Magnetostriction in Rare Earth Elements Measured with Capacitance Dilatometry

Home , Magnetostriction

Diplomarbeit

zur Erlangung des akademischen Grades eines Magisters der Naturwissenschaften

Betreuer: Dr. habil. Martin Rotter

eingereicht von: Alexander Barcza Matrikelnummer: 9902823

Mai 2006 ii

Eidesstattliche Erkl¨arung

Ich, Alexander Barcza, geboren am 24.01.1981 in Sankt P¨olten, erkl¨are,

1. dass ich diese Diplomarbeit selbstst¨andig verfasst, keine anderen als die angegebe- nen Quellen und Hilfsmittel benutzt und mich auch sonst keiner unerlaubten Hilfen bedient habe,

2. dass ich meine Diplomarbeit bisher weder im In- noch im Ausland in irgen- deiner Form als Pr¨ufungsarbeit vorgelegt habe,

Wien, am 27.05.2006 Unterschrift iii Kurzfassung

Die vorliegende Diplomarbeit wurde im Zeitraum von September 2005 bis bis Mai 2006 verfasst. Die Arbeiten wurden teilweise an der Universität Wien, an der Technischen Universität Wien und am National High Magnetic Field Laboratory (NHMFL) Tallahassee, Florida durchgeführt. Das Thema der Arbeit ist die Mes- sung der thermischen Ausdehnung und der Magnetostriktion der Selten-Erd-Metalle Samarium und Thulium. Die Kapazitätsdilatometrie ist eine der empfindlichsten und deswegen am meisten eingesetzten Methoden, um diese Eigenschaften zu messen. Mit dieser Methode ist es möglich, relative Längenänderungen im Bereich von 10−7 zu messen. Ein hoch entwickeltes Silber-Miniatur-Dilatometer wurde aus den Einzel- teilen zusammengebaut und an einer Serie von Standardmaterialien getestet. Das neue Dilatometer wurde auch in magnetischen Feldern von bis zu 45 T am weltweit führenden NHMFL eingesetzt. Ein kurzer Uberblick¨ über die Technik zur Erzeugung hohe Magnetfelder, deren Vorteile und Grenzen wird gegeben. Weiters wird die Kühltechnik und die Stromversorgung des Forschungszentrums beschrieben. Erstmalig durchgeführte Messung der Magnetostriktion an Sm in ultra hohen Mag- netfeldern werden präsentiert. Der magnetoelastische Effekt wurde an allen drei Achsen eines Einkristalls gemessen. Sowohl longitudinale, als auch transversale Mag- netostriktion wurde gemessen. Der Einfluss der Temperatur auf die Magnetostrik- tion wurde ebenfalls untersucht. Der Spin-Flop Ubergang,¨ der in hohen Feldern passiert (33 T), wurde durch ein Modell beschrieben. Weiters wurde die thermische Ausdehnung und die Magnetostriktion von Thulium gemessen. Dieses Element zeigt eine große magnetische Anisotropie, die eine Fix- ierung im Magnetfeld schwer macht. Ein spezielles experimentelles Setup machte eine Messung der Magnetostriktion bis 9 T in c-Richtung des Kristalls möglich. Dieses Magnetfeld wurde mit einem üblichen supraleitenden Magneten erzeugt. Es wurde ein großer Effekt bei 3.5 T gemessen, der von einem magnetischen Phasenübergang kommt. iv Abstract

The following diploma thesis was done from September 2005 to May 2006 partly at the Technical University of Vienna, the University of Vienna, and at the Na- tional High Magnetic Field Laboratory (NHMFL) in Tallahassee, Florida. The work presented here is concerned with the measurement of thermal expansion and magnetostriction of the rare earth elements Samarium and Thulium. The capacitance dilatometry is one of the most sensitive, and therefore most common methods to measure these quantities. With this method it is possible to detect relative length changes as small as 10−7. An highly developed silver miniature dilatometer was assembled from the individual parts, and tested in a series of standard materials measurements. The new dilatometer was also used in an ultra high static magnetic ﬁeld up to 45 T at the worldwide leading magnet facility (NHMFL). A short overview about the technique for producing such high ﬁelds, its advantages, and limits is given. Peripheral devices of the NHMFL, such as the cooling system, and the power supply, are described.

First time measurements of the magnetostriction of the rare earth element Sm in ultra high magnetic fields are presented. The magnetoelastic effect was measured along all three axes of a single crystal. Longitudinal and transversal magnetostriction was measured. The temperature influence on the magnetostriction is investigated. A model for calculating the spin flop transition, which occurs in high magnetic fields (33 T), is applied to the experimental data. The thermal expansion and magnetostriction of Thulium were measured. This element shows a large magnetic anisotropy, which made it difficult to fix it in magnetic fields. A special experimental setup made it possible to measure the magnetostriction of Tm along the c-axis in fields up to 9 T. This fields were reached with a standard superconducting magnet. A large magnetostriction effect was observed at 3 T, indicating a magnetic phase transition. v Acknowledgments

My special thanks belong to Dr. habil. Martin Rotter, who suggested the topic of this diploma thesis. With his passion for making excellent science and solving interesting problems, he always encouraged me during this work. From the start on he did everything to support me. Many experiments were done at the Technische Universit¨at Wien with the help of Ass. Prof. Herbert M¨uller, to whom I want to dedicate my thanks. His excellent knowledge about the method of measuring magnetostriction and thermal expansion helped me a lot.

I also want to thank A. Lahner and whole workshop of the Technische Universit¨at Wien, who manufactured the dilatometer. All requests, or suggestions for technical improvements were granted. Furthermore I want to thank Dr. Mathias D¨oerr, Mag. D. Le, E. Jobiliong, Prof. J. Brooks, Dr. Bruce Brandt, Dr. Andreas Kreysig, Dr. S. Hannahs, A. Devishvili, and A Marry Ann for the cooperation, and the interesting and fruitful experiments at the NHMFL (Tallahassee, Florida).

I want to express my gratitude for all the support and love I got from my parents, and the rest of my family. I also want to say thank you to my girlfriend Maga(FH) Michaela Steurer for supporting me. Many thanks to all my friends. vi Contents

1 Theory 1

1.1 Thermalexpansion ...... 2

1.1.1 Isotropicthermalexpansion ...... 3

1.1.2 Anisotropic thermal expansion ...... 4

1.2 Magnetostriction ...... 5

1.2.1 Magnetostriction in the standard model of rare earth magnetism 5

1.2.2 Crystalﬁeld...... 7

1.2.3 Exchangestriction ...... 8

2 Experimental details 11

2.1 Measurementmethods ...... 11

2.2 Thecapacitancedilatometry ...... 12

2.3 Anewkindofdilatometer ...... 14

2.3.1 Assembling the capacitance dilatometer ...... 14

2.3.2 Preparing the dilatometer for measurement ...... 19

2.4 Calibration and testing of the dilatometer ...... 19

2.4.1 Force control and enhancement ...... 26

2.5 Generation of static high magnetic ﬁelds ...... 28

2.5.1 Thecoolingsystem ...... 28

2.5.2 PowerSupply ...... 29

2.5.3 TheFloridaBitterMagnet...... 30

vii viii CONTENTS

2.5.4 TheHybridMagnet...... 31

2.6 Measurements in high magnetic ﬁeld ...... 32

3 Results and discussion 35

3.1 Samarium ...... 35 3.1.1 The crystal and magnetic structure of Sm ...... 35

3.1.2 ThermalexpansionofSm ...... 36 3.1.3 MagnetostrictionofSm...... 36

3.1.4 Results...... 38 3.1.5 The inﬂuence of temperature ...... 41 3.1.6 Aspin-ﬂoptransitioninSm ...... 44

3.1.7 Calculations for the spin-ﬂop transition ...... 47 3.1.8 CalculationofmagnetizationofSm ...... 47

3.1.9 Calculation of the magnetostriction of Sm ...... 49 3.2 Thulium...... 58 3.2.1 Crystalstructure ...... 58

3.2.2 ThermalexpansionofTm ...... 58 3.2.3 MagnetostrictionofTm ...... 59

4 Summary and conclusion 63

APPENDIX 67

A AKramersGroundStateDoublet ...... 67 Chapter 1

Theory

This chapter is concerned with the theoretical background of thermal expansion and magnetostriction. It gives a short and compact overview over the main principles and formulas needed to explain these effects. To avoid confusion about the notation of the different physical values, the following table explains the symbols used, and their definitions [4, 5]:

ǫµ(T, H): Lagrange ﬁnite-strain coordinates describing the deformation of a single crystal from some state chosen as the ’reference conﬁguration’.

µ ∂ǫµ(T,H) α (T, H)= ∂T : Thermal Expansion Coeﬃcients

µ αµ (T, H)= ∂ǫ (T,H) : Magnetostriction Coeﬃcients Hν ∂Hν cµν: Elastic constants sµν: Elastic compliances

= Note that for the components ǫµ of the strain tensor ǫ and for the elastic constants cµν etc. Voigt’s abbreviated notation is used (i.e. µ = 1, 2, 3, 4, 5, 6 denote 11, 22, 33, 12, 13, 23 respectively). The elastic compliances sµν are related to the elastic constants by (see e.g. [23])

6 γν cµγ s = δµν (1.1) γ=1 X The thermal expansion coeﬃcients αµ are related to the second derivatives of the = Helmholtz free energy F (ǫ, T, H) by [5, 23]

1 2 CHAPTER 1. THEORY

sµν ∂2F αµ = − (1.2) V ∂T∂ǫν ν X The temperature dependence of the thermal expansion is given by the second derivative of the free energy 1.2 with respect to temperature and strain. If the free energy can be written as the sum of the free energy of several subsystems (magnetic electrons, lattice contribution, conduction electrons), then the same is possible for the thermal expansion coeﬃcients and the strains.

sµν ∂F + F + F ǫµ(T, H) = − m ph el V ∂ǫν ν µ X µ µ = ǫm(T, H)+ ǫph(T )+ ǫel(T ) (1.3)

In evaluation of Eq. 1.3 usually the temperature/ﬁeld variation of sµν can be neglected in comparison to the variation of the derivative of the free energy.

1.1 Thermal expansion

Thermal expansion is the change of the length or volume of a substance with respect to a change of temperature. Most materials, no matter if gases, liquids, or solids, change their dimensions if the temperature changes. Thermal expansion can be positive (expansion of the material) as well as negative (contraction of the material, for example water below 4 ◦C). Anisotropic thermal expansion leads to a change of shape. There are special designed materials which have almost no thermal expansion (this is known as the Invar-eﬀect [26]). For some needs also substances with large thermal expansion have been developed. The fundamental, and phenomenologically most used equation to determine the thermal expansion of a solid is:

Lfinal − Linitial = α(Tfinal − Tinitial) (1.4) Linitial This gives the length change of a sample in one direction with varying temperature.

In Eq. 1.4 Linitial is the length of the sample at the temperature Tinitial, and Lfinal is the length at Tfinal. Owing the fact, that different materials expand different, it is necessary to introduce a material specific coefficient. This is α, the linear thermal expansion coefficient. 1.1. THERMAL EXPANSION 3

Thermal expansion is as important for daily life as it is for science, engineering, construction work, and many more. In the following section the main principles of thermal expansion are explained as well as some important formulas and equations, necessary to explain experimental results.

1.1.1 Isotropic thermal expansion

If a material changes its length equally in all directions with respect to temperature, this is called isotropic thermal expansion. If the volume of a sample is considered, instead of its length, then the volumetric expansion of matter is described by its isotropic expansion coeﬃcient β.

1 ∂V β = (1.5) V ∂T P In equation 1.5 V is the volume at a temperature T at constant pressure P. The unit of β is K−1 and it gives the relative change of volume per Kelvin. The isotropic thermal expansion coeﬃcient has the following thermodynamic relation to the free energy F:

∂lnV ∂lnV ∂P ∂P ∂2F ∂S β = = − = κ = κ = κ (1.6a) ∂T ∂P ∂T T ∂T T ∂V ∂T T ∂V P T V V T

with 1 ∂2F = V 2 (1.6b) κT ∂V

In Eq. 1.6a S is the entropy. The temperature dependence of the isothermal com- pressibility κT is usually neglected compared to the volume derivative of the entropy in evaluation of Eq. 1.6a [8]. This is a good approximation for smaller temperatures.

If the free energy and the entropy can be written as a sum of these contributions it is possible to do the same with β. Thus the isotropic expansion coeﬃcient consists of diﬀerent parts, such as the phonon contribution, electronic contribution, and magnetic contribution. This is a good approximation if there is no coupling between the single components such as phonon-electron coupling, or if it is negligible. In Eq. 1.7 the single contributions are denoted as r in the superscript. 4 CHAPTER 1. THEORY

∂Sr β = βr = κ (1.7) T ∂V r r T X X

Often it is interesting to know the relative volumetric expansion of a material, rather then the expansion coeﬃcient. This can be calculated from the expansion coeﬃcient through integration, or summation of the linear expansion along the three axes:

V − V T ǫ = 0 = βdT = ǫ + ǫ + ǫ (1.8) v V a b c 0 ZT0

In equation 1.8 V is the volume at the temperature T, and V0 is the volume at temperature T0 . The linear thermal expansion along the three axes is denoted as

ǫα (α is a,b,c). The reference volume V0 is often taken at T=0 K because then all contributions to the thermal expansion vanish.

1.1.2 Anisotropic thermal expansion

Anisotropic thermal expansion occurs if not only the volume of the substance changes but also its shape. That means that the solid expands in each direction by a diﬀerent amount. In order to determine the anisotropic expansion coeﬃcients it is important in which direction of the material the thermal expansion is measured. A great advantage of x-ray, and neutron scattering is that all three directions in space can be measured at once (seen Sec. 2.1 for an overview). If a solid expands anisotropically there could be a contraction in one direction, and an expansion in the other one, or any other possible combination. In order to describe anisotropic behavior, it is necessary to introduce the thermal expansion tensor βij.

∂ǫ β = β = ij (1.9) ij ji ∂T σ

Here ǫij is one of the components of the symmetric Lagrange strain tensor. The maximum number of independent tensor-components is 6 and therefore Voigt’s ab- breviation notation is used. 1.2. MAGNETOSTRICTION 5

ǫa = ǫ1 = ǫ11 (1.10)

ǫb = ǫ2 = ǫ22 (1.11)

ǫc = ǫ3 = ǫ33 (1.12)

ǫ4 =2ǫ23 =2ǫ32, ... (1.13) (1.14)

The relative volumetric expansion can be calculated with Eq. 1.15, which is the sum over volumetric expansions in the three principal axes.

∆V 3 = ǫ (1.15) V λ Xλ=1 1.2 Magnetostriction

An easy and demonstrative explanation for the effect of magnetostriction of a solid is the change of shape or volume of a solid in a magnetic field. For practical reasons magnetostriction is often described with phenomenological models, that are concerned with the crystal symmetry. To explain the magnetoelastic phenomenon on an atomic level it is not sufficient to deal with crystal symmetry. A microscopic model has to be used. The main sources of magnetostriction in rare earth compounds are the crystal field and exchange striction. Whereas the former is a widely accepted and long studied theory the later has been less investigated. Experimen- tally it is difficult to separate the two mechanisms. The following chapter gives a very short overview over the two main contributions to magnetostriction in rare earth compounds.

1.2.1 Magnetostriction in the standard model of rare earth magnetism

The theory presented here in short is based on [17] and was generalized to arbitrary magnetic and crystal structures by Rotter et. al. [8]. The approach made here is valid for systems with small magnetoelastic energy compared to the magnetic and elastic energy. Any dynamic coupling between the lattice and the crystal ﬁeld, or magnetic 6 CHAPTER 1. THEORY exchange is neglected. Assuming that the principal sources of magnetostriction in rare earth elements are the crystal ﬁeld, and exchange striction, the overall magnetic Hamiltonian for such a system can be written as:

H = Hcf + Hex + HZe + Eel (1.16) m = m Hcf = Bl (i, ǫ)Ol (Ji) (1.17) i Xlm, 1 = H = − JαJ (ǫ, i − j)Jβ (1.18) ex 2 i αβ j ij X,αβ HZe = − gJ µBJiH (1.19) i X where the Hamiltonian in Eq. 1.16 is the sum of crystal ﬁeld Hcf , two ion exchange

Hex, and Zeeman HZe parts. The term Eel is the elastic energy, and thus contains all the energy contributions from the rest of the system.

m In Eq. 1.17 the Bl are the crystal field parameters, which describe the strength m α of the crystal electric field, and the Ol (Ji) are the Stevens operators. The Ji in equation 1.18 is the α component of the angular momentum operator of the ion i, = and J αβ is the two ion interaction tensor. H is the magnetic field and gJ , and is the Landeéfactor. The crystal field Hamiltonian depends on the position of one single atom i which is in a certain electronic environment. The two ion interaction tensor = J αβ (ij) depends on the position of atom i and j. With this equations, and some further analysis it is possible to get an expression for the magnetostrictive strain α ǫm. Starting point is the expansion of Eq. 1.17 and Eq. 1.18 into first order Taylor series with respect to the strain.

m = m m = m α m m Bl (i, ǫ)Ol (Ji) ≈ Bl (i, ǫ= 0)Ol (Ji)+ ǫ Bl(α)(i)Ol (Ji)+ ··· (1.20) i i i Xlm, Xlm, Xlm,

α = = = 1 = 1 = 1 α − Ji J(ij, ǫ)Jj ≈ − Ji J(ij, ǫ= 0)Jj − ǫ Ji J (ij)Jj + ··· (1.21) 2 2 2 (α) ij ij ij X X X with the magnetoelastic constants

= = = ∂ J (ij, ǫ) J (ij)= (1.22) (α) ∂ǫα " #ǫ=0 1.2. MAGNETOSTRICTION 7

= ∂Bm(i, ǫ) Bm (i)= l (1.23) l(α) ∂ǫα " #ǫ=0 In most cases the expansion is limited to ﬁrst order in strain, which is suﬃcient to explain linear phenomenon.

By deﬁnition the magnetic free energy Fm is given by

Fm = −kBT ln Z (1.24) with the partition sum

Z = Tr{e−H/kB T } (1.25)

Here kB denotes the Boltzmann constant and Tr the trace of a quantummechanical operator. Inserting the Hamiltonian H (1.16) into (1.24) and (1.25), calculating the α derivative of the magnetic free energy Fm with respect to the strains ǫ and using equation (1.3) yields the ﬁnal result

α α α ǫm = ǫcf + ǫex (1.26) 1 ǫα = − sαβBm (i)hOm(J )i (1.27) cf V l(α) l i T,H i Xβ, = α 1 αβ ǫ = + s hJi J (ij)Jji H (1.28) ex 2V (β) T, ij Xβ, The magnetic contribution to the strain can be calculated as the sum of the crystal ﬁeld contribution and the exchange part. From equations 1.27 and 1.28 it is clear, that the whole temperature and ﬁeld dependence of the magnetoelastic strain can be calculated from thermal expectation values of the Stevens operator equivalents ′ m γ γ hOl (Ji)iT,H and static correlation functions hJi Jj i.

1.2.2 Crystal ﬁeld

To understand the crystal ﬁeld mechanism and the exchange striction it is easier to explain them using the model system shown in Fig. 1.1. The subﬁgure a shows two negative charges (rare earth ions) surrounded by four positive charges. The crystal 8 CHAPTER 1. THEORY

field splitting ∆cf of the 4f ground state is due to Coulomb’s law. At temperatures below the crystal field splitting only low energy states are populated which leads to a deformation of the charge density of the 4f electrons. This leads to an increase or decrease of force between the 4f electrons and the positive charge which causes deformation, that is called crystal field striction. That mechanism is shown on the right side of Fig. 1.1. Note that for this mechanism no long range order is necessary. If the temperature is lowered below the Neél temperature the moments order antiferromagnetically. Crystal field theory says that the Stevens factors (αJ , βJ ,γJ ) govern the magnetic anisotropy of a system. If the Stevens factor is positive (Sm3+, Er3+, T m3+, Y b3+), the magnetic easy axis is vertical in the model system in fig. 1.1. It is horizontal, if the Stevens factor is negative ((Ce3+, Pr3+, Nd3+, T b3+, Dy3+, 3+ Ho ). Consider a rare earth with negative α. Below TN the moments order antiferromagnetically and deform the charge density. This has an influence on the distance between the positive and negative charges, and leads to spontaneous magnetostriction below TN , as indicated by the small arrows in figure 1.1. In a magnetic field normal to the magnetic moments (easy axis), the moments are turned. Due to spin orbit coupling the charge density also changes its shape and again leads to magnetostriction. Now the effect is opposite to the case described in subfigure b in Fig. 1.1.

1.2.3 Exchange striction

In systems with zero orbital momentum (L = 0) the 4f charge density is spherical symmetric. There is no crystal field and the magnetostriction is mainly dominated by the exchange striction. In such systems the orientation of the magnetic moments is not determined by the crystal field but by anisotropies such as the dipolar interaction. The two ion interaction depends on the distance between these ions. Below the Néel temperature the exchange energy (Heisenberg, RKKY, etc.) of the system may be lowered by a change of the distance between the ions (see Fig. 1.1). A change of the distance between ions is nothing else than magnetostriction, which is referred to as exchange striction. This effect may only occur for ordered magnetic moments, which is in contrast to the crystal field mechanism. This is one possibility to separate the two mechanisms from each other. Furthermore, in systems with L = 0 (for example Gd3+) only exchange striction is possible. 1.2. MAGNETOSTRICTION 9

Crystal Field Exchange - Striction

kT> a cf kT< cf d + + + +

e- e- e- e- T>T N

+ + + + b e Ce, Pr Nd, Dy Ho ααα<0 T

c f H

Figure 1.1 The crystal-ﬁeld (a-c) and exchange (d-f) mechanism for magnetoelastic strains. The description is in the text. 10 CHAPTER 1. THEORY Chapter 2

Experimental details

This chapter gives a short overview of the methods used to measure magnetostriction. The capacitance dilatometry is described in more detail, and the complete procedure to assemble a dilatometer is outlined. Some of the presented results where measured in static high magnetic fields. Therefore the generation of such high fields is outlined, as well as the experimental dispositions necessary. The common attribute of all the methods is, that they all have to give access to a length change of a sample that is caused by a temperature change, or a change of the magnetic field, or both.

2.1 Measurement methods

The measurement of thermal and magnetic expansion can be done with various methods which all have advantages and disadvantages. It is possible to divide the methods into two groups - microscopic (e.g. x-ray, and neutron scattering) and macroscopic (e.g. capacitance dilatometry, interferometry) methods. Microscopic methods such as x-ray diffraction and neutron scattering are able to determine the atomic positions in the crystallographic unit cell. If the device can be modified to measure at different temperatures and magnetic fields these methods can be used to obtain thermal expansion and magnetostriction. One great advantage is to get these values from a powder sample, because the lattice parameters can be determined with scattering methods. However, these measurements require relatively great prearrangement, and high precision experiments can only be accom- plished at large-scale research facilities. The application of an external field is often

11 12 CHAPTER 2. EXPERIMENTAL DETAILS a problem.

The most important representative of the macroscopic methods is the capacitance dilatometry. It is an extensively used method with very high sensitivity (theoret- ically it is possible to resolve ∆l/l changes that are as small as 10−9). The main principle is the change of capacitance between two capacitor plates, which can be related to a length-change of a sample. This can be measured with high precision capacitance bridges. To get all 9 components of the magnetostriction tensor (refer to Sec. 1.1.2) it is necessary to have a single crystal. Another possibility to measure length changes of solid samples is to use strain gages that are glued to the sample-surface. These gages detect the length-change via a change in resistivity of the gage material. One problem is that the probe has to have a very flat surface, and it must be a few millimeters in length in order to glue the gages to it. An optical method to detect thermal and magnetic expansion is the interferometry. A change in the distance between two optical flats results in different interference patterns. They can be used to determine the value of the length change. To get an overview of the common methods refer to [8]. The results in the present work were solely gained with the capacitance dilatometry which is explained in the next section.

2.2 The capacitance dilatometry

The roots of capacitance dilatometry date back to 1961, when White [28] combined experiences of a two-terminal capacitance method for measuring thermal expansion with Thompsons three-terminal method [24] for capacity measurements using a ra- tio transformer bridge. He achieved a hitherto unreachable resolution of 10−7mm. Afterwards the two-terminal method was scarcely used again. Whites design principles of absolute and relative dilatometers were adopted and improved by a number of different authors. It led to absolute thermal expansion measurements on a number of reference metals like Cu, Ag, Au, and Al. Green [11], Chanddrasekhar, and Fawcett [9] were among the first to use the dilatometers for magnetostriction measurements. Tilford and Swenson used an inverted configuration of Whites dilatometer to measure the thermal expansion of solid Ar, Kr, and Xe. The reproducibility of the dilatometer was increased by replacing the oxygen- 2.2. THE CAPACITANCE DILATOMETRY 13

Figure 2.1 A schematic drawing of a capacitance dilatometer.

free copper reference rods with silicon [19] or sapphire [7]. However, using for the whole dilatometer Si or quartz with metal-plated electrodes is problematic and has not been widely adopted except in pulsed magnetic fields. Subrachmanyam and Subramanyam [22] went back to a copper dilatometer for the use of samples dif- fering in length, however, still samples should have parallel surfaces. Sparavigna et al. [3] designed an apparatus for simultaneous measurement of thermal expansion and thermal diffusivity still keeping the main features of Whites dilatometer design. The idea of separating the sample from the displacement sensor was fas- cinating, because samples of different lengths and shapes can be used. The drift of capacitance with time could be reduced by the use of sapphire isolation washers instead of epoxymylar isolation [18]. In the dilatometer used here the tilted plate principle [6, 10] is applied.

All these developments improved the method from the very beginning to the present state of the art. The great advantage of the capacitance method is, that it is one of the most sensitive methods for measuring small length changes in solids. In practice the accuracy is limited frequently by mechanical sample quality and dilatometer eﬀects. A schematic drawing of a dilatometer can be seen in Fig. 2.1. It is a direct dilatometer, which means that a expansion of the sample located between the capacitance plates leads to an increase of the distance between the capacitor plates. The picture shows the principal assembling of a capacitance dilatometer. It consists mainly out of two parts, namely the upper and the lower capacitance plate housing and the capacitance plates, which are located inside the housing. It is crucial for the measurement that the housing and the capacitance plates are isolated against each other. 14 CHAPTER 2. EXPERIMENTAL DETAILS 2.3 A new kind of dilatometer

For the present diploma thesis a new dilatometer was used. It was part of this work to assemble and test it. The improvement of this device is the possibility to adjust and measure the force that acts onto the sample inside the dilatometer. The single steps from the individual parts to the first measurements are described in the next section. The device is a miniature capacitance dilatometer, based on the tilted plate principle. Recent developments made it possible to miniaturize the device down to diameters of 20 mm, which is important for a wide range of applications. However, to reach the highest possible sensitivity and reproduceability, special effort is needed in the design of the capacitance dilatometer and the sample preparation. Therefore a dilatometer has been developed by combining the experience of the group at the TU with already published methods. The device was built at the TU’s workshop. It has tilted plates, which means that the condensator plates are not necessarily parallel to each other. This circumstance effects the formulas for calculation of the distance between the plates (see Sec. 2.4). A similar dilatometer was used to study phase transitions in rare earth metals. The samples used in this research field are small (a few mm3) because it is difficult to grow big single crystals on the one hand, and on the other hand the materials are expensive. This needs to be taken into account, when such a dilatometer is built. Another reason to restrict the dimension of the device is that it has to be put into a magnet. The space inside a magnet is limited and only in a certain range there is a homogeneous field. The outer diameter of the dilatometer is 20 mm (both in transversal and longitudinal operation mode) and it fits inside most standard magnets. The construction is schematically shown in Fig. 2.1.

2.3.1 Assembling the capacitance dilatometer

The dilatometer-parts were mainly manufactured in the workshop at the Technical University of Vienna by A. Lahner and his group. Silver, Cu, Cu-Be alloy, and sapphire are used to built it. The capacitance plate holder itself is made out of silver and also the capacitance plates are out of silver. All bolts and nuts are made out of brass because of the better mechanical strength and the fact that they have to be rather small but are heavily used parts. Sapphire is used for isolation and compen- sation. It is preeminently qualiﬁed for parts that should not change their shape in 2.3. A NEW KIND OF DILATOMETER 15

Figure 2.2 The photo shows all parts that are necessary to build the dilatometer. The plate- housings and the capacitor plates are made out of silver. All bolts, screws, and nuts are made out of copper. a magnetic ﬁeld and during thermal stress. Former dilatometer constructions used insulating glue, to ﬁx the capacitance plates in the housings, which might lead to capacitance drift.

The assembling was done in several steps which are described now. Figure 2.2 shows all the individual parts of the dilatometer. One capacitance plate has a bore in the center and the other one has none. In the further text the capacitance plate with the bore is referred to as the upper plate and the other one is the lower plate. This comes from the position during sample preparation, where the lower plate lies on the table, and the upper one is mounted onto the lower plate. In the following text refer to the description of the parts, that is shown in ﬁgure 2.2. The best best way to start with the assembling is to input the sapphire washers, that have no hole, into the deepening of the lower plate-housing. Then screw the brass-bolt as far as it will go into the lower capacitance plate (the one that has only one threaded hole). Lay the capacitance plate inside the housing. Take care that it is in the center of the housing. Turn the housing and the plate upside down and impose the Cu-Be-disk onto the bolt. Then add the Kapton foil, and one nut onto the bolt. Tighten the nut with tweezers while keeping the plate in the center of the housing. It is easy to keep the plate centered if one uses a piece of folded paper that is stuck in-between the slit, that is between the plate and the housing. If that step is ﬁnished, one can test the electrical isolation with an ohmmeter. There must not be an electrical contact between the capacitance plate and any part of the housing. 16 CHAPTER 2. EXPERIMENTAL DETAILS

Figure 2.3 The completely assembled halves Figure 2.4 View from the other side. The left of the dilatometer from the top. The brass half (lower part) has a counter sunk tip in the screws hold the capacitor plates in their po- center of the capacitor plate. On the right sitions. There are three screws for the upper side the upper part of the dilatometer can be part and only one to hold the lower plate. The seen. The capacitor plate is ring-shaped with capacitor plates are insulated with sapphire a brass shield in the center, which isolates a disks from the housings. sample from the plate.

The procedure for the upper half is similar. The sapphire washers with a hole in the middle (the diameter of the hole is about 2 mm) are laid into the three sunken bore holes of the upper plate-housing (the one with a cylindrical brass shield around the center). Then the three threaded bolts are screwed inside the upper capacitance plate as far as they will go. After this it is put into the plate-housing. Turn it around. Again on each threaded bolt one Cu-Be-disk, one capton foil, and one nut is put. To keep the plate centered it is again recommended to use a piece of paper in the slit, like it was done when the lower plate was built. Fix the nuts with tweezers and test the electrical isolation with an ohmmeter.

After this the dilatometer should look like the parts in Fig. 2.3 and Fig. 2.4. Now the surfaces have to be grounded. The plates and the housing must have a surface that is as plain as possible to avoid fringe effects at the measurement, and to reach high capacitance values. There are special bores in the upsides of the two housings in which threaded bolts have to be screwed. Use three long threaded brass bolts (they are not shown in Fig. 2.2). Impose the brass cylinder with the three transfixions onto the housing very carefully (see the photo in Fig. 2.5). Screw the nuts onto the bolts at the upside of the brass cylinder and put the assembled parts inside the bigger brass cylinder (a photograph can be seen in Fig. 2.6). To fix the dilatometer in a certain position inside the bigger cylinder, it has two screws on the side, that go inside. These screws are used to set the smaller inner cylinder inside the outer 2.3. A NEW KIND OF DILATOMETER 17 cylinder. This setup is used to abrade the dilatometer on a rotating disk grinder. Two different sandpapers were used to get the surface as smooth as possible. The downside surfaces of both halves should be plain after that procedure. It is important to control the position of the capacitance plates. It should still be centered in the housing.

In Sec. 2.4 the calibration procedure is described. For this and all further calculations it is important to know the capacitance value when the capacitance plates are parallel to each other. This is a time-consuming, but necessary step. To tune the distance between the plates, the dilatometer has two pivot bearings. They are located at the lower plate-housing (the parts are described in Fig. 2.7). To set up the bearings, push the excentric bolts into the side-holes of the dilatometer. Then put the sapphire disks into the top-holes. Finally add the pivot bearings onto them. The more diﬃcult part of the work follows now. The capacitance plates have to be set into the parallel position. For this purpose a microscope with a visible scale was used. In principle the distance between the plates could be tuned with the excentric bolts. They have slits on one side. So they can be turned using a screwdriver. A turn of the bolts moves the pivots up and down. Hence the distance between the plates is changed. Better results are achieved when the excentric bolts are not turned, but the length of the pivots is changed directly. This can be done by grinding them shorter with sandpaper. To alter the length of the pivot bearings, they are inserted into the hole of a special brass cylinder, and abraded with sandpaper. The procedure is similar to the one used for the grinding of the dilatometer halves. However, it has to be done very carefully, because once they are too short, they can not be used anymore. The pivots are then again inserted into their bores and the resulting distance between the capacitance plates is checked. To achieve parallelism along the entire plates, a placeholder in-between the plates is necessary. As a placeholder a small silver cylinder was used. Before inserting the silver cylinder into the dilatometer, a sapphire disk is laid onto the tip in the center of the lower capacitance plate. On top of the sapphire disk the silver cylinder is then placed. The dilatometer is closed by putting the upper half onto the lower half. It is ﬁxed with a nut. The Ag cylinder gives one the possibility to set the plates into the parallel position. It also guarantees that the plates do not touch. The length of the Ag cylinder has to be changed too, in order to set the plates parallel. This gives a total number three changeable lengths: two lengths of the pivot bearings and one of 18 CHAPTER 2. EXPERIMENTAL DETAILS

Figure 2.6 The smaller cylinder with Figure 2.5 The imposed brass cylinder one half of the dilatometer is put inside is used to hold the dilatometer-surface the larger cylinder and tightened with parallel to the surface of the bigger screws. A rotating disk grinder is used brass cylinder which can be seen on the to grind and polish the surface of the right side of the picture. capacitor plate and the housing.

Figure 2.7 To set the pivots into their proper positions, ﬁrst insert the sapphire disks into the top holes. Then push the excentric cylinders into the side holes. Finally add the pivots at the top of both. the Ag cylinder. One possible approach is the following: change the lengths of the two pivot bearings, until the slit between the housings on the side where the pivot bearings are, is constant. Then alter the length of the Ag cylinder which is placed inside the dilatometer, until the slit between the dilatometer halves is constant.

If all three parts (the two pivots, and the Ag cylinder) have the right length, the distance between the capacitance plates is constant along the entire plate area. Now the capacitance has to be measured with this setup (for this dilatometer: 4.75 pF at a plate-distance of about 0.2 mm). Before the dilatometer was used for new measurements it was tested on well known materials such as Ag, Cu, Nb, Pb, Boronsilicate. The results were compared with literature [1]. 2.4. CALIBRATION AND TESTING OF THE DILATOMETER 19

2.3.2 Preparing the dilatometer for measurement

One further advantage of the dilatometer is, that the samples used only need to have two plain surfaces. Samples should be at least 1 × 1 × 1mm3 large for an accurate measurement. If they are smaller the dilatometer is diﬃcult to handle. The upper limit is 4 × 4 × 4mm3. Samples bigger than this would touch the brass shield which isolates the sample from the rest of the dilatometer. As mentioned in Sec. 2.3.1, the lower plate has a conical, counter-sunk tip in its center. On top of this tip a circular sapphire disk is loosely placed. The sample is placed onto that sapphire disk. The purpose of this tip and disk is to hold the usually irregularly shaped sample with maximum contact in a ﬁxed position. The second purpose of the sapphire disk is to isolate the sample from the capacitance plate. On top of the sample a Ag cylinder is placed. This is used as a spacer. For every sample the proper Ag cylinder has to be used. The length of the Ag silver spacer has to be between two boundary values. On the one hand it must not be too short, in order to prevent the plates from touching each other. If this is the case, it is physically impossible to measure a capacitance. On the other hand it must not be too long, because then the measured capacitance is very low. This means loss of sensitivity. The capacitance is the higher the smaller the distance between the plates is. For the used dilatometer the length of the sample and the silver spacer together should be approximately 5 mm. If a suitable cylinder is prepared and added onto the sample, the upper part of the dilatometer (the one with the hole) is imposed onto the lower one. The two parts are screwed together using a bolt, a spring, and a nut. After this the dilatometer is mounted on the holder in the desired setup. Figure 2.8 shows the two possibilities for the measurement. The wires for the measurement are soldered at the dilatometer holder, and have to be soldered to the threaded bolts, that connect the wires to the capacitance plates.

2.4 Calibration and testing of the dilatometer

The capacitance dilatometry as it is used here is a relative measurement method. It therefore needs calibration. The length changes of the samples always have to be compared to a reference-material which is silver in this case. If Ag is the probe as well, the signal due to thermal expansion should be zero. For high precision measurements the Ag signal can be subtracted in order to get the eﬀective length change of the material. The calibration is performed using the tilted plate capaci- 20 CHAPTER 2. EXPERIMENTAL DETAILS

Figure 2.8 In the left part of the figure the dilatometer is mounted in the transverse position. That means that the expansion of the sample is measured perpendicular to the field direction. On right side the longitudinal setup can be seen. This is used to measure the magnetostriction along the magnetic field direction. tance formula [6, 10]. The contributions to the change of capacity are:

• change of the sample length

• change of the radii of the capacitor plates

• change of the length of the silver plate holders,

• which is partly compensated by the change of thickness of the capacitor plates and the sapphire washers

With the following formula and computer supported software it is straight forward to calculate the resulting gap d(T) between the capacitor plates from a measured change in capacitance, as a function of temperature [6, 10]:

2 2 2ǫ0 1 − 1 − γo 1 − 1 − γo C(T )= Ao(T ) 2 − Ai(T ) 2 (2.1) d(T ) " pγo pγo # with,

r k(T ) γ = o − 1 , (2.2) o b d(T ) r k(T ) γ = i − 1 , (2.3) i b d(T ) 2.4. CALIBRATION AND TESTING OF THE DILATOMETER 21

where ro is the outer plate radius and ri the inner plate radius. The b is the distance between center of capacitor and pivot.

∆l k(T )= k(T ) 1+ Ag−Lit (T ) (2.4) 0 l

Where k(T0) is the pivot distance at T0 = 300K. The term in the square brackets considers, that also the bearings expand with temperature. In this term ∆lAg−Lit/l(T ) is the thermal expansion of Ag from literature [27].

∆l 2 A (T )= A (T ) 1+ Ag−Lit (T ) (2.5a) i i 0 l

2 is the inner capacitance area (ri π), and

∆l 2 A (T )= A (T ) 1+ Ag−Lit (T ) (2.5b) o o 0 l 2 is the outer capacitance area (roπ). The terms in the square brackets in Eq. 2.5a and 2.5b stand for the linear thermal expansion of Ag. This expansion changes the area of the capacitor plates, and therefore the capacitance. Because d(T ) apppears in the term of C(T ) Eq. 2.1 and in γ, Eq. 2.2 and Eq. 2.3, Eq. 2.1 has to be solved numerically with respect to d(T ). There is also the possibility of fringe effects when such a construction is used. These effects are usually very small as long as the capacitance changes are small. This holds for normal metals such as Ag or Cu. They are considered as correction terms in the software that is used to calculate the gap from the capacitance. The correction simply enlarges the effective plate radius and therefore giving a larger capacitance. An estimation made by W. C. Heerens [12] shows that the necessary corrections are not crucial. Once the gap d(T,H) between the capacitance plates is calculated for each temperature or magnetic field it can be related to the values at the initial circumstances, i.e. zero field or 4 K. In order to eliminate all influences of the expansion of silver we use the normalized gap e(T):

d(T ) e(T )= ∆l (2.6) 1+ l Ag−Lit(T ) 22 CHAPTER 2. EXPERIMENTAL DETAILS

This quantity should be constant in a silver calibration measurement in an ideal dilatometer. The thermal expansion, or the magnetostriction of any solid sample is calculated in the following way:

∆l ∆e − ∆e ∆l Sample (T,H)= Sample Ag−Sample (T,H)+ Ag−Lit (T,H) (2.7) l lSample l where ∆e = e(T )−e(T0). With an index Sample it is the measurement of the sample and with an index Ag-Sample it refers to the measurement of an Ag calibration sample. If a dilatometer is newly built it is necessary to perform some consistency checks and measurements. In the first place it was tested if the dilatometer works correct without any sample. This is done by simply moving the capacitance plates away from each other with a special device shown in Fig. 2.9. It is a standard micrometer screw that was cut into two parts. The part with the nonius is kept and the counter part is not used. On the front end of the screw there is a stick with a tip mounted. This part of the micrometer screw is surrounded by brass plates. On the front plate the dilatometer can be mounted. In the upper part of the dilatometer housing there is a hole through which the tip of the micrometer screw can foraminate. The tip goes inside the dilatometer until it hits the other half of the dilatometer. If the stick is screwed further inside, it forces the halves apart. A maybe more instructive photo of this calibration device is shown in figure 2.9. With this device it is easy to check if the forced gap between the dilatometer halves agrees with the calculated gap. In the first step the tip, that is mounted on the adapted micrometer screw, is pushed inside the hole on the upper housing of the dilatometer with a certain stepwidth. The capacitance is measured with the capacitance-bridge, and saved to a file. This gives the data shown in Fig. 2.10. The software calculates then the gap between the plates with the formulas in Eqs. 2.1 to 2.5b. These values are stored to a file. Then both values, namely the forced gap and the calculated gap, can be compared. They are shown in Fig.2.11. The x-axis is the distance that is was measured with the micrometer screw, and the y-axis is the calculated gap. The dotted line accords to the measured points and the continuous line is a 45◦ line. The total length-change was 0.295 mm, which is rather large compared to the length-changes measured at typical thermal expansion, or magnetostriction experiments. 2.4. CALIBRATION AND TESTING OF THE DILATOMETER 23

Figure 2.9 This device was used to perform ﬁrst measurements with the new dilatometer. With the adapted micrometer screw, it is possible to simulate the length change of a sample by simply driving a stick into the dilatometer.

12 -4 3,0×10

10 absolute length change = 0.295 mm

8 -4 2,0×10 6 capacity (pF)

× -4 2 1,0 10 calculated gap (m)

0 0 0,05 0,1 0,15 0,2 0,25 0,3 change of the plate-gap (mm) 0,0 -4 -4 -4 0,0 1,0×10 2,0×10 3,0×10 Figure 2.10 The device from ﬁg. 2.9 was forced gap (m) used to test the dilatometer. A stick at the end of the micrometer screw spreads Figure 2.11 The change of the gap which the dilatometer apart stepwise. At each is produced in ﬁg. 2.10 can be calculated point the capacitance is recorded. This is using the capacitance change, and the for- also done in the opposite direction, when mula form Eq. 2.1. the stick goes out. 24 CHAPTER 2. EXPERIMENTAL DETAILS

-5 6,0×10

-5 5,0×10

-5 4,0×10

-5 Figure 2.12 If silver is used as the sample, the 3,0×10 gap between the dilatometer halves should vary -5 2,0×10 only slightly. This is due to the fact, that all ex-

change of the gap (m) -5 pansions should be compensated. The gap change 1,0×10 between the two capacitor plates in a temperature 0,0 0 50 100 150 200 250 300 range from 4 to 300 K can be seen in this ﬁgure. T (K)

Further tests were made with standard materials such as Ag, Cu, Pb, Nb, and Borosilcate. Silver was used to determine the zero-signal of the method. If a piece of silver is inside the dilatometer the thermal expansion signal should be very small, because the thermal expansion of Ag from literature is subtracted from the measured signal. The resulting change of the gap should be zero. The reason for choosing these materials is the following. The dilatometer is mainly made out of silver, so in order to determine the zero signal it is necessary to measure the capacitance change, when a piece of Ag is inside the dilatometer. As Ag also expands and contracts during the measurement the capacitance signal increases, if a material is inside the dilatometer that expands less than Ag. This is because the distance between the capacitance plates decreases, if the silver expands more, than the piece of the material that is measured. In order to test the performance under extreme conditions, very ’hard’ and very ’soft’ substances were chosen for this tests (soft and hard should indicate, that they have either small, or large thermal expansion coeﬃcients). Figures 2.13 to 2.16 show the thermal expansion of the selected materials. All data is compared with literature [1]. The values from literature are indicated by circles. Lead is the softest material and therefore shows the largest expansion. In a temperature range of approximately 300 K, the relative length change is approximately 7 × 10−3. Copper expands less (3 × 10−3). The element with the least expansion is borosilicate with a relative length change of 8 × 10−4. This is a factor ten less, than Pb. The dilatometer works better with materials that show large thermal expansion. The error between data in literature and experiment with this dilatometer is less than 1% in the case of Pb, and Cu. This relative error gets larger when low 2.4. CALIBRATION AND TESTING OF THE DILATOMETER 25

-3 -3 8,0×10 4,0×10

-3 -3 6,0×10 3,0×10

-3 l/l

l/l -3 4,0×10 2,0×10 ∆ ∆

-3 -3 2,0×10 1,0×10

0,0 0,0 0 50 100 150 200 250 300 0 50 100 150 200 250 300 T (K) T (K)

Figure 2.13 Thermal expansion of Lead in the Figure 2.14 Thermal expansion of Cu in the temperature range from 4 to 300 K compared temperature range from 4 to 300 K compared with literature (circles). with literature (circles).

-3 1,0×10 -3 1,5×10 -4 8,0×10

-3 -4 1,0×10 6,0×10 l/l l/l ∆

∆ -4 4,0×10 -4 5,0×10 -4 2,0×10

0,0 0,0 0 50 100 150 200 250 300 0 50 100 150 200 250 300 T (K) T (K)

Figure 2.15 Thermal expansion of Nb in the Figure 2.16 Thermal expansion of borosilicate temperature range from 4 to 300 K compared in the temperature range from 4 to 300 K com- with literature (circles). pared with literature (circles). 26 CHAPTER 2. EXPERIMENTAL DETAILS expansion materials such as Nb, and borosilicate are measured.

2.4.1 Force control and enhancement

To improve the possibility to control the force, that acts on the sample, a new part for the dilatometer was built. It is called a force enhancement bow. It consists out of a Cu-Be bow, that has a threaded bolt in the middle of it. The bow embraces the dilatometer and the screw ends in a sink on the dilatometer holder. A Cu-Be disk is inserted between the screw and the dilatometer holder. It acts as a spring and allows measurements with different spring constants, if the thickness of the Cu- Be disk is changed. If the bolt is screwed further inside, the force that acts on the dilatometer is increased. If it is on the other hand screwed out, the force is decreased. To measure the capacitance change at a certain length change of the sample, the calibration device that is mentioned in Sec. 2.4 is used. This additional information gives the possibility to eliminate the signal that is due to sample movement. This can occur if, for example, the sample turns in the magnetic field. The assumption made m here is that the measured overall length change ∆Li (H) is the sum of the ’real’ magnetostriction ∆Lms(H) and the length change that comes from a movement of F the sample ∆Li (H) in the dilatometer, as denoted in Eq. 2.8. According to Hook’s law (see Eq. 2.9) the force that acts on a spring is proportional to the deformation F of it, as long as the deformation is small. With that is possible to express ∆Li (H) in terms of the force and the spring constant, which leads to Eq. 2.10 and Eq. 2.11. Elimination of F(H) in this equations gives Eq. 2.12. Disturbing signals, that are due to a movement of the sample, are eliminated. This method was not studied to great extent, and further experiments testing the limits of this method are necessary. Originally it was planned to use the device to control the force, that acts onto the Tm sample. Thulium shows a large anisotropy, which leads to a turn of the sample in a magnetic field. During the tests of the device another solution for measuring the magnetostriction of Tm was found (see Sec. 3.2).

m ms F ∆Li (H) = ∆L (H) + ∆Li (H) i =1, 2, 3 (2.8)

F = Fi + K∆L (2.9) 2.4. CALIBRATION AND TESTING OF THE DILATOMETER 27

m ms F (H)−F1 ∆L1 (H) = ∆L (H)+ K1 (2.10) m ms F (H)−F2 ∆L2 (H) = ∆L (H)+ K2 (2.11)

F − F + K ∆Lm(H) − K ∆Lm(H) ∆Lms(H)= 2 1 2 2 1 1 (2.12) K2 − K1

Eddy currents

Eddy currents occur if a conducting material is moved in a static magnetic field, or if the material is kept fixed in a varying magnetic field. Due to Lenz’s rule another magnetic field is induced by the eddy currents. It has the opposite direction to field change. In the case of a variable magnetic field, the eddy currents are proportional to the variation of the magnetic field with time. Eddy currents have influences on capacitance dilatometry measurements. The dilatometer is mainly made out of Ag, which has good electrical conductivity, and therefore these currents play a role. Three main sources of errors can be named. (i) First an inhomogeneity of the magnetic field inside the magnet can give a force onto the dilatometer. (ii) If the dilatometer is tilted with respect to the magnetic field, the magnetic moment, that arises from the eddy currents is not parallel to the applied magnetic field. This gives a torque on the dilatometer. The effects (i), and (ii) are proportional to (∂B/∂t)B. (iii) Another possibility for disturbing signals can come from the fact, that there are currents in the upper and the lower plate, which run in the same direction. This gives a repelling force, that spreads the dilatometer halves apart. It is proportional to (∂B/∂t)2. All these effects interfere with the measurement and give disturbing signals. However, it is possible to correct the data with respect to these errors. It is possible to evaluate the disturbing signal at the highest, and lowest field measured. As far as they are linear with respect to the magnetic field, a linear disturbing signal can be calculated. This is then subtracted from the measured signal when the field increases, and added to the data taken when the field is decreased. 28 CHAPTER 2. EXPERIMENTAL DETAILS 2.5 Generation of static high magnetic fields

The behavior of matter in static high magnetic fields (above 30T) is not well studied till now. The simple reason for this is that the generation of such high magnetic fields is very difficult and hence restricted to only few laboratories around the world are using them. The National High Magnetic Field Laboratory (NHMFL) in Talla- hassee, Florida is the worldwide leading facility in magnet technique and production of high magnetic fields. Among other departments there is the Magnet Design & Technology division which constantly develops new magnet systems. Every magnetic field used in their experiments is generated through induction in conductors. The main problems arising, are the formation of heat and the enormous energy consumption. Nowadays it is possible for the scientists and engineers to have special magnet systems that deliver static magnetic fields up to 45T which is the world current record.

2.5.1 The cooling system

One of the main issues the NHMFL has to deal with is the maintenance of power and water for their facilities. Due to very high current in the Bitter magnets the systems have to be cooled by water in order not to destroy the magnet system and to reach such high magnetic fields. The strategy of the NHMFL is to provide the cooling system with two separated closed cooling water cycles. To avoid corrosion, the water in the inner cycle is purified and distilled (resistivity is 10 M Ω). The water contained in the inner cycle is pumped with high pressure through the resistive magnets carrying the heat away from the magnet. Water with an initial temperature of about 10 ◦C is heated up to 60 ◦C within seconds. The thermal contact to the second water loop is established in a heat exchanger outside the laboratory. Flowing next to each other heat from the warm water is transferred to the chilled water-cycle which therefore heats up. The inner-cycle loses heat and flows back to the magnets. Very efficient pumps bring the water from the outer-cycle to the cooling towers, which are supplied with huge fans that transport evaporated water into the air. In a second step the water falls down a wall like a waterfall which also amplifies the cooling. After this procedure the water from the outer cycle has a temperature that is below 60 ◦C and is ready to take up heat again. In fig. 2.17 one can see the cooling tower and the fans which are located at the top of the building. 2.5. GENERATION OF STATIC HIGH MAGNETIC FIELDS 29

Figure 2.17 Water from the outer cooling cycle is pumped to the cooling towers where it is chilled through forced draft evaporation. The fans at the top of the building enhance the cooling eﬀect. Each of the four fans is driven by a 75 HP motor.

2.5.2 Power Supply

As a consequence of the use of resistive magnets at the laboratory the power consumption is very high. The city of Tallahassee has an power consumption of 500MW. To meet the clients power demands the electricity company is obligated to keep 10%- reserves. The NHMFL benefits from this regulation because its power consumption lies within this reserve. The magnets consume a maximum of 40MW of electrical power which is more or less in the order of the reserves of the company and about 10 % of the city of Tallahassee. The laboratory has a special agreement with the electricity company allowing the laboratory to save the infrastructural costs which usually add up to the total costs of electricity. This agreement includes the condition that in case of very high power requirements of the regular customers the laboratory has to shut down its power consuming magnets within a few minutes. The laboratory is supplied by 12,5 kVwhich is standard ”high voltage” provided by the electricity company. In a nearby electric power transformation substation the voltage is changed to 460 V(alternating current) which is rectified by semiconductor based rectifiers and smoothed by a capacitor-bank. This process is highly complex and needs a advanced technical parts to give the correct energy for the measurements. The effort done during the transformation process of the power results in one of the cleanest, most stable power sources worldwide which is essential for the scientific work done at the institute. Four power lines distribute the power to the various magnet-systems, which are lo- 30 CHAPTER 2. EXPERIMENTAL DETAILS cated in the immediate vicinity. To operate the 33T magnets it is sufficient to use two of the four lines whereas the 45T hybrid-system needs all four of them. As a result of the power consummation it is only possible to run two 33T magnets at the same time, or the hybrid-magnet alone. To guarantee the optimal workload the administrators defined three working shifts. The first is from 8am to 4pm, the second from 4pm to midnight and the third from midnight to 8am. The costs to operate the magnets are about 4 million $ a year and can be looked up the financial section of the NHMFL annual report.

2.5.3 The Florida Bitter Magnet

The Florida Bitter Magnet is a magnet system that was mainly developed at the NHMFL and is used there. These magnets consist out of copper spirals that are cooled by water. The insert can be seen in Fig. 2.18 and is mainly made out of copper. Due to the fact that the current decreases with increasing radius (J ∝ 1/R) a special design is chosen to compensate that. There are four single coils that add up to the complete system. The two inner coils consist out of Cu-Ag alloy (higher conductivity due to silver), the next coil consists out of Cu-Be (to increase mechanical stability), and the outer coil is made out of Cu-Zr which again improves the resistance to mechanical stress. The complete system can be seen in Fig. 2.19. Such magnets are called resistive magnets because materials with finite electrical resistivity are used in contrast to superconducting magnets. They are built out of superconducting materials as the name says. The highest available field reachable with a resistive magnet is 33T inside the bore of the magnet. This bore is a very important factor because it gives the crucial space that can be used for experiments. For the measurements presented here a Bitter magnet with 33T and 32mm bore was used. The power consumption of such a device is 20MW. Almost all of this energy is transformed into heat and has to be carried away from the magnet. This is done with ultra pure water from the cooling cycle mentioned in Sec. 2.5.1. The magnet is located on an elevated platform and is easy to access to load it from the top. Vibrations that are a disturbing side-effect and arise from the cooling water are damped by a vibration insulating platform. The whole magnet device is about 180 cm high and has a diameter of about 50 cm. 2.5. GENERATION OF STATIC HIGH MAGNETIC FIELDS 31

Figure 2.18 The Florida-Bitter-Magnet consists only out of one Windung which has sev- Figure 2.19 The picture shows the assembled eral parts of diﬀerent constitution to minimize device. The main parts are the coils that are current drop and maximize stability. Water is shown in ﬁgure 2.18. pumped through the holes inside the plates to cool them.

2.5.4 The Hybrid Magnet

Based on the construction of the Bitter magnet there is an even more powerful system used at the NHMFL. The Hybrid magnet combines a resistive magnet with a superconducting magnet to a very new and powerful magnet system that unites the two magnets giving rise to the worlds strongest static magnetic field. The resistive insert is more or less the same system as used in the 33 T magnets and described in Sec. 2.5.3. The outsert consists of two connected superconducting coils. The inner coil is a Nb3Sn superconducting coil and the outer a NbTi coil. The superconducting outsert parts are completely separated from the insert not only concerning the vessels but also the cooling system and the power supply. To make the coils superconducting it is necessary to fill the vessel in which they are placed with liquid Helium (4 K). The cryo-system is an extra machine which is located about five meters away from the magnet system. This cryo-system also houses the current junctions for the hybrid which are cooled with liquid Helium too. The NHMFL is the only laboratory in the world that has such a hybrid magnet, and it can be only used as the only device active in the laboratory. Both of single magnets, the resistive insert and the superconducting outsert, need so much electrical power that it is only possible for the NHMFL to run either the hybrid magnet or two smaller magnets. Once the superconducting coils are loaded they are put to persistant mode and do not need electrical power any more. This procedure takes about three hours and is performed 32 CHAPTER 2. EXPERIMENTAL DETAILS every time the magnet is switched on. During this procedure various tests are made in order to assure the faultless functionality of the complete system and even to estimate the remaining lifetime of the coils. For all further measurements that are done within one shift the magnetic field only varies between 11 T and 45 T. This is important because all the results presented below are measured within this restricted magnetic field range. The exceptions are runs that were started early in the morning, which include the the range from 0 T to 11 T, when the superconducting outsert was turned on, or later in the evening, which are from 11 T to 0 T, when the outer coil was ramped down.

2.6 Measurements in high magnetic ﬁeld

Technical details concerning the NHMFL and its magnets are described in section 2.5. That section should give important details concerning experimental setup, necessary for the measurements in high fields. One important parameter that lead to improvement of the dilatometer is the diameter of it. To cool the samples and the dilatometer, a cryostat was used that surrounded the dilatometer. The crystat is double-walled steel cylinder with a wider (diameter about 50 cm) upper part and a tube like lower part (diameter about 40 mm). It is filled with liquid Helium. In the middle of the cryostat there is a bore where the sample is placed. The narrow bore is the limitation for the size of the dilatometer. Of course the dilatometer was not put directly into the cryostat. This would lead to corrupted results due the fact that Helium is a dielectric fluid and would give the wrong capacitance. Instead it is plugged to the ’sample stick’. This is a long steel tube, that carried the cables, that connected the dilatometer with the data acquisition, and held the dilatometer in a well defined position. The assembled device was then put into the ’sample-can’, which again is a tube that surrounded the sample-stick. This had to be done very carefully because the sample-can had a bore that was not much wider than the diameter of the dilatometer. The best way to do this is to slide it in with slight turns to accommodate the stick inside the sample-can. To measure temperature dependend effects it is crucial to avoid physical contact between the dilatometer and the can, because this leads to a heat flow between them. A very simple but effective construction was applied. One part of the dilatometer holder is a circular shaped circuit-board. On the edge of it short outwise bent metal pins 2.6. MEASUREMENTS IN HIGH MAGNETIC FIELD 33

Figure 2.21 To keep the dilatometer safe from Figure 2.20 The dilatometer is mounted at harm and prevent liquid Helium to get be- the end of a long steel tube. The wiring is in- tween the capacitor plates, The tube from fig- side the tube to protect it. This tube with the ure 2.20 is itself inserted into a wider tube and dilatometer on one end is then inserted into a then evacuated to get a stable thermal envi- larger tube which can been seen in figure 2.21. ronment. were mounted, that guaranteed maximum space between the wall of the sample-can and the dilatometer, and minimum thermal contact. The sample can was evacuated to additionally reduce heat flow. This was done before every measurement to get a reproduceable environment and set a constant exchange-gas pressure. After the can was evacuated it was put into the cryostat. The experiment was controlled via software from this point on. The software mainly used at the NHMFL is LabView on Apple computers. The following parameters were measured: the capacity, the field, the sample temperature, the temperature at the sample-can, and the time. One measurement takes about 160 minutes at a field ramp of 0.2T/min. 34 CHAPTER 2. EXPERIMENTAL DETAILS Chapter 3

Results and discussion

3.1 Samarium

3.1.1 The crystal and magnetic structure of Sm

In order to analyze magnetostriction data it is necessary to know more about the crystal structure of Sm. The crystal and the magnetic structure have been determined by x-ray and neutron scattering. Early x-ray measurements by Daane [2] determined the crystal structure as an rhombohedral one, namely R3¯m with three atoms in the unit cell. The crystal structure is somehow complicated and can be envisaged as a nine-layer stacking sequence ABABCBCAC. . . of closed packed hexagonal layers. There is a second structural domain which is turned by 180 deg about the c-axis (so the stacking is ACACBCBAB. . . ). Two thirds of the atoms have a surrounding similar to that found in the hexagonal-stacking sequence, ABAB. . . , and one third has neighboring like in the cubic closed packed, ABCABC. . . stacking. For simplicity the sites are denoted as the hexagonal and cubic sites respectively, in the further text. Sm has the following hexagonal lattice parameters: a=3.621 A˚ and c=26.25A.˚ Koehler et. al. [13] performed neutron diﬀraction measurements on a Sm single crystal. They used the isotope 154Sm to lower the high absorption cross section of natural Sm. They proved the assumption which was derived from electrical resistivity and speciﬁc heat measurements, that there are magnetic-order phase transitions at 106 and 13.8 K. Below 106 K the magnetic moments at the hexagonal sites order in ferromagnetic sheets. The second transition can be assigned to the antiferromagnetic ordering of the cubic sites. The distance between two cubic

35 36 CHAPTER 3. RESULTS AND DISCUSSION sites layers is 9A.˚ The large distance is a reason for exchange interaction occurring only within the same plane. The single layer structure can be seen in Fig. 3.1b.

The spin-sequence within one layer is + + −− along the hexagonal a1-axis. The magnetic structure of Sm can be seen in Fig. 3.1.

3.1.2 Thermal expansion of Sm

In Fig. 3.2 the zero field thermal expansion of Sm is shown [15]. It was measured along the three axes and the volumetric thermal expansion ǫv was calculated from them. TN is the Néel temperature and T1 the ordering temperature of the magnetic moments at the quasi cubic sites. The spontaneous magnetostriction at these temperatures is very small. The different expansion of the axes shows the anisotropy.

On cooling, the c axis contracts at TN , whereas the a and b axes expand.

3.1.3 Magnetostriction of Sm

In this section the forced magnetostriction of the rare earth element Sm will be discussed. The measurements were done on two different high quality Sm single crystals to exclude influences of crystal growth effects. One of the crystals was grown at the University of Birmingham (UK) and the second one at the Ames Laboratory (USA). The Sm single crystal prepeared in Birmingham was grown by solid state crystal growth techniques. It was sealed in a tantalum container to prevent the material from weight loss through evaporation. The quality was checked by back reflection x-ray Laue technique. The size of the crystal is 1.6 × 1.7 × 1.2 mm3. A strain annealing procedure was used to grow the single crystal at Ames Laboratory. Its high quality was again assured by the Laue x-ray method. The shape of the Sm piece is nearly cubic, 2.28 × 2.34 × 2.38 mm3. It is not necessary to protect the crystals from air as they are not sensitive to it. The measurements presented here were performed at the NHMFL in Tallahassee from November to December 2005 in fields up to 45T. The magnet design is described in 2.5.4 and the capacitance dilatometry in 2.2. For the experiments three different capacitance dilatometers were used. They differ only slightly in their construction and represent different states of the dilatometer development. They were also used to exclude device specific effects on the measurement. To prevent the crystal from turning in the field it was glued to the head of a screw, which was grounded blank 3.1. SAMARIUM 37

Figure 3.1 (a)High-temperature magnetic structure involving only the hexagonal sites of the crystal. closed circles indicate hexagonal sites (h), and open circles cubic sites (c). The hexagonal sites are coupled ferromagnetic with layers perpendicular to the c-axis. The arrows next to the plane show the direction of the moments. In the picture one half of the unit cell is shown (the other half has all moments turn around 180◦). (b) Low-temperature magnetic structure involving cubic sites. Lower picture shows the antiferromagnetism within one layer. The arrows in the upper part show the direction of the moments along the ~a2 direction. Again only one half of the cell is shown. The other half is constructed by translating the lower half about ~cM /2 and reversal of the direction of all moments. 38 CHAPTER 3. RESULTS AND DISCUSSION

Figure 3.2 Anisotropic thermal expansion of Sm single crystal in zero magnetic field. The volume effect (∆V/V ) at the ordering temperatures TN and T1 is very small. before, to have a flat surface. Epoxy-glue was used. The screw with the crystal on top of it was put into the dilatometer through the bore in the upper housing (see Fig. 2.3) and screwed with a brass nut. Then the other half of the dilatometer was imposed and the two halves were screwed together. It was then mounted on the dilatometer-holder, the wires soldered to the plug, and then connected to the sample-stick. The rest of the procedure is described in Sec. 2.6.

3.1.4 Results

Magnetostriction along the a-axis

In Figure 3.3 the data, which was obtained from measurements with the magnetic field applied along the c-axis, is shown. Note, that the magnetostriction is measured parallel to the a-axis. That means that the dilatometer was used in the transverse setup (see Fig. 2.8). Arrows in the picture denote whether the field was ramped up, or down. So if one follows the direction of the arrows, one also follows the cycle of the magnetic field. This cycle starts at magnetic fields of 11 T, goes up to 45 T, and back to 11 T. Note that not every measurement was done in this wide magnetic field range. The data in figure 3.3 is only sown in a field from 20 T to 40 T. At a field of about 3.1. SAMARIUM 39

H || c H || c 2 K 9 K

-5 2×10 i i

l/l) -5 l/l) ∆ 2×10 ∆ = ( = ( i i ε ε

20 25 30 35 40 20 25 30 35 40 µ Η(Τ) µ Η(Τ) 0 0

Figure 3.3 The ﬁgure shows the magnetostriction signal along the a-axis of the crystal. The magnetic ﬁeld was applied parallel to the c-axis. The left picture was obtained at 2 K, and the right one at 9 K. It is clearly visible that the hysteresis decreases with increasing temperature.

33 T the transition starts, if the field goes up (from 11 T to 45 T). This value changes when the field is ramped down (transition starts at 30 T and ends at 25 T). At a temperature of about 2 K the hysteresis of the data is large. Another point which should be mentioned is, that the data does not end at the values it started. There are several possibilities why the data shows such a behavior. One possible explanation are Eddy currents which spread the dilatometer apart at non zero field. In principle the data is corrected owing that fact, but in some cases this was not straight forward, and can lead to errors. Another possibility is that the sample has turned during the measurement. This means that it is not in the initial position after the measurement. However, this kind of irreversibility is not seen in the data at higher temperatures and might therefore be a single artifact for this measurement. The signal recorded during the ramping up of the field lies almost exactly at the values when the field was ramped down. There is no hysteresis visible.

Magnetostriction along the b-axis

The magnetostriction along the a- and the b-axis are very similar. The diﬀerences are likely because of not identical experimental circumstances. The left picture in ﬁgure 3.4 shows the magnetostriction signal along the b-axis of the Sm single crystal at a temperature of 2 K. The right side shows the data at 9 K. The hysteresis is 40 CHAPTER 3. RESULTS AND DISCUSSION

H || c H || c 2 K 9 K

-5

i i 2×10 × -5 l/l) 2 10 l/l) ∆ ∆ = ( = ( i i ε ε

20 25 30 35 40 20 25 30 35 40 µ Η(Τ) µ Η(Τ) 0 0

Figure 3.4 Magnetostriction of a Sm single crystal with magnetic ﬁeld applied along the c-axis. The length change of the crystal was measured along the b-axis. On the left side the eﬀect is shown at a temperature of 2 K. The right picture is the data obtained at 9 K. The hysteresis gets smaller when temperature is increased. This was also observed along the a-axis (see Fig. 3.3).

clearly visible in the 2 K data. When the temperature is increased and comes close to the transition temperature for the quasicubic sites the hysteresis disappears and the forced magnetostriction gets larger. At 2 K it is about 3 × 10−5 and at 9 K it is approximately 7 × 10−5. The same is true for the striction along the a-axis.

Magnetostriction along the c-axis

To measure the magnetostriction along the c-axis, the longitudinal setup has to be used. A photograph of the dilatometer in this setup can be seen in figure 2.8. A plot of the data can be seen in figure 3.8. The data obtained during the experiment is much more noisy than the data along the a-, and b-axis. It is not clear what are the reasons for this. One possible explanation is that the noise in the data comes from the vibrations of the cryostat. The cryostat vibrates because the cooling water for the magnet is pumped through the system with high pressure. The vibrations resulting from the water flow is transferred to the cryostat. It is not possible to exclude a contact between the dilatometer and the cryostat, and therefore such vibrations have negative influence on the measurement. The data at 4 K shows a shoulder at about 30 T when the field was ramped down. The data was also recorded during the measurement and shows an instability in this region. This might be an explanation for this shoulder. From the data it is not possible to say anything about the hysteresis. An experiment at higher temperature, like it was carried out for the 3.1. SAMARIUM 41 basal components, was not done.

3.1.5 The inﬂuence of temperature

Temperature plays a crucial role in magnetism and all related phenomenons. There- fore also magnetostriction is influenced by temperature. For the first time it was possible to change the temperature during the experiments in such high fields. The main problem that prevented experimenters from contolling temperature was the size of the dilatometers used. If the radius of the device is too big, it touches the walls of the surrounding tube. This contact leads to heat transport to the dilatometer. Therefore the temperature of the dilatometer is more or less the temperature that the surrounding has. This is the temperature of liquid He (4.2 K). Great effort was made to miniaturize the dilatometer, which resulted in an outer diameter of 20 mm. The small diameter reduced the likeliness that the dilatometer touches the walls. In addition thin metal bolts, that were mounted at the dilatometer-holder should keep the distance between the dilatometer and the tube-walls. To increase the temperature a resistive heater was used. On the other hand the system was cooled by pumping out the He gas, which decreases temperature.

An impressive experiment is plotted in ﬁgure 3.5. The sample was heated above the transition temperature T1 (T1=13.8 K). This is the temperature at which the moments at the quasicubic sites of Sm order. At a temperature of 20 K there is no phase transition seen any more. If the Sm is cooled down again, a transition is measured. This experiment was done during one measurement-cycle, so there is no need to change the setup for temperature-dependence experiments.

To analyze the influence of temperature on magnetostriction it is instructive to look at the figures 3.6 to 3.8. In this figures the data at different temperatures is plotted in one graph. For the c-axis only data at 2 and 4.2 K is available. The magnetoelastic strain in the basal plain shows an interesting effect on heating. From the data it seems that at higher temperature the magnetostriction is reversed at fields higher than 30 T. After the phase transition the shrinking of the a-, and b-axis is reversed by the magnetic field. Along the b-axis the effect is measured up to 45 T. A common attribute to the basal components is that the hysteresis almost vanishes at 9 K. 42 CHAPTER 3. RESULTS AND DISCUSSION

phase transition i ε =

i −4 10 K 1 ×10 l/l ) ∆

( 20 K

no phase transition 20 25 30 35 µ Η 0

Figure 3.5 The Sm sample was heated from 10 K to 20 K when the magnetic ﬁeld was ramped up. On the way down the temperature was set to 10 K again. This shows that there is no phase transition observable at temperatures above the odering temperature T1 (T1 ≈ 14 K).

a-axis, field in c-direction b-axis, field in c-direction

2K 2K 4.2K 4.2K 9K 9K i i

l/l) l/l) × -5 ∆ ∆ 2 10 × -5

= ( 2 10 = ( i i ε ε H || c H || c

10 20 30 40 10 20 30 40 µ Η(Τ) µ Η(Τ) 0 0

Figure 3.6 Magnetostriction along a-axis Figure 3.7 Magnetostriction along a-axis at 2, 4.2, and 9 K. The hysteresis decreases at 2, 4.2, and 9 K. Note that at ﬁelds above with increasing temperature. 30 T the magnetoelastic eﬀect is reversed. 3.1. SAMARIUM 43

c-axis, field in c-direction

4.2K 2K i

l/l) × -5 ∆ 2 10 = ( i ε

10 20 30 40 µ Η(Τ) 0

Figure 3.8 Magnetostriction along c-axis at 2 and 4.2 K. The shoulder in the curve at 4.2 K is likely to be eﬀected by instable temperature in this region, caused by experimental circumstances.

Comparison with older data

In 2004 Rotter et al. already tried to find a phase transition at the NHMFL in Sm [15]. It was possible to see the start of the magnetoelastic effect and get an idea about the fields at which it happens. This preliminary experiments motivated the measurements at higher field. Of course the data from the different experiments should be compareable and this is done in Fig. 3.9 to 3.10. In all the figures the data respresented by the squares is from the experiments done in 2004 and the pluses indicate recent data. The first big difference of course is that the old data is only available up to 33 T but these were limitations due to the fact that the hybrid system was not available for these experiments. The second obvious difference is the greater noise that is inherent to the new data. This can mostly be explained by the strong vibrations during the measurement. Because of the high currents in the magnets they have to be cooled by water all the time. The water goes with high pressure through the coils of the resistive insert of the hybrid. The bore in the center of the hybrid is even smaller than that of the purely resistive magnets. That means that the dilatometer is more likely to touch the walls of its adjacency which is mainly the sample can. This contact between the dilatometer and the can leads to a transmission of the vibrations onto the dilatometer. All the other parameters where the same such as cables, shielding, capacitance bridge, and so on. Besides this, the accordance between the data gathered two years ago and the recent data is impressive. If one considers that the experiments were carried out 44 CHAPTER 3. RESULTS AND DISCUSSION

0 0 4.2K 4.2K -1e-05 -1e-05 H || c H || c

-2e-05 -2e-05 -5 -5 1×10 1×10 -3e-05 -3e-05 a/a) b/b) ∆ ∆ = ( = (

a -4e-05 -4e-05 b ε ε

-5e-05 -5e-05

-6e-05 -6e-05

0 10 20 30 40 0 10 20 30 40 µ Η(Τ) µ Η(Τ) 0 0

Figure 3.9 This picture shows data from experiments that were carried out in 2004 [15](squares). The older data was averaged over the complete magnetic field range. The crosses indicate the new data. On the left side the magnetostriction data along the a-axis is shown. The agreement between the two data sets is very good, if one takes into account, that the experimental circumstances were changed dramatically. The right side illustrates the data along the b-axis of Sm. The shoulder at approximately 15 T in the old data is not confirmed by recent data. The temperature is 4.2 K and the magnetic field is applied along the c-axis.

at a completely different magnet system, with different dilatometers and a partly different experimental setup, the data-sets agree very well. The old data is the average between the signal when the magnetic field was ramped up to 33 T and the signal when it was ramped down to 0 T again, so no hysteresis effects can be seen. In Fig. 3.10 also the new data is only part of the measurement. The reason for this is that when the magnetic field was ramped down an unexplainable peak appeared which has definitely no physical meaning. Especially in this figure (Fig. 3.10) the accordance between the data-sets is good, altough the new data are more noisy.

3.1.6 A spin-ﬂop transition in Sm

In the last section the experimental data for the magnetic phase transition of Sm was presented and discussed. The aim of this section is to give an explanation on a microscopic level. McEwen [16] reported about the magnetization of Sm. They measured parallel and perpendicular to the c-axis at 4.2 and 20 K. From their data they proposed a spin-ﬂop transition. 3.1. SAMARIUM 45

-4 1×10 4.2K H || c

-5 8×10

-5 2×10 -5 c/c) 6×10 ∆ = ( c

ε -5 4×10

-5 2×10

0 0 10 20 30 40 50 µ Η(Τ) 0

Figure 3.10 The graph shows two diﬀernet data sets, which were obtained 2004 by Rotter et al. [15] and 2005. Squares represent the data from Rotter et al. and crosses the new data.

The spin-ﬂop transition

The basic mechanism of a spin flop transition can be seen in figure 3.12. The process can be divided into three parts related to the orientation of the spins. The lower part of the picture shows the spins at the quasicubic sites of Sm. They are separated by two hexagonal layers which are denoted by the line between the arrows. The magnetic structure is described in more detail in section 3.1.1. The graph in the upper part shows the magnetization resulting from these spins. In the beginning when the field is low the spins at the quasicubic sites of Sm order antiferromagnetically. The lengths are equal, and therefore the magnetization is zero. When the field is increased, the moment which lies antiparallel to the field is reduced, and a net-magnetization is established. This state still refers to part I in the figure 3.12. In this state the spins remain collinear due to the magnetic anisotropy. With increasing field the net-magnetization increases, as the antiparallel moment decreases. At the critical field the so called spin-flop occurs. This is when the former antiparallel spin turns into the direction of the field. The result is a spin configuration similar to that in part II of the picture. The c-components of both magnetic moments point now in the same direction, and do not compensate each other any more. This leads to a jump in magnetization. As the field is increased the moments are rotated into the direction of the magnetic field (see part III in the figure 3.12). At high fields the moments are parallel to each other and the field direction. The magnetization at this field saturates. This is shown in part III of the 46 CHAPTER 3. RESULTS AND DISCUSSION

Sm H || c T=4.2K

a ∆ a/a=ε

b Figure 3.11 This summarizes the ∆b/b=ε magnetostriction data of Sm with i ε

applied field along c. The longitudi- = −5 i ×10 nal and transverse components are 2 l/l ) shown. The data at lower fields was ∆ ( taken from Rotter et al. [15]. The comparison of the two data sets is c described in the text. The basal ∆c/c=ε components show a negative effect at a field of about 33 T. The c-axis expands at the same magnetic field. In the lower part of the figure the volume effect of the magnetostriction is a b c ∆V/V=(ε +ε +ε )/3 shown. The volume change is simply the sum of the length changes along the a-, b-, and c-axis. Compared to the striction along the three axes, the volume change is small at the 0 10 20 30 40 spin-flop transition. µ Η (Τ) 0 3.1. SAMARIUM 47

Figure 3.12 The spin-flop transition shows the following development of the spins with increasing external field. At a critical magnetic field the spin which direction is antiparallel to the magnetic field makes a “flop”. Then the spins are turned until they reach the saturation magnetization.

scheme.

3.1.7 Calculations for the spin-ﬂop transition

In order to interpret the magnetostriction data the spin-flop transition of the moments at the quasicubic sites was modelled. This was done with McPhase, a mean field program, for calculating physical properties of rare earth elements [20]. A mean field model was adopted.

3.1.8 Calculation of magnetization of Sm

3+ The magnetism of the quasicubic Sm ions (S=5/2, L=5, J=5/2, gJ = 5/2) was described by a doublet |±i, which is split by the magnetic ﬁeld. Single ion anisotropy was considered (diﬀerent saturation moments in plane and out of plane) [21]. The 48 CHAPTER 3. RESULTS AND DISCUSSION

Hamiltonian consists of a sum of the two ion interaction and the Zeeman term: J aa(ij)0 0 1 bb H = − Ji  0 J (ij) 0  Jj − gJ µBJiH (3.1) 2 ij i X  0 0 J cc(ij)  X     here i,j run over all quasicubic Sm atoms. We consider a simple antiferromagnet with two sublattices and nearest neighbor interactions only. The mean ﬁeld equations for such a problem are:

c c H (i)gJ µBJ hJ ci = J c tanh eff sat (3.2) i T,H sat kT J cc(ij) Hc (i)= Hc + hJ ci (3.3) eff g µ i T,H j J B X c In Eq. 3.2 the hJi iT,H are the thermal expectation values for the angular moment c operator of the Sm ions in sublattice i (i=1,2). Jsat is the c-component of the saturated angular moment operator. J aa(ij) and J cc(ij) are the coupling constants between the two antiferromagnetic sublattices (see table 3.1 for their values). Hc is c the c-component of the applied ﬁeld, and Heff (i) is the c-component of the eﬀective

ﬁeld at ion i. The gJ denotes the Lande´efactor and µB is the Bohr magneton. c c Equations 3.2 and 3.3 are solved iteratively for Heff (i) and hJi iT,H until self consistency is reached. The result gained from this calculation is the magnetization. The magnetization per Sm ion at the quasicubic sites is then given by:

1 M = g µ hJ i (3.4) N J B i T,H i X here i runs over all atoms and N is the number of atoms (quasicubic and hexagonal).

Equations 3.2 to 3.3 are valid for part one in figure 3.12 and for applied fields not parallel to c. This is when the spins are aligned antiferromagnetically along c. For part II the structure is non-collinear and equation 3.2 has to be substituted (see appendix A). For such a noncollinear anisotropic problem, mean field components in the plane have to be considered. J aa(ij) Ha (i)= Ha + hJ ai (3.5) eff g µ i T,H j J B X The single ion Hamiltonian has to be diagonalized by numerical methods and the α thermal expectation values hJi iT,H have to be calculated for each iteration step 3.1. SAMARIUM 49

coupling constants (meV) saturation moments (µB) aa bb a b JNN =JNN -0.1879 gJ Jsat=gJ Jsat 0.5117 cc c JNN -0.2851 gJ Jsat 0.5174

Table 3.1 Nonzero components of the two ion interaction tensor between the two sublattices. and saturation moments for Sm at the quasicubic sites. Only nearest neighbors (NN) are considered.

in the self consistent mean ﬁeld procedure. Details are described in the appendix (see A). For this problem the exchange striction is assumed to be diagonal with only a small anisotropy. This is also mirrored by the values in table 3.1.

A comparison of the calculated magnetization with the measured one can be seen in figure 3.13. The accordance is very good for the data at 4.2 K. The ordering temperature for the quasicubic sites of Sm is calculated to be 10 K, whereas it is 13.8 K from experiment. This discrepancy is likely to give not so good agreement of magnetization data between experiment and calculation for the data at 20 K. It is also possible to predict a saturation field of approximately 70 T for the moments at the quasicubic sites. This motivates further experiments at pulsed field magnets, which can reach up to 100 T.

3.1.9 Calculation of the magnetostriction of Sm

A short introduction in magnetostriction of rare earth elements has been given in section 1.2.1. With the background described there and some element specific information it is possible to calculate the magnetostriction of Sm. One crucial information from experiment is that the magnetostriction along the hard axes (a-, and b-direction) is very small compared to the easy axis. This is in contradiction to crystal field striction, which should give a larger signal for fields along the hard magnetization axes, than along the easy axis (see Fig. 1.1 b and c) in Sec. 1.2.1). Therefore it is assumed, that the governing mechanism in Sm is exchange striction. The field dependence of the magnetoelastic strain ǫα is described within the exchange striction model by the expression (see Sec. 1.2.1):

1 αβ ∂Jcc(ij) c c ∂Jaa(ij) a a ǫα = s hJi JjiT,H + hJi Jj iT,H (3.6) 2V ∂ǫβ ∂ǫβ Xβ,ij 50 CHAPTER 3. RESULTS AND DISCUSSION

0,2 || c-axis 4K (exp) ⊥ c-axis 4K (exp) || c-axis 4 K (calc) ⊥ c-axis 4K (exp) 0,15 || c-axis 20 K (calc) || c-axis 20K (exp) /f.u.)

B 0,1 µ M (

0,05

0 0 20 40 60 80 100 µ 0H (T)

Figure 3.13 Experimental data of magnetization [16] is fitted using McPhase. Accordance between theory (solid lines) and experiment (symbols) is very good. For 4.2 K with magnetic field along (circles) and perpendicular (squares) to the c-axis the agreement is within a few percent. The applied model does not well describe the ordering temperature for the moments at the quasicubic cub sites TN and the magnetization at 20 K. The reason for this might be that at higher temperatures crystal field states are excited, which are not considered by the model.

αβ α α In Eq. 3.6 V denotes the volume, s the elastic compliances. The hJi Jj iT,H are α the static two-ion correlation functions (Ji denotes the α components of the angular momentum operator of ion i, and the brackets hiT,Hdenote the thermal expectation value). Without loss of generality the b-component can be neglected, because Sm is hexagonal and many properties in the hexagonal ab-plane are nearly isotropic. The ﬁeld and temperature dependence of the strain is mainly due to variation of α α the correlation function hJi Jj iT,H (see ﬁgure 3.14).

Rewriting equation 3.6 and combining the constants to Kαβ yields a simpliﬁed form:

c c a a ǫα = KαchJi JjiT,H + KαahJi Jj iT,H (3.7) with the constants Kαγ deﬁned as:

αβ αβ s ∂Jaa(ij) s ∂Jcc(ij) Kαa = Kαc = (3.8) 2V ∂ǫβ 2V ∂ǫβ Xβ Xβ The ﬁeld dependence of the magnetoelastic strain is described by the two-ion static 3.1. SAMARIUM 51

4 0

-0,5 2 T T H, H, > > j j a c

J 0 J

i -1 i a c J J < < -1,5 -2 4 K 4 K -2 -4 0 20 40 60 80 100 0 20 40 60 80 100 µ H (T) µ 0 0H (T) Figure 3.14 The two-ion static correlation function for the a- and c-component of the angular momentum operator. The magnetic ﬁeld is applied along c-axis. correlation functions which are calculated by McPhase and shown in ﬁgure 3.14.

With the knowledge of the constants Kαγ , it would be possible to calculate the magnetostriction. In the following it will be shown evidence, that in Sm metal the Kαγ for the a-, and c-component have the same value, but diﬀerent signs.

First experimental evidence for Kαc = −Kαa

One important point for getting this result is the following assumption.

sat a c J ≈ Jsat ≈ Jsat (3.9)

This is possible without making a significant error, because the anisotropy determined in magnetization experiments [16] is small. That can be seen from table 3.1. To get a model for the magnetoelastic strain in Sm, the lower part of figure 3.12 is helpful. Until the critical field is reached the spins have no other component then the c-component, hence the a-, and b-component are zero. The model-lattice for Sm can be considered as two sublattices. One with the spins pointing in the positive c-direction (J ↑), and the other one in the negative (J ↓). The spins in the “positive lattice” are already saturated, because they are parallel to the magnetic field. Hence: 52 CHAPTER 3. RESULTS AND DISCUSSION

↑ ↑ . . . J = Jsat (3.10) ↓ . . . J↓ (3.11)

The contribution of the two antiferromagnetic quasicubic sublattices to the total magnetization can be written as:

g µ M = J B J↑ + J↓ (3.12) 6 hence M J↓ = − Jsat (3.13) 6gJ µB The correlations function are:

c c ↑ ↓ hJi JjiT,H = J J (3.14) a a hJi Jj iT,H =0 (3.15)

The a-component is zero, because the the spins only point in c-direction, as mentioned before. With Eqs. 3.13 and 3.10 the correlation function in part I of Fig. 3.12 can be expressed as a function of magnetization M:

sat c c sat 2 J hJi JjiT,H = − J + M (3.16) 6gJ µB Equation 3.16 relates the static correlation function to the magnetization and is valid for the antiferromagnetic alignment of the spins at the quasicubic sites of Sm. It is only valid for fields below the critical field (hence only in part I of fig. 3.12).

The equation for part II and III in ﬁgure 3.12 is also derived using the knowledge one gets when looking at the spins in Fig. 3.12. At a critical ﬁeld the spins in the “negative” sublattice turn around (they have now a positive c-component, and a negative a-component). Simultaneously the spins from the “positive” sublattice tilt a little. It is not possible any more to characterize the two sublattices with respect to the sign of the c-component of the spin. To distinguish them they are arbitrarily called lattice 1 and lattice 2. For the angular momentum the following expressions are valid: 3.1. SAMARIUM 53

hJ1,ai = −hJ2,ai (3.17) hJ1,ci = hJ2,ci = hJci (3.18)

In equations 3.17, and 3.18 the number in the superscripts denotes the lattice. Note that the c-components of both lattices have the same value, and that the a-components diﬀer in sign. Hence the sum of the a-components is zero. The contribution of the quasicubic atoms to the total magnetization can then be written as:

g µ M = J B hJci (3.19) 3

There angular momentum in a-direction cancels out, as it is written in Eq. 3.17. The movement of the spins in part II and III, which is now considered, can be described approximately as a circular motion, because the single ion anisotropy is small (see Eq 3.9). The radius of the circle is constant, and following expression is valid for each sublattice:

a 2 c 2 2 hJ i + hJ i ≈hJsati (3.20)

The correlation function for the a-direction can be written as:

a a a a a 2 hJi Jj iT,H = hJ1J2iT,H = −hJi iT,H (3.21) inserting the result into Eq. 3.20 gives:

a a a 2 2 c 2 hJi Jj iT,H = −hJi iT,H = −hJsati + hJ i (3.22)

Using equation 3.19, the correlation function in c-direction is related to the magnetization M in part II of Fig. 3.12 by:

M 2 1 hJcJci = hJci2 = (3.23) i j T,H g µ 9 J B

Inserting the expressions for the correlation functions into Eq. 3.6 leads to the result: 54 CHAPTER 3. RESULTS AND DISCUSSION

Jsat ǫ = K − Jsat 2 + M . . . part I of Fig. 3.12 (3.24) α αc 6g µ J B M 2 1 ǫ = −K hJ i2 + (K + K ) . . . part II of Fig. 3.12 (3.25) α αa sat g µ 9 αa αc J B Equations 3.25 and 3.25 relate the magnetostriction to the magnetization. Both properties were measured and give information for evaluation of the spin-flop transition. The first addend in Eq. 3.25 is a constant, and the second one is proportional the M 2. This relation can now be used to compare it with the experimental data. In all measurements that were done, the magnetostriction signal above magnetic field of 33 T, is almost constant, whereas the magnetization increases in the same region. This means that the result of Eq. 3.25 must be constant. As just mentioned the first addend is constant. To give the expected result, also the second addend must be constant. As far as the magnetization is not constant, which can be seen from figure 3.12, the factor must be zero! Hence:

Kα = Kαa = −Kαc (3.26) using this result in Eq. 3.7 leads to an expression for the magnetoelastic strain:

c c a a ǫα = Kα hJi JjiT,H −hJi Jj iT,H (3.27)

Second experimental evidence for Kαc = −Kαa

In order to explain a negligible magnetostriction for ﬁelds along the hard axes the two ion magnetoelastic constants have to be anisotropic. In order to see this we note that for example the ﬁeld dependence of the magnetoelastic strain ǫa is described within the exchange striction model by the expression (compare [8])

c c a a ǫα = KαchJi JjiT,H + KαahJi Jj iT,H (3.28)

The Kαc and Kαa are deﬁned in Eq. 3.8. When two antiferromagnetically aligned moments are turned into the a-direction by c c the external magnetic ﬁeld, the correlation function hJi JjiT,H is negative and ap- c 2 a a proximately equal to −|hJi iT,H| . On the other hand hJi Jj iT,H is positive and equal 3.1. SAMARIUM 55

Kα −5 ǫa −1.05 × 10 −5 ǫb −0.91 × 10 −5 ǫa 1.13 × 10

Table 3.2 In the table the values for Kα (α = a,b,c) are shown. They correspond to the scaling factor in the ﬁgures 3.15 to 3.17

a 2 to |hJi iT,H| . The magnetization measurements [16] indicate, that the magnitude of the moments does not change significantly, when these are turned into the basal a 2 c 2 plane, i.e. |hJi iT,H| + |hJi iT,H| = const., Therefore the small field dependence of the magnetostriction (3.6) along the hard axes can only be explained, if the Kαγ in equation 3.28 fulfill approximately

Kα = Kαa = −Kαc (3.29)

Inserting this result into Eq. 3.6 again leads to Eq. 3.27. Thus there are two evidences, that the temperature and ﬁeld dependence of the magnetostriction of Sm can be expressed by the static correlation functions multiplied by a constant term.

Comparison between model and experiment

Having such a relation 3.29, makes it possible to fit experimental magnetostriction data to calculated correlation functions, via the constant Kα. The outcome of such fitting procedures can be seen in Figs. 3.15 to 3.17. The figures show the calculated magnetostriction as solid lines and the experiment as circles. The results from the computational calculation are scaled to the experiment. The scaling factors are the

Kα, which are listed in table 3.2. Calculations for ﬁelds along and normal to the c-axis are shown. There was no magnetostriction experiment along the b-axis with ﬁelds along c-axis. This is the reason for the missing experimental data in Fig. 3.16. 56 CHAPTER 3. RESULTS AND DISCUSSION

magnetostriction along a-axis 4K a

l/l) H || c-axis (exp) ∆ -5 -2×10 H || c-axis (calc) =( a H || a-axis (exp) ε H || a-axis (calc)

0 20 40 60 80 100 µ 0H (T)

Figure 3.15 The ﬁgure compares theory (lines) versus experiment (circles). The magnetostriction is shown along c-, and a-axis.

magnetostriction along b-axis 4 K b l/l)

∆ -5 H || a-axis (calc) -2×10

=( H || c-axis (exp) b

ε H || c-axis (calc)

0 20 40 60 80 100 µ 0H (T)

Figure 3.16 The comparison of theory versus experiment along the b-direction. There is no experimental data with ﬁeld along a-direction. 3.1. SAMARIUM 57

H || a-axis (calc) H || a-axis (exp) -5 c -2×10 H || c-axis (calc) l/l) ∆ =( c magnetostriction along c-axis ε 4 K

0 20 40 60 80 100 µ 0H (T)

Figure 3.17 Magnetostriction of Sm along the b-axis, theory versus experiment. 58 CHAPTER 3. RESULTS AND DISCUSSION

Figure 3.18 The magnetic phase diagram of Tm constructed from isothermal magnetization (squares), magnetoresistance (diamonds), and resistance in a constant ﬁeld (ﬁlled circles). [14]

3.2 Thulium

3.2.1 Crystal structure

For the present measurements a Tm single crystal was used. The crystal is about 2 × 2 × 2 mm3 big. The crystal structure of Tm is hexagonal close packed (hcp, with a stacking sequence of ABABAB. . . ), as in most heavy rare earth metals. The magnetic structure of elementary Tm was first obtained with neutron diffraction by Koehler et al. in 1962 [25], who found that below the Neèl temperature (TN = 56 K)the magnetic structure is sinusoidally modulated along c-axis, with the spins constrained along the c-axis. The magnetic structure is incommensurate. As the temperature is reduced the modulation of the spins becomes a square wave which is commensurable with the lattice. At 4 K Tm is a ferrimagnet. The magnetic phase diagram [14] of Tm can be seen in Fig. 3.18.

3.2.2 Thermal expansion of Tm

The thermal expansion of Tm was measured with the capacitance dilatometer in the temperature range from 4 to 320 K. The resulting length change of the material can 3.2. THULIUM 59

-2 1×10

-3 Tm 8×10 c H=0 T ε

c -3 6×10 l/l) ∆ =( c × -3

ε 4 10

-3 2×10

0 0 50 100 150 200 250 300 T (K)

Figure 3.19 The thermal expansion of a Tm single crystal along the c-axis from 4 to 320 K. be seen in ﬁgure 3.19. Over the whole temperature range the single crystal expands about 1%.

3.2.3 Magnetostriction of Tm

The magnetostriction of Tm was measured at the Institut für Festköerperphysik at the Technische Universität Wien in collaboration with Dr. Herbert Müller. Former measurements done by A. Lindbaum showed an effect at 3 T, but the results were not reproducible because it was believed that the crystal turned in the magnetic field. This can happen if the crystal is not cut precisely along one certain axis, but cut with a small angular deviation. The easy axis (the crystallographic axis in which the magnetic moments turn easy) is along the c-axis. The magnetoelastic effect (magnetostriction) was always measured along the easy axis of the crystal. If there is a difference the macroscopic surface that is believed to be the c-axis and the real c-axis, the crystal turns itself into the magnetic field like a bar magnet.

A lot of effort was put into the experiments to determine the magnetostriction of Tm. The experimental method was improved in order the be sure that the signal is purely the magnetostriction and not any other effect. The problem with Tm was its great anisotropy which made experiments in magnetic fields difficult. The first measurements always showed a big hysteresis which was interpreted as a sample turn. If the sample turns during the measurement it moves the two capacitance 60 CHAPTER 3. RESULTS AND DISCUSSION

-2 3,0×10 H || c × -2 2,5 10 4 K

-2 2,0×10

c/c) -2 ∆ 1,5×10 =( c ε -2 1,0×10

-3 5,0×10

0,0 0,0 2,0 4,0 6,0 8,0 µ 0H (T)

Figure 3.20 Due to great anisotropy of Tm it is believed that the crystal turns in the dilatometer during the measurement which leads to a hysteresis that is seen in the ﬁgure

plates apart which results in a disturbing signal. An indication for such a sample turning is an increase of the signal in the saturated field range above 4 T, and the hysteresis of the magnetostriction signal. In Fig. 3.20 both can be clearly seen. After the phase transition there is still an apparent elongation of the sample which is not there anymore when the field is ramped down. There is no convenient physical explanation for this behavior of Tm. From the beginning on the crystal was glued onto the grounded head of a brass screw with Loctite glue. This did not lead to the desired results as explained above (the hysteresis did not vanish). To increase the force that acts on the crystal the nut that holds the dilatometer halves together was tightened even more. This was done with a torque wrench to have compareable and controlable experiments. The solution to that problem was a very simple idea. The crystal is rotated into the z-axis direction by the magnetic field on the one hand. On the other hand it is held back by the dilatometer. There is a certain field, at which the force that is produced by the housing of the dilatometer is not sufficient to prevent it from turning. So an easy way to release some force is to let the whole dilatometer move in the magnetic field. To do so, the plug was not plunged into its counterpart anymore, but hung loose at the end of the sample-stick (explained in Sec. 2.6), fixed with wires. The advantage of this configuration is that if the crystal rotates there is not counter-force of the dilatometer anymore. The whole dilatometer moves and the capacitance plates stay in their position. If magnetostriction occurs the whole dilatometer with the sample inside is already oriented in the field and the signal is 3.2. THULIUM 61

-3 -4 2,0×10 5,0×10 H || c H || c -4 4,0×10 -3 4 K 4 K 1,5×10

-4 3,0×10 a/a) c/c) -3 ∆ ∆ 1,0×10 -4 =( =( c a 2,0×10 ε ε

-4 -4 5,0×10 1,0×10

0,0 0,0 0,0 2,0 4,0 6,0 8,0 0,0 2,0 4,0 6,0 8,0 µ µ H (T) 0H (T) 0

Figure 3.22 Magnetostriction along the a-axis Figure 3.21 The magnetostriction of Tm after with ﬁeld applied along the c-axis. The signal the improvement of the experimental setup. is by a factor three smaller than the magne- The arrows indicate whether the ﬁeld is tostriction along the c-axis which is the easy ramped up or ramped down. axis. not disturbed. The result is shown in Fig. 3.21. 62 CHAPTER 3. RESULTS AND DISCUSSION Chapter 4

Summary and conclusion

The present work presents thermal expansion and magnetostriction measurements on the rare earth metals Sm and Tm. A new capacitance dilatometer was assembled and tested. For the tests the thermal expansion of well known materials was measured and compared to the data available from literature. The metals Ag, Cu, Nb, Pb, boronsilicate were used. The results are satisfactory and the deviation from literature is less than 1%. The conclusion of this analysis is, that the dilatometer operates in a wide range of expansion values. This makes it serviceable for almost all solid materials. The application of the dilatometer in ultra high fields (45 T) at the National High Magnetic Field Laboratory yielded high quality magnetostriction data. The magnetostriction of the rare earth element Sm was measured on a single crystal. The experiments were done in the longitudinal and transversal operation mode. The small size of the dilatometer made it possible to change temperature, which was not possible with former constructions. The dilatometer was also used to measure the magnetostriction of a Tm single crystal in c-direction. Samarium has a close packed hexagonal like crystal structure with two different lattice positions. Due to the local symmetry of these position sites, they are referred to as hexagonal and quasicubic sites. The different surrounding leads to two transition temperatures, which corespond to magnetic ordering at the hexagonal (106 K) and the quasicubic sites (13.8 K). McEwen [16] et al. measured magnetization of Sm along the c-axis and perpendicular to it. From their data they proposed a spin flop transition of the magnetic moments at the quasicubic sites. In order to study the magnetoelastic effect of this spin flop, magnetostriction measurements on a single crystal of Sm were done. A mean field model was applied to describe magnetiza-

63 64 CHAPTER 4. SUMMARY AND CONCLUSION tion and magnetostriction (using the program McPhase). A signiﬁcant anisotropy aa bb cc in the two ion coupling (JNN =JNN =6 JNN ) describes the experimental magnetization data [16] best. However, the single ion anisotropy is small. The agreement between calculated and experimental magnetization along the c-axis is very good. A small deviation from the calculated magnetization can be seen along the a-axis. Magnetization-data at 20 K does not match the calculation, although the magnitude is correct. The experimental values lie above those calculated from the model. A possible reason for this might be the neglect of excited crystal electric ﬁeld states in the model, which play a role at higher temperatures.

In order to arrive at an interpretation of the magnetostriction data of Sm, theoretical considerations were made. From crystal field striction theory the magnetostriction “with applied field” along the hard axes (a- and b-axis in the case of Sm) should give a large signal. The fact, that the measured magnetostriction along the hard magnetization axes is very small compared to the signal along the easy axis leads to the assumption, that at low temperature the crystal field striction effect is small in Sm. The main mechanism responsible for the magnetostriction in Sm must be the exchange interaction.

There are two striking evidences, that made it possible to simplify the general expressions for the exchange striction. (i) In the first place, the magnetostriction signal in fields above the transition field is constant. (ii) Secondly, the magnetostriction along the hard magnetization axes (a- and b-axis) is very small. These two facts lead to an expression for the magnetostriction of Sm, that contains only the static tow ion correlation functions and one magnetoelastic constant. This constant can in principle be calculated from the elastic compliances and the derivatives of the magnetic two ion interaction with respect to strain. As neither of them are available for Sm, the magnetoelastic constant was determined for the three different strains ∆a/a, ∆b/b, ∆c/c by comparison to experimental data. All components of the magnetostriction-tensor, except the b-direction with field along the a-direction, were compared with the calculated magnetostriction. The agreement between the model and experiment is very good considering the limits of the method. Both, transversal as well as longitudinal components are well described. The volume effect of the forced magnetostriction in Sm is small (approximately 1× 10−5). However, there is a surprisingly large spontaneous magnetostriction visible from thermal expansion measurements near the Néel temperature of the quasicubic sites (TN ≈ 14 K). This spontaneous magnetostric- 65 tion is an order of magnitude larger than the forced magnetostriction at 4 K and can not be explained by the exchange striction model. Crystal field effects probably play a role, at higher temperatures and contribute to the spontaneous magnetostriction.

Thulium is a ferrimagnet at 4 K in zero magnetic field with a hexagonal close packed crystal structure. At about 3 T a magnetic phase transition from a ferri- to a ferromagnetic state is observed. Due to a large magnetic anisotropy, it is difficult to measure the magnetostriction at this transition of Tm with the capacitance method. A torque in field disturbs the measurement, and leads to a wrong signal. Effort was made to improve the method. An enhancement of the force onto the sample with a bow has been proposed. On the way of improving the method, another solution was found. The dilatometer was not fixed on the holder-stick, which keeps the dilatometer focused inside the magnet, anymore. The force, that acted onto the Tm before, was now transferred onto the whole dilatometer, which could turn now in the field. The magnetostriction of Tm could be measured with very small hysteresis. It is of the order of 1.5 × 10−3 along c, and 3 × 10−4 along the a-axis, both with field along c. The magnetostriction components are positive, which results in a large volume effect. The magnetostriction of Tm is about two orders of magnitude larger than in Sm.

Outlook

The calculation of the magnetization of Sm suggests saturation in fields above approximately 70 T. Such high fields can only be reached with pulsed magnets. A re-measurement of the magnetostriction using higher fields would give the possibility to test the applied model. Complementary measurements of the magnetization in high fields would give direct access to the saturation magnetization of Sm. Further investigation of the large discrepancy between forced and spontaneous magnetostriction have to be done. It should be possible to apply a similar mean field model to explain the magnetostriction data of Tm. A comparison with experimental data could give insight into the mechanisms of the magnetostriction of this rare earth element. Further improvements of the capacitance dilatometry concerning the control of the force are necessary. 66 CHAPTER 4. SUMMARY AND CONCLUSION APPENDIX

A A Kramers Ground State Doublet

The crystal field ground state of a magnetic ion often can be approximated by a doublet. In this description the crystal field anisotropy enters by defining the saturation moment of this doublet in a,b and c direction: denoting the two states of the doublet by |± > the nonvanishing matrix elements of the angular momentum operator can be abbreviated by

⋆ < ±|Ja|∓ >= A A = A A... saturation moment in a direction ⋆ < ±|Jb|∓ >= B B = −B B... saturation moment in b direction ⋆ < ±|Jc|± >= ±C C = C C... saturation moment in c direction (1) (2)

then the single ion Hamiltonian H = Hcf − gJ µBHJ can be written as

CH −AH − BH α β H = g µ c a b = (3) J B ⋆ −AHa + BHb −CHc ! β −α !

This Hamiltonian may be diagonalized yielding the 2 eigenvalues λ±,∆ = λ+ − λ−:

2 ⋆ λ± = ± α + β β (4) p and the eigenvectors

−β|+ > +(α − λ±)|− > |λ± >= (5) 2 ⋆ |α − λ±| + β β

p 67 68 APPENDIX using Boltzmann statistics (Z = exp(−∆/2kT ) + exp(∆/2kT )) the expectation values of the magnetic moment can be calculated as

exp(−λ /kT ) < M >= <λ |g J|λ > ± (6) ± J ± Z ± X with

⋆ −2Aℜ[β (α − λ±)] <λ±|Ja|λ± >= 2 ⋆ (7) |α − λ±| + β β

⋆ 2 −Bβ β + B|α − λ±| <λ±|Jb|λ± >= 2 ⋆ (8) |α − λ±| + β β

2ℜ(βC)(α − λ±) <λ±|Jc|λ± >= 2 ⋆ (9) |α − λ±| + β β ℜ denotes the real part of a complex number. The magnetic energy U can be calculated (cP is therefore also straight forward to calculate):

exp(−λ /kT ) U = λ ± (10) ± Z ± X List of Figures

1.1 Crystal-ﬁeld and exchange mechanism for magnetoelastic strains. . . 9

2.1 A schematic drawing of a capacitance dilatometer...... 13

2.2 Individual parts of the capacitance dilatometer ...... 15

2.3 The completely assembled halves of the dilatometer from the top. . . 16

2.4 View of the dilatometer from the downside...... 16

2.5 Dilatometer with smaller brass cylinder ...... 18

2.6 Insertion of the dilatometer into the cylinder for grinding...... 18

2.7 Insertion of the pivots into the lower dilatometer half...... 18

2.8 Dilatometer in the transverse and longitudinal mounting position . . 20

2.9 The calibration device for the dilatometer ...... 23

2.10 Measuring the change of the capacitance versus the plategap. . . . . 23

2.11 Comparison of the calculated gap change with the actual gap change. 23

2.12 The gap change in the dilatometer using a silver sample...... 24

2.13 ThermalexpansionofLead...... 25

2.14 ThermalexpansionofCu...... 25

2.15 ThermalexpansionofNb...... 25

2.16 Thermal expansion of borosilicate...... 25

2.17Coolingtower...... 29

2.18 OnediskofaBittermagnet...... 31

2.19 ApictureofaBittermagnet...... 31

69 70 LIST OF FIGURES

2.20 Thedilatometeronthesteeltube...... 33

2.21 Inserting the dilatometer into the tube...... 33

3.1 ThemagneticstructureofSm...... 37

3.2 ThermalexpansionofSm...... 38 3.3 Magnetostriction of Sm along the a-axis with ﬁeld parallel c-axis. . . 39 3.4 Magnetostriction of Sm along the b-axis with ﬁeld parallel c-axis. . . 40

3.5 Magnetostriction was only obtained below 20 K...... 42 3.6 Magnetostriction along a-axis at various temperatures...... 42 3.7 Magnetostriction along b-axis at various temperatures...... 42

3.8 Magnetostriction along c-axis at various temperatures...... 43 3.9 Comparison with other data along the a, and b-axis ...... 44

3.10 Comparison with other data along the c-axis ...... 45 3.11 Summary of the magnetostriction of Sm at 4 K with ﬁeld along c-axis. 46 3.12 Aspin-ﬂoptransition...... 47

3.13 Magnetization data: theory vs. experiment...... 50 3.14 Two-ion static correlationfunction...... 51

3.15 Magnetostriction of Sm along the a-axis, theory versus experiment. . 56 3.16 Magnetostriction of Sm along the b-axis, theory versus experiment. . 56 3.17 Magnetostriction of Sm along the c-axis, theory versus experiment. . 57

3.18 MagneticphasediagramofTm...... 58 3.19 ThermalexpansionofTm...... 59

3.20 Magnetostriction data when the crystal turns...... 60 3.21 Magnetostriction data of Tm, when the method was improved. . . . . 61 3.22 Tm magnetostriction along the a-axis with ﬁeld along c-axis...... 61 Bibliography

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