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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. 2 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