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Magnetostriction in Rare Earth Elements Measured with Capacitance Dilatometry

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Magnetostriction in Rare Earth Elements Measured with Capacitance Dilatometry Magnetostriction in Rare Earth Elements Measured with Capacitance Dilatometry 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 Universit¨at Wien, an der Technischen Universit¨at Wien und am National High Magnetic Field Laboratory (NHMFL) Tallahassee, Florida durchgef¨uhrt. Das Thema der Arbeit ist die Mes- sung der thermischen Ausdehnung und der Magnetostriktion der Selten-Erd-Metalle Samarium und Thulium. Die Kapazit¨atsdilatometrie ist eine der empfindlichsten und deswegen am meisten eingesetzten Methoden, um diese Eigenschaften zu messen. Mit dieser Methode ist es m¨oglich, relative L¨angen¨anderungen 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 f¨uhrenden NHMFL eingesetzt. Ein kurzer Uberblick¨ ¨uber die Technik zur Erzeugung hohe Magnetfelder, deren Vorteile und Grenzen wird gegeben. Weiters wird die K¨uhltechnik und die Stromversorgung des Forschungszentrums beschrieben. Erstmalig durchgef¨uhrte Messung der Magnetostriktion an Sm in ultra hohen Mag- netfeldern werden pr¨asentiert. 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 m¨oglich. Dieses Magnetfeld wurde mit einem ¨ublichen supraleitenden Magneten erzeugt. Es wurde ein großer Effekt bei 3.5 T gemessen, der von einem magnetischen Phasen¨ubergang 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 mag- netostriction 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 ele- ment 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 magnetostric- tion 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 con- stants 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 deriva- tive 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 elec- trons, 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 ne- glected 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.
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