Magnetism and lattice dynamics under high pressure studied by nuclear resonant scattering of synchrotron radiation / von Rainer Lübbers [original]

Magnetism and Lattice Dynamics

under High Pressure Studied by

Nuclear Resonant Scattering

of Synchrotron Radiation

Dem Fachbereich Physik

der Universität - GH - Paderborn

zur Erlangung des Grades eines Doktors der

Naturwissenschaften (Dr. rer. nat.)

vorgelegte

Dissertation

von

Rainer Lübbers

Paderborn, im März 2000

Gutachter: Prof. Dr. G. Wortmann

Prof. Dr. W.B. Holzapfel

Tag der Einreichung: 9.3.2000

Tag der mündlichen Prüfung: 16.05.2000

Contents

1 Introduction 7

2 Synchrotron Radiation 9

2.1 Basic Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Insertion Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3 Properties of Synchrotron Radiation . . . . . . . . . . . . . . . . . . 13

3 Nuclear Resonant Scattering 15

3.1 Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.1.1 The 57Fe Nucleus . . . . . . . . . . . . . . . . . . . . . . . 15

3.1.2 Mössbauer Spectroscopy . . . . . . . . . . . . . . . . . . . . 16

3.1.3 Excitation with Conventional Sources . . . . . . . . . . . . . 17

3.2 Nuclear Forward Scattering . . . . . . . . . . . . . . . . . . . . . . 19

3.2.1 Mathematical Treatment . . . . . . . . . . . . . . . . . . . . 20

3.2.2 Speed-up of the Exponential Decay . . . . . . . . . . . . . . 22

3.2.3 Quantum Beats . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.2.4 Coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.2.6 Measurement of Isomer Shifts . . . . . . . . . . . . . . . . . 30

3.3 Nuclear Inelastic Scattering . . . . . . . . . . . . . . . . . . . . . . 31

3.3.1 Basic Features . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.3.2 Data Evaluation . . . . . . . . . . . . . . . . . . . . . . . . 33

4 Experimental Details 39

4.1 Nuclear Resonance Beamline at ESRF . . . . . . . . . . . . . . . . . 39

4.1.1 High-Resolution Monochromator . . . . . . . . . . . . . . . 41

4.1.2 Fast Detectors . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.1.3 Focusing Elements . . . . . . . . . . . . . . . . . . . . . . . 42

4.2 High-Pressure Technique for NFS . . . . . . . . . . . . . . . . . . . 44

4.3 High-Pressure Technique for NIS . . . . . . . . . . . . . . . . . . . 46

5 NFS in RFe2Laves Phases 51

5.1 Structure and Magnetism of RFe2Laves Phases . . . . . . . . . . . . 51

5.1.1 Crystal Structure . . . . . . . . . . . . . . . . . . . . . . . . 51

5.1.2 Magnetism in R-Fe Compounds . . . . . . . . . . . . . . . . 53

5.1.3 Magnetic Phase Diagram . . . . . . . . . . . . . . . . . . . 55

5.2 Special Features of NFS Spectra . . . . . . . . . . . . . . . . . . . . 60

5.2.1 Texture Effects . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.2.2 Complex Hyperfine Interactions / External Field . . . . . . . 61

5.2.3 Thickness Distributions . . . . . . . . . . . . . . . . . . . . 63

5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.3.1 YFe2.............................. 65

5.3.2 GdFe2.............................. 69

5.3.3 TiFe2.............................. 73

5.3.4 ScFe2.............................. 73

5.3.5 Measurement of Isomer Shifts . . . . . . . . . . . . . . . . . 77

5.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.4.1 Volume Dependence of Magnetic Ordering Temperatures Tm.82

5.4.2 Intersublattice Coupling Under Pressure . . . . . . . . . . . . 86

6 Lattice Dynamics in Iron Under Pressure 89

6.1 Iron Under Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . 89

6.2 Lattice Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

6.2.1 Debye Model . . . . . . . . . . . . . . . . . . . . . . . . . . 92

6.2.2 Grüneisen Parameter . . . . . . . . . . . . . . . . . . . . . . 93

6.2.3 Elastic Coefficients and Aggregate Velocities . . . . . . . . . 94

6.3 Nuclear Inelastic Scattering in Iron . . . . . . . . . . . . . . . . . . . 97

6.3.1 NIS Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

6.3.2 Extracted Phonon DOS . . . . . . . . . . . . . . . . . . . . . 99

6.3.3 Derived Properties . . . . . . . . . . . . . . . . . . . . . . . 100

6.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

6.4.1 Grüneisen Parameter . . . . . . . . . . . . . . . . . . . . . . 110

6.4.2 Aggregate Velocities and Shear Modulus . . . . . . . . . . . 111

6.4.3 The α-εPhase Transition . . . . . . . . . . . . . . . . . . . . 113

7 Summary 115

Chapter 1

Introduction

High-pressure experiments with conventional Mössbauer Spectroscopy (MS) are well

established since the early days of this method [PBD61, Hol75]. Up to now, high-

pressure MS delivers very useful information in many scientific fields such as solid-

state physics, geophysics or mineralogy [TP90, MH98]. The study of magnetism is

one of the most prominent subjects for high-pressure MS. Here, both the magnetic

ordering temperatures and local moments can be studied. When diamond-anvil cells

(DACs) are used, MS is possible at pressures well above 100 GPa (corresponding to

1 Mbar) [PTJ97], allowing a huge variation of the lattice parameters.

The use of synchrotron radiation (SR) for Mössbauer Spectroscopy was proposed

in 1974 [Rub74], the first successful experiments were performed a decade later by

the group of E. Gerdau using nuclear Bragg reflection from an YIG single crystal

[GRW85]. Polycrystalline samples could be investigated with the method of Nuclear

Forward Scattering (NFS) introduced in 1991 by Hastings et al. [HSB91, BSH92]. It

became immediately evidentthat the NFS method, which is the time analog of the clas-

sical Mössbauer effect, is especially suited for high-pressure experiments with DACs

by taking advantage of the laser-like collimation of SR.

The extremely fruitful symbiosis between high-pressure research and synchrotron

radiation is documented by an enormous progress of x-ray diffraction studies in the

Mbar range [LLH96, LTW99]. Other SR methods benefitted also very much from

this development, enabling high-pressure studies, for instance, with x-ray absorption

[BFM98], inelastic x-ray scattering [KMM97] and x-ray fluorescence [BSS99].

The subject of this thesis are high-pressure studies with two new methods applying

Nuclear Resonant Scattering of SR. These methods are:

(a) Nuclear Forward Scattering of SR for the study of hyperfine interactions, based on

recoilless absorption and emission of gamma quanta similar to conventional MS, and

(b) Nuclear Inelastic Scattering (NIS) of SR, which uses the energy transfer connected

with the inelastic nuclear absorption of gamma quanta. This new method, introduced

in 1995 by M. Seto et al. [SYK95] and W. Sturhahn et al. [STA95], allows a direct

experimental determination of the phonon density of states (DOS) in a solid.

In both cases specially modified diamond-anvil cells are used. The NFS studies are

concerned with the magnetism in RFe2Laves phases, the NIS studies with the phonon

DOS in the α- and ε-phase of metallic iron.

To (a): Magnetic intermetallic compounds of rare earth elements (R) with iron-group

transition metals have been intensively studied since decades both for a principal

understanding of their magnetic properties as well as for technological application.

In this thesis the magnetic properties of RFe2Laves phases (R=Y, Gd, Sc) are stud-

ied with NFS at pressures up to about 100 GPa. Due to their simple crystallographic

structures, these RFe2systems are considered as model systems for magnetism in R-

Fe compounds. The application of external pressure allows for a systematic study of

the large variety of magnetic properties in these systems, which are normally caused

by the different constituents R [BCK90]. The present work is a continuation of the

first high-pressure NFS study of the α-εtransition in metallic iron, performed by H.F.

Grünsteudelinthe frameworkofhis PhDthesis atthe Universityof Paderborn[Grü97].

To (b): The inner core of the Earth is composed almost entirely of iron or iron-rich

alloys [Buk99]. This geophysical aspect is one of the many motivations to study the

phase diagram of iron with its various isomorphs in a wide pressure and temperature

range. The hcp high-pressurephase of iron(ε-Fe)is considered the most relevantphase

for the solid inner core [YAC95]. While a considerable amount of data on the pres-

sure and temperature phase diagram of iron is available [And97], there is much less

experimental information on the lattice dynamics of ε-Fe. Here we determine experi-

mentally with NIS up to 42 GPa the phonon DOS in the α-phase and, for the first time,

in the ε-phase of iron. From these data a variety of thermodynamic parameters were

derived, such as Debye temperatures, Grüneisen parameters, the vibrational contribu-

tions to entropy and specific heat. Of particular geophysical interest are the derived

sound velocities.

The thesis is organized as follows: The special properties of SR are described in

chapter 2. Chapter 3 provides an introduction to the basic features of Nuclear Res-

onant Scattering. A considerable part of the thesis work was concerned with the devel-

opment of new diamond-anvil cells and gasket materials. These experimental aspects

are outlined in chapter 4 together with the experimental set-up of the Nuclear Res-

onance Beamline at the European Synchrotron Radiation Facility (Grenoble). The

experimental results of NFS high-pressure studies of RFe2systems and the derived

magnetic properties are presented and discussed in chapter 5. The NIS study of the

phonon DOS in iron and the derived thermodynamic properties are contained in chap-

ter 6. The final chapter summarizes the results and gives an outlook to future NFS and

NIS experiments.

Chapter 2

Synchrotron Radiation

The investigation of magnetic and dynamic properties of RFe2systems and metallic

iron in this thesis was performed with Nuclear Resonant Scattering (NRS). The rapid

development of this new spectroscopy and its applicability at very high pressures is

made possible by the special properties of synchrotron radiation (SR) at third genera-

tion sources like the European Synchrotron Radiation Facility (ESRF) in Grenoble or

the Advanced Photon Source (APS) at the Argonne National Laboratory, Chicago.

The aim of this chapter is to introduce briefly the basic features of SR followed by a

short description of radiation sources. For further details the reader is referred to the

literature [Jac75, IFF92].

2.1 Basic Features

Synchrotron Radiation is electromagnetic radiation which is emitted by charged rela-

tivistic particles on a curved trajectory. It is no longer used at circular particle accel-

erators (the actual synchrotrons) but at storage rings, where the particles are stored at

constant energy E. In this case the radiated power is [Jac75]

e2c

6πε0

(

m0c2

)

R2(2.1)

with particle charge eand rest mass m0, vacuum permittivityε0and radius of curvature

Due to the factor m4

0in the denominator, it is obvious that only light particles such

as electrons or positrons are used for the generation of SR. In a magnetic field B, the

particles can be forced to follow a circular trajectory with radius

[

336 E

[

GeV

]

[

]

(2.2)

<< vc

Figure 2.1: Emissionof radiationfrom a radiallyaccelerated electronfor non-relativistic

(left) and relativistic (right) velocities [Wil96].

The energy loss during one cycle of a relativistic electron is

∆E

[

keV

46 E4

[

GeV

]

[

]

(2.3)

For the storage ring at the ESRF with E

6GeV and R

4m, one obtains 4.9 MeV

per cycle. The resulting total radiation power along the ring is 490 kW, when a typical

beam current of 100 mA is assumed.1

The radiation pattern of a radially accelerated electron is shown in figure 2.1 for ve-

locities v



cand v



c. The radial distribution for the non-relativistic case has the

well-known dipole character, whereas for v



cone has to perform a Lorentz trans-

formation. The radiation in the laboratory frame is emitted into a small cone with an

energy-dependent opening angle

m0c2

(2.4)

As a consequence, the radiation seen by a stationary observer is a short pulse originat-

ing from the passing electron. This leads to a wide distribution of the energy spectrum,

which is characterized by the critical energy [IFF92]

3¯hcγ3

22 keV

(

E[GeV]

)

[

]

(2.5)

below which half of the radiation power is emitted. The critical energy for a bending

magnet at the ESRF is Ec= 20.5 keV.

1This value represents the minimum power for re-acceleration of the electrons. With additional

Insertion Devices (see section 2.2) an actual power of about 5 MW is necessary at ESRF to keep up the

beam [ESR96].

Intensity

For a quantitative description of SR sources, the three physical parameters brightness,

flux and brilliance are defined [IFF92]. Brightness is the numberof photons per second

which are emitted in a solid angle of 1 mrad2within an energy bandwidth ∆E

1%. Integration over the vertical opening angle of the beam gives the flux. If the

brightness is normalized to the size of the electron beam (in mm2), one obtains the

brilliance which is given by

brilliance

photons/s

solid angle [mrad2]



source size [mm2]



0.1% bandwidth (2.6)

This parameter is a measure of the spatial photon density in the beam and therefore

most important for high pressure experiments.

2.2 Insertion Devices

Very high brilliance SR is generated by insertion devices, which consist of periodic

magnetic structures installed in the straight sections of a storage ring (see figure 2.2).

Theirpurpose isto forcethe particlesona sinusoidaltrajectoryandenhance theemitted

radiation without perturbing the stored electron beam. The optical properties of an

insertion device are characterized by the undulator parameter K: the ratio between the

maximum deflection angle αin the device and the opening angle Θof the SR

eλ0B0

2πmec

934



λ0

[

]



[

]

;

(2.7)

where λ0is the period length and B0the peak value of the magnetic field. With respect

to the parameter K, one can distinguish two basic types of insertion devices: wigglers

and undulators.

Wigglers

A wiggler is characterized by a large period length and a high magnetic field (K



1).

Due to the large deflection angle α, the radiation of each wiggle adds up incoherently

and the intensity from N magnetic periods is 2N times the intensity of a single bending

magnet. A wiggler can be regarded as a set of stacked bending magnets and has there-

fore similar characteristics: a broad energy distribution and an emission in a relatively

large solid angle.

Figure 2.2: Schematic sketch of an insertion device, consisting of a magnetic array

which deflects the electrons on a sinusoidal trajectory; modified after [Wil96].

Undulators

For undulators the parameter Kis



1. The angle αis small and the radiation of suc-

cessive periods adds up coherently. Interference leads to sharp peaks in the spectral

brilliance at

λn

λ0

2nγ2



γ2φ2



;

; :::

(2.8)

where φis the angle to the undulator axis. The intensity is increased by a factor (2N)2

and on the axis (φ

0) only the odd harmonics (n = 1,3,...) can be observed. The

spectral width of the peaks depends on the number of periods N:

∆λ



(2.9)

The energy position of the harmonics can be tuned by changing the gap between the

magnetic poles. A reduction of the gap increases the magnetic field and the parameter

K, and shifts the peaks to smaller energies. Table 2.1 shows the characteristics of the

twoundulators installed at the Nuclear Resonance Beamline ID18. Undulator 1 is used

for nuclear scattering experiments with the 57Fe resonance.

Undulator 1 Undulator 2

period λ02.28 cm 3.4 cm

magnetic field B00.12 T 0.16 T

undulator parameter K 0.256 0.857

energy 14.4 keV 6..8/21.5..27 keV

power (100 mA) 0.05 kW 0.28 kW

source size (h x v) 812 x 54 µm2812 x 54 µm2

divergence (h x v) 28 x 4 µrad 28 x 4 µrad

beam size @30 m (h x v) 1.8 x 0.6 mm21.8 x 0.5 mm2

Table 2.1: Characteristics of undulators at the Nuclear Resonance Beamline ID18 at

ESRF [ESR97].

2.3 Properties of Synchrotron Radiation

The important properties of SR with respect to NRS experiments are summarized be-

low:



time structure:

The energy loss of the particles in a storage ring is compensated in radio fre-

quency cavities. Since only particles with the right phase relative to the radio

frequency νrf are re-accelerated, they are grouped together in bunches. Depend-

ing on the operation mode at the ESRF storage ring (circumference l = 844 m,

νrf = 352 MHz), the time window between two adjacent bunches (i.e. two ra-

diation pulses) ranges from 2.8 µs (= l/c) with only one single bunch to 2.8 ns

(= 1/νrf) for complete filling. The nuclear scattering experiments discussed in

this thesis are mostly performed during 16-bunch-mode with a time window of

176 ns. The pulse duration is 20 ps for bending magnets and 100 ps for insertion

devices [ESR92].



high brilliance:

The intense laser-like collimated beam of undulators results in short data accu-

mulation times even in high pressure experiments, where only very small sample

amounts (about 1µg at 100 GPa) are available.



polarization:

SR is 100% linearly polarized in the plane of the storage ring. This feature

allows the excitation of certain nuclear sublevels in NRS, when the orientation

of internal hyperfine fields is well defined (e.g. for a magnetic sample in an

external magnetic field as discussed in chapter 3.2.5).



broad energy distribution:

The broad energy distribution of SR from the infrared to the hard x-ray re-

gion allows a wide range of energy tunability using appropriate undulators and

monochromators. This tunability is essential forNRS experiments on two differ-

ent energy scales. First, by variation over several keV, the resonance energies of

many different nuclei can be provided. Second, the tunability in the meV range

around a nuclear resonance has opened up the new spectroscopic branch of Nu-

clear Inelastic Scattering which enables the investigation of lattice dynamics.

Chapter 3

Nuclear Resonant Scattering

In this chapter a survey of Nuclear Resonant Scattering techniques is given, consid-

ering both elastic Nuclear Forward Scattering (NFS), the analogue of conventional

Mössbauer Spectroscopy, and Nuclear Inelastic Scattering(NIS)forthe study of lattice

dynamics. We start with a few remarks on57Fe and classical Mössbauer Spectroscopy.

3.1 Fundamentals

3.1.1 The 57Fe Nucleus

The probe for Nuclear Resonant Scattering (NRS) experiments in this thesis is the57Fe

nucleus with a natural abundance of 2%. The nuclear level scheme is shown in figure

3.1.

The nuclear level with spin I = 3/2 has an excitation energy Eγ= 14.413 keV and a

relatively long lifetime τ0of 141 ns. The transition to the ground state can proceed by

1/2-

12.7 ns

141 ns

3/2 14.413 keV

5/2 136.46 keV

57Fe:

Figure 3.1: Energy level scheme for 57Fe. The nuclear spin I is marked on the left, the

energy relative to the ground state on the right.

the emission of a photon with energy Eγor an internal conversion process, in which

the excitation energy is transferred to a shell electron (mostly from the K-shell). Since

Eγis larger than the binding energy (7.11 keV for the K-shell), the electron is removed

from the atomic shell and triggers a cascade of fluorescence radiation, mainly Kαwith

6.4 keV and Kβwith 7.1 keV (relative weight Kα/Kβ



7 [Sco74]). The ratio of con-

version processes to direct γ-transitions for 57Fe is α

2. The linewidth Γ0of the

transition is Γ0= 4.7 neV.

When built into a solid crystal, the nucleus interacts with its electric and magnetic

surrounding. These hyperfine interactions lead to a shift and/or splitting of the nuclear

levels. The type and size of hyperfine interactions yield information about structural

and magnetic properties of the investigated sample (see e.g. [Gib76]).

The rest of the chapter will show that this information is obtained in very different

ways, depending on the type of radiation source, which is used for excitation of the

nuclear levels.

3.1.2 Mössbauer Spectroscopy

InconventionalMössbauer Spectroscopy(MS),nuclearlevels areexcited withradioac-

tive sources, which emit a resonant photon with energy Eγafter a preceding nuclear

decay. Since the transitions in source and absorber have approximately the same small

linewidth Γ0, a recoil during the emission and/or absorption process makes resonance

fluorescence1impossible:

recoil energy ER

2mc2



2meV



Γ0(3.1)

The effect of recoilless nuclear resonance fluorescence in solids was first discovered in

191Irby Rudolf Mössbauer [Mös58a]: if the nucleus is bound in a crystalline structure,

the solid as a whole can take up the recoil momentum leading to a negligible recoil

energy. The probability for such a process increases at lower temperatures and is given

by the so-called Lamb-Mössbauer factor fLM.

Soon after this discovery, it was shown that the Mössbauer Effect can provide a pow-

erful spectroscopic tool [Mös58b]: based on an idea of Moon [Moo51], the relative

velocity v between radioactive source and sample is varied inducing a Doppler shift

)

Eγbetween the corresponding resonance energies. A detector behind the

sample measures the transmission as a function of the Doppler shift (see figure 3.2).

The resulting energy spectrum gives information on the nuclear level splitting in the

sample. For 57Fe, with energy splittings smaller than 500 neV, the necessary peak ve-

locities of 10 mm/s are easily accessible. The importance of iron in many scientific

branches and the large Lamb-Mössbauer factor even at high temperatures (fLM



1Excitation of a nuclear level with a photonresulting from the de-excitation of a nuclear level of the

same kind.

Figure 3.2: Principle of conventional Mössbauer spectroscopy.

at 300 K) have ensured that 57Fe is the most commonly used isotope for MS. As an ex-

ample, figure 3.3 shows the Mössbauer spectrum of α-iron at ambient conditions. The

typical six line pattern is due to Zeeman splitting of the nuclear levels in a magnetic

hyperfine field of 33 T.

Velocity (mm/s)

Relative Transmission (%)

100

-6 -4 -2 0 2 4

-Iron

Figure 3.3: Conventional Mössbauer spectrum of α-iron at ambient conditions. The

relative energy between source and absorber is given in mm/s (1 mm/s = 48 neV). The

source was 57Co in a rhodium matrix.

3.1.3 Excitation of Nuclei with Conventional X-ray Sources

The observation of nuclear resonance excitation with conventional (= non-radioactive)

x-ray sources is extremely difficult. The reason is the large linewidth of such sources:

even with a highly sophisticated crystal monochromatorthe linewidth of the excitation

(



1meV) exceeds the width of the nuclear resonance by more than 5 orders-of-

magnitude. A normal transmission experiment like in MS is impossible. The reso-

nant absorption process can be verified only by the detection of the re-emitted photon.

For the joint treatment of absorption and subsequent re-emission, the description shifts

to a scattering point of view, where the time constant of the scattering is determined

by the lifetime τ0of the excited state. The Lamb-Mössbauer factor fLM denotes the

probability of elastic scattering without recoil.

Scattering Geometry

In a real scattering experiment, an ensemble of nuclei is involved, resulting in a super-

position of different scattering responses. The scattering characteristics depend very

much on the phase relationship between these different contributions. In the absence

of a constant phase in space, one obtains incoherent scattering, which is randomly dis-

tributed in all directions (in 4πsolid angle). This behaviour occurs e.g. in all inelastic

scattering events, where the creation or annihilation of a phonon is involved. Internal

conversion leads also to incoherent scattering. Coherence (= constant relative phase)

between differentnuclei is obtained for elastically scattered photons and the respective

phase shift ∆ϕdepends on the spatial arrangement of the particles and the scattering

geometry. The superposition leads to constructive interference for forward scattering

(∆ϕ= 0) and for scattering from single crystals under a Bragg angle (∆ϕ



2π). In

every other direction the scattering responses cancel out.

However, the detection of nuclear fluorescence radiation is extremely hampered by the

huge amount of electronic non-resonant scattering, caused by the linewidth mismatch

between source radiation and nuclear resonance. Two ways to solve this problem were

proposed already in 1962 by Seppi and Boehm [SB62]:



"The use of a very intense x-ray source with improved energy resolution due to

monochromatization with high-index reflections.



The background problem might be solved by taking advantage of the instanta-

neous character of atomic (= non-resonant) scattering as compared to the rel-

atively long lifetime of the nuclear excited states. Through the use of a pulsed

X-ray beam and a properly gated detector it should be possible to observe only

nuclear excitation events in the sample."

The technical realization of these suggestions was far beyond the scope of that time. It

was not until the beginning of the nineties that the combination of high-resolution

monochromators (∆E

5meV) and x-ray detectors with sufficient time resolution

(



1 ns) and dynamic range (>105) led to the first observation of Nuclear Reso-

nant Scattering in the forward direction [HSB91]. Before that, S. Ruby [Rub74] first

proposed the use of synchrotron radiation for such experiments and E. Gerdau et al.

[GRW85] had performed the first Nuclear Resonant Scattering experiments with SR

under a Bragg angle. The Gerdau group used an electronically forbidden reflection of

an Yttrium Iron Garnet single crystal to suppress the non-resonant scattering. How-

ever, the rapid development of NRS for hyperfine spectroscopy is connected with the

utilization of powder samples in forward scattering experiments. The special prop-

erties of SR open up a variety of new possibilities. For the extreme conditions of a

high-pressure experiment, the high brilliance of synchrotron radiation is especially ad-

vantageous when compared to radioactive sources, which emit their radiation in 4π

solid angle.

3.2 Nuclear Forward Scattering

The principle of NFS is presented in figure 3.4. In the 16-bunch mode at ESRF (see

chapter 2) a synchrotron pulse of 100 ps length strikes the sample every 176 ns. The

countrate after the high-resolution monochromator (HRM) is about 109Hz in a band-

width of 5 meV. A few ns after the excitation, when all electronic scattering processes

(time constant 10

;

15 s) are completed, the detector starts to count nuclear resonant

events as a function of time after excitation (



100 Hz).

For the hypothetical time spectrum of an isolated 57Fe nucleus, a simple exponential

decay ∝exp

(

;

τ0

)

is expected with the lifetime τ0as time constant. This is indi-

cated by the straight line in the logarithmic scale in figure 3.4. However, NFS of an

ensemble of nuclei is a collective effect and the influence of coherence is reflected by

a modulation of the time spectra.

SR Fast detector

sample (57Fe)

log I

time

176 ns

(16 bunch mode)

nonresonant events

(1 GHz)

delayed events

(100 Hz)

mono

Figure 3.4: Principle of Nuclear Forward Scattering, showing the schematic set-up at

the top and a simplified time spectrum at the bottom of the figure.

The two main coherence effects are (more details in sections 3.2.2 and 3.2.3):



Speed-up of the exponential decay: With increasing number of 57Fe nuclei,

the probability for multiple scattering increases, resulting in a speed-up of the

exponential decay.



Quantum beats: In case of a hyperfine splitting of the nuclear levels, all pos-

sible transitions are excited by the broadband synchrotron pulse. The interfer-

enceof there-emittedradiationcomponents withdifferentresonance frequencies

leads to beats in the time spectrum, which are determined by the level splitting.

3.2.1 Mathematical Treatment

The interaction of radiation with matter can be treated very generally with the concept

ofa complex indexof refraction ˜n

(

)

. Consideringthe solutionof thewave equationin

adispersive medium, theamplitudeEtr ofanelectromagneticwave transmittedthrough

such a medium of thickness dis related to the incident wave Eiby

Etr

(

;

i˜n

(

)

kdEi

(

)

;

(3.2)

where kis the wave vector in vacuum and ωthe photon frequency. In case of a scalar

index of refraction, ˜n

(

)

can be expressed by the nuclear scattering amplitude f

(

)

for elastic nuclear scattering [Jac75]:

˜n

(

4π

k2ηsf

(

)



2π

k2ηsf

(

)

(3.3)

Here ηsdenotes the volume density of elastic scatterers and fthe nuclear scattering

amplitude. The approximation in equation 3.3 is valid in the x-ray region, because

˜n

(

)

differs not very much from 1. Analogously to the case of atomic scattering, the

nuclear scattering amplitude fcan be decomposed into a non-resonant Thomson term

f0and a resonant contribution f

called anomalous scattering with real and

imaginary part [HT94]. For nuclear scattering the Thomson term is proportional to

Ze2

mpc2, involving the large proton mass mp; it is therefore negligible.

The resonant term for a single nuclear resonance is given by [Smi96]:

(

) =

;

8πσ0Γ0

¯h

(

;

ω0

)

;

iΓ0

2(3.4)

;

8πσ0x

| {z }

8πσ0

;

| {z }

(3.5)

with x

¯h

(

;

ω0

)

Γ0

;

σ0

2π

(

)

2Ie

2Ig

-8 -6 -4 -2 0 2 4 6 8

Energy ( )

-600

-400

-200

200

400

600

Scattering amplitude ( )

f''

57Fe

Figure 3.5: Real f

and imaginary f

part of the nuclear scattering amplitude for 57Fe

in units of the classical electron radius re



;

15 m. For the graph, fLM is taken

as unity.

The maximum resonance cross section σ0contains the conversion coefficient αand

the nuclear spins Igand Ieof the ground and excited states, respectively. Since the

resonant wave length (



1 Å) is large compared with the size of the nuclei, the angular

dependence of the nuclear scattering amplitude is very weak. Hence, we can take

fin equation 3.4 as the forward scattering amplitude with a very sharp frequency

dependence. Combining equations 3.2 and 3.3, the transmitted wave Etr can be written

as2

Etr

(

) =

exp

(

;

ikd

)

exp

(

;

iληsdf

(

))

(

)

(3.6)

exp

(

;

ikd

)

exp

(

;

iληsdf

(

))

| {z }

phase shift

exp

(

ληsdf

(

))

| {z }

attenuation

(

)

(3.7)

The first exponential term on the right-hand side accounts for the phase shift of the

unperturbed wave after travelling the distance d. Modifications of the incident wave

are due to forward scattering: while the real part f

of the scattering amplitude causes

a frequency dependent phase shift, the imaginary part f

determines the attenuation.

Figure 3.5 shows f

and f

in units of the classical electron radius re.

Exactly at resonance, the attenuation reaches a maximum whereas the phase shift is

zero. The maximum values of f

and f

are almost two orders-of-magnitude larger

than the electronic scattering factor, which can be estimated as Z



rewith Z

26 for

the number of electrons in an iron atom. Far off resonance, f

drops in value according

to E

;

1and is large in comparison to f

, which drops according to E

;

2Here and inthe followingequationsforEtr, the electronic attenuationterm exp

(

;

µed

)

is omitted

for clarity, because it shows no time or frequency dependence in the investigated energy band.

So far, we treated only the transmitted wave amplitude Etr

(

)

. The finally measured

physical quantity is the transmitted intensity Itr. As mentioned in chapter 3.1.2, con-

ventional MS monitors the transmitted intensity for varying photon energies. This can

be obtained from equation 3.7 by taking the square modulus:

Itr

(

) =

Etr

exp

(

2ληsdf

(

))

(

)

(3.8)

exp



;

ηsdσ0fLM



(

)

(3.9)

exp



;



(

)

(3.10)

with χ

ηsdσ0fLM (3.11)

χis called effective thickness. Exactly at resonance (x

0) the transmitted intensity

is Itr

(

ω0

exp

(

;

)

(

ω0

)

.3The information about the real part of the scattering

amplitude is lost in a conventional Mössbauer experiment. An increasing thickness

χbroadens the Lorentzian shape (given for χ

1 using the approximation exp

(

)



(

)

) of an absorption line to a more complicated shape4according to equation 3.10.

In the presence of several nuclear resonances due to hyperfine splittings, the overall

absorption is obtained from the superposition of the absorption effects of the single

resonances considering the respective transition intensities.

The effect of a large effective thickness χand a multiple nuclear resonance on the NFS

spectra will be discussed in the next two sections.

3.2.2 Speed-up of the Exponential Decay

Ina NFSexperimenttheincidentradiationis a shortsynchrotronradiationpulse, which

has a uniform amplitude distribution within the energy band selected by the HRM:

(

;



exp

(

iωt

)

. We get the frequency-dependent transmitted wave by insert-

ing this relation into equation 3.7 and substituting χfrom equation 3.11:

Etr

(

) =

E0exp

(

ωt

;

))

exp



ixχ



exp



;



(3.12)

(

Efs

(

)

(3.13)

3The definition of χin equation 3.11 is commonly used in conventional MS and has its origin in

this relation. In the literature on NFS (e.g. [HSB91], [BSH92]), the effective thickness is defined as

4, which gives a simpler presentationof most of the formulas on NFS as we willsee later in equation

3.12. However, the priority is on the side of MS and for easier comparison we will use the introduced

definition3.11 for both spectroscopies.

4The actually measured transmission spectrum in MS arises from the folding with the line shape of

the source.

The transmitted wave Etr can be regarded as a superposition of the incident wave Ei

and the forward scattered wave Efs. However, in the frequency domain these two

components can not be decomposed experimentally. To obtain the time-dependent

representation of Etr we have to integrate over all frequency components in equation

3.12. Froma mathematical point of view this is a complex Fourier transformation. The

result can be expressed in the form [KAK79]

Etr

(

)

∝



∆

(

)

;

exp



iω0ττ0

;



(

χτ

)

χτ



;

(3.14)

where τis the time in units of the nuclear lifetime τ0and J1is the first-order Bessel

function. The influence of the incident amplitude Eiis reflected by the δ-function at

0 which can be separated from the delayed nuclear scattering at τ

0. Taking the

square of the modulus leads to the time dependent intensity (for τ

0):

Itr

(

)

∝exp

(

;

)

τJ2

(

χτ

)

(3.15)

For thin samples with small χor during a short time after excitation, this expression

can be approximated by [BSH92]

Itr

(

)

∝χ

2exp

[

;

(

)

]

(3.16)

This equation describes an exponential decay with an accelerated decay rate

(

)

which leads to the so-called speed-up (see figure 3.6). Whereas the absorption effect

in conventional MS increases with χ(see equation 3.10), the time-dependent NFS

intensity for thin samples is proportional to χ2. This enhanced sensitivity of nuclear

0 500 1000



ime (n

10-6

10-4

10-2

100

102

Intensity (a.u.)

=10

=20

Figure 3.6: Calculated NFS spectra for different sample thickness according to equation

3.15. The thin sample with χ

1 gives an exponential decay represented by a straight

line in a logarithmic scale. For sample thicknesses of χ

10 and χ

20 dynamical beat

modulations are visible, with the first minimum shifted to earlier times for χ

20.

scattering was shown experimentally in investigations of surface structures containing

down to one monolayer of 57Fe [NMR98].

For thick samples, equation 3.16 is not a valid approximation and the time dependence

shows beat modulations according to the Bessel function with larger separation of the

minima at later times.

The exact positions of the minima are determined by the zero points of J1

(

χτ

)

and

the first minimum t1is related to the effective thickness χby

χt1

τ0



)





2040ns (3.17)

3.2.3 Quantum Beats

So far, we have considered a single resonance line with degenerated nuclear levels. In

the presence of hyperfine interactions, the degeneracy is removed and several nuclear

resonances can occur. According to the type of nuclear transition (M1 for 57Fe) the

particular transitions have different polarization properties and the nuclear scattering

amplitude must be written in matrix form [Smi96]:



;

(

)



;

const.



∑

Γ0

¯h

(

;

ω0

;

)

;

iΓ0

2G2

(

;

)

;

(

)

(3.18)

where ω0

;

jdenotes the resonance frequency of transition j. The square of the Clebsch-

Gordon coefficients Gprovides the probability of the respective transition which de-

pends on the magnetic quantum numbers mgand meof the ground and excited states.

;

is the polarization factor for the polarization state of incident radiation (s) and

forward scattered radiation (s

). As shown by U. Bergmann, the formalism with a

complex index of refraction (see equations 3.2 and 3.3) can be extended to scattering

amplitudes in matrix form [Ber94].

In the following the simplified case of two resonance lines with the same probability

and polarization is discussed. When the energy separation ∆Ehf

¯h∆ωbetween the

two lines is large compared to their effective width, the resulting time spectrum is well

approximated by multiplying the single-line pattern by a cos2term representing the

interference of the two lines [Smi96]:

Itr

(

)

∝exp

(

;

)

τJ2

(



χτ

)

cos2



∆ω

χΓ0

8∆E



(3.19)

The first term in the argument of cos2describes the quantum beat modulation deter-

mined by the line splitting. The beating period Tqb is given by

Tqb

2π

∆ω

880ns

∆Ehf

[

Γ0

]

86ns

∆Ehf

[

]

(3.20)

The second term in the argument of cos2reflects the influence of the interaction be-

tween the two transitions. It depends on χand ∆Ehf, and causes a small shift ∆tqb of

the quantum beats to shorter times:

∆tqb

(

∆Ehf

[

Γ0

])



35ns

(

∆Ehf

[

])



34ns (3.21)

According to equation 3.19 the resulting time spectrum forthe two-linecase is a multi-

plicative combination of Bessel beats and quantum beats. The Bessel beat modulation

for this case is the same as for the single-line case if the sample is twice as thick (i.e.

when it has the same effective thickness per resonance line). This is demonstrated in

figure 3.7, where the time spectrum of a single resonance with χ

10 forms the enve-

lope of a quantum beat pattern originating from a splitting of 1 mm/s and χ

20. The

first half period is shortened by 6.8 ns, the time shift given in equation 3.21 for χ

20.

As the effective thickness gets larger, the approximation 3.19 fails due to the increas-

ing overlap between the tails of the two lines, causing the lines to become strongly

asymmetric [BSH92]. In this regime, the NFS spectrum can be calculated by numer-

ical computation of the Fourier transform of equation 3.7 using the modified nuclear

scattering factor of equation 3.18.

0 500 1000

Time (ns)

10-6

10-4

10-2

100

102

Intensity (a.u.)

= 20, E = 1 mm/s

hf

= 10, E = 0 mm/s

hf

Figure 3.7: Calculated NFS spectrum for a hyperfine splitting of ∆Ehf = 1 mm/s (solid

line) and χ

20 according to equation 3.19. The period of the quantum beat modulation

is 86 ns. The dashed line indicates the spectrum for the single-line case with χ

10.

3.2.4 Coherence

As seen in the last section, the interferenceof two (ormore)differentnuclear scattering

amplitudes Aand Bcan lead to quantum beat modulations in the forward scattered

intensity

2ABcos

(

∆ωt

;

∆ϕ

)

(3.22)

However, modulations of the time spectra due to interference effects occur only if the

scattering components have a constant phase difference ∆ϕin time and space (temporal

and spatial coherence). Otherwise the interference term cancels out and Iis just the

sum of the two intensities A2and B2.

The requirement for temporal coherence is fulfilled, when the width of the SR pulse

tSR is small compared to both the lifetime τ0of the excited state and the beat period

Tqb. This yields a well-defined time zero for all excitation processes. Both conditions

are fulfilled for 57Fe (tSR



100 ps

;

τ0

141ns

;

∆Ehf

10mm

)

Tqb

8ns). If

two hyperfine transitions belong to the same single nucleus with a split excited state,

spatial coherence is not necessary and temporal coherence alone can result in beat

modulations. The observation of such single-nucleus quantum beats in NFS was first

reported by [BCR96]. It gives information on the excited-state splitting only and does

not require recoilless scattering.5

However, the full information about both excited and ground state splittings can be

extracted only if the interference of transitions from different nuclei is included. This

requires spatial coherence of an ensemble of nuclei and therefore elastic (recoilless)

scattering. Depending on the kind of separation of elastic scatterers parallel or perpen-

dicular to the beam propagation direction, one can distinguish between longitudinal

and transverse coherence, respectively.

longitudinal coherence:

In NFS the phase shift between two scatterers, lying "behind each other" in the beam

propagation direction, is zero: the geometrical phase difference kx, due to their dis-

tance x, is compensated by the temporal phase ωt, caused by the time t, the wave field

needs to propagate between the two scatterers in vacuum: ωt

kct

kx.

transverse coherence:

When two scatterers are separated perpendicular to the beam direction, a geometrical

phase difference ∆ϕhas to be considered. It is determined by the length difference ∆l

between the scattering paths from a source point zsto a detector point zd, as shown in

figure 3.8. Since the phase difference ∆ϕ

k∆lvaries for different detector and source

points, the integration over the full source and detector size leads to a blurring of a

5Quantum beats following the excitation of single nuclei are well known from time-dependent per-

turbedangularcorrelation(TDPAC) experiments[SH72]. InTDPAC thenuclearsublevelsare populated

by the decay of a radioactive parent state. The beats are then measured with a time reference set by ob-

serving the gamma photonfrom the parent state decay.

source

sample

detector

Figure 3.8: Measuring geometry and transverse coherence for an example of two scatterers,

separated by a distance d.

quantum beat pattern. When the phase differences involved are larger than



π, the

beat pattern vanishes and the two scattering components add incoherently. Extending

this discussion, Baron et al. [BCG96, Bar99] calculated the phase differences for a

Gaussian distribution of source size (σ0) and detector size (σd). They defined an ef-

fective transverse coherence length Ltr as the minimum transverse separation between

parts of the sample necessary to ensure that their responses add incoherently:

Ltr

2π

σt;σ2



σ0





σd



2(3.23)

For a typical experiment at ESRF with S

40m,D

1m,σ0

50µm and σd

5mm

the value of Ltr is dominated by the effect of a finite detector size:

σ0

50µm

;

σd

σ0

50µm

;

σd

5mm

Ltr

;

source



10µm source effect

Ltr



30nm total effect

The value given for the pure source effect is calculated for the ideal case of a point

detector (σd

0) on the optical axis.

Summarizing the effects of longitudinal and transverse coherence on the NFS spec-

tra, it should be mentioned that coherent and incoherent superpositions of scattering

components can not be distinguished for completely homogenous samples. Inhomo-

geneities lead to a noticeable incoherent superposition, when parts of the sample with a

certain property (say "blue") are as large as the sample thickness in the beam direction

and separated from regions with property "red" by more than Ltr perpendicular to the

beam. An example for such an inhomogeneity was observed for metallic iron under

pressure. In the transition region of the α

;

εphase transformation, the two crystallo-

graphic phases are spatially separated and produce a complex mixture of coherent and

incoherent behaviour [Grü97]. Incoherent superposition is also obtained for thickness

distributions in powder samples. Here the thickness varies on a length scale compara-

ble to the grain size which is usually in the range of µm and therefore much larger than

Ltr. The effect of these distributions on the NFS spectra is discussed in more detail in

chapter 5.2.3.

3.2.5 Examples

Figure 3.9 presents NFS spectra of non-magnetic stainless steel and magnetic iron, the

latter measured without and with an external polarizing field of 0.6 T. The three spectra

serve as a finger print for different magnetic states of iron atoms. For comparison, the

corresponding conventional Mössbauer spectra are shown in the right panel.

Figure 3.9: The left panel presents NFS spectra of (a) non-magnetic stainless steel (χ



5) and (b,c) magnetic α-iron (χ



50). The dashed line in (a) indicates the natural decay

∝exp

(

;

τ0

)

. Part (c) shows the time spectrum for a totally polarized iron foil in an

external field perpendiculartothe σ-polarizationand the directionof the incomingbeam.

The corresponding Mössbauer spectra (schematic) are shown in the right panel. The

solid lines are fits made with the CONUSS program package [SG94].

The time spectrum of the non-magnetic sample in figure 3.9a shows a straight line

which indicates a simple exponential decay. It is modified by thickness effects, re-

sulting in a speed-up of the collective nuclear decay compared to the natural decay

(dashed line). For magnetic α-iron with random orientation of the quantization axis,

all 6 possible transitions between the 3/2 excited and 1/2ground state are excited by the

broadband (meV) SR pulse. The corresponding decay spectrum in figure 3.9b shows

a complex modulation, arising from the superposition of all transition frequencies. It

is dominated by the beating of the two intense outer transition lines with an energy

difference of 10.7 mm/s (110 Γ0), corresponding to Tqb

8ns.

The complex time spectrum of a non-polarized sample can be simplified, when the

polarization of the SR is exploited to select certain transition lines in a magnetically

aligned absorber. For the nuclear M1 transition in 57Fe the polarization dependence

corresponds to a classical dipole oscillator: a linear magnetic dipole oscillation along

the quantization axis for ∆m

0 transitions and a right hand / left hand circular oscil-

lation about the quantization axis for ∆m

1/∆m

;

1 transitions (see figure 3.10).

For the iron sample measured in figure 3.9c the axis of magnetization is parallel to the

Bvector of the incident SR, and only the two ∆m = 0 transitions of the 57Fe resonance

contribute to the NFS spectrum. The magnetic hyperfine field can be determined with

high accuracy from the periodic sequence of beats (here Bhf

33T,∆E= 6 mm/s and

Tqb



14 ns), modified only by thickness effects (χ= 25/line), resulting in the present

case in a Bessel minimum at 85 ns. Since in the case of the magnetically polarized

absorber, the two ∆m = 0 transitions belong to different ground and excited states (i.e.

different nuclei), the beating in the NFS spectrum demonstrates the spatial coherence

of the NFS process.

1/2

3/2

1 42 53 6

+1 0 -1 +1 0 +1

1/2-

3/2

ImI

Figure 3.10: M1 transition scheme of 57Fe in the presence of a magnetic field acting

on the nucleus. The polarization properties for emission along the quantization axis

are marked below. For emission perpendicular to the quantization axis, the ∆m



transitions show linear polarization.

3.2.6 Measurement of Isomer Shifts

It is obvious from the previous sections that only frequency differences can be mea-

sured with NFS. In contrast to a conventional Mössbauer source, the SR provides no

reference energy, relative to which the center of gravity of a transition line pattern can

be determined. However, informationabout the isomer shift can be obtained withNFS,

when an additional reference sample is placed in the beam. In this case, the nuclear

scattered photons of reference and investigated sample can interfere and the energy

difference of the two components can be extracted from the resulting beat pattern.

Consequently, the isomer shift is given relative to the employed reference sample. In

orderto get simplebeat patterns, a reference sample with a single-linetransitionshould

be used.

An application of this method using the 151Eu resonance (21.5 keV) is presented in

[PLS99] for a pressure-induced valence transition in EuNi2Ge2.

3.3 Nuclear Inelastic Scattering

In the previous sections we focused on elastic (=recoil-free)scattering of resonant pho-

tons (E

Eγ) by an ensemble of nuclei. The corresponding probability was given by

the Lamb-Mössbauer factor fLM. Incident photons with E

Eγwere only treated with

respect to the non-resonant background they produce. However, Nuclear Resonant

Scattering is also possible for incident photons with E

Eγ; in this case the interac-

tion necessarily involves the recoil of a 57Fe nucleus with energy transfer from or to

the crystal lattice in order to fulfil the resonance condition. This inelastic fraction of

nuclear scattering, which was considered as the lost part (1 - fLM) in conventional MS

and also in NFS, turned out to be the basis of a new spectroscopic tool for the study of

lattice dynamics.

The necessary condition for this new pathway is the energy-tunability of SR, which

gives the possibility to measure the energy dependence of Nuclear Inelastic Scattering

(NIS) over an energy range of



100 meV, which is covered by lattice vibrations. In

contrast to other methods like inelastic neutron, x-ray and Raman scattering, NIS does

not deal with phonon dispersion relations but, complementary to that, gives direct ac-

cess to the density of phonon states (DOS). This topic will be discussed in the next two

subsections.

3.3.1 Basic Features

The first NIS experiments were carried out in 1994 [SYK95, STA95]: The energy E

of an incoming SR beam is tuned about



100 meV around a nuclear resonance mon-

itoring the time-delayed inelastically scattered radiation. Since the inelastic scattering

process is spatially incoherent, the scattered photons are emitted in 4πsolid angle

and can be detected perpendicular to the incident beam (inset of figure 3.11a). The

measured intensity depends on the probability that the energy of a phonon matches

the energy E

;

Eγ. The three main parts of a NIS spectrum are shown schematically

in figure 3.11a. The central peak involves no energy transfer to the lattice and con-

tains mostly fluorescence radiation after elastic absorption at E

Eγ. Photons with

Eγcan excite the nuclear resonance after annihilation of a phonon. The high-

energy sideband for E

Eγcorresponds to phonon creation. Figure 3.11b shows the

energy spectrum of elastic nuclear scattering, which is measured in forward direction.

Here, scattering appears only when the energy of incident radiation coincides with the

energy of the nuclear transition. The obtained data provide the instrumental function

of the high-resolution monochromator, because the width of the nuclear transition in

this scale is negligible. Furthermore, the peak gives a precise reference for the energy

position of the nuclear resonance.

As introduced in section 3.1.1 for 57Fe, the scattered photons follow the de-excitation

of the nuclei via two different channels: the radiative channel with



11% probabil-

-40 -20 0 20 40

E - E (meV)

NIS

NFS

phonon

creation

Inelast. Scatt. Intensity

Forward Scatt. Intensity



phonon

annihilation

NFS

NIS

HRM sample

60-60

Figure 3.11: a) The inset sketches the measuring geometry for a scattering experiment,

where a high-resolution monochromator (HRM) is tuned around the nuclear resonance.

The scattered intensity in forward direction (NFS) and perpendicular to the incoming

beam (NIS) is monitored as a function of energy relative to Eγ. The plot in a) displays a

schematic NIS spectrum with elasticpeak at E

Eγand energy sidebandscorresponding

to phonon creation for E

Eγand phonon annihilation for E

Eγ. b) shows the inten-

sity of forward scattered radiation calculated for an instrumental function with Gaussian

shape and 5 meV FWHM.

ity and the internal conversion channel with 89% probability. Thus, the dominating

part of detected photons are products of internal conversion, mostly Fe-Kαx-rays with

6.4 keV. 6Therefore, the allowed momentum transfer is not specified by the experi-

mental set-up: The exact location of the detector relative to the incident beam does not

matter, because the angular distribution of the atomic emission is an entirely atomic

property and does not depend on the specific way of nuclear excitation. Phonons with

any momentum, which is allowed by the dispersion relations for a particular energy

transfer, contribute equally to NIS. Consequently, Nuclear Inelastic Scattering pro-

vides an ideal "momentum-integrated" tool for the study of lattice dynamics and does

not require single crystalline samples.

6In practice, the relative amount of detected fluorescence radiation is even higher, since the detector

efficiency is much better for 6.4 keV than for 14.4 keV (see section 4.1.2).

The properties of NIS can be summarized as follows:



NIS provides direct access to the phonon DOS. In contrast to coherent inelastic

neutron scattering [Squ78] no theoretical model has to be used (see also chapter

6.2). The experiments can be done with polycrystalline samples.



NIS benefits from the large cross section of Nuclear Resonant Scattering, which

is in the case of57Fe about five orders-of-magnitudelarger than the relevant neu-

tron cross section (σnuclear



;

18cm2and σneutron



;

23 cm2

[Squ78]). In combination with the small size of SR beams, this allows to study

very tiny samples (



1mg).



NIS has as an intrinsic energy reference. The phonon-related energy transfer

is easily determined by the energy of the incident SR relative to the nuclear

resonance. An energy analysis of the scattered particle is not necessary.



Since NIS is only sensitive to lattice vibrations where resonant nuclei are in-

volved, a partial phonon DOS of the sample is measured. This implies certain

limitations to the class of accessible materials but leads on the other hand to the

useful feature of isotope selectivity, which can simplify the data especially for

large molecules [KAO97].



The background problem of other inelastic scattering techniques can be largely

overcome in NIS by the detection of time-delayed photons. Within the limit

of the detector background (10 mHz), every detected photon originates from a

nuclear scattering process. Like in NFS, this feature is especially valuable in

high-pressure experiments.

The extractionofthephononDOSis discussed in thenextsubsection following[STA95,

CRB96, CS99].

3.3.2 Data Evaluation

The photons monitored in an NIS experiment follow a preceding nuclear absorption

process. According to [STA95] the measured intensity I

(

)

is proportional to the

absorption probability S

(

)

per unit of energy and to the effective number ηeff of 57Fe

nuclei in the sample7:

(

const



ηeff



(

)

with

∞

;

∞

(

)

(3.24)

7In the following,the energy Eis given relative to Eγ.

Assuming a quasi-harmonic lattice with well-defined phonon states, S

(

)

can be ex-

panded in terms of n-phonon contributions [STA95, SS60]:

(

fLM δ

(

)

|{z }

Sel

(

)

fLM

∞

∑

1Sn

(

)

| {z }

Sin

(

)

;

(3.25)

with the elastic part Sel

(

)

and the inelastic part Sin

(

)

. The relative weight of the

inelastic part is (1-fLM). The phonon DOS g

(

)

is proportional to the single-phonon

term in the expansion 3.25 and the multiphonon contributions Sn

(

)

for n



2 are

obtained by convolution of Sn

;

(

)

with S1

(

)

(

) =

ERg

(

)

(

;

kBT

)

;

(3.26)

(

) =

∞

;

∞

;

(

;

)

(

)

dε

;



(3.27)

Since the ratio of the integrated n- and (n-1)-phonon terms is given by

;

ln fLM

[CR98], the multiphonon contribution (n



2) for metallic iron with fLM



8 is less

than 12% of the total inelastic part.

The different steps in the extraction of g

(

)

from I

(

)

are demonstrated in figure 3.12

for the example of a Debye-like DOS (see chapter 6.2.1). The Debye temperature ΘD

was taken as 420K, resulting in a Lamb-Mössbauer factor of 0.8 at room temperature.

The spectrometer resolution was included by convolution with a Gaussian of 5 meV

FWHM.

1. In general, S

(

)

can not be calculated from I

(

)

by a normalization of equa-

tion 3.24, since the effective number of resonant nuclei ηeff is not constant over

the measured energy range. The reason for this is the sharp increase in atten-

uation of the incident SR at the nuclear resonance. The nuclear part of the

attenuation length for iron metal is 0

09µm at Eγand 0.36 m in the sidebands

of a NIS spectrum, whereas the electronic part is constantly 20 µm [STA95].

Therefore, ηeff strongly decreases at Eγand the ratio between the elastic peak

and the remaining spectrum is not well-defined. A possibility to solve this nor-

malization problem was provided by [STA95]. The procedure makes use of the

general property of S

(

)

, that the first moment

(

)

EdEis equal to the recoil

energy ERof a free nucleus [Lip60, Lip95]. Since the damped elastic part of

(

)

is assumed to be symmetric around E

0, it has no effect on the first mo-

ment and hence the inelastic (n-phonon)part of I

(

)

can be normalizedcorrectly

(

)

Inorm

(

)

in figure 3.12):

extract n-phonon contribution

removal of elastic peak

normalization of I(E) with sum rule

Sn(E)Inorm (E)

calculation of phonon DOS

n=1

1-ph.

n=2

0.0

0.002

0.004

0.006

0.0

0.01

0.02

0.03

Energy (meV)

0.02

0.04

0.06

g(E)

0.0

0.002

0.0 04

0.006

log (S)

in (E)

3-ph.

2-ph.

-10 0 -80 -60-40 -20 0 20406080100

0.08

Figure 3.12: 1) Simulated NIS spectrum for an ideal Debye solid with ΘD

420K

36meV

kB. The inelastic part is normalized using a sum rule of Lipkin [Lip60, Lip95].

2) After removal of the elastic peak, the remaining inelastic part is the sum ∑Sn

(

)

single- and multi-phonon contributions. Integration of this spectrum gives 1

;

fLM.3)

The multi-phonon contributions Sn

(

)

for n



2 can be separated by a recursive proce-

dure according to equations (3.25) and (3.27). 4) The phonon DOS g

(

)

, convoluted

with the Gaussianresolutionfunction, can be extracted from S1

(

)

with equation(3.26).

∞

;

∞

(

)

EdE

const



∞

;

∞

ηeff S

(

)

EdE

const



ηel

eff

∞

;

∞

Sel

(

)

EdE

| {z }

ηin

eff

∞

;

∞

Sin

(

)

EdE

|{z }

const



ηin

eff



)

Inorm

(

) =

∞

;

∞I

(

)

EdE I

(

)

(3.28)

2. After adjusting and removing the elastic peak, the resulting spectrum is equiv-

alent to the n-phonon part Sin

(

)

of the absorption probability. The integration

gives directly the recoil fraction 1

;

fLM (see equation 3.25). Consequently, fLM

can be determined withoutany knowledgeabout the number of 57Fe nuclei in the

sample. In conventional MS and NFS, the recoil-free fraction fLM is contained

in the effective thickness χ(see equation 3.11) and can be determined only with

specific information about ηeff.

3. Sin

(

)

is decomposed into the different n-phonon contributions Sn

(

)

(n=1,2,...)

with an iterative procedure [STA95], shown in the flow diagram in figure 3.13:

the iteration is started with a first approximation S



Sin

fLM using the value

for fLM from the above mentioned integration. Then S

2is calculated with equa-

tion 3.27 and subtracted from S

1to get the next approximation S

1. This new S

is used to calculate the contributions S

2and S

3. Forevery furtheriterationstep n,

the subtraction of the multi-phonon terms S

(

;

)

ifrom the initial approximation

1leads to a new set of S

(

)

. This procedure has to be repeated until S

(

)

1gets

negligible small.

4. Finally, the phonon DOS g

(

)

can be calculated from the obtained S1

(

)

with

equation 3.26.

Start iteration with normalised inelastic S

Calculation of ' with eq (*) and subtraction from 'SS

in in

(*)

2 1

1 123

S' -S'

123

S' -S''

-S''

S' S'

S'' S'' S''

SSSS

n+1 n+1

Next iteration with new ''S

Stop, when is negligibleS

n(n)

(*)

(*) (*)

(*)

(n) (n) (n) (n)

Figure 3.13: Flow diagram of the recursive procedure to extract the different multi-

phonon contributions from the normalize inelastic spectrum Sin. The procedure is fin-

ished, when the last multi-phonon part S

(

)

1becomes negligible. The "(*)" sign stands

for the application of equation 3.27.

Using the Bose occupation factor nB

(

;

(

exp

(

kBT

)

;

)

, equation 3.26 for

the single-phonon term S1

(

)

can be rewritten:

0: S1

(

) =

ERg

(

)

(

;

)

phonon annihilation

;

(3.29)

0: S1

(

) =

ERg

(

)

(

;

)

phonon creation

(3.30)

Thus, the annihilation part of the NIS spectrum is proportional to the occupation of the

phonon states nB

(

;

)

and vanishes at low temperatures. The creation part is propor-

tional to (nB

(

;

1) and remains finite even at T

0 leading to a recoilless fraction

fLM

(

)

1. An incident x-ray photon can gain energy only from an existing

phonon, whereas an energy loss is possible via two ways: increasing the energy of

an existing phonon or creating a new one. The intensity ratio between the high- and

low-energy sideband of an NIS spectrum is

(

;

(

;

)

kBT

(3.31)

Chapter 4

Experimental Details

This chapter presents some experimental and technical details about Nuclear Resonant

Scattering under pressure. The first section introduces the Nuclear Resonance Beam-

line at ESRF including a short description of its main components. The second section

gives a detailed presentation of the high-pressure technique, which was developed as a

part of this thesis.

4.1 Nuclear Resonance Beamline at ESRF

The principleset-up of the NuclearResonance Beamline ID18 at ESRF is shown in fig-

ure 4.1. For experiments with the 57Feresonance, the 6 GeV electron storage ring was

operated in 16-bunch mode providing a time window of 176 ns between the bunches.

The bunch purity, defined as the relative amount of photons from spurious bunches, is

better than 10

;

9. The synchrotron radiation (SR) from an undulator is tuned with its

first harmonic to 14.4 keV. The radiation bandwidth of 300 eV is monochromatized by

a high-heat-load Si(1 1 1) double crystal monochromator (PM) down to a bandwidth

of 3 eV and then further to about 5 meV by a "nested" high-resolution monochromator

(HRM, see subsection 4.1.1). During these monochromatization steps the total flux is

reduced from 1014 Hz to about 109Hz. This beam was used to excite the 14.413 keV

levels of the 57Fe nuclei in the absorber, schematically shown in figure 4.1 within a

diamond-anvil cell (DAC).

In a NFS experiment the detector, an avalanche photo diode (APD, see subsection

4.1.2) behind the sample measures the nuclear scattered intensity of SR as a function

of timeafterexcitation. The time-modedetection allows an easy discriminationagainst

electronically scattered radiation, which exceeds the nuclear scattering by a factor of

105but occurs almost instantaneously within the width of the SR bunches (100 ps).

For a NIS experiment the HRM is tuned around the nuclear resonance monitoring

the intensity of Fe-Kαx-ray fluorescence (6.4 keV) following the de-excitation of nu-

NFS

NIS

DACHRMPMSR Source

5 meV3eV300 eVE:

10 Hz

10 HzFlux: 14 9

5 meV

500 neV

10 Hz

delayed

prompt

APD

Figure 4.1: Schematic set-up of the Nuclear Resonance Beamline ID18 at ESRF. SR

source with storage ring and undulator, PM: pre-monochromator, HRM: high-resolution

monochromator, DAC: diamond-anvil cell, APD: avalanche photo diode as fast detector

for NFS and NIS. The corresponding energy bands and estimated SR flux are denoted

below. Downstream of the sample these specifications are given separately for prompt

and delayed (i.e. nuclear scattered) photons.

clei after internal conversion. The corresponding APD is placed perpendicular to the

incident beam and close to the sample in order to cover a large solid angle. After feasi-

bility studies with NIS under high pressure at the Nuclear Resonance Beamline ID18,

the data with highest quality was obtained at the ESRF beamline ID22. This beam-

line can be used as a second station for nuclear scattering experiments since 1998.1

The principle set-up is the same as at ID18 in figure 4.1. During our experiments at

ID22, a second 1.5 m undulator was operated at 14.4 keV. However, the main step

from the stage of feasibility studies to accurate measurements on a 0.5 µg sample up

to 42 GPa was made possible by the use of focusing elements in the beamline optics to

concentrate the incident flux on the small sample dimensions.

The key components of the set-up: the high-resolution monochromator, the fast detec-

tors and the focusing elements are discussed in the next subsections.

1The operation of ID22 for nuclear scattering experiments is limited to periods with special timing

modes like 16-bunch mode. For most time of the year it is used for experiments with micro-focusing,

-imaging and -diffraction [ESR97].

4.1.1 High-Resolution Monochromator

The task of the HRM is to reduce the energy resolution of the incoming SR to a small

bandwidth of



5 meV around the nuclear resonance. This can be achieved with a large

Bragg angle ΘBof a crystal reflection (h k l). According to Bragg’s law

2dsinΘB

nλ

;

(4.1)

a given variation ∆Θ around a large value of ΘBleads to a minimum variation ∆λ in

energy. For d

l2with silicon (a = 5.43 Å) and 57Fe (λ= 0.86 Å),

the largest possible value is ΘB



for the Si(9 7 5) reflection. Usually, such

high-index reflections have a small angular acceptance. This small acceptance can be

adapted tothedivergence oftheincomingbeamby anadditionalasymmetricreflection.

The corresponding "nested" set-up of a HRM was first proposed by Ishikawa et al.

[IYI92] and is shown in figure 4.2 for a HRM used at ESRF. The combination of two

Si(9 7 5) and two Si(4 2 2) reflections leads to an energy resolution of 4.4 meV. A

second nested HRM at ESRF is operated with the combination Si(12 2 2)/Si(42 2) and

has an energy resolution of about 6.4 meV.

Si (9 7 5)

b=-1

Si (9 7 5)

b=-1

Si (4 2 2)

b = -10

Si (4 2 2)

b = -0.1

E = 4.4 meV

Figure 4.2: Set-up of a high-resolution monochromator with "nested" design. The two

high-resolutionSi (9 7 5) crystals are placed between two asymmetric Si(4 2 2) crystals,

which adapt the emittance of the incident SR beam to the small angular acceptance of

the inner crystals. The asymmetry parameter b is defined as b

;

sinΘin

sinΘout with

the angle Θin (Θout) between the incoming (outgoing)beam and the crystal surface.

4.1.2 Fast Detectors

The detector system in a nuclear scattering experiment has to meet severe demands.

Even with a HRM, the total flux in a high-pressure NFS experiment is in the MHz

range, whereas the delayed count rate may be as low as 1 Hz or less. The detector

needs to have a good time resolution (nanoseconds) and quantum efficiency, low noise

and a highdynamic range. The firstNFSmeasurements were performedwith scintillat-

ing crystals [HSB91] but since 1994 silicon-based avalanche photodiodes (APD’s) are

most commonly used. The reason for their success is the high dynamic range (>106)at

very low background rates of 0.01 Hz. The time resolution is better than 1 ns. With an

active area of 10x10mm2the quantum efficiency is about 12% at 14.4 keV and 90%

at 6.4 keV. More details can be found in [Kis91, BR94, Bar95].

4.1.3 Focusing Elements

The two kinds of focusing elements which were used for the NIS experiments are

briefly discussed in the next two subsections. With the combined use of these elements

a flux of 3



109Hz on a spot size of 100µm



100µm was achieved.

Compound Refractive Lens (CRL)

In general, lenses for electromagnetic radiation in the x-ray regime (E

10 keV) are

difficult to realize, since the index of refraction n

;

δfor these energies varies not

much from unity. Consequently, the achievable focal length ffor a single refracting

device is rather large. A new approach [SKS96] is to stack several single lenses to a

so-called compound refractive lens (CRL) to obtain focal lengths in a reasonable range

of 1-10 m. The principleset-up of a CRL is shown in figure 4.3. Since nis smaller than

unity, one has to use a concave shape of the lens to obtain a focusing property.2For

a single lens (upper part of the figure), the focal length is f1

2δ, when Rdenotes

the radius of curvature. For typical values of R

2 mm and δ



;

6(Be

at 14.4 keV) the focal length is f1



60 m. When Nlenses are stacked behind each

other (N=number of holes in the lower part of the figure), the focal length is reduced

to fN

2δN; a typical value is f30

2mfor N

30.

Since the focusing of SR is equivalent to the imaging of a SR source, it is governed

by the well-known Gaussian lens formula 1

di, where dsand diare the

distances from the lens (focal length f) to the source and to the image, respectively.

The geometrical demagnification m

dsis determined by dsand f:m

(

;

)

Besides the focal length, an important and inherent property of CRL’s is the photoab-

sorption of SR by the lens material. Since CRL’s with typical thicknesses in the order

of mm show very high absorption, the use of a low-Z material is mandatory. Beryllium

(Z=4) is the most often used material which has in addition appropriate mechanical

properties for high-precision machining. The large spherical aberration in the above

mentioned example of cylindrical holes is reduced in recent developments by the ma-

chining of holes with parabolic shape [Ric98b]. It should be mentioned that the pre-

sented CRL’s lead only to focusing in one dimension. A two-dimensional focusing

could be obtained with two crossed refractive lenses. However, the typical acceptance

2Or in a reciprocal view, one uses a convex lens of air as the refractive medium.

f= R

n=1

n=1-

2

f= R

N2N

Figure 4.3: Schematic view of a compound refractive lens (CRL) for x-rays. The upper

part shows the principle of a single concave lens with cylindrical shape, the radius of

curvature R, the refractive index 1

;

δand the focal length f1. The lower part shows the

example of Nstacked lenses with focal length fN.

of a CRL (0.2 mm) is much smaller than the horizontal beam size (1 mm). For an

optimal performance, a CRL for vertical focusing can be combined with a focusing

monochromator for the horizontal direction.

Focusing Monochromator

The schematic set-up of a focusing monochromator (FM) is depicted in figure 4.4.

Usually, a FM consists of a conventional Si(1 1 1) crystal (A) and a bent Si(1 1 1)

crystal (B). Whereas the first crystal preserves the incident beam direction of the com-

plete FM, the focusing property is due to the second crystal. For this purpose, the thin

second crystal is mounted on a "U-shaped" profile, which can induce a well-defined

curvature of the Si crystal by a spread of the two legs.

The relation between the bending radius Rbof the curved crystal and the focusing

properties is given by [FCH98]

2dsdisinΘB

2dsmsinΘB

;

(4.2)

where ds,diand mare defined as above for CRL’s; ΘBis the Bragg angle.

side view

front view

Figure 4.4: Schematic sketch of a focusing monochromator witha conventionalSi crys-

tal (A) and a bent Si crystal (B).

In practice, the minimum bending radius is determined by the fracture limit of the

crystal material. For typical conditions of a nuclear scattering experiment with an

incident energy of 14.4 keV, ΘB



,ds

30mand di

3m, the FM is operated

with Rbvalues of about 750 mm.

4.2 High-Pressure Technique for NFS

Diamond-anvil cells (DAC’s) have been successfully used in conventional MS for al-

most two decades [CTW82]. A general overview over DAC’s was provided by Jayara-

man [Jay83]; various applications for MS are reviewed in [PT89, TP90, PT96]. The

Mbar borderline for conventional MS was reached recently by M.P. Pasternak et al.

[PTJ97].

Since MS experiments are performed in the same transmission geometry as NFS,

DAC’s for conventional MS can also be used for NFS experiments. In order to fully

exploit the possibilities of nuclear scattering, a new DAC was developed on the basis

of previously used cells (see e.g. [Hes97]).

This DAC (called DAC #1 in the following)is shown in figure 4.5. The central part is a

pair ofdiamonds contained in a cylindricalpress consisting of a piston (A in figure4.5)

and a cylinder (B). The diamond flats are aligned and adjusted with a half sphere (C)

and an xy-stage (E), respectively. The force is generated by 8 screws (G) and balanced

by a backingplate (F). The inner partof the cell withthe sample (K) containedin a hole

in the gasket (I) together with ruby chips (J) and pressure-transmitting medium (L) is

shown on the right. The powder samples were grinded under inert atmosphere (N2or

Ar) and mixed with epoxy as the pressure transmitting medium. Depending on the

envisaged pressure range, the sample diameter within the Ta90W10 gasket varied from

250 µm (50 GPa) to 80 µm (100 GPa). The pressure was monitored before and after

each measurement using the ruby fluorescence method [FPB72] with the non-linear

calibration formula given in [MBS78].

10 mm

40°

NFS

NIS

Figure 4.5: Sketch of the diamond anvil cell DAC #1 made of Cu0

98Be0

02 and em-

ployed for NFS in transmission geometry and for NIS. The two large openings in the

cylinder are 17 mm x 10 mm (h x v). The resulting opening angle perpendicular to the

beam is 40



and can be used for the insertion of two magnetic arrays (see figure 4.6)

or for the detection of NIS. A: piston, B: cylinder, C: half sphere, D: diamond seats

(also made of Cu0

98Be0

02), E: xy-stage, F: backing plate, G: 8 threads, H: diamonds, I:

gasket, J: ruby chips, K: sample, L: pressure-transmitting medium.

The special features of DAC #1 for NFS are:



Use of a nonmagnetic Cu0

98Be0

02 alloy which allows the application of an ex-

ternal magnetic field. Good mechanical stability for the alignment of diamonds

with small flat sizes down to 0.2 mm was achieved by electroplating the piston

(A) with a thin layer of hard chromium. The resulting surface is very stable and

could be machined with high precision to a mechanical tolerance of 5

;

10µm

relative to the cylinder (B).



In order to apply an external field by an array of permanent magnets, the cylinder

(B) of the DAC has two large openings (17 mm x 10 mm, h x v) as shown in

figure 4.6. Each side of the magnetic array consists of two Nd2Fe14B magnets

and one iron pole piece. With a distance of 10 mm between the iron poles,

homogenous fields up to 0.75 T were obtained.

The nuclear resonant count rates for high-pressure NFS experiments depend on the

sample size (i.e. the pressure), the degree of enrichment in 57Fe and the electronic

absorption factor of the sample. At ESRF we achieved between 1 Hz for GdFe2at

105 GPa and 40 Hz for YFe2at 10 GPa (both enriched to 30% with 57Fe). This leads

to accumulation times for one spectrum between 120 min and 15 min, respectively.3

3The values were obtained withoutfocusing optical elements. The beamsize was



1mm2.

Nd Fe B Fe

NFS



ext

Figure 4.6: Schematic view of the experimental set-up with DAC #1 and an array of

Nd2Fe14B magnets and iron pole pieces which "guide" the magnetic flux to the sample.

The magnetizationdirectionsare marked by arrows. The distance between the iron poles

is 10 mm and homogenous fields up to 0.75 T were obtained.

For a pure 57Fe-foil at ambient conditions, the countrate is in the range of kHz and a

NFS spectrum can be taken within a few seconds.

After every run of NFS measurements at ESRF, the DAC’s remained at the highest

pressures (105GPaforYFe2and GdFe2, 51 GPa forScFe2) and additionalexperiments

with conventional MS were carried out in Paderborn [Lu00]. Since DAC #1 has an

opening angle of 25



in the direction of the beam, energy-dispersive XRD experiments

are possible on the same sample. These measurements were conducted at beamline F3

at HASYLAB [Rei00] and yield important information on the crystal structure and the

lattice parameter.

4.3 High-Pressure Technique for NIS

Since inelastically scattered radiation is emitted in 4πsolid angle, an NIS experiment

can not be performed in a simple transmission geometry4.For

57Fe, the radiation

consists with 89% probability of x-ray fluorescence photons with 6.4 keV.

The detection of low-energy photons with 6.4 keV in a high-pressure experiment is ex-

tremely hampered by the surrounding parts of the DAC. This is illustrated in figure 4.7

4Actually, the very first NIS experiment [SYK95] was performed in forward direction, where both

NFS and NIS events were counted.

40°

25°

NFS

NIS

brass

Cu Be

NIS

0.020.98

Figure 4.7: Possibilities for the detection of NIS in a high-pressure experiment with a

standard settingof the diamond anvils. The scattered radiation in the forward cone (25



)

is strongly absorbed by the diamond anvil and superimposed by NFS. A detection of

x-rays perpendicular to the beam requires a low-Z gasket material.

where a diamond pair is shown together with its setting: (i) The emitted radiation in

the forward opening cone of a DAC is strongly absorbed in a standard diamond anvil

of 2.5 mm height (transmission below 0.01% for 6.4 keV and 50% for 14.4 keV; see

figure 4.8) and superimposed by elastic forward scattering. (ii) The scattering perpen-

dicular to the incoming beam is completely blocked by conventional gasket materials

like Ta90W10, Inconel or steel. However, the transmission of x-rays through the gas-

ket can be significantly enhanced, when gasket materials with low atomic number Z

are chosen. This approach is well-known from XRD [KB76] and x-ray absorption

spectroscopy (XAS)[IGS78] and requires large radial openings of the employed DAC.

The most commonly used low-Z gasket materials are lithium hydride, beryllium and

boron powder mixed into epoxy. Among these materials, beryllium (bulk modulus

110 GPa [You91]) allows for the highest pressures. The transmission of 6.4 keV pho-

tons through a typical Be gasket with 1 mm radius is about 70% (see bottom of fig-

ure 4.8). XAS experiments using Be in a DAC are reported by the Paderborn group

up to 26 GPa at the LIII-edge (5.9 keV) of Praseodym [RL94, RL95]. In recent XRD

experiments with a high-strength Be gasket, Hemley et al. reached pressures above

200 GPa [HMS97].

An experimental set-up for the high-pressure NIS experiments in this thesis was devel-

oped in three main steps:

01 2 3 4 5

Thickness (mm)

Diamond

Beryllium

0.2

0.4

0.6

0.8

1.0

rel. Transmission

23.8 keV ( Sn)

119

23.8 keV ( Sn)

119

14.4 keV ( Fe)

6.4 keV (Fe-K )



6.4 keV (Fe-K )



0.2

0.4

0.6

0.8

1.0

Figure 4.8: Relative transmission of x-rays through diamond (top) and beryllium

(bottom) for different photon energies, calculated with absorption cross sections from

[MMH69].

NIS on α-iron up to 10 GPa

The firstNISexperiments(June 1997)underhighpressure wereperformedwithDAC #1

and two APD detectors perpendicular to the incoming beam. Since the detector boxes

were larger than the openings in the DAC cylinder, they were placed outside the DAC.

With an active area of 10 x 10 mm2, 15 mm apart from the sample, each APD covered

an opening angle of 37



. The flat size of the diamonds was 1.0 mm. The gasket was

made of pure epoxy, hardened at 100



C. This gasket material, however, became unsta-

ble above 10 GPa. In several test experiments with a 150µm thick Be foil, the gasket

broke, beginning with cracks in the outer part, due to the brittleness of metallic Be.

After the NIS experiment at 10 GPa, it turned out that the mechanical control of the

high-resolution monochromator had been defective: a reliable determination of the

energy position was not possible and the data could not be used for further evaluation.

NIS on ε-iron at 24 GPa

Fora second runofNISexperiments (June1998)on 57Fe, an optimizedDAC (DAC#2)

was developed. This cell is shown in figure 4.9a. The principle set-up is similar to

DAC #1, but DAC #2 is bigger and has a larger opening (26 x 23 mm2, h x v) perpen-

dicular to the cell axis. The detectors can now be mounted inside the DAC, 5 mm apart

from the sample. In order to fully exploit this detector acceptance, the lateral diamond

setting (brass ring in DAC #1) was completely omitted. Instead, the diamonds were

fixed with epoxy. This configuration allows an opening angle of 85



, as shown in fig-

ure 4.9b. However, for opening angles > 60



the opening cone includes transmission

through up to



2 mm of diamond. This leads to an almost complete absorption of

6.4 keV photons and to 40% absorption of 14.4 keV photons.

In this run, diamonds with 0.5 mm flat were used and a cylindrical Be disk with 2 mm

diameter and 250 µm initial thickness served as the gasket. In order to avoid a breaking

of the Be, the disk was annealed at 800



C to reduce the brittleness. In addition, the

disk had to be radially supported by an aluminium ring of 0.5 mm thickness. The

maximum pressure was 24 GPa, enough to obtain the pure ε-phase of iron. However,

this run suffered from a very low countrate of inelastically scattered photons and the

statistical accuracy was not sufficient for data evaluation.

10 mm

0.1 mm

0.4 mm

Be-gasket:

APD

b) c)

APD 85°

Figure 4.9: a) Sketch of DAC #2 with mounted detector boxes for the detection of NIS. b)

Optimized diamond setting without radial support. The diamonds are fixed with epoxy which

allows an opening angle of 85



. c) Sketch of the Be gasket with two conical drillings adapted

to the dimensions of the diamonds.

NIS on α- and ε-iron up to 42 GPa

After these two attempts, which showed the principle feasibility of the present high-

pressure techniques, a successful experiment was conducted in April 1999 at ESRF.

NIS spectra were measured of α-Fe and ε-Fe at pressures up to 42 GPa. Two high-

pressure cells of type DAC #2 were used for the whole series, utilizing diamonds with

flat diameter of 1 mm for α-Fe and beveled diamonds with outer diameter of 0.4 mm

(bevel 7



, inner diameter 0.3 mm) for ε-Fe.

A reasonable nuclear count rate of



4 Hz in the phonon wings of the NIS spectra at

42 GPa was made possible by the use of focusing optical elements (section 4.1.3) and a

newBe gasketwithoutanadditionalradialsupportandcapable ofhigherpressuresthan

previously achieved. The modified gasket is shown in figure 4.9c. It has a cylindrical

shape with two conical drillings adapted to the dimensions of the diamonds [She98].

The thickness of the central part is



0.1 mm and was manufactured by acid etching5.

When the gasket is pressed between the diamond anvils, the much thicker outer part

remains unaffected and acts as a radial support for the inner part. A hole of 90 µm

was drilled 6into the gasket after prepressing to 40-50 µm. The iron foil (95% 57Fe)

was placed in the hole together with ruby chips and an ethanol/methanol/watermixture

(13:4:1) as pressure transmitting medium.

For pressures up to 11 GPa a different approach for the gasket shape was made. It

turned out that cracks at the edge of the Be gasket do not occur, when thin cylindri-

cal Be disks are used, which have the same diameter as the diamond flat (thickness

0.15 mm, diameter 1 mm). In this case, the stress differences between outer and inner

parts are minimized and the drilled gasket hole remains stable under pressure. Due

to the larger sample volume (diameter 0.25 mm), the count rate for pressures up to

11 GPa was



7 Hz. However, this approach is not applicable for smaller diamond

flats.

5Phoenix Precision, Hudson, MA (USA)

6Elemental beryllium and BeO are extremely toxic, when incorporated. Every machining has to

be done with appropriate safety precautions. For the drilling procedure, the gaskets were fixed with

adhesive and covered with mineral oil.

Chapter 5

NFS in RFe2Laves Phases

The investigation of RFe2compounds in this thesis is part of a series of Diploma-

and Ph.D.-theses using x-ray diffraction (XRD) [Web95, Rei00], x-ray absorption

spectroscopy (XAS) [Nes98, SGG98], x-ray magnetic circular dichroism (XMCD)

[GSW98], Mössbauer spectroscopy (MS) [Rup99a, Lu00, Str00, SW99] and resis-

tivity measurements [Str00] for the structural, electronic and magnetic characteriza-

tion of RM2Laves phases under pressure. From the introduction of Nuclear Resonant

Scattering in chapter 3 it is obvious that, disregarding all methodical differences, NFS

and MS give in principle the same kind of information on the physical properties of a

sample. Consequently, the present work – actually the first systematic NFS study of

samples with complex (magnetic + electric) hyperfine interactions – was performed in

close collaboration with the accompanying Mössbauer studies.

We start witha briefintroductionofthe basic propertiesofRFe2Laves phases followed

by the main topic of this chapter: the NFS experiments for the investigation of mag-

netism in RFe2(R=Y, Gd, Sc, Ti) up to the Mbar range. Special emphasis is devoted

in section 5.2 to the particular properties of the data evaluation in NFS. A compari-

son with the MS studies will be given throughout the presentation of the experimental

results in section 5.3.

5.1 Structure and Magnetism of RFe2Laves Phases

5.1.1 Crystal Structure

The Laves phase structures C15 and C14 are adopted by a large number of binary

intermetallic compounds of composition RM2. Depending on the ratio of the ionic

radii of R and M as well as on the averaged conduction electron number the C15

or C14 structure appears [BCK90] (see figure 5.1). The cubic C15 phase (MgCu2,

Pearson symbol cF24) is a common structure for RM2intermetallics with Ras the

C15 C14

Fe RFe(2a)Fe(6h)

Figure 5.1: C15 and C14 structures of RFe2Laves phases.

larger metal ion like trivalent Y and Ln (lanthanide) metals andM=3dmetals like

Mn, Fe, Co, Ni, and also simple metals like Al. The hexagonal C14 phase (MgZn2,

Pearson symbol hP12) is formed, for instance, with R as a trivalent, tetravalent or

pentavalent d-metal like Sc, Ti, Hf, Ta.

In the C15 structure, all R and all M sites are crystallographically equivalent. The R

sites build up a diamond lattice, whereas the M atoms form tetrahedra connected via

their corners to fill up the empty interstices. The R sites have local cubic symmetry,

whereas the M sites have local C3 symmetry. In the C14 structure, the R sites form a

hexagonal lattice, resembling in some aspects a hcp lattice. The interstices are filled

again by M tetrahedra, sharing now common planes and corners. The two M sites, 6h

and 2a, are crystallographically and magnetically different.

For both C15 and C14 structures the R and M sites are similarly coordinated: the M

sites have 6 M and 6 R nearest neighbours and the R sites have 12 M nearest neigh-

bours. This (joint)coordinationnumber of 12 makes the C15 and C14 structure similar

to the fcc and hcp lattice of monoatomic metals, e.g. to γ-Fe and ε-Fe, the fcc and hcp

allotropes of iron. Due to their simple structure, the RFe2Laves phases are considered

as model systems for Fe magnetism in intermetallic compounds. 1

1For comparison, the technological relevant compound Nd2Fe14B has a tetragonal structure with 68

atoms per unit cell. The 6 crystallographically different Fe sites have 8-12 nearest Fe neighbours with

mean Fe-Fe distances ranging from 2.54 to 2.70 [Dep87].

5.1.2 Magnetism in R-Fe Compounds

The electronic structure and magnetism of R-Fe compounds is governed by three kinds

of electronic states [Ric98a]:



R 4f states forming a subsystem of localized magnetic moments,



Fe 3d band states which both participate in metallic bonding and magnetic order,



nearly free conduction electron states responsible for metallic bonding and cou-

pling of the two different magnetic subsystems (R 5d and Fe 4s).

The relevant energies for the electronic and magnetic interactions 2within the two

subsystems are the on-site Coulomb correlation energy Uand the bandwidth W, rep-

resenting the intra- and interatomic interaction between electron states, respectively.

The balance between these energies decides whether the electrons remain localized or

whether itinerant band states are formed. In the extreme cases we have a system with

well-definedlocal momentsin the strongcorrelationlimit(U



1)and a Paulipara-

magnet in the weak correlation limit (U



1). The localization threshold is about



;

10 for almost empty or almost completely filled shells, and U



1 for

half-filling [Ric98a].

Forrareearth4fstates withnegligibleoverlapbetween neighbouringatoms,W4f



0.5-

0.2 eV is small and decreases from Ce to Lu, whereas U4fvaries between 5 and

10 eV. Consequently, all rare earth metals at ambient conditions have localized 4f

states. This property is prevailed in R-Fe intermetallics with the exception of Ce com-

pounds, where 4f contributions to the chemical bonding have been found for example

in CeFe2[BCK90]. The magnetism of localized moments is usually described within

the Heisenberg model, assuming an interaction Hamiltonian which is proportional to

the dot product of the spins:

Hex

;

2Jij



(5.1)

The so-called exchange parameter Jij represents the coupling strength between the

moments of two different atoms labelled iand j. Due to the absence of direct ex-

change the coupling between atomic rare earth moments is mediated by conduction

electrons. When considering a solid it is necessary to sum the exchange over all

pairs of atoms which contribute to the interaction. In many cases one is only inter-

ested innearest-neighbourinteractionsand the Heisenberg Hamiltonianis simplifiedto

Hex

;

2J∑



Sjwhere the sum is only over nearest neighbours with the correspond-

ing value of J. Obviously, J

0 provides ferromagnetic and J

0 antiferromagnetic

ordering.

2Unless not mentioned explicitely, these interactions are considered at low temperatures.

A contrasting picture to the 4f states holds for transition metals with incompletely

filled 3d shells. The wave functions are spatially more extended than those of R4f

states. Thus, U3d



1-3 eV is considerably smaller than U4f. Even more important

is the absence of closed shells outside the 3d shell. This leads to a large overlap of

neighbouring 3d wave functions. Consequently, W3d



;

5 eV is much larger than

the related 4f value and 3d band states are formed. Even within the band picture the

degree of localization varies depending on U

W. The spatial extension of the electron

states and possible magnetic moments increases with decreasing U

A precondition for a net magnetic band moment is the imbalance of spins caused by

the intraatomic exchange interaction between the electrons. The larger the exchange

energy the greater the exchange splitting ∆(



8eV for Fe) between the spin up and

spin down half bands. The schematic band structure density of states D

(

)

with an

exchange splitting ∆of the half bands is shown in figure 5.2. It is easily seen that the

magnetic 3d moment µ3ddepends on both, the exchange splitting ∆and the density of

states at the Fermi level D

(

)

. The Stoner criterion I



(

)

1 for the stability of

a ferromagnetic phase combines these two parameters: the strength of the intraatomic

exchange interaction, described by the Stoner parameter I, and a high density of states

(

)

D(E)



Figure 5.2: Schematic band structuredensityof statesD

(

)

, showingexchange splitting

∆of the spin up and spin down half bands.

Sublattice Coupling

In RFe2systems the two types of magnetic sublattices are "pasted" together by inter-

atomic 3d-5d coupling which appears to be antiferromagnetic and originates from the

interplay of hybridization and spin polarization between a nearly filled 3d shell (Fe)

and a nearly empty 5d shell (R) [Ric98a]. It can produce significant conduction elec-

tron and spin density at the R sites, even when they possess no 4f moment. This is

demonstrated in YFe2with an induced moment of -0.45 µBat the Y site [ADM86].

However, for R=Y the magnetic moment is due to 4d polarization and 3d-4d coupling

which is completely analogous to the above mentioned 3d-5d coupling for the Lan-

thanides where band structure calculations [LBC94] yield a Gd 5d moment of -0.54 µB

in GdFe2.

The connecting link for the magnetic exchange between the 3d and 4f subsystems is

the intraatomic 4f-5d interaction in the R atoms. Hereby, the 4f spin moments polarize

locally the 5d electrons via ferromagnetic exchange interaction. However, the direc-

tion of the 4f total moment depends on the size and direction of the orbital moment.

According to Hund’s third rule this gives (with the above antiferromagnetic3d-5d cou-

pling) ferromagneticordering of Fe and R moments for the light rare earth (J=L-S) and

ferrimagnetic ordering for the heavy rare earth (J=L+S). The alignment of the two spin

moments leads to an energy reduction of E4f5d



10-100 meV, depending on the size

of the 4f spin moment [BNJ91]. Canted spin structures can occur when the crystal

field energy is larger than E4f5d.

A measure for the R-Fe interaction is the coupling constant JRFe appearing in the

nearest-neighbour Heisenberg-type Hamiltonian Hex

;

∑2JRFeSRSFe with R and Fe

spins SRand SFe, respectively. It can be obtained with high-field magnetization exper-

iments. For GdFe2a value of JRFe

;

22K

9meV is found [LBC94]. How-

ever, despite the high spin moment at the R sites, the driving force of magnetic order

in RFe2compounds is the 3d spin polarization. This is reflected by the much larger

coupling constant JFeFe, which was determined with spin waves in inelastic neutron

scattering for YFe2to be 24.4 meV [PCR99]. Since this value is very close to that of

α-Fe (24.0 meV), and trivalent Y has no open dor fshells, the authors of [PCR99]

consider YFe2as "diluted iron in the cubic C15 structure". The R

;

Rcoupling JRR

is small compared to JFeFe. This was shown by resistivity measurements in GdFe2,

which revealed an ordering temperature of 140 K for the Gd sublattice [SW99].

5.1.3 Magnetic Phase Diagram

The impact of different constituents Rwith different size and coupling to the Fe sys-

tem on the magnetic properties of RFe2compounds is shown in figure 5.3. The figure

presents the ordering temperature Tmand the magnetic hyperfine field Bhf; the lat-

ter related for RFe2systems to the magnetic moment µFe by µFe

µB

Bhf

[BCK90]. In order to compare materials with different crystal structure, the Fe

;

distance dFe

;

Fe is chosen as the common parameter for the x-axis. For C15 sys-

tems, this parameter depends in a simple way upon the lattice constant aaccording

to dFe

;

4. For the C14 structure, the distances between iron atoms in 6h and

2a sites are slightly different 3and the average is taken for representation in figure 5.3.

3The Fe(2a) atoms have 6 nearest Fe(6h) neighbours with equal distance whereas the Fe(6h) atoms

have 4 nearest Fe(6h) neighbours and 2 Fe(2a) neighbours with different distances (∆dFe

;



04Å

for TiFe2).

100

200

300

400

500

600

700

800

900

1000

B (T)

2.25 2.30 2.35 2.40 2.45 2.50 2.55 2.60 2.65 2.70

1.0

1.5

0.5

C15 (fm)

T (K)

C15

C14 (fm,afm)



Fe

Fe-Fe distance ( )Å

,Gd

-Fe

12 NN

C14

-Fe

8NN

-Fe

12 NN

Figure 5.3: Magnetic ordering temperatures (a) and hyperfine fields at 4.2 K (b) versus

Fe-Fe distance for different RFe2Laves phase compounds; compiled from [BCK90].

Full symbols are for fm ordering and open symbols for afm ordering. The solid line

represents calculated values for Tmaccording to equation 5.2. The arrows indicate the

low-pressure behaviour of Tm, derived from resistivitystudies[BB73] and of Bhf derived

from NMR studies [DRM86]. The proportional behaviour of Bhf and the iron moment

µFe, marked on the right axis of (b), is given in [BCK90].

Magnetic Ordering Temperatures

AllRFe2C15 compoundsorderferro-orferrimagnetically(seerightpartoffigure5.3a).

The influence of the R

;

Fe couplingis reflected by the increase of TCfrom535 K forY

without intrinsic moment to a maximum of 790 K for Gd withJ=S=7/2. In analogy

to a similar compilation of TCvalues in R2Fe14Bcompounds [SRF84], the increase of

TCin RFe2can be described by a mean-field approach

3kBTC

aFeFe

aRR



(

aFeFe

;

aRR

)

4aRFeaFeR



;

(5.2)

aFeFe

ZFeFe AFeFe SFe

(

SFe

)

;

aRR

ZRRARR G

;

aRFe aFeR

ZRFe ZFeR SFe

(

SFe

)

GA2

RFe

with the iron spin SFe, the de Gennes factor G

;

)

(

)

for the Rspin

and the number of nearest neighbours ZFeFe

6, ZRR

6, ZRFe

12 and ZFeR

The variables axy represent the magnetic interaction energy between the xand yspins.

The respective coupling parameters Axy are related to the above mentioned exchange

parameters Jxy but are usually treated as empirical. They are assumed to be identical

for all Rand can be estimated from



the value of TC

535K in YFe2with G

AFeFe

8meV,



the ordering temperature 140 K of the Gd sublattice in GdFe2with G

ARR

19meV,



the ordering temperature TC

790K in GdFe2

ARFe

37meV,

using SFe

µFe

2µB



75. The calculated values for the other rare earths from

equation 5.2 are indicated by the solid line in the right part of figure 5.3a, connecting

the TCvalues of C15 type RFe2systems. As mentioned above, the CeFe2compound

has non-systematic magnetic properties with respect to the remaining RFe2series.

The C14 structure is preferred for smaller constituents R(see left part of figure 5.3a).

ScFe2is ferromagnetically ordered with the Fe moments of both 6h and 2a sites di-

rected alongthe c-axis. GoingfromScFe2to TiFe2, theorderingtemperaturedecreases

drastically from 542 K to 283 K and TiFe2shows an interesting antiferromagnetic

structure: The 6h sites are ferromagnetically ordered within the planes, the moments

oriented along the c-axis; the 6h planes, however, couple antiferromagnetically, leav-

ing the 2a sites in a non-magnetic position [BCK90].

Magnetic Hyperfine Fields

The behaviour of the magnetic hyperfine fields for C15 compounds in figure 5.3b in-

dicates that also the Fe moment itself is affected by different Rneighbours. Although

the qualitative dependence on the Rspin moment is similar to Tm, the effect is much

smaller. Magnetization measurements in the YxGd1

;

xFe2series yield a continuous in-

crease ofµFe from1.45 µBin YFe2to 1.6 µBinGdFe2[BU79]. Consequently, the effect

of transferred hyperfine fields from the Rsites can be neglected at ambient pressure.

Similar to the volume-dependence of Tm, also the hyperfine fields show a considerable

decrease for smaller Fe-Fe distances reaching a "low-moment" state with µFe

7µB

in TiFe2.

The direction of the hyperfine fields relative to the crystal lattice and hence to the elec-

tric field gradient (EFG) is governed by the structural properties of the Laves phases

and by the direction of magnetization

M. As elaborated in detail by Genin et al.

[GBB81] for the C15 structure, the direction of the main axis of the EFG is deter-

mined by the angle βbetween

Mand the particular trigonal axis of the iron atoms at

thefourtetrahedracorners ([111],[¯

111],[1¯

11],[11¯

1]). This leads to fourmagnetically

ineqivalent Fe sites for an arbitrary direction of

M. The number of different Fe sites

can be less than four, if one considers directions of

Mwith certain symmetries. Since

the effect on the energy levels is given by f

(

)=(

3cos2β

;

)

2 the simplest case is

given for a magnetization along the [100] direction, as in the case of Rions like Dy

and Er with large 4f orbital moments. Then all Fe sites remain magnetically equiva-

lent and the magnetic hyperfine field and the EFG form the "magic" angle, β



(



0). When the magnetization is along the [111]direction, which is the case

in RFe2C15-phases with non-magnetic Rmetals like Y and Lu [BCK90], there are two

magnetically non-equivalent Fe sites in the ratio 3:1; the majority site with an angle

β1



, the minority site with β2



. GdFe2with a large S = 7/2 spin moment

has no simple axis of magnetization. Genin et al. propose that

Mlies between the

[111] and [110] direction [GBB81]. This case of a not well-known or well-defined

magnetization direction is approached by 4 sublattices with varying angles β. Since

the dipolar contribution to the hyperfine field shows the same angular dependence of

the angle β, there is usually a clear correspondence between f

(

)

and the magnitude

of the hyperfine field [GBB81, BCK90].

The magnetic properties of the hexagonal C14 RFe2compounds are also strongly in-

fluenced by their crystal structure with two inequivalent Fe sites 6h and 2a in a ratio

3:1. For the 2a sites the main axis of the EFG is oriented parallel to the magnetiza-

tion which points in the [001] direction (



). The Fe atoms on the 6h sites

experience an EFG which is perpendicular to the c-axis (



Regarding figure5.3, the overall variation of magnetic propertiesin RFe2Laves phases

with decreasing dFe

;

Fe can be summarized as follows:

1. Suppression of the Fe moment from a relatively localized high band-momentvia

a more itinerant low-moment to a non-magnetic state.

2. Decrease ofthe magneticexchange couplingreflected bythe decreasingordering

temperatures.

3. Change of the magnetic ordering type from ferro- to antiferromagnetism.

4. Preference of the hexagonal C14 structure for smaller volumes.

The features described in 1.-3. are analogous to the behaviour of γ-Fe on CuAu sub-

strates, where the lattice parameter can be adjusted by a varying Au-concentration

[KSF95] (indicated on the top of figure 5.3).

Following this argumentation the motivation for high-pressure experiments is obvious:

Is the decrease of the Fe

;

Fe distance the main reason for the strong variation of

magnetic properties in RFe2i.e. is dFe

;

Fe a "universal" parameter for the description

of iron magnetism in these compounds? The application of high pressure gives the

unique possibility to answer this question for particular samples without influence of a

changing number of conduction electrons due to different constituents R.

For the present NFS high-pressure studies, the samples YFe2, GdFe2and ScFe2were

chosen. YFe2is representing the case of a non-magnetic Ratom in the C15 systems.

The influence of a magnetic 4f sublattice can be studied by comparison with GdFe2.

Furthermore, these two compounds have similar lattice constants at ambient pressure

and also theelectronconcentrationdoes not changewhen replacingyttriumby gadolin-

ium. The bulk moduli are 133 GPa (YFe2) and 104 GPa (GdFe2) [Rei00]. The low-

pressure behaviour of Tm(from resistivity experiments [BB73]) and Bhf (from NMR

[DRM86]) are indicated by arrows in figure 5.3. For the envisaged pressure range of

100 GPa (=1Mbar) the reductionof dFe

;

Fe is as large as 10% covering the whole range

of Fe-Fe distances shown in the figure.

The possible occurrence of a pressure-induced structural transition from C15 to C14 is

subject of a close collaboration with XRD experiments of G. Reiss [Rei00].

The compounds ScFe2and TiFe2are studied as C14 reference systems, the latter only

at ambient pressure. In the Sc1

;

xTixFe2alloy series, a transition from ferro- to antifer-

romagnetism can be reached at x = 0.65 [NY85, NY86]. Since the substitution of Sc

by Ti is accompanied, beside a change in the averaged conduction electron number, by

a decrease of the lattice parameters, we expected that this magnetic transition can be

also induced in pure ScFe2by the application of pressure.

5.2 Special Features of NFS Spectra

Before presenting the high-pressure spectra of RFe2(R=Y,Gd,Sc) Laves phases we

discuss typical features of the NFS for systems with (i) texture effects (magnetic +

crystallographic, section 5.2.1), (ii) a combination of magnetic and electric hyperfine

interactions (section 5.2.2) and (iii) thickness distributions (section 5.2.3).

5.2.1 Texture Effects

Texture effects in NFS spectra are due to the interplay between preferred orientations

of magnetic and/or crystallographic axes in the sample and the polarization of the

SR. They are inevitable in a high-pressure cell and we show, as the simplest case,

the impact of a magnetic texture on the NFS spectrum of an 57Fe foil, as used for

calibration purposes in conventional MS and NFS. Figure 5.4 shows the NFS spectrum

of this foil assuming (a) no texture and (b) a magnetic texture for the fit done with the

CONUSS program package [SG94]. It is evident that the fit without texture is only a

poor representation of the data, whereas the fit including texture delivers a much better

adjustment.

NFS MS

Time (ns) Velocity (mm/s)

-iron

20 60 80 100120 140 16040 -6 -4 -2 0 2 4 6

poly

-iron

texture

Intensity

Transmission

Figure 5.4: NFS spectrum of magnetic α-iron shown together with a fit assuming (a)

a random orientation of the magnetic hyperfine axis and (b) a magnetic texture (30%)

within the sample plane. The corresponding spectra (schematic) for conventional MS

are shown in the right panel.

In conventional MS, the texture can be considered by one additional parameter, de-

scribing the deviation of the averaged magnetization in the absorber plane from the

magic angle 54



. In the case of a magnetic iron foil, the direction of preferred mag-

netization normally lies predominantly in the plane of the foil, which results in an

averaged polarization angle larger than 54



The situation in NFS is different due to the well-defined polarization vector

σof the

SR. For a polycrystalline sample with different orientations of domains with magneti-

zation

M, the averaging procedure for all combinations of

Mand

σis done by the eval-

uation software. Deviations from a completely random orientation can be accounted

for by a texture coefficient Ctex, describing the percentage of Fe atoms with a certain

preferred magnetization direction

Mtex. In general, the orientation of

Mtex with respect

σis defined by another two parameters. For the above mentioned iron foil, the num-

ber of free parameters is reduced to one, namely Ctex, by the assumption of a random

distribution of

Mtex within the sample plane. The fit analysis in figure 5.4b yields a

percentage of 30% polarization.

5.2.2 Complex Hyperfine Interactions / External Field

So far, we considered the simple case of pure magnetic hyperfine interactions in a

57Fe-foil. However, all investigated compounds in this thesis exhibit a combination of

magnetic and electric interactions leading to more complicated spectra. The related

features are presented in the following example for NFS spectra of YFe2measured

at ambient pressure at the BW4 Beamline of HASYLAB in 2-bunch mode, providing

a time window of 480 ns (figure 5.5). The sample with 30% enrichment in 57Fe was

prepared by mixing the grinded material with paraffin. The area density of 12 mg/cm2

leads to an effective thickness of χ



40. The spectra were recorded at 300 K without

external field and at 77 K with a polarizing field of 3 T, provided by a superconducting

solenoid.

The NFS spectrum of YFe2without a polarizing field was fitted, according to sec-

tion 5.1, with two sites in the ratio 3:1 with Bhf

;

1= 18.6 T and Bhf

;

2= 17.9 T and a

quadrupole splitting ∆EQ

eQVzz

;

)

35 mm

s(β1



and β2



).4Sim-

ilar to the case of α-Fe without external field, a small magnetic texture (Ctex

20%)

has to be taken into account. This texture is actually not expected for a powdered

sample of a cubic compound embedded in paraffin, but may be due to a remaining

magnetization from previous experiments in external fields.

With the application of an external field Bext = 3 T at 77 K, the magnetization in the

polycrystalline YFe2sample is fully aligned in the same configuration as in the case

4Since an inversion of the energy scale for hyperfine interactionshas no effect on the time spectrum,

the sign of the electric quadrupole splitting can not be determined with this NFS spectrum. For the

Laves phase compounds the negative sign is well established (e.g. [BCK90],[Rup99a]) and therefore

indicated here in brackets for consistency. For the sign determination with NFS see section 5.3.5.

Intensity

Bext =0T

Bext =3T

0 50 100 150 200 250 300

Time (ns)

YFe2

Figure 5.5: NFS spectra of YFe2, measured at BW4 (HASYLAB) without (300 K) and

with(77 K) external field of 3 T. The solidlinesare fits assuming a magnetic texture with

Ctex

20% for the upper and 100% polarizationfor the lower spectrum. The dashed line

at the bottomindicatesthe calculated spectrum for uniform effective thicknessof χ

40.

of α-Fe in figure 3.9c:

Bext is perpendicular to the direction

kand polarization

Eof

the SR. A similar "simple" periodic beat structure appeared, now with a modulation

frequency corresponding to the smaller splitting of the ∆m = 0 transitions in YFe2.

Since the direction of the EFG is determined only by the orientation of the crystal

axes, the angle between quadrupole interaction and hyperfine field is now randomly

distributed. This situation is adopted in the CONUSS program package by the adjust-

ment withone value Beff forthe magnetic field and a gridof 48 subspectra with varying

angles β. The fitted value of Beff = 18.0 T leads to Bhf =B

eff -B

ext = -21.0 T at 77 K.

The external field lowers the hyperfine field at the nucleus because the hyperfine field

is antiparallel to the Fe band moment which is aligned by the external field.

According to the estimated effective thickness of χ



40 (χ

20 per resonance line)

a Bessel beat minimum should appear in the polarized NFS spectrum at 105 ns as

indicated by the calculated dashed line in figure 5.5. However, this calculation is only

valid for samples with homogeneous thickness such as properly rolled metallic foils.

The absence of a Bessel beat can be explained by a variation of the local thickness in

a powder sample. In this case the superposition of scattering intensities (see also the

coherence aspects in chapter 3.2.4) with minima at different places leads to a blurring

of the Bessel beats. For the actual fit of the YFe2data a variationof 15



7µmhas to be

assumed where



7 denotes half the width of a Gaussian distribution. Thus, the final

fit consists of subspectra with Bessel minima ranging from 70 ns (22 µm) to 200 ns

(8 µm). The impact of this effect is much less pronounced for the unpolarized case

since the effective thickness per resonance line is smaller and varies for the different

hyperfine transitions.

From the experience with grinded powder samples in paraffin, e.g. from X-ray absorp-

tion spectroscopy, such a huge thickness distribution of 15



7µm was unexpected and

thoughtto beunrealistic. However, as willbe shown inthe next section, the distribution

of different thicknesses depends strongly on the spatial resolution of the observation

method.

5.2.3 Thickness Distributions

With respect to the homogeneity of a sample, rolled foils with well-defined uniform

thickness are perfectly suited for many kinds of experiments with transmission geom-

etry. When such foils are not available the grinding of powder materials is the usual

way of sample preparation. In order to avoid effects of severe milling on the structural

and magnetic properties of a sample (studied for YFe2in [LAB95]) typical grain sizes

are in the range of about 1-5 microns as revealed by a microscopical inspection of the

present YFe2sample.

The resulting thickness distribution depends on the spatial resolution of the observa-

tion since integration over large (compared to the grain size) sample areas leads to an

averaging. This effect is demonstrated by a computer simulation of an artificial sam-

ple obtained by random distribution of spherical grains in a virtual "paraffin volume"

of 250µm



250µm



250µm (see figure 5.6). The number of different spheres with

diameters 1-5 µm (see table 5.1) is chosen in a way that every species of spheres oc-

cupies 1/5 of the total volume. In order to match the area density of the real sample

(12 mg/cm2, average thickness of YFe215 µm) a total number of 453838 spheres was

used.

sphere diameter (µm) number ratio

1 360000 1

2 72000 2

;

3 13333 3

;

4 5625 4

;

5 2880 5

;

Table 5.1: Distribution of different sphere diameters for the computer simulation of an

artificial sample in figure 5.6.

-100 -50 0 50 100

Distance ( m)

Thickness ( m)

Thickness ( m)



norm. Distribution

0 5 10 15 20 25 30 35 40

Figure 5.6: (a) Thickness of an artificial sample (mean thickness 15 µm) "seen" with

a spatial resolution of 0.1 µm (solid line) and 25 µm (dashed line). The sample was

obtained by random distribution of 453838 spheres in a volume of

(

250µm

)

3. The used

sphere diameters are listed in table 5.1. (b) Normalized thickness distributions for the

corresponding curves in (a). The thin lines represent Gaussian fits with a FWHM of

13 µm and 2.5 µm, respectively.

The resultingthicknesses arecalculatedwith0

1µmresolutionfora linearchain (length

200 µm) in the central part of the artificial sample (solid line in figure 5.6a). Within

this small spatial resolution the corresponding thickness distribution (figure 5.6b) is

very broad. It can be approximated by a Gaussian with FWHM of 13 µm. When the

thickness is averaged with a resolution of 25 µm (dashed line in figure 5.6) the sample

appears much more homogenous and the width of the distributiondecreases drastically

to a FWHM of 2.5 µm.

According to chapter 3.2.4 the spatial resolution of NFS is given by the transverse co-

herence length Ltr. Since this value is in the order of 10-100 nm for usual experimental

conditions, the large thickness distribution, necessary to explain the blurring of Bessel

beats, can be made plausible by the above simulation.

Although thickness distributions can be accounted for in the analysing software, they

renderthe data evaluation more difficultand tend to mask the influence of hyperfine in-

teractions. They could be reduced in powder samples with a larger thickness compared

to the grain size, keeping the effective thickness constant by the use of non-enriched

material. However, this pathway will reduce the available count rates in a NFS experi-

ment and is not desirable for high-pressure experiments.

5.3 Results

The high-pressure NFS investigations, presented in the following, were carried out

at room temperature (294 K) using the experimental set-up described in chapter 4.

Hereby, the magnetic properties of YFe2and GdFe2were studied up to a maximum

pressure of 105 GPa (=1.05 Mbar). As explained in the previous section 5.2.2, the

sophisticated influence of an arbitrary magnetic texture can be overcome by inducing

a well-defined polarization in an external magnetic field. On the other hand, the aver-

aging procedure for the angle βin the polarized case destroys the information on the

relative orientation of hyperfine field and electric field gradient. Therefore, the NFS

spectra were recorded as often as possible with and without an external field. In special

cases (GdFe2) this provides additional information on the magnetic ordering type.

5.3.1 YFe2

Fig. 5.7 shows NFS spectra of YFe2at various pressures with a polarizing field in the

same configuration as explained above. The simple beat pattern at ambient pressure,

now measured at 294 K with Bext = 0.75 T, is similar to that measured at 77 K with

Bext = 3.0 T (see figure 5.5) and can be analysed in the same way. The fit reveals a

hyperfine field of 18.6 T and a quadrupole splitting of (-)0.35 mm/s. With increasing

pressure we observe a decrease of the magnetic hyperfine fields, which can be tracked

by the increase of the beat period in the time spectra. The damping of the beat struc-

tures can be explained by a concomitant increase of the quadrupole interaction since

the random orientation of the two hyperfine interactions leads to a broadening of the

∆m

0 transition lines. This is best demonstrated in the spectrum at 50 GPa, where

the simple magnetic beat pattern is already strongly modified. Going further up with

pressure, there is a drastic change in the spectrum at 71 GPa, being now dominated by

a quadrupole interaction with a period of 120 ns, corresponding to (-)0.74 mm/s split-

ting, and slightly modified by a magnetic site with a hyperfine field of about 10 T. The

spectrum at 105 GPa arises solely from a quadrupole interaction with a beat period of

100 ns, corresponding to (-)0.87 mm/s splitting.

The non-magnetic origin of the beat pattern above 70 GPa is confirmed by NFS mea-

surements without external field (see figure 5.8): the spectra up to 43 GPa show a com-

plicated magnetic pattern whereas at 92 GPa and 105 GPa they have the same simple

shape than with polarizing field. Additional studies with this pressure cell cooled down

at 105 GPa in a closed-cycle He cryostat revealed the complete absence of magnetic

interactions down to temperatures as low as 15 K. This NFS spectrum is shown at the

bottom of figure 5.8 and indicated with a pressure of 115 GPa*. The pressure-increase

at low temperatures depends on the special properties of the pressure cell (such as

manufactured material, force generating mechanism or size). For this type of DAC

and pressure range it was found to be about 10% [Rup99a] leading to an estimated

YFe2, B = 0.75 T

ext

Intensity

0 50 100 150

Time (ns)

Intensity

0 50 100 150

Time (ns)

25 GPa

50 GPa

71 GPa

105 GPa

0GPa

10 GPa

20 GPa

15 GPa

Figure 5.7: NFS spectra of YFe2at variouspressures, measured at 294 K with an exter-

nal field of 0.75 T. The solid lines are fits which are explained in the text.

pressure of 115 GPa at 15 K.

The numerical evaluation of the series of NFS spectra was made along the guidelines

given in section 5.2, namely assuming a

1. small magnetic texture for the "unpolarized" spectra,

2. a random orientation of the quadrupole interaction with respect to the oriented

magnetic hyperfine field for the polarized case,

3. a large thickness distribution of typically d



8µm.

YFe2,B =0

ext

Intensity

0GPa

19 GPa

33 GPa

92 GPa

105 GPa

0 50 100 150

Time (ns)

0 50 100 150

Time (ns)

43 GPa

0 50 100 150

Time (ns)

115 GPa*

15 K

Figure 5.8: NFS spectra of YFe2at various pressures, measured without external field.

The spectrumat 115GPa* wasmeasured ina cryostatsystemat 15K. The corresponding

pressure value is estimated from earlier experiments [Rup99a]. The solid lines are fits

which are explained in the text.

4. At higher pressures the effects of a pressure gradient are adapted by the use of

up to three magnetic sites with slightly different hyperfine fields Bhf and angles

β.

5. For all subspectra the same isomer shift and quadrupole splitting was assumed.

NFS

YFe2

Bhf (T)

EQ(mm/s)

0 20406080100120

Pressure (GPa)

-1.0

-0.8

-0.6

-0.4

-0.2

Figure 5.9: Experimental results for the magnetic hyperfine field (top) and quadrupole

splitting (bottom) in YFe2obtained from the NFS spectra with (circles) and without

(diamonds) external field. Full symbols for Bhf represent average values for all magnetic

sites whereas open symbols stand for the total average, including also the non-magnetic

sites. The "*" indicates the results for an estimated pressure of 115 GPa obtained at

15 K. For comparison the values from a Mössbauer study [Rup99a] are also given. The

solid lines represent linear fits to the data in different pressure regimes divided by the

vertical dashed line at 40 GPa.

The resulting values for the hyperfine fields Bhf and the quadrupole splitting ∆EQ

are shown in figure 5.9 together with corresponding results from a Mössbauer study

[Rup99a]. Both data sets are in very good agreement.

The pressure dependence of Bhf exhibits a different behaviour in the range below

and above 40 GPa. Below this point, Bhf shows a small linear pressure derivative of

;

086

(

)

T/GPa (dlnBhf

;

0046

(

)

GPa

;

1) which agrees with NMR stud-

ies at 4.2 K up to 1 GPa [DRM86] (dlnBhf

;

0042

(

)

GPa

;

1). This slope

increases drastically to -0.43(5) T/GPa at higher pressures as indicated by the dashed

line in figure 5.9. This increase is induced by the appearance of a non-magnetic site in

the NFS spectrum at 43 GPa (15%) which becomes dominant at 71 GPa (80%). How-

ever, the NFS spectrum at 50 GPa with external field could not be analysed with the

occurrence of a non-magnetic site. Most probably, the influence of a non-magnetic site

is hidden by the averaging procedure for the angle βand the thickness. The external

field may also induce a defined transferred hyperfine field at the non-magnetic site.

Nevertheless, the mean value Bav agrees well with the corresponding value from MS

at 50 GPa which is obtained in the same way including also the non-magnetic site. By

extrapolating the pressure dependence to Bav = 0, a pressure of about 75 GPa is found

where the magnetic ordering temperature is suppressed to room temperature.

The behaviour of the quadrupole interaction ∆EQshows also a change at 40 GPa. The

linear decrease with d∆EQ/dP = -0.009(1) mm/s GPa in the low-pressure regime is

reduced to -0.004(1) mm/s GPa at higher pressures.

Having in mind that the intermetallic compounds RFe2tend to prefer the hexagonal

C14 structure at smaller volumes (see section 5.1), the above mentioned results sug-

gest the occurrence of a structural phase transition C15

C14 in YFe2. Especially

the appearance of a non-magnetic site above 40 GPa can be hardly explained in the

framework of the cubic C15 structure with only one species of equivalent Fe sites. In-

deed the XRD experiments by G. Reiss show additional lines in the diffraction pattern

above 20 GPa which belong to the hexagonal C14 phase [Rei00].

5.3.2 GdFe2

Two similar series of NFS spectra up to 105 GPa were taken for GdFe2with and

without external field (see figures 5.10 and 5.11).

At a first glance, the spectra at low pressure are similar to those of YFe2and show a

relatively simple beat pattern in the polarized case and a complicated pattern without

external field. However, in contrast to YFe2, the effective hyperfine field at the Fe

nuclei is obtained by adding the external field of 0.75 T to the hyperfine field for the

unpolarized case. This finding indicates that the Fe moment (µFe

6µB) is aligned

antiparallel to the external field whereas the much larger Gd momentis aligned parallel

to minimize the potential energy. Hence, the applied field confirms the ferrimagnetic

ordering type in GdFe2.

For higher pressures up to 53 GPa one observes an increased damping of the magnetic

beat pattern similar to YFe2. Going furtherto pressures above 71 GPa, the NFS spectra

show a more pronounced pattern again. Since this pattern, especially at 105 GPa, has

no longer the uniform shape as at low pressures, a principle change of the magnetic

behaviour can be deduced. Unambiguously, the sequence of NFS spectra demonstrates

that GdFe2remains magnetically ordered well above 300 K in the whole pressure

range.

The evaluation of the NFS spectra was performed in analogy to YFe2, but considering

the differences due to the ferrimagnetic ordering and an unknown easy magnetization

direction in the unpolarized sample (see section 5.1.2) by the following constraints:

GdFe2, B = 0.75 T

ext

Intensity

0 50 100 150

Time (ns)

Intensity

0 50 100 150

Time (ns)

0GPa

5GPa

17 GPa

28 GPa

53 GPa

71 GPa

87 GPa

105 GPa

Figure 5.10: NFS spectra of GdFe2at various pressures, measured at 294 K with an

external field of 0.75 T. The solid lines are fits which are explained in the text.

1. Since the magnitude of the external field is weak, it is supposed not to disturb the

ferrimagnetic ordering type throughout the high-pressure series. Therefore, the

value of Bext is always subtracted from the measured effective hyperfine fields.

2. The unpolarized spectra are fitted with 4 magnetic sublattices having different

angles βbetween hyperfine field and main axis of the EFG in the range 25

;



. Free parameters are the relative weight of the sublattice and the respective

hyperfine field Bhf.

3. Due to the lack of a defined direction of the magnetization axis and the large

GdFe2,B =0

ext

Intensity

0 50 100 150

Time (ns)

Intensity

0 50 100 150

Time (ns)

0GPa

8GPa

20 GPa

30 GPa

36 GPa

45 GPa

53 GPa

105 GPa

Figure 5.11: NFS spectra of GdFe2at various pressures, measured without external

field. The solid lines are fits which are explained in the text.

number of free parameters introduced in the previous point, the spectra are fitted

without the assumption of a magnetic texture.

The numerical results for Bav and ∆EQare plotted in figure 5.12. Again, both hyperfine

parameters indicate a drastic change of the magnetic and/or structural properties at a

certain transition pressure, here 53 GPa. The low-pressure region up to 20 GPa is char-

acterized by a linear decrease of Bav with dBav

;

175

(

)

T/GPa. Between 20

and 53 GPa, the slope is slightly smaller before the value of Bav drops from 14.7 T at

53 GPa to 11.1 T at 71 GPa. Beyond this jump a modest linear decrease is observed

with dBav

;

035

(

)

T/GPa reaching 9.9 T at 105 GPa.

NFS

GdFe 2

Bavg (T)

0 20406080100120

Pressure (GPa)

-1.6

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

EQ(mm/s)

Figure 5.12: Experimental results for the averaged magnetic hyperfine field Bav (top)

and quadrupole splitting ∆EQ(bottom) in GdFe2obtained from the NFS spectra with

(circles) and without (diamonds) external field. For comparison the values from a Möss-

bauer study [Lu00] are also presented (triangles), including preliminary results above

20 GPa. The solid lines represent linear fits to the data of Bav in different pressure

regimes.

The quadrupolesplitting ∆EQshows no pronouncedpressure dependence up to 53 GPa.

Above 53 GPa, however, the spectra can only be modeled with a drastically increased

electric field gradient with ∆EQ= -1.3(2) mm/s at 71 GPa.

The agreement with results from MS [Lu00] is very good for pressures below 20 GPa.

At higher pressures the MS results indicate also a drastic change of the magnetic prop-

erties: a drop of the average hyperfine field and a sharp increase of the EFG. However,

the transition pressure lies between 20 and 28 GPa whereas for NFS it is above 50 GPa.

This discrepancy is not due to uncertainties in the pressure determination, because the

experiments at 28 GPa, 50 GPa and 105 GPa were performed on the same sample for

MS and NFS. The differences could be explained when the observed transitionshows a

time-dependence: The NFS experiments were carriedout within 1-2 hours afterthe ad-

justment and calibration of a new pressure while the MS experiments were performed

within 1-2 months afterwards.

In analogy to YFe2the drastic changes of the hyperfine parameters may be caused by

a structural transformation from the C15 to the C14 phase. Complementary XRD ex-

periments revealed diffraction patterns with hexagonal classification at high pressure,

here again above 50 GPa [Rei00], as found also in the NFS studies. This structural

transition was observed in a completely independent high-pressure run. The XRD ex-

periments were carried out in a 30-60 min time scale, which is comparable to the NFS

studies. Comparison of XRD and NFS with the MS results demonstrates that the C15

C14 transition is slow, at least at 300 K.

5.3.3 TiFe2

The effect of pressure and temperature on the magnetic properties of hexagonal TiFe2

(C14) was studied with conventional MS up to 20 GPa [LHZ96]. For the series of

NFS experiments TiFe2was measured at ambient pressure and various temperatures

as a reference sample for antiferromagnetic ordering with a non-magnetic 2a site. Ac-

cording to the ordering temperature TN



285 K[LHZ96] the NFS spectrum at 294 K

shows no magnetic beat pattern (figure 5.13). The evaluation of the magnetic spectrum

at 20 K revealed the same set of parameters as the mentioned MS experiments: (i) a

magnetic (6h) and a non-magnetic (2a) site in the ratio 3:1, (ii) a hyperfine field of

9.4 T for the 6h sites, (iii) a quadrupole splitting ∆EQ= -0.4 mm/s, (iv) two angles

β6h



and β2a



and (v) an asymmetry parameter η

4 for the 6h sites.

Intensity

0 50 100 150

Time (ns)

0 50 100 150

Time (ns)

300 K 20 K

TiFe2,B =0

ext

Figure 5.13: NFS spectra of TiFe2at ambient pressure for 294 K and 20 K measured

without external field. The solid lines are fits which are explained in the text.

5.3.4 ScFe2

In ScFe2with hexagonal C14 structure a series of NFS spectra was measured up to

51 GPa. Since the magnetization in ScFe2does not saturate up to about 4 T [NY86]

the sample could not be fully polarized with the available external field of 0.75 T and

Intensity

0 50 100 150

Time (ns)

Intensity

0 50 100 150

Time (ns)

0GPa

6GPa

11 GPa

15 GPa

21 GPa

26 GPa

35 GPa

51 GPa

ScFe2,B =0

ext

Figure 5.14: NFS spectra of ScFe2at various pressures, measured at 294 K without

external field. The solid lines are fits which are explained in the text.

the NFS spectra are recorded only without external field (figure 5.14). The spectrum

at ambient pressure exhibits the familiar shape with two pronounced beatings around

50-80 ns, indicating a well-defined hyperfine field at all Fe sites. At higher pressures

a strong decrease of the magnetic hyperfine fields is observed. Above 26 GPa the

beat pattern changes drastically and the spectrum at 51 GPa indicates the absence of

magnetic order at 294 K. The minimum is caused by thickness effects.

The spectra are fitted according to the hexagonal structure of ScFe2with two subspec-

tra in a ratio 3:1 representing the 6h and 2a sites with β6h



and β2a



(see

section 5.1.2). Due to the small influence of the quadrupole interaction ∆EQon the

NFS

ScFe 2

Bhf (T)

0 1020304050607080

Pressure (GPa)

-0.8

-0.6

-0.4

-0.2

EQ(mm/s)

Figure 5.15: Experimental results for the magnetic hyperfine fields Bhf (top) and

quadrupole splitting ∆EQ(bottom) in ScFe2obtained from the NFS spectra at 294 K.

Full symbols for Bhf represent average values for all magnetic sites. The open symbols

stand for the total average, including also the non-magnetic 2a sites. For comparison the

values from Mössbauer studies [Lu00, Rup99a] are also given (triangles). The solid line

represents a linear fit to the data in the low pressure regime.

NFS spectra, this parameter was chosen to be identical for both sites. In order to ob-

tain a reasonable modeling of the data, a relatively strong magnetic texture has to be

considered (35%). For pressures above 21 GPa a distribution of hyperfine fields with

a width of 2% (21 GPa) to 5% (35 GPa) of the mean value was assumed. The results

for ∆EQand the magnetic hyperfine fields Bhf are plotted in figure 5.15.

At ambient pressure, the 6a and 2a sites show the same hyperfine field Bhf of 17.7 T.

The low-pressure region up to about 20 GPa is characterized by a linear decrease of

Bhf following dBhf

;

160

(

)

GPa, identical for both sites. At 35 GPa the

hyperfinefield of the 2a sites drops to zero whereas the field at the 6h sites is reduced to

about 10 T. This behaviour is similar to the situation in TiFe2and indicates a pressure-

induced transition of the ordering type in ScFe2from ferro- to antiferromagnetism

exhibiting a reduced hyperfine field at the 6h sites and a non-magnetic 2a site.

This decrease of the magnetic hyperfine fields is not simply due to a decrease of the

ordering temperature Tm, i.e. an increase of T

Tm. This is proved by additional exper-

iments at high pressure and low temperature, which reflect a decrease of the magnetic

Intensity

0 50 100 150

Time (ns)

0 50 100 150

Time (ns)

294 K

240 K

130 K

50 K

ScFe2,P=51GPa

0 50 100 150 200 250 300

Temperature (K)

Bhf (T)

Figure 5.16: (top) NFS spectra of ScFe2at 51 GPa



and various temperatures, measured

without external field. The solid lines are fits which are explained in the text. (bottom)

Correspondingresults for the magnetic hyperfine fields Bhf. Full symbolsrepresent aver-

age valuesfor all magnetic sites. The open symbolsstand for the totalaverage, including

also the non-magnetic 2a sites.

Fe moment. 5Figure 5.16 shows NFS spectra at 51 GPa



and various temperatures

down to 50 K. 6The "slow" beat pattern at 50 K reveals a value of Bhf = 4.5 T. Fol-

lowing the linear relation between Bhf and µFe in [BCK90], this hyperfine field corre-

sponds to µFe

3µB. The evaluation could not be performed with a fixed ratio of 3:1

for the two subspectra, but revealed a coexistence (ratio about 1:1) of a magnetic and

a non-magnetic site (as observed in TiFe2).

5Mössbauer studies at 77 K indicate a reduction of Bhf from 19.7 T at 0 GPa to 10.4 T at 40 GPa

[Rup99a]. This corresponds to a moment reduction from 1.3µBto 0.7µB.

6The pressure of 51 GPa



was measured at 294 K. Similar to the experiments withYFe2at 15 K, the

"*" indicates the occurence of a pressure increase at low temperatures, estimated to be 10% at 50 K.

5.3.5 Measurement of Isomer Shifts

The objectiveof this section is to demonstratethe feasibility ofmeasuring isomer shifts

Swith the 57Fe resonance using a reference sample as introduced in chapter 5.3.5. The

knowledge of Sgives information on the electronic charge density at the nucleus, a

higher charge density leads to a smaller S. An ideal reference sample should exhibit a

relatively sharp single nuclear resonance and an effective thickness comparable to the

effective thickness per resonance line of the sample.

During the different runs of NFS experiments several attempts have been made to

measure the isomer shift of YFe2at different pressures with an enriched K4Fe

(

)

reference (powder) sample, embedded in paraffin. Although this reference provides a

sharp single resonance line and an adjustable thickness, the NFS spectra of the com-

bined samples could not be analysed: the random thickness distribution of both sample

and reference leads to undefined experimental conditions.

Stainless steel foils with uniform geometrical thickness were available for us only in

non-enriched form and their use as a reference sample is therefore hampered by the

strong electronic absorption; for instance a 50 µm (25 µm) thick foil has an effective

thickness of χ

15 (7.5) and an absorption of more than 90% (70%) at 14.4 keV.

Three examples of isomer shift measurements are shown in figure 5.17a for YFe2at

0 GPa, 25 GPa and 92 GPa measured together with stainless steel absorbers (Bext

75 Tfor 0 and 25 GPa). The spectrum at 92 GPa was obtained in the last runat ESRF

(April 1999) using focusing optics (see 4.1.3) and an already prepared DAC from MS

studies performed in Paderborn [Rup99a]. For the fits of the NFS spectra the results

fromaccompanying measurements of thepure YFe2sample and the stainless steel foils

are combined with the isomer shift as the only free parameter. The fitted values for the

isomer shift of YFe2relative to stainless steel are 0.03(4) mm/s, -0.21(4) mm/s and

-0.51(4) mm/s for 0, 25 and 92 GPa. It should be mentioned that the sign of Scan not

be determined definitely from NFS alone. For YFe2at 0 GPa the positive sign was

confirmed by conventional MS whereas the negative sign at 25 and 92 GPa follows

from the well-known negative pressure derivative of S(describing an increase of the

s-electron density with reduced volume).

Once the sign of Sis fixed, the combined spectrum of YFe2at 25 GPa and the stainless

steel absorber provides the possibility to determine the (negative) sign of the EFG with

NFS. This special case is due to the fact that the asymmetry of the energy spectrum at

25 GPa (see right panel of figure 5.17a) is governed by the sign of the EFG providing

a clear distinction between positive and negative values. Furthermore, this energy

spectrum at 25 GPa demonstrates, when compared with the spectrum at 0 GPa, the

increase of ∆EQand the decrease of Bhf (see figure 5.9).

The results for Sare plotted in figure 5.17b relative to the isomer shift at ambient pres-

sure; they agree quite well with values from MS [Rup99a]. The accuracy for the mea-

surement of isomer shifts with NFS depends very much on the specific combination of

YFe2vs. stainless steel

Intensity

0 50 100 150

Time (ns)

0.0

0.2

0.4

0.6

0.8

1.0

-4 -3 -2 -1 0 1 2 3 4

Energy (mm/s)

GPa

25 GPa

92 GPa

0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

020406080100120

Pressure (GPa)

-0.8

-0.6

-0.4

-0.2

0.0

Isomer shift (mm/s)

NFS

YFe2

Figure 5.17: (a) NFS spectra of YFe2measured together with stainless steel (SS) as

reference sample. For details of the fits, see text. The corresponding energy spectra

are shown in the right panel with stainless steel (grey line) set to zero. (b) Results for

the isomer shift of YFe2from these NFS spectra (circles). For comparison with the

results from MS [Rup99a] (triangles) the values for both series are given relative to

the respective isomer shift at ambient pressure. The uncertainty of the MS results is

estimated to 0.01 mm/s [Rup99b].

sample and reference. It improves significantly with the number of beating periods in

the time spectrum caused by the energy difference of the nuclear resonances in sample

and reference. This is strikingly demonstrated by experiments with the 151Eu reso-

nance, where a large difference of about 10 mm/s between the two valence states Eu2

and Eu3

and a corresponding beating period of 6 ns in the time window of 100 ns

allows a precise determination of isomer shifts [PLS99]. For the case of 57Fe with

rather small differences in S, the precision could be enhanced by NFS measurements

in a longer time window.

5.4 Discussion

In this section we discuss the pressure-induced changes of the magnetic ordering be-

haviour and Fe moment formation in the investigated Laves phases. Figure 5.18 sum-

marizes, as a function of the Fe-Fe distance dFe

;

Fe, the NFS results for the mag-

netic ordering temperatures Tmand the averaged hyperfine fields Bav of YFe2, GdFe2,

and ScFe2. In addition, Tmvalues for YFe2from a XRD study of the magneto-

volume anomaly [Rei00], and Tmand Bhf values from MS on TiFe2[LHZ96] are

presented. The structural transformations from C15 to C14 are indicated by vertical

dashed (YFe2) and dashed-dotted (GdFe2) lines [Rei00].

All the present RFe2systems show a considerable decrease of the magnetic hyper-

fine fields and, concomitantly, of the magnetic Fe moments.7In table 5.2 the volume

coefficients ΓBhf :

;

dlnBhf

dlnVat low pressure are listed for the present RFe2

compounds8, indicating that the initial moment reduction becomes more pronounced

at smaller dFe

;

Fe spacings going from YFe2to ScFe2and TiFe2. For YFe2and ScFe2

the moment reduction is accelerated at higher pressures reaching a coexistence of a

low-moment and a non-magnetic state. The latter properties are very similar to those

of TiFe2at ambient pressure and low temperatures [LHZ96] and can be attributed

to the hexagonal C14 structure with antiferromagnetic ordering between Fe atoms in

neighbouring 6h planes and non-magnetic Fe atoms in the intermediate 2a sites.

The volume coefficients ΓTm:

;

dlnTm

dlnVof the magnetic ordering temperature

at ambient pressure are presented in the last column of table 5.2.9Regarding the be-

haviour at higher pressures in figure 5.18, the continuous variation of Tmin YFe2is

of particular interest: Tmincreases, despite the moment reduction, from TC

535K

at ambient pressure (dFe

;

Fe = 2.60 Å) to 660 K at 15 GPa (2

52Å) and then drops to

294 K around 75 GPa (2

34Å) and finally to below 15 K at 115 GPa (2

27Å). This

pressure dependence is illustrated in figure 5.18a by a dashed line.

Similar volume effects on TCand µFe are observed for "negative pressure" in hydro-

genated YFe2[BD76]. In isostructural YFe2H4with an 8% increase of the lattice

parameter, the ordering temperature is lowered to 308 K, whereas the iron moment is

enhanced by 25% from 1

45µBto 1

83µB. This topic is discussed in more detail in the

next subsection.

Among the studied RFe2systems, ScFe2represents the link between YFe2and TiFe2.

The properties of ScFe2at ambient conditions, hexagonal structure and ferromagnetic

ordering, are approached by YFe2at pressures of about 20 GPa (2

50Å). With further

decreasing Fe

;

Fe distance bothYFe2and ScFe2become antiferromagnetic(dFe

;

7Precise values for the moment reduction were obtained from additional MS studies at low temper-

atures [Rup99a].

8The negative sign in the definition of ΓBhf (and later ΓTm) provides the same sign of pressure and

volume coefficients

9ΓTmis also called the magnetic Grüneisen parameter.

12 N

-Fe

8 N 12 N

C15

C14

Zr Yb Pr

-Fe -Fe

B (T)

T (K)

100

200

300

400

500

600

700

800

900

1000

C15C14

2.25 2.3 2.35 2.4 2.45 2.5 2.55 2.6 2.65 2.7

Fe-Fedistance(A)

Bav (T)

Figure 5.18: Summary of NFS results for the magnetic ordering temperature Tmand

averaged hyperfine fields Bav at 294 K (TiFe2at 78 K) as presented in the previous sec-

tions. Full symbols are for fm ordering and open symbols for afm ordering. The two TC

values for YFe2marked with "*" were obtained from XRD experiments measuring the

magneto-volume anomaly [Rei00]. All compressibilites and the C15

C14 phase tran-

sition for YFe2and GdFe2(vertical dashed and dashed-dotted lines in (b), respectively)

were provided by G. Reiss [Rei00].

dFe

;

Fe µFe TmK0dBhf

dp ΓBhf dT

dp ΓTm

(Å) (µB) (K) (GPa) (T/GPa) (K/GPa)

YFe22.60 1.45 535 133 -0.086(5) -0.61(3) 6a1.5

GdFe22.61 1.6 790 104 -0.175(10) -0.84(3) 3a0.4

ScFe22.48 1.4 542 153 -0.160(10) -1.38(3) - -

TiFe22.39 0.7 285 193 -0.258b-5.30 -3b-2.0

α-Fe 2.49 2.2 1043 168 -0.055c-0.3 0c0

Table 5.2: Magnetic properties of RFe compounds and α-Fe at ambient pressure; ΓBhf :

;

dlnBhf

dlnV,ΓTm:

;

dlnTm

dlnV. The pressure and volume coefficients of Bhf are for

room temperature, except TiFe2(78 K). References: dFe

;

Fe and K0are from [Rei00], µFe and

Tmfrom [BCK90], afrom [BBM74], bfrom [LHZ96], cfrom [WBI72].

;

40Å) and are finally transformed to a non-magnetic state at dFe

;



;

35Å.ForYFe2the latterfindingis verifiedbylow-temperatureexperimentsat115GPa.

Thus, the complete variety of magnetic properties in RFe2systems with non-magnetic

constituents Rcan be reproduced in a single compound, namely YFe2, by the applica-

tion of external pressure. This means that the Fe

;

Fe distance dFe

;

Fe is a universal

parameter for the iron magnetism in RFe2. This even holds for the structural phase

transformation C15

C14 with decreasing dFe

;

Fe.

A different behaviour is observed in GdFe2which also exhibits the C15

C14 transi-

tion but remains magnetically ordered well above room temperature up to the highest

pressures. This is clearly caused by the magnetic Gd sublattice; details are discussed

in section 5.4.2.

5.4.1 Volume Dependence of Magnetic Ordering Temperatures Tm

According to a survey on band structure theory of magnetism in 3d-4f compounds

by M. Richter [Ric98a], the theoretical treatment of Curie temperatures is one of the

most demanding tasks in the field of first principles theories of magnetic properties.10

Beside the principal problems of electronic band structure calculations delivering only

groundstate propertiesat T = 0 K, physical intuitionis needed to parameterize the rele-

vant excitations at elevated temperatures: (i)single-particle spin-flipStoner excitations

and (ii) collective spin fluctuations, which are of many-particlenature. The importance

ofspin fluctuations is reflected by the fact that the energy of a Stoner excitation is equal

to the exchange splitting ∆. In pure Fe, however, this splitting (1.8 eV) is far too large

10A recent theoretical study of Y-Fe compounds revealed a TCof 650 K for YFe2at ambient pressure

[SJ98] which is about 20% higher than the experimental value of 535 K.

to explain the measured Curie temperature of 1043 K. Whereas the Stoner theory with

the parameter Iexplicitely includes only intraatomic exchange (which leads to the ex-

change splitting ∆), the energyspectrum of collective spin fluctuationsdepends mainly

on interatomic interaction between neighbouring spins.

In this section the main pressure effects on the intra- and interatomic interactions and

their influence on the magnetic ordering temperatures are discussed. Two "classical"

models are presented, which describe these effects on magnetic ordering temperatures

in magnetic 3d systems.

The most obvious pressure effect in transition metal compounds is the enhanced over-

lap of neighbouring 3d wave functions. Following Heine’s band structure calculations

[Hei67], this leads to an increased 3d bandwidth

W∝d

;

(5.3)

where dis the interatomic distance. Furthermore, the concomitant lowering of D

(

)

induces a reduction and, finally, a suppression of the magnetic moment µFe. For RFe2

systems, this effect is directly reflected by the decrease of the magnetic hyperfine fields

with decreasing volume (see table 5.2 and figure 5.18b). Since the energy gain due to

the spin polarizationof the Fe 3d band is proportionalto µ2

Fe (see equation 5.1), the mo-

ment reduction should destabilize the magnetically ordered phase and, therefore, result

in a decrease of TC. This is also suggested by the mean-field approach in equation 5.2,

which simplifies for pure Fe magnetism to

3kBTC

2aFeFe

6AFeFe SFe

(

SFe

)

(5.4)

)

TC∝AFeFe µ2

(5.5)

Interestingly, the opposite behaviour is observed for YFe2, namely an increase of TC

from 535 K at 0 GPa to 660 K at 15 GPa. This result can be explained by a pressure-

induced increase of the interatomic coupling AFeFe between the Fe moments, which

overcompensates the effect of moment reduction. Using the simple formula 5.5 and

assuming the same volume dependence for µFe and Bhf (table 5.2) one obtains a rough

estimation for the increase of AFeFe in YFe2at ambient pressure:

ΓTC

;

dlnAFeFe

dlnV

;

2dlnµFe

dlnV(5.6)

) ;

dlnAFeFe

dlnV

ΓTC

;

2ΓBhf



7 (5.7)

For higherpressures, one has to takeinto account that the magnetic 3d moments couple

via direct exchange with overlapping electronic orbitals. From the Pauli exclusion

principle, it is obvious that parallel alignment of neighbouring spins must become

unfavourable at high compression. Consequently, the exchange energy and TCtend to

decrease at higher pressures leading to an antiferromagnetic state.

Bethe-Slater Curve

A qualitativediscussion of the pressure effecton the exchange coupling and TCis given

in the literature in terms of the Bethe-Slater curve (figure 5.19). Here, an interatomic

exchange interaction between two localized moments is expressed by the interaction

parameter Jas a function of dFe

;

2r3d, where r3dis the mean 3d-shell radius at

ambient pressure. Barbara et al. [BGG73] interpreted their results on dT

dpin YFe2

(5 K/GPa), Y6Fe23 (0.1 K/GPa) and R2Fe17 (-9.8 K/GPa) by the distance dependence

of the exchange interaction along this curve.

The similarity between the shape of the Bethe-Slater curve and the TCdependence of

YFe2in figure 5.18a suggests that this intuitive interpretation can be extended to the

present results which are obtained for YFe2just by the application of external pressure.

Since the Bethe-Slater curve is based on the presence of localized magnetic moments,

this implies that the Fe band moment in YFe2at low pressures exhibits a high degree

of localization. With increasing interatomic interaction at high pressures, reflected by

the suppression of µFe, the picture of localized moments is not valid any more. The

following model takes the band character of the 3d moment into account.

d-2r(A)

0.5

-200

-100

100

200

300

Exchange energy (relative)

0.7

-Fe

YFe2

YFe (115 GPa)

YFe

623

YFe

217

Ni-V

0.9 1.1 1.3 1.5

Figure 5.19: Bethe-Slater curve describing the distance dependence of the exchange

interactionwith Fe-Fe distance dFe

;

Fe and r3d

55 following[BGG73]. The position

of variousiron compoundsis marked by arrows. Further informationon the Bethe-Slater

curve can be found in [Jil91, BCK90].

Wohlfahrt Model

A more quantitative description of the pressure effect on TCwas introduced by Wohl-

fahrt [Woh81, WW81, BBM74] based on the Stoner-Wohlfahrt theory of itinerant

ferromagnetism. Neglecting the influence of the sp-band, TCis assumed to be propor-

tional to the so-called degeneracy temperature TFwhich depends on the shape of the

electron density of states D

(

)

at the Fermi energy:

(

;

)

(5.8)

With the abbreviation ¯



(

)

, including Iand D

(

)

from the Stoner criterion,

1 leads again to ferromagnetism.

As elaborated in [Woh81] the following pressure dependence can be derived from

equation 5.8 using the compressibility κ:

3κTC

;

κα

TC(5.9)

where α

IbT2

(5.10)

or in terms of volume:

ΓTC

;

(5.11)

Here, Iis the effective and Ibthe bare intraatomic interaction between the itinerant

electrons (I

1). The main assumptions for the derivation of equation 5.9 are a

proportional relationship of TFand bandwidth Wunder pressure as well as an electron

density of states D

(

)

∝1

The first term on the right-hand side in equation 5.9 is proportional to TCand hence

dominant for ferromagnets with high TC. This is the case for YFe2at ambient pressure,

where 5

3κTC

7K/GPa is close to the experimental value of dT

6K/GPa

[BB73]. The corresponding value for αis about (220 K)2. However, the overall pres-

sure dependence for YFe2(with negative ΓTCfor relative volumes V



85) can

not be described with a constant value for α. According to equation 5.11, the volume

coefficient ΓTCchanges its sign, when αapproaches a critical value

αc

3T2

(5.12)

A rough estimation of TC



700K at the reversal of the ordering temperature in YFe2

can be obtained from the inspection of figure 5.18a (dashed line), leading to αc



(900 K)2. Neglecting the weaker volume dependence of intraatomic correlations I

[Woh81], the necessary increase of αmust be attributed to a fourfold increase of TF.

Taking into account the above assumption TF∝W, this enormous increase of TFand

αis not predicted in the framework of equation 5.9.

Hence, the Wohlfahrt theory can model the volume dependence of TCin YFe2only

at low pressures. Considering the underlying simplifications, a real quantitative de-

scription can not be expected. This holds especially for the electron density of states

(

)

, which is by far more complex than assumed in the Wohlfahrt model. This is

demonstrated in a band structure calculations of YFe2[MS85, ADM86], which also

reveals an induced moment of 0.45 µBat the Y site, in accordance with NMR studies

[DRM86].

5.4.2 Intersublattice Coupling Under Pressure

In contrast to YFe2, the behaviour of GdFe2is governed by the interplay of two sepa-

rate magnetic sublattices (3d band magnetism and localized 4f moments). In GdFe2at

ambient pressure, the Fe sublattice contributes dominantly to the ordering temperature

of 790 K, reflecting a large coupling parameter JFeFe. The higher TCin comparison

to YFe2(535 K) is due to the exchange JFeGd between 3d and 4f moments. The cou-

pling parameter JGdGd is smaller and can be estimated from the ordering temperature

TGd

140K of the Gd sublattice [SW99] (see also the discussion on pages 54-56).

Identifying the behaviour of YFe2as the pressure dependence of pure iron magnetism

in RFe2compounds, one must conclude from the results presented in figure 5.18 that

in GdFe2the Gd sublattice takes over the leading role in the Fe moment formation

above 50 GPa.

Since the Gd 4f moment can be regarded as pressure-independent, this statement has

implicationson the twocoupling constants JGdGd and JFeGd above 50 GPa: JGdGd must

be large enough for a magnetically ordered Gd sublattice at room temperature and the

intersublattice coupling JFeGd must be strong enough to induce a magnetic moment at

the Fe sites. The observed average hyperfine field of about 10 T in GdFe2at 105 GPa

is too large to be simply transferred from the 6 neighbouring Gd atoms [BCK90].

For GdFe2under pressure, resistivity measurements revealed a strong increase of TGd

to about 180 K at 7 GPa [Str00]. The corresponding volume coefficient dT

dlnV

;

1040

(

300

)

K can be used for an extrapolation to 50 GPa (



25% volume reduction),

indicating that TGd reaches room temperature at that pressure.11

As mentioned in section 5.1.2 the interaction between the Gd and Fe sublattices is

mediated by a hybridization between Fe 3d and Gd 5d states, where the latter are

spin polarized by the 4f moments. When pressure is applied, the 3d-5d overlap will

11Another example for a pressure-induced increase of 4f-4f spin coupling was given in a recent NFS

study by the Paderborn group. Here, the magnetic ordering temperature in EuTe was found to increase

from 10 K at ambient pressure (NaCl phase) to almost 100 K at 22 GPa (CsCl phase) [LPH99, RW99,

LRW00].

strongly increase leading to an enhanced coupling constant JFeGd. This qualitative

volume dependence was experimentally verified by Liu et al. for a variety of Er-Fe

compounds with different molar volumes [LBC94]. This study was also accompa-

nied by theoretical calculations of a hypothetical GdFe2with the lattice parameters of

ErFe2, corresponding to a reduction of the GdFe2lattice parameter by 1.5%. The cou-

pling parameter JFeGd was found to be almost 10% larger than for the uncompressed

lattice. Hence, also JFeGd can be considered to increase strongly in the investigated

pressure range.

Summarizing, the observations for pure iron magnetism in YFe2and the pressure-

induced increase of the coupling parameters JGdGd and JFeGd support the conclusion

that the magnetic strength of the Gd sublattice surpasses that of the Fe sublattice above

50 GPa and becomes the driving force for the Fe moment formation. A more quanti-

tative discussion requires the determination of ordering temperatures in GdFe2at high

pressures, a demanding task for future NFS experiments. It should be further remem-

bered that we are concerned, above 50 GPa, with GdFe2in the C14 structure.

Chapter 6

Lattice Dynamics in Iron Under

Pressure

This chapteris concernedwith the study oflattice dynamics in the bcc and hcp phase of

iron at pressures up to 42 GPa. For this challenging task the new technique of Nuclear

Inelastic Scattering of synchrotron radiation is employed, which has been introduced

in chapter 3.3. In the first part (chapter 6.1), the general high-pressure behaviour of

elemental iron is discussed considering also its geophysical relevance. Some principle

features and equations of lattice dynamics which are necessary for later discussion are

summarized in the second part. The rest of the chapter is devoted to the experimental

results and their discussion.

6.1 Iron Under Pressure

The inner core of the Earth consists almost entirely of iron or an iron-rich alloy. This

geophysical aspect is one of the many motivations to study the phase diagram of iron

with its various allotropes in a wide pressure and temperature range (see figure 6.1).

This holds especially for the hcp high-pressure phase of iron (ε-Fe), which is con-

sidered to be the most relevant phase for the inner core [YAC95] with pressures of

300-400 GPa and temperatures between 4000 K and 8000 K. At ambient pressure and

temperature, iron is ferromagnetic and crystallizes in the bcc structure (α-Fe). The

phase transformation from α-Fe to ε-Fe around 13 GPa was first reported by Bancroft

et al. [BPM56]. As evidenced later, ε-Fe is nonmagnetic down to 30 mK [CTW82].

Depending on the employed pressure-transmitting medium, the α-εtransformation ex-

hibits a broad coexistence range of both phases [BB90, TPJ91]. The transition interval

(the pressure range between onset and completion of the transition) increases with the

shear strength of the pressure medium from less than 0.2 GPa in helium to 15 GPa in

Al2O3.

Liquid

Melting

uncertainty

Solid

OC-IC



(?)

' (?)





100 200 300

2000

4000

6000

Pressure (GPa)

Temperature (K)



Figure 6.1: High-pressure phase diagram of iron, modified from [SMW96], including

establishedphases α(magnetic bcc), γ(fcc), δ(nonmagnetic bcc), and ε(hcp) as well as

two new phases β(proposed dhcp) and α’ (proposed bcc), not yet established. For a re-

view on recent experimental work and the discussionabout the new phases see [And97].

The arrow ("OC-IC") at



330 GPa indicates the pressure at the inner-core/outer-core

boundary.

Beside an enormous amount of experimental work related directly to the α-εphase

transformation, e.g. with x-ray diffraction [MBT67], shock-waves [BH74] and x-ray

absorption [WI98], there is also recent theoretical work on this martensitic phase tran-

sition [MJ96, BW97, EEB98].

While a considerable amount of crystallographic data on the pressure/temperature

phase diagram of Fe is available, there is, due to the difficult access, less experimental

informationon the lattice dynamics of iron under pressure, especially in the hcp phase.

6.2 Lattice Dynamics

In general, the lattice dynamics of solids are described by phonons, which are char-

acterized by their dispersion relations ωj

(

)

for different branches jand by their den-

sity of states (DOS) g

(

)

. The dispersion relations are usually measured with single-

crystalline samples using inelastic neutron scattering (see e.g. [AM76]). Correspond-

ing spectra for α-Fe at ambient conditions and for γ-Fe at ambient pressure and 1428 K

can be found in [MSN67] and [ZS87], respectively. The phonon DOS g

(

)

can be cal-

culated from nearest-neighbour force constants [GR66], which are obtained by fitting

the experimental results with a Born-von-Karman model [MSN67] (see figure 6.2).

However, experimental work with inelastic neutron scattering under pressure is diffi-

cult due to the requirement of large sample volumes (> 10 mm3) and the limited flux of

existing neutron sources [KBS95]. Exploratory measurements at pressures up to 7 GPa

are reported for Fe3Pt using a sample volume of 25 mm3[KBS95]. A set of transverse

acoustic phonon frequencies has been measured for Ge up to 9.7 GPa [KBB97]. Infor-

mation on the phonon dispersion is also accessible from the rapidly developing field of

inelastic x-ray scattering with synchrotron radiation. Here, first high-pressure studies

are performed for CdTe at 7.5 GPa [KMM97]. Optical vibrational modes at the zone-

center (Γpoint)can be investigated with Raman spectroscopy. Corresponding pressure

experiments are reported by Olijnyk et al. for Zr metal up to 16 GPa [OJ97].

In contrast to inelastic scattering with neutrons and x-rays, Nuclear Inelastic Scattering

(NIS) gives direct access to the phonon DOS without the necessity of measuring the

full dispersion relations. Since NIS requires only polycrystalline samples, the experi-

0 5 10 15 20 25 30 35 40 45 50

-Fe

nergy (meV)

0.00

0.02

0.04

0.06

0.08

0.10

0.12

g(E) (1/meV)

Force constant model

Dispersion relation (single crystal)

Figure 6.2: (top) Dispersion relation for α-Fe at ambient conditions from inelastic neu-

tron scattering [MSN67]. (bottom) The solid line represents the derived phonon DOS.

The dashed line shows the Debye approximation, calculated from equation 6.5 with

ΘD=ED

kB= 420 K.

ments are not hampered by pressure-induced structural phase transformations, which

limit neutron or x-ray scattering studies, requiring single crystals.

The principles and terminology of particular aspects of lattice dynamics are discussed

below:

6.2.1 Debye Model

According to [AM76] the normalized phonon DOS g

(

)

can be expressed in the gen-

eral form as

(

2π

)

3∑

(

;

¯hωj

(

))

;

(6.1)

whereVis the volume of the unit cell and the integral is taken within the first Brillouin

zone. Similar to the case of the electronic DOS, this equation can be written in an

alternative form [AM76]:

(

2π

)

3¯h

3∑

∇ωj

(

)

;

(6.2)

where the integration is now taken over that surface in the first Brillouin zone on which

¯hωj

(

)



E. Due to the periodicity of ωj

(

)

, there will be a series of discontinuities in

(

)

, reflecting the fact that the group velocity appearing in the denominator of equa-

tion 6.2 must vanish at some frequencies. As in the electronic case, the discontinuities

are related to van Hove singularities.

In the Debye approximation gD

(

)

of the phonon DOS, the same linear dispersion

relation ω

vkis assumed for the three acoustic branches of ωj

(

)

with an average

sound velocity v. In addition, the integral in equation 6.1 is replaced by an integral

over a sphere of radius kD, chosen to fulfil the normalization condition for gD

(

)

(

) =

(

2π

)

(

;

¯hvk

)

k(6.3)

2π2

k2δ

(

;

¯hvk

)

dk (6.4)

αE2

;



;

(6.5)

with ED

¯hvkDand α

2π2¯h3v3

(6.6)

For the above transformations the relations

4π

k2dk,E

¯hvkand dk

¯hv have been used. The cut-off energy EDis determined by the normalization

(

)

;



6π2¯h3v3



;

(6.7)

Consequently, the complete phonon DOS gD

(

)

is determined by a single parameter,

namely ED. It is often convenient to define a Debye temperature ΘD

kBas a

measure of the temperature, above which all phonon modes begin to be excited and

below which modes begin to be "frozen out". Both, EDand ΘD, can be also regarded

as measures of the rigidity of the crystal. For iron at ambient conditions a value of

ΘD= 420 K can be derived from high-temperature specific heat data [AM76]. The

Debye approximation for the phonon DOS of α-Fe is shown in figure 6.2 togetherwith

the experimental data derived from neutron scattering. Under pressure, the phonon

energies and hence the Debye temperature ΘDshould increase, since a normal solid

becomes more rigid under compression. The pressure dependence of ΘDis usually

given in terms of the Debye-Grüneisen parameter γD, which is discussed in the next

subsection.

6.2.2 Grüneisen Parameter

The theory of harmonic lattice vibrations fails in the description of important physical

phenomena such as thermal expansion and the volume dependence of elastic coeffi-

cients. It is evident that these effects of anharmonicity in the interaction energy should

have a decisive impact on the equations of state of condensed matter under strong

compression.

Usually, the harmonic theory is expanded for a quasi-harmonic lattice by taking into

account anharmonicities only through mode Grüneisen parameters

γi

;

dlnωi

dlnV(6.8)

for the volume dependence of the lattice mode frequencies ωi. In most cases, the Mie-

Grüneisen approximation is used which implies that all mode Grüneisen parameters γi

can be replaced by one common average value γ, which depends only on the volume

(see [AM76, Hol96]). In the Debye model, where all mode frequencies scale linearly

with the cut-off energy, one obtains

γD

;

dlnΘD

dlnV

(6.9)

Within the Debye approximation, the parameter γDbecomes identical to the thermo-

dynamic Grüneisen parameter

γth

αVKTV

;

(6.10)

withthe volumeexpansion coefficientαV, the isothermal bulk modulus KT, the volume

Vand the isochoric heat capacity cV. Thus γth is composed of individual measurable

physical quantities, each of them varying significantly with temperature. However, the

ratio of these properties is almost independent of temperature. Usually one observes

γth

2, and generally γth decreases as the volume decreases.

Importantgeophysicalsignificance of γDinthe case ofironresults fromtheLindemann

law in the form

dlnTm

dlnρ



γD

;



;

(6.11)

which relates the melting temperature Tmto the density ρ.1Equation 6.11 can be

used to extrapolate existing melting data up to 200 GPa [Boe93] to the pressure of the

outer-inner core boundary at 330 GPa. Furthermore, γis needed for the evaluation of

shock-wave data up to 240 GPa [BM86] and their extrapolation to 330 GPa.

Estimates on the temperature profile of the Earth’s core rely on the assumption that

the boundary between the solid inner core and the liquid outer core is at the melting

temperature of the core material. Given that the core is composed of an iron-nickel

alloy with small amounts (< 10%) of lighter elements, and accounting for the fact that

impurities tend to lower melting temperatures, the Tmvalue for pure iron at or near

330 GPa would place an upper limit to the boundary temperature [Buk99].

A further important aspect of γis connected with the heat of crystallization at the

solid-liquid core boundary which depends on γ[AD97]. This continuously generated

thermal energy is the driving force for the Earth’s magnetic field, which is produced

by convection in the electrically conducting outer core [Ols97]. Furthermore, the core

provides sufficient heat to influence even mantle convection and hence tectonic mo-

tions, earthquakes and volcanism [Buk99].

6.2.3 Elastic Coefficients and Aggregate Velocities

Another significant aspect of lattice dynamics is the propagation of sound waves in

matter. In the case of iron, the geophysical relevance of sound waves under pressure is

obvious since seismic waves are the most important probes for the internal structure of

the Earth (see figure 6.3). Of actual interest are the sound velocities in the ε-phase of

iron, since seismic wave experiments indicate an anisotropy of the sound velocities in

the Earth´s inner core [Cre92] and a differential rotation of the inner core with respect

to the adjacent liquid outer core [SR96].

1The representation 6.11 of the Lindemann law is developed for example in [And95] assuming a

Debye-like solid and a Grüneisen parameter γDfollowing equation 6.9. In its original form (see e.g.

[Gsc64]), the Lindemann law includes a special "melting Debye temperature" Θm

Dand thus a special

Grüneisen parameter γm. However, as shown in [Gsc64], the approximation γm



γDis valid for a large

number of elements.

1000 km

2900 km

5080 km 330 GPa

360 GPa

140 GPa

40 GPa

6371 km

Depth

Upper mantle

Lower mantle

Outer core (liquid)

Inner core (solid)

Pressure

Figure 6.3: Schematic structure of the Earth’s interior with different layers as derived

from seismological observations (adapted from [Hol98b]). Whereas the mantle is made

up of minerals and rock, the liquidouter core and solidinner core consist mainly of iron.

An estimation for the boundary temperature between outer and inner core is provided by

the melting temperature of iron.

Sound waves are intimately related to the elastic properties of solid materials. In or-

der to compare studies of sound velocities and elastic coefficients, this relationship is

briefly discussed in the next paragraphs following [SAS73].

Elastic coefficients are defined in terms of the response of a crystal to an applied stress.

In themost general case, a 6x6 matrixCwith36 elastic stiffness coefficientscij is used.

Its inverse is called elastic compliance S. In the absence of body torques both matrices

are symmetric and the number of independent elastic coefficients reduces to 21 in the

lowest-symmetry case of a triclinic crystal. For higher crystal symmetry, this number

is further reduced: for a hexagonal system to five and for a cubic system down to three.

The bulk modulus Kand the shear modulus Gof polycrystalline solids can be calcu-

lated from Cand Swith various averaging schemes. The Voigt approximation is based

on the assumption that the stress is uniform throughout the sample and gives:

(

c11

c22

c33

(

c23

c13

c12

)

;

(

c11

c22

c33

)

;

(

c23

c13

c12

(

c44

c55

c66

)

;

whereas the Reuss scheme assumes an uniform strain providing:

= (

s11

s22

s33

(

s23

s13

s12

)

;

(

s11

s22

s33

)

;

(

s23

s13

s12

(

s44

s55

s66

)

The Voigt and Reuss averaging schemes determine upper and lowerlimits forthe poly-

crystalline moduli. Their arithmetic mean is usually used as the most probable values

for Kand G.

For cubic systems, symmetry aspects lead to c11

c22

c33,c12

c23

c31,c44

c55

c66. For hexagonal systems one obtains c11

c22,c13

c23,c44

c55,c66

(

c11

;

c12

)

The so-called aggregate velocities can be calculated from K,Gand the density ρfor

longitudinal or compressional waves: v2





;

(6.12)

and for transversal or shear waves: v2

(6.13)

The commonly used notation with subscripts pfor "primary" and sfor "secondary"

originates from seismology because the longitudinal waves are faster and detected ear-

lier than the shear waves.

The velocities vpand vsare simply related to the mean sound velocity v in the Debye

model (equation 6.5), because the sum over three acoustic branches in equation 6.1

with a mean sound velocity v can be replaced by the sum over one longitudinal branch

with sound velocity vpand two transverse branches with sound velocity vs:

(6.14)

6.3 Nuclear Inelastic Scattering in Iron

Now we turn to the experimental results which were obtained with Nuclear Inelastic

Scattering (NIS). A tiny 57Fe sample of 0.5 µg was pressurized in a diamond-anvil cell

as explained in chapter 4. In the first part of this section we present the measured NIS

spectra and the extraction of the phonon DOS. The second part covers the derivation

of related properties and their discussion.

6.3.1 NIS Spectra

The NISspectra were measured forα-Fe (0 GPa, 6 GPa and 11 GPa) and ε-Fe (20 GPa,

32 GPa and 42 GPa) at room temperature (294 K). At 20 GPa and higher, the addi-

tionally measured NFS spectra revealed the non-magnetic phase of iron, confirming a

complete structural transition to ε-Fe. The energy spectra are shown in figure 6.4 after

normalization with Lipkin’s sum rule (equation 3.28). They consist of a central peak

originating from elastic scattering and sidebands resulting from inelastic scattering

with concomitant annihilation (left side) and creation (right side) of phonons. Accord-

ing to equation 3.31 the relative weight ofthe twosidebands is given by exp

(

kBT

)

Under pressure, the main spectral features are shifted to higher energies, reflecting an

increase of lattice vibration frequencies; consequently, the asymmetry gets more pro-

nounced at higher pressures.

The spectrum at ambientpressure is shown with thecorrespondingspectrum calculated

with the phonon DOSfromneutron scattering [MSN67] using equations 3.25, 3.26 and

3.27. For comparison it is convoluted with the present spectrometer function. The two

data sets are shown in absolute scale without any adjustable parameter, demonstrating

good agreement between the two methods.

As outlined in section 3.3.2, NIS spectra of iron are dominated by single-phonon scat-

tering. The small contribution of multi-phonon scattering is clearly visible at the high-

energy side of the spectra (indicated by a little arrow in the spectrum at ambient pres-

sure). It amounts less than 12% of the total intensity of the sidebands.

The spectra shown in figure 6.4 are added from 20-30 single energy scans of the high-

resolution monochromator (HRM) in a range of about



110 meV around the nuclear

resonance. Before summing up, the energy scale of every single scan is calibrated with

the zero position from the resolution function, measured in forward direction.

All spectra at high pressure were measured with a nested HRM composed of two chan-

nelcut Si(42 2) and Si(122 2) crystals. The spectrumat ambientpressure was recorded

in an earlier beamtime with a similar monochromator but using Si(4 2 2)/Si(9 7 5) re-

flections. The corresponding energy resolutions are obtained by fitting the measured

resolution functions with a Gaussian shape (see table 6.1). For ambient pressure and

above 20 GPa (ε-Fe)the observed resolutionmatches the expected values of



4.4 meV

0.000

0.002

0.004

0.006

0.000

0.002

0.004

0.006

-80-60-40-200 20406080

Relative energy (meV)

0.000

0.002

0.004

0.006

-80 -60 -40 -20 0 20 40 60 80

Relative energy (meV)

0GPa

6GPa

11 GPa

20 GPa

32 GPa

42 GPa

-Fe -Fe

Probability density of recoil (1/meV)

Figure 6.4: Energy dependence of Nuclear Inelastic Scattering in α-Fe (left) and ε-Fe

(right) at various pressures. The spectra are normalized according to equation 3.28. The

intensityof theinelasticsidebandsisproportionalto(1-fLM); itdecreases withincreasing

pressure by about 50%, reflecting the change in the recoil-free fraction, fLM, from 0.80

to about 0.90 (see table 6.2). The solid line at 0 GPa is calculated from neutron data

[MSN67], convoluted with the spectrometer function. The little arrow in the spectrum

at ambient pressure indicates the small contribution of multi-phonon scattering.

and



6.4 meV, respectively. However, the spectra at 6 GPa and 11 GPa show higher

values of 7.9 meV and 9.8 meV. The reason for this resolution broadening is not com-

pletely clear. A defective HRM adjustment can be rejected, because all spectra are

measured with the same set-up in the sequence 20 GPa, 32 GPa, 6 GPa, 11 GPa,

42 GPa. One possible reason for the energy broadening is the interplay between focus-

ing optics, pre-monochromator and the HRM, since the spot size of the concentrated

synchrotron radiation (100 µm x 100 µm) matches the sample size for 20 - 42 GPa, but

is considerable smaller than the sample at 6 - 11 GPa (diameter 250 µm).

Pressure (GPa) FWHM (meV)

0 4.6

6 7.9

11 9.8

20 6.5

32 6.5

42 6.4

Table 6.1: Energy resolution for the NIS experiments obtained as the FWHM (full width at

half maximum) of a Gaussian which is adjusted to the forward scattered energy spectrum.

6.3.2 Extracted Phonon DOS

The procedure to extract the phonon DOS g

(

)

from the NIS spectra was introduced

in detail in section 3.3.2. For this purpose the spectra were normalized and multi-

phononcontributionsas well as thecentral elastic peakwere subtracted. The remaining

spectrum represents the single phonon contribution, from which g

(

)

was extracted

(equation 3.26)2.

In order to verify the normalization of the resulting phonon DOS, the areas were cal-

culated. They showed a slight variation around unity with a mean deviation of 3.1%.

In a previous NIS study of the temperature-dependent behaviour of elemental iron

[CRB96], a similar deviation of 2.3% was found and attributed to statistical errors.

The uncertainty of the normalization was corrected by renormalizing each g

(

)

The resulting phonon DOS for α-Fe and ε-Fe are shown in figure 6.5. The DOS at am-

bient pressure is again well described by the solid line representing the data from neu-

tron scattering [MSN67]. A strong increase of all spectral features with pressure is ob-

vious. The resolved high-energymaximum ofthe DOS, originatingmainly fromlongi-

tudinal acoustic (α-Fe) and longitudinal acoustic and optical (ε-Fe) phonon branches,

is shifted from 35 meV in α-Fe at ambient pressure to 51 meV in ε-Fe at 42 GPa. The

observed changes in the phonon spectra reflect mostly the reduced volume of the unit

cell; modifications resulting from the structural transition are small for mono-atomic

lattices (see e.g. [AAM93]) and are not resolvable with the present energy resolution.

The present resolution, however, is sufficient to derive a set of related parameters from

the integral properties of the phonon DOS.

2In general, it is possible to extract the phonon DOS individuallyfrom both sides of the NIS spectra

(see figure 3.12). In practice, the (left) annihilation part is usually discarded due to its lower statistical

accuracy.

0.00

0.01

0.02

0.03

0.04

0.05

0.06

g(E) (1/meV)

-Fe

0.00

0.01

0.02

0.03

0.04

0.05

0.06

g(E) (1/meV)

0 1020304050607080

Energy (meV)

0.00

0.01

0.02

0.03

0.04

0.05

0.06

g(E) (1/meV)

-Fe

0 1020304050607080

Energy (meV)

0GPa

6GPa

11 GPa

20 GPa

32 GPa

42 GPa



Figure 6.5: Phonon DOS g

(

)

for α-Fe and ε-Fe derived from the NIS spectra in fig-

ure 6.4. The solidline represents the DOS from neutron scattering[MSN67], convoluted

with the spectrometer function.

6.3.3 Derived Properties

The phonon DOS g

(

)

is of fundamental importance for the study of lattice dynamics.

Its knowledge provides information on the lattice rigidity as well as thermodynamic

properties according to the following equations and figures. The results are summa-

rized in table 6.2. In order to examine the reliability of the derived parameters, they are

compared with those from neutron data of α-Fe at ambient pressure (first line in table

6.2).

100

PressureV/V0fLM (area) fLM (DOS)

∆x2

Fvib Uvib cVSvib Dv

(GPa) (10

;

3Å2) (meV/at.) (meV/at.) (kB/at.) (kB/at.) (N/m) (km/s)

neutron 1 - 0.801(4) 4.15 5.9 83.6 2.71 3.07 173 3.54

0 1 0.791(6) 0.802(4) 4.13(8) 7.4(10) 83.4(15) 2.70(3) 3.00(3) 185(12) 3.57(10)

6 0.967 0.848(12) 0.852(8) 3.00(16) 23.1(20) 87.0(30) 2.58(6) 2.52(6) 279(25) 4.10(25)

11 0.945 0.855(12) 0.870(8) 2.61(16) 29.6(20) 89.4(30) 2.52(6) 2.36(6) 322(25) 4.16(25)

20 0.862 0.883(6) 0.882(4) 2.35(8) 31.5(10) 90.0(15) 2.51(4) 2.31(4) 320(15) 4.82(15)

32 0.828 0.893(6) 0.893(4) 2.12(8) 38.1(10) 91.8(15) 2.44(4) 2.12(4) 365(15) 5.00(15)

42 0.805 0.898(6) 0.897(4) 2.03(8) 40.8(10) 93.0(15) 2.42(4) 2.06(4) 388(15) 5.14(15)

Table 6.2: Properties of iron at T

294K: Lamb-Mössbauer factor fLM derived from the area of the inelastic spectrum and from the

density of phonon states g

(

)

, mean-square displacement

∆x2

; lattice contribution to Helmholtz free energy Fvib, internal energy

Uvib, isochoric specific heat cVand entropy Svib; mean force constant Dand mean sound velocity v, derived from the low-energy

part of g

(

)

. For details see text. V/V0: relative volume of unit cell obtained from the compressibilitydata for α-Fe [MBT67] and

ε-Fe [MWC90] withV0

093 cm3/mol. The uncertainties given in brackets represent statistical standard deviations. The error in

the recoil fraction 1 - fLM was estimated by the normalization uncertainty of g

(

)

101

Lattice Rigidity

First, the Lamb-Mössbauer factor fLM, describing the elastic (recoil-free) fraction of

nuclear scattering, can be calculated from g

(

)

using an equation from Singwi and

Sjölander [SS60]:

fLM

exp

;

∞

(

)

;

kBT

;

kBTdE

(6.15)

As mentioned in chapter 3.3.2, fLM can be also determined from the area under the

normalized NIS spectrum (

;

fLM). The values for both procedures are listed in

table 6.2 and show rather good agreement. Due to the possibility to renormalize the

obtained g

(

)

, the fLM(DOS) values from equation 6.15 are less influenced by experi-

mental error [CRB96]. They are used to calculate the mean-square displacement

∆x2

;

(

fLM

)

;

(6.16)

using the wave vector k = 7.31 Å

;

1of the 14.413 keV quanta.

The Lamb-Mössbauer factor fLM(DOS) from g

(

)

and the mean-square displacement

are plotted in figure 6.6. In α-Fe fLM shows a steep increase from 0.802(4) at ambient

pressure to 0.870(8) at 11 GPa. After the phase transition the slope is strongly reduced

and a value of 0.897(4) is reached at 42 GPa (V/V0= 0.805). Since the recoil-free

fraction fLM can not exceed the value of 1, the pressure effect on the lattice dynamics

from 0 GPa to 42 GPa is more pronounced in the mean-square displacement

∆x2

which is reduced by 50% from 4.13(8)



;

3Å2to 2.03(8)



;

3Å2.

The hardening of the crystal lattice is also reflected by the behaviour of the mean

force constant D. Following [CS99, KCR98], D

(

)

can be obtained along the direction

kof the incident x-ray beam using the mass mof the vibrating atoms:

(

¯h2

∞

(

)

E2dE

(6.17)

In the investigated pressure range, Dshows a strong increase from 185(12) N/m to

388(15) N/m.

The mean sound velocity v can be determined fromthe low-energypart of g

(

)

, where

one can assume a linear relation between the phonon frequency ωand the lattice k-

vector. With an average slope v

kfor the three acoustic branches, a quadratic

energy dependency is obtained (see equation 6.5 and [KCR98]).

102

0.75

0.8

0.85

0.9

0.95

1.0

fLM

100

200

300

400

500

D(N/m)

3.5

4.0

4.5

5.0

5.5

6.0

v(km/s)

(10-3 A2)



0 1020304050

Pressure (GPa)

Figure 6.6: Lamb-Mössbauer factor fLM from g

(

)

, mean-square displacement

∆x2

mean force constant Dand mean sound velocity v. The dashed lines are guides to the

eye. The vertical lines indicate the borders of the α-εtransformation.

103

The low-energy parts up to 15 meV are shown in more detail in figure 6.7 together

with a quadratic fit g

(

αE2to the experimental data. In order to increase the

number of data points for the fit, the phonon DOS from both sides of the NIS spectra

is used. According to equation 6.6, the mean sound velocity v was derived from αand

the corresponding volume Vfor each pressure (see fig 6.6).

In contrast to fLM the values of v indicate a discontinuity at the phase transition. When

the transition pressure is approached, a jump by 16% from 4.16(25) km/s at 11 GPa to

4.82(15) km/s at 20 GPa is observed. Within the ε-phase the change is smaller and v

increases only to 5.14(15) km/s at 42 GPa.

0 GPa

6 GPa

11 GPa

20 GPa

32 GPa

42 GPa

-15 -10 -5 0 5 10 15

Energy (meV)

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

0.016

g(|E|) (1/meV)

Figure 6.7: Plot of g

(

)

up to 15 meV for each pressure together with a quadratic fit to

the experimental data. For clarity the spectra are successively shifted by 0.008 meV

;

104

Thermodynamic Properties

The vibrationalcontributionto a varietyof thermodynamicpropertiescan be calculated

from g

(

)

in the quasi-harmonic approximation, beginning with the Helmholtz free

energy Fvib and deriving the internal energy Uvib, the specific heat cVand the entropy

Svib [JM85]:

Fvib

3kBT

∞

(

)

2kBT

;

2kBT

dE (6.18)

Uvib

;



∂F

∂T



∞

(

)

EeE

kBT

;

1dE (6.19)



∂U

∂T



3kB

∞

(

)

(

kBT

)

2eE

kBT

(

kBT

;

)

2dE (6.20)

Svib

;



∂F

∂T



3kB

∞

(

)

(

kBT

)

kBT

;

1(6.21)

;



2kBT

;

2kBT

i

Using the Bose occupation factor nB

(

;

(

exp

(

kBT

)

;

)

, the equation for

Uvib simplifies to

Uvib

∞

(

)



(

;

)



(6.22)

This equation elucidates the definition of g

(

)

dE as the number of phonon states

within the energy interval [E,E

dE], which has to be multiplied with the energy

(

;

)

of the corresponding harmonic oscillators to obtain the internal

energy Uvib.

The pressure dependence of the thermodynamic parameters at 294 K (figure 6.8) in-

dicates a discontinuos behaviour at the α-εphase transition reflecting the different

thermodynamic properties of the two phases. Whereas the pressure-induced increase

of Fvib and Uvib demonstrates the energy gain of the lattice vibrations, the decrease of

cVand Svib can be interpreted in terms of a decreasing effective crystal temperature at

high pressure. This effective crystal temperature can be defined as the ratio T

ΘDof

the real temperature Tand the Debye temperature ΘD. The latter is discussed in the

next section.

105

Fvi b (meV/atom)

100

Uvib (meV/atom)

0 1020304050

Pressure (GPa)

1.8

2.0

2.2

2.4

2.6

2.8

3.0

3.2

Svib (kB/atom)

2.2

2.4

2.6

2.8

3.0

cV(kB/atom)





Figure 6.8: Thermodynamic properties of iron at T

294Kderived from g

(

)

: lattice

contribution to Helmholtz free energy Fvib, internal energy Uvib, isochoric specific heat

cVand entropy Svib. The dashed lines are guides to the eye.

106

Debye Temperature ΘD

One of the basic features of the Debye model is the representation of the phonon DOS

(

)

by a single parameter ΘD(see page 92). This allows an easy discussion of the

volume dependence of the phonon DOS with the widest impact on the related proper-

ties. Since the actual g

(

)

for a real solid is generally not completely Debye-like, it

can not be described by a single parameter. However, in the absence of experimental

data on g

(

)

, the measured temperature dependence of thermodynamic properties is

usually approximated assuming a Debye-like behaviour, e.g. for cV

(

)

, and within

this approximation a single parameter ΘDis obtained. In principle, these Debye tem-

peratures are related to the energy part of the phonon DOS, which determines the

behaviour of the corresponding thermodynamic property. A well-known example is

the Debye temperature derived from the specific heat cV

(

)

in different temperature

regions: whereas the low-temperature part of cV

(

)

is governed by low-energy vi-

brations, the high-temperature part includes information about the complete phonon

spectrum (see e.g. [Gsc64]). The two resulting Debye temperatures are usually called

low-temperature limit and high-temperature limit of ΘD.

In the case of a measured phonon DOS, a Debye temperature can be assigned directly

to the function g

(

)

. Using certain properties of the Debye DOS gD

(

)

, three different

possibilities to determine ΘDdirectly from the experimental g

(

)

are obvious: the

extraction of ΘDfrom

1. the low-energypartof g

(

αE2fromwhich also the meansound velocity was

derived. Accordingto equation 6.7, ΘDcan be calculated by kBΘD

)

;

These values represent the low-temperature limit of ΘD(see ΘD(lt) in table 6.3);

2. the average phonon energy Eph, i.e. the first moment of the phonon DOS, which

is given in the Debye model by ΘD

)

Eph

kB. The corresponding values

represent the high-temperature limit of ΘD(see ΘD(ht) in table 6.3);

3. the cut-off energy in the phonon DOS, determined by longitudinal branches. Al-

though this value can be easily estimated from g

(

)

without further calculation,

the exact determination is arbitrary and much more influenced by the energy

resolution than the above mentioned procedures. Therefore, this possibility is

discarded in the following.

Within the studied pressure range the Debye temperatures in table 6.3 and figure 6.9

show an increase by 45% and 55% for ΘD(ht)and ΘD(lt), respectively. The differences

between the twovaluesreflect deviationsofthe measured phononDOSfromthe Debye

model. In α-Fe the results for ΘD(lt) are about 30 K higher than for ΘD(ht) (except

11 GPa). This behaviour is well-known for normal solids, especially with respect to

measurements of the specific heat cV, which are often represented within the Debye

model, but with the use of a temperature dependent ΘD.3

3However, the phonon DOS itself does not change with temperature and the presentation with a

107

0 1020304050

Pressure (GPa)

(K)

400

500

600

700

800 

D(lt)

(ht)

Figure 6.9: Pressure dependence of Debye temperatures, determined from the low-

energy part of g

(

)

(ΘD(lt)) and the first moment of g

(

)

(ΘD(ht)).

In ε-Fe the difference between the two ΘDvalues is about 100 K. Consequently, the

deviations from the Debye model are much more pronounced in this high-pressure

phase. Since the Debye model includes only acoustic vibrations with ω

vk, the

deviations can be attributed to the existence of optical branches in the hcp phase.

AnthirdsetofDebye temperaturescanbe derivedfromtherecoilless fraction fLM(DOS)

in table 6.2 using the formalism of conventional Mössbauer spectroscopy [Gib76].

These values (ΘD(fLM) in table 6.3) lie in between the above mentioned limits.

Pressure V/V0Eph ΘD(ht) ΘD(lt) ΘD(fLM)

(GPa) (meV) (K) (K) (K)

neutron 1 27.2 421 460 439

0 1 28.0(8) 433(12) 467(10) 440(5)

6 0.967 33.3(16) 515(24) 542(25) 523(12)

11 0.945 36.4(16) 563(24) 555(25) 563(12)

20 0.862 36.5(8) 565(12) 663(15) 597(10)

32 0.828 38.7(8) 599(12) 697(15) 632(10)

42 0.805 40.7(8) 630(12) 723(15) 646(10)

Table 6.3: Mean phonon energy Eph and Debye temperatures ΘDfor iron derived from

various procedures as explained in the text.

variable ΘDcan be removed to a large extent by the use of a reasonably adjusted "pseudo-Debye"

model [Hol94, Hol98a].

108

Temperature Behaviour

Considering the usual quasi-harmonic approximation, the phonon DOS g

(

)

can be

treated as temperature-independent. Consequently, once g

(

)

is determined at room

temperature, the calculation of the complete temperature dependence of thermody-

namic properties is possible [JM85]. This is demonstrated in figure 6.10 for fLM

(

)

∆x2

(

)

,Fvib

(

)

,Svib

(

)

,Uvib

(

)

and cV

(

)

at 0 GPa and 42 GPa.

At T

0K the crystal lattice is not completely frozen leading to a Lamb-Mössbauer

factor fLM

(

)

1 and to a zero-point motion with

∆x2

0. The latter value

decreases under pressure from 1



;

3Å2at 0 GPa to 1



;

3Å2at 42 GPa.

Furthermore, the Helmholtz free energy and the internal energy are identical at ab-

solute zero. The value Fvib

(

Uvib

(

)

increases from 41.8 meV/atom to

42 GPa

0 GPa

0.4

0.5

0.6

0.7

0.8

0.9

1.0

fLM

100

150

200

250

Uvib (meV/atom)

0 100 200 300 400 500 600 700 800

Temperature (K)

cv(kB/atom)

0 100 200 300 400 500 600 700 800

Temperature (K)

(10-3 A2)

-250

-200

-150

-100

-50

100

Fvib(meV/atom)

Svib(kB/atom)

D D

Figure 6.10: Temperature dependence of thermodynamic properties in iron at 0 GPa

(solid lines) and 42 GPa (dashed line) derived from the phonon DOS g

(

)

. The cor-

responding Debye temperatures ΘD(ht) are depicted at the bottom. Svib represents the

temperature derivative of Fvib and cVthe derivative of Uvib.

109

60.5 meV/atom at 42 GPa. At high temperature (T

ΘD) the internal energy Uvib

approaches the classical behaviour Uvib

3kBT. This is also reflected in the specific

heat cV

∂U

∂T

)

Vwherethe classical Dulong-Petitvalue of3kB/atomis approached

at T



ΘD.

6.4 Discussion

In the following discussion, we concentrate on the Debye temperature ΘDand its vol-

ume dependence, namely the Grüneisen parameterγD. Furthermore, the present results

for the sound velocity are compared with recent values from ultrasonic and elasticity

studies as well as theoretical calculations. Finally, the impact of the derived thermody-

namic properties on the α-εphase transition is discussed.

6.4.1 Grüneisen Parameter

The volume dependence of the phonon DOS can be discussed within the Debye ap-

proximation with a common average value γDreplacing the mode Grüneisen parame-

ters γidefined in equation 6.8. Thus, γDis related to the average phonon energy Eph

and can be expressed with the corresponding Debye temperature ΘD(ht) :

γD

;

dlnΘD

(

)

dlnV

(6.23)

The left part of figure 6.11 shows the volume dependence of ΘD(ht) together with a fit

according to equation 6.23, including the three points for α-Fe and ε-Fe, respectively.

The resulting Grüneisen parameters are γD(α-Fe)=4.3(5) and γD(ε-Fe)=1.5(2).

A similar evaluation for the low-temperature values ΘD(lt) is depicted in the right part

of figure 6.11. The corresponding values γD

;

lt(α-Fe)=3.4(5) and γD

;

lt(ε-Fe)=1.3(2)

indicate that the volume dependence of the low-energy modes is less pronounced than

the average volume dependence.

Both values, γDand γD

;

lt, for α-Fe are larger than commonly observed values: The

equation for the thermodynamic Grüneisen parameter γth

αVKTV

cVleads to a

value for ambient conditions of γth

(

)

= 1.60 (αV= 35.1



;

1[AM76], KT=

163 GPa [MBT67], V0= 7.093 cm3/mol [MBT67] and cV

3 J/(K mol) [Kuc82]).

Ramakrishan et al. [RBK78] measured γth up to 3.3 GPa and found γth

(

66 and

6, using the form

(

)(

)

(6.24)

However, a direct comparison of NIS results with values from ambient or lower pres-

sure is difficult since the NIS results for γDrepresent average values for the pressure

110



400

500

600

700

800

D(ht) D(lt)

0.750.80.850.90.951.0

V/Vo



0.80.850.90.951.0

V/Vo

Figure 6.11: Volume dependence of the Debye temperatures ΘD(ht) and ΘD(lt). The

solid lines represent fits to the experimental data using equation 6.23.

ranges of 0 - 11 GPa (α-Fe) and 20 - 42 GPa (ε-Fe). As shown for example forthe γ-α

transition in metallic Ce, the Grüneisen parameter can exhibit a strong increase near a

high-pressure phase transition [RK80].

A more detailed discussion of the volume dependence of γ, e.g. with equation 6.24 is

not possible with the present amount of data. Future NIS studies with furtherimproved

flux of SR may provide a larger number of data points in both phases to give a more

detailed picture of the high-pressure behaviour of the Grüneisen parameter, especially

at the α-εphase transition.

6.4.2 Aggregate Velocities and Shear Modulus

In order to explain sound velocity data for seismological observations, several recent

theoretical investigations deal with the elastic coefficients of hcp iron under strong

compression [SC95, SMW96, SSC99]. Due to the difficult experimental access to hcp

iron there are little possibilities to compare such calculations with laboratory experi-

ments. Only recently, a new experimental method was developed measuring the lattice

strain at high pressure with radial x-ray diffraction [HMS97]. When interpreted in

terms of elastic coefficients, this method delivered results on the elastic coefficients in

Fe up to 52 GPa [SMH98] and to 211 GPa [MSS98]. However, the interpretation of

lattice-strain experiments is controversely discussed and their re-examination is rec-

ommended in [SSC99].

The connecting link between seismology and elastic coefficients are the longitudinal

(vp) and transversal sound velocity (vs), which are determined by the bulk modulus

K, the shear modulus Gand the density ρ(equations 6.12, 6.13). Although these

aggregatevelocities can not be obtaineddirectlyfromthe momentum-averagedphonon

DOS, they can be determined indirectly when the equation-of-state data is used to

provide not only the unit cell volume but also the bulk modulus Kat high pressure.

111



aggregate velocity (km/s)



[MSS98]

[SJS98]

[SMW96]

calculations:

experiments:

[SC95]

[SSC99]

this work

020

40 60 80 100

ressure (GPa)

Figure 6.12: Pressure dependence of the aggregate velocities vpand vstogether with

data from the literature. The experimental data points were obtained with lattice-strain

(full squares, diamonds) and ultrasound (open square) studies.

In this case, the results for v permit the determination of the shear modulus Gand

the aggregate velocities vpand vsfrom the three equations 6.12, 6.13 and 6.14. The

resulting values are listed in table 6.4 and plotted in figure 6.12.

The NIS results for ambient pressure (G

81GPa, vp

87km/s, vs

20km/s,

57km/s) are in very good agreement with tabulated values, which were obtained

as the mean of the corresponding Voigt and Reuss averages (see section 6.2.3) from

Pressure V/V0ρvKG vpvs

(GPa) (g/cm3) (km/s) (GPa) (GPa) (km/s) (km/s)

neutron 1 7.86 3.54 163 79 5.84 3.17

0 1 7.86 3.57(10) 163(4) 81(4) 5.87(6) 3.20(10)

6 0.967 8.13 4.10(25) 205(6) 109(12) 6.57(15) 3.66(25)

11 0.945 8.32 4.16(25) 241(6) 115(12) 6.89(15) 3.72(25)

20 0.862 9.12 4.82(15) 292(8) 172(10) 7.56(12) 4.34(15)

32 0.828 9.48 5.00(15) 369(8) 191(10) 8.11(12) 4.49(15)

42 0.805 9.76 5.14(15) 434(8) 207(10) 8.53(12) 4.61(15)

Table 6.4: Elastic properties for iron under pressure. The volume/density data and the bulk

moduliKwere taken from the compressibilitydata of α-Fe [MBT67] and ε-Fe [MWC90]. The

mean sound velocity v is from NIS. The shear modulus Gand the aggregate velocities vpand

vsare derived from equations 6.12, 6.13 and 6.14. The uncertainty of Gand vsis primarily

induced by the error in v, the uncertainty of vpprimarily by the error in K.

112

single crystal elastic coefficients [SW71]: G

(

)

GPa, vp

(

)

km/s, vs

(

)

km/s, v

(

)

km/s.4

At high pressure, the present results agree with lattice-strain experiments of Singh et

al. [SMH98] in both iron phases and with theoretical studies in ε-Fe by Söderlind et

al. [SMW96] and Stixrude et al. [SC95]. However, values from recent ultrasound and

lattice-strain experiments by Mao et al. [MSS98] are about 20% smaller whereas the

theoretical work by Steinle-Neumann et al. [SSC99] tends to give higher aggregate

velocities.

6.4.3 The α-εPhase Transition

With respect to the α-εphase transition in iron, the present NIS values for the vibra-

tional entropySvib can be used to determine the slope of the phase boundary in the T

;

phase diagram at T

294K(see figure 6.13).



Pressure (GPa)

Temperature (K)

1000

500





10 20 30

1500

Figure 6.13: Phase diagram of Fe showing enlarged the phase boundary between the

α- and ε-phase around 294 K, which can be determined with the Clausius-Clapeyron

equation 6.25.

The Clausius-Clapeyron equation

dTtr

dPtr

∆V

∆S(6.25)

relates the slope dTtr

dPtr of transition temperature Ttr versus transition pressure Ptr

to the volume and entropy differences between the two phases. The value for ∆Vat

13 GPa and 294 K can be calculated from the literature data [MBT67, MWC90] and

amounts to ∆V

;

344 cm3/mol. For an estimation of ∆S, the values of Svib for

both iron phases are extrapolated to 13 GPa as shown in figure 6.14. Hereby, the

small electronic contribution to the entropy is neglected. With the value of ∆Svib

4The values in brackets give the mean deviation of the tabulated data sets.

113

(

)

kB/atom a slope of

dTtr

dPtr

;

260

(

)

GPa (6.26)

is obtained for the phase boundary at 294 K.

Due to the sluggish nature ofthe α-εtransitionand the strong variationof the transition

pressure for different pressure-transmitting media, values for dTtr

dPtr are difficult to

obtain with x-ray diffraction. Consequently, the results in the literature exhibit a large

variation. A collection of different values given by Manghnani et al. [MMN87] covers

a range of -169 K/GPa up to -455 K/GPa, which is consistent with the NIS result.

0 1020304050

S = 0.16(2) k /atom

ressure (GPa)

1.8

2.0

2.2

2.4

2.6

2.8

3.0

Svibr

vibr

(kB

/atom)

3.2

Figure 6.14: Extrapolated pressure dependence of the vibrational entropy Svib for α-Fe

and ε-Fe. For the transition pressure at 13 GPa an entropy difference ∆Svib

(

)

atom between the two phases is obtained.

The results presented in this chapter represent an extensive characterization of the lat-

tice dynamics of elemental iron under pressure. A geophysical relevance of the work

can be derived fromthe first measurement ofthe phonon DOS in ε-Fe, which is a major

component of the Earth’s inner core.

For the derived thermodynamic properties, it should be emphasized that, in contrast

to other (e.g. calorimetric) measurements, pure vibrational parts of internal energy,

specific heat and entropyare provided. They are freeof possible electronic or magnetic

contributions and are therefore extremely useful for testing theoretical calculations

of thermodynamic parameters responsible for the equation-of-state [SMW96] or the

melting temperature of iron [AGP99].

114

Chapter 7

Summary

In the framework of this thesis the new methods of Nuclear Resonant Scattering of

synchrotron radiation have been successfully applied to the investigation of magnetic

properties and lattice dynamics under pressure. For this purpose a new generation of

diamond-anvil cells has been developed according to the specific properties of syn-

chrotron radiation and the individual needs for Nuclear Forward Scattering (NFS) and

Nuclear Inelastic Scattering (NIS).

The NFS experiments were performedon magnetic Laves phases with the composition

RFe2(R = Y, Gd, Sc) at pressures up to 115 GPa. This pressure range allowed the study

of iron magnetism in these model systems with a large variation of interatomic Fe-Fe

distances. The competing variation of exchange interactions and Fe band moments

is reflected by a systematic change of the magnetic ordering temperatures and of the

magnetic ordering type. The latter varied from ferromagnetism with well-localized Fe

moments via antiferromagnetism with more itinerant Fe moments to a non-magnetic

state. This behaviour is similar to the observations in elemental iron (γ-Fe), when the

lattice parameter is changed. We conclude from comparative studies on ScFe2that the

antiferromagnetic state can only be obtained after a pressure-induced structural phase

transformation from the cubic C15 to the hexagonal C14 structure.

It is further demonstrated that the variety of differentmagnetic phenomena in the RFe2

series withnon-magneticR atoms can be reproducedby a single model system, namely

YFe2, when exposed to high pressure. A comparison with GdFe2, exhibiting large Gd

4f moments, indicated a strongly increased interaction between the Fe and the Gd

sublattices under pressure, which leads to a stabilization of the Fe moment.

The objective of the second part of the thesis was the pioneering application of NIS

for the study of phonons in iron under pressure. With a new high-pressure technique,

based on a Be gasket for sufficient transmission of the Fe Kα

;

βx-ray fluorescence,

the phonon density of states in the ε-phase of iron was experimentally determined

for the first time. From NIS spectra measured in α-Fe and ε-Fe at various pressures

up to 42 GPa, a variety of thermodynamic parameters, such as Debye temperatures,

115

Grüneisen parameters, the Helmholtz free energy and the vibrational contributions to

the specific heat and the entropywere derived. Since ε-Feis the main component ofthe

Earth’s solid inner core, the derived sound velocities have direct geophysical impact.

The present results can be further used to test theoretical ab initio calculations, which

describe the physics of the Earth’s core.

Outlook

The success of the present NFS and NIS studies is intimately connected with the use

of synchrotron radiation from 3rd generation sources. Further developments can be

expected from improvements in the performance of existing synchrotron facilities and

from the instrumentation of the beamlines, where for high-pressure studies the use of

focusing optics is most important. In this field, there is already a strong (and fruit-

ful) competition between different groups performing Nuclear Resonant Scattering at

ESRF (Grenoble), APS (Argonne, Chicago) and SPring8 (Tsukuba, Japan).

The use of focusing devices was essential for the NIS studies of iron under pressure.

The NFS experiments in this thesis were, however, performed without focusing optics.

Further high-pressure NFS studies at ESRF would benefit very much from a tenfold

increased photon flux on the sample, possible with the focusing elements already in

use. Such a gain in photon flux may lead to major improvements of NFS studies in two

directions:



confinement of the measuring time from the order of hours to 5-10 minutes,

allowing a systematic study of magnetic ordering temperatures under pressure,

and



diminishing the sample dimensions by an order of magnitude, providing access

to pressures well above 200 GPa.

From the geophysical point of view, it is obvious that higher pressures than the present

42 GPa are desirable for NIS experiments on iron; the pressure of the Earth’s core

varies from 140 GPa to 360 GPa. An actual approach to such high pressures was

obtained in recent experiments at APS, where a collaboration between the Geophysical

Laboratory (H.K. Mao et al., Washington), APS beamline scientists (W. Sturhahn et

al.) and the Paderborn group reached 153 GPa in a NIS study of iron [MXS00].

For both spectroscopic applications, NIS and NFS, high-pressure experiments at ele-

vated temperatures are envisaged. Although relevant temperatures for the Earth’s core

(6000-8000 K) are far beyond the experimental possibilities, the feasible temperature

range up to about 1300 K with electrical heating [DST00] is already sufficient to test

theoretical forecasts of temperature-dependent lattice dynamics and to study ordering

temperatures of magnetic systems of interest, like the RFe2systems with magnetic

rare-earth constituents.

116

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128

Danksagung

Die Arbeit mit Synchrotronstrahlung ist nichts für Einzelkämpfer. Ich möchte daher

allen danken, die mich auf meiner Reise ins Innere der Erde unterstützt haben:

Mein besonderer Dank gilt Prof. Gerhard Wortmann. Sein großes Engagement für das

Forschungsprojekt hat entscheidend zum Gelingen der Arbeit beigetragen. Es hat viel

Spaß gemacht, mit ihm zu arbeiten und ich denke, wir waren ein gutes Team. Herzlich

danken möchte ich ihm auch für seine ortskundigen Führungen in Chicago und zum

Grab des Wildschütz’ Jennerwein.

Bei den verbliebenen Gruppenmitgliedern Kirsten Rupprecht und Hubertus Giefers

bedanke ich mich für die Unterstützung vor und während der Strahlzeiten und das

gute Arbeitsklima. Kirstens Ordnungssinn im gemeinsamen Büro machte das Leben

leichter, genauwie diesehr amüsantenHamburgermahlzeitenmitHubertusinChicago.

Ichdankeauch denehemaligenMitgliedernderArbeitsgruppefürihreHilfe. Marianne

Pleines und Hans-Josef Hesse waren meine Mitstreiter bei den NFS-Experimenten in

Grenoble. Meinem Weggefährten aus Diplomzeiten, Matthias Strecker, danke ich für

Diskussionen über den Magnetismus als solchen und für viele Jahre guter Zusammen-

arbeit. Mein großer Rückhalt im Mössbauerlabor war Jiangang Lu; seine Spektren

waren mir bei den NFS-Auswertungen eine wichtige Stütze. Viel Spaß bei den Ex-

perimenten in Hamburg hatte ich mit Jens Dumschat. Genau wie Stephan I. Györy,

Frank Nessel und Günther Nowitzke gehört er noch zu meinem früheren Leben mit

Röntgenabsorptionsspektrospie.

Wie schon zu Zeiten der Diplomarbeit, war die Zusammenarbeit mit der Hochdruck-

gruppe von Prof. Wilfried Holzapfel hervorragend. Mit Prof. Holzapfel hatte ich

einen äußerst kompetenten Diskussionspartner in vielen Bereichen der Hochdruck-

physik, besonders auf dem Gebiet der Gitterdynamik. Gerd Reiss und Felix Porsch

danke ich für die gute Kooperation bei den Laves-Phasen. (Was hätte ich wohl ohne

Eisenabstände unter Druck gemacht?) Andreas Schiwek hat freundlicherweise einige

Kapitel dieser Arbeit durchgesehen. Thomas Tröster, Werner Sievers und Wilfried

Bröckling waren immer gute Ansprechpartner, wenn es um neue Entwicklungen in der

Hochdrucktechnik ging.

129

Der Beitrag von Herrn Franz Risse und seinen Mitarbeitern der mechanischen Werk-

statt bei der Planung und Ausführung dieser Entwicklungen ist gar nicht hoch genug

einzuordnen.

Die Arbeit an internationalen Forschungseinrichtungen wie dem ESRF in Grenoble

kann nur erfolgreich sein, wenn man eine engagierte und hilfsbereite Mannschaft

vor Ort antrifft. Hanne und Hermann Grünsteudel, Olaf Leupold, Joachim Metge,

Alessandro Barla, Sasha Chumakov, Alfred Baron und Rudolf Rüffer waren so eine

Mannschaft, mit der die Strahlzeiten auch noch viel Spaß machten. Hanne und Her-

mann waren dabei fast wie eine französische Familie für mich. Hermann und Sasha

danke ich zudem für die besonders erfolgreiche Zusammenarbeit, die zur Veröffent-

lichung in Science führte.

Der Kölner Gruppe von Prof. Mohsen Abd-Elmeguid danke ich für viele interessante

Stunden beim ESRF mit anregenden Diskussionen.

Obwohl die Zinn-Experimente nicht in dieser Arbeit auftauchen, danke ich dem Team

um Wolfgang Sturhahnbei APS in Chicago für die freundlicheAufnahme in der neuen

Welt.

Meiner lieben Mutter und meinen Schwestern mit ihren Familien habe ich es zu ver-

danken, dass ich die Verbindung zur Oberfläche der Erde nicht verlor. Sie haben

mich in allen Phasen der Arbeit treu unterstützt. Das Gleiche gilt auch für meine

Schwiegereltern, die mir während meiner Reisen und auch sonst so oft den Rücken

frei gehalten haben.

Mein letzter und herzlichster Dank gilt meiner wunderbaren Frau Nataly. Sie musste

viel Verständnis dafür aufbringen, dass man für eine Reise ins Innere der Erde erst

zweimal drumherum fliegen muss. Ihr und unseren beiden Töchtern Johanna und

Pauline ist diese Arbeit gewidmet.

130

Publications

1. J. Röhler and R. Lübbers, The Valence of Elemental Praseodym in the Collapsed

"Pr IV"-Phase from LIII X-Ray-Absorption up to 260 kbar, Physica C 235-240,

1031 (1994).

2. J. Röhler and R. Lübbers, X-Ray Absorption Study of the "Pr III" - "Pr IV"

Transition in Elemental Praseodymium, Physica B 206 & 207, 368 (1995).

3. H. Welker, H.F. Grünsteudel, G. Ritter, R. Lübbers, H.-J. Hesse, G. Nowitzke,

G. Wortmann and H.A. Goodwin, Combined Mössbauer and EXAFS Study of

the High-spin/Low-spin Transition in [Fe(II)(bpp)2](BF4)2,Conf. Proc. Vol.50

(SIF Bologna), (1996) p. 19-22.

4. R. Lübbbers, G. Nowitzke, H.A. Goodwin and G. Wortmann, X-Ray Absorption

Study of the High-spin/Low-spin Transition in [Fe(II)(bpp)2](BF4)2,J. Phys. IV

France 7, C2-651 (1997).

5. R. Lübbers, J. Dumschat, G. Wortmann and E. Bauer, Temperature and Pressure

Induced Valence Transitions in YbCu5

;

xGaxStudied by Yb-LIII XANES, J. Phys.

IV France 7, C2-1021 (1997).

6. G. Nowitzke, H. Meier, R. Lübbers, J. Dumschat and G. Wortmann, EXAFS

Study of Oxygen Distortions in Gd2CuO4as Function of Temperature and Pres-

sure, J. Phys. IV France 7, C2-1023 (1997).

7. J. Dumschat, R. Lübbers, I. Felner, G. Lucazeau and G. Wortmann, (Tb,Ce)-

LIII XAS High-pressure Study of Tetravalent BaTbO3and SrCeO3,J. Phys. IV

France 7, C2-1019 (1997).

8. H.-J. Hesse, R. Lübbers, M. Winzenick, H.W. Neuling, G. Wortmann, Pressure

and Temperature Dependence of the Eu Valence in EuNi2Ge2and Related Sys-

tems Studied by Mössbauer Effect, X-ray Absorption and XRD, J. Alloys and

Compounds 246, 220-232 (1997).

131

9. R. Lübbers, H.-J. Hesse, H.F. Grünsteudel, R. Rüffer, J. Zukrowski and G. Wort-

mann, Probing Magnetism in the Mbar Range, ESRF Highlights 1996/97, p.

56-57

10. R. Lübbers, H.-J. Hesse, H.F. Grünsteudel, R. Rüffer, J. Zukrowski and G.

Wortmann, High-Pressure Studies of Magnetism by Nuclear Scattering of Syn-

chrotron Radiation, Proc. 2nd Russian-German Workshop on Synchrotron Ra-

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Neutron Techniques" (Gordon & Breach, 1999).

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Synchrotron Radiation: Principles and Applications", edited by E. Gerdau and

H. de Waard, Baltzer, 1999, in print (33 pages).

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132