
-40
-30
-20
-10
0
10
20
30
-5
-10
-15
x (mm)
At negative z there is a positive force that pushes the paperclip toward the magnet. At positive x the
force is negative, again pushing the paperclip toward the magnet.
Below is the situation if we place two magnets together separated with 30 mm.
0.5
20
40
15
30
0.3
0.2
0.1
20
2
dB/ dx (T/m)
0.4
B (T)
50
10
0
-10 -25
-15
-5
5
15
-20
-30
0
-25
-15
-5
5
15
x (mm)
25
35
45
B×dB/ dx (T /m)
0.6
25
35
45
10
5
0
-5
-25
-15
5
15
-10
-40
-15
-50
-20
x (mm)
If the two magnets are placed antiparalell, the situation is slightly different:
36
-5
x (mm)
25
35
45
0.4
20
40
15
0
-25
-15
-5
5
15
25
-0.2
35
45
20
2
dB/ dx (T/m)
30
0.2
B (T)
50
10
0
-10 -25
-15
-5
5
15
-20
-30
-0.4
25
35
45
10
5
0
-5
-25
-15
-5
5
15
25
35
45
-10
-15
-40
-0.6
B×dB/ dx (T /m)
0.6
-50
-20
x (mm)
x (mm)
x (mm)
The magnetic forces are the same regardless of whether the magnets are coupled parallel or antiparallel.
Paperclips usually have a small, but significant magnetic memory. This can be verified with the
following exercise:
A
B
C
A paperclip is magnetised, and a second one can hang onto it (A). As the magnet is removed, there
is still a small hysteresis that make the paperclips hold to each other (B). If the magnet is turned
around, and brought back, the magnetisation is reversed, and as it becomes zero, the second
paperclip will fall down.
This system, though simple can be used to test various properties of magnetisation. Try for
example to first magnetise two paperclips, and then bring them together. Most likely, they will not
attract each other, but this can be explained from the magnetisation curves.
37
2 Mössbauer spectroscopy
2.1 Introduction
Mössbauer spectroscopy is among very few nuclear methods that are named after a person. Among
the reasons, is that the technique was discovered rather late (1957) and a specific person (Rudolf
Mössbauer) found the effect when it was considered theoretically impossible. The technical name
for Mössbauer spectroscopy is “Recoil-free resonance emission and absorption of gamma-quanta”
gives better details of what the method is all about. Resonance emission and absorption means that
it is an absorption technique, where a decay of a state is absorbed resonantly in e.g. a sample.
gamma-quanta means that it has to do with the high energy nuclear states. These high energies are
usually associated with recoil, meaning that resonance conditions are not fulfilled. Mössbauer’s
discovery was that this recoil could be neglected to some degree in solid materials, giving the
conditions for resonance absorption. The basic beauty of technique arises through the uncertainty
relationship:
ΔE =
h
Δt
(2-1)
Where ΔE is the energy resolution of the technique, and Δt is the time-uncertainty, which in terms
of Mössbauer spectroscopy is the lifetime, τ, of the excited state of the nuclei. For 57Fe, the first
exited state with energy E0 = 14.4 keV has lifetime of τ = 140 ns, which gives energy resolution of
5·10-9 eV. With such resolution, the interactions between the nuclei and the electron structure of the
atom can be probed, giving access to some quantities that are out of reach for other techniques. To
modify the source energy a simple Doppler movement can be used where the energy becomes E(v)
= E0(1+v/c). For velocities of the order of cm/s, the energy of the source is modified by 100 times
the energy resolution, giving the possibilities to probe the nuclei energy levels.
In Mössbauer spectroscopy, the convention is use Doppler velocity scale instead of energy, and
the preferred unit is mm/s. In this unit, the natural line-width in 57Fe becomes
ΓN = ch /( E 0τ ) = 0.097 mm/s.
2.1.1 Experimental setup
Fig. 27 shows a block diagram of a Mössbauer measurement
38
Function
generator
Single-channel
analyser
Multi-channel
analyser
Drive controller
Spectroscopic
amplifier
Mössbauer drive
Detec
tor
Pre-amplifier
Fig. 27: Block diagram of an Mössbauer experiment in transmission mode.
The function generator sends out a triangular signal to the drive control unit and to the multichannel analyser (MCA). Usually, the function generator gives three signals, the analogue output to
the drive controller and a signal sending a pulse at the start of each period, and a signal sending 512
pulses within a period to let the multichannel analyser know when to start measuring in each
velocity bin.
The common type of drive system is a double-loudspeaker type 1 . The first loudspeaker drives
the source, while the other loudspeaker measures the velocity. A feedback between the two
loudspeakers makes the drive run according to the input signal.
There are many different types of detectors used, but the most common is a proportional
counters with 1-2 bar Xe gas. Kr gas detectors have been used especially for 57Fe Mössbauer
spectroscopy, but these are a bit more complicated to operate due to the Kα edge.
These type of gas detectors are more or less insensitive to the 122 keV radiation when 57Co is
used, but may have very limited resolution (~5 keV when a strong source is used). The output of the
preamplifier consists of long lived pulses (~150 µs), and pulses arriving at similar times will come
on top of each other. The Spectroscopic amplifier uses only the rise time, and delivers ~10 µs
pulses. These are sorted by the single channel analyser to pick up the 14.4 keV radiation, that is sent
to the MCA.
2.2 The f-factor
The f-factor, is the probability that the radiation is emitted without recoil. To investigate this, one
needs to take a look at the recoil energies involved.
When an energy level decays, sending a γ-ray in specific direction of energy Eγ, the nuclei is
given a recoil (ER) with momentum in the opposite direction. The momentum of the γ is Eγ/c, and
the momentum of the nuclei is (2mER)½, where m is the mass of the nuclei. As Eγ = E0 - ER ≈ E0 (if
ER << E0), we find
E 02 Fe −3
ER =
≈ 10 eV
2mc 2
57
1
(2-2)
Sometimes called Kankeleit drives after the person that first applied them. Kankeleit research group was very
interested in the technical development within Mössbauer spectroscopy, and he was the supervisor of G. Klingelhöfer
that developed the Mössbauer drives on the Mars exploration Rovers.
39
Which in the case of 57Fe is orders of magnitudes higher than the energy resolution. If, however, m
does not mean the mass of the nuclei, but is the mass of the crystal as a whole (at least 106 atoms)
the recoil becomes negligible, and the conditions for resonance absorption are fulfilled.
Rudolf Mössbauer initial work was on the attempt to high temperature of source and absorber to
compensate partly for the recoil. This is illustrated in Fig. 28.
Low
temperature
E
2ER
High
temperature
~kT
E
2ER
Fig. 28: Illustration showing Rudolf’s Mössbauer initial attempts to reach resonance conditions by increasing the
temperature.
As the temperature is increased, the thermal motion results in some nuclei’s emitting radiation with
higher energy and others absorbing at lower energies. This would in principle increase the
conditions for nuclear resonance absorption. On the contrary, Mössbauer found stronger absorption
at lower temperatures, indicating that some of the emitted and absorbed radiation was without
recoil.
It can be shown that the probability of a recoil free emission of gamma-quanta is can be written
as
⎛ Eγ2
⎞
f = exp(−k 〈 x 〉 ) = exp⎜ − 2 2 〈 x 2 〉 ⎟
⎜ h c
⎟
⎝
⎠
2
2
(2-3)
where k is the wave vector of the gamma radiation ( k = p / h = Eγ /(hc) and 〈 x 2 〉 is the average
square amplitude of the nuclei. In order to proceed from here, one needs to have some model for the
vibration spectrum of the solid. The simplest model is due to Einstein and assumes the solid to be
composed of a large number of independent linear harmonic oscillators each vibration at a
frequency ωE. Here the mean square amplitude can be written as
〈x2 〉 =
〈 E〉
Mω E2
The mean energy of the oscillators is given by the mean quantum number of the oscillators as
40
(2-4)
1⎞
⎛
〈 E 〉 = hω E ⎜ 〈 n〉 + ⎟
2⎠
⎝
(2-5)
where the thermal average of 〈n〉 can be written in terms of the Planck distribution function.
〈 n〉 =
1
(2-6)
exp(hω E / kT ) − 1
Combining, one obtains
f
⎛ E
⎛ θ ⎞⎞
= exp⎜⎜ − R coth ⎜ E ⎟ ⎟⎟
⎝ 2T ⎠ ⎠
⎝ kθ E
Einstein
(2-7)
where θ E ≡ hω E / k is the Einstein temperature.
The Einstein model is on the other hand a rather crude model of the solid, and one usually
applies the Debye model where the characteristic temperature is θD. After formally similar
calculations as above
⎛ − 3E
R
f = exp⎜
⎜ 2kθ D
⎝
⎛1 ⎛ T
⎜ +⎜
⎜ 4 ⎜⎝ θ D
⎝
⎞
⎟⎟
⎠
2
θD / T
∫
0
⎞⎞
x
⎟⎟
dx
ex −1 ⎟⎟
⎠⎠
(2-8)
This equation can be approximated analytically at low and high temperatures as
T <<θ D
⎛ E
f = exp⎜⎜ − R
⎝ kθ D
f
⎛ 3 π 2T 2
⎜⎜ + 2
θD
⎝2
⎛ − 6 E RT ⎞
⎟⎟
= exp⎜⎜
2
⎝ kθ D ⎠
T ≥θ D / 2
⎞⎞
⎟⎟ ⎟
⎟
⎠⎠
(2-9)
The high temperature approximation shows that the resonance area drops exponentially with
temperature, with a slope that can be used to determine the Debye temperature. The low
temperature approximation, shows that even at T = 0 K, f < 1. If the recoil energy is too large, this
will hamper detection of resonances and the highest observed Mössbauer transition has transition
energy around 150 keV (188Os). Fig. 29 shows the f-factors calculated from Eq. (2-8) for 57Fe and
119
Sn.
41
1.0
200
0.9
200
0.8
300
0.8
300
0.7
400
0.7
400
0.6
500
0.6
500
0.5
600
0.5
600
0.4
700
0.4
700
0.3
800
0.3
800
0.2
900
0.2
900
0.1
1000
0.1
1000
Sn f -factor
0.9
119
57
Fe f -factor
1.0
0.0
0.0
0
200
400
600
800
1000
0
200
Temperature (K)
400
600
800
1000
Temperature (K)
Fig. 29: f-factors for 57Fe and 119Sn for the Debye temperatures indicated.
2.3 Mössbauer isotopes
For a “good” Mössbauer isotope 2 , there has to be several conditions fulfilled. (1) There has to exist
a low lying ground state, so that a transition with a reduced recoil energy is possible. This
eliminates many of the lighter elements where this is not the case. (2) The Mössbauer state has to
have reasonably long lifetime, so one can observe the interactions of interest. (3) There has to exist
a source material, that is not too difficult to generate and use. (4) The internal conversion factor has
to be lower than 1. The internal conversion factor, α, is defined as the fraction that is emitted as
electrons instead of γ-rays. Fig. 30 shows diagram of isotope with useful Mössbauer states.
Fig. 30: Mössbauer isotopes (purple). Most used isotopes are marked with a blue box (from Gütlich, 2004).
The most applied Mössbauer isotope is 57Fe. This is because of many different aspects of it coming
together, and the importance of iron in nature. In the coming chapters, 57Fe will be used as an
example for most of the processes illustrated. However, to illustrate what can be regarded as
constants and what is not, 119Sn will also be introduced to show other examples. Table 11 shows the
properties of some of the Mössbauer isotopes.
2
The definition of a “good” Mössbauer isotope may vary.
42
Table 11: Parameters of Mössbauer transitions.
Isotope
E0
(keV)
57
14.41
67.4
93.32
13.26
9.35
89.36
23.83
37.15
35.46
57.6
39.58
21.54
25.66
8.42
6.23
82.33
73.03
77.35
59.54
Fe
Ni
67
Zn
73
Ge
83
Kr
99
Ru
119
Sn
121
Sb
125
Te
127
I
129
Xe
151
Eu
161
Dy
169
Tm
181
Ta
191
Ir
193
Ir
197
Au
237
Np
61
Spin/parity
(excited →
ground)
3/2- → 1/25/2- → 3/21/2- → 5/25/2+ → 9/2+
7/2+ → 9/2+
3/2+ → 5/2+
3/2+ → 1/2+
7/2+ → 5/2+
3/2+ → 1/2+
7/2+ → 5/2+
3/2+ → 1/2+
7/2+ → 5/2+
5/2- → 5/2+
3/2+ → 1/2+
9/2- → 7/2+
1/2+ → 3/2+
1/2+ → 3/2+
1/2+ → 3/2+
5/2- → 5/2+
Natural
abunda
nce (%)
Lifetime
(ns)
2.14
1.16
4.1
7.76
11.55
12.7
8.58
57.25
7
100
26.4
47.82
18.99
100
99.988
37.4
62.6
100
0
141.1
7.5
13417
4256
212
29.7
25.7
5.05
2.16
2.7
1.44
13.7
41.1
5.6
9810
5.48
9.15
2.73
96.66
Linewidth
(mm/s)
Internal
conv.
factor
Max. cross
section, σ0
(10-20 m2)
0.194
0.78
0.00032
0.0070
0.199
0.149
0.645
2.10
5.14
2.51
6.91
1.34
0.374
8.33
0.00644
0.8739
0.59
1.87
0.0685
8.2
0.12
0.54
1100
17.9
0.42
5.12
10.5
13.3
3.7
15
30
2.9
220
46
12.4
6.5
4.3
1.1
257
72
12.2
0.761
118
14.4
141
20.6
27.2
21
20
23
95
31.2
170
1.35
3.06
3.9
33
In the case of 57Fe, there are three ways to feed the Mössbauer state 57*Fe. Using EC from 57Co (T½
= 271 d), β- decay from 57Mn (T½ = 85 s) and coulomb excitation of 57Fe (cf. Fig. 31).
Mn (T½ = 85 s)
57
Co (T½ = 271 d)
57
EC
β+
Fe (τ = 6 ns)
57**
57*
Fe (τ = 140 ns)
57
Fe
136 keV, 5/2-
14.4 keV, 3/21/2-
Fig. 31: 57Fe parent isotopes
For applications in the laboratory, 57Co is the only choice. Sources of this material are reasonably
easy to generate by neutron irradiation of stable 56Co. Commonly used sources contain 57Co in
43
metal matrix (Pd and Rh are commonly used) and have strength of up to 1.85 GBq (50 mCi). They
have a useful laboratory lifetime of several years.
2.4 Resonance conditions and thickness effects
We consider first a single line emitter that sends out radiation originating from a state with
separation of E0, and move it at a velocity v in the direction of observation. The energy profile
observed will be
I ( E , v) =
f s ΔE /( 2π )
( E − E 0 (1 + v / c)) 2 + (ΔE / 2) 2
(2-10)
Where ΔE is the natural line-width given by the uncertainty relationship (see Eq. (2-1)), and fs is the
fraction of the radiation which is emitted without recoil. The lineshape described by Eq. (2-10) is
called Lorentzian or Breit-Wigner lineshape and can be derived directly from the solution of the
Schrödinger equation for decaying state. The full width at half maximum (FWHM) = ΔE. The
stationary absorber has a resonance cross section per nuclei described by σ(E, v).
σ ( E , v) =
σ 0 f a (ΔE / 2) 2
(2-11)
( E − E 0 ) 2 + (ΔE / 2) 2
where σ0 is the maximum Mössbauer cross section (= 2.57·10-18 m2 for 57Fe). In transmission
experiment, the radiation that will be absorbed is described by the so-called “transmission integral” 3
∞
T (v) = ∫ I ( E , v)(1 − exp(−σ ( E )na ) )dE
(2-12)
-∞
where n is the number of atoms per unit area and a is the fraction of which is the isotope that gives
resonance 4 (2.17% in the case of 57Fe). We define the dimensionless thickness parameter as
t ≡ σ 0 f a na
(2-13)
Inserting (2-10) and (2-11) into (2-12) gives
∞
f s ΔE /(2π )
T (v ) = ∫
2
2
− ∞ ( E − E 0 (1 + v / c )) + ( ΔE / 2)
⎛
⎛
t (ΔE / 2) 2
⎜1 − exp⎜ −
⎜ ( E − E ) 2 + (ΔE / 2) 2
⎜
0
⎝
⎝
⎞⎞
⎟ ⎟dE
⎟⎟
⎠⎠
(2-14)
By making change of variables, and representing the line-width in terms of Doppler velocity, ΓN =
cΔE/E0, Eq. (2-14) becomes
3
The name transmission integral seems strange, as this the integral yields the absorbed radiation, but this is the
convention.
4
One could just as well defined n the number of nuclei’s of the right isotope, but this convention has been applied in
Mössbauer spectroscopy, and is followed here.
44
fsΓN
T (v ) =
2π
⎛
⎛
t (ΓN / 2) 2
⎜1 − exp⎜ −
⎜ u 2 + (Γ / 2) 2
⎜
N
⎝
⎝
∞
1
∫−∞ (u − v) 2 + (ΓN / 2) 2
⎞⎞
⎟ ⎟du
⎟⎟
⎠⎠
(2-15)
In the case of thin absorber (t << 1), Eq. (2-15) can be solved
Tthin (v) =
fsΓN
2π
∞
t (ΓN / 2) 2
1
∫−∞ (u − v) 2 + (ΓN / 2) 2 u 2 + (ΓN / 2) 2 du
(2-16)
f tΓ
= 2 s N2
v + ΓN
This line describes a Lorentzian line with line-width ΓE corresponding to twice the natural linewidth (ΓE = 2ΓN = 0.194 mm/s) and resonance area that is proportional to t
∞
Athin = ∫ Tthin (v)dv =
πf s tΓE
(2-17)
2
−∞
In the general case, Eq. (2-14) has to be solved numerically. Fig. 32 shows calculations of
absorption pattern 1-T(v) for different values of t.
1.2
1.1
1
1
Relative transmission
Transmission
0.9
0.8
0.7
0.6
t = 0.3
t = 1.0
t = 3.0
t = 10
t = 30
t = 100
0.5
0.4
0.8
0.6
t = 0.3
t = 1.0
t = 3.0
t = 10
t = 30
t = 100
0.4
0.2
0
0.3
-2
-1
0
Velocity (mm/s)
1
-2
2
-1
0
Velocity (mm/s)
1
2
Fig. 32: Effect of thickness factor on the lineshape in Mössbauer spectroscopy.
With increased thickness factor, the intensity, or line area increases, but one also notes changes in
the line-shape, especially at the bottom of the line, where it saturates. Thickness effects have two
effects that have to be taken into account. First of all, the area of the resonance line is no longer
proportional to t, but can be expressed as
45
∞
A = ∫ T ( v ) dv ≈
−∞
ΓE f s πt −( t / 2)
e
( I 0 (t / 2) + I 1 (t / 2))
2
(2-18)
where I0 and I1 are the modified Bessel functions of the first kind. Useful approximations of this
expression, valid up to t ~ 3 are
A~
t
~ t (1 − t / 8)
1+ t / 4
(2-19)
The line-width increases due to thickness effects, and can be approximated with
Γ(t ) ~ ΓE (1 + t / 8)
(2-20)
If the iron site has a Mössbauer spectrum with more than 1 line, the thickness factor of every line is
a weighted fraction of all lines. This is particularly important to note when estimating thickness
effects in natural samples.
Due to inhomogeneous iron sites in natural samples, vibrations in the experimental setup,
thickness effects, the lines observed are broader than the theoretical experimental limit. In
normalized form, the background normalized absorption Lorentzian line is usually given as
L (v ) = 1 −
Γ
A
2
2π v + (Γ / 2) 2
(2-21)
With A representing its area, usually divided by 100 and given in units of mm/(s*%)
2.5 Second order Doppler shift
The relativistic energy of a nuclei is given by
⎛ v⎞
E (v ) = E 0 ⎜ 1 + ⎟ 1 − ( v / c ) 2
⎝ c⎠
(2-22)
The movement of the source in a Mössbauer experiment is of the order of few cm/s, and relativistic
effects do not need to be taken into consideration. However due to lattice vibrations, the v2 term
does not cancel and leads to shift of resonance lines that can be described by the average velocity of
the nuclei
δ SOD = −
E0 v 2
2c 2
.
(2-23)
Again, one has to make use of a model of the lattice, and in case of the Debye model one obtains
with the energy shift represented in terms of Doppler velocity
46
δ SOD
3θ /T
9kT ⎛⎜ 1 θ D ⎛ T ⎞ D x 3 dx ⎞⎟
.
=−
+⎜ ⎟
2m Fe c ⎜ 8 T ⎜⎝ θ D ⎟⎠ ∫0 e x − 1 ⎟
⎝
⎠
(2-24)
Fig. 33 shows the profiles obtained here.
0.0
-0.1
200
300
-0.1
300
-0.3
400
-0.4
500
Sn SOD
200
-0.2
600
-0.5
700
-0.6
119
57
Fe SOD
0.0
-0.1
800
-0.7
900
-0.8
1000
400
-0.2
500
-0.2
600
-0.3
700
800
-0.3
900
-0.4
-0.9
1000
-0.4
0
200
400
600
800
0
1000
200
Temperature (K)
400
600
800
1000
Temperature (K)
Fig. 33: Second order Doppler shift for 57Fe and 119Sn.
The second order Doppler shift allows one only to determine the Debye temperature by making low
temperature measurements, opposite to what is the case with the f-factor.
2.6 Interactions between the nuclei and the electron density
The energy of the nuclear charge ρ(r) in the electrical potential caused by the electrons V(r) can be
written as the volume integral
E = ∫ ρ (r )V (r )dr
(2-25)
The potential can be written as a Taylor expansion
V (r ) = V (0) + r∇V (0) + ∑
i, j
xi x j ∂ 2V (0)
+ ...
2 ∂xi ∂x j
(2-26)
Limiting us to only three terms, Eq. (2-26) becomes
E = E (1) + E ( 2 ) + E (3) + ...
(2-27)
With
E (1) = ∫ ρ (r )V (0)dr
E ( 2 ) = ∫ ρ (r )r∇V (0)dr
E ( 3)
x 2 ∂ 2V (0)
dr
= ∫ ρ (r )∑ i
2 ∂xi2
i
47
(2-28)
Where we have used the fact that we can choose coordinate system such that the mixed terms in
∂ 2V (0)
= 0, i ≠ j . The first term becomes constant E(1) = -V(0)Ze. As we can choose the
E(3), xi x j
∂xi ∂x j
zero point of the energy arbitrarily, this term does not give rise to any interactions. The second term
vanishes as the nuclear charge is not a dipole and we choose the origin of the coordination system
as the centre of the nucleus charge distribution. The third term contains hence all the information
we are interested in. It can be re-written as
E ( 3 ) = Eδ + E Q
Eδ =
1 ∂ 2V (0)
ρ (r )r 2 dr
∑
2
∫
6 i ∂xi
EQ =
1 ∂ 2V (0)
ρ (r )( xi2 − r 2 / 3)dr
∑
2
∫
2 i ∂xi
(2-29)
Which can be understood in such a way, that we take the spherical symmetric part out. The first
term is called the isomer term and gives size to the isomer shift, while the second is called the
quadrupole term, and gives rise to the quadrupole shift/splitting of resonance lines.
2.7 The isomer shift
As ∇ 2V = 4πe | ψ (0) | 2 where | ψ (0) | 2 is the electronic charge density at the nucleus and define
∫r
ρ (r )dr ≡ r 2
(2-30)
2πe | ψ 2 (0) | 2
r
3
(2-31)
2
the term Eδ in Eq. (2-29) can be written as
Eδ =
The value of | ψ (0) | 2 will only depend on the absorber and source, as the electronic configuration
does not change with the excitement of the nuclear level, and r 2 will depend only on the ground
state or the excited state.
Εea
Εes
Excited
state
γ
E0
Ground
state
E0
Εgs
Source
Εga
Absorber
Fig. 34: Schematic level diagram for isomer shift
The contribution of (2-31) will shift energy levels the amount δ as
48
δ = ( E ea − E ga ) − ( E es − E gs ) =
2π
| ψ a ( 0) | 2 − | ψ s ( 0) | 2
3
(
)( r
2
e
− rg2
)
(2-32)
where the subscripts a and s refer to the absorber and source, respectively and the subscripts e and g
refer to the excited and ground state, respectively. Assuming that the nucleus has a spherical charge
distribution of Ze of radius RN, the integral
r2 =
3Ze 2
RN
5
(2-33)
which gives
re2 − rg2 =
6ZeR g ΔR
3Ze 2
Re − R g2 ≅
5
5
(
)
(2-34)
where ΔR = Re - Rg. For 57Fe, it turns out that ΔR is a negative number (ΔR/R = -9·10-4). The
classical view of the 57Fe nucleus (nuclear spins Ig = 1/2, Ie = 3/2 for the ground state and excited
state respectively) is that the excited state is formed by neutron being expelled from the nucleus and
into orbit around the nucleus, and that the charge distribution becomes smaller.
In the case of 57Fe, one can see from Eq. (2-32) that higher charge density at the nucleus site
results on lowering of the isomer shift (as ΔR is negative). The charge density at the nucleus arises
from s-electrons, and their number does not change with different bindings of the Fe atoms.
However, the 3d electrons shield the nucleus from s-electrons. This results in higher shielding with
increased number of 3d electrons, and consequently lower isomer shift. Fig. 35 shows the general
trends of isomer shift for different valence and spin states of Fe.
49
Isomer shift (mm/s)
Fig. 35: Range of isomer shifts for Fe in different valence and spin states (from Gütlich, 2004).
Some of the trends seen in this diagram can be qualitatively explained. Consider first high spin Fe.
Fe3+ and Fe2+. The only difference is that Fe2+ has one additional electron in 3d orbital. This
additional electron causes additional shielding for s-electrons at the nucleus and correspondingly
lower electron density at the nucleus and we would expect higher isomer shift, which is the case. In
this way, the trend between the high spin species can be understood. Within each valence state,
there is an apparent increased isomer shift with spin state that can also be explained in similar way.
2.7.1 Thickness factors in multiphase systems
When there are two or more phases represented in a sample, each with its own thickness factor ti
(corresponding to different concentrations n and fa) and resonance position δi, Eq. (2-16) becomes
fsΓN
T (v ) =
2π
∞
1
∫−∞ (u − v) 2 + (ΓN / 2) 2
⎛
⎛
t i (ΓN / 2) 2
⎜1 − exp⎜ − ∑
2
2
⎜
⎜
⎝ i (u − δ i ) + (ΓN / 2)
⎝
⎞⎞
⎟⎟ ⎟du
⎟
⎠⎠
(2-35)
In the thin absorber approximation, the spectrum will be composed of a sum of Lorentzian lines,
each one of area proportional to its thickness factor. If the resonances are far apart (δi - δj >> ΓN, i ≠
j), Eq. (2-35) can be written as
T (v ) =
fsΓN
2π
∞
1
∫−∞ (u − v) 2 + (ΓN / 2) 2
⎛
⎛
t i (ΓN / 2) 2
⎜
⎜
−
−
1
exp
∑i ⎜
⎜ (u − δ ) 2 + (Γ / 2) 2
i
N
⎝
⎝
⎞⎞
⎟ ⎟du
⎟⎟
⎠⎠
(2-36)
or a sum of transmission integrals. The area of each line has its own thickness factor. This result is
important when evaluating the relative concentration of small components. Small components will
have area fraction proportional to their concentration while the dominating lines may be
underestimated due to thickness effects.
50
2.7.2 Alloy broadening
Many effects lead to broadening of lines in Mössbauer spectroscopy. Among the important is
e.g. alloying that lead to distribution in the environment of the nuclei’s in the sample. These will
generally be uncorrelated, and the resulting lineshape a convolution between Lorentzian and
Gaussian lineshape. The resulting lineshape is called a Voigt profile 5 , and has additionally a width
factor σ as the width of the Gaussian distribution. Fig. 36 shows an example of this in the case of
stainless steel.
Relative emmission (arb. units)
CEMS measurement
of Stainless
steel
Lorentsian
fit
Voight
profile fit
-2
-1
0
1
2
Velocity (mm/s)
Fig. 36: CEMS measurement of stainless steel at room temperature analysed with Lorentzian line (top) and an
Voigt profile (bottom).
Visually it is clearly seen that the Lorentzian line fits the spectrum much more poorly than the
Voigt profile. The peak is overshot, and there is an overestimate of the area outside the peak at +1
and -1 mm/s. Similar effects are often observed when spectra of natural samples are analysed with
Lorentzian lines, where impurities lead to broadening of lines.
2.7.3 Cosine broadening
In order to maximize the counting statistics, it is beneficial to place the source as close to the
absorber as possible. However, then some of the g detected will not get the full momentum from the
drive, and this effect is called “Cosine broadening”, cf. Fig. 37.
5
the Voigt profile is a spectral line profile named after Woldemar Voigt (2 September 1850 – 13 December 1919) and
found in all branches of spectroscopy in which a spectral line is broadened by two types of mechanisms, one of which
alone would produce a Gaussian profile (usually, as a result of the Doppler broadening), and the other would produce a
Lorentzian profile [Wikipedia].
51
source
absorber
ds
da
θ
d
Fig. 37: Illustration of the cosine effect
If the source is at a velocity vd, the γ gets a momentum in that direction corresponding to vd(1-cosθ).
Generally, there will be a distribution of θ values P(θ), and the average velocity transferred is
v = vd ∫ P(θ )(1 − cosθ )dθ . The line broadening can be estimated as the width of the distribution,
and an empirical law that is often used is ΔΓ = 0.083·vr·a1.78, where a is the aspect ratio of the
experiment, defined as a = (ds + da)/d. and vr is the position of the resonance. This relationship is
illustrated in Fig. 38.
Line broadening ΔΓ/vr
0,05
0,04
0,03
0,02
0,01
0
0
0,2
0,4
0,6
0,8
aspect ratio (d a +d s )/(2d)
Fig. 38: Line broadening due to the cosine effect.
The source usage is roughly a2/4. For high statistic counting one accepts 3% usage of the source,
giving aspect ratio of 0.35 and broadening factor of 12.5%. For a resonance located at 8 mm/s this
means additional broadening of 0.1 mm/s. For typical sources and absorbers (ds = 6 mm, da = 14
mm) this means that the distance between the source and the absorber is of the order of 3 cm. If
quality is needed, one can go down to 1% source usage, and obtain 0.04 mm/s broadening of
resonances at 8 mm/s by letting the source be 5 cm away from the absorber.
2.8 Quadrupole splitting
∂ 2V (0)
of Eq. is called the electrical field gradient tensor or the EFG tensor. It is
∂xi2
an important quantity in the description of the nucleus environment, and comes up in different
nuclear methods within solid state physics. It is easy to show that if all elements of Vii = C are the
same, EQ becomes zero. This is an important finding, as it means that if the environment of the
The tensor Vii ≡
nucleus is the same in three arbitrary directions, the quadrupole interaction vanishes. This is in
particular true for cubic lattice sites.
52
As before, we should expect
EQ =
1
∑Vii ∫ ρ (r)( xi2 − r 2 / 3)dr
2 i
∑V
= 4πe | ψ (0) | 2 . However, here we can allow us the
i
ii
(2-37)
simplification that | ψ (0) | 2 ≠ 0 for spherically electron distributions only, and in Eq. we weight the
asymmetric part. Hence, we can require that the EFG tensor is traceless. It is the convention to let
order the values of the EFG tensor as V zz ≥ V yy ≥ V xx . . Then the EFG tensor can be described in
terms of two independent variables, Vzz and the asymmetry parameter, η
η=
V yy − V xx
(2-38)
V zz
With the definitions above, one can see that 0 ≤ η ≤ 1 . In the principal axis system, the quadrupole
moment of the nucleus is
Q=
1
ρ (r )(3z 2 − r 2 )dr
e∫
(2-39)
and the Hamiltonian for the interaction can be written
ˆ =
H
Q
(
eQ
V xx Iˆ 2x + V yy Iˆ 2y + V zz Iˆ 2z
2 I (2 I − 1)
eQVzz ⎛ ˆ 2 ˆ 2 η ˆ 2 ˆ 2 ⎞
=
⎜ 3I z − I + (I + + I − ) ⎟
2 I (2 I − 1) ⎝
2
⎠
)
(2-40)
which can be solved, giving the eigenvalues
EQ =
eQVzz 3m 2 − I ( I + 1)
1+η 2 / 3
2
4 3I − I ( I + 1)
(2-41)
with m = -I, …, I. The quadrupole moment is a measure of the shape of the nucleus charge
distribution. For cigar shaped nuclei, Q > 0, while for pancake shaped nucleus, Q < 0. Nuclei’s with
spin I = 0, ½ do not have quadrupole moments.
2.8.1 Quadrupole splitting in 57Fe
57
Fe has I = 1/2 ground state, that does not have quadrupole moment, and a exited state with I = 3/2
that has positive quadrupole moment, or a cigar like nuclear distribution. Inserting values into
(2-41), the exited state splits up into two states,
53
eQVzz
1+η 2 / 3
4
eQVzz
1+η 2 / 3
E Q ( I = 3 / 2, m = ±3 / 2) =
4
E Q ( I = 3 / 2, m = ±1 / 2) = −
(2-42)
This leads to a splitting of resonance lines of
ΔE Q = E Q ( m = ± 3 / 2 ) − E Q ( m = ± 1 / 2 ) =
eQV zz
1+η 2 / 3
2
(2-43)
The lineshape can be written as
L (v ) =
⎞
1
1
ΓA ⎛⎜
⎟
+
2
2
2
2
4π ⎜⎝ (v − δ − ΔEQ / 2) + (Γ / 2)
(v − δ + ΔEQ / 2) + (Γ / 2) ⎟⎠
(2-44)
combining isomer shift and quadrupole splitting.
2.8.2 Calculating the strength of the quadrupole interaction
It is possible to calculate the magnitude of the quadrupole interaction to some level. The quadrupole
interaction originates from two contibutions, the lattice and the electronic configuration of the atom.
The lattice contribution can be evaluated in a point-charge model. The resulting EFG is then a sum
over the lattice atoms,l, as
EFG = (1 − γ ∞ )
∂2
∂xi x j
⎛ 1
⎜
⎜ 4πε
0
⎝
epl ⎞
⎟
⎟
l ⎠
∑ r −r
l
⎛ 3 xl2 1
⎜ 5 − 3
r
⎜ r
epl ⎜ 3xl y l
= (1 − γ ∞ )∑
⎜
r5
l 4πε 0 ⎜
⎜ 3 xl z l
⎜ r5
⎝
3xl y l
r5
2
3 yl
1
− 3
5
r
r
3 yl zl
r5
3 xl z l ⎞
⎟
r5
⎟
3 yl zl ⎟
⎟
r5
⎟
3 z l2 1 ⎟
−
r 5 r 3 ⎟⎠
Where pl is the charge of the lattice atom located at rl. Shielding effects of the outer shell electrons
are taken into account by the Sternheimer antishielding factor (1-γ∞) which has the value of 10.43.
2.9 Magnetic hyperfine interactions
If the nucleus is in a magnetic field, the resonance lines will split up due to Zeeman splitting,
described with the Hamiltonian
ˆ = − g β IˆB
H
N N
hf
(2-45)
where gN is the nuclear g-factor, depending on the nuclear state, and βN is the nuclear magneton (βN
= 5.0505·10-27 J/T). With quantization after the z-axis, Eq. (2-45) gives energy levels according to
54
E = − g N β N mBhf
(2-46)
Bhf is called the magnetic hyperfine field. It has the unit of Tesla, but exists only within the nucleus,
and is therefore not measurable quantity with magnetic methods. The magnetic field at the nucleus
site has at least four contributions
Bhf = Bext + BC + BL + BD
(2-47)
Bext is the external magnetic field, BL comes from orbital motion of the electrons and BD is the
contribution from the magnetic moment of the spins of the electrons outside the nucleus. The
(usually) dominating term is BC, the so-called Fermi contact term. It originates from spin
polarization of the s-electrons and can be expressed as:
BC =
2
16πμ B
3
A nonzero contribution from ψ s↑ − ψ s↓
2
⎛ψ ↑ 2 − ψ ↓ 2 ⎞
⎜ s
⎟
s
⎠
s − electrons ⎝
∑
(2-48)
comes from the uneven shielding of the 3d electrons. If
the 3d electrons are polarized ↑, we expect ψ s↓ to be larger, and hence BC negative. The contact
term is a measure of the polarization of 3d electrons that is proportional to the magnetization of the
material.
2.9.1 Magnetic hyperfine splitting in 57Fe
The magnetic hyperfine field results in splitting of resonance lines according to Eq. (2-46). The
constants gNβN can be given in units of mm/(Ts) as γ1/2 = -0.119015 mm/(Ts) for the ground state
and γ3/2 = 0.067975 mm/(Ts) for the excited state. Fig. 39 shows a level diagram for 57Fe.
55
+3/2γ3/2Bhf
I = 3/2
+1/2γ3/2Bhf
-1/2γ3/2Bhf
-3/2γ3/2Bhf
E0
-1/2γ1/2Bhf
I = 1/2
+1/2γ1/2Bhf
1 2 3 4 5 6
Fig. 39: Level diagram showing in 57Fe. Below is shown the six line pattern resulting from this interaction and
the conventional way of label the lines.
The excited state splits into four different sublevels and the ground state to two. Of the 8 possible
transitions, only six are allowed ( Δm I = 0,±1 ) resulting in spectrum split into 6 lines. The splitting
between the outermost lines will be ½(3γ3/2 - γ1/2)Bhf. Typical hyperfine fields in Fe3+ (full
polarization) is of the order of Bhf = 50 T, resulting in hyperfine splitting of 16 mm/s. This splitting
is much higher than the quadrupole interaction and isomer shifts, and is usually readily observed in
the Mössbauer spectra.
2.9.2 Line intensities and single crystal measurements
The intensity ratio of each line of Mössbauer spectroscopy is found as the matrix element
represented by Clebsch Gordon coefficients multiplied with the appropriate angular dependence
between the direction of the magnetic hyperfine field and the direction of γ-radiation. For the I = 1/2
→ 3/2 case (57Fe) the intensity ratios given in Table 12 are obtained
Table 12: Transition probabilities for I = 1/2 → 3/2 transition. θ is the angle between the γ-direction and the
magnetic hyperfine field. Relative (not normalized) line areas for the cases indicated are shown.
Transitions (mg, me)
-1/2 → -3/2, +1/2 → +3/2
-1/2 → -1/2, +1/2 → +1/2
-1/2 → -3/2, +1/2 → +3/2
Lines
Relative intensity
1, 6
2, 5
3, 4
3(1+cos2θ)
4sin2θ
1+cos2θ
Polycrystalline
3
2
1
B || γ
B⊥γ
3
0
1
3
4
1
Though the relative intensities of Mössbauer lines vary there is no change in the absolute area. Pure
bulk α-Fe breaks up into domains, with random alignment of the magnetic hyperfine field, and one
observes the polycrystalline case. Alignment of the hyperfine field relative to the direction of the γrays is obtained by setting external hyperfine field onto the sample. As has been seen above, a field
in the range of few hundreds of mT is usually sufficient to obtain full polarization.
56
For paramagnetic materials, angular dependence is also observed. Here the two transitions,
usually labelled π and σ transitions depend on the angle between the principal axis of the EFG
tensor as shown in Table 13
Table 13: Transition probabilities for quadrupole doublets.
Transitions (mg, me)
π: ±1/2 → ±3/2
σ: ±1/2 → ±1/2
Line
δ + ΔEQ/2
δ - ΔEQ/2
Relative intensity
3+3cos2θ
5-3cos2θ
Polycrystalline
4
4
By monitoring the relative line intensity relative to the EFG axis it is possible to determine the sign
of Vzz.
2.9.3 Combined magnetic hyperfine and quadrupole interactions
Usually, all types of the interactions that have been described above are observed. The isomer shift
did not depend on the quantization axis, but both the quadrupole interaction and the magnetic
hyperfine field interaction introduced quantization axis, that is not necessarily the same axis in both
cases. Often the quadrupole interaction is much smaller than the magnetic hyperfine field
interaction, and can be treaded as perturbation. In the case of axial symmetry the perturbation can
be described with the angle β between the hyperfine field and the principal axis of the EFG tensor
and the energy levels shift according to
E M ,Q ( I , m) = −γ I Bhf m + (−1) |m|+1 / 2 (eQI V zz / 8)(3 cos 2 β − 1)
(2-49)
Where Q3/2 = Q and Q1/2 = 0. For the Mössbauer spectrum, this means that lines 1 and 6 move ε in
energy and lines 2-4 move –ε in energy, where e is defined as the quadrupole shift
ε ≡ (eQV zz / 8)(3 cos 2 β − 1)
(2-50)
A level diagram for combined interactions in 57Fe is shown in Fig. 40 and the position of resonance
lines in the Mössbauer spectrum are given in Table 14.
57
ε (Vzz > 0)
I = 3/2
E0
I = 1/2
-ε
+ε
1 2 3 4 5 6
Fig. 40: Level diagram for 57Fe for combined interactions. The diagram is drawn for positive Vzz.
Table 14: Resonance position for combined interactions
Line
Resonance position
1
δ + ½(−3γ 3 / 2 + γ 1 / 2 ) Bhf + ε
2
δ + ½(−γ 3 / 2 + γ 1 / 2 ) Bhf − ε
3
δ + ½(γ 3 / 2 + γ 1 / 2 ) Bhf − ε
4
δ + ½(−γ 3 / 2 − γ 1 / 2 ) Bhf − ε
5
δ + ½(γ 3 / 2 − γ 1 / 2 ) Bhf − ε
6
δ + ½(3γ 3 / 2 − γ 1 / 2 ) Bhf + ε
We note that formally, a quadrupole doublet is identical to sextet with Bhf = 0 and 2ε = ΔEQ.
2.9.4 Distribution analysis
In many cases, it is not possible to assign a definite lineshape to the Mössbauer spectra, and a
general form of hyperfine parameter distribution has to be applied. If L6(v;Bhf, δ, Γ) is the lineshape
of a Lorentzian sextet, a magnetic hyperfine field distribution P(Bhf) leads to a lineshape S(v) as
S (v) = ∫ L6 (v; Bhf , δ , ε , Γ) ⋅ P( Bhf )dBhf
(2-51)
Different methods exist for determining P(Bhf) exist. Two main concepts are usually applied: (1)
Calculation of hyperfine parameter distributions directly from the experimental data using matrix
methods with some smoothing conditions (e.g. the Window method [B. Window, J. Phys. E 4
(1971) 401] or the Hesse-Rübartsch method [J. Hesse and A. Rübartch, J. Phys. E 7 (1974) 526.]).
(2) Making assumptions about the distribution, possibly based on knowledge of the system, and
applying line shapes based on the convolution of the natural Lorentzian shape with a hyperfine
parameter distribution (e.g. introducing number of Gaussian broadened lines [D. G. Rancourt and J.
Y. Ping, Nucl. Instr. Meth. B. 58 (1991) 85] or using folding of relaxation line-shapes with particle
58
size distribution functions [M. F. Hansen, C. B. Koch and S. Mørup, Phys. Rev. B. 62 (2000)
1124]).
In the former case, it may be difficult to estimate whether features observed in the obtained
distribution are significant or not and in the latter case, obviously, if the assumptions on the physics
are incorrect, the results may also be incorrect.
A method that is somewhat in-between these two concepts is to simulate P(Bhf) with reduced
number of linear segments [H. P. Gunnlaugsson, Hyp. Int. 167 (2006) 851]. Fig. 41 shows typical
spectrum of small particle system, characteristic for the inward asymmetry of individual lines.
Relative Transmission
1,01
1,00
0,99
0,98
0,97
0,96
0,95
-10
-8
-6
-4
-2
0
2
4
6
8
10
Velocity [mm/s]
Fig. 41: Room temperature Mössbauer spectrum of an annealed soil from Salten Skov, consisting mostly of small
maghemite (γ-Fe2O3) particles. The solid line is obtained by analysis with the Hesse-Rübartsch method.
Fig. 42 shows the hyperfine field distribution functions obtained by the Hesse-Rübartsch method
and the method of linear segments.
Probabiity (arb. units)
0,005
Hesse Rübartsch
method
Linear segments
0,004
0,003
0,002
0,001
0,000
-0,001
0
10
20
30
40
50
60
Magnetic hyperfine field (T)
Fig. 42: Hyperfine distributions obtained from the Hesse-Rübartsch method (solid line) and the linear segment
method (dashed line). The errors calculated in the linear segment model are 3σ coupled errors.
The benefits of the linear segments method is that error estimates can be performed with relative
ease giving confidence in the obtained distribution function.
2.10 Calibration of Mössbauer spectra
The common method is to use foil of pure α-Fe at room temperature and define all shifts of lines
relative to the centre of the spectrum of α-Fe. α-Fe in one laboratory should not differ significantly
from α-Fe in another laboratory, and this should give a good reference. The spectrum obtained is a
series of numbers. First, we velocity profile takes during a single period, the signal twice through a
given velocity as indicated in Fig. 43.
59
15
Velocity (mm/s)
10
5
0
0
100
200
300
400
500
600
-5
-10
-15
channel/time
Fig. 43: Velocity of a source as a function of the channel number.
The resulting spectrum is recorded twice in the 512 channels, and the conversion electron
Mössbauer spectrum of α-57Fe is shown in Fig. 44. The sample is 700 nm of 57Fe grown on Al2O3.
4000
3500
counts
3000
2500
2000
1500
1000
500
0
0
100
200
300
channel
400
500
600
Fig. 44: Unfolded spectrum of α -Fe.
The first task of the calibration is to fold the two half’s together to obtain one spectrum. There is no
general rule of how this is done, ideally, the spectrum should be folded in such a way that one
would sum up channels 1 and 512; 2 and 511; … 256 and 257. Here it is possible to define 513 as
the folding parameter as the sum of the channel numbers used. Due to delays in the electronics,
imperfections of the drive unit and other reasons, the folding parameter is never exactly 513, and
not even an integer. Obtaining a single spectrum is therefore usually done by interpolating, and
requiring that the difference of the two half’s is minimized. The spectrum after folding is shown in
Fig. 45.
60
8000
7000
counts
6000
5000
4000
3000
2000
1000
0
0
50
100
150
channel
200
250
300
Fig. 45: Folded spectrum of α-Fe.
The number of channels has been halved, and the statistical noise is slightly less. One should note,
that it is not clear from the spectrum of α-Fe whether the velocity profile or period starts with a
negative or positive slope (Fig. 43 assumes positive slope). In this case, the velocity scale would
have to be inverted. The spectrum of hematite is often used as a standard (calibration with two
sextets), but the asymmetry in the spectrum of hematite allows for determining whether one has to
“flip” the spectrum.
The rest of the calibration is to find a transformation of the channel numbers to a velocity. This
is usually done by finding the numbers C and Z so that vi = C(i – Z), where C is called the
calibration constant, Z the zero velocity and i the channel number. These numbers are found so the
spectrum fits the theoretical spectrum of α-Fe, that has a hyperfine field of 32.9 T, quadrupole shift
of zero and isomer shift of zero (by definition). The final spectrum together with analysis is shown
in Fig. 46.
counts over background
7
6
5
4
3
2
1
0
-15
-10
-5
0
Velocity (mm/s)
5
10
15
Fig. 46: Folded and analysed spectrum of α-Fe.
Some notes on this spectrum: As this is a thin layer, the magnetisation is in the plane, as reflected
by a almost 3:4:1 area ratio. There is some oxidation seen as a misfit at ~0 mm/s.
2.11 Application of Mössbauer spectroscopy
In this section, few examples of Mössbauer spectroscopy will be described with examples
61
2.11.1
Diffusion broadening
If the probe nuclei jumps in a crystal structure, with frequency f, the Mössbauer spectrum will show
broadening
ΔE = 2hf (1 − exp(ikr) )
(2-52)
Where k is the k-vector of the radiation and r is the jump vector. In crystals, the exponent should be
summed over all possible jump directions. The jump frequency can be related to the macroscopic
diffusion constant D usually given in units of cm2/s. For macroscopic diffusion, one imagines
starting with a concentration of probe atoms as a delta function at the origin. At time t, one observed
a 3 dimensional Gaussian distribution due to random jumps within the lattice that can be written as
P( R) =
(
1
exp − R 2 /(4 Dt )
3/ 2
8( Dtπ )
)
(2-53)
To relate these two quantities, one needs information on the jump mechanism in the crystal. Fe
diffuses through interstitial sites in Si. Placing an iron at origin, it can jump in one of four [111]
directions. The distance between interstitial sites is l = 2.35 Å. Looking at one specific Cartesian
direction, each jump can be regarded as a random variable taking the atom the distance ±l/3½. After
Nj jumps, the distribution of Fe atoms along this direction will be a Gaussian shape with width
l(Nj/3)½. The macroscopic diffusion equation along one direction is also of Gaussian shape with
width (2Dt)½. If we set the time to 1 sec and replace Nj with the jump frequency (jumps per second)
we find the relationship f = 6D/l2. Then the broadening in units of mm/s becomes
ΔΓ =
12hcD
E0 l 2
Neglecting the angular dependent part.
Fig. 47 shows results obtained using radioactive probes in Si
62
(2-54)
Fig. 47: Left: 57Mn Mössbauer spectra obtained after implantation of radioactive 57Mn (T½ = 1.5 min) into two
different types of Si single crystals. Right: Line broadening of the interstitial line as a function of temperature
compared to Arrhenius law.
The interstitial line is furthest to the left in the Mössbauer spectra. At the lowest temperatures, it is
not possible to see significant line broadening. At 567 K in p++ Si (interstitial iron as Fe+) a similar
line broadening is observed as in n-type Si (interstitial iron uncharged) at 636 K, showing that the
diffusion of the positively charged Fe was much faster than neutral. Comparison to Arrhenius law
(right) assuming diffusivity to follow
D = D0exp(-Ea/kT)
(2-55)
gave both pre-exponential factors (D0) and migration energies (Ea).
In the case described here above, the angular dependence was not observed, due to large solid
angles used in the experimental setup. Diffusion of substitutional Fe in Al has been studied in some
details with Mössbauer spectroscopy (see Mantl et al., Phys Rev. B 27 (1983) 5313). Fig. 48 shows
the lattice structure
Fig. 48: The fcc lattice for substitutional diffusion of Fe. The labels refer to different just that are needed and
their rate (from Mantl et al., Phys Rev. B 27 (1983) 5313).
Here, each diffusion step involves introduction of a vacancy, its jump and interchange with the
Fe atom. To describe the jump mechanism, five partly correlated rate constants had to be
63
introduced. The broadening shows asymmetry (cf. Fig. 49) depending on which direction relative to
the lattice one is looking at.
Fig. 49: Mössbauer spectra of Fe in Al recorded at the temperatures and directions indicated (from Mantl et al.,
Phys Rev. B 27 (1983) 5313).
The broadening in different crystal directions was determined at 923 K and compared to a model
based on Eq. (2-52) and the rate constants. Fig. 50 shows a comparison.
Fig. 50: Diffusional broadening of the Mössbauer resonance for 57Fe in Al at 923 K as a function of observation
direction. The lines show the calculated anisotropy (from Mantl et al., Phys Rev. B 27 (1983) 5313).
With the method, it was possible to verify that the jump mechanism was to a NN neighbour in one
of the <110> directions and that NNN jumps could be neglected.
2.11.2
Superparamagnetism
Superparamagnetism of small particles follows the same principles as in magnetism as described in
section 1.5, where the relaxation rate is described by
64
⎛ KV ⎞
⎟
⎝ kT ⎠
τ = τ 0 exp⎜
(1-54)
Transmission (arb. units)
At low temperatures or for large particles, τ is much slower than the measurement time in
Mössbauer spectroscopy τM which is related to the Larmor time of the nucleus, and one observes
well resolved six line pattern. When τ << τM on the other hand, a paramagnetic spectrum (doublet or
a singlet) is observed. In the intermediate range, complicated line pattern is observed.
0.9
30
0.6
3
0.3
1.5
0.01
1.0
-15 -12 -9
-6
-3
0
3
6
9
12 15 -15 -12 -9
-6
-3
0
3
6
9
12 15
Velocity (mm/s)
Fig. 51: Calculated Mössbauer spectra with the method of Blume and Tjoin. The numbers indicate the relative
relaxation rate.
For natural samples, the particle size distribution is usually broad, and can be examined by
changing the temperature. At low temperature, clear sextets will be observed and at higher
temperatures, doublets will be observed. Due to the exponential nature of the relaxation rate (Eq.
(1-54)), natural samples will show mixture of doublet components and sextets. The fraction of
particle sizes showing complicated spectra is negligible. Fig. 52 shows a tropical soil sample from
Kabete, Kenya.
65
Fig. 52: Mössbauer spectra of tropical soil samples from Kabete, Kenya, recorded at the temperatures indicated
(from Hansen et al., Proc. It. Phys. Soc., 50 (1995) 805).
Let’s first study the Mössbauer spectra of superparamagnetic particles at temperatures above the
blocking temperature and in an external magnetic field. The energy of the particle can be described
as
E = −μ ⋅ B
(2-56)
where µ is the magnetic moment of the particle. For nanoparticles, the moment of the particle as a
whole is of the order of 103-105 Bohr magnetrons, and we can describe the magnetisation with a
Brillouin function (Eq. (1-14)) with J → ∞ . This gives the solution
⎛ μB ⎞
μ = μ L⎜
⎟
⎝ kT ⎠
(2-57)
where L(y) is the so-called Langevin function
L( y ) = coth ( y ) − 1 / y
For small values of y, L(y) ~ y/3 and for high values, L(y) ~ 1-1/y. Fig. 53 shows a plot of this
function.
66
(2-58)
1.0
L(y )
0.8
0.6
1-1/y
y /3
0.4
0.2
0.0
0
1
2
3
4
5
y
Fig. 53: The Languvin function.
For kT/µB < 1/2 the observed magnetic hyperfine field becomes
⎛
kT ⎞
⎟ + B ext
B obs = B 0 ⎜⎜1 −
μBext ⎟⎠
⎝
(2-59)
This formula shows two important results. First note, that in this approximation, the value inside the
brackets is from 0.5 to 1.0, and for external magnetic field of few Tesla, we may observe 50 T
splitting of the spectrum, or many times the external magnetic field. This distinguishes the system
under study from paramagnetic systems, where the field would be expected to be of the order of the
external magnetic field. Secondly, it gives an independent way of measuring the particle size if the
magnetisation of the sample is known. The value of Bind = |Bobs-Bext|, over 1/Bext gives a line that
crosses the y axis at B0 and has a slope of – B0kT/µ. If the volume magnetisation of the material is
known, the volume of the particle can be determined through µ = MV. Fig. 54 shows a series of
spectra of nano-sized α-Fe particles.
67
Fig. 54: Mössbauer spectra of α-Fe particles recorded at the temperatures and external magnetic field indicated
(From [S. Mørup, 1994]).
The value of the induced field, Bind as a function of inverse applied magnetic field is shown in Fig.
55.
Fig. 55: Induced magnetic hyperfine field of the spectra shown in Fig. 54 as a function of the inverse applied
magnetic field.
From this information, the average particle size can be calculated.
Below the blocking temperature, one needs to take into consideration the alignment toward the
easy axis. In the case of uniaxial anisotropy, (i.e. one easy axis, and the magnetic moment can be
68
pointed in positive or negative direction), the anisotropy energy can be described in terms of the
angle θ between the magnetisation vector and the easy axis as
E = KV sin 2 θ
(2-60)
where the easy directions are with magnetic moment along θ = 0 and θ = π. Below the
superparamagnetic blocking temperature, the Mössbauer spectra are magnetically split. However,
the minimum of Eq. (2-60) is not sharp, and the magnetisation vector will have finite probability of
being away from the minimum. The observed magnetic splitting will be the thermal average of
cos θ in the minimum. For kT << KV, the observed field can be written as
kT ⎞
⎛
Bobs = B0 ⎜1 −
⎟
⎝ 2 KV ⎠
(2-61)
Thus, we would expect a plot of Bobs/B0 as a function of temperature to be linear function with
intercept at the y-axis at 1 and has slope –k/(2KV). Fig. 56 shows this correlation in terms of
nanoparticles of magnetite.
Fig. 56: The value of Bobs/B0 as a function of temperature for 6-12 nm particles of magnetite.
2.11.3
Applications in metallurgy
The effect on impurities in added to α-Fe can be studied with Mössbauer spectroscopy. Adding few
percentages may not change the crystal structure, but changes the Fe environment. In the BCC
crystal structure each Fe site has 8 NN of Fe. Changing one of these neighbours to an impurity atom
causes changes in the hyperfine parameters accordingly.
According to the Wertheim model, it is assumed that the quadrupole shift is zero, and that
changes in the magnetic hyperfine field (Bhf) and the isomer shift (δ) can be attributed to n impurity
69
nearest neighbours (NN). Assuming that the impurity atoms are randomly distributed throughout
the lattice the probability P(n) of having n impurity atom NN neighbours can be evaluated with the
binomial distribution. Writing
Bhf (n, c) = Bhf ,0 (1 + an + kc)
(2-62)
Where Bhf,0 is the hyperfine field of pure α-Fe, and a and k describe the effects on the number of
NN neighbours and the concentration c, respectively. Similarly for the isomer shift
δ (n, c) = nδ NN + δ c c
(2-63)
Where δNN and δcc describe the effects on the isomer shift caused by number of NN neighbours and
the concentration c, respectively. Then the lineshape can be written as
8
∑ P(n, c) L ( B
6
n =1
hf , 0
(n, c), δ (n, c), Γ)
(2-64)
Relative transmission (arb. units)
where L6 is a Lorentzian sextet, Γ the line width and P(n,c) is the probability of an Fe site with n
NN impurity atoms. Generally for α-Fe, one has to take the number of NNN impurity atoms into
account (6 in total), which gives more complicated results.
Fig. 57 shows the spectra obtained for Ti impurities for the concentrations indicated.
19%
11.2%
5.7%
0%
-8
-6
-4
-2
0
2
4
6
Velocity (mm/s)
Fig. 57: RT CEMS spectra obtained at RT on Fe-Ti samples with the concentrations indicated. The spectra have
been analysed with modified Wertheim model (from Gunnlaugsson et al., J. Alloys Comp. 398 (2005) 33).
70
At 5.7 At.% concentration of Ti, the there is clear indication of satellite lines with reduced
hyperfine field compared to pure a-Fe. At the highest concentration, line in the central region is
observed, originating from a Fe2Ti phase.
The alloy parameters obtained in the analysis are given in Table 15
Table 15: Alloy parameters obtained from the analysis of the spectra in Fig. 57. b and δNNN are the dependences
of NNN impurity neighbours on the magnetic hyperfine field and isomer shift, respectively.
α-Fe-Ti
-0.078(2)
-0.038(3)
-0.11(1)
-0.019(5)
-0.01(2)
0.20(5)
Parameter
a
b
k
δNN (mm/s)
δNNN (mm/s)
δc (mm/s)
The ratio of b/a corresponds roughly to the ratio of 3d orbital overlap between NN and NNN. Ti
does not contribute to the magnetism of the material, and this is reflected in the negative values of
a, b and k. The isomer shift increases with the concentration of Ti in the sample. As Ti is larger than
Fe, the lattice expected to expand, resulting in decreased 3d orbital shielding at the nucleus and
hence higher isomer shift.
2.12 Setups used
2.12.1
Transmission Mössbauer spectroscopy (TMS)
The setup that is usually applied in laboratories around the world is the transmission mode
illustrated in Fig. 58.
source
absorber
detector
v
Fig. 58: Illustration of Mössbauer experiment in transmission mode.
The source is moved on a velocity drive relative to the absorber, and the transmission through the
absorber monitored with a detector.
2.12.1.1
Optimal absorber thickness
If the absorber is too thin, it will contain too few 57Fe atoms to give a good signal. On the other
hand, if it is too thick, it will absorb most of the radiation, and give rise to a bad signal. The signal
over background (S/B) is (almost) proportional to the thickness, d. However, the background counts
are proportional to exp(-d/d0), where d0 is the thickness where 1/e of the radiation is transmitted,
and depends on the composition of the absorber. Then the signal over noise (N = B½) becomes
proportional to dexp(-d/(2d0)), that is maximized when d = 2d0.
71
Absorption coefficients, k, are usually given in units of mass/area, and to get the thickness one
needs to know the density, ρ, and find d0 = k/ρ.
2.12.2
Backscatter Mössbauer spectroscopy (b-MS)
In Backscatter Mössbauer spectroscopy (b-MS) one measures the gamma’s and the X-rays emitted
from the sample. The technique is especially useful in measurements of samples that may not be
destroyed to form powder samples for transmission experiments (paint works or archaeological
samples) and where sample preparation is not possible (robotic measurements on Mars). The setup
is illustrated in Fig. 59
source
sample
v
detector
Fig. 59: Illustration of a measurement in backscattering geometry.
Compared to transmission, the technique is not as effective, and many additional issues have to be
taken into consideration. The solid angle covered by detectors can never reach completely 2π. Due
to fluorescence scattering from the sample (Compton scattering of the 122 keV radiation in the case
of 57Co), there will always be a background contribution and good energy resolution of the detectors
has to be applied to get better statistics.
The method allows for limited depth selectiveness. The re-emite14.4 keV radiation can for
typical sample of basalt composition escape from a depth of 100-150 µm. However, in the case of
57Fe Mössbauer spectroscopy, for each transition where a γ is emitted, there are 9 cases where
internal conversion takes place, where an electron from the inner shells is emitted. In these cases, xrays are emitted, and the 6.4 keV X-ray radiation can be used to measure the Mössbauer effect. In
this case, these escape from a depth of 50 – 75 µm.
2.12.2.1
The NIMOS II Mössbauer spectrometer
Three NIMOS II spectrometer where launched to Mars in 2004, two on the NASA’s Mars
Exploration Rovers, and one on the unsuccessful ESA’s Beagle II lander. The design of the
spectrometer is shown in Fig. 60.
72
Fig. 60: Illustration of the NIMOS II Mössbauer spectrometer.
The aspect ratio of the spectrometer is close to 0.5, which leads to considerable cosine broadening,
but is a trait-off between effectiveness and resolution. The detectors have energy resolution of the
order of 1 keV, and selected due to their large area. New detector development allows for even
larger detectors with 300 eV resolution, scheduled for the ESA´s ExoMars mission in 2011.
2.12.3
Conversion electron Mössbauer spectroscopy (CEMS)
In the decay of the Mössbauer state of 57Fe, there are number of electrons emitted. Fig. 61 illustrates
the process.
Fig. 61: Electrons emitted in the decay of 57Fe. The figure must be from DeGrave et al., 2006.
In only 9% of instances, there is emission of 14.4 keV g-ray, otherwise, there emitted electrons and
X-rays. It is possible to measure the Mössbauer effect by measuring only on the emitted electrons.
For this there are many different ways, either using channeltrons or wire chamber but Fig. 62 shows
the detection via parallel plate avalanche detector (PPAD).
73
source
CEMS detector
Detector
house
Sample
v
Electrode
Fig. 62: Diagram over the setup for a parallel plate avalanche detector, often called CEMS detector.
The electrons released in the decay of 57Fe are accelerated in the gap between the sample and the
electrode. They cause ionization of a suitable counting gas, and give a measurable peak in the
electronics.
Suitable gases are among others butane, CO2, acetone around 20-25 mbar, plate distance of 2-4
mm and voltage difference of about one kV. The effect in a CEMS detector of this type depends
mostly on the concentration of 57Fe and the amount of heavier nuclei’s that produce photoelectrons. For pure 57Fe, it can be as high as 1000 mm/(%s), but for natural basalt (10 wt.% natural
Fe) it is of the order 10 mm/(%s).
The main benefit of a CEMS detector is the surface sensitivity. The electrons can only escape
of the order 0.25 µm, depending on the sample composition. This means that with CEMS, one can
(easily) measure on 1013 57Fe atoms, which would be unrealistic with transmission methods. Due to
the surface sensitivity, a natural sample being measured is of the order of few µg, giving additional
possibilities.
2.12.4
Radioactive Mössbauer spectroscopy
In radioactive Mössbauer spectroscopy, the radioactive element is inside the sample, and a single
line absorber (Fig. 63) or a resonance detectors (Fig. 64) are used to measure the Mössbauer
spectrum.
Sample/source
Single line
absorber
detector
v
Fig. 63: Source experiment utilizing single line absorber.
74
sample/source
CEMS detector
Detector
house
Single line
absorber
v
Fig. 64: Source experiment utilizing resonance detector or CEMS detector.
The benefits of radioactive Mössbauer spectroscopy can be many, and the most important is the fact
that the 57*Fe impurities can be studied at a 10-5% level concentration where they do not interact. A
useful transmission method uses concentrations of the order of 1%.
It should be noted that in the setups where the source is moved relative to an absorber, the
velocity (in the direction of the absorber) of the source will be proportional to the energy needed to
make the resonance, or proportional to the isomer-shift. In source experiments, this is opposite, and
the velocity scale in source experiment therefore proportional to negative isomer shifts, which
means that one needs to think before interpreting spectra using usual methods.
2.12.4.1
Resonance detectors
Resonance detectors are CEMS detectors with a single line absorber material. They are
characterised by three quantities, the area factor A0, the counting efficiency ε and the line-width Γd.
One can find the area of a resonance Ai from the area factor as Ai = A0·fi·pi, where fi is the f-factor of
the resonance and pi the site population. A good CEMS detector has A0 ~ 1100 mm/(s%), and a
line-width Γd ~0.4 mm/s, giving the resonance roughly at a factor 2fA0/(πΓd) = 13 over the
background for a usual 57Co source (f ~ 0.7). As the counting method is basically a surface sensitive
CEMS detection, the counting efficiency is usually low, typically ~ 3% off resonance. It is often
enhanced by making sandwich like structures.
The trick of a resonance detector, is to have a layer containing as much 57Fe as possible.
However, when 57Fe atoms is close to each other, they will interact, leading to broadening.
Commonly used detectors make use of Stainless steel enriched in 57Fe.
The main benefits in applying resonance detectors instead of a usual absorption techniques, is
that they can withstand count rates of more than 104 counts/s. This would correspond to placing
them in front of a GBq 57Co source, a situation which a common proportional counter would have
difficulties with.
2.12.4.2
57
Co
Radioactive experiments using 57Co are rarely done today. A lot of work on different systems was
done in the late 70’s to the early 90’s, and the data from these experiments is of as good use today,
and the need to repeat them is limited. 57Co is not considered a nice material to work with due to its
long lifetime (T½ = 270 d). Contamination with a 270 days isotope would be uncomfortable as it
would take years for it to decay from an experimental setup.
As a nuclear probe, it has several features of interest. It primarily decays with one of the inner
electron being removed (electron capture), which results in highly excited electronic state of the
57*
Fe daughter nuclei. In metals the electronic configuration comes at rest within nanoseconds.
75
There are however materials where this takes longer, and one make studies on how the electronic
state comes at rest.
The drawbacks may be the long lifetime of the probe, as if any reactions take place during the
lifetime (e.g. surface diffusion), these will hamper the usefulness of the probe.
2.12.4.3
In-Beam Mössbauer spectroscopy (IBMS)
In recent years, experiments with Coulomb excited 57*Fe have been performed at the Hahn Meitner
Institute in Berlin. Here the primary beam is 110 MeV Ar that hit a foil containing 57Fe. The energy
is set just under the Coulomb barrier, and 57*Fe is recoil implanted into the sample. The setup used
at HMI is illustrated in Fig. 65.
v
sam
ple
CEMS detector
Fe
57*+
110 MeV Ar+ beam
Fe foil
57
Fig. 65: Setup for In-Beam Mössbauer spectroscopy used at the Hahn Meitner Institute. The setup is actually
doubled, having sample material and detector on both sides.
Here it is necessary to apply resonance detectors. The primary beam is pulsed, and when it hits the
target, all sorts of short lived radiation is emitted and detected. One has to avoid counting these as
they would only add to the background and give less counting statistics and count only for times
from few ns to several hundreds of ns. Resonance detectors are able to have timing resolution of the
order of ns. At the HMI, counting statistics of the order of 10-20 counts/s are reached.
2.12.4.4
57
Mn
Useful beams of 57Mn for Mössbauer studies are produced at the ISOLDE facilities at CERN and at
the RIKEN institute in Japan. At ISOLDE/CERN, these are produced by proton induced fission in a
UC2 target. Clean beams are obtained following element selective Laser ionisation and mass
separation. Beam intensities as high as 5·108 ions/s are obtained, and can be implanted into samples
with 60 keV energy.
76
Radioactive
Laboratory
1-1.4 GeV Protons
Robot
GPS
HRS
Control
room
REX-ISOLDE
Experimental Hall
New Extension
Fig. 66: Machine layout of the ISOLDE facilities at CERN. There are two target areas (GPS and HRS) and then
the beam is magnetically separated before distribution to users.
At the RIKEN facilities, Japan, a fragmentation of primary 59Co Beam is used. The intensities are
not as high as at ISOLDE/CERN, of the order of 5·105 57Mn/s, and due to short lived impurities
created in the nuclear reaction, one needs to wait for few minutes until a measurement can start.
2.13 Synchrotron Mössbauer spectroscopy
Using Synchrotron radiation to make Mössbauer spectroscopy is the newest major addition in
Mössbauer spectroscopy. The first experiments where done in the early 90´s, and the techniques and
the underlying theory have been under development since. For references, there are good overview
papers written in Hyperfine Interactions volume 123/124 from 2000.
There are several things that can be done with synchrotron Mössbauer spectroscopy (S-MS) that
can not be done with traditional Mössbauer spectroscopy using radioactive sources and vice verse.
Synchrotron radiation is generally a polarized source, and there is plenty of it, but it does not have
energy resolution corresponding to the natural line-width.
There are two techniques discussed in this section and their potential, the so-called “Nuclear
Forward Scattering (NFS)” and “Nuclear inelastic scattering (NIS)”. There is still not a generally
accepted naming convention in S-MS, so these techniques can be used under different names in the
literature, NIS is sometimes called NRIXS (= Nuclear Resonance Inelastic X-ray Scattering).
2.13.1
Setup for S-MS
A typical setup (ESFR) is shown in Fig. 67.
77
Fig. 67: Setup for high pressure S-MS studies at EFRS.
The synchrotron beam is sent through three types of monochromators to enhance the energy
resolution, a primary monochromator (PM), a high resolution monochromator (HRM) and finally a
focusing monochromator (FM). Each one is usually a silicon plate that reflects according to Braggs
law the wanted energy in one direction. Energy resolution better than few meV is usually obtained,
still orders of magnitude more than the natural line-width of the Mössbauer isotope. The beam hits
the sample, and there is one detector in the forward direction for NFS and two beside for NIS
studies.
The main facilities for S-MS are at APS at Argonne National Laboratory, USA; ESRF in
Grenoble, France; two facilities in Japan (Spring-8 and KEK-AR) and HASYLAB in Hamburg,
Germany. Other facilities (2002) may not have suitable setups or timing for doing nuclear
resonance studies [E. E. Alp, W. Sturhahn, T. S. Toellner, J. Zhao, M. Hu and D. E. Brown, Hyp.
Int., 144/145 (2002) 3].
To date (2002) the technique is limited to isotopes with transition energies less than 30 keV,
though there is hope future development into higher energy transitions. These are Fe, Kr only at
APS, Eu, Sn, Dy, Ta only at ESRF, K only at Spring-8 and Ni only at HASYLAB.
2.13.2
Nuclear Forward Scattering (NFS)
When the intense pulsed beam hits the sample, the Mössbauer state is populated in short time
(usually ns), and the decay measured in the forward direction as a function of time. For a single line
emitter, the usual exponential decay is measured in 57Fe case of τ = 140 ns. When the absorber is
thick, one observes a so-called dynamical beat pattern due to resonance absorption and emission in
the sample as illustrated in Fig. 68.
78
Fig. 68: Mössbauer transmission spectra – left column, and synchrotron radiation scattering spectra in energy
and time domain – middle and right columns, respectively, for the case of a single resonance in a thin target –
upper panel, and in a thick target – lower panel (taken from G. V. Smirnov, Hyp. Int. 123/124 (1999) 31).
For a sample with quadrupole splitting, one observes the disappearing of the intensity at some times
in the same way as two waves with different frequency, and the beat period is proportional to the
quadrupole splitting as Δt = h / ΔEQ . This effect may be on top of the dynamic beat pattern for thick
samples as illustrated in Fig. 69.
Fig. 69: Mössbauer transmission spectra – left column, and synchrotron radiation scattering spectra in energy
and time domain – middle and right columns, respectively, for the case of a quadrupole split resonance in a thin
target – upper panel, and in a thick target – lower panel (taken from [G. V. Smirnov, Hyp. Int. 123/124 (1999)
31).
For magnetic compounds the situation is similar, but here one has to take into account that the
incoming beam is polarized, and not all states may be populated.
S-MS gives similar information as the conventional Mössbauer spectroscopy, except for the
isomer shift that is not directly obtainable. Furthermore, the relatively complicated beat pattern may
be too complicated to allow distinction of many spectral components, as one frequently can work
with in Mössbauer spectroscopy in the energy domain. The main benefits are the polarized source,
and the intensity, that allows recording of a time spectrum in matter of minutes.
For magnetic systems, one has to take into consideration the hyperfine field direction relative to
the polarisation of the source. In this way, it is possible to selectively excite only ΔmI = 0 or ΔmI =
±1. With circular polarised sources, it is possible to excite only ΔmI = +1 or -1 transitions.
79
2.13.3
Nuclear Inelastic Scattering (NIS)
In Nuclear Inelastic Scattering (NIS) one shifts the energy of the incoming beam and measures the
integrated time delayed gamma radiation from the target. This gives information on the phonon
density of states, as the nuclei can absorb the “off energy” γ if it gets the rest of the energy from the
lattice. Mössbauer spectroscopy is the only method that can derive such information for one atomic
species in the lattice, and such information is of great interest for comparison to calculations and in
the study of effects phase transitions.
Fig. 70: Left: counts as a function of energy for crystalline (c-) and amorphous (a-) forms of Fe2Tb. Left: the raw
data showing statistics better than 104. Right: the derived density of states (taken from [E. E. Alp, W. Sturhahn,
T. S. Toellner, J. Zhao, M. Hu and D. E. Brown, Hyp. Int., 144/145 (2002) 3]).
The crystalline phase shows specific vibration energies, while due to the distribution of
coordination number in the amorphous phase, are spread out, leading to increase in both low and
high energy modes.
Obtaining the density of state function g(E) from the absorption probability S(E)
g (E) =
E
⎛ E
⎞
tanh⎜
( S ( E ) + S (− E )) ⎟
ER
⎝ 2kT
⎠
(2-65)
gives many information on the system. Among the things that can be calculated are the recoil free
fraction (f), second order Doppler shift (δSOD), Force constants, specific heat, vibrational entropy,
Debye sound velocity, vibration amplitudes and, actually the temperature of the sample can be
calculated.
80
3 Mössbauer spectroscopy of volcanic material
3.1 Introduction
Iron is the fourth most common element in the crust on Earth. Therefore, it is found in various
minerals. The mineral assemblage of a sample can tell us about its formation process, and which
modification it has been subjected to. One of the more important properties of Fe in geological
context is the fact that it is found in two valence states, as Fe2+ and Fe3+. Fresh basaltic rock
contains mostly Fe in the valence state 2+. Interactions with Earths atmosphere lead to oxidation of
this iron toward the valence state 3+. How this oxidation takes place depends on the thermal history
of the sample, and to what degree it has been subjected to water. The Mössbauer spectrum can give
direct information on the iron containing minerals, their valence state and hence tell the story of the
sample under investigation.
The aim of these notes is to give an introduction to iron containing minerals in basalts and
related materials and their Mössbauer spectra. This is in order to be able to interpret the spectra and
find out what information can be extracted from the analysis of them. With this introduction, it
should be possible to take a look at the Mössbauer spectra obtained from Mars and see what story
they can tell us about the planets evolution.
A very simplified picture of the mineralogy of basalt is shown in Fig. 71.
Plagioclase (50 wt%)
Pyroxene (30 wt %)
Olivine (10 wt %)
Iron oxides (8 wt %)
Other (2 wt %)
Fig. 71: Typical distribution of minerals in basalt.
The composition of these main components is shown in Table 16.
Table 16: The composition of the main mineral components in basalts
Mineral
Plagioclase
Pyroxene
Olivine
Iron oxides
Other
General composition
(Na, Mg, Ca, K, …)SiO4
(Fe, Mg, Ca, ...)SiO3
(Fe, Mg)2SiO4
Titanomagnetite (Fe3-xTixO4),
ilmenite (FeTiO3),
hematite/maghemite (α/γ -Fe2O3)
Chromite (Fe,Cr)2O3, sulphites
(Fe1-xS), ...
Only the plagioclases are the major non iron containing minerals, so despite the fact that iron
constitute only 10-12 wt.% of basalt, the determination of the iron mineralogy gives information on
roughly half of the total mineralogy. Chromite and iron sulphites are usually not in amounts that can
be seen directly in bulk spectra, and may require separation in order to be visible in Mössbauer
81
spectra. In these notes, the main iron containing minerals will be described and how their properties
are illustrated in their Mössbauer spectra.
Among the iron oxides are the magnetic minerals that will determine the magnetic properties of
the basalt. Their properties (amount, composition, size) depend on various variables, such as the
cooling rate of the basalt, chemistry, external magnetic field during solidification and later stages
thermal or chemical alteration.
A typical chemical analysis of basalt is shown in Table 17 together with a chemical analysis of
the rock Adirondack as measured by the Spirit Mars Exploration Rover (MER).
Table 17: Elemental composition (represented as oxides) of basalts from Iceland (GF, LW and ML) and the rock
abraded Adirondack rock as measured by the MER Spirit at Gusev crater on Mars (from [Gellert et al., Science
305 (2004) 829]). Iron is given as ferrous oxide. (n.d. = not determined).
Wt. %
SiO2
TiO2
Al2O3
FeO
Mn3O4
MgO
CaO
Na2O
K2O
P2O5
SO3
Cl
Cr2O3
Volatiles
Sum
GF
47.7
2.20
15.3
12.3
0.21
7.87
9.72
2.10
0.31
0.28
n.d
n.d
n.d
1.34
99.33
LW
47.0
3.03
14.1
13.8
0.22
4.86
10.9
2.48
0.25
0.35
n.d
n.d
n.d
2.42
99.37
ML
45.3
2.06
15.5
11.7
0.2
7.77
10.5
2.20
0.45
0.22
n.d.
n.d.
n.d.
2.97
98.87
Adirondack
45.5
0.46
10.9
17.2
0.41
11.9
7.51
2.70
0.11
0.60
2.06
0.23
0.37
n.d.
97.29
The three terrestrial samples show some of the differences seen between samples on Earth. The Mg
content of the LW sample is considerably low, and this may explain that this sample does contains
very little olivine if any. The GF and LW contain high amount of olivine and are classified as
olivine basalts 6 . The main difference between the Adirondack class of samples and the terrestrial
samples is lower amount of Al (on earth, Al is the third most common element after Si and O, while
on Mars, Fe is more common than Al) and Ti, higher Fe and Mg content and detectable amount of
S. Mars is generally a sulphur rich planet, and this has led to speculations that much of it is in iron
compounds.
3.2 Paramagnetic minerals
For the paramagnetic minerals, there are two systems of importance, silicates and oxides. The
chemical formula for the silicates can be written in a very general way as:
MxSiOy
(3-1)
where M stands for metal ion(s) and iron can be one of them and y is usually between 3 and 4. In
many silicates, there are OH groups that are not taken into account here. This allows us to break
6
Olivine basalts are commonly identified from morphological (more weathered) and colour (often greenish), containing
olivine.
82
them up into SiO3 and SiO4 based silicates that show slightly different Mössbauer properties. Fig.
72 shows the general tendencies in this system.
Quadrupole splitting (mm/s)
4,0
Fe2+
Fe3+
3,5
3,0
SiO4 based
2,5
2,0
1,5
SiO3 based
1,0
0,5
Oxides
0,0
0,0
0,5
1,0
1,5
Isomer shift (mm/s)
Fig. 72: Isomer shift and quadrupole splitting for silicates.
The greatest distinguishing is between Fe3+ and Fe2+ minerals, but there is also a weaker trend based
on the quadrupole splitting from oxides to SiO3 and SiO4 based silicates. Table 18 shows the room
temperature Mössbauer parameters of selected silicate minerals. High spin Fe(III) has spherically
symmetric 3d electrons and the quadrupole splitting arises only from the lattice contribution, while
the extra electron in Fe(II) necessarily gives additional asymmetry.
Table 18: Mössbauer parameters of selected silicate minerals at room temperature.
δ (mm/s)
(Mg,Fe,Mn)2SiO4
(Mg,Fe)3Al2(SiO4)3
Fe ox.
State
2
2
1.16-1.18
1.31
ΔEQ
(mm/s)
2.75-3.02
3.53-3.56
Ca3(Fe,Al)2(SiO4)3
3
0.41
0.58
(Mg,Fe,Mn)SiO3
2
Ca(Fe,Mg)SiO3
2
1.15-1.18
1.12-1.16
1.16
2.35-2.69
1.91-2.13
2.15
(Fe,Mg,Mn)7Si8O22(OH)2
2
Mineral name or
series
Olivine
Garnet group
(pyropealmandine)a
Garnet group
(andradite)
Orthopyroxene
Formula
Clinopyroxene,
diopsitehedenbergite
Cummingtonitegrunerite
Anthophyllite
Notes
M1
M2
M1
1.14-1.18
2.76-2.90 M1,M2,M3
1.05-1.11
1.58-1.68 M4
(Fe,Mg)7Si8O22(OH)2
2
1.12-1.13
2.58-2.61 M1,M3
1.09-1.11
1.80-1.81 M4
Actinolite
Ca2(Fe,Mg)5Si8O22(OH)2
2
1.15-1.16
2.81-2.82 M1,M3
1.13-1.16
1.89-2.03 M2
Epidote
Ca2(Al,Fe,Mn)AlOH.AlO.Si2O7.SiO4 3
0.34-0.36
2.01-2.02
Stauroliteb
(Fe,Mg)(Al,Fe)9O6(SiO4)8(O,OH)2
2
0.97
2.30
Gillespitec
BaFeSi4O10
2
0.76
0.51
a
Coordination number 8, bCoordination number 4, tedrahedral symmetry, cCoordination number 4 in square
planar symmetry.
83
3.2.1 Olivine
Olivine has the general formula (Mg, Fe)2SiO4 and ideally only ferrous iron. The two end members
are called Forsterite (Mg2SiO4) and Fayalite (Fe2SiO4). It is among the first minerals to precipitate
from the magma, and the Mg/Fe ratio gives a temperature measurement.
Fig. 73: Temperature-composition diagram for the forsterite-fayalite system.
When a melt at a composition x is cooling, solid olivine of composition x1 will start to form, and the
resulting liquid will become more iron rich. When the liquid continues to cool, more iron rich
olivine will form. This effect is easily seen in backscatter scanning electron microscopy (SEM) of
olivine grains in basalt as illustrated in Fig. 74.
Fig. 74: Backscatter SEM picture of a olivine grain from basalt. The centre part (C-1) is darkest, indicating
higher Mg/Fe ratio increasing toward the edge of the grain (C-3). The resulting liquid has become so iron rich
that pure magnetite has precipitated close to the rim. Such fronts are called diffusion rims.
The iron is situated on both M1 and M2 crystallographic sites, but the difference between these sites
is too small to be observed with Mössbauer spectroscopy. The quadrupole splitting in Mössbauer
spectra at room temperature shows a slight dependence on the composition of the olivine, 2.88
mm/s for pure fayalite and about 3.00 mm/s for pure forsterite (extrapolated). Determining olivine
84
composition from Mössbauer spectra requires on the other hand samples dominated in olivine,
where overlap of lines is not too significant.
Oxidation of olivine can lead to the formation of single domain magnetite (SD) particles. This
can happen via:
6 ⋅ FeMgSiO 4 + O 2 → 2 ⋅ Fe 3O 4 + 6 ⋅ MgSiO 3
( olivine )
( magnetite )
(3-2)
( enstatite )
This transformation of olivine can take place in various ways, depending on the original
composition of the basalt, partial pressure of oxygen in the magma and temperature which will
determine the end products. Indeed the formation of quartz and hematite is possible. If this process
takes place at high temperatures (~1000oC) during the solidification of the magma, the SD
magnetite may dominate the magnetic properties of the basalt. Fig. 75 shows a SEM image
comparison between high and low temperature oxidation of olivine.
E
C
A
B
Fig. 75: SEM backscatter images showing olivine particles in basalt. (Left) Low temperature oxidation of
Olivine, Scale is missing, but the image is roughly 200 µm across: (A) Olivine showing compositional gradient,
(B) Olivine particle showing magnetite formation as veins, (C) titanomagnetite particle, showed also contrast
enhanced in the inset figure. (Right) High temperature oxidation of olivine grain: (A) plagioclase, (B) pyroxene,
(D) titanomagnetite, (E) partially exsolved olivine particle. The inset figure shows that the interior is filled with
voids and submicron magnetite.
Pallasites are a class of interesting meteorite samples that have formed in asteroids on the
transition zone between a liquid core and silicate mantle. They are characterised by olivine
imbedded in a metal matrix as shown in Fig. 76.
85
Fig. 76: Picture of a Pallasite sample from the Esquel meteorite (obtained from [http://www.meteorites.tv/]). The
yellow/brownish grains are semi-transparent olivine crystals.
As the liquid has cooled at extremely slow rate, the precipitation of olivine has taken place at
1890oC over extended times. For this reason, the olivine crystals in Pallasite are almost pure
forsterite.
3.2.2 Pyroxenes
Pyroxenes are among the most common mineral group in the Earths crust, and the most common
group containing iron. The chemical composition of pyroxenes can be expressed by a general
formula XYZ2O6, where X represents Na+, Ca2+, Mn2+, Fe2+, Mg2+, and Li+ in the M2
crystallographic site; Y represents Mn2+, Fe2+, Mg2+, Fe3+, Al3+ Cr3+ and Ti4+ in the M1 site and Z
represents Si4+ and Al3+ in the tetrahedral sites of the chain.
Crystallographically, pyroxenes are subdivided into clinopyroxenes and orthopyroxenes in a
CaSiO3 (wollastonite) – MgSiO3 (enstatite) – FeSiO3 (ferrosilite) ternary diagram shown in Fig. 77.
CaSiO3
Dipside
Hedenbergite
Augite
}Clinopyroxenes
Pigeonite
Enstatite
Ferrosilite
MgSiO3
}Orthopyroxenes
FeSiO3
Fig. 77: MgSiO3-CaSiO3-FeSiO3 ternary diagram for pyroxenes.
Generally, one observes two sites in synthetic pyroxenes. Fig. 78 shows results from
Mg0.81Fe0.19SiO3 (From [S.G. Eeckhout et al., Am. Min. 85 (2000) 943.]).
86
Fig. 78: Mössbauer spectra of synthetic pyroxene of composition Mg0.81Fe0.19SiO3 recorded at the temperatures
indicated (from [S.G. Eeckhout et al., Am. Min. 85 (2000) 943]).
The quadrupole splitting at room temperature of the M1 site (ΔEQ ~ 2.6-2.7 mm/s) is higher than the
M2 site (ΔEQ ~ 2 mm/s). At and above room temperature, the sites almost overlap, and in natural
samples, where other components contribute to the spectra, it is usually impossible to determine
both sites, and one usually makes use of one broadened quadrupole split component.
Fe in the M2 site dominates the spectrum of pyroxene in basalt, and the M1 site may not be
visible due to overlap with olivine. In such cases, one has to take care in the interpretation of spectra
as the olivine component may represent a mixture of the M1 site of pyroxene and the true olivine.
3.2.3 Ilmenite
Ilmenite has the formula FeTiO3. The structure is derived from rhombohedral structure of hematite
(see later) with alterning layers of Fe2+ and Ti4+ along the c direction as illustrated in Fig. 79.
87
Iron Titanium(IV) Trioxide
c
O-2
Fe2+
Ti4+
Fig. 79: The crystal structure of ilmenite.
The local structure of the iron site is very close to octahedral, giving rise to low quadrupole splitting
(ΔEQ ~ 0.75 mm/s). For this reason, it is easily detected in Mössbauer spectra of basalt as the right
leg has intensity far away from overlapping lines.
Ilmenite orders antiferromagnetically with TN = 57 K. Fig. 80 shows the Mössbauer spectrum
recorded at 5 K.
Fig. 80: Mössbauer spectrum of ilmenite recorded at 5 K (from [Grant et al., Phys. Rev. B 5 (1971) 5]).
The magnetic hyperfine field was found to be 4.3 T, and the quadrupole interaction ΔEQ (2ε) =
+1.44 mm/s, suggesting magnetisation along the c axis.
3.2.4 Other important paramagnetic minerals
When bulk spectra are measured, the list above is almost complete of what can be observed. The
most important additional spectral component is due to Fe(III) in paramagnetic compounds. There
is a wide range of such compounds that have very similar Mössbauer parameters, and can not be
88
Absorption (%)
distinguished with any certainty. Usually they are just labelled mineralogically unspecific Fe(III). In
un-weathered basalt, Fe(III) in pyroxenes is the most likely candidate for dominating this
component but various Fe-hydroxides, chlorite, pyrite, are possible candidates. The isomer shift in
all cases is characteristic for ferric iron (δ ~ 0.3-0.45 mm/s), and the quadrupole splitting usually
below 1 mm/s. In specific cases, such as in epidote and jarosite (K,Na,H3O)(Fe,Al)(OH)6(SO4)2, the
quadrupole splitting is high, and the Mössbauer spectrum can be used for identification of the
mineral.
Another important group is Fe in basaltic glass in rapidly quenched basalt or ash. It is of interest
as it gives information on the oxidation state of the basalt before mineral precipitation takes place
and can be related to the conditions in the magma processes. Fig. 81 shows typical Mössbauer
spectra of synthetic basalt glass.
fO2 = 10-9
fO2 = 10-5
fO2 = 0.2
Velocity (mm/s)
Fig. 81: Room temperature Mössbauer spectra of basalt glass quenched from 1300oC at the oxygen pressures
(fugacity) indicated (adapted from [Helgason and Gunnlaugsson, Raust 2 (2004) 55], not original).
The lineshape deviates from the usual Lorentzian lineshape due to the amorphous nature of the iron
sites, and this is in this case simulated with two Lorentzian doublets in for the Fe(II) component. At
low oxygen fugacity Fe(III) component is barely visible, and the spectrum dominated by Fe(III). At
the highest oxygen fugacity (usual atmosphere) the spectrum is dominated by Fe(III) and Fe(II) is
barely visible. In the intermediate range, all possible combinations are possible. Typical average
room temperature Mössbauer parameters are given in Table 19.
Table 19: Typical average room temperature Mössbauer parameters from the analysis of basalt glass.
Basalt
glass
Fe(II)
Fe(III)
δ
(mm/s)
1.05(1)
0.42(1)
ΔEQ
(mm/s)
1.83(7)
1.13(7)
Iron sulphites (Fe1-xS) are also important ingredient in basalts, though they are more common as
weathering products. Pyrite (FeS) is a low spin Fe(II) compound, and would in analysis of bulk
spectra contribute to the mineralogically unspecific Fe(III) component. Terrestrial basalt does not
89
contain much of sulphur to change this picture, but Mars is sulphur rich planet, and
misinterpretation a possibility there.
3.3 Magnetic minerals
Magnetic minerals in Mössbauer spectroscopy are generally those that order magnetically below a
Curie or Néel temperature and show a magnetically hyperfine split spectrum with six lines. For this
reason, minerals such as goethite (α-FeOOH) and hematite fall under this category though they can
hardly be separated from a bulk sample using hand magnets.
3.3.1 Magnetite
Magnetite has the inverse spinel structure Fe3+[Fe2+, Fe3+]O4. The additional electron on the
octahedral sites changes sites rapidly compared to the timescale of Mössbauer spectroscopy, so for
Mössbauer spectroscopy, the structure can be written as Fe3+ Fe 22.5+ O 4 . From this, one would
expect the spectrum to have one sextet originating from Fe3+ on tetrahedral sites characteristic for
Fe3+, and double as large sextet due to Fe3+ and Fe2+ on octahedral sites with parameters that are
somewhat an average of Fe2+ and Fe3+ characteristic parameters. Fig. 82 shows the Mössbauer
spectrum of natural magnetite and Table 20 lists the hyperfine parameters.
[
]
1,02
Relative Transmission
1
0,98
0,96
0,94
0,92
0,9
Exprerimental
Simulation
A
B
0,88
-10
-8
-6
-4
-2
0
2
4
6
8
10
Velocity [mm/s]
Fig. 82: Room temperature spectrum of natural magnetite.
Table 20: Mössbauer parameters obtained from the analysis of a room temperature spectrum of natural
magnetite.
Natural
magnetite
Bhf (T)
δ (mm/s)
ΔEQ (mm/s)
Γ16 (mm/s)
Γ25 (mm/s)
Γ34 (mm/s)
Area (%)
A
B
48.73(3)
0.275(4)
-0.006(6)
0.33(1)
0.27(1)
0.21(1)
34(1)
45.53(3)
0.667(4)
-0.001(6)
0.51(3)
0.42(3)
0.33(2)
65(2)
90
The magnetic hyperfine field for the A site, is in good agreement with characteristic values for a
Fe3+ oxide at T << TC. The value for the B site, is somewhat in-between the values for Fe2+ and
Fe3+. The same applies for the isomer shift values. The quadrupole shift is not distinguishable from
zero, as one could expect for cubic sites. The line-width of the B site is slightly larger than the linewidth of the A site, and one can see that the line-width of the outer lines is larger than the inner
lines. This latter may be due to experimental conditions (cosine broadening and thickness effects) or
the fact that impurities that effect the hyperfine field slightly have most influence on the outer lines.
3.3.1.1 The Verwey transition in magnetite
Relative absorption
At temperatures below 120 K, the Fe3+ and Fe2+ order and electron hopping halts in pure magnetite.
This transition is called the Verwey transition and the transition temperature (TV) for the Verwey
transition temperature. This results in slight change in the lattice from cubic to monoclinic. The
Mössbauer spectrum of magnetite below the Verwey transition temperature is relatively
complicated, and one needs at least five sextets to get a reasonable description of the data. The most
remarkable change in the Mössbauer spectra is that the disappearance of the B-line as is illustrated
in Fig. 83.
Velocity (mm/s)
Fig. 83: High negative velocity part of the spectrum of magnetite, just below and above the Verwey temperature.
The Verwey transition temperature depends weakly on impurity atoms in the magnetite, but
strongly on the level of non-stochiometry. For non-stochiometric magnetite (Fe3-δO4) the Verwey
temperature reduces drastically dTV/dδ = - 1000 K [Aragón et al., J. Appl. Phys. 57 (1985) 3221]
and can be used to estimate the level of non-stochiometry. The Verwey temperature can also be
observed in magnetisation measurements where a peak is seen in the susceptibility corves around
the Verwey transition temperature.
3.3.2 Titanomagnetite
Magnetite in terrestrial basalt is usually found in the form of titanomagnetite, Fe3-xTixO4. For low Ti
substitution (x < 0.2), Ti4+ can be assumed to occupy octahedral sites with only ferric iron on
tetrahedral sites. As a consequence some of the Fe is isolated on the octahedral sites as Fe2+. The
spinel formula can be written as Fe 3+ Fe 22.−54+x , Fe 32+x , Ti 4x+ O 4 . This means that the Mössbauer
[
]
91
spectrum consists of three components, which the relative intensity of depends on the parameter x.
Fig. 84 shows a series of spectra of titanomagnetite (from [H. Tanaka and M. Kono, J. Geomagn.
Geoelectr. 39 (1987) 463–475]).
Fig. 84: Room temperature Mössbauer spectra of titanomagnetite with the x-values indicated.
For x = 0.04, one can see a shoulder on the right side of the first B sextet line. This is the first line of
the so-called C sextet originating from isolated Fe2+ on octahedral sites. This feature becomes
stronger with increasing x, and simultaneously, the relative area of the B sextet is reduced. For x <
0.25, the hyperfine parameters of the three lines can be represented by an empirical law, that gives a
reasonable estimate of the value of x from Mössbauer spectroscopy. The dependency of the
hyperfine parameters is given in Table 21
92
Table 21: Empirical model for the hyperfine parameters of titanomagnetite as a function of x that can be us for 0
< x < 0.2.
Bhf (T)
δ (mm/s)
ε (mm/s)
Relative area
κ = 2+f-x(2-f)
f = 0.95
A
49.2-6x
0.29
0.0
1/κ
B
45.6-6x
0.67
0.0
(1-2x)(1+f)/κ
C
41-3x
0.86
0.0
3fx/κ
For higher values of x it is impossible to find a reasonable line model description of the spectra, and
one needs to use unphysical magnetic hyperfine field distributions in order to get some information
out of the spectra.
There are three models applied in describing the ion distribution on tetrahedral and octahedral
sites for the whole range in titanomagnetite. These are summarized in Fig. 85.
Fig. 85: Different models of the cation distribution in titanomagnetite. The distribution in the end members is
well known, but there is inconclusive date in the intermediate range. (a) The Akimoto model that assumes the
distribution to follow combination of the end members with Fe2+ entering tetrahedral sites at low x. (b) The NéelChevallier model, that assumes combination of the end members with no Fe2+ on tetrahedral sites at low x. (c)
The more complicated O’Reilly-Banerjee model. Probably from [Reilly, 1984].
The three models suggest different saturation magnetisation as a function of x, as illustrated in Fig.
86.
93
Saturation magnetisation pfu Fe3-xTixO4
A
NC
OB
Composition, x
Fig. 86: Saturation magnetisation as a function of x for titanomagnetite compared to experimental data and
models of (A) Akimoto, (NC) Néel-Chevallier and (OB) O’Reilly-Benerjee. Adapted from [Reilly 1976].
Relative transmission (arb. units)
Due to discrepancy in experimental data it is from these consideration not possible to state whether
The Akimoto model or the O’Reilly-Banerjee is better description. However, the O’Reilly-Banerjee
model is more consistent with Mössbauer data. Fig. 87 shows selected spectra from the Roza
profiles.
x ~ 0.4
x ~ 0.5
x ~ 0.6
x ~ 0.7
-10
-5
0
5
10
Velocity (mm/s)
Fig. 87: Room temperature Mössbauer spectra of titanomagnetite containing samples from the Roza profiles.
Middle part has been removed to enhance the magnetic part.
As x increases, one observes broadening of lines due to distribution in Fe environments. Up to x ~
0.5, one can see clearly the Fe2.5+ line. At higher x values, the spectrum can only be analysed in
94
terms of magnetic hyperfine field distribution. Both the average magnetic hyperfine field and the
average isomer shift have been used to estimate x.
Titanomagnetite can oxidize to pureI magnetite according to:
6 ⋅ Fe 3− x Ti x O 4 + x ⋅ O 2 → (6 − 4 x) ⋅ Fe 3 O 4 + 6 x ⋅ FeTiO 3
( titanomagnetite )
( magnetite )
( ilmenite )
(3-3)
This can take place at high temperatures during the solidification of the basalt, and is then called
oxy-exsolution. Low temperature oxidation can also take place after solidification leading to the
same kind of process, in this case called solvus exsolution. In both cases, the formation of ilmenite
takes place along the [111] planes in the original titanomagnetite host, and the products are easily
recognised in backscatter SEM images as triangular like structures as indicated in Fig. 88, and such
a particle is also seen in the inset of the left frame of Fig. 75.
50 µm
Fig. 88: Backscattering SEM image of partially oxidized (darker area) titanomagnetite particle from basalt
sample. The triangular shaped ilmenite/magnetite lamellae show different generations in the oxidation process.
In Mössbauer spectra, the effects of such a process are most easily seen from the spectral changes in
magnetic separation, where ilmenite, though paramagnetic, may be enhanced in the separation
process.
3.3.3 Hematite
Hematite (α-Fe2O3) has a rhombohedral structure. Part of the structure is illustrated in Fig. 89.
95
c
O-2
Fe3+
Fig. 89: Simplified diagram of the hematite structure.
The structure can be seen as sheets of oxygen atoms in the planes perpendicular to the c-axis. The
iron sites are in-between these layers. The iron sites have oxygen atom arranged as triangles above
and below. The opening of the triangles is different and the Fe atom is shifted from the centre
position toward the oxygen triangle with larger opening. With some imagination it is possible to
convince oneself that there is only one type of iron atoms (all have the same local surrounding) and
that there is an axial symmetry around the c-axis.
Hematite is perfect antiferromagnetic below the so-called Morin transition temperatuITM) which
is TM = 260 K for pure hematite and canted antiferromagnet above TM. Below TM, Fe atoms in the cdirection are coupled with each other, and above TM the couplings within the Fe planes. The canting
angle is less than 0.1o leading to a saturation magnetisation of 0.42 Am2/kg at room temperature.
The Morin transiti“n temperature shows interesting Mössbauer features. Due to the symmetry,
the principal component of the EFG is along the c axis and the asymmetry constant is η = 0. Below
TM the angle β between Bhf and the EFG axis is 0o and close to 90o above TM. The quadrupole
interaction is then written as
T <TM
eQeVzz
=
eQeVzz
2
2
2ε =
(3 cos β − 1) T >TM
eQeVzz
4
= −
4
The dependence of the hyperfine parameters of hematite on temperature is shown in Fig. 90.
96
(3-4)
Bhf(T)/Bhf(0)
(A)
2ε (mm/s)
T/TN
0.4
(B)
0.2
0.0
-0.2
T (K)
δ (mm/s)
(C)
T (K)
Fig. 90: Hyperfine parameters of hemati“e as a function of temperature. The isomer shi“t in (C) is not given
relative α-Fe (obtained from [Greenwood and Gibb, 1971] but not original).
As predicted by theory, the quadrupole interaction is halved and changes sign at TM. From this it is
possible to calculate the value of eQVZZ = +0.8 mm/s = +3.84 eV. Using the numerical values of Q
and e, this corresponds to VZZ = +1.83·1021 V/m2, which is reasonably close to the value obtained by
point charge calculations (+1.39·1021 V/m2, following the recipe given in section 2.8.2).
Impurities and particle size can change the Morin transition temperature. Impurities can both
lead to increase and decrease in TM, and small particle phenomena lower TM and for particle size
below 20 nm, the Morin transition is absent.
3.3.4 Maghemite
Maghemite, γ-Fe2O3 has a spinel structure and only Fe3+ with vacancies on octahedral sites. The
[
]
spinel formula can be written Fe 3+ Fe 53+/ 3 , V1 / 3 O 4 . The Mössbauer parameters for the two sites are
almost identical, and can not easily be distinguished by means of Mössbauer spectroscopy, and
97
usually one broadened sextet is used to describe the site. Only by applying strong magnetic field, it
is possible to see clearly the two different sites (see
Fig. 91: Mössbauer spectrum of 6 nm maghemite particles recorded at 5 K (a) without and (b) with an applied
field of 5 T (from S. Mørup, 1994).
Pure maghemite inverts to hematite at relatively low temperatures. Fig. 92 shows a temperature
series illustrating this.
98
Absorption (%)
Velocity (mm/s)
Fig. 92: Spectral series of a sample of maghemite, γ-Fe2O3 recorded at the temperatures indicated. The spectrum
at bottom was recorded after the high temperature measurements.
Already at 207oC, it is possible to see that the lineshape does not correspond to a single sextet. At
450oC, it is clearly seen that a sharp lined sextet is growing into the spectrum, and the
transformation is complete at 500oC. The room temperature spectrum at the bottom shows that
hematite has formed.
Due to the instability of pure maghemite, it was not considered as a major component in basalts.
In a general study on the Mössbauer properties of Icelandic basalt, led by Sigurður Steinþórsson and
Örn Helgason, they noticed that the area ratio of the two legs of magnetite was often very far from
the B/A ratio of 2 suggested by the structure. Fig. 93 shows a spectrum from their study.
99
Fig. 93: Room temperature Mössbauer spectrum of sample containing mixture of maghemite and magnetite
(from [S. Steinþórsson et al., Min. Mag. 56 (1992) 185])
The first suggestion was that this was due to non-stochiometry or partial oxidation of the magnetite.
However, to explain the spectrum they needed δ ~ 0.2, and found the Verwey transition not
significantly different from 120 K as expected from stochiometric magnetite. This led them to the
conclusion that the discrepancy in the B/A area ratio was due to an additional spectral component.
This additional component was found to have negligible quadrupole shift, excluding the possibility
of hematite. The lattice constant as determined by means of X-ray diffraction (XRD) of nonstochiometric magnetite decreases almost linearly with level of non-stochiometry from 8.40 Å for
pure magnetite to 8.34 for δ = 1/3 or pure maghemite. Their conclusion were that lattice constants
indicating non-stochiometric magnetite could be interpreted as due to mixture of maghemite and
magnetite despite the fact that the component showed excellent thermal stability.
Their conclusions have later on been supported by saturation magnetisation measurements and
analogue work that has shown that impurities in the maghemite increase it’s thermal stability to
above the Curie temperature.
The only basic difference between non-stochiometric magnetite and mixture of magnetite and
maghemite is the presence of a Verwey transition temperature in the latter case. Much of the
literature refers to non-stochiometry, though there is no proof that it is not due to mixture of
magnetite and maghemite.
The effect of forming maghemite has been for long time noted in backscatter SEM images. Fig.
94 shows a typical picture of a Fe-Ti oxide particle in a sample containing mixture of maghemite
and magnetite.
100
Fig. 94: SEM backscatter image of a particle showing the effect of maghemitisation.
The formation of maghemite leads to crack like features in the Fe-Ti oxide particles that do not
seem to follow the crystal directions as clearly as for magnetite/ilmenite exsolutions. Similar effect
is seen where oxidation has taken place in the particle illustrated in Fig. 88.
3.4 The FeO-Fe2O3-TiO2 ternary diagram
Many of the features and transformation discussed in preceding sections, can be viewed graphically
using the FeO-Fe2O3-TiO2 ternary diagram illustrated in Fig. 95.
Ti4+
TiO2
rutile
FeTiO3
ilmenite
Fe2TiO4
rutile
Fe2+
FeO
wüstite
tita
n
tita
nom
agn
oh
em
ati
t es
etit
es
Fe3O4
magnetite
Fe3+
α,γ-Fe2O3
hematite/maghemite
Fig. 95: The FeO-Fe2O3-TiO2 ternary diagram.
Each corner point represents an ion, and the distance along each axis, how much of the ion is
present. The chemical composition can then be deduced by measuring the distance along the edges
and add O-2 to balance the charge.
Fresh rapidly cooled basaltic lava contain titanomagnetite with x ~ 0.6, and this common
composition is often the starting point of the magnetic mineral in the basalt. Fig. 96 shows spectra
of titanomagnetite with approximately this composition.
101
Relative transmission (arb. units)
Roza
DC-2
1258.5
Bulk
sample
Magnetic
seperate
-10
-5
0
5
10
Velocity (mm/s)
Fig. 96: Room temperature Mössbauer spectra of sample containing titanomagnetite with x ~ 0.6.
The bulk spectrum contains very little of the magnetic mineral and in the magnetic separate, the
magnetic particles have been pulled out illustrating the magnetic phase better. In this case, it is not
possible to see individual lines due to variations in the Fe environments. Under oxidizing conditions
like the Earths and Mars atmosphere, titanomagnetite is unstable, and eventually, all the iron would
be oxidised to Fe3+. However, this can be a lengthy process. Upon oxidation, there are two common
transformations; solvus-exsolution as mentioned above, and direct oxidation or so-called
maghemitisation, illustrated in Fig. 97.
FeTiO3
ilmenite
Fe2TiO4
rutile
ulvöspinel
FeO
wüstite
TM60
Maghemitization
Solvusoxidation
Fe2O3
Fe3O4
magnetite
Fig. 97: Possible transformations of TM60 titanomagnetite.
While solvus oxidation is the common transformation, maghemitisation is seen at least in rapidly
cooled basalt, where the original titanomagnetite is in sub-micron assemblies in pyroxene matrix.
The general method to deduce the composition of a sample in this range is by applying combination
102
of X-ray diffraction to deduce the spinel phase lattice constant and Curie temperature
measurements, as illustrated in Fig. 98.
Fig. 98: (a) Spinel lattice constants and (b) Curie temperatures in the FeO-Fe2O3-TiO2 ternary diagram (from
[W. Xu et al., Geophys. Res. Lett. 23 (1996) 2811]).
Relative transmission
(arb. units)
The lattice constant shows a general tendency to be smaller upon higher oxidation due to relatively
smaller size of the Fe3+ ion. The Curie temperature is highest for the pure Fe3+ oxide, but is lowered
as more of weaker magnetic Fe2+ or diamagnetic Ti4+ is added to the structure.
This identification method may though be hampered as if partial solvus-oxidation takes place,
one may end up with some metastable ulvöspiunel hampering accurate determination of the lattice
parameter, and the samples may show thermal instability hampering Curie temperature
measurements.
Typical Mössbauer spectrum of titanomaghemite is shown in Fig. 99.
KJ-4
bulk sample
-10
-5
0
5
10
Velocity (mm/s)
Fig. 99: Room temperature Mössbauer spectrum from the Kjalarnes magnetic anomaly.
The sample originates from the Kjalarnes magnetic anomaly. The rocks are intrusive rocks that
have been quenched and contain titanomaghemite with Fe3+/Fe2+ ~ 0.64 and atomic Ti/Fe ratio ~
0.25. The slight Fe2+ character of the magnetic part of the spectrum can be seen as a relative
broadening of the left most sextet feature relative to the right hand side.
Despite the fact that clear lines are not seen in the Mössbauer spectra, the average parameters of
magnetic hyperfine field Bhf and isomer shift δ can be evaluated by analysing the spectra in
103
terms of simple distribution functions. The magnetic hyperfine field at room temperature will
depend on the relative strength of the exchange interaction. The exchange interaction is strongest
for the species containing only Fe3+, and get lowered when the Fe3+ is exchanged by Ti4+ and Fe2+,
and therefore have a functional dependence resembling that of the Curie temperature. The average
isomer shift will shows the greatest dependence on the valence state of iron. This is shown in Fig.
100.
0
Fe2TiO5
15
Fe2TiO4
30
37
42
〈Bhf〉 (Τ)
45
47
FeO
Fe2O3
Fe3O4
Fe2TiO4
Fe2TiO5
0.8 0.7 0.6
0.5
0.4
〈δ〉 (Τ)
FeO
Fe3O4
Fe2O3
Fig. 100: Average room temperature parameters of magnetic hyperfine field and isomer shifts for
titanomaghemites (From Gunnlaugsson et al., Hyp. Int. (2008).)
3.5 Other important magnetic minerals
The above list is not complete list over rock forming minerals, but is “almost” complete when
dealing with bulk samples of basalt. Additional minerals can be separated by special techniques but
have in bulk samples too low intensity to be taken into consideration. Other minerals can be formed
upon weathering and/or chemical alteration, some of which are given a small description below.
3.5.1 Goethite
Goethite (α-FeOOH) is not a common mineral in basalt rocks, but can form as weathering rinds
with interactions with water. More often, Goethite is formed after the minerals of the rocks have
been dissolved in water. Fe2+ is soluble in water, but can oxidize readily to Fe3+ that is insoluble and
forms precipitates, among them is Goethite.
Pure Goethite is antiferromagnetic and has a relatively low Neel temperature of 384 K and for
this reason the magnetic hyperfine field is close to 34 T at room temperature. The iron sites are
close to octahedral symmetry and have a quadrupole shift (2ε) of -0.26 mm/s.
Goethite is dehydrated readily by warming, forming hematite. In the presence of biological
material, the formation of maghemite can though be favoured.
3.6 Analysis of Mössbauer spectra
The art of analysing Mössbauer spectra is a large field and takes a lot of training to be a specialist
in. At some stages, it involves making some educational assumptions on fitting components to get
the basic information out.
104
The general task is to build up a list of fitting components, where each component is a model
based on the hyperfine parameters. For doublets for example, there are usually four independent
parameters that control the line positions, (δ and ΔEQ) line-width (Γ) and area (usually in mm/s/%).
For sextets, there are generally six line-width parameters (Γi, i = 1, … 6), however, one usually
treats them as three independent pairs, Γ16, Γ25, and Γ34. In natural samples, it is often impossible to
determine the line width of the inner pairs with any accuracy due to overlapping of lines. There it is
possible to apply the proportional increase in line width with velocity and assume Γ25 =(Γ16 +
Γ34)/2. The area ratio of lines is usually taken as 3:2:1 unless there is any reason to expect deviation
from these values (e. g. orientation or thickness effects). In the end, sextets can be described with as
few as six independent parameters.
Having obtained a list of fitting components or a model in the parameter vector p, m(v; p), the
task is to minimize the chi-square (χ2) defined as
χ2 =
Nv
Nv − N p
( m ( vi , p ) − d i ) 2
∑
di
i =1
Nv
(3-5)
where Nv is the number of velocity steps (usually 256 or 512), and di is the data count in channel i.
Error analysis can be done by studying the χ2, where all permutations in p that fulfil Δχ2 < 1/Nv are
within 1σ error.
3.6.1 Quantitative analysis of Mössbauer spectra
It may be of interest to relate the relative areas obtained from the analysis of Mössbauer spectra to
the amount of iron in the individual phases or, if the stochiometry is known, relate that to the
amount of the mineral. The area of each component has then to be related to the amount of Fe in the
phase, and there are two effects that have to be taken into consideration (1) Recoil free fractions and
(2) thickness effects.
One can obtain the relative site populations, pi, of component labelled i from it’s area, using
pi =
Ai / f i
∑ Ai / f i
(3-6)
i
Which requires knowledge of the recoil free fractions. Table 22 shows the f-factors at room
temperature for the common minerals in basalt.
Table 22: f-factors at room temperature for the common minerals in basalt (taken from [De Grave & Alboom 18
(1991) 337]).
Mineral
f at room
temperature
0.84
0.84
0.83
0.74
0.72
0.87
0.65
Hematite
Maghemite
Magnetite
Olivine
Pyroxene
Fe(III)
Ilmenite
105
The general trend that the f-factors for ferric compounds are ~15% larger than for ferrous
compounds is seen from this data.
The effect of thickness may be a significantly worse problem. Basalt containing ~10 wt.% Fe
and with a usual absorber of ~50 mg/cm2, there would be a thickness factor of t ~ 3 if all the
resonances where in a single line. Generally the highest intensity features will be underestimated
and/or the lowest intensity features underestimated. It is often not practical to use extremely thin
absorbers, and the best way would be to use an approximation for the transmission integral and take
thickness effects into consideration. A more simple way is often possible in basalts using the
saturation magnetisation. Usually the iron oxides are only seen as small features at high positive and
negative velocities, representing the overestimated fractions. The level of overestimation can be
estimated by calculating the expected saturation magnetisation:
σ S ,bulk = wt.%(Fe) ⋅ ∑ piσ S ,i
(3-7)
i
In basalts where a good determination of area fractions is possible, this method has been shown to
give excellent results.
3.6.2 Examples of analysis of Mössbauer spectra:
3.6.2.1 Beach sand from Skagen
Beach sand from Denmark contains sand that originates from grinding of granite rocks from
Norway and Sweden. Granite is a rock type that has been metamorphosed at high temperatures and
pressures and often re-crystallized. A bulk spectrum of sand from Skagen is shown in Fig. 101
1,01
Relative Transmission
1
0,99
0,98
0,97
Exprerimental
Simulation
S1
D1
D2
0,96
0,95
0,94
0,93
0,92
-10
-8
-6
-4
-2
0
2
4
6
8
10
Velocity [mm/s]
Fig. 101: Room temperature spectrum of beach sand from Skagen.
The spectrum can be analysed in terms of three components, a sextet (S1), and two doublets (D1
and D2). The parameters obtained are shown in Table 23.
106
Table 23: Hyperfine parameters obtained from the analysis of the room temperature spectrum of bulk sand
sample from Skagen shown in Fig. 101. The numbers in the parenthesis represent the 1σ coupled error in the last
digit.
Skagen,
S1
Bulk
Bhf (T)
50.7(1)
0.38(2)
δ (mm/s)
ΔEQ (mm/s) -0.24(3)
0.54(5)
Γ16 (mm/s)
0.40(6)
Γ34 (mm/s)
Area (%)
39(3)
D1
D2
1.28(1)
3.50(2)
0.99(2)
0.78(3)
0.37(2)
0.58(6)
36(2)
24(2)
The parameters of the sextet are in reasonable agreement with hematite having some substitution of
impurities that lower the hyperfine field. Small particle effects seem to be excluded by the fact that
no significant paramagnetic Fe(III) component is observed.
The D1 component has a huge quadrupole splitting that enables the determination of this
component as due to garnet. Garnet is usually found in metamorphic rocks and only in small
quantities in volcanic rocks. The parameters of the D2 component suggest the mineral ilmenite.
This suggests the origin of the material from volcanic material, and that the ilmenite could be due to
transformation of magnetite.
To test the last hypothesis, a magnetic separate was measured. The spectrum is shown in Fig.
102.
1,01
Relative Transmission
1
0,99
0,98
0,97
Exprerimental
S1
Magnetite
0,96
0,95
-10
-8
-6
Simulation
D2
-4
-2
0
2
4
6
8
10
Velocity [mm/s]
Fig. 102: Room temperature spectrum of a magnetic separate of beach sand from Skagen.
The line at -6.7 mm/s is characteristic for pure magnetite, and this component can be added to the
analysis without further considerations. This implies the presence of the A-line of magnetite that in
the model is included automatically. Additionally the spectrum is analysed with a sextet and a
doublet that resembles S1 and D2 from the analysis of the bulk spectrum. The total absence of the
D1 component, suggests that the garnet is contained in separate particles from the magnetic phase
.
107
Table 24: Hyperfine parameters obtained from the analysis of the room temperature spectrum of magnetic
separate of a sand sample from Skagen shown in Fig. 102. The numbers in the parenthesis represent the 1σ
coupled error in the last digit, parameters with omitted errors were not included as fitting variables.
Skagen,
magnetic sep.
Bhf (T)
δ (mm/s)
ΔEQ (mm/s)
Γ16 (mm/s)
Γ34 (mm/s)
Area (%)
S1
50.99(5)
0.398(5)
-0.16(1)
0.41(2)
0.31(2)
25(2)
mt-A
mt-B
49.09(5)
45.69(3)
0.301(6)
0.664(4)
0
0
0.32(3)
0.42(3)
0.24(2)
0.31(2)
52(2)
D2
1.03(1)
0.78(1)
0.50(2)
22(1)
The magnetite component has parameters that are in reasonable agreement with table values. The
S1 component has quadrupole shift that is significantly different from the quadrupole shift of the
hematite component in the analysis of the bulk spectrum. This may be due to the fact that there is
the presence of small amounts of maghemite, and we do only observe the average of the hyperfine
parameters. The D2 component is in good agreement with the ilmenite seen above.
The magnetic particles seem to originate from titanomagnetite that has undergone exsolution to
pure magnetite and ilmenite. The fact that the weakly magnetic hematite is extracted in the
magnetic separate, suggests that it is associated with the magnetite, possibly as a surface oxidation
on magnetite grains.
These results show the origin of the material in a rather clear-cut way, but also suggest that one
could make additional measurements and analysis. To test the hypothesis of a surface oxidation of
the magnetite, one could perform CEMS measurements, and one could include maghemite into the
analysis and magnetite in a very small amount to the analysis of the spectrum of the bulk sample.
3.6.2.2 Mixture of components
To illustrate some of the characteristics described above, the analysis of the spectra of basalt is
shown below.
108
Relative transmission/emission (arb. units)
CEMS of
magnetic separate
Magnetic separate
Bulk sample
-12
-8
-4
0
4
8
12
Velocity (mm/s)
Fig. 103: Mössbauer spectra of a basalt sample from Iceland (from [Gunnlaugsson et al., Phys. Earth Planet.
Int., 154 (2006) 276]). The bar diagram shows the fitting components, from top: maghemite, magnetite, hematite,
ilmenite, Fe(III), pyroxene and olivine.
The bulk spectrum is virtually impossible to analyse due to the large number of components and
overlap of lines. This is especially the case for the sextet components. To get a clearer picture of
those, a magnetic separate was obtained using hand magnets. To distinguish features even further, a
CEMS measurement of the same sample was obtained. The spectra were analysed simultaneously,
i.e. assuming the presence of the same components in each spectrum with varying amounts. This
technique allow for the determination of hyperfine parameters with much greater accuracy than
would otherwise be possible. The results are shown in Table 25.
Table 25: Hyperfine parameters and area fractions obtained from simultaneous analysis of the spectra in Fig.
103. The numbers in the parenthesis represent 1σ coupled error in the last digit. Parameters with omitted errors
where not included as fitting variables. Labels used are Bhf: Magnetic hyperfine field, δ: isomer shift, ΔEQ:
Quadrupole shift (= 2ε) or splitting, Γ: Line-width, for sextets only the line-widths of the inner lines (3 and 4) and
outer lines (1 and 6) is presented.
Bhf (T)
δ (mm/s)
ΔEQ (mm/s)
Γ16 (mm/s)
Γ34 (mm/s)
Abulk (%)
Amag (%)
ACEMS (%)
Amag/Abulk
ACEMS/Amag
Magnetite Magnetite-A
B
49.2
46.5
0.29
0.67
0
0
0.46(3)
0.53(5)
0.33(3)
0.37(4)
12(2)
38(3)
31(2)
3.1(4)
0.82(7)
Hematite
51.46(8)
0.374(7)
-0.2
0.43(5)
0.30(4)
12(2)
11(2)
12(2)
1.10(2)
1.1(2)
Maghemit
e
49.5(1)
0.282(9)
0
0.52(6)
0.37(5)
8(2)
17(3)
18(2)
2.1(5)
1.1(2)
109
Pyroxene
(+2)
Olivine
(+2)
Fe(III)
Ilmenite
1.158(4)
2.059(9)
1.164(5)
2.93(2)
0.41(1)
0.73(2)
1.00(2)
0.78(3)
0.42(2)
0.33(2)
0.51(3)
0.40(5)
32(2)
14.8(8)
13.9(7)
0.46(3)
0.94(7)
11.6(8)
5.8(5)
6.3(5)
0.50(6)
1.1(2)
19.1(9)
6.9(6)
11.3(7)
0.36(4)
1.6(2)
4.0(6)
4.7(6)
5.0(6)
1.2(3)
1.1(2)
The presence of the B-line of magnetite at v ~ -7 mm/s is a clear indicator for the presence of pure
magnetite in the sample. This implies the presence of the A line of roughly half the intensity. This is
clearly not the case, as the A-line appears much more intense than the B-line. This indicates that
there are one or two other contribution, from maghemite and/or hematite. It turns out that a single
ferric sextet is not enough to explain the spectrum, which shows that both maghemite and hematite
are present. These two components overlap significantly, and a clear solution is not possible. One
way of dealing with this is to set the quadrupole shifts of these components to empirical values, and
this assumption allows for a reasonable analysis of the three spectra.
The parameters of the paramagnetic components are in reasonable comparison to literature
values. The line width of the pyroxene is slightly larger than the line width of olivine, as could be
expected from the fact that pyroxene has two sites, and a broader composition range. The line width
of the Fe(III) component is even larger, suggesting that there are more than one components that
make up this line and/or small particle effects.
The area ratio of the silicate phases and Fe(III) is approximately reduced by factor two in the
magnetic separate. This is, however, not true for the ilmenite, which indicates that it is closely
associated with the magnetite, most probably due to exsolution lamellas in the original
titanomagnetite host. Similarly, hematite is not reduced significantly, showing that it is associated
with the magnetic particles. The magnetic components increase strongly in the magnetic separate.
The main difference in the CEMS measurement is the increase of the Fe(III) component, that gives
the opportunity to determine its parameters with greater accuracy.
3.7 Mössbauer spectra from Mars
The Mössbauer spectra having relevance to the planet Mars can be subdivided into three classes: (1)
SNC meteorites, (2) Mars Exploration Rovers and (3) Analogue material. Before distinguishing
each of these, some of the main aspects of the geology of Mars are described.
3.7.1 Geology of Mars
A map based on MOLA 7 data is shown Fig. 104.
7
MOLA stands for Mars Observer Laser Altimeter onboard the Mars Global Surveyor spacecraft (1997-2006). Using
the time it took for a laser pulse to be reflected of the surface, a global topography map was generated.
110
Fig. 104: MOLA topography of the surface of Mars.
Mars has two kinds of surfaces, southern heavily cratered highlands, and relatively flat northern low
lying planes, divided by the so-called dichotomy boundary. Some unknown processes in the early
history of Mars have caused this.
The conventional naming of geological periods on Mars is given in Table 26.
Table 26: Geological periods on Mars
Age (109 yr)
0-3
3-3.6
3.6-4-56
Geological period
Amazonian
Hesperian
Noachian
In very broad terms, the southern highlands are of Noachian age, the northern plains of
Hesperian/Amazonian age. Young localized features (very few craters) are of Amazonian age.
Global mineralogical information’s are few, but two aspects are described below: Global
magnetic field and TES 8 /OMEGA 9 mineralogical maps.
8
The Thermal Emission Spectrometer (TES) is an instrument on board Mars Global Surveyor. TES collects two types
of data, hyperspectral thermal infrared data from 6 to 50 micrometers (μm) and bolometric visible-NIR (0.3 to 2.9 μm)
measurements. TES has six detectors arranged in a 2x3 array, and each detector has a field of view of approximately 3 ×
6 km on the surface of Mars. The TES instrument uses the natural harmonic vibrations of the chemical bonds in
materials to determine the composition of gases, liquids, and solids [Wikipedia].
9
OMEGA is a visible to Infrared mineralogical mapping spectrometer onboard the Mars Express spacecraft (2004-).
OMEGA is building up a map of surface composition in 100 metre squares. It will determine mineral composition from
the visible and infrared light reflected from the planet's surface in the wavelength range 0.5-5.2 microns. As light
reflected from the surface must pass through the atmosphere before entering the instrument, OMEGA will also measure
aspects of atmospheric composition [ESA’s homepage].
111
3.7.1.1 Magnetic field of Mars
The Mars Surveyor spacecraft (1997-2005) had magnetometer to investigate the magnetic field
from Mars and interactions of the solar wind with the planet. The most detailed maps came from
100 km height during airobraking, while global maps were generated from 400 km orbit.
Fig. 105: The remanent magnetic field on Mars and Earth. Based on data from [Acuña et al., Science, 284: (1998)
790; Purucker et al., Geophys. Res. Lett., 27 (2000) 2449].
Mars does not have a dipole field as Earth does and the magnetic field observed at 400 km height
by spacecrafts is due to crustal remanence. This crustal remanence is concentrated south of the
dichotomy boundary, around the oldest surface of Mars. The generally accepted explanation of this
is that Mars only had dipolar field in Early Noachian. Note that in comparison to Earths magnetic
field, the strength of the Martian remanence intensity is an order of magnitude stronger. Model
calculations suggest that the Martian magnetic anomalies originate from rocks having remanent
magnetisation of the order of 20-40 A/m, about factor 10 stronger than mid ocean ridge basalts
(MORB)’s.
3.7.1.2 Mineralogical Maps
Mineralogical maps have been generated from the results of the Thermal Emission Spectrometer
(TES) onboard Mars Global Surveyor and the OMEGA spectrometer onboard Mars Express. Both
instruments make use of near-infrared to infrared reflection to identify minerals on the surface. The
TES instrument detected among others hematite (α-Fe2O3) on Mars. Hematite is reasonably easy to
detect, as it has a characteristic double absorption band in around 20 µm (cf. Fig. 106).
112
Fig. 106: Left: Distribution of hematite on Mars from TES data. Right: Absorption of well crystalline hematite.
The hematite seems to be found mainly on the oldest terrain on Mars with an exceptionally high
concentration at a place called Meridian planum (red area in Fig. 106). As hematite is often a
signature for aqueous processes, the Meridiani planum was selected as the landing site for the Mars
Exploration Rover Opportunity.
Pyroxenes and olivine are detected by characteristic absorption bands in the near-infrared (Fig.
107)
Fig. 107: Reflection spectra of clinopyroxene and olivine (from Sunshine and Pieters, 1983).
Pyroxenes have two main absorptions, at ~1 µm and 2.3 µm. There are small differences in the
positions of the absorption bands between orthopyroxene and clinopyroxene, allowing for
distinction between them. Olivine has a broad absorption feature at ~1.1 µm.
From this, several mineralogical maps over the surface of Mars have been made. Many can be
seen at http://jmars.asu.edu/data/. Clinopyroxene is often labelled as “High-Ca pyroxene” in these
maps.
113
Fig. 108: Global distribution of clinopyroxene based on TES data (from Bandfield, J.L., Global mineral
distributions on Mars, J. Geophys Res., 107, 2002).
Mafic minerals have highest concentration in the southern hemisphere. This does not necessarily
mean that they are not presented at other places, but the signal may be obscured by dust covering.
Dust covering is estimated from absorption at ~0.4 µm, and the lack of higher wavelength
absorption bands. Fig. 109 show the TES estimate of surface dust covering.
Fig. 109: Global distribution of surface dust from TES data. Blue areas show the least dust covering and green to
red increased covering (from Bandfield, J.L., Global mineral distributions on Mars, J. Geophys Res., 107, 2002).
Despite the fact that the whole planet can be covered with dust during global dust storms, and one
would expect even covering of settled dust, the dust seem to have high concentration around the
youngest volcanic features on Mars and the northern plains.
The OMEGA instrument onboard ESA’s Mars express can detect minerals having bands in the
0.5 - 5 µm region with a high resolution where among others, pyroxenes and olivine have
absorption bands. Fig. 110 shows a representative example.
114
Fig. 110: OMEGA mineral map around Syrtis Major (from Mustard et al., 2005).
Some characteristic patterns are seen in the figure. The most cratered surface, contains mostly
orthopyroxenes (greenish areas), and these seem to be overlaid by clinopyroxenes (blue areas).
Some have suggested that the transition from orthopyroxene to clinopyroxene can be used for
relative age determination, but it is not clear whether these differences are local or global features.
Olivine is found scarcely in rims of impact craters and usually together with orthopyroxene units.
3.7.2 SNC meteorites
The SNC (Shergotties, Nakhlites, Chassignites) are a class of basaltic meteorites believed to have
originated from Mars, and brought to earth via meteorite impacts on Mars. They have igneous
composition. The ages at which the SNC meteorites crystallised on Mars have been determined in a
variety of ways including Rb-Sr, Sm-Nd and 39Ar-40Ar dating techniques. The crystallisation ages
fall within 5 groups.
Table 27: Crystallisation ages of SNC meteorites. Ga=109 years, Ma=106 years.
Meteorite(s)
ALH 84001
Chassigny
Nakhlites
lherzolitic shergottites
basaltic shergottites
Crystallisation ages
4.5 Ga
1.35 Ga
1.3 Ga
180 Ma
165-475 Ma
115
From Table 27 it can be seen that ALH84001 is a fragment of Noachian crust while other
meteorites date from the Amazonian system. The basic problem in using the SNC meteorites is that
we have little knowledge of there geological context, still a variety of information on the parent
body can be derived from their chemistry and isotope content. Fig. 111 shows selected Mössbauer
spectra of samples from Nakhla and Zagami (basaltic shergotty).
Fig. 111: Mössbauer spectra of the magnetic fractions of (a) Nakhla at 295 K, (b) Zagami at 295 K and (c)
Zagami at 80 K (from Madsen et al., Hyp. Int. 95 (1995) 291).
The magnetic phase in both cases is titanomagnetite with x ~ 0.1 (Nakhla), and x ~ 0.7 (Zagami)
which are consistent with Nakhla being slowly cooled sample that crystallized within the crust
while Zagami is surface rocks.
3.7.3 Mars Exploration Rovers
The Mars Exploration Rovers (MERs), Spirit and Opportunity have backscatter Mössbauer
spectrometer on the robotic arm. Fig. 112 shows an overview of them.
116
Mars Exploration
Rovers
MINI-TES infrared
Spectroscopy (5-29 µm)
3D Camera
(13 spectral bands)
Antenna
Magnets
Solar panels
Instrument arm: Microscopic imager (resolution ~
30 µm/pixel), Rock abrasion tool, APXS
(elemental analysis) and Mössbauer spectrometer
Fig. 112: Illustration of Mars Exploration Rovers.
The main instruments are MINI-TES, that can reveal the mineralogy in similar way as the TES
instrument from orbit and thermal properties of the surface material. The Pancam can take 3D
images in 13 different spectral bands (440 nm to 1000 nm). Alpha-Particle-X-ray Spectrometer
(APXS() uses radioactive source to obtain elemental composition of the surface material by
measuring the characteristic X-rays from the exposure. The instrument arm contains also a rock
abrasion tool (RAT), microscopic imager (MI) and the Mössbauer spectrometer.
3.7.3.1 Spirit landing site
Spirit landed in Gusev crater (see Fig. 113) on 4. January 2004.
Gusev crater approx 150 km in diameter
[MOLA image 1350 km in width]
Fig. 113: MOLA shaded relief showing Gusev crater (box).
Gusev crater was believed to possibly contain sediments from an inflow channel to the south. Fig.
114 shows the view from the landing site.
117
Adirondack
rock
Fig. 114: View toward West from Spirits Landing site (credits NASA/Cornell/JPL).
No immediate signs of sedimentary rocks were seen at the landing site that looked more like a lavaplain. Few days into the mission, Spirit recorded the first Mössbauer spectrum of the reddish soil at
the landing site (Fig. 115).
(Mg,Fe)2SiO4
Pyroxene, (Mg,Fe)SiO3
[Klingelhöfer et al., 2004]
Fig. 115: The first Martian Mössbauer spectrum
The remarkable feature of the spectrum is the relatively low content of Fe3+. When terrestrial red
soil is investigated, one commonly observes Fe(III) as the dominating species, but this was not the
case here. This resulted in the hypothesis that the soil was composed of basaltic sand grains (~100200 µm in diameter), covered with much finer red dust particles (~2-3 µm) (cf. Fig. 116).
118
Basaltic sand
100-200 µm
particles
+
5% wt. red dust
~2 µm particles
=
Red mixture
Mössbauer spectrum
unaltered
Fig. 116: Effect of mixing dust with sand-sized particles (from Gunnlaugsson et al., (2004)).
The dust particles cover each and every sand sized particle resulting in red particles. On the other
hand, 14.4 keV Mössbauer radiation penetrates of the order 150 µm, so the Mössbauer spectrum
obtained will be 95% unaltered compared to the dust-free particles. This led to the idea that it was
the dust that might contain information on the aqueous history of the planet. Unfortunately, none of
the scientific instruments onboard were specifically able to investigate the dust alone, except maybe
the magnets. These findings are described in section 3.7.4.
Fig. 117 shows the Mössbauer spectrum of the rock Adirondack.
(Mg,Fe)2SiO4
(Mg,Fe)SiO3
-Typical olivine
basalt
- Approx. 15% of
Fe in magnetite:
Fe3O4 ~ 2.8 wt.%)
[Klingelhöfer et al., 2004]
Fig. 117: First Mössbauer spectrum of a rock target measured by Spirit sol 33.
The spectrum is dominated by lines assigned to olivine and has resembles the spectra of terrestrial
olivine basalt. There are still several things in the spectrum that are of interest. The total amount of
paramagnetic Fe3+ (Fe(III))is not substantially high and even lower that is typical for terrestrial
olivine basalt. The magnetic part can be assigned to magnetite, possibly mixed with additional
fraction of maghemite and/or hematite, but there is little if any evidence for ilmenite. This seems to
exclude titanomagnetite as a possible precursor for the magnetite. According to formula (3-3), the
area fraction between ilmenite and magnetite would be expected to be around 0.5 for x = 0.6 which
would here give 7% line of ilmenite, that should be easily detected. Based on the amount of olivine
a process described by formula (3-2) can be suggested. Another feature of rock targets from the
floor of Gusev crater, is the difference between 14.4 keV spectra and 6.4 keV spectra (see Fig. 118).
119
Relative emission (arb. units)
14.4 keV
spectrum
14.4 keV Mazatzal
spectrum
Laguna
Hollow
(soil)
rattet
(rock)
Ol./Pyr.
= 1.16
Ol./Pyr.
= 1.76
6.4 keV
spectrum
6.4 keV
spectrum
Ol./Pyr.
= 1.52
-12
-8
-4
Ol./Pyr.
= 3.75
0
4
8
-12
12
-8
-4
0
4
8
12
Velocity (mm/s)
Fig. 118: Mössbauer spectra of soil and rock target from Gusev crater (from Rasmussen et al., Hyp. Int. (2005)).
The outstanding difference in these spectra is the relative amount between Fe in olivine and
pyroxene. This suggests very inhomogeneous surface layers of the rocks, causing enhancement of
olivine in the surface layer. The questions why and how, may have been answered with analogue
studies.
As Spirit has progressed further into the Colombia hills, it has encountered more oxidized rocks
and containing hydrated minerals such as Goethite in the rock target dubbed Clovis (see Fig. 119
and Fig. 120).
Fig. 119: Pancam image of the rock target dubbed Clovis (NASA/JPL/Cornell).
120
Fig. 120: (left) Observed and fitted Mössbauer spectra (14.4 keV) acquired by MER Spirit at Gusev crater, rock
Clovis in the Columbia Hills, in three temperature windows as indicated. These spectra were fitted using a
distribution for the relaxed goethite sextet and three discrete sites: a hematite sextet, an Fe3+ and a Fe2+ silicate
doublet (from Klingelhöfer et al., Hyp. Int., 166 (2005) 549).
The detection of Goethite is rather firm due to the specialty of its parameters, and suggests aqueous
alteration. Despite the rather broad features of the Goethite spectra, it was not possible to see
indications of Superparamagnetism. In the last two years, Spirit has investigated the feature named
Home Plate shown in Fig. 121..
Fig. 121: The Inner Basin of the Columbia hills, looking southward from the summit of Husband hill.
Approximate true colour image on Sol 594 (from Squyres et al., Science 316 (2007) 738).
Home Plate is a plateau of light toned layered rocks within Columbia hills in Gusev crater, the most
extended layered bedrock encountered by Spirit on Mars. It has been interpreted to consist of
121
materials emplaced in a volcanic explosion, showing interactions of magma with water. The top
layer may have been reworked by the Aeolian processes.
Fig. 122: The northern edge of Home Plate (a), showing the coarse-grained lower unit (b) and the fine-grained
upper unit (c). False colour image obtained on sols 748-751 (from Squyres et al., Science 316 (2007) 738).
Fig. 123: Image taken by the HiRISE (High Resolution Imaging Science Experiment) camera onboard Mars
Reconnaissance Orbiter of Home Plate. Thin white line denotes Spirit’s traverse and locations of targets are
indicated (from Mariek et al. EPSL (2008) submitted.
122
Fig. 124: Mössbauer spectra of targets from Home Plate (Schröder et al., LPSI (2006)).
The Mössbauer spectra of samples within Home Plate, show high magnetite content, possibly
originated from oxidation of olivine, and Fe(III) component, almost anti-correlated with the amount
of pyroxene. The mineralogy of the samples from Home Plate together with variations in elemental
content has been used to understand the magma-water interactions at the site (Mariek et al., EPSL
(2008) submitted).
3.7.3.2 Opportunity landing site
Opportunity landed at Meridiani planum, where TES data suggested the presence of hematite. The
first images taken showed layered rocks.
Fig. 125: One of the first images taken by the MER Opportunity on Mars, showing light toned outcrop rocks.
123
The first Mössbauer spectrum showed soil similar to soils investigated on the opposite side of the
planet, showing dominant lines of olivine and pyroxene, and very little if any clear indications of
hematite.
A
B
Fig. 126: The first Mössbauer spectrum from Eagle crater on Mars. The inlet images show (A) the Mössbauer
spectrometer in action and (B) close-up of the soil investigated before deployment of the Mössbauer spectrometer
(~3 cm across) (image credits NASA/Cornell/JPL).
Among the first close-up images of the soil, it was found to contain small spherules, few mm
across. These spherules became nicknamed “blueberries”, not because they were blue, but as in the
first false colour presentation of them, they were represented in blue colour. The soil showed
interesting properties as the Mössbauer spectrometer pressed against the surface (Fig. 127).
Fig. 127: MI images (~3 cm across) of the soil at Eagle crater before (left) and after (right) deployment of the
Mössbauer spectrometer showing the disappearance of the spherules in a part of the image field (image credits
NASA/Cornell/JPL).
124
Apparently the spherules could be pressed into to soil. This indicated that these particles were lying
loosely, mostly on top of the soil, and could be pressed directly underneath the soil surface. This
fact was used to explain the strange reflection of the surface material observed after landing.
Fig. 128: Hematite signature (red) close to the Opportunity landing site. Bounce marks (blue areas in the lower
part of the image) where the lander airbags touched the ground showed lack of hematite signatures (image
credits NASA/Cornell/JPL).
Areas where the airbags of the lander touched the ground, showed lack of hematite signature.
Assuming that the process was the same as with the deployment of the Mössbauer spectrometer, the
spherules were the ideal candidates for the hematite signature. First, Opportunity investigated the
outcrop rocks, and here the mineral jarosite was identified by Mössbauer spectroscopy.
Fig. 129: Mössbauer spectrum of El Capitan, showing jarosite. Velocity range is not given, but is approximately
+- 4 mm/s.
Though jarosite is among paramagnetic Fe3+ minerals, it has unusually high quadrupole splitting
that enables reliable detection of it. Jarosite is a common mineral, often seen as yellow oxidation
product on Fe-S rich soils. Fig. 130 shows a MI picture of the outcrop rocks after the rock abrasion
tool had been used.
125
Fig. 130: Microscopy Imager picture (~3 cm in diameter) showing the pattern left by the Rock abrasion tool on
the right.
First of all, it was found that the rocks were extremely mechanically weak and filled with vugs, seen
as linear features of voids within the rock. In the lower part of the image, one can see a spherule that
has been cut by the RAT. This, together with chemical and morphological nature of the rocks
suggested that these were modified river sediments that are currently being eroded. The spherules
are harder material, and are left over on the surface. A location on a relatively flat area of the rocks
was found, to study the spherules in some details (Fig. 131).
Fig. 131: Mössbauer spectra taken of the “Blueberry bowl”, a location containing concentration of the spherules,
and the bare rocks surface beside it.
The signal from the “Blueberry bowl” showed sextet signal, that could be related to hematite. The
definite identification came from comparing spectra obtained at temperatures above and below the
Morin transition temperature (see section 3.3.3).
3.7.4 Magnet results
The Mars Exploration Rovers included variety of magnetic targets to study various aspects of the
magnetic dust suspended in the Martian atmosphere.
126
Both the Viking and the Pathfinder missions included magnetic targets to investigate magnetic
particles suspended in the Martian atmosphere. The results from these mission, illustrated the
importance of studying the dust accumulated in some details, making use of the APXS and
Mössbauer capabilities of the rovers. Fig. 132 shows an overview of these targets.
Fig. 132: Overview of the magnetic properties experiment on the Mars Exploration Rovers. Top-left: Capture
and Filter magnets. Top-right: Sweep magnet, located beside the colour calibration target (sun-dial). Lowerright: The RAT magnets, located within the rock abrasion tool (RAT).
The Capture and Filter magnets aimed at collecting as much material as possible, the Capture
magnet as strong as possible, while the Filter magnet is weaker. Fig. 133 shows the design of these
magnets.
Fig. 133: Design of the MER Capture and Filter magnets (from M. B. Madsen, et al., J. Geophys. Res., 108 (2003)
8069).
127
The force on a magnetic particle is given by Eq. (1-57). In a low field, the force on a magnetic
particle can be approximated as F ∝ B∇B . The capture of magnetic particles can be enhanced by
either maximizing the magnetic field B from the magnet or the magnetic field gradient. The Capture
magnet has ring magnets with alternating magnetic field giving high magnetic field gradients at the
surface of the magnet, but a limited extend of the magnetic field. The Filter magnet is made from a
single block of magnetic material, giving magnetic field that extends further, but of lesser surface
strength.
Fig. 134 shows the surface values of magnetic field and magnetic field gradients of the two
magnets.
Fig. 134: Values of the magnetic field and the magnetic field gradient on the surface of the MER Capture and
Filter magnets (from M. B. Madsen, et al., J. Geophys. Res., 108 (2003) 8069).
The design of the Sweep magnet is shown in Fig. 135.
128
Fig. 135: Design of the Sweep magnet (from M. B. Madsen, et al., J. Geophys. Res., 108 (2003) 8069).
The Sweep magnet was designed in such a way that magnetic particles will be expelled from the
area within the ring structure. Examination of the magnetic field gradient close to the surface (Fig.
136) shows upward pointing field gradient in the central region. This would result in upward
pointing force from this region and result in only non-magnetic particles to settle within the ring.
Fig. 136: Vector plot of the magnetic field gradient above the Sweep magnet (from M. B. Madsen, et al., J.
Geophys. Res., 108 (2003) 8069).
The results from the Sweep magnet (Fig. 137), showed that in the start of the mission, no significant
amount of dust settled within the ring of the magnet.
129
Fig. 137: A: Picture of the Sweep magnet on Spirit sol 73. B: Optical reflection spectrum of selected areas on the
Sweep magnet on Spirit sol 73 (from Bertelsen et al., Science 305 (2004) 827).
The dust on the ring, shows reflection spectrum that is very similar to the red soil on Mars,
reflecting very little light in the blue end of the spectrum (~480 nm) and more in the green (~530
nm) and red part (~600 nm). Outside the magnet, a reddish pattern is observed, most probably
showing an incomplete covering of the aluminium surface of the magnet. Within the ring, the
reflection is what is expected of clear aluminium surface, indicating that no dust particles have
settled there.
This is a very important result, as it illustrates that all particles are magnetic to some degree on
Mars. This means that the dust accumulating onto the magnets is representative for the dust in
general and can be used to study that. Fig. 138 shows the Mössbauer spectrum obtained of the Filter
magnet.
101,0
Filter
3-pt smoothed data
magnet
Data (raw, folded)
0.4 %
100,8
100,6
Intensity (%)
100,4
100,2
100,0
99,8
99,6
Gusev
soil
4
6
8
10
4.0 %
99,4
-10
-8
-6
-4
-2
0
2
Velocity (mm/s)
Olivine
Magnetite Fe(III)
Fe(III)
oxide
-12
-8
-4
Pyroxene
0
4
8
12
Velocity (mm/s)
Fig. 138: 14.4 keV Mössbauer spectrum obtained of the dust accumulated to the Filter magnet on Spirit between
sols 244-258 compared to a representative spectrum of Gusev soil (adapted from Bertelsen, Hyp. Int (2005)).
130
There are several observations here that are of importance: First of all, the spectrum shows
primarily Fe2+ containing phases. Note that the spectrum consists of two major legs, concentrated at
around 2.5 mm/s and -0.5 mm/s. Paramagnetic Fe2+ species, contribute to both these features in
equal amounts, while paramagnetic Fe3+ species contribute only to the -0.5 mm/s feature. The
intensity difference between these two legs suggests that Fe(III) constitute only between around
40% of the total spectral area.
The magnetic part of the spectrum shows an apparent lack of intensity in lines 2 and 5,
suggesting that the magnetic moments are aligned along the gamma direction, as one would expect
for ferri or ferro magnetic phases, and at variance with hematite as the dominating magnetic
species.
Later in the mission, dust devil activity ensured extended lifetime of the missions by clearing
dust from the solar panels. This resulted in significant changes in the dust pattern on the Capture
magnet on both rovers (Fig. 139).
Fig. 139: Pictures of the Capture magnets on Spirit (A) and Opportunity (B) (from M. B. Madsen et al., J.
Geophys. Res. (2009) manuscript in preparation.
On Spirit sol 240 and Opportunity sol 168, dust had accumulated at a steady rate onto the magnets
and minor differences were observed between the rovers. Between Spirit sol 240 and 505 and
Opportunity sol 168 and 328, dust clearing episodes took place, and the dust pattern changed
drastically, showing the presence of dark material, clearly aligned along the field lines of the
underlying magnet and showing formation of magnetic chains. Further into the missions, this
pattern got obscured by new material accumulating onto the magnets.
Fig. 140 shows representative spectra taken after these events.
131
Fig. 140: 14.4 keV Mössbauer spectra obtained on the Capture magnets on Spirit (A) and Opportunity (B) (from
M. B. Madsen et al., J. Geophys. Res. (2009) manuscript in preparation).
Here, the major fraction of the spectra is in the sextet components, and based on the asymmetry in
the left part of the spectrum, the presence of magnetite is evident.
3.7.5 Mars analogues
The Mössbauer spectra from Mars can give us wealth of information about the iron mineralogy at
the landing sites. What this can tell up about the processes taking place can be studied in details by
analogue material in the laboratory.
3.7.5.1 Rocks of terrestrial olivine basalt
Olivine basalt is widespread in e.g. Iceland. It is often characterised by greenish appearance and
more weathered surfaces. The former is due to the greenish colour of olivine and the latter from the
fact that olivine dissolves easily in water, and basalt containing olivine weathers more quickly than
olivine poor basalt. Typical Mössbauer spectra are shown in Fig. 141.
Fig. 141: Typical spectra of samples of olivine basalt (from Rasmussen et al., Hyp. Int. (2006)).
132
The oxidation of olivine (formula (3-2)) leads to the formation of iron oxide. Mössbauer spectra
show therefore a negative correlation between the amount of Fe in olivine and iron oxides. This
correlation is often seen within a single flow, due to natural variations in oxidation state on a few
centimetre scale. Fig. 142 shows typical samples.
Fig. 142: Olivine-magnetic phases correlation of olivine basalt samples from the HRI series (Rasmussen et al.,
Hyp. Int (2006)), and anomalously magnetic rocks from Iceland (Gunnlaugsson, PEPI (2006)).
The HRI series has oxidized at low temperatures, and samples from this series do not show any
anomalous magnetic properties. The remanent magnetisation was in all cases found to be of the
order of Mr,mag = 100Mr/wt.%(mag) ~ 300 A/m. However, for samples, were high temperature
oxidation has taken place a different relationship is observed as indicated in Fig. 143.
Fig. 143: Remanent magnetisation of olivine basalt (GF’s and MM samples) that has been oxidized at high
temperatures, compared to olivine poor basalt (LW).
For highly magnetic rocks, a slope of Mr,mag = 100Mr/wt.%(mag) ~ 3000 A/m is observed
suggesting single domain magnetic properties.
Another thing that characterises samples of olivine basalt that have been oxidized at high
temperatures is the difference in the olivine content on the surface and in the bulk (cf. Fig. 144).
133
D
A
E
B
20 µm
Fig. 144: Left: Room temperature CEMS and Transmission spectrum of a powder sample from the Stardalur
magnetic anomaly. Right: SEM backscatter image of the same sample. A: plagioclase, B: pyroxene, D iron oxides
(contrast enhanced in the upper-right corner) E: remains of the olivine.
Consider the line complex at v ~2.5 mm/s. In the CEMS spectrum, the line is asymmetric toward
the right, suggesting relatively higher amount of olivine. On the other hand, the transmission
spectrum indicates that these lines are of roughly the same intensity. A possible interpretation of
this is by considering the SEM image. When generating the powder, cracks will go through the
voided areas exposing particles of type E in the CEMS spectrum.
The data from Gusev Crater on Mars, does not show as clearly the correlation between Fe in
olivine and Fe in magnetic phases.
134
70
(A)
60
6.4 keV
data
50
1:1
Fe in olivine (%)
40
70
14.4 keV
data
(B)
60
50
1:1
40
30
30
Mazatzal
Humphrey
Adirondack
20
10
Mazatzal
Humphrey
Adirondack
20
10
0
0
0
70
10
20
Calculated
interiour
data
1:1
(C)
60
30
50
40
30
0
70
10
30
Terrestrial
bulk
data
(D)
60
20
50
40
1:1
30
Mazatzal
Humphrey
Adirondack
20
10
20
HRI samples
30% < A(Pyr.) < 35%
10
0
0
0
10
20
30
0
10
20
30
Fe in oxides (%)
Fig. 145: Correlation between the area fraction of olivine and oxides of rock samples investigated by Spirit in
Gusev crater, (A) based on 6.4 keV data, (B) 14.4 keV data and (C) interior data. (D) Terrestrial data for
comparison.
Neither the 6.4 keV data nor the 14.4 keV data seem to show this expected tendency. One possible
explanation is that the samples have different origins and stem from different flows, but
inconsistencies within the same rocks exclude this explanation. However, calculating the interior
composition by subtracting the surface contribution (6.4 keV spectra) from the 14.4 keV spectra,
seems to give the expected correlation.
Until now, terrestrial samples that show high olivine enhancement in the 6.4 keV spectra have
not been found (Fig. 146).
Olivine/Pyroxene ratio
4.0
3.5
Adirondack (Mars)
Average
STI-60 (Earth)
high temperature
oxidation
3.0
2.5
2.0
1.5
1.0
HRI-04 (Earth)
low temperature oxidation
0.5
Bulk
level
0.0
0
25
50
75
100
125
150
Sampling depth (µm)
Fig. 146: Olivine/pyroxene ratios of different terrestrial samples and Adirondack rock from Mars.
The oxidation of olivine at high temperatures offers a simple explanation of the presence of
magnetic anomalies on Mars.
135
3.7.5.2 Dust analogues
Just as with rock analogues, a “good” dust analogue that represents all aspects of the Martian dust
does not exist.
Danish soil from Salten Skov has been used in wind-tunnel simulations at the Mars Simulation
Laboratory at the Aarhus University. This dust is a reasonable optical analogue, very good size
analogue, but poor Mössbauer and chemical analogue. However, it exists in waste amounts and is
easy to apply to study the effects of dust under Martian wind conditions.
In these experiments, it has been found that dust behaves in a different way under the conditions
on Mars. This (among others) has resulted in the interest of facilities to test whole spacecraft
components under Martian conditions, and a ESA financed facility is under construction in Århus.
Aarhus Mars wind
tunnel facilities
Under
construction:
Current:
Fig. 147: Århus Mars Wind-tunnel facilities, the one under construction and the current, roughly to scale.
Samples with similar or the same mineralogical content are lacking. Such samples are unstable
in our moisture atmosphere, and disappear in the order of max hundreds of years. Attempts have
been made to generate dust analogues by crushing rock samples of olivine basalt, and size separate
them. The so-called Gufunes dust is one of these analogues, from the same lava flow material as
some of the HRI samples. Fig. 148 shows a series of Mössbauer spectra of this material
136
Relative transmission (arb. units)
Ilmenite
Fe(III)
Pyroxene
Olivine
〈d 〉 ~ 4 µ m
〈d 〉 ~ 25 µ m
〈d 〉 > 125 µ m
-6
-3
0
3
6
Velocity (mm/s)
Fig. 148: Spectra of samples from Gufunes dust
Olivine/pyroxene ME ratio
The magnetic fraction of these samples is very small, and hardly observable in the images. The
central part shows though spectra that are similar to soil samples from Gusev crater with the
addition of ilmenite. The remarkable observation in these samples is that the ratio between olivine
and pyroxene is strongly dependent on the average particle size of the sample (Fig. 149)
2,8
2,6
2,4
2,2
2,0
1,8
1,6
1,4
1,2
1,0
1
10
100
1000
Average particle size (µ m)
Fig. 149: Fe in Olivine/Pyroxene ratio as determined by Mössbauer spectroscopy as a function of average
particle size (from Mølholt et al. Hyp. Int. (2008)).
This indicates that olivine is easier to break down into small particles and it hence dominates in the
small particle separates. Still, very interestingly, the determination of the same ratio with Visual to
Near-Infrared (VNIR) optical spectroscopy, which measures the same properties as the OMEGA
instrument, shows lacking olivine in the optical spectra (Fig. 150).
137
Optical Fe oliv./pyr. ratio
2,5
2,0
1,5
1,0
0,5
0,0
0,0
0,5
1,0
1,5
2,0
2,5
Mössbauer Fe oliv./pyr. ratio
Fig. 150: Comparison of the ratio between olivine and pyroxene determined by VNIR measurements and
Mössbauer spectroscopy.
These interesting findings indicate that olivine may be much more widespread on Mars than
inferred from optical detection from orbit. It has been suggested that this is due to the semitransparent properties of olivine. When the particle size is small, the light scattered from an olivine
particle may originate from the particle beneath, and lead to underestimate of the olivine content.
Other authors have seen the same or similar tendency, but by applying Mössbauer spectroscopy,
the magnitude of this underestimation could be quantified. Table 28 shows some of the chemical
tendencies in this system.
Table 28: Weight percentage of oxides in selected Gufunes samples and samples measured at Gusev crater. Fe is
given calculated as ferric oxide.
GN_A
GN_Ny_A
GN_B2
GN_Ny_B
GN_C
GN_Ny_C
Gusev soil
Adirondack
RATted
〈d〉
[µm]
8
8
11
11
23
23
-
SiO2 TiO2 Al2O3
[%] [%]
[%]
47,61 1,36 16,78
47,63 1,37 16,53
46,85 1,38 14,23
47,09 1,36 14,41
46,82 1,24 11,36
46,62 1,20 10,40
45,8 0,81 10,0
45,4
0,45
10,9
Fe2O3 FeO MnO MgO CaO Na2O
[%] [%] [%] [%] [%]
[%]
10,65 0,00 0,161 8,59 11,78 2,30
10,95 0,00 0,170 8,86 11,69 2,24
12,26 0,00 0,188 10,52 11,91 1,91
12,18 0,00 0,187 10,48 11,92 1,96
13,27 0,00 0,213 12,87 12,49 1,51
13,87 0,00 0,223 13,76 12,49 1,34
17,6
0,31
9,3
6,1
3,3
20,0
-
0,38
11,9
7,42
2,7
K2O P2O5
[%] [%]
0,18 0,14
0,18 0,13
0,15 0,12
0,15 0,12
0,11 0,09
0,10 0,08
0,41 0,84
0,06
0,54
Some of the tendencies observed in the terrestrial samples, assuming that the RATted surface of
Adirondack represents the largest particle size and soil containing some of the smaller particles.
This is at clearly the case with Fe, Mg and Ca, and indicates that the soil forming process in Gusev
crater has been mechanical abrasion of the rocks over long periods.
References
[Dunlop and Ösdemir, 1997]: David J. Dunlop and Ôzden Özdemir, Rock Magnetism: Fundamental
and frontiers, Cambridge University Press, 1997: Among the best books on rock magnetism,
recommended for details in application of magnetic methods in geology. Available in The Mars
Library.
138
[Gütlich, 2004]: In August 2005, Prof. Philipp Gütlich of the Institut für Anorganische Chemie und
Analytische Chemie at Johannes Gutenberg-Universität Mainz delivered eight hours of lectures on
the topic "Mössbauer Spectroscopy – Principles and Applications." This PowerPoint presentation,
in PDF format, was used in his lectures and contains much information about the Mössbauer effect.
http://ak-guetlich.chemie.uni-mainz.de/Moessbauer_Lectures_web.pdf
[S. Mørup, 1994]: S. Mörup, Mössbauer Spectroscopy and its Applications in Material Science,
DTU 1994: Excellent lecture notes on Mössbauer spectroscopy. Originals for copying available
from HPG.
[Reilly, 1976]: Extended article on rock magnetism. Explains many matters in a very good way,
tough many issues discussed are out of date. Originals for copying available from HPG.
[Greenwood and Gibb, 1971]: N. N. Greenwood and T. C. Gibb, Mössbauer Spectroscopy,
Chapman and Hall, London, 1971. Old but excellent book on Mössbauer spectroscopy. Goes into
the details of many different systems and is often the start point of any significant study. Available
for inspection at HPG.
[Crangle, 1991]: J. Crangle, Solid State Magnetism, Van Nostrand Reinhold, New York, 1991.
Very good book at general level on magnetism. Uses though nomenclature that only complicates
matters and typesetting that results in unclear formulation.
[Wikipedia]: www.wikipedia.org, contains articles on many of the subjects treated here, there are
probably errors in the articles, but there are usually references to the original work.
139
Appendix 1: Constants
Numerical values:
Atomic mass unit
Avogadro’s number*
Bohr magneton
Boltsmann constant
Electron charge
Speed of light
Permeability of vacuum
Planks constant
Nuclear magneton
Amu
NA
µB
K
e
c
µ0
h
βN
1.66·10-27
6.02·1023
9.27·10-24
1.38·10-23
1.60218·10-19
2.99792·108
4π·10-7
1.0546·10-34
5.0505·10-27
kg
Am2
J/K
A·s
m/s
kg·m/(A2s2)
J·s
J/T
Constants relevant for 57Fe Mössbauer spectroscopy:
Maximum Mössbauer cross section
Ground state magnetic hyperfine splitting
Excited state magnetic hyperfine splitting
Transition energy
Lifetime of Mössbauer state
Quadrupole moment
Sternheimer antishielding factor
σ0
γ1/2
γ3/2
E0
τ
Q
(1-γ∞)
140
2.57·10-22
-0.119015
0.067975
14.41
140
0.21·10-28
10.43
m2
mm/(T·s)
mm/(T·s)
keV
ns
m2
Appendix 2: Mössbauer parameters
Some paramagnetic iron compounds
Mineral name or series
Actinolite [1]
Formula
Ca2(Fe,Mg)5Si8O22(OH)2
δ
T
(K)
ΔEQ
(mm/s)
Notes
(mm/s)
295
1.15-1.16
2.81-2.82
M1,M3
1.13-1.16
1.89-2.03
M2
Akaganeite [1]
β-FeOOH
295
0.38
0.37
0.55
0.95
Anthophyllite [1]
(Fe,Mg)7Si8O22(OH)2
295
1.12-1.13
2.58-2.61
M1,M3
1.09-1.11
1.80-1.81
M4
Clinopyroxene, diopsitehedenbergite [1]
Ca(Fe,Mg)SiO3
295
1.16
2.15
M1
Cummingtonitegrunerite [1]
(Fe,Mg,Mn)7Si8O22(OH)2
295
1.14-1.18
2.76-2.90
M1,M2,M3
1.05-1.11
1.58-1.68
M4
1.16(2)
2.78(2)
M1,M2,M3
1.06(2)
1.51(2)
M4
Grunerite [2]
(Fe0.93,Mg0.07)7Si8O22(OH)2
295
Epidote [1]
Ca2(Al,Fe,Mn)AlOH.AlO.Si2O7.SiO4
295
0.34-0.36
2.01-2.02
Ferriphosphate [1]
Fe(PO4)3*4H2O
295
1.21
2.50
1.23
2.98
Ferrosulphate [1]
FeSO4*7H2O
295
80
1.27
1.39
2.83
3.08
Garnet group (andradite)
[1]
Ca3(Fe,Al)2(SiO4)3
295
0.41
0.58
Garnet group (pyropealmandine) [1]
(Mg,Fe)3Al2(SiO4)3
295
1.31
3.53-3.56
Gillespite [1]
BaFeSi4O10
295
0.76
0.51
Ilmenite [3]
FeTiO3
295
1.07
0.68
3+
Jarosite [4]
[(K, Na, X)(Fe , Al)3 (SO4)2(OH)6]
290
0.37-0.38
1.25-1.29
iron trifluride [3]
FeF3
295
0.49
0
Lepidocrocite [3]
γ-FeOOH
295
0.3
0.55
Molysite (hydrated) [3]
FeCl3*6H2O
295
0.45
0.97
Olivine [1]
(Mg,Fe,Mn)2SiO4
295
1.16-1.18
2.75-3.02
Orthopyroxene [1]
(Mg,Fe,Mn)SiO3
295
1.15-1.18
2.35-2.69
M1
1.12-1.16
1.91-2.13
M2
0.31
0.42
0.61
0.61
Pyrite [3]
FeS2
295
80
141
[2]
Siderite [3]
FeCO3
295
80
1.23
1.37
1.8
2
Staurolite [3]
(Fe,Mg)(Al,Fe)9O6(SiO4)8(O,OH)2
295
0.97
2.3
Tripotassium
Hexafluoroferrate [3]
K3FeF6
295
0.42
0.38
Ulvöspinel [3]
FeTiO4
295
1.07
1.85
Wüstite [3]
Fe1-xO
(x = 0.07)
(x=0.93)
295
0.81
0.48
295
0.86
0.78
[1] Greenwood and Gibb, 1971
[2] J Linares et al., J. Phys. C: Solid State Phys., 21 (1988) 1551
[3] S. Mørup, 1994
[4] K. Nomura et al., Hyp. Int., 166 (2005) 657.
Some magnetic iron compounds
Compound
Formula
Hematite [1]
α-Fe2O3
Maghemite [1]
γ-Fe2O3
Magnetite [1]
Fe3O4
δ
2ε
(mm/s)
T
(K)
Bhf
(T)
295
80
4.2
295
80
4.2
295
-0.20
+0.40
+0.40
0.00
0.00
0.00
0.00
0.00
(mm/s)
Goethite [1]
α-FeOOH
Akaganeite [1]
β-FeOOH
80
4.2
295
80
4.2
4.2
Lepidocrocite [1]
γ-FeOOH
4.2
51.7
0.37
54.2
0.48
54.2
0.49
49.9
0.32
52.6
0.44
52.6
0.44
49.2
0.29
46.0
0.67
at least 5 sextets
at least 5 sextets
38.1
0.37
50.1
0.48
50.6
0.48
48.9
0.50
47.8
0.48
47.3
0.48
45.8
0.47
Feroxyhite [1]
δ-FeOOH
4.2
52.5
0.45
0.00
Ferrihydrite [1]
~Fe4O6*H2O
4.2
46-50
0.45
0.00
metallic iron [1]
α-Fe
Troilite [1]
Ilmenite [2]
FeS
FeTiO3
295
80
4.2
295
4.2
32.9
33.9
34.0
31.5
4.8
0.00
0.12
0.12
0.81
1.22
0.00
0.00
0.00
-0.64
1.43
[1] S. Mørup, 1994
[2] Grant et al., Phys. Rev. B 5 (1971) 5
142
-0.26
-0.26
-0.26
-0.02
-0.24
-0.80
0.00
Index
Clebsch Gordon coefficients, 56
clinopyroxene, 86, 113, 114, 115
Mössbauer parameters, 83
Clovis, 121
cobalt
anisotropy constant, 23
Curie temperature, 17
coercivity, 29, 30, 31, 32, 33, 35
Colombia hills, 120
Columbia hills, 121
Columbia Hills, 121
cummingtonite
Mössbauer parameters, 83
Curie
temperature, 16, 17, 18
Curie constant, 13, 17
Curie temperature, 17, 18
demagnetisation, 8
demagnetisation factor, 23
dichotomy boundary, 111, 112
Domain magnetism, 22
Domain walls, 24
Eagle crater, 124
earths magnetic field, 7
easy directions, 22
El Capitan
Mössbauer spectrum of, 125
electron charge, 7, 140
enstatite, 85, 86
epidote, 89
Mössbauer parameters, 83
exchange integral, 22
exchange interactions, 16, 21
fayalite, 84
susceptibility, 13
Fe(III), 107, 109, 110, 118, 119, 123, 131
f-factor, 105
feldspar
susceptibility, 13
feroxyhite
Curie temperature, 17
Mössbauer parameters, 142
ferrihydrite
Mössbauer parameters, 142
ferrimagnetism, 20
Ferrimagnetism, 20
actinolite
Mössbauer parameters, 83
Adirondack, 82, 119
oxide composition, 82, 138
akaganeite
Mössbauer parameters, 142
Akimoto model, 93, 94
Alpha-Particle-X-ray Spectrometer (APXS),
117
Amazonian, 111, 116
amphiboles
susceptibility, 13
Analysis of Mössbauer spectra, 104
anisotropy constant, 22, 23
anthophyllite
Mössbauer parameters, 83
antiferromagnetism, 19
Århus Mars Wind-tunnel facilities, 136
atomic mass unit (amu), 7, 140
Avogadro’s number, 7, 140
basalt, 31, 72, 74, 81, 82, 84, 85, 104
elemental composition, 82
mineralogy, 81
olivine basalt, 82, 119, 132, 133
Mössbauer spectrum, 132, 134
olivine-magnetic phase correlation, 133
remanence, 133
SEM image, 134
Biot and Savart law, 8
biotite
susceptibility, 13
blueberries. See spherules
Blueberry bowl, 126
Bohr magneton, 7, 9, 140
Boltsmann constant, 7, 140
Breit-Wigner lineshape. See Lorentzian
profile
Brillouin function, 12, 66
calcite
susceptibility, 13
Capture magnet, 127, 128
after dust clearing, 131
CEMS, 51, 70, 73, 74, 75, 108, 109, 110, 134
detector, 74, 75
chlorite, 89
chromite, 81
143
ferrites
spinel. See spinels
ferromagnetism, 15
ferrosilite, 86
f-factor, 105
Filter magnet, 127, 128, 130
Mössbauer spectrum of, 130
forsterite, 84
in pallasite, 86
susceptibility, 13
gadolinium
Curie temperature, 17
garnet, 107
Mössbauer parameters, 83
gillespite
Mössbauer parameters, 83
glass, 89
Mössbauer parameters, 89
Mössbauer spectrum, 89
goethite, 90, 121
saturation magnetisation, 21
Goethite, 104, 120, 121
Mössbauer parameters, 142
Neel temperature, 19
granite, 106
greigite
saturation magnetisation, 21
grunerite
Mössbauer parameters, 83
Gufunes dust, 136
Mössbauer spectrum of, 137
oxide composition of, 138
Gusev crater, 117, 119, 120, 121
Gusev soil
oxide composition of, 138
hematite, 32, 81, 85, 90, 108, 109, 110, 112,
113, 119, 121, 123, 124, 125, 126, 131
anisotropy constant, 23
Curie temperature, 17
f-factor, 105
magnetisation curve, 33
Mössbauer par, 97
Mössbauer parameters, 142
on Mars, 112
saturation magnetisation, 21, 33
SD/MD radius, 26
Hesperian, 111
high-Ca pyroxene. See clinopyroxene
HiRISE, 122
Home Plate, 121, 122, 123
Hund’s rules, 10
Husband hill, 121
illite
susceptibility, 13
ilmenite, 87, 88, 101, 107, 108, 109, 110, 119
susceptibility, 13
composition of, 81
crystal structure, 88
f-factor, 105
Mössbauer parameters, 142
Mössbauer spectrum, 88
Neel temperature, 19, 88
intrusive rocks, 103
iron
Curie temperature, 17
saturation magnetisation, 21
SD/MD radius, 26
isomer shift, 48, 49, 50, 54, 56, 57, 79
basalt, 109
for 57Fe, 49
h, 97
in alloys, 69, 70
in FeTi alloy, 71
in source experiments, 75
magnetite, 91
of ferric iron, 89
of metallic iron, 61
of silicates, 83
titanomaghemite, 103, 104
titanomagnetite, 95
jarosite, 89, 125
Kjalarnes magnetic anomaly, 103
Landé g-factor, 11
Langevin function, 66
lepidocrocite
Mössbauer parameters, 142
Neel temperature, 19
Lorentzian profile, 44, 45, 46, 50, 51, 58, 70,
89
maghemite, 31, 81, 97, 99, 100, 101, 108,
109, 110, 119, See
anisotropy constant, 23
Curie temperature, 17
f-factor, 105
inversion, 32, 98
inversion of, 99
144
Mars Exploration Rover, 110, 116, 117, 126,
See MER
Oppertunity, 113
Mars Exploration Rovers, 116
Mars Express, 112
Mars Global Surveyor, 112
Mars Reconnaissance Orbiter, 122
mass density, 9
MER, 82
Meridiani planum, 113, 123
metallic iron
anisotropy constant, 23
Mössbauer parameters, 142
metamorphic, 106, 107
microscopic imager (MI), 117
Microscopy Imager (MI), 126
MINI-TES, 117
MOLA, 110, 111, 117
molecular field. See Weiss field
molecular field coefficient. See Weiss
constant
montmorillonite
susceptibility, 13
MORB, 112
Morin tra, 96
Morin transition, 97
Morin transition temperature, 126
Nakhla, 116
Nd-Fe-B magnets, 30
Neel temperature, 19
Néel-Chevallier model, 93
nickel
anisotropy constant, 23
Curie temperature, 17
Noachian, 111, 112, 116
nonstochiometric magnetite, 100
non-stoichiometric magnetite, 100
Nuclear magneton, 140
O’Reilly-Banerjee model, 93, 94
olivine, 84, 85, 87, 109, 110, 113, 114, 115,
119, 120, 123, 124
composition of, 81
f-factor, 105
in pallasite, 86
Mössbauer parameters, 83
optical detection of, 113
oxidation of, 85
SEM image, 84
magnetisation curve, 33
Mössbauer parameters, 142
Mössbauer spectrum, 98, 99
saturation magnetisation, 21, 33
SD/MD radius, 26
maghemitization, 101, 102
magnetic dipole, 8
magnetic field
external, 9, 11
internal, 15
magnetic field H, 6
Magnetic forces, 34
magnetic hyperfine field (Bhf), 55
57
Fe, 55, 56
Magnetic hyperfine interactions, 54
magnetic moment, 9
of an ion, 9
magnetic separate, 102, 107, 108, 109, 110
magnetisation, 9
curve, 31
mass, 9
measurements, 30
saturation, 21
volume, 9
magnetite, 20, 31, 32, 84, 85, 90, 91, 99, 100,
101, 107, 108, 109, 110, 119, 123
anisotropy constant, 23
coercivity, 32
crystal structure, 20
Curie temperature, 17
f-factor, 105
magnetic coupling, 21
Mössbauer parameters, 90
Mössbauer spectrum, 90, 91
observed hyperfine, 69
reduced saturation magnetisation, 31, 32
saturation magnetisation, 21
SD/MD radius, 26
susceptibility, 18
Verwey transition, 18
Magnetite
Mössbauer parameters, 142
Magnetostatic energy, 23
Mars, 81, 110, 111
Geology, 110
magnetic field, 112
Mars analogues, 132
145
in NFS, 79
quantum number, 9
quarts
susceptibility, 13
quartz, 85
relaxation time, 29, 30
Resonance detector. See CEMS detector
rock abrasion tool (RAT), 117
Rock abrasion tool (RAT), 126
saturation magnetisation, 32, 106
scanning electron microscopy. See SEM
self energy, 23
SEM, 84, 85
serpentinite
susceptibility, 13
shape anisotropy, 29
siderite
susceptibility, 13
silicates, 82, 83
Mössbauer parameters, 83
Single domain particles, 26
single domain/multi domain critical radius, 26
Skagen, 106, 107
Sm-Co magnet
anisotropy constant, 23
SNC, 115, 116
SNC meteorites, 110
speed of light, 7, 140
spherules, 124
spherules, 124
spherules
MI image, 124
spherules, 125
spherules, 125
spherules, 126
spherules, 126
spherules
Mössbauer spectrum, 126
spinel, 20, 90, 91, 97
spinels, 20
Spirit, 117, 118, 120, 121, 122, 130
staurolite
Mössbauer parameters, 83
sulphite, 89
sulphites, 81
superparamagnetism, 29
susceptibility, 9, 13, 18, 21
above the Curie temperature, 17
oxidized, 134
susceptibility, 13
OMEGA, 111, 112, 115
Opportunity, 123, 125
orthopyroxene, 86, 113, 115
Mössbauer parameters, 83
orthopyroxenes
susceptibility, 13
oxidation, 81
pallasite, 85
Pancam, 117
Paperclip magnetism, 35
permeability of vacuum, 7, 140
phyrotite
SD/MD radius, 26
plagioclase, 85
composition of, 81
SEM image, 134
plagioclases, 81
Planks constant, 7, 9, 140
pyrite, 89
susceptibility, 13
pyroxene, 85, 86, 87, 89, 109, 110, 113, 114,
120, 123, 124
composition of, 81
f-factor, 105
optical detection of, 113
SEM image, 134
pyrrhotite
saturation magnetisation, 21
quadrupole
doublet, 58
interaction, 52, 56, 57
shift, 57
metallic iron, 61
Wertheim model, 69
splitting, 52
ilmenite, 88
olivine, 84
pyroxene, 87
silicates, 83
splitting in 57Fe, 53
term, 48
quadrupole doublet
line intensity, 57
quadrupole moment, 53
quadrupole moment, 53
quadrupole splitting, 54, 83
146
below the Curie temperature, 18
ferrimagnetic material, 18
Sweep magnet, 127, 128, 129, 130
Syrtis Major, 115
TES, 111, 112, 113, 114, 117, 123
Thermal Emission Spectrometer. See TES
titanomaghemite, 103
average isomer shift, 104
average magnetic hyperfine field, 104
lattice constants, 103
Mössbauer spectrum, 103
titanomagnetite, 85, 91, 92, 93, 101, 102, 108,
110, 119
composition of, 81
in SNC, 116
Mössbauer parameters, 93
Mössbauer spectrum, 92, 102
oxidation of, 102
saturation magnetisation, 21, 94
SD/MD radius, 26
Titanomagnetite
oxidation of, 95
titanommaghemite
Curie temperatures, 103
total angular momentum, 9
total spin, 9
troilite
Mössbauer parameters, 142
ulvöspinel
Neel temperature, 19
Verwey transition, 18, 91, 100
vibrating sample magnetometer, 30
Voigt profile, 51
water
susceptibility, 13
Weiss
constant, 16
field, 15
theory, 15, 16, 17, 19, 22
wollastonite, 86
Zagami, 116
Zeeman splitting, 54
χ2, 105
147