research collection · 2020. 7. 1. · systeme beinhalten ndal3, ralxga1¡x (r=seltenerdion) and...

Research Collection

Doctoral Thesis

Effect of pressure on the thermodynamics of rare earthcompoundsthe use of pressure for magnetic cooling

Author(s): Strässle, Thierry

Publication Date: 2002

Permanent Link: https://doi.org/10.3929/ethz-a-004385227

Rights / License: In Copyright - Non-Commercial Use Permitted

This page was generated automatically upon download from the ETH Zurich Research Collection. For moreinformation please consult the Terms of use.

ETH Library

https://doi.org/10.3929/ethz-a-004385227http://rightsstatements.org/page/InC-NC/1.0/https://www.research-collection.ethz.chhttps://www.research-collection.ethz.ch/terms-of-use

DISS. ETH NO. 14707

EFFECT OF PRESSURE ON THE THERMODYNAMICSOF RARE EARTH COMPOUNDS:

THE USE OF PRESSURE FOR MAGNETIC COOLING

A dissertation submitted to the

SWISS FEDERAL INSTITUTE OF TECHNOLOGY ZURICH

for the degree of

Doctor of Natural Sciences

presented by

THIERRY STRÄSSLE

Dipl.!Phys.!ETH Zurich

born 05.06.1972

citizen ofKirchberg SG

accepted on the recommendation of

Prof.!A.!Furrer, examinerProf.!K.A.!Müller, co-examinerProf.!B.!Batlogg, co-examiner

2002

Effect of Pressure on the Thermodynamics

of Rare Earth Compounds:

the Use of Pressure for Magnetic Cooling

Dissertation

Thierry Strässle

Laboratory for Neutron Scattering

Swiss Federal Institute of Technology, Zürich (ETH)

CH-5232 Villigen

FINALÃVERSION

27.6.2002

Abstract

A new method for the adiabatic cooling by application of pressure has been recentlyproposed by Müller et al. [1]. The present thesis thoroughly investigates thismethod. It represents the first comprehensive work in this field.

As well as any other (magnetic) adiabatic cooling technique, the cooling resultsfrom a change in the magnetic entropy of the system under investigation. As opposedto the well known method of adiabatic demagnetization (for paramagnets) or to themagnetocaloric effect (for ferro- and antiferromagnets), the entropy change does notresult from the application of an external magnetic field, but from the applicationof external pressure.

The original idea of realizing the entropy change by a pressure-induced struc-tural phase transition [1] has been extended by other mechanisms equally result-ing in a change of magnetic entropy. These include: pressure-induced magneticphase transitions, pressure-induced changes in the degree of 4f − sd hybridizationin Kondo systems, pressure-induced valence transitions and pressure-induced spin-fluctuations. The above mechanisms have all been verified experimentally with anexperimental setup allowing for the direct observation of the cooling effect. For themodelling of the effect, microscopic properties of the respective system were used,which have been determined by neutron diffraction, inelastic neutron scattering andmeasurement of the specific heat and the magnetic susceptibility.

Rare-earth compound are among the most interesting systems for the study of theabove mentioned effect. In these compounds the magnetic entropy is governed by thecrystal-field splitting of the rare earth ion, which for the ground-state J multipletis of the order of a few 10 K. The large value of total angular moment J allowsfor comparatively large entropy changes. The systems studied within this workinclude: CeSb, Ce1−xLaxSb, HoAs, Ce3Pd20Ge6 and EuNi2(Si,Ge)2. Measurementson a single crystal sample of CeSb revealed a directly observed cooling of -2 Kupon a release of 0.52 GPa uniaxial pressure at ∼ 20 K. The cooling is a resultfrom a pressure-induced magnetic phase transition. In HoAs the observed coolingeffect also results from a pressure-induced magnetic phase transition. An additionalcontribution to the effect is caused by the structural distortion of the system underuniaxial pressure (-0.35 K per 0.3 GPa uniaxial pressure release at ∼ 6 K). In theKondo compound Ce3Pd20Ge6 a combination of decreased 4f -conduction electronhybridization and structural distortion upon pressure is found responsible for theobserved cooling (-0.75 K per 0.3 GPa uniaxial pressure release at ∼ 4 K). Finally thepressure-induced valence transition in EuNi2(Si,Ge)2 revealed an observed coolingof -0.52 K upon 0.48 GPa hydrostatic pressure release at ∼ 60 K.

iii

While the release of uniaxial pressure may be realized to a fair amount quasi-adiabatically, the realization of adiabatic conditions under hydrostatic pressure be-comes considerably more difficult.

The studies on the above compounds revealed values for the entropy changeinduced by moderate pressure (a few kbar), which are well comparable with those re-sulting from the magnetocaloric effect induced by large magnetic fields (a few Tesla).Quantitative comparison of the directly observed adiabatic temperature changeswith the model calculations yielded differences, which could well be explained interms of the non-adiabaticity of the experimental setup. Aim of the present studywas to verify experimentally the proposed technique of magnetic cooling by use ofpressure and to extend the class of potential refrigerating materials. The variety ofunderlying mechanisms resulting in a cooling effect may be regarded as a specificquality of this very technique.

The above mentioned compounds are all well studied and characterized at ambi-ent pressure. Nevertheless corresponding experimental data at elevated pressure israre and must often be extrapolated from data at ambient conditions. The appliedextrapolation schemes do commonly not account for all the parameters affected bypressure. For instance, the application of pressure is often reduced to mere geometricfactors neglecting the electronic and magnetic properties of the solid.

Hence the second part of the thesis is devoted to the interplay between thestructure and the electronic or magnetic properties of solids. Elastic and inelasticneutron scattering under pressure proved to be a powerful experimental technique inthis respect. The investigated systems include NdAl3, RAlxGa1−x (R=rare earth)and Cs(Mn,Mg)Br3. The effect of pressure on the crystal-field splitting of NdAl3has shown that even at moderate pressures (a few kbar) the observed changesmay indeed not be explained by mere geometric factors anymore. On the otherhand in RAlxGa1−x the realized structure is found to be dictated by the electronicproperties, while in Cs(Mn,Mg)Br3 the magnetism directly influences the localstructure of the compound via the mechanism of exchange-striction.

[1] K.A.Müller et al. Cooling by adiabatic pressure application in Pr1−xLaxNiO3.Appl.Phys.Lett. 73, 1056 (1998)

iv

Zusammenfassung

Eine neue Methode der adiabatischen Kühlung mittels Druck wurde kürzlich vonMüller et al. [1] vorgeschlagen. In der vorliegenden Doktorarbeit wird dieseKühlmethode eingehend untersucht. Sie stellt die erste umfassende Arbeit aufdiesem Gebiet dar.

Wie in anderen (magnetischen) adiabatischen Kühlmethoden, rührt die Kühlungvon einer Änderung der magnetischen Entropie des zur Kühlung verwendeten Sys-tems her. Im Gegensatz zur wohlbekannten Kühlung durch adiabatische Demag-netisierung (in Paramagneten) oder zum magnetokalorischen Effekt (in Ferro- oderAntiferromagneten), wird die Entropieänderung nicht durch Anlegen eines äusserenmagnetischen Feldes, sondern durch Anlegen von Druck erzwungen.

Die ursprüngliche Idee, die Entropieänderung durch einen druckinduziertenstrukturellen Phasenübergang zu realisieren, wurde durch weitere Mechanismenergänzt, welche ebenfalls zu einer Entropieänderung führen. Diese beinhalten:druckinduzierte magnetische Phasenübergänge, druckinduzierte Veränderung der4f−sd Hybridisierung in Kondo-Systemen, druckinduzierte Valenzübergänge, sowiedruckinduzierte Erhöhung der Spinfluktuationen. Die oben aufgeführten Metho-den der Kühlung wurden direkt experimentell nachgewiesen mit Hilfe einer eigensaufgebauten Versuchsanordnung. Zur Modellierung des beobachteten Effektes wur-den mikroskopische Eigenschaften des entsprechenden Systems herangezogen. Diesewurden mittels Neutronendiffraktion, Neutronenspektroskopie, sowie Messungen derspezifischen Wärme und der Magnetisierung bestimmt.

Seltenerdverbindungen haben sich für eine druckinduzierte magnetische Kühlungals besonders geeignet herausgestellt. In diesen Verbindungen wird die mag-netische Entropie durch die Kristallfeldaufspaltung des Seltenerdions bestimmt.Diese beträgt für das Grundzustandsmultiplett des Seltenerions einige 10 meV.Die grossen Gesamtdrehimpulse J dieser Ionen führt zu vergleichsweise grossen En-tropieänderungen. Die untersuchten Systeme beinhalten: CeSb, Ce1−xLaxSb, HoAs,Ce3Pd20Ge6 and EuNi2(Si,Ge)2. Messungen an CeSb Einkristallen zeigten eine di-rekt beobachtbare Kühlung von -2 K beim Entspannen von 0.52 GPa uniaxialemDruck bei ∼ 20 K. Die Kühlung rührt von einem druckinduzierten magnetischenPhasenübergang her. Der in HoAs beobachtete Kühleffekt ist ebenfalls Folge einesdruckinduzierten magnetischen Phasenübergangs. Darüberhinaus wird hier der Ef-fekt auch von einer druckinduzierten strukturellen Verzerrung hervorgerufen (-0.35 Kper -0.3 GPa uniaxialem Druck bei ∼ 6 K). Im Kondosystem Ce3Pd20Ge6 geht dieKühlung einher mit einer Reduktion der 4f − sd Hybridisierung und einer struk-turellen Verzerrung des Systems unter Druck (-0.75 K per -0.3 GPa uniaxialemDruck bei ∼ 4 K). Schliesslich wurde für den druckinduzierten Valenzübergang in

v

EuNi2(Si,Ge)2 eine Kühlung von -0.52 K per -0.48 GPa bei ∼ 60 K beobachtet.Während uniaxialer Druck in genügender Weise unter quasi-adiabatischen Be-

dingungen entfernt werden kann, stellt das adiabatische Lösen von hydrostatischenDruck ein sichtlich schwierigeres Problem dar.

Die Untersuchungen an den oben genannten Verbindungen ergaben, dass sichin geeigneten Systemen mittels mässigem Druck (einige kbar) Entropieänderungenrealisieren lassen, die mit jenen des magnetokalorischen Effektes unter hohen Mag-netfeldern (einigen Tesla) vergleichbar sind. Die durch den Vergleich der effektiven,beobachteten adiabatischen Temperaturänderungen mit den Modellrechnungenresultierenden Differenzen, konnten durch die nicht vollständig adiabatischenBedingungen der Versuchsanordnung erklärt werden. Ziel der vorliegenden Arbeitwar der experimentelle Nachweis der magnetischen Kühlung mittels Druck und dasAuffinden weiterer geeigneter Kühlsubstanzen. Die grosse Anzahl unterschiedlicherMechanismen, die letztendlich zu einer Kühlung führen, kann als eine Spezialitätdieser neuartigen Kühlmethode bezeichnet werden.

Die oben erwähnten Verbindungen wurden bereits eingehend unter Normal-druck untersucht. Hingegen sind nur wenige Daten über deren Eigenschaften unterDruck verfügbar. Oft werden Extrapolationen zur Abschätzung der Eigenschaftenunter Druck herangezogen. Diese berücksichtigen häufig nur einen bestimmtenGesichtspunkt des untersuchten Systems. So wird beispielsweise der Effekt vonexternem Druck gerne auf rein geometrische Faktoren reduziert ohne Beachtung derelektronischen oder magnetischen Eigenschaften des Systems.

Im zweiten Teil der vorliegenden Arbeit wird daher das Wechselspiel zwischenStruktur und elektronischen oder magnetischen Eigenschaften in Festkörpern unter-sucht. Die elastische und inelastische Neutronenstreuung unter Druck hat sich dabeials eine sehr wertvolle Untersuchungsmethode herausgestellt. Die untersuchtenSysteme beinhalten NdAl3, RAlxGa1−x (R=Seltenerdion) and Cs(Mn,Mg)Br3.In der Tat hat die Untersuchung der Kristallfeldaufspaltung in NdAl3 bereitsbei vergleichsweise niedrigem Druck (einige kbar) gezeigt, dass die veränderteKristallfeldaufspaltung nicht bloss durch geometrische Effekte erklärt werden kann.Andererseits scheinen in RAlxGa1−x elektronische Eigenschaften für das Auftretenbestimmter Kristallstrukturen verantwortlich zu sein. In Cs(Mn,Mg)Br3 schliesslichkonnte der direkte Einfluss magnetischer Eigenschaften auf die lokalle Struktur desSystems beobachtet werden.

[1] K.A.Müller et al. Cooling by adiabatic pressure application in Pr1−xLaxNiO3.Appl.Phys.Lett. 73, 1056 (1998)

vi

Contents

1 Introduction - what it is all about 1

2 Thermodynamic Theory 5

2.1 General thermodynamics of adiabatic cooling . . . . . . . . . . . . . 5

2.2 Magnetocaloric effect and adiabatic demagnetization . . . . . . . . . 7

2.3 Barocaloric effect and elastic heating/cooling . . . . . . . . . . . . . 8

2.4 Entropy and its pressure dependence . . . . . . . . . . . . . . . . . . 9

2.4.1 Two definitions for the entropy . . . . . . . . . . . . . . . . . 10

2.4.2 Lattice entropy . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.4.3 Electronic entropy . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4.4 Magnetic entropy in rare-earth compounds - the crystal field 13

2.5 Implementations of the barocaloric effect . . . . . . . . . . . . . . . . 19

3 Experimental Techniques 25

3.1 Setups for the barocaloric measurements . . . . . . . . . . . . . . . . 25

3.1.1 Thermometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.1.2 Force transmitting system . . . . . . . . . . . . . . . . . . . . 27

3.1.3 Uniaxial setup . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.1.4 Hydrostatic setup . . . . . . . . . . . . . . . . . . . . . . . . 29

3.1.5 Relaxation models . . . . . . . . . . . . . . . . . . . . . . . . 32

3.1.6 Comments on the adiabaticity of the setups . . . . . . . . . . 35

3.2 Neutron scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.2.1 Neutron diffraction . . . . . . . . . . . . . . . . . . . . . . . . 37

3.2.2 Neutron spectroscopy . . . . . . . . . . . . . . . . . . . . . . 41

3.3 Macroscopic measurements . . . . . . . . . . . . . . . . . . . . . . . 44

3.3.1 Macroscopic measurements under pressure . . . . . . . . . . . 44

4 Experimental observation of the barocaloric effect 47

4.1 CeSb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.1.2 Principle of the magnetically driven BCE in CeSb . . . . . . 52

4.1.3 Direct observation of the intensive BCE . . . . . . . . . . . . 54

4.1.4 Model for the BCE in CeSb . . . . . . . . . . . . . . . . . . . 55

4.1.5 BCE under hydrostatic pressure . . . . . . . . . . . . . . . . 57

4.2 The diluted system Cex(La,Y)1−xSb . . . . . . . . . . . . . . . . . . 58

4.2.1 The BCE in Ce0.85(La,Y)0.15Sb . . . . . . . . . . . . . . . . . 58

4.2.2 Spin-cluster behavior in Cex(La,Y)1−xSb for x ≤ 0.85 . . . . 62

vii

4.3 HoAs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.3.2 Inelastic neutron scattering measurements of the CEF . . . . 654.3.3 Barocaloric Effect . . . . . . . . . . . . . . . . . . . . . . . . 664.3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.4 The missing BCE in YbAs . . . . . . . . . . . . . . . . . . . . . . . . 724.5 Ce3Pd20Ge6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.5.2 Direct observation of the intensive BCE . . . . . . . . . . . . 754.5.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.6 EuNi2(SixGe1−x)2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 794.6.2 Preparation and characterization of the sample . . . . . . . . 794.6.3 Estimation of the BCE in EuNi2(Si0.15Ge0.85)2 . . . . . . . . 814.6.4 Direct observation of the BCE in EuNi2(Si0.15Ge0.85)2 . . . . 824.6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5 Neutron Scattering Studies under Pressure 875.1 Phase diagram of the pseudo-ternary rare-earth compounds RAlxGa2−x 88

5.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 895.1.2 Sample Preparation . . . . . . . . . . . . . . . . . . . . . . . 905.1.3 Pressure-induced structural phase transition in NdAlxGa2−x 905.1.4 CEF in ErAlxGa2−x and implications on the chemical phase

diagram of the RAlxGa2−x family . . . . . . . . . . . . . . . 945.1.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

5.2 Pressure dependence of the CEF in NdAl3 and CEF of PrAl3 . . . . 1045.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1045.2.2 Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . 1045.2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055.2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

5.3 Dimer exchange-striction in CsMn0.28Mg0.72Br3 . . . . . . . . . . . . 1105.3.1 Formalism of exchange-striction . . . . . . . . . . . . . . . . . 1105.3.2 Dimer excitations in CsMnxMg1−xBr3 . . . . . . . . . . . . . 1115.3.3 Evidence for exchange-striction in CsMn0.28Mg0.72Br3 . . . . 1125.3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

6 Concluding remarks (BCE) 117

viii

Chapter 1

Introduction- what it is all about

Ever since Warburg (1881) first observed the heat evolution in iron upon the ap-plication of a magnetic field, the so-called magnetocaloric effect (MCE) attractedscience. The effect is a consequence of the variation in the total entropy of a solidby the magnetic field. It was first Debye (1926) and Giauque (1927) who proposedthe principle of adiabatic cooling or more specifically the technique of adiabatic de-magnetization. In the first stage, an external magnetic field is applied isothermally(the system is in contact with a heat sink), thus reducing the magnetic entropy 1 ofthe system. In the second stage, the magnetic field is removed adiabatically (thesystem is isolated from the heat sink). In order to keep the entropy unchanged, thesystem is forced to lower the temperature.2

Progress in the theoretical as well as the experimental characterization of themagnetothermal properties of materials has renewed the interest in the investigationof the MCE for two reasons: (i) the MCE can yield information on magnetic phasetransitions not obtainable by other experimental techniques, (ii) its potential for theimplementation of magnetic cooling machines (magnetic refrigerators) [1]. Thesedays, magnetic refrigeration proved to be one of the most efficient cooling tech-niques in a wide range of temperatures. For the time before the rise of 3He dilutioncryostats, adiabatic demagnetization represented the only technique allowing tem-peratures below 4.2 K down to the milli Kelvin range (Giauque (1927)). Nowadays,magnetic cooling at room temperature and even higher has been demonstrated fora wide class of rare-earth compounds [2, 3]. Up to now, all magnetic refrigeratorssuffer the drawback of needing large magnetic fields of a few Tesla in order toachieve cooling effects in the Kelvin range.

Despite the fact that adiabatic cooling has to be associated by no means with anentropy change induced by an external magnetic field, its implementation by vari-

1A definition of the magnetic entropy and of other contributions to the total entropy of a solidis given in section 2.4.

2The term adiabatic demagnetization is generally used in context with the application of a mag-netic field on a paramagnet, whereas for the MCE the field is applied on a ferro- or antiferromagnet.Both effects result in magnetic cooling and are referred in a wider context to as adiabatic coolingtechniques (compare Fig.2.7).

1

00

TT-∆TTemperature T

Ent

ropy

S Si

Sf

∆T

Figure 1.1: Principle of adiabatic cooling: due to an external influence the entropyof the system gets changed (Si → Sf ).

ation of a different external thermodynamic variable has hardly been investigated.Müller et al. [4] proposed to implement adiabatic cooling by the application of pres-sure in the vicinity of a pressure-induced structural phase transition in a rare-earthcompound. A pressure-induced structural phase transition will change the pointsymmetry at the rare-earth ion. In general this will result in a different splitting ofits 2J+1-fold degenerate ground-state J-multiplet by the crystal field (CEF), whichgoverns the thermodynamic properties of the system at low temperatures.3 Henceexternal pressure p may well serve to change the magnetic entropy in the very sameway as an external magnetic field does in the MCE. In analogy to the magnetocaloriceffect, the associated effect has been given the name barocaloric effect (BCE).

An example for the principle of adiabatic cooling in general and more specificallyfor the BCE is illustrated schematically in Figure 1.1. The system is assumed initiallyin a state i of large CEF splitting with low degeneracy of its lowest CEF levels.Therefore the magnetic entropy will increase only slowly with temperature (solid line,Fig. 1.1). The application or removal of external pressure can bring the system viaa pressure-induced structural phase transition into a state f of small CEF splittingand high degeneracy of its CEF energy levels. In state f the system happens to havea higher entropy than in state i at the same temperature (dashed line, Fig. 1.1).Thus if the transition i → f is performed adiabatically the system must cool down by∆T in order to remain its entropy unchanged (adiabatic). Even though no magneticordering in this system is present, the effect is solely a consequence of the variation inthe magnetic entropy. Hence the BCE leads to what may be called pressure-inducedmagnetic cooling.

The present thesis project aimed to provide a detailed understanding of thisnew cooling technique, which before has been verified experimentally on a singlecompound only [4]. The use of an external applied pressure instead of an exter-nal magnetic field as the thermodynamic variable driving the entropy change andhence cooling, may open new technical applications. The large magnetic fields of a

3The crystal field (crystalline electric field) is introduced in section 2.4.4.

2

Socket

Piston

Sample

40 50 60 70

5.6

5.8

6.0

Time t [s]

Tem

pera

ture

T [K

]

Pressure release

Figure 1.2: Schematic setup for the measurement of the BCE (under uniaxial pres-sure) and resulting temperature evolution (HoAs, ∆p = −0.3 GPa). The tem-perature of the sample is logged by a thermocouple, while pressure is applied andreleased.

few Tesla needed for the efficient cooling by the MCE must be considered the maindrawback of magnetic refrigeration these days which hinders its broad, technical ap-plication in everyday life. On the other hand the application of a few kbar pressureon a solid is comparatively easy to achieve even outside of the laboratory. This be-comes especially true for the application of uniaxial pressure generated, e.g. by use ofpiezo actuators. In what follows the technical problems associated with an efficientrealization of the MCE or BCE are though not tackled with priority. The main focusis set on the exploration of suitable refrigerating materials and the modelling of theobserved cooling effects on the basis of microscopic parameters. In this connectionother pressure-induced mechanisms were found which equally well result in an effec-tive change of the magnetic entropy, as for instance pressure-induced magnetic phasetransitions [5]. Many rare-earth compounds show pronounced pressure-dependencesin their magnetism and thus may be suitable candidates for the BCE. In the orderedstate a large splitting of the CEF levels due to the internal magnetic field occurs(Zeeman effect). Thus cooling can be realized by a transition from the ordered statei (with low entropy) into the disordered state f (with high entropy) in the verysame way as in the structurally driven case.4 Other mechanisms leading to a BCEhave also been considered throughout this work, these include: effects of pressure onKondo systems, pressure-induced valence transitions and pressure-dependent spin-fluctuations.

In either case detailed knowledge about the microscopic properties of thematerial is indispensable for the explanation of the BCE and in the search forbest-suited BCE materials. Elastic and inelastic neutron scattering together withmacroscopic measurements of the specific heat and the magnetization proved tobe the experimental techniques of choice in order to study the BCE. The effectitself can be directly measured in a simple experimental setup as shown in Figure 1.2.

4In this sense the magnetically driven BCE can also be considered as an internal MCE usingthe molecular field of the solid instead of an external magnetic field.

3

To the knowledge of the author the present thesis represents the first compilationof experimental work dedicated to the BCE and pressure-induced magnetic cooling.The chapters are organized as follows: In the first chapter a short introductionto the basic thermodynamic theory necessary for the understanding of the BCE isgiven. The notation is followed in as close as possible analogy to the MCE. Specialissues valid for rare-earth compounds are addressed. The second chapter explainsthe experimental methods used in the course of this work. Special importance ispaid to methods that may not be common to the reader, these include the setup forthe measurement of the BCE and high pressure techniques. Experimental results ofthe barocaloric effect are summarized in the third chapter. Eventually the fourthchapter resumes neutron scattering experiments under pressure on compounds whichwere considered as potential refrigerating materials or which constituted systems wellsuited to test the extrapolation schemes carried out in chapter three.

4

Chapter 2

Thermodynamic Theory

In what follows the thermodynamics of adiabatic cooling are recalled and issues spe-cific for the barocaloric effect are discussed with a focus on the microscopic propertiescausing this effect. Comprehensive introductory reviews about the magnetocaloriceffect and about the crystal field in rare-earth compounds can be found in [1] and [6],respectively. Some theoretical aspects specific to measured systems introduced laterin the experimental part of this work are given at the respective places of chapters 4and 5. These include the resonance-level model for the description of the magneticentropy in Kondo lattice compounds, and the theory of exchange-striction for dimersystems. Models involving measurement techniques of the BCE are given in section3.1.

2.1 General thermodynamics of adiabatic cooling

All adiabatic cooling techniques (i.e. adiabatic demagnetization, MCE and BCE)are based on the same thermodynamic principle and hence suffer the same generalthermodynamic restrictions. They make use of the fact that the total entropy Sof a system is a thermodynamic state function and thus depends on three externalthermodynamic variables, namely besides temperature T on pressure p and magneticfield H

S = S(T, p,H) (2.1)

with

dS(T, p,H) =

(∂S

∂T

)

p,H

dT +

(∂S

∂p

)

T,H

dp+

(∂S

∂H

)

T,p

dH. (2.2)

Apparently, varying only one of the external variables1 p or H, denoted as thecontrol variable X, and keeping the temperature T and the other variable K (H orp) constant, leads to a change in entropy (2.2) (Fig. 2.1, step 1→ 2)

∆S(T,X,K)X1→X2 = S(T,X2,K)− S(T,X1,K) (2.3)

=

∫ X2X1

(∂S(T,X,K)

∂X

)

T,K

dX. (2.4)

The extensive quantity ∆S upon isothermal2 change of H is often given the name

1p for the BCE, H for the MCE and adiabatic demagnetization

5

Temperature

Ent

ropy

S(T,p,H

)X1

S(T,p,

H) X2

∆Tad

∆S

1

2

3

∆Tad

∆S

1

2

3

A

B∆S = const.

Figure 2.1: A schematic of adiabatic cooling. The adiabatic temperature change∆Tad strongly depends on the slope of the S(T, p,H) function (isothermal entropychange ∆SX1→X2 kept constant).

extensive MCE, hence in the general case we may call ∆S the extensive caloric effect.Subsequent adiabatic3 change of X from X2 back to X1 leads to a change in

temperature ∆Tad as the total entropy must be conserved (Fig. 2.1, step 2→ 3)

S(T,X2,K) = S(T +∆Tad, X1,K). (2.5)

Accordingly the intensive quantity ∆Tad is given the name intensive caloric effect.Recalling the second law of thermodynamics

(dS(T,X,K)

dT

)

X,K

=

(C(T,X,K)

T

)

X,K

(2.6)

with C denoting the total heat capacity of the system, ∆Tad may be written (2.4)

∆Tad(T,X,K)X2→X1 =

∫ X1X2

(T

C(T,X,K)

∂S(T,X,K)

∂X

)

T,K

dX. (2.7)

The intensive and the extensive caloric effect are coupled via the adiabatic equa-tion (2.5)

S(T,X2,K) = S(T +∆Tad, X1,K) = (2.8)

= S(T,X1,K) +

(∂S(T,X,K)

∂T

)

X,K

∆Tad +O(∆T2ad) (2.9)

For ∆Tad follows in 1st order (i.e. assuming S ∝ T )

∆Tad =∆S(T,X,K)X1→X2∂S(T,X,K)/∂T

(2.10)

2dT = 03dS = 0

6

and with C = T ∂S∂T (2.6)

∆Tad = ∆S(T,X,K)X1→X2T

C(T,X,K). (2.11)

The same result is obtained by linearization of (2.7) (i.e. assuming C ≡ const).These strong assumptions are hardly justified in any real system, nevertheless (2.11)has some important consequences valid for the non-linear cases too:

• ∆Tad ∝ ∆SX1→X2 ; large intensive caloric effects require large extensive caloriceffects

• ∆Tad ∝ T/C; the intensive caloric effect is large for either very low T as C → 0(T → 0) or for high T as C → const (T high enough).

The fact ∆Tad ∝ 1/C is also depicted in Figure 2.1. While ∆S is the same forboth illustrated cases, the large slope in case B reduces ∆Tad considerably. Theseconsiderations are valid for all adiabatic cooling techniques. For more quantitativeconclusions the specific relation of the entropy on the control variable X (i.e. ∂S/∂Xin (2.4)) must be considered.

2.2 Magnetocaloric effect and adiabatic demagnetiza-tion

In the magnetocaloric effect and in the adiabatic demagnetization, the externalmagnetic field H acts as the control variable X and the isothermal entropy changeis related to the change in bulk magnetization M . Hence the extensive (2.4) and theintensive (2.7) MCE can be written in M(T, p,H) using the Maxwell relation4 [1, 7]

(∂S

∂H

)

T,p

=

(∂M

∂T

)

p,H

(2.12)

resulting in

∆S(T, p,H)H1→H2 =

∫ H2H1

(∂M(T, p,H)

∂T

)

p,H

dH (2.13)

∆Tad(T, p,H)H2→H1 =

∫ H1H2

(T

C(T, p,H)

∂M(T, p,H)

∂T

)

p,H

dH. (2.14)

In practice the lack of analytical forms for M(T, p,H) and C(T, p,H) makes ananalytical integration of these equations for the MCE (which involves a magneticphase transition) impossible. In the case of adiabatic demagnetization, analyticalforms for both the magnetization and the heat capacity for paramagnets at lowtemperatures exist and analytical expressions for ∆S and ∆Tad can be worked out [8].

4This Maxwell relation results from the differential of the Gibbs potential dG = −SdT −MdH

(dG = −SdT + V dp with V → −M , p → H) and the relation(∂u∂y

)x= ∂

2f∂y∂x

= ∂2f

∂x∂y=

(∂v∂x

)y

valid for the exact differential df of f(x, y) with df = u(x, y)dx + v(x, y)dy and u(x, y) =(∂f∂x

)y,

v(x, y) =(∂f∂y

)x[G→ f ].

7

For the MCE on the other hand, numerical integration of (2.13) and (2.14) requiresvery precise experimental data of the magnetization and the heat capacity in functionof both, temperature and magnetic field. Hence the numerical evaluation of (2.13)and (2.14), although completely defining the MCE, may result in an accumulationof errors as large as 20-30 % [9]. Another often applied approach for an indirectdetermination of the MCE is based on the second law of thermodynamics (2.6) [1]

S(T, p,H) =

∫ T

0dS(T, p,H)dT =

∫ T

0

C(T, p,H)

TdT (2.15)

and

∆SH1→H2 =

∫ T

0

C(T, p,H1)− C(T, p,H2)T

dT. (2.16)

The intensive MCE ∆Tad may be then calculated numerically from S(T, p,H1) andS(T, p,H2) by use of the adiabatic equation (2.5). Note however, that this approachrequires heat capacity measurements down to the lowest temperatures and similarlymay result in large accumulated errors [9, 3].

Nevertheless the calculation of the extensive MCE via the (H,T )-dependenceof the magnetization or via the (H,T )-dependence of the heat capacity has provento be a valuable first check for materials expected to show a MCE. The procedureis fast and standard setups for the measurement of C and M may be used. Withto the ongoing discussion as to what extent indirect measurements of the MCEare valid [10, 11], the direct measurement of the MCE has gained in importanceand recently much more literature on direct measurements is available ([12, 13, 14],[1] and references therein). This tendency is partly influenced by the discovery ofmaterials showing the so called giant MCE [15, 16, 17]. These materials involvefirst order magnetic transitions (which in turn are strongly coupled to simultaneousstructural phase transitions) resulting in huge ∆S. Note that for any first ordertransition the entropy becomes discontinuous and the Maxwell relation ∂S/∂M =∂H/∂T must no longer be used. In these cases the indirect determination of theMCE is restricted to the integration of C (2.16) and the complicated nature of thephase transition often does not allow the analysis of the MCE on the basis of generalmacroscopic thermodynamics [18].

2.3 Barocaloric effect and elastic heating/cooling

In the barocaloric effect and in elastic heating/cooling the entropy change ∆S iscaused by external pressure p being the control variable X. In principle the Maxwellrelation analogous to (2.12) may be applied5, i.e.

(∂S

∂p

)

T,H

= −(∂V

∂T

)

p,H

(2.17)

5This Maxwell relation is deduced in the same way as for (2.12), i.e. by means of the differentialof the Gibbs potential.

8

and (2.4), (2.7) become

∆S(T, p,H)p1→p2 = −∫ p2p1

(∂V

∂T

)

p,H

dp (2.18)

∆Tad(T, p,H)p2→p1 = −∫ p1p2

(T

C(T, p,H)

∂V (T, p,H)

∂T

)

p,H

dp. (2.19)

From these equations the change of volume appears as the source for the BCE (inanalogy to the change of bulk magnetization in the MCE). It is noteworthy to addressa few important remarks at this point. Although (2.18) and (2.19) explicitly includethe thermal expansion coefficient α = 1/V (∂V/∂T ) they may equally well includemagnetic terms in an utterly implicit manner. The exact incorporation of these termsdepends on the nature of the pressure effect and requires a microscopic description ofthe thermodynamics involved. Additionally, in contrast to (magnetic) transitions in-duced by external magnetic fields (e.g. metamagnetic transitions), pressure-inducedtransitions often involve a change in structure of the system under investigation.Note that structural phase transitions are often of first order and thus the use ofthe Maxwell relations is strictly not allowed. The continuation of the formalism ofmacroscopic thermodynamics on general grounds must be given an end already atthis point, except for the special case of elastic heating, which will be addressed insection 2.4.2. In this case pressure does not alter the magnetic properties of thesystem and the problem can be handled analytically. It thus may be regarded as theanalogue to adiabatic demagnetization.

The important fact, that the application of pressure may well influence the mag-netic properties of the system implies a somewhat more difficult macroscopic ther-modynamic description which shall not be discussed here. On the other hand thevery same fact allows the BCE to alter exactly the same quantity of a solid as anexternal field does in the MCE, namely the magnetic entropy. Thus the next sec-tions will discuss the microscopic thermodynamics involved in the BCE on the basisof the total entropy of the respective system. Once the entropy S(T, p,H) may bemodelled the extensive as well as the intensive BCE can be calculated (2.16),(2.5).In finishing the discussion about macroscopic thermodynamics it must not be for-gotten, that (2.15) and (2.16) remain valid for the BCE too. As in the case of theMCE these equations allow in principle the indirect determination of the BCE giventhat accurate data on the specific heat under pressure is available (see 3.3.1).

2.4 Entropy and its pressure dependence

In order to discuss how pressure affects the thermodynamics of a solid the entropymust be split into terms of different origin. In a magnetic solid with localized mo-ments the total entropy is the sum of the lattice SL, electronic SE and magnetic SMentropy:

S = SL + SE + SM (2.20)

All three contributions are functions of both temperature and pressure. The mag-netic entropy is generally also a function of the magnetic field. The electronic con-tribution SE is not directly affected by the magnetic field, however in 4f magnetismSE and SM may be coupled via 4f -conduction electron hybridization, so that they

9

cannot be strictly separated anymore. In 3d magnetism the separation between SMand SE is inherently not straight forward. In solids with a coupling between themagnetic and the chemical lattice, SL may show also a dependence on magneticfields. Before discussing the three contributions separately two definitions for theentropy are recalled.

2.4.1 Two definitions for the entropy

Communication theory

M events are considered with the respective probabilities p1, . . . , pM (∑

m pm = 1).The degree of non-predictivity, uncertainty or choice is an intuitive concept whichcan be assigned a number called the statistical entropy [19]:

S(p1, . . . , pM ) = −kM∑

m=1

pm ln pm (2.21)

where k is a multiplicative factor. The statistical entropy is maximal forM equiprob-able events S(1/M, . . . , 1/M) = k lnM .

Statistical mechanics

Considering a system with Hamiltonian H the canonical partition function Z andfree energy F at temperature T is defined as [20]

Z(T,H) = tr(e−H/kT ), (2.22)F (T,H) = −kT lnZ (2.23)

with k the Boltzmann factor. The entropy is then given by

S = −(∂F

∂T

)

p,H

. (2.24)

For a Hamiltonian resulting in a set of discrete energy levels Ei both definitionsbecome equivalent and maximal entropy is found for a fully degenerate ground-state.

2.4.2 Lattice entropy

The lattice entropy results from the occupation of vibrational states of the latticeby phonons. The associated internal energy is

UL =

∫ ∞

0dEρ(E)n(E)E (2.25)

with ρ(E) the phonon density of states and n(E) the Bose-Einstein distribution

n(E) = (eE/kT − 1)−1. (2.26)

The lattice specific heat and entropy are derived from C = ∂U/∂T and S =∫C/TdT

(2.6). In the Debye approximation

ρ(E) =

{∝ E2 , E ≤ kΘD0 , E > kΘD

(2.27)

10

the specific heat CL is then

CL(T ) = 9Nk

(T

ΘD

)3 ∫ ΘD/T

0

x4ex

(ex − 1)2 dx→12π4

5Nk

(T

ΘD

)3for T ¿ ΘD

(2.28)with N the total number of atoms in the system and ΘD the Debye temperature.Thus at low temperatures the lattice specific heat and entropy S =

∫C/TdT are

often modelled by a simple T 3-law

CL = βT3 (2.29)

SL =β

3T 3. (2.30)

The application of pressure generally leads to an increase in the stiffness of thelattice and the phonon density of states is shifted to higher energies (see inset ofFig. 2.2). In the Debye approximation this shift results in a scaling of ΘD withvolume V

ΘD(V ) = ΘD(V0)

(V0V

)Γ(2.31)

with Γ the (phononic) Grüneisen parameter defined by this relation.6 The shift ofthe phonon spectrum to higher energies results in a decrease of the lattice entropyas “the phonons may now choose on fewer states” (2.21). Pressure hence generallyresults in a decrease of lattice entropy if applied isothermally and in an increase oftemperature i.e. heating if applied adiabatically.

The elastic heating may also be calculated in the framework of macroscopicthermodynamics starting from (2.19)

∆Tad =

∫ p1p2

(T

C(T, p,H)

∂V

∂T

)

p,H

dp =

∫ p1p2

(T

C(T, p,H)αV

)

p,H

dp (2.32)

using the thermal expansion coefficient α = 1/V (∂V/∂T ). The Grüneisen parametermay be introduced via the relation (αBV = ΓC)V,H with B the isothermal bulkmodulus. Assuming (C)V ≈ (C)p and Γ, B constant with pressure one finds

∆Tad =

∫ p1p2

TΓ

Bdp ≈ T Γ

B∆p. (2.33)

Figure 2.2 compares the elastic heating of copper (ΘD = 316 K, dΘD/dp =3.2 K/GPa [21]) derived from the Debye model (2.28) and derived from the macro-scopic approximation (2.33). The difference results from the strong assumptionsmade by both models: the Debye model assumes ρL ≈ E2; the macroscopic model as-sumes (C)p = (C)V and linearizes twice. Nevertheless the calculations demonstratethe order of magnitude of elastic heating that can be expected under hydrostaticapplication of pressure (i.e. typically ∆Tad/∆p < 20 mK/0.1GPa for T ≤ 20 K).

6In a more general approach every vibrational mode η can be assigned a separate Grüneisenparameter Γη.

11

0.20

0.15

0.10

0.05

0.00

100806040200

T [K]

DT

ad [K

]

Energy

Den

sity

of s

tate

sp2>

p1p 1

kQ2kQ1kT

Figure 2.2: Calculated elastic heating of copper at an adiabatic hydrostatic pressurechange ∆p = 0.1 GPa (solid line: Debye model, dashed line: macroscopic approxi-mation). Inset: effect of pressure on the phonon density of states (Debye model).

2.4.3 Electronic entropy

The electronic heat capacity of a solid may be discussed within the theory of freeelectrons. The Pauli exclusion principle requires that only those electrons for whichcorresponding free states are available, can be thermally excited, i.e. only electronswithin an energy range EF − kT . . . EF and EF . . . EF + kT . They contribute to aninternal energy of

UE =

∫ EF0

D(E)[1− f(E)](EF − E)dE +∫ ∞

EF

D(E)f(E)(E − EF )dE (2.34)

with EF the Fermi energy, D(E) the electron density of states and f(E) the Fermi-Dirac distribution. The corresponding heat capacity follows from ∂U/∂T [22]:

CE =∂UE∂T

≈ 13π2D(EF )kT = γT , (kT ¿ EF ). (2.35)

Hence for the entropy S =∫C/TdT one can write (2.6)

SE = γT. (2.36)

In practice γ often does not agree well with the value calculated by the free electronmodel. Departures origin from band effects, interactions with phonons and interac-tions with other itinerant or localized electrons and may be expressed by an effectivemass m ≡ mfreeγ/γfree. Compounds with considerably enhanced effective mass arecalled heavy-fermion compounds (γ & 100 mJ/K2mol). In this context lanthanideand actinide ions with an almost empty or almost full f -shell play a special role.Here the conduction electrons may interact, i.e. hybridize, very strongly with thef -electrons leading to an increase of γ by as much as three orders of magnitude.

12

This effect is referred to as the Kondo effect, corresponding compounds are calledKondo lattice compounds. Most prominent ions for the Kondo effect are Ce, Yb andU, but also in Pr strong hybridization is often observed.

In the frame of the free electron model pressure results in a decrease of D(EF )and hence in a lower CE as the reciprocal lattice gets mechanically dilated.

7 Onthe other hand one generally observes an increase of the hybridization in Kondocompounds and hence larger γ values with the application of hydrostatic pressure(see [23] and section 4.5).8

2.4.4 Magnetic entropy in rare-earth compounds - the crystal field

The crystal-field (CEF) interaction is the manifestation of the quantum mechanicalStark effect on a single ion caused by the electric field of neighboring ligand ions.While for 3d ions this interaction turns out to be larger than the spin-orbit coupling,it is found to be smaller for 4f ions. Hence for 4f ions the CEF interaction can betreated as a perturbation on the 2J +1-fold degenerate energy states |J〉 with totalangular momentum J combined from the spin and the orbital momentum. Formetallic rare-earth compounds the CEF interaction results in a total splitting of theground-state J-multiplet of typically a few 10 K.9 Hence the CEF is the dominatingfactor defining the magnetic and thermodynamic properties in rare-earth compoundsat low temperatures (in paramagnetic state).

In what follows the CEF is treated as a perturbation within the ground-stateJ-multiplet only. Admixtures from excited J-multiplets are neglected. This approx-imation is valid for most metallic compounds, where generally the CEF splitting ofthe ground-state J-multiplet is found small compared to the distance to the first ex-cited J-multiplet, which accounts to more than 2700 K (except for Sm3+ and Eu3+).The corresponding formalism is referred to as the Stevens notation.

Description of the crystal field

The energy of an ion with unfilled 4f -shell in a crystal field is given by the Hamil-tonian

ĤCEF = e∑

i

V (ri) (2.37)

with V the electrostatic potential of its surrounding charge distribution ρ(r) andi running over all 4f electrons. At the rare-earth site the CEF potential fulfills∆V = 0 and hence may be written as a multipole series

V (r) =∑

n,m

cmn rnY mn (θ, φ) (2.38)

in the spherical harmonics Y mn .10 Stevens [24] demonstrated that the polynomial

terms in the cartesian coordinates ri can be replaced by polynomial terms Ômn in

7direct a ↓ ⇒ reciprocal k ↑ ⇒ D(EF ) ↓ ⇒ γ ↓8Also note that a 4f -conduction electron hybridization generally leads to a decrease of the

magnetic entropy discussed in the next section.91 meV = 11.6 K10with l4f ≤ 3 follows n ≤ 6

13

Jz, J− and J+ retaining all transformation properties and acting on the unfilled4f -shell as a whole. The Hamiltonian (2.37) can thus be written as

ĤCEF =∑

n,m

Amn 〈rn〉χnÔmn =∑

n,m

Bmn Ômn (2.39)

with Ômn and χn the Stevens operators and coefficients respectively [25], 〈rn〉 theaverage of the n’th radial moment of the 4f electron [26] and Amn (B

mn ) ∈ C the

CEF parameters reflecting the charge distribution ρ(r), i.e. the CEF potential V (r).The point symmetry at the respective rare-earth ion limits the terms appearing

in (2.39) [27]. E.g. for cubic and hexagonal11 symmetry one finds (Bmn ∈ R)

ĤcubCEF = B04[Ô04 + 5Ô

44

]+B06

[Ô06 − 21Ô46

], (2.40)

ĤhexCEF = B02Ô02 +B04Ô04 +B06Ô06 + |B66 |Ô66. (2.41)

The diagonalization of ĤCEF is now straight forward and yields 2J+1 energy eigen-states |n〉 which can be assigned to irreducible representations of the respective pointgroup.

Crystal-field entropy

The splitting of the ground-state J-multiplet into 2J + 1 energy levels, i.e. into theenergy eigenvalues Ei of (2.39), causes an entropy according to (2.21) and (2.24)

SCEF = −Nk∑

i

pi ln pi , pi =1

Ze−Ei/kT , Z =

∑

i

e−Ei/kT (2.42)

with Z the partition function, N the total number of rare-earth ions in the systemand pi the thermal population factors of the energy levels. The associated free energyand heat capacity are

FCEF = −NkT lnZ, (2.43)

CCEF = T∂SCEF∂T

= Nk

∑

i

(EikT

)2pi −

(∑

i

EikT

pi

)2 . (2.44)

The CEF contribution to the total heat capacity causes an anomaly known as theSchottky anomaly which is absent in compounds with non-magnetic rare-earth ionssince they have no effective splitting from the CEF as J = 0 (Fig. 2.3). Hencefor rare-earth compounds SCEF can be associated with the magnetic entropy SM .It follows from (2.42) that SCEF grows with the number of thermally activatedenergy levels. For a singlet ground-state system SCEF vanishes at absolute zero,SCEF → 0 (T → 0). However for a system with an m-fold degenerate ground-state one finds SCEF → Nk lnm 6= 0 (T → 0) in apparent disagreement with thethird law of thermodynamics. Especially for all rare-earth ions with an odd numberof 4f electrons (Kramers ions), the time-reversal invariance of ĤCEF requires atleast twofold degenerate energy levels with SCEF →≥ Nk ln 2 (T → 0) (Kramers11point groups D6h,D3h,C6v,D6,C6h,C3h,C6 only [27]

14

108

6

42

CC

EF(

Pr)

[J/

K m

ol]

1208040

T [K]

80

60

40

20

C [

J/K

mo

l]

30025020015010050

T [K]

PrAl3 LaAl3

Figure 2.3: The CEF splitting in PrAl3 (JPr = 4) causes a pronounced Schottkyanomaly in the specific heat, whereas LaAl3 (JLa = 0) shows no CEF splitting andhence no Smag (data taken from Mahoney [28]).

degeneracy [29]). In these cases S → 0 (T → 0) is retained by further symmetrybreaking perturbations on Ĥ with T → 0 preventing a truly degenerate ground-state. The most prominent of these mechanisms include: magnetic ordering of therare-earth ion sublattice, magnetic ordering of the nuclear spins or a Jahn-Tellerdistortion.

Point-charge model and its extensions

In principle the CEF parameters can be calculated on the basis of the charge distri-bution ρ(r). In the point-charge model (PCM) ρ(r) is defined by the positions Riand effective charges qi of the neighboring ligand ions [6]

ρ(r) =∑

i

qiδ(Ri − r). (2.45)

Generally the CEF parameters can then be expressed by

Bmn = amn γ

mn 〈rn〉χn (2.46)

with reduced CEF parameters amn reflecting the charge distribution independent ofgeometry and rare-earth and geometric coordination factors γmn calculated by thePCM. However the computation of the CEF parameters from microscopic theoryturns out to be a difficult task and the application of the above model is of lim-ited success even in the case of rare-earth salts [6]. Morisson [30] has introducedan extension of the original PCM, which corrects the free Hartree-Fock 4f–radial-moments 〈rn〉 → 〈rn〉/τn for the situation of ions embedded in solids and whichtakes into account the shielding due to the outer 5s2 and 5p6 shells of the rare-earth

15

ion by scaling Bmn → (1−σn)Bmn . τ and σn are phenomenological parameters of therare-earth ion tabulated in [30]. Equation (2.46) reads then

Bmn = amn γ

mn 〈rn〉

(1− σnτn

)χn. (2.47)

The application of these corrections in the case of insulators leads to a muchbetter agreement with the experimental observation of the CEF parameter. How-ever in the case of metals the conduction electrons further screen the CEF potential.In the Thomas-Fermi theory of screening (strictly valid in the free-electron approx-imation only) this effect is implemented by substituting the Coulomb potential bya screened Coulomb potential (i.e. the Yukawa potential) with a screening factor k0proportional to the square root of the density of states at the Fermi level D(EF )[31, 32]. Even with these extensions the PCM generally fails to predict the parame-ters quantitatively for metallic compounds. However it has been proven to be helpfulin accounting for general tendencies and dependencies of the CEF parameters upona change in chemical structure or upon the substitution of ions [33]. For metals withstrong screening it is valid to consider the electric potential caused by the nearestneighbors of the rare-earth only. Within this limit the effect of the conduction elec-trons can be discussed equally well by either introducing a screening parameter k0or by reducing the effective charge of the ligand and thus decreasing the reducedCEF parameters amn . E.g. in cubic symmetry with lattice constant a one finds [34]

ãn(k0) = an(0)

(1− k

20a2

ηn

); η4 = 14, η6 = 22 (2.48)

with an(0) the unscreened reduced CEF parameter. In considering only the nearestneighbors of one kind analytical forms of Bmn can be found for the cubic [6] andhexagonal [35] case. As Bmn represents the n’th moment of the CEF potential onegenerally finds

Bmn ∝ d−(n+1) (2.49)with d the interatomic distance between the 4f - and the ligand charges. It becomesobvious from (2.49) that the estimation of the Bm2 parameters is least reliable as thisparameter probes the charge distribution over a considerably larger region than Bm4and Bm6 .

In the context of barocaloric cooling the extended point-charge model has beenfound to be useful with respect to estimating trends in the change of the Bmn causedby a pressure-induced change in geometry (i.e. distortion or structural phase transi-tion): in the first step based on the initial geometry, microscopic parameters, e.g. qi,k0, σ and τ , are found to parametrize the observed CEF splitting at p = 0 . Thenin the second step these parameters are fixed and the CEF is recalculated using thegeometric parameters Ri for p > 0. It remains to be noted that this procedure doesnot account for any change in the electronic properties of the system (e.g. screeningof the charges), which however is also expected to occur under pressure (sections2.4.3 and 5.2).

Bilinear exchange and interaction with magnetic fields

So far interactions of the rare-earth ion with an external magnetic field or withthe molecular field in case of spontaneous magnetization have not been considered.

16

These interactions lead to further splitting of the energy levels due to the Zeemaneffect and result in a decrease of the magnetic entropy (2.42). Assuming isotropicbilinear exchange and spontaneous magnetic order of the rare-earth sublattice only,the magnetic interaction can be written within the mean-field approximation [36, 37]:

Ĥmag = −gJµB(Hex +Hmf ) · Ĵ+O(M2) (2.50)

where gJ stands for the Landé factor of the respective rare-earth ion and Hex andHmf denote an external magnetic field and the molecular field, respectively.

12 Thelatter must obey the self-consistency equation

Hmf = λM = gĴµBλ〈J〉 (2.51)

with λ the mean-field parameter and M the magnetic moment of the rare-earth ion.The mean-field parameter is related to the magnetic susceptibility χ of the systemvia

χ =χ0

1− λχ0, (2.52)

where χ0(T ) = χ0(T,Bmn ) denotes the single-ion susceptibility defined by the CEF

([36, 37] and section 3.2.2). At the ordering temperature T ∗, the magnetic suscep-tibility χ diverges, so that λ must fulfill

λ = χ−10 (T∗, Bmn ). (2.53)

Figure 2.4 illustrates the Zeeman splitting resulting from diagonalization of Ĥ =ĤCEF + Ĥmag in function of the effective molecular field Heff = Hex +Hmf actingon the Ce3+ ion in CeSb together with the magnetic entropy SM . Apparently,Zeeman splitting can lead to a drastic reduction in SM , especially for systems withspontaneous magnetization as the associated molecular fields are often found to bebigger than conventionally applicable by external magnetic fields.

Biquadratic exchange

In analogy to the two-ion bilinear exchange discussed in the previous section, higher-degree pair interaction, i.e. interactions in higher powers of the spins

J(n)ij (Si · Sj)n , n ≥ 2 (2.54)

may also be relevant in some systems.13 In rare-earth compounds quadrupolar terms(n = 2) may origin from indirect Coulomb interactions and exchange interactionswhere conduction electrons with d-character are dominant or from the couplingbetween the quadrupoles of the 4f -shell and the lattice [36]. The latter case is asso-ciated with a magnetoelastic interaction and the corresponding macroscopic latticedistortions are described by a cooperative Jahn-Teller distortion. The incorporationof quadrupolar exchange can have a large impact on the thermodynamics of thesystem. In addition to spontaneous magnetization, i.e. an ordering of the dipolar

12The non-operator term O(M2) in (2.50) is often omitted as it only causes a shift in absoluteenergy. However it becomes important, if different magnetic models are compared to one another.13For convenience the general spin operator is denoted by S whereas J = L ± S is used for the

combined total angular moment.

17

-4

-3

-2

-1

0

1

2

3

4

CE

F le

vels

Ei [

me

V]

150100500

magn. Field H [kOe]

0.70

0.65

0.60

0.55

0.50

0.45

0.40

En

trop

y S [R

]

150100500

magn. Field H [kOe]

Hm

f(T=

5K)

Hm

f(T=

5K)

T=5K

Figure 2.4: Left: Zeeman splitting in CeSb (J = 5/2) for an effective magneticfield along the cubic axis. Right: magnetic entropy at T = 5 K in function of themagnetic field. The molecular field Hmf for this compound corresponds to about160 kOe = 16 T (TN = 16.1 K). The upturn in Smag above 170 kOe results fromthe convergence of the second with the third excited energy level.

moment, an ordering of the quadrupolar moment also evolves. The different mul-tipole moments are found to affect each another resulting in a kinematic couplingbetween the two order parameters. Levy [38] has illustrated the effect of biquadraticexchange on a simple spin-one system considering the Hamiltonian

Ĥ = −J∑

〈i,j〉

(Ŝi · Ŝj + α(Ŝi · Ŝj)2) (2.55)

with α denoting the ratio between biquadratic and bilinear exchange and takingonly nearest neighbor interactions into account. The phase diagram resulting froma mean-field approximation with J > 0, i.e. ferromagnetic alignment of the spins, isshown in Figure 2.5. Regions with both quadrupolar and dipolar moment and regionswith only quadrupolar moment are found. The most striking feature concerns thepresence of first order transitions for α > 2/3. Note that within the mean-fieldapproximation first order transitions are absent in systems with bilinear exchangeonly.

Biquadratic exchange is also found to trigger first order transitions in more com-plicated systems. Here the CEF interaction often plays a crucial role in silent featuresof the coupling between dipolar and quadrupolar moment. The operator equivalencemethod used by Stevens for the description of the CEF may be used for biquadraticexchange too. The resulting mean-field Hamiltonian for isotropic biquadratic ex-change is then given by [36, 38]

ĤQ = −λQ(〈Ô02〉Ô02 + 3〈Ô22〉Ô22

)(2.56)

18

= 0 = 0

¹ 0 ¹ 0

¹ 0 = 0

¹0¹0

a

kT/J

2/3 1

Figure 2.5: Phase diagram for a spin-one system described by (2.55). Thin andthick lines denote continuous and first order transitions. 〈M〉 and 〈Q〉 stand for thedipolar and quadrupolar moment, respectively (after Levy [38]).

with λQ the quadrupolar mean-field parameter and with the Stevens operators

Ô02 = 3(Ĵz)2 − Ĵ(Ĵ + 1), (2.57)

Ô22 =1

2

[(Ĵ+)2 + (Ĵ−)2

]. (2.58)

2.5 Implementations of the barocaloric effect

Effects of pressure on the different parts of the total entropy have been discussedin the previous sections. The effect on the lattice entropy is the most familiar one.However one may not want to call it a barocaloric effect and instead reserve the termbarocaloric in the context with a pressure-induced change of the magnetic entropy.

In this sense the caloric effect caused by a pressure-induced change of the elec-tronic entropy must not be called a BCE too. However, as the change of the Som-merfeld parameter γ (2.36) often involves a change in the magnetic entropy (e.g. bya pressure-induced enhancement or suppression of spin-fluctuations) the effect maywell be called barocaloric.

Concerning the magnetic entropy and specifically the CEF entropy in the caseof rare-earth compounds, it turns out that the very same is influenced by a diversityof factors introduced in the previous sections. These include

• the specific type of rare-earth ion,

• the point-symmetry at the rare-earth site,

• the charges of the ligands and the rare-earth ion14,

• screening of the charges and 4f -hybridization due to conduction electrons,14In most cases the rare-earth ion is trivalent.

19

• Zeeman splitting due to an internal or an external magnetic field.

On most of these factors pressure has a direct influence. The corresponding pres-sure effects open various mechanisms of how the BCE may be implemented. Inwhat follows these mechanisms are summarized. Some of them have been observedexperimentally and references to the corresponding sections of chapter 4 are given.

point-symmetry of the rare-earth site The point-symmetry of the rare-earthion site may be directly changed by a pressure-induced structural phase tran-sition or a pressure-induced distortion of the actual structure. This type ofBCE may be called structurally driven BCE. In general the high-symmetricalstate shows higher magnetic entropy than the low-symmetrical state. Rep-resentatives of this type of BCE are the rare-earth nickelate Pr1−xLaxNiO3(chapter 4), RAlxGa2−x (section 5.1) and to some respect HoAs (section 4.3)and Ce3Pd20Ge6 (section 4.5).

15

Zeeman splitting Large internal magnetic fields may be provided by the molecu-lar field in the magnetically ordered state. The change from the paramagnetic(high entropy) to the magnetically ordered state (low entropy) may be ac-complished by a pressure-induced magnetic phase transition. Many rare-earthcompounds show pronounced shifts of the ordering temperature by pressure.Magnetically critical systems may be triggered between the magnetic and non-magnetic state with moderate pressure. The effect is referred to as magneti-cally driven BCE, representatives are the BCE in CeSb (section 4.1) and HoAs(section 4.3).

screening and hybridization by conduction electrons Pressure is known toaffect the degree of 4f -conduction electron hybridization in many of the heavy-fermion systems. The gain in electronic entropy (see section 2.4.3) is therebycoupled to a loss in magnetic entropy as the CEF gets smeared out (see sec-tion 4.5 on Ce3Pd20Ge6 for the resonance level model).

type of rare-earth ion Although the rare-earth ion type cannot be physically re-placed by the application of pressure, a pressure-induced valence transitionmay well result in a similar effect. E.g. pressure may drive a divalent Eu2+

ion, which behaves like a trivalent Gd3+ ion (J = 7/2), to a trivalent Eu3+

(J = 0 → SM ≡ 0). The associated change in the CEF and hence magneticentropy can be dramatic. Thereby the difference in ionic radius between corre-sponding valence states makes valence transitions prominent for large pressuredependences. The associated effect may be called valence driven BCE.

In cases involving phase transitions it becomes important to further distinguish twocategories of systems.

1. Systems non-critical in p, where the respective phase transition takes place pri-marily in function of temperature and where the phase transition temperatureT ∗ is only found to be shifted by pressure. For every pressure, correspond-ing temperatures for either phase can be found. In these systems a change ofpressure from p1 to p2 only results in a pressure-induced phase transition for

15R=rare earth

20

T

S∆Tad

p2

p1

p

T

high-Sphase

low-Sphase

p2p1

T

S∆Tad

p2

p1

p

T

high-Sphase

low-Sphase

p2p1

T*(p)

criticalin p

∆Tad

p1

p2

T

S

p

T

∆T

(p)

high-Sphase

low-Sphase

p2p1

T*(p)

non-criticalin p

∆T(p)

Figure 2.6: Schematics for systems non-critical in p and critical in p. Pressure onlyshifts the transitions temperature T ∗ in non-critical systems, whereas in criticalsystems one of the two phases completely vanishes. For non-critical systems themagnitude of dT ∗/dp represents another limiting factor for the intensive BCE ∆Tad.

temperatures within the interval T ∗(p1) ≤ T ≤ T ∗(p2). For many systems atlow temperatures it is this constraint, and not the size of the extensive BCE∆S nor the slope in S(T ), which limits the intensive BCE ∆Tad (compare con-clusions from (2.11) and Fig. 4.12). The corresponding (p, T )-phase diagramand S(T )-curves are illustrated in the upper part of Figure 2.6.

2. Systems critical in p. In this case only one phase is found for certain pressuresirrelevant of temperature. Here the pressure-induced transition is not limitedto a certain temperature interval (or limited to one side only) and ∆S and theslope of S(T ) are the dominating factors limiting ∆Tad (2.11) (Fig. 2.6, lowerpart).

Hence systems critical in p are preferred for a large BCE. Unfortunately criticalsystems are not nature’s favorites and are a lot less frequent than non-critical ones.

In the case of non-critical systems, the pressure dependence of the transition

21

temperature dT ∗/dp becomes a crucial factor. The equation of Clausius-Clapeyronfor first order transitions

dT ∗

dp=

∆S

∆V(2.59)

and Ehrenfest’s equations for continuous transitions

dT ∗

dp=V T∆α

∆C,

dT ∗

dp=

∆α

∆κ(2.60)

relate the pressure dependence of the transition temperature with discontinuitiesin macroscopic thermodynamic quantities (κ stands for the isothermal compress-ibility). In principle the two equations allow to calculate dT ∗/dp given that allnecessary parameters are known accurately enough. At this point however, it isimportant to apply (2.59), (2.60) in the proper thermodynamic context and to re-member the remarks already addressed in section 2.3. Both equations are correct inthe paramagnetic state only, where any effect from a change in the magnetism of thesystem can be ignored. The Clausius-Clapeyron16 and Ehrenfest’s equations17 arebased on the comparison of differences in the Gibbs free energy G and their slopes,respectively at constant magnetization M . However G is a thermodynamic statefunction and thus generally depends on M too:

dG = −SdT + V dp+HdM + . . . (2.61)

Strictly, both equations can only be applied for the structurally driven BCE andthe valence driven BCE and only if the corresponding phase transitions take placewithin the paramagnetic state (dM = 0, Heff = 0).

The diversity in mechanisms leading to a BCE may be regarded as a specificfeature of this kind of caloric effect that makes the BCE worthwhile to be studiednot only from an applicative but also from a fundamental point of view.

All adiabatic cooling techniques and underlying mechanisms discussed in thischapter are schematically summarized in Figure 2.7.

16Clausius-Clapeyron: dG1 = −S1dT + V1dp ≡ dG2 = −S2dT + V2dp → dT/dp17Ehrenfest: ∂

∂TdG1 ≡

∂∂TdG2 → dT/dp,

∂∂pdG1 ≡

∂∂pdG2 → dT/dp

22

adia

batic

cool

ing

mag

netic

fiel

d

pres

sure

SM

chan

ge o

f spi

nal

igne

men

tad

iaba

tic d

emag

netiz

atio

nin

par

amag

netic

sal

ts

SM

+SL

sam

e as

abo

ve +

cou

plin

g of

mag

netic

and

chem

ical

sub

latti

cegi

ant M

CE

SL

elas

tic h

eatin

g/co

olin

gsh

ift o

f pho

non

spec

trum

SE

enha

ncem

ent/s

uppr

essi

onof

spi

n-flu

ctua

tions

BC

E n

ot b

ased

on

SM

SM

+SE

chan

ge o

f 4f-

cond

.e-

hybr

idiz

atio

nin

hea

vy-f

erm

ion

syst

ems

BC

E in

hea

vy-f

erm

ion

syst

ems

SM

SM

shift

/enh

ance

men

t/sup

pres

sion

of m

agne

tic tr

ansi

tion

MC

E

shift

/enh

ance

men

t/sup

pres

sion

of s

truc

tura

l tra

nsiti

onst

ruct

ural

ly d

riven

BC

E

shift

/enh

ance

men

t/sup

pres

sion

of m

agne

tic tr

ansi

tion

mag

netic

ally

driv

en B

CE

shift

/enh

ance

men

t/sup

pres

sion

of v

alen

ce tr

ansi

tion

vale

nce

driv

en B

CE

ther

mod

ynam

icpr

inci

ple

cont

rol v

aria

ble

entr

opy

affe

cted

mec

hani

smcl

assi

ficat

ion

Figure 2.7: Classification of adiabatic cooling techniques discussed in this chapter.

23

Chapter 3

Experimental Techniques

This chapter discusses the experimental techniques, setups and data analysis toolsused throughout this work. Emphasis is given to issues which may not be common tothe reader, these include the setups for the BCE experiments and the high-pressuretechniques applied to neutron scattering methods and to macroscopic measurements.For an introduction to more general aspects of neutron scattering methods or macro-scopic measurements the reader is referred to the extensive literature available.

The reader may feel free to continue reading chapter 4 about the experimentalresults while using this chapter as a convenient reference.

3.1 Setups for the barocaloric measurements

Two different setups for the measurement of the BCE on single crystals (uniaxialpressure) and on powder samples (hydrostatic pressure) have been developed. Thetwo setups do not only differ in mechanical aspects but also in the thermometryapplied.

3.1.1 Thermometry

Aside a carefully designed mechanical apparatus to apply pressure, a fast and accu-rate thermometry to observe the BCE is of utmost importance. The demands onthe thermometry can be summarized as follows:

• The sensor must be as small as possible to account for the typical sampledimensions and to allow for a prompt response.

• Hydrostatic pressure must have as little as possible influence on the sensor.

• The sensitivity of the sensor must be large enough to guarantee good relativeaccuracy.

• To account for the dynamic nature of the measurement, the temperature mustbe tracked at high repetition rates.

Among all the different temperature sensors, thermocouples have some superiorfeatures. They are an excellent choice to meet the two first requirements. Ther-mocouples are known to be only a little dependent on external hydrostatic pressure

25

Samplein Cryostat

ReferenceCu block

in liquid nitogen

AmplifierMultimeterComputer

Metal #1Metal #2

∆T

U

Samplein Cryostat

AmplifierMultimeterComputer

∆T

UP

P

Figure 3.1: The two basic setups for the thermometry used for the measurement ofthe BCE. Left: direct circuit suitable for relative T measurements; thermal instabil-ities at the voltage measuring point P result in noise. Right: reference circuit allowsabsolute T measurements with no noise from thermal instabilities at P .

[39], whilst sensors based on changes in the electrical resistivity cannot be used forhydrostatic pressure experiments at all.1 For a comprehensive introduction to thetheory and the applications of thermocouples see e.g. [40].

Thermocouples consists of two wires of two different metals connected to a junc-tion at one end. The latter may be spot-welded to very small dimensions and henceallows fast response times. Thanks to the Seebeck effect a voltage in function ofthe temperature difference between the junction and the measuring point P occurs(Fig. 3.1, left). The sensitivity S(T ) ≡ ∆V/∆T , known as the Seebeck coefficient, isa polynomial function in T . For most types of thermocouples corresponding tableswith Seebeck coefficients can be found in literature [39]. It is important to statethat with thermocouples principally only temperature differences can be measured.Hence in the direct circuit (Fig. 3.1, left) thermal instabilities at the voltage mea-suring point P result in noise and hinder an accurate measurement even in the caseof a relative temperature measurement. Hence on the right side of Figure 3.1 anasymmetric reference junction is added to the primary thermocouple (reference cir-cuit). Intermediate temperatures along the thermocouple do not matter. Thereforethe resulting voltage is a direct (absolute) measure for the temperature differencebetween the sample and the reference independent of temperature variations at P .

An inherent disadvantage of thermocouples is their relative small and hard tomeasure signals (typically a few 10 µV/K) and their loss in sensitivity with decreasingtemperature. Fortunately, a wide variety of different thermocouples for differentapplications are known, with some of them suitable at the lowest temperaturestoo [39]. Among the most commonly used thermocouples are the Chromel-Alumelcouples2 with a useful working range between ca. 20 K and 1400 K. Although thistype can hardly be used for temperatures below 20 K, two-core thermocouples with aprotective ceramic tube are available from stock. Mechanically these thermocouples

1The inverse diameter of the wire is directly proportional to the resistivity and subject to changewith pressure.

2also referred to as type-K thermocouples

26

easily sustain high hydrostatic pressure (even in non-liquid environments). Thesmallest available standard diameter is 0.5 mm.

At low temperatures Au/Fe-Chromel thermocouples3 are one of the best choices.They allow the measurement down into the sub-K ranges. Unfortunately thesethermocouples are not manufactured with protective tubes and hence they can beused for the uniaxial BCE setup only.

For all the measurements presented in this work the voltage signal of the ther-mocouple(s) had to be amplified before being fed into a commercial HP multimeter.A linear, analog amplifier with a 37x and 9700x gain factor has been used for thispurpose. The data of the multimeter is either instantly logged by a computer ata rate of 20 meas/sec. or firstly stored in the internal memory of the multimeterat rates of up to 300 meas/sec. and later read out by the computer. The voltageswere then converted to temperatures by means of calibration against the cryostattemperature sensors (for the reference circuit) or according to published Seebecktables (for the direct circuit).

3.1.2 Force transmitting system

All BCE measurements have been carried out in a standard ILL He-cryostatequipped with a specially designed force transmitting system. This system con-sists of an adapted cryostat insert shown in Figure 3.2. Technical details are givenin [41]. Pressure is produced with a hydraulic (oil) hand-pump and converted touniaxial force by a hydraulic cylinder mounted on top of the cryostat insert. Theuniaxial force is thence transmitted via a hollow plunger onto a piston and the sam-ple (uniaxial setup) or onto the piston of a pressure cell (hydrostatic setup). Thegeometry and the dimensions of the plunger are chosen in order to minimize effectsof thermal expansion and misalignment of the acting force. Sealing rings at the topof the plunger ensure a proper sealing of the sample chamber while adding only alittle extra friction to the plunger.

The hydrostatic pressure ph of the oil is measured with a manometer. Assuminglittle frictional losses of the system the force F of the plunger is given by the effectivearea Ah of the hydraulic cylinder via

F = Ah · ph. (3.1)

The system is designed to transmit forces up to about Fmax = 10 kN. With Ah =640 mm2 this corresponds to a hydrostatic pressure in the hand-pump of aboutpmaxh = 150 bar.

4 The friction force of the plunger system accounts to less than 20 Nand may hence be neglected.

Originally the system was designed for neutron diffraction studies under uniax-ial pressure. Thus the hydrostatic pressure may be kept to a less than 10 % lossover a period of 24 h thanks to two extra valves positioned between the manometerand the hydraulic cylinder and between the manometer and the hand pump, respec-tively. A further bypass-valve installed in the hand pump allows for the fast releaseof the oil pressure. The force of the plunger was found to be released within lessthan 0.01 s as determined by measurements of the BCE and the elastic heating of

3Au-(0.02-0.03)at.Fe vs Ni-10Cr41 bar ≡ 105 Pa; 1 GPa ≡ 100 kg/mm2.

27

Figure 3.2: Schematic drawing of the force transmitting system with hydraulic hand-pump A, two extra valves B, manometer C, flexible hydraulic tube D, hydrauliccylinder E, plunger F , force retaining tube G, sample chamber H, bottom I, pistonsK and sample L (from [41]).

samples at room temperature with the two extra valves fully opened. An additionalCernox thermometer is attached at the very bottom of the device in order to com-pare the temperature of the sample against the temperature of the heat exchanger.Within the experimental accuracy no difference between the two sensors was ob-served. Hence an effective heat flow from the end of the plunger through the samplecan be ruled out for a constant heat exchanger temperature of the cryostat.

3.1.3 Uniaxial setup

The setup for the measurement of the BCE under uniaxial pressure is describedin this section. The samples studied under uniaxial pressure include: CeSb,Ce0.85(La,Y)0.15Sb, HoAs, YbAs and Ce3Pd20Ge6. Figure 3.3 depict this setup.The single crystal is placed on a socket. The uniaxial force of the force transmittingsystem is directed via an additional piston onto the sample. Piston and socket aremade of zirconia. Polycrystalline stabilized zirconia (ZrO2) is characterized by verysmall thermal conductivity. The heat flow from the sample to the piston and to thesocket is hence minimized insuring quasi-adiabatic conditions for the measurementof the intensive BCE. The effective uniaxial pressure ps acting on the sample is given

28

Piston

Socket

Sample The

rmoc

oupleSpacer

5 mm

Figure 3.3: Setup for the measurement of the BCE under uniaxial pressure.

by (3.1)

ps = F/As = phAhAs

(3.2)

with As being the sample area. Measuring single crystal samples of a typical dimen-sion such as 3 ·3 ·3 mm3, the maximum applicable pressure is generally restricted bythe mechanical limits of the single crystal and not by the force transmitting system.

Maximal pressures can only be applied along symmetry directions of a crystal.Hence only crystals with nice, clean cleavage planes can be used. Smallest defectsin the structure of the crystal, e.g. cracks, steps, twins, etc., drastically reduce themechanical limit of the crystal and lead to early sample breakage. Shear stressesoriginating from a misalignment of the acting force must also be avoided. In order tofurther minimize the risk of breaking the sample due to microscopic imperfectionsof the sample surface, dust particles, etc., a thin Teflon disk (0.1 mm) is placedbetween the sample and the socket and between the sample and the piston. Inthe course of this work thin foil of tin (0.02 mm) has been proven to work equallywell. Experiments without the use of these soft spacers have shown that these areessential and should also be used for neutron diffraction experiments despite theircontribution to the background intensity.

The thermometry for the uniaxial setup turns out to be a lot simpler than for thehydrostatic setup. The thermocouple is glued directly onto one of the free surfaces ofthe crystal. GE Insulating Varnish from General Electrics proved to be best suitedfor this purpose. This glue shows good thermal conductivity and keeps being stickydown to the lowest temperatures. For all experiments carried out with this setup thethermocouple voltage was measured by using the reference circuit (section 3.1.1).

3.1.4 Hydrostatic setup

The hydrostatic setup was used for the BCE measurements on polycrystalline bulksamples of CeSb and EuNi2(Si0.15Ge0.85)2. The basic ideas for the design of thissetup are borrowed from experience with hydrostatic clamp pressure cells as used

29

Teflon cap

Cu/Be ring

Pb foil

Pressuremedium

Chr

omel

-Alu

mel

Sample

Pressure matrix

Pressure cell

Screw

Piston

Coa

xial

cab

le (

Cu)

∆T

4.8 mm

16 mm

20 m

m

Figure 3.4: Setup for the measurement of the BCE under hydrostatic pressure.

for neutron scattering measurements under pressure described later in section 3.2.1.However the need to change pressure in situ at low temperatures requires somespecial measures.

The hydrostatic setup is depicted in Figure 3.4. The sample is placed in apressure matrix, which is bottom loaded into the pressure cell and fixed with ascrew. All parts are made of stainless steel as this setup is primarily designed foruse at low temperatures where the thermal conduction of stainless steel becomesfairly poor. The sample is placed within a solid pressure transmitting medium andhas no contact with the walls of the pressure matrix. A Teflon cap and a copper-beryllium (Cu/Be) ring ensure the sealing of the pressure cell and stop the pistongetting blocked by the pressure transmitting medium. Indeed the piston was neverfound to be blocked after the experiments and could be removed by bare hands. Inorder to minimize friction between the (solid) pressure transmitting medium and thewalls of the pressure matrix, a thin foil of Pb (0.02 mm) is placed between the two.The temperature is measured inside the sample. For this purpose a small hole of0.52 mm is drilled into the sample. A Chromel-Alumel thermocouple (0.5 mm diam)with a protective tube is used to track the temperature. Good thermal contact withthe sample is ensured by some Apiezon grease put between the sensor and the sample.The thermocouple is fed out via holes through the pressure matrix (0.52 mm diam)and the pressure cell (0.6 mm diam) and fixed with Araldit glue onto the pressurematrix. Inspection of the thermocouple after the experiment has found that it wasremained in the correct position at all times. The temperature was measured usingthe direct circuit for the thermometry with the voltage measuring point at the outsideof the massive pressure cell (section 3.1.1).

30

0.12

0.10

0.08

0.06

0.04

0.02

0.00

-∆T

ela

[K]

120100806040200

T [K]

Figure 3.5: Signal of the elastic cooling of lead after a pressure release of p =−0.30(5) GPa. Below the ductile-brittle transition at about 60 K, the signal vanishesas the lead cannot transmit the pressure anymore.

The reason a solid pressure transmitting medium was used was due to priorexperience that at high-pressures almost all known liquid media are already frozenat moderate temperatures. In fact Fluorinert, which is one of the most often usedpressure transmitting media5, is known to become solid at room temperature forp & 1 GPa and freezes at around 150 K for p = 0 into an amorphous glass. Hencethe use of a ’liquid’ pressure transmitting medium would bring in many additionalunknowns, which may all affect the BCE signal of the sample. On the other hand,the p − T phase diagrams of solid pressure media are generally known and presentless ’surprises’. Solids are comparatively harder than liquids and correspondinglyresult in less elastic heating/cooling superimposed on the BCE of the sample. Thepreparation and sealing of the pressure cell can be done in a much more controlledmanner. On the other hand they contribute to less hydrostatic conditions for thesample and therefore a considerably broad pressure distribution cannot be ruled out.At low temperature a solid medium also results in more friction with the sample andthe walls, which is superimposed on the BCE from the sample.

Lead constitutes a solid pressure transmitting medium which already has beenused successfully for neutron scattering experiments under hydrostatic pressure [42].Lead is comparatively soft and thus transmits pressure well. In these experimentspressure was applied at room temperature and clamped before the pressure cell wascooled to base temperature. To the knowledge of the author, lead has not beenused as a pressure medium in conjunction with an in situ change of the pressureat low temperatures before. Hence preliminary experiments have been carried outon a pressure matrix filled only with lead. The elastic cooling of lead was trackedin function of temperature. As can be seen from Figure 3.5 the elastic coolingvanishes at around 60 K which was later found to coincide with the ductile-brittletransition of lead. Like most other metals, lead becomes mechanical brittle if thetemperature is low enough and thereafter cannot transmit the pressure anymore.

5Fluorinert contains no hydrogen and hence is especially well suited for neutron scattering mea-surements (see section 3.2).

31

For this reason a pressure medium other than a metal must be used. An analogoustest experiment on sodium chloride has shown the elastic cooling to decrease aboutlinearly with decreasing temperature as expected (see section 2.4.2) and hence hasconfirmed the applicability of in situ pressure changes at cryogenic temperatures.After both experiments on CeSb and EuNi2(Si0.15Ge0.85)2, inspection of the pressurematrix has shown that the sodium chloride has become non-porous and has flownnicely around the sample.

3.1.5 Relaxation models

In all the measurements the observed intensive BCE ∆T is followed by a relaxationback to the initial temperature due to heat flow between the sample and the envi-ronment, i.e., due to the non-adiabaticity of the system. In experiments performedat high measurement rates it is further observed that the temperature does notdrop down instantaneously but continuously. The analy

research collection · 2020. 7. 1. · systeme beinhalten ndal3, ralxga1¡x (r=seltenerdion) and...

Documents